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The role of context and gender in predicting success in a modified laboratory course

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THE ROLE OF CONTEXT AND GENDER IN PREDICTING SUCCESS IN A
MODIFIED LABORATORY COURSE
BY
KERON SUBERO
A dissertation submitted to the Graduate School in partial fulfillment of the
requirements for the degree
Doctor of Philosophy
Major Subject: Physics
Minor Subject: Applied Statistics
New Mexico State University
Las Cruces, New Mexico
December 2010
UMI Number: 3448957
All rights reserved
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"The role of context and gender in predicting success in a modified laboratory
course," a dissertation prepared by Keron Subero in partial fulfillment of the
requirements for the degree, Doctor of Philosophy, as been approved and accepted by
the following:
^**%.
HJoijaotQ
Linda Lacey
Dean of the Graduate School
J J^a-r»^
1
<Cs>-
Stephen E. Kanim, Chair
Chair of the Examining Committee
Date
Committee in charge:
Dr. Stephen E. Kanim, Chair
Dr. Dennis L. Clason
Dr. William R. Gibbs
Dr. Heinrich Nakotte
Dr. Dominic A. Simon
Dr. Theodore B. Stanford
11
ACKNOWLEDGEMENTS
To my wife Katurah, this is the result of your perseverance and support.
To my kids for growing up and supporting each other when I could not be there Especially to Anisa for fathering Zolani and Nya.
Dr. Steve Kanim, thank you for going above and beyond always, for putting your care
and time into this research, and for shaping my view on everything.
Thanks to everyone on my committee for all of your help at various stages in this
dissertation.
Thanks to my parents for giving me simple mores that guided me through many
rough times.
iii
VITA
Nov. 4, 1978.
Born in Port of Spain, Trinidad and Tobago.
1997-2001
B.Sc. North Carolina Central University.
2001-2003
Graduate Program , Oregon State University.
2004-2010
Ph.D. New Mexico State University.
PROFESSIONAL SOCIETIES
American Association of Physics Teachers
PUBLICATIONS
S. Kanim and K. Subero, "Introductory labs on the vector nature of force and
acceleration" Am. J. Phys., 78 (5), 461-466, 2010.
PRESENTATIONS
K. Subero, "What is proportional reasoning, anyway?" October 19, 2007, APS Four
Corners Section Meeting, Flagstaff, AZ.
K. Subero, "Promotion of vector use in introductory mechanics laboratories."
July 24, 2006, AAPT Summer Meeting, Syracuse, NY.
FIELD OF STUDY
Physics Education Research
IV
ABSTRACT
THE ROLE OF CONTEXT AND GENDER IN PREDICTING SUCCESS IN A
MODIFIED LABORATORY COURSE
BY
KERON SUBERO
Doctor of Philosophy
New Mexico State University
Las, Cruces, New Mexico, 2010
Dr. Stephen Kanim, Chair
We designed and implemented curriculum intended to be used by students in an
algebra-based introductory physics laboratory course. Our curricular goal was to
foster, through observations in the lab, a coherent framework in students'
understanding of general principles presented in the introductory mechanics course,
while addressing known student difficulties. The research that guided our curriculum
development efforts, however, was previously implemented in an intervention setting
which was quite different from ours, and was conducted on students enrolled in
calculus-based physics courses who were generally academically better prepared than
our students. We describe the development of laboratory materials, designed to fit the
specific curricular constraints of a lab course at NMSU. We present some results from
post-testing of our labs, which were not as favorable as results obtained by
v
researchers at other institutions implementing similar curricula in their courses.
We attempted to quantify differences in preparation among our introductory
physics student populations who use these laboratory materials. We developed a short
proportional reasoning pretest, which we found to be a relatively efficient predictor of
student success in our courses. We investigated the effect of context variations on
performance by various student populations on this pretest, and found that the effect
of context variation was not the same for all of our student populations. Results from
our calculus-based population showed a small but significant increase in performance
when we modified the context of our pretest, while the performance of our algebrabased population showed very little sensitivity to the variation in pretest context.
Finally, when considering students' gender, we found in both algebra-based and
calculus-based physics courses that female students were significantly affected by
context variation, while male students' performance remained relatively unchanged
when we varied our pretest context.
vi
TABLE OF CONTENTS
Page
LIST OF TABLES
xiii
LIST OF FIGURES
xv
CHAPTER 1: OVERVIEW
1
Introduction
1
Research questions
5
Context for investigation
7
Student population at NMSU
10
Introductory physics courses at NMSU and the students enrolled
in them
12
Predicting success in these labs
15
Epistemology
16
Summary
17
CHAPTER 2: MODIFICATION OF THE LABORATORIES
18
Background
18
Structure of the labs at NMSU
19
Example of curricular modification: Two-dimensional
motion
21
Motivation
21
Tutorial on motion in two dimensions
24
Laboratories using long-exposure digital photography
25
Equipment
26
vii
Page
Two-dimensional motion lab
29
Procedure for finding acceleration direction
30
Introductory pencil-and paper exercises
31
Analysis of long exposure photographs of moving obj ects
31
Central force motion: Spherical pendulum
32
Parabolic motion: Hovercraft on a ramp
32
General two-dimensional motion: Roller coaster
34
Newton's Second Law lab
35
Introductory pencil-and paper exercises
36
Lab exercises connecting force to acceleration
36
Physical pendulum
37
Constant velocity: Hovercraft on a level surface
37
Linear motion: Block on a ramp
38
Other labs using long-exposure digital photography
40
Energy
40
Rotation
41
Measurements of effectiveness of the labs
45
Results on questions about acceleration in two dimensions
46
Results on questions relating force direction to acceleration
direction
47
Discussion
50
viii
Page
Conclusion
52
CHAPTER 3: PROPORTIONAL REASONING AS A PREDICTOR
OF SUCCESS IN PHYSICS
55
Introduction
55
Description of statistics used
57
Test statistics
57
Correlation Coefficients
59
Review of previous studies
62
Predictors of physics grade, mathematics and education
reform
62
Mathematics as a predictor
64
Piagetian ability as a predictor
66
Paper-and-pencil tests of Piagetian ability
69
Various other factors used as predictors
70
Summary of previously used predictors
75
Lawson test questions as predictors of success
76
Results of using questions from the Lawson test as a
pretest
78
Proportional reasoning pretests
80
Performance on pretests
81
Performance on final exam
83
Pretest version and correlation
85
IX
Page
Fall 2009 pretest versions
87
Working memory capacity
90
Description of working memory capacity
90
Previous use of working memory as a predictor
93
Working memory capacity as a predictor at NMSU
94
Description of our N-back test
95
Scoring of our N-back test
97
Results of our N-back test
98
Spatial working memory test
Results using spatial working memory capacity
Summary of working memory results
Conclusion
100
102
102
103
CHAPTER 4: CONTEXT DEPENDENCE OF PERFORMANCE ON
PROPORTIONAL REASONING TASKS
105
Two perspectives on student difficulties
105
Example of context-dependent performance
109
Background to study at NMSU
113
Method
114
Results for all students
115
Difference in performance by course
117
Discussion of performance by course
Gender effects on performance
122
124
x
Page
Results of NMSU study separated by gender
125
Fall 2009 pretest versions
129
Pretest correlation with final exam
135
Discussion and conclusion
137
CHAPTER 5: CONCLUSION
140
Introduction
140
Summary of findings
141
Implications for instruction
144
APPENDICES
A. Motion in two dimensions tutorial at the University of
Washington
149
B. Selected labs and associated homework exercises
152
C. Selected final exam questions
200
D. Lawson Test of Scientific Reasoning
203
E. 15 Question subset of Lawson pre-test
208
F. Three versions of proportional reasoning pretest
212
G. Correlation coefficients between Lawson subcomponents,
pretest and
final
H. Various scatterplots of final exam score vs measures of working
memory capacity
216
218
I. Various tables showing differences in performance on pretests by
two introductory physics populations
221
J. Various tables showing gender separated differences in performance
on pretests by two introductory physics populations
225
xi
Page
K. Fall 2009 pretest versions
229
L. Correlation matrix of TA ratings of students with other measures
of student success
234
REFERENCES
235
xii
LIST OF TABLES
Page
Table
2.1
Student performance on matched pre/post questions
49
2.2
Comparison of performance for sections with two
instructional emphases
49
Correlations among Lawson test subcomponents
and final exam score for Physics 211 students
79
Correlations among Lawson test subcomponents
and final exam score for Physics 221 students
79
Descriptive statistics of difference in student performance
on two pretest versions
116
T-test summary of difference in student performance
on two pretest versions
116
Descriptive statistics of differences in performance on original
pretest questions by students in two intro physics courses
221
T-test summary of differences in performance on original
pretest questions by students in two intro physics courses
222
Descriptive statistics of differences in performance on modified
pretest questions by students in two intro physics courses
223
T-test summary of differences in performance on modified
pretest questions by students in two intro physics courses
224
Descriptive statistics of difference in student performance
on two pretest versions within each course
120
T-test summary of difference in student performance
on two pretest versions within each course
120
Descriptive statistics of gender separated differences in student
performance on two pretest versions for Physics 211
225
3.1
3.1
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
xiii
Page
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
T-test summary of gender separated differences in student
performance on two pretest versions for Physics 211
225
Descriptive statistics of gender difference in performance
on two pretest versions for Physics 211 students
226
T-test summary of gender difference in performance
on two pretest versions for Physics 211 students
226
Descriptive statistics of gender separated differences in student
performance on two pretest versions for Physics 215
227
T-test summary of gender separated differences in student
performance on two pretest versions for Physics 215
227
Descriptive statistics of gender difference in performance
on two pretest versions for Physics 215 students
228
T-test summary of gender difference in performance
on two pretest versions for Physics 215 students
228
Gender separated correlation coefficients of pretests score
with final exam score
136
Correlations among TA's rating of students with
students' final exam scores and pretest scores
234
xiv
LIST OF FIGURES
Page
Figure
1.1
Five bulbs Question
2
1.2
Distribution of ACT math scores for NMSU students vs All high
school seniors
11
1.3
Distribution of class and maj or of introductory physics courses
13
2.1
Blinkie Circuit
28
2.2
Equipment for motion in two dimensions lab
29
2.3
Photograph of pendulum moving in an ellipse
33
2.4
Long-exposure photograph for the toy hovercraft
34
2.5
Long-exposure photograph of the roller-coaster
35
2.6
Photograph of a single swing of the physical pendulum
38
2.7
Photograph of the wooden block on an inclined ramp
39
2.8
Roller coaster on steep and shallow tracks
41
2.9
Photograph of plywood disk rotating with constant angular velocity...
42
2.10
Photograph of plywood rotating disk with constant angular
acceleration
43
2.11
Plywood disk with variable moment of inertia
44
2.12
Questions on two-dimensional motion
47
3.1
Sample scatterplots showing different correlations
60
3.2
Scatterplots showing R-squared
62
3.3
Stages of Piagetian development
67
xv
Page
3.4
Scatterplots of score on final vs pretest score
78
3.5
Distribution of student scores on proportional reasoning pretests
82
3.6
Distribution of final exam scores
84
3.7
Scatterplots of final score vs pretest score for different courses
86
3.8
Model of memory processing cycle
91
3.9
Model of inner workings of working memory
92
3.10
N-back slide - question 3
96
3.11
N-back slide - question 5
97
3.12
Scatterplot of final score vs working memory score
99
3.13
Scatterplot of final score vs time to take working memory test
218
3.14
Scatterplots of final score vs working memory correct response rate...
219
3.15
Spatial working memory test slide
101
3.16
Scatterplot of final score vs spatial working memory score
219
3.17
Scatterplot of final score vs time to take spatial working memory
test
220
4.1
Two time of flight questions with different context cuing
108
4.2
Three contexts of question on predicting trajectory
Ill
4.3
Distributions of students' scores on two pretest versions
121
4.4
Student performance on each set of paired questions
122
4.5
Distributions of students' scores on two pretest versions separated
by gender for the algebra-based physics course
126
xvi
Page
4.6
4.7
Distributions of students' scores on two pretest versions separated
by gender for the calculus-based physics course
127
Frequency counts of correct responses on students' paired
pre-post test scores on two versions of ratio questions
130
4.8
Diagram of two interlocked gears
132
4.9
Student performance on gear question
132
4.10
Student Fall 2009 performance on two pretest versions
134
xvn
CHAPTER 1: OVERVIEW
Introduction
Physics education research (PER) has evolved as a field of research conducted
in many physics departments worldwide out of a desire to systematically study issues
related to students' learning of physics. Much of the work in the field has consisted
of pinpointing and correcting reasoning or conceptual difficulties on specific topics
which were repeatedly observed to affect students' learning of physics.
The impetus behind many early physics education investigations was a
dissatisfaction with the physics knowledge of students after completing traditionally
taught physics courses.1' There was evidence that traditional instruction invariably
resulted in students using formula-driven problem solving strategies and that students
often left physics courses with a largely unmodified, incorrect view of the physical
world.5 In response many physics education researchers have modified curricula to
include inquiry-based and other interactive-engagement techniques.5 These
modified curricula have led to significant gains in students' physics content
knowledge.
Oftentimes, curriculum development is thought of as the primary purpose of
o
all physics education research. However, McDermott points out that researchers in
PER can be categorized by their perception of the various teaching processes
occurring in physics classrooms. She identifies researchers who use the context of
1
students studying physics to focus on understanding human cognition as one group;
those researchers focusing on physics subject matter as another; and those who use
the context of physics as a means to focus on instructional strategies as a third.
Regardless of focus, results from PER influences and often forms the basis of
curriculum development in physics.
PER has been reasonably successful to date in finding specific topical
difficulties that are encountered by students enrolled in undergraduate physics
courses, and in designing curricula based on this research that successfully addresses
these difficulties.10 An example of a research question that elicits common student
difficulties is the 'five bulbs question' shown in Figure l.l. 11 Students are asked to
rank the brightness of the five bulbs, assuming all bulbs are identical and the batteries
are ideal and identical.
B
A
T
c
T
(j)
D
E
(j)
Figure 1. 1. 'Five bulbs Question' asked by McDermott and Shaffer. "Students are
asked to rank the brightness of five identical bulbs connected to identical ideal
batteries.
Only about 15% of students in introductory calculus-based courses correctly
answer this question. This result does not depend strongly on whether the question is
2
asked before or after relevant standard instruction in electric circuits. Students'
incorrect answers suggest several widely held incorrect beliefs about electric current
in circuits, for example that current is 'used up' as it passes through resistive elements
in a circuit (consistent with a student ranking of bulb B brighter than bulb C); or that
a battery acts as a constant current source (consistent with a student ranking of bulb A
brighter than bulb D or E). By uncovering and categorizing these common incorrect
models or misconceptions, researchers have formed a basis on which to develop more
effective curriculum. Research-based instruction typically refers to curricular design
based on investigations into student understanding and incorporating instructional
strategies that actively involve the learner in the construction of correct physical
models.
These researchers have shown that scores can be significantly improved on
conceptual questions such as this if students are carefully guided through the
reasoning necessary to develop a model that is consistent with physical laws. For
example, on the five bulbs question, 15% of students in an introductory calculusbased electricity and magnetism course, who have completed two 50-minute
recitation section exercises (instead of more typical recitation section problemsolving instruction) from Tutorials in Introductory Physics, correctly answer the 5bulbs question. For comparison, 70% of graduate students at the University of
Washington give a correct ranking on this question.
When curricula developed at one institution and with one group of students is
adopted at another institution, it has been shown that improvements can be obtained
3
that are similar to those reported from the institution that conducted the research and
developed the materials. For example, at the University of Colorado, 15% of students
answer the 5-bulbs question described above correctly after traditional instruction,
whereas upon completion of the relevant Tutorials, 57% of students correctly answer
this question upon post-testing.12 These values are very similar to the ones observed
at the University of Washington.
Based on these results, it is tempting (especially for physicists) to treat student
populations as essentially interchangeable, much in the same way that the results of
physics experiments do not depend on which particular batch of electrons are being
manipulated. With this assumption, when implementation of curriculum falls short
of reported results, it is assumed that the problem lies in the implementation.
However, in the past 8 years researchers have established correlations between
student performance on various pretests given at the beginning of instruction with the
gains in conceptual understanding that these students make during physics
instruction. For example, Meltzer55reports that student performance on a
mathematics pretest is significantly correlated with normalized gain in student
performance on a standard conceptual test on electricity (CSE), while Coletta
reports that students' reasoning ability correlates strongly with their improvement in
score on a popular test of the concept of a force (FCI).
These results have implications for the degree to which we might expect
curriculum developed at one institution and with one student population to be as
effective at improving student understanding when it is implemented at a different
4
institution or with another student population. Many initial PER investigations were
primarily conducted in calculus-based physics courses at Research I institutions with
selective admission requirements. These results are often applied at less selective
institutions, and with more poorly prepared students. For example, Kanim14 reports
that while about three-quarters of all PER research on topics involving introductory
physics curricula is done using students enrolled in calculus-based introductory
physics courses, this population accounts for only about one-third of the total
introductory physics population.
At NMSU we have reported using curriculum materials similar to that used
elsewhere yet our success rates were less than to be expected. ' Similarly,
differences in performance were also noted by our colleagues at Cal State Fullerton.
Anecdotally, instructors describe differences in algebra-based and calculus-based
student populations or between pre-med and pre-engineering students and we were
curious to find out if there were measurable differences between these populations
that might help explain differences in performance. This led us to pay close attention
to the success of curricular development among various introductory physics
populations as is explored in greater detail in chapters three and four.
Research Questions
Among the guiding questions we asked ourselves throughout the planning and
implementation of the research presented in this dissertation were:
5
1) Can we modify our laboratory course to include exercises which significantly
improve our introductory physics students' conceptual understanding of
physics, while at the same time maintaining faculty and student expectations
of laboratory-like activities?
2) Can we help develop an expert-like, coherent view of physics formalism in
our students through curricular modifications?
3) How do the anecdotal differences in algebra-based and calculus-based
introductory physics populations play out in terms of measurable differences
in reasoning abilities and in performance on conceptual physics problems?
4) Are there limits to how successful our curriculum are expected to be, and what
factors influence students successfully understanding the material presented?
5) What is the role of student epistemology in determining student performance
on typical tests?
6) Do various styles of questions produce differing effects on different student
populations?
These overarching questions, constrained by the practical issues of integrating
our research into the normal conduct of the laboratory courses, led to the following
questions that we attempted to answer in the course of our investigation:
1)
Through curriculum developed to emphasize a vector treatment of displacement,
velocity and acceleration, can we achieve results on post-test questions
indicating student understanding of acceleration as vector quantity, comparable
to research-based materials used at other institutions?
6
2)
Can we further develop materials that lead to measurable improvements in
students' understanding of Newton's second law as a vector equation?
3)
Can we design a pretest, based on previous research, that would give us some
indication of which among our students would be most likely to succeed in our
laboratory course, and which among our students would be most likely to have
difficulty with the materials presented?
4)
Is pretesting of scientific reasoning ability context-dependent? And if so, does
variation in the context of questions asked on this pretest influence student
performance?
5)
Are there differences in the level of context-dependence of student performance,
for various subsets of our student populations at NMSU?
Context for Investigation
At NMSU, unlike at many other universities nationally, there are no recitation
sections associated with the introductory physics courses. The lecture meets for about
150 minutes per week and forms the main arena for student learning of physics. A lab
that is closely aligned with each lecture course is offered, however, some majors
(such as engineering technology, mechanical engineering and mathematics) do not
require their students to take these labs. As such, students' physics content knowledge
is largely left up to individual instructors' lectures. In comparison, at the University
of Washington a typical engineering student taking an equivalent introductory physics
7
course is required (along with lecture) to enroll in a recitation section as well as take
the associated laboratory course, for a total of over 300 minutes of physics instruction
per week. In light of this discrepancy of time dedicated to physics learning, we
decided to modify the structure of our lab course to make more efficient use of the
time our students spent with physics.
Prior to the year 2000, the introductory physics labs at NMSU were
traditional.1 In an effort to improve students' conceptual understanding of physics, the
Tutorials in Introductory Physics16 (from henceforth referred to as the Tutorials) were
introduced in both first semester (mechanics) and second semester (electricity and
magnetism) introductory physics lab courses as the basis of our lab instruction. The
Tutorials, while designed to be used in a recitation section (which we do not have at
NMSU), is arguably the most widely adopted research-based curriculum in use at
many universities today.
The Tutorials were developed at the University of Washington through an
iterative development sequence of: 1) systematic investigation of student
understanding, 2) application of what is learned from this research as a guide towards
the development of new curriculum and 3) design, testing and revision of
17
instructional material based on classroom use.
For instance, if through exploratory
testing of physics content it is observed that students misunderstand some topic, in
subsequent semesters the curricular materials used are revised to try to address this
1
A more detailed description of the differences between traditional and inquiry-based
labs is given at the beginning of chapter 2.
8
issue. In light of these results, the effectiveness of the curricular modification is
ascertained and it is determined which, if any, .further modifications are necessary.
This cycle is repeated until there is satisfaction that the curricular materials
successfully address student misunderstandings. The structure, focus and instructional
strategies used in the labs we describe in chapter two of this dissertation, owe a great
deal to the Tutorials16 and to the extent that we could, we also copied this iterative
developmental sequence.
At NMSU, while these tutorials were somewhat successful in improving
students' understanding of some concepts, it was felt that many of the tutorials relied
on pencil-and-paper exercises that were less appropriate for a laboratory than for the
recitation sections they were designed for. In addition it was found that some of the
labs, such as the Motion in Two Dimensions tutorial shown in appendix A, were not
successful in improving our students' content knowledge. It is against this backdrop,
beginning in Spring 2002 with a few laboratory exercises each semester, that we
began to replace the tutorial exercises with labs which we modified for use within our
lab sections at NMSU.
As part of a collaborative project with Michael Loverude at California State
University Fullerton and Luanna Gomez at Buffalo State College, we designed
laboratories intended to strengthen students' conceptual understanding of topics in
introductory mechanics. Funding from the National Science Foundation's Course,
Curriculum, and Laboratory Improvement (CCLI) program allowed us to design,
implement, test, and improve a set of inquiry-based labs intended for students in the
9
algebra-based course. In addition, we conducted research into student understanding
of some of the topics associated with these labs in concert with the curriculum design.
A total of 17 labs were developed. At New Mexico State University, we use about 12
of these labs for the mechanics semester.
Our curriculum goal was to develop a set of labs that improved student
conceptual understanding, and that allowed students to connect the rules and
equations that they were learning in their lecture to physical events in the laboratory.
We wanted to tailor the labs to suit the needs of students in the algebra-based course,
and a secondary goal of our project was to better understand the differences in needs
between students in the algebra-based and calculus-based courses. In practice, at
NMSU, we use almost exactly the same sequence of labs for both populations, as we
have seen them to be more beneficial than the labs that they have replaced for both
student populations.
Student Population at NMSU
We believe that our descriptions of the curriculum development we have
implemented and our efforts towards predicting success in our labs are not complete
without at least some understanding of the student population we serve at NMSU.
New Mexico State University is the state's land grant university. We are a minoritymajority institution: 39% percent of our students are Hispanic. Three-fourths of our
students are from New Mexico, which is one of only four states considered
10
Distribution of ACT Math scores for NMSU
freshmen compared to High School seniors
nationwide.
NMSU incomin
freshmen
i.o%0.8%
All High School
Seniors
0.6%
0.4%
4
0.2%-
1L
0
2
4
6
8
10
12
14
Vy*—
16
18
20
22
24
26
28
30
32
34 36
ACT Math Score
Figure 1.2. Relative distribution of ACT math scores for All High School Seniors
nationally compared to ACT scores for NMSU incoming freshmen.
1R
minority-majority in the nation. About 43% of incoming freshmen at NMSU qualify
for Pell grants based on income. For comparison, at the University of Washington, a
i
io% veloper of PER-based curricula used nationally, 16% of all undergraduates
i ilify for Pell grants. The distribution of ACT scores for incoming freshmen at
] 1SU is a close match to the distribution for all high school seniors taking the ACT.
] 4SU has an acceptance rate of about 96%, which places it on the high end of
acceptance rates of colleges and universities nationwide.19
11
The results we report in this dissertation will primarily be from students
enrolled in labs associated with the first semester of introductory physics courses. As
mentioned previously, at NMSU, students do not automatically enroll in a lab and
some majors require the lecture but do not require the lab. For grading purposes,
then, the mechanics labs we describe in this dissertation are stand-alone 1-credit
courses. The lab coordinator generally communicates with the lecture instructors to
ensure that the lab sequence is aligned with the lecture, in order to best support the
lecture courses.
About one-third of the students enrolled in our lab courses have taken high
school physics. Roughly 90% of students enrolled in the labs are simultaneously
enrolled in the corresponding lecture course.
There are four such first semester
introductory physics courses offered at NMSU - Physics 211, Physics 213, Physics
215 and Physics 221 - only two of which (Physics 211 and Physics 215) are offered
every semester.
Introductory Physics Courses at NMSU and the Students Enrolled in Them
The bulk of data presented in this dissertation is from students enrolled in the
lab sections associated with two main courses Physics 211 and Physics 215. Physics
211 is a standard, algebra-based course with a non-calculus treatment of mechanics.
Knowledge of simple algebra and trigonometry is required of students enrolled in this
12
Distribution of Class and Major of three
introductory physics courses
Physics 211
•
Freshmen
Sophomores
Ujniore
Engineering
Technology
Seniors
Biology
Agriculture
Education
Mcrobiology
Biochemistry
Animal
Sdercc®
Crvil
Engineering
fwfechanical
Engineering
Chemical
Engineering
• • • • •
Other
Physics 22:
25»
9SW
0%
(M
Freshmen
Soonomores
Juniors
•
Hology
Seniors
Physics 215
2M4_
Freshmen
Soononwes
Juiiors
Electrical
Engineering
Seniors
Other
Engineering
Figure 1.3 Spring 05 histograms of the relative distributions of Class (left column)
and Major (right column), of students enrolled in three of our introductory physics
courses - Physics 211, Physics 221 and Physics 215.
13
course. About two-thirds of Physics 211 students are juniors and seniors (see Figure
1.3). These students major in a variety of fields, however, depending on semester and
whether or not Physics 221 is offered, biology majors usually make up the slight
majority of Physics 211 students.
Physics 221 is an algebra-based physics course where special emphasis is
given to applications in the life sciences, and is also recommended for students
preparing for the physics part of the MCAT. Most majors requiring their students to
take an algebra-based physics course do not distinguish between Physics 221 and
Physics 211. The context of the physics material presented within the course is
usually biology-oriented, thus this course tends to attract majors with an interest in
biology. This course is not offered every semester, and class sizes are relatively small.
Students enrolled in the two algebra-based lecture courses (Physics 211 and Physics
221) are enrolled in the same lab sections (called Physics 211 labs). We do not
distinguish between these two groups in the results reported from the labs.
Physics 215 is a calculus-based introductory mechanics course primarily made
up of various engineering majors (see Fig. 1.3). Calculus is a prerequisite for students
enrolled in this course. In contrast to the algebra-based courses, about two-thirds of
Physics 215 students are freshmen and sophomores. This course is offered every
semester and is usually large-enrollment - above 80 students per semester.
Physics 213 is a calculus-based course for physics and chemistry majors.
Calculus is also a prerequisite for this course, however students can also be
concurrently enrolled in calculus and still be allowed to register. Physics 213 is only
14
offered in the Fall semesters (as opposed to Physics 211 and Physics 215 which are
offered every semester) and class sizes are relatively small - around 20 students.
Because of its small student population, in many cases throughout this dissertation,
we omit results from students enrolled in this course. These students usually perform
markedly differently from the rest of our introductory physics student populations.
Predicting Success in These Labs
A second theme of this dissertation, discussed in chapter three, centers on
predicting success in these laboratories through the use of pretests. The grading and
structure of the laboratory course is similar to many lectures in the sense that weekly
homework scores, participation, and scores on a final exam all contribute to students'
lab course grade. Aspects such as homework scores and participation are difficult for
researchers to control for and influence, especially for lab sections graded and run by
multiple instructors. As such, we believe the uniform final exam to be the most
unbiased estimate of what students actually know upon leaving our lab course. For
the purposes of this dissertation, then, success in the laboratory course is primarily
measured by students' scores on the final exam of the lab course.
We will explore some of the factors which guided the development of tests
used by other researchers seeking to predict success in their courses, with particular
emphasis on the role of Piaget's idea of cognitive development, in influencing items
on historically used predictive tests. We will briefly describe how predictive some of
15
these tests have been at different institutions. We will also detail a short diagnostic
test we have developed, which we have found to be reasonably predictive of student
success in our laboratory course.
Epistemology
In chapter four of this dissertation we explore two views of the source of
differences in student performance on physics problems such as our short diagnostic
test. The view that has guided the development of many of the assessments used as
predictors of success to date, portrays differences in students' incoming ability as the
reason for varying degrees of success. Some researchers portray these tests as good
measures of scientific reasoning ability.
Other researchers21 believe that what
students 'know' is highly context dependent and rarely transferred intact from one
instance to the next, thus tempering their reliance on such tests in characterizing
students. In chapter four we take a look at how student epistemology plays into
performance on questions appearing on a commonly used assessment of proportional
reasoning ability. Further, we take a look at how different student populations are
affected by the contexts in which the questions are presented. A summary of our
findings in this dissertation can be found in chapter five.
16
Summary
This dissertation was motivated by a desire to improve our students'
knowledge of physics within the constraints of our course structure. As explored
further in chapter two, our initial results from curriculum changes were not as we
would have hoped.
This led us to consider if curriculum developed based on
difficulties observed and interventions attempted with one student population might
not be as successful with a different population as a result of student preparation
among other factors. We explored factors that other researchers have identified as
pertinent to student success in physics, and highlight our experience at NMSU using
these in accounting for different success rates among our students. Finally we
observed the interplay between students' epistemologies, the use of these predictive
factors, and students' successful acquisition of physics knowledge.
17
CHAPTER 2: MODIFICATION OF THE LABORATORIES"
Background
Before the year 2000 the labs at NMSU could be described as traditional labs.
Traditional labs as described in PER22include labs during which students are asked to
complete sequential steps, much like a recipe in a cookbook, performing
measurement and error analysis with an eye towards verification of quantitative
relationships. At the end of traditional labs students typically complete a lab report
following a prescribed format. Researchers have shown however
that students
develop little conceptual understanding of relevant physical phenomena as a result of
these traditional lab experiences.
The
Tutorials
in
Introductory
Physics
(commonly known as the Tutorials) developed at the University of Washington were
implemented at NMSU after the year 2000 to improve students' conceptual
understanding in introductory physics courses.
As measured by analysis of
homework and exam questions, this intervention was moderately successful for some
of the topics covered in the tutorials, while for others there was little improvement in
students' understanding of the materials presented. While today the second semester
course in our introductory physics sequence (which covers topics in electricity,
magnetism and optics) continues to use the Tutorials, we have been modifying the
laboratory materials for our mechanics course since 2002.24
" Substantial portions of this chapter are reprinted with permission from Am. J. Phys.
78(5), 461-466 (2010). Copyright 2010, American Association of Physics Teachers.
18
Structure of the Labs at NMSU
The focus of the labs we describe in this chapter is on developing a solid
conceptual understanding, on making connections between observations and physics
formalism, on promoting reasoning based on observation, and on strengthening
scientific reasoning skills. Students download each week's lab worksheet from the
course website before coming to the lab room, but we do not expect them to have
read the lab. The first page of every lab summarizes the underlying concepts and
provides a summary of the lab goals and procedures. Typically, the topics of the labs
have previously been discussed in the course lecture. At the beginning of the lab
period, students are given a pretest that we use to gauge their understanding of the
concepts and procedures related to the lab material and that have been helpful in
designing the lab exercises. The pretest takes less than 10 minutes. Some teaching
assistants then give students a brief summary of the physics that they will be using in
the lab and of any procedural issues that are anticipated; others just let students start
immediately after the pretest.
The labs follow a guided inquiry format. Students work in groups of 3 or 4 to
answer the questions on the lab worksheet. The teaching assistant circulates from
group to group, helping with procedures and asking students questions to assess their
understanding of the material. Students are often asked to predict what they expect to
happen before performing experiments, and they are expected to interpret their
observations in light of physics principles.
19
As in the Tutorials, students are
occasionally presented with student dialogues and are asked to critique the reasoning
presented in these dialogues.
In some places, students work on developing the
procedural skills (for example, drawing free-body diagrams or subtracting vectors)
that are associated with the lab topic. Often the physics principle that is under study
is only presented after students have done experiments related to that principle; in
these cases they are asked to verify that the observations they have made are
consistent with this principle. The lab worksheets are typically 6-10 pages long.
Selected relevant labs and their associated homework are included in appendix B.
Although the lab period is 3 hours long we have tried to design the labs so that most
students are done in 2 - 2.5 hours. Each lab section has one teaching assistant and
there are usually around 20 students in a section.
There are no lab writeups. Each lab has an associated homework assignment
that is intended to reinforce the underlying concepts, to require students to extend
these concepts to other contexts, and to give students practice with the associated
procedural skills. These homework assignments are turned in at the beginning of the
lab at the following meeting.
For research purposes, success in our lab course is primarily measured
through their scores on our lab final exam. The lab final is given during the last lab
meeting and has roughly 36 questions (3 per lab), and counts for 20% of the students'
grade. Most of the questions are qualitative and focus on concepts that have been
explored during the lab meetings. Students may take up to the full 3-hour lab period
to complete the final: In practice, most are done in half an hour.
20
In the next section, we give a detailed example in the context of 2-dimensional
acceleration, of the modifications we have made to the introductory mechanics labs at
NMSU, and of our attempts to measure the effectiveness of these changes.
Example of Curricular Modification: Two-Dimensional Motion
Motivation
Physics instructors would like to have students leave an introductory physics
course with an appreciation of the laws of physics as powerful unifying principles. A
more typical outcome, unfortunately, is that students come to the conclusion that
physics is a collection of specialized rules that apply to specific cases, and that there
is a lot to remember. The Motion in Two Dimensions and Newton's Second Law labs
we describe in this chapter (also in appendix B) were motivated by a desire to
improve our students' conceptual understanding of Newton's second law as an
equation relating two vector quantities. In the past few years, physics education
researchers have documented25'26'27 introductory mechanics students' difficulties with
vector addition and subtraction and with acceleration and force as vector quantities.
In particular, at NMSU, we had found that students in our introductory algebra- and
calculus-based courses typically could not (1) reason qualitatively about the
magnitude and direction of vector sums and differences; (2) find the direction of an
acceleration vector based on a change in velocity; or (3) reason about the relative
magnitudes of individual forces in a free-body diagram based on knowledge about the
21
direction of the acceleration.
Without an understanding of how the direction of
acceleration is obtained, and without recognizing how the acceleration direction
constrains the relative magnitudes of the forces acting on a body, it is hard to imagine
that students form a sense of Newton's laws as unifying principles. Instead, they are
left with rules that apply in individual circumstances but that do not connect to one
another.
For example, motion in a line and motion in a circle become separate
situations with different equations that govern them.
In a study of concept interpretation in physics, Reif and Allen
outline five
steps necessary to apply the definition of acceleration to the motion of a particle: (1)
Identify the velocity v of the particle at the time of interest; (2) Identify the velocity
v' of the particle at a slightly later time t' ; (3) Find the small velocity change
Av = v' — v of the particle during the short time interval At — t' — t ; (4) Find the
ratio Av/At and (5) Repeat the preceding calculation with a time t' chosen
progressively closer to the original time /. They note that understanding and applying
the definition of acceleration has many complexities that are hidden by the seemingly
simple definition a = Av/At - f o r example the procedure for subtracting vectors.
Reif and Allen suggest that the procedural knowledge necessary to interpret
the definition of acceleration is usually not adequately taught. They recommend that
the presentation of operational definitions of concepts such as acceleration be
followed by the opportunity for practice in a variety of special cases. This emphasis
on operational definitions is reiterated by Arons and by Shaffer and McDermott. '
22
As part of their study, Reif and Allen asked students in an introductory
physics course for scientists and engineers qualitative questions about the directions
and relative magnitudes of accelerations in 13 different situations. Even though these
students had been using acceleration as part of the course for at least two months,
none of them were able to answer more than 3 of the 13 questions correctly. In
answering the same questions, physics faculty members rarely used the definition of
acceleration, and often used information about forces to answer questions that were
purely kinematical. Both experts and novices had difficulty applying the definition of
acceleration to objects moving in a curved path. They tended to use case-specific
knowledge about acceleration rather than working from a more general definition,
and both groups often used this case-specific knowledge incorrectly. Shaffer
found
similar difficulties: Asked to indicate the direction of the acceleration of a swing at
five points along its path, none of 124 students in a calculus-based introductory
physics course answered correctly for all points. Of 22 physics teaching assistants
asked the same question, only 3 drew the acceleration correctly for all 5 points.
Similar results were obtained on physics comprehensive exams for graduate students.
Our content goals for these two labs, which form the main basis for this
chapter were for students to (1) gain practice at using the definition of acceleration to
describe the direction and to compare the magnitudes of accelerations for objects in
two-dimensional motion; (2) recognize that Newton's second law requires that the
direction of acceleration and of net force be the same for objects in two dimensional
motion; and (3) be able to use kinematical knowledge to make inferences about the
23
relative magnitudes of forces acting on an object in two-dimensional motion. Our
epistemological goal was to have the labs contribute to students' sense of the
coherence of the kinematics and dynamics they were learning.
Tutorial on Motion in Two Dimensions
In order to address the conceptual and procedural difficulties described, the
implementation of the Tutorials in the labs at NMSU in 2000 included a tutorial on
two-dimensional motion called Motion in Two Dimensions that is included in
appendix B. This tutorial was based on extensive research conducted by the Physics
•in
Education Group at the University of Washington.
McDermott and Shaffer report
that, after relevant lecture instruction, only about 20% of students could consistently
find the direction of acceleration for an object moving at constant speed in two
dimensions. After completion of the Motion in Two Dimensions tutorial, about 80%
recognized that the acceleration was perpendicular to the trajectory at all points. For
an object moving with increasing speed on a two-dimensional trajectory, about 60%
could sketch the direction of acceleration after tutorial instruction. However, for
questions involving vertical motion (for example, questions about the acceleration of
a pendulum bob) only about 15% could answer correctly after tutorial instruction.
While there is clearly room for improvement, these post-test results are better than the
pretest results for physics graduate students working as teaching assistants in the
introductory course.
24
We wanted to replace the Motion in Two Dimensions tutorial in our lab
sequence for three reasons. First, the tutorial consisted entirely of pencil-and-paper
exercises, and so did not fit well with student or instructor expectations, and lacked
the opportunity for students to reason based on observation that we would like as part
of all of our labs. Second, we wanted to be able to use the labs with students in the
algebra-based course, and a significant portion of the tutorial is devoted to exploring
what happens to the direction of the acceleration vector in the limit of small time
intervals, a topic more appropriate to the calculus-based course. Finally, our posttesting revealed that our students were not performing as well after the tutorial as
students at the University of Washington.
Laboratories Using Long-exposure Digital Photography
We wanted to create a set of laboratory exercises that went beyond the penciland paper exercises of the tutorial, allowing students to start with observations of
actual motion. At the same time, we wanted to emphasize the procedural steps
necessary to find the acceleration of an object as described previously, and to allow
students to practice these steps as part of the laboratory. In addition, we hoped to use
the same lab procedures in subsequent labs to allow students to connect the
acceleration vector for an object moving in two dimensions to the net force acting on
that object at an instant in time. As an overall objective, we hoped to offer students
25
lab experiences that would allow them to think conceptually about Newton's second
law as a vector equation.
We decided to incorporate long-exposure digital photography as a means of
connecting actual motions with representations of that motion that were amenable to
graphical analysis. In the Motion in Two Dimensions lab described in this section,
and in the other labs described in this chapter, students use inexpensive digital
cameras to record the motion of objects with attached blinking light emitting diodes
('blinkies'). These photographs can then be used to make inferences about the
velocity and acceleration of the objects at different points along their paths.
Here we give details about the equipment we use and describe the sequence of
exercises that we developed for a two dimensional motion lab. In the next section we
discuss the lab on Newton's second law. We will also briefly describe a more recent
introduction of the same digital photography techniques into labs on conservation of
energy and on rotational motion. Finally, we present some pre- and post-test data
from questions related to the concepts underlying these labs.
Equipment
The labs we describe here require a digital camera that has a manual setting or
a 'shutter priority' setting.31 We use older cameras
that we buy used online for
about $60. Students photograph a moving flashing light emitting diode (LED) using
exposure times of between one and four seconds. A similar photographic technique
has been described by Terzella et al. for measuring the acceleration of gravity.
26
Because we intend the lab to be used in the algebra-based course, we restrict the lab
to finding average accelerations over short time intervals. By using flashing LEDs
with only short 'off periods we have students create photographs with light streaks
that look very much like motion diagrams.
Although we now have one camera for each lab group, we have run the labs
with as few as three cameras for 20-24 students without causing significant 'wait
times' since the photographs do not take very long to make and download. We have
instructions for how to use the cameras at each lab station: Generally students are not
familiar with using a shutter priority setting, but have no difficulty downloading the
pictures and sending them to the black and white laser printer. The cameras use AA
batteries, which often need to be recharged during a one-week lab cycle. Software at
each lab station (currently Adobe Photoshop Elements) allows students to invert the
photographs before they print them, so that the dark background is printed as a light
grey and the light streaks are printed as dark lines. This saves toner and makes it
easier to write on the prints. Additional equipment includes cables for downloading
the pictures from the camera, an inexpensive 'gooseneck' lamp at each lab station (so
that students can read and write while other groups are taking photographs), and
several tripods.
We tried a variety of flashing light sources34 before deciding to build our own
so that we could choose a flash rate and control the fraction of each period that the
LED was on (the duty cycle). Each blinkie has an on/off switch, a second switch to
change the flash rate, and a 'trimpot' variable resistor to control the duty cycle.
27
In
the future we will probably replace the variable resistor with two fixed resistors that
set the duty cycle at about 90%. The circuit is powered by two CR2032 3-volt disc
batteries installed in a holder on the back of the board. We have not yet had to
replace a battery except when the circuits were accidentally left on. The circuits are
about 2.4 cm wide and 4.8 cm long.
A +6V
555
Timer
- ^
I.C.
1 3
C1TJC2]_
VLHD
Y
S2
4r0V
Figure2.1. Blinkie circuit. Switch SI turns the circuit on; S2 increases the time
constant. The potentiometer controls the fraction of time in each cycle that the LED
is on.
For the Motion in Two Dimensions lab, the circuit is mounted on a toy
hovercraft, on a Pasco roller coaster cart, and in a PVC plumbing pipe cap used as a
spherical pendulum bob, as shown in Figure 2.2.35
28
1&
^W-^\
f
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the change in velocity for a short time interval. Finally, we want them to recognize
that this direction is the direction of average acceleration.
Procedure for Finding Acceleration Direction
As described by Van Heuvelen,
the procedure for using motion maps begins
with a drawing of velocity vectors along the object's path. The change-in-velocity
vectors for points of interest are then found by subtracting the vector just before that
point from the vector just after that point, and this allows a qualitative determination
of the direction of acceleration for that point. We have modified this procedure for
our labs. A long-exposure photograph of an object with a blinkie attached shows the
path of the object. The photograph is printed, and students draw vectors with tails at
the beginning of each light stripe and heads at the end. Each vector they have drawn,
then, represents the displacement during the time that the blinkie was on. Since each
of these vectors represents a displacement for the same duration, their lengths are
proportional to the average speed of the object, and they can also be used as average
velocity vectors for the time intervals that the blinkie is on. Two adjacent vectors can
be used to determine the change in (average) velocity from one light stripe to the
next: we assign this change-in-velocity vector (and the associated average
acceleration) to the space between adjacent vectors. The lengths of these vectors
allow a qualitative comparison of the magnitudes of the acceleration for various
points along a path.
30
Introductory Pencil-and-paper Exercises
In the first section of the lab the operation of the blinkie circuit is described
and an example photograph is given. A drawing of what a blinkie trail might look
like for a hovercraft that is kicked so that it changes direction and speed is shown, and
students are asked to make inferences about the speed of the hovercraft based on
measurements taken from the drawing. Students draw scaled vectors to represent the
hovercraft's velocity before and after the kick.
They are guided through the
procedure for graphical vector subtraction and then determine the change-in-velocity
vector for the hovercraft.
The direction of the acceleration of the hovercraft is
determined for the kick. Students then find the change-in-velocity vector for a portion
of a drawing based on a blinkie photograph, and are asked some qualitative questions
about this vector.
Analysis of Long-exposure Photographs of Moving Objects
Following this extended introduction, students take long-exposure digital
photographs of three different objects in two-dimensional motion and determine the
acceleration direction for each motion following the procedure described in section i.
Because we have only one station for each experiment and there are up to 8 lab
groups, we let groups take these photographs in any order. The three motions - of a
spherical pendulum, a hovercraft on a ramp, and a toy roller coaster - are intended to
provide practice with the procedure for determining an acceleration, and to illustrate
that the same procedure and definition are applicable across a range of motions. For
31
each of these motions students are asked to print out an inversion of the photograph
they obtain, and to find the acceleration vector for various points along the path. We
briefly describe details of each exercise.
Central force motion:
Spherical Pendulum.
A spherical pendulum
approximately 2 meters long is suspended from the ceiling. The LED protrudes
through a hole in the center of the bottom surface of the pendulum bob. The camera
is placed on the floor (about 1.5 meters below the pendulum bob) pointing up towards
the suspension point of the pendulum. Students push the pendulum bob so that it
moves in an ellipse. Figure 2.3 is a photograph of this motion, with an example of the
vector construction we expect our students to perform in order to find the direction of
the change in velocity. The point of suspension can be seen in the photographs, and
when asked to give a description for the acceleration direction that is true for all
points on the path, students generally state that it is toward the center of the ellipse.
Parabolic Motion: Hovercraft on a Ramp. Students push a toy hovercraft
so that it travels in a parabola along a piece of plywood with one edge raised so that it
is at about a 4° angle. The camera is mounted on a tripod that is placed at the bottom
of the plywood ramp. The board has a grid painted onto it so that the pictures that are
taken have horizontal and vertical reference lines. A long-exposure photograph taken
of this motion is shown in Figure 2.4.
For this exercise, the camera is further from the top of the plywood ramp than
from the bottom, so a light stripe in the photograph will be smaller for a hovercraft in
32
motion near the top of the ramp than it will be for a hovercraft moving at the same
speed near the bottom. When the vector generated from a light stripe at one location
is subtracted from one generated at another location this introduces an error in
magnitude and in direction for the change-in-velocity vector. In general, these errors
W
.'
r\tMr.+
is*rk*-if
Aw
4/
<••?-
«*'-"
Figure 2.3. Photograph of spherical pendulum moving in an ellipse. The blinkie flash
rate is about 10 Hz, and the exposure is 1.5 seconds, less than the time of a complete
orbit to avoid overlap of stripes. Superimposed on the photograph are examples of
vector differences based on the length and direction of the light streaks just before
andjust after the point of interest.
33
Figure 2,4. Long-exposure photograph for the toy hovercraft moving in a parabolic
path. The blinkie flash rate is about 3 Hz, and the exposure is 3 seconds.
are small enough that they do not prevent the students from concluding that the
acceleration direction is down the ramp, parallel to the grid lines in the photograph.
We intend to modify this lab so that the camera is mounted above the plywood and is
pointed directly down to minimize this distortion.
General two-dimensional motion: Roller coaster. In this exercise, a toy
roller coaster cart travels down a flexible track and is launched into a container filled
with Styrofoam peanuts. The cart has a blinkie circuit mounted on the top with the
LED placed through a hole in the side of the cart close to the center of mass.
34
mmmm»:mmmmM
Figure 2.5. Long-exposure photograph of a roller coaster cart traveling down the
track and then being launched into a catch basin. The blinkie flash rate is about 20
Hz, and the exposure is 2.5 seconds. Students are asked to find the acceleration
direction for points A—Eas a homework exercise.
The resulting photograph (see Figure 2.5) Is the basis for the initial homework
exercises associated with this lab.
Since the lab generally takes less than 2.5 hours9 many student groups choose
to do this part of the homework in the lab room together. Subsequent homework
exercises give students practice with graphical vector subtraction with finding a
ehange-in-velocity vector from two velocity vectors, and with determining the
direction of an acceleration vector for objects moving along various paths.
N©wtom's §®<s©ondl Law Lalb
The lab that follows the Motion in Two Dimensions lab, called Forces,
introduces weight, normal forces, tension, and friction. The directions of these forces
3
and the factors affecting these forces are explored, and students are given some
practice drawing free-body diagrams. The fifth lab, Addition of Forces, provides
practice with vector addition, with reasoning about the relative magnitudes of forces
for situations where the net force is zero, and culminates in exercises using the force
table. The lab we describe in this section, Newton's Second Law, is the sixth lab in
the sequence, and requires students to reason about relative force magnitudes for
situations where the acceleration is not zero.
Introductory Pencil-and-Paper Exercises. The initial exercises for the lab
are again pencil-and-paper. Students use a photograph of the two light stripes
adjacent to the bottom of the roller coaster to determine that there is a large change in
velocity and therefore a large acceleration upward at that point. They draw a freebody diagram of the coaster at the bottom of the track, and are guided through the
reasoning required to compare the magnitudes of the normal force on the cart and the
weight at that point.
Lab Exercises Connecting Force to Acceleration. In addition to the
equipment mentioned earlier, for the Newton's Second Law lab, blinkie circuits are
also mounted onto a block of wood cut to fit a 1.2-meter Pasco track, and onto a thin
board that is used as a physical pendulum. The Pasco track is mounted on a hinge so
that we can adjust its angle to the horizontal.
In this lab, students take photographs of four different motions: circular
motion with changing speed, linear motion with acceleration both in and opposite to
36
the direction of motion, and motion with no acceleration. For each motion, they
determine the direction of acceleration and draw a free-body diagram for the object at
one or more points, and use the direction of the acceleration to reason about the
relative magnitudes of the forces in the free-body diagrams. Our intention is to
emphasize that Newton's second law is a unifying principle that applies to all
motions, and serves as a common theme for much of what they have studied up to
this point.
Physical Pendulum. The physical pendulum is a board approximately 2
meters long with a blinkie circuit at one end. From a photograph of the swing (see
Figure 2.6) students determine the direction of the acceleration at two points. They
draw a free-body diagram of the blinkie circuit for these points, and are asked to
determine the relative lengths of the tension and the weight vectors that results in a
net force direction consistent with the acceleration.
Constant Velocity: Hovercraft on a Level Surface. The toy hovercraft is
pushed across a plywood board that is level on the floor. Students take a 2-second
photograph of the hovercraft, and determine that there is no change in velocity and
therefore no net force. From a free-body diagram of the hovercraft, they reason about
the relative magnitudes of the forces on the hovercraft.
37
Figure 2.6. Photograph of a single swing of the physical pendulum. Students
determine that the acceleration is not toward the center of the circle for points where
the speed of the blinkie is changing.
Lnnnear M©ttt©ms Bltaxek ©m a Manrnpo In the last two exercises In this lab,
students photograph a blinkie attached to a block that slides down a 1.2-rneter long
Pasco ramp. The angle of the ramp is such that the block speeds up when it slides
with the wood surface In contact with the rarnp but slows down when it Is turned over
so that a cork surface on the other side of the block is In contact with the ramp (see
Figure 2.7). For these two cases, students determine the relative magnitudes of the
three forces in the free-body diagram consistent with a net force in the direction of the
3
eieration. They then compare the forces for 1
T>* *
Igure 2.7.
The
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Other Labs Using Long-exposure Digital Photography
In this section we briefly describe two additional labs that we have developed
in the past couple of years that use the blinkie circuits and long-exposure
photographs. These are the seventh and ninth labs in the sequence, Energy, and
Rotational Motion (also see appendix B).
Energy. The Energy lab is intended to introduce kinetic and potential
energies and the conservation of mechanical energy. Again starting with a
photograph of the roller coaster, students observe that the length of the light streaks is
about the same for locations that are at the same height. They measure the lengths of
light streaks at different heights, and from this find that the speed of the coaster
doubles when the height below the release point quadruples. Expressions for kinetic
and potential energy are introduced, and the measured results are compared to what
would be obtained if the potential energy lost from the release point to a point below
it on the track was all converted into kinetic energy.
Students then look at factors that influence the speed of the roller coaster as it
travels down a ramp. Two tracks are set up with different slopes and shapes but with
the same starting and ending heights, and students are asked to predict whether the
steeper track will result in a higher terminal speed.
In addition, they are asked
whether adding mass to the cart would result in a higher terminal speed. Students
then photograph the coaster as it travels down the shallow track (see Figure 2.8),
40
down the steep track, and down the steep track again with added mass, and the
lengths of the light streaks at the bottom of the tracks are compared.
Figure 2.8. Long-exposure photograph for the roller coaster cart traveling down the
lesser steep of two tracks.
At this point the idea of energy conservation is introduced and the results
obtained up to this point are discussed in light of energy conservation. We make use
of energy bar charts to emphasize the idea of a conserved quantity,37 and students
construct these bar charts with different reference heights.
Rotation. The Rotation lab is an introduction to the kinematics of rotation
and includes a qualitative introduction to moment of inertia. The lab uses two
plywood disks, each about 25 centimeters in diameter, that rotate on bearings. One
disk has 4 LEDs, one at the center and the other three at different radii, all driven by
41
the same blinkie circuit. Students measure the flash rate of the blinkie circuit, and
then spin this disk and photograph it using a one-half second exposure.
Figure 2.9. Long-exposure photograph for a plywood disk rotating at approximately
constant angular velocity. In addition to the LED in the center, there are 3 LEDs at
different radii, all controlled by the same blinkie circuit.
From this photograph they verify that the arc length of each light streak is
proportional to its distance from the LED at the center of the disk, and that the
angular displacement during one blinkie period is the same for the three rotating
LEDs. (See Figure 2.9) They then calculate the angular velocity of the disc.
The second disk has a blinkie at the center of the disk and one near the outside
of the disk. A groove in the edge of the disk allows students to wind a string attached
42
to a weight around the disk. When the weight is released, the disk rotates with
Increasing angular velocity. Students photograph the disk and measure the chan
m — AB/M from one blinkie cycle to the next (See Figure 2.10).
Figure 2.10.
Long-exposure photograph for a plywood disk rotating
approximately constant angular acceleration. The torque is created by a
falling weight attached to the string visible at lower right.
!##%*«
•
'
W
•
v
• <0fo&®&-
Figure 2.11. Plywood disk with four removable lOOg cylindrical masses inserted into
four holes near the outer edge of the disk. Students compare the angular acceleration
of the rotating disk for this configuration of masses, to a configuration where the
masses are inserted into the inner four holes (vacant in this picture).
From this measurement they calculate the (roughly constant) angular
acceleration of the disk. This disk has four holes a small distance from the center and
four more holes near the outside. For the first photograph, brass cylinders are
inserted in the Inner holes. The experiment is repeated with the same falling mass but
with the brass cylinders moved to the outer holes (see Figure 2.11). Students again
calculate the angular acceleration of the disk and compare their result to what they
obtained previously. These results are used to motivate a qualitative introduction to
Measurements of Effectiveness of the Labs
In this section, we report results from the pretesting and limited post-testing
that we have conducted in order to measure the labs' effectiveness. As mentioned in
chapter one, free-response pretests are given at the beginning of each lab that give us
some sense of students' initial understanding of that lab topic. We use questions on
the multiple-choice lab final as a crude measure of their understanding at the end of
the semester. As part of the lab development, we have also used student responses to
homework questions to gauge lab effectiveness.
We encourage students to work
together on the homework with the caution that what they turn in should be their own
work. In practice, it is often hard to tell whether a correct homework response
reflects individual or collective understanding of the topic.
There are only a few questions on nationally administered assessments that
ask for information about force based on knowledge of acceleration. We report
below, results from two of these questions asked on one of our final exams. Most
questions about force and acceleration on widely-used
assessments (such as the
Force Concept Inventory) are for one-dimensional motion, or are for one component
of parabolic motion, and do not test for student understanding of force and
acceleration as vector quantities. Because the focus of the acceleration labs we have
described is on the vector nature of force and acceleration, most of the assessment
questions we have asked are questions that we have written, or are questions that have
45
been asked as part of other research projects into student understanding of twodimensional kinematics and dynamics. '
Results on Questions About Acceleration in Two Dimensions
In Fall 2009, when questions about the direction of acceleration for an object
in two-dimensional motion were asked on a pretest, about 4% of our students
answered correctly, which is consistent with pretest data reported by Shaffer and
McDermott.27 On the Mechanics Baseline Test, the question shown in Figure 2.12a
asks for the direction of the acceleration of the block when it is at the position shown.
Reported results39 range between 12% and 18% correct after standard university
instruction. About 39% of 135 students in our algebra-based course and 51% of 88
students in our calculus-based course answered this correctly on the lab final.
Nagpure38 gives results for a question about a dog that is speeding up as it moves
along a curved path (Figure 2.12b).
After tutorial instruction using a modified
version of the Motion in Two Dimensions tutorial at the University of Maine, 44% of
the students in their algebra-based course and 69% of the students in the calculusbased course answered correctly. We asked a multiple-choice version of this question
on our lab final: About 20% answered correctly in both algebra- and calculus-based
sections. For a similar question about an object changing both direction and speed,
Shaffer and McDermott report27 about 60%) of students in a calculus-based course
answering correctly after tutorial instruction.
46
11a
lib.
lid.
r"7
11c.
Steel
cable
,200 m/'s
FJevator
going up
at constant
speed
350 m/s
Figure 2.12a. Question from Mechanics Baseline Test about acceleration for twodimensional motion. Figure 2.12b. Question about acceleration direction for a dog
speeding up. Figure 2.12c. Question about direction of net force for an object
changing direction and speed. Figure 2.12d. Question from Force Concept Inventory
requiring reasoning about forces based on information about kinematics.
Results on Questions Relating Force Direction to Acceleration Direction.
One question on the Force Concept Inventory40 asks for a comparison of
forces based on information about acceleration: For an elevator moving upward at a
constant speed (Figure 2.12d), 61% of students recognized that the tension in the
cable would be equal to the weight of the elevator after standard university-level
instruction. We asked this question on our lab final exam: About one-third of the 223
students in both courses answered correctly. For an object that is speeding up as it
moves in a curved path (Figure 2.12c), about one-quarter of the students in our lab
courses correctly chose the direction of the net force (about one-third gave a direction
toward the center of the curve). We have obtained similar results after extensive
47
practice in the lecture portion of a course at NMSU. We do not have data from other
institutions to compare this to.
For Spring 2009, there were 6 questions that were asked both as free response
questions as part of the pretests and as multiple-choice questions on the lab final.
(Five of these questions were about the rotation and energy labs, because these labs
were introduced in the 2008/2009 academic year for the first time and we wanted to
see whether they were working.) The distractors for the multiple-choice questions are
based on answers given on the pretest.
Students were not given answers to the
pretests, nor were the pretests discussed after they were administered. In Table 2.1
we give results from the four questions from the Spring 2009 exam (one question
based on each of the four labs that have been described in this chapter). Questions 16 in appendix C are the multiple-choice final exam versions of these questions. No
data are shown for students enrolled in the physics course for majors because it is not
offered in spring semester.
In addition, we show some data that compares results from students enrolled
in different lecture sections of the calculus-based course in Fall 2008, one taught by
one of the lab designers (SK) with an instructional focus that was well-matched to the
labs, and with homework assignments that included conceptual free-response
questions as well as quantitative problems. The other section was taught with less
emphasis on graphical techniques or on qualitative analysis, and used online
homework that was primarily quantitative. Questions 7, 8, and 9 in appendix C are
48
Table 2.1. Student performance on matched pre/post questions for spring 2009.
Algebra-based sections
Calculus-based sections
Pretest
Lab Final
Pretest
Lab Final
Net Force direction (Question 1):
22% (N = 50)
44%(N = 71)
19%(N = 79)
45% (N = 85)
Energy (Question 2):
26% (N = 65)
61%(N = 71)
47% (N = 83)
75% (N = 85)
Energy (Question 3):
52% (N = 65)
82%(N = 71)
58% (N = 83)
87% (N = 85)
Energy (Question 4):
14%(N = 65)
39%(N = 71)
22% (N = 83)
47% (N = 85)
Rotation (Question 5):
41% (N = 63)
65%(N = 71)
43% (N = 60)
71%(N = 85)
Rotation (Question 6):
41%(N = 63)
62%(N = 71)
40% (N = 60)
69% (N = 85)
Table 2.2. Comparison of student performance on Newton's Second Law questions
for lecture sections with different instructional emphases.
Lecture
not aligned
with lab
(N = 46)
Lecture
aligned
with lab
(N = 40)
Newton's Second Law (Question 7):
41%
65%
Newton's Second Law (Question 8):
15%
38%
Newton's Second Law (Question 9):
63%
55%
the questions related to the Newton's second law lab taken from the lab final exam
for Fall 2008. In Table 2.2 we compare results for these three questions for students
in the section taught by the designer ("aligned") with students in the other section
("not aligned").
49
Discussion
Student performance in general on the lab final exam is disappointing. In
Spring 2009, the average score for students in the algebra-based lab sections was 40%
and for students in the calculus-based lab sections it was 47%. (We grade on a
curve!) In contrast, for the 10 students in our calculus-based course for majors, the
average score was 78%. While we believe that the labs are improving our students'
conceptual understanding of the material, there are still a large number of students
who are not benefiting as much as we would like. We do not know whether there are
significant improvements that can be made to student performance by improving the
labs, or whether the constraints inherent in a 1-credit course with only one meeting
per week means that only incremental gains can be achieved. We will continue to
assess student performance and to modify the labs where appropriate.
Generally, however, there is improvement in student performance from pretest
to final, as can be seen in Table 2.1. It is almost certainly true that not all of this
improvement is due to the lab, as students are also learning about and applying the
related physics concepts in their lecture course. Nonetheless, we believe that for our
students these labs have been at least as successful in promoting student conceptual
understanding as the more traditional labs and tutorial that they replaced.
When the goals and instructional focus of the lecture portion of the course is
aligned with the goals and instructional focus of the lab, students generally do
somewhat better on the lab questions. (There are, however, exceptions, such as the
results from question 9 shown in Table 2.2) Final exam scores averaged 60% for
50
students enrolled in one of the lab developers' lecture section; the average score was
50% for students from the other section.
We believe that this reinforcement is
important in both directions, and that the lab experiences contribute strongly to
student understanding of the material in the lecture course. One of the test sites for
this lab development project, Chicago State University, has chosen to intersperse
portions of the labs with their lecture instruction, and this seems to be a very
promising model for promoting student understanding.
When looking at student understanding of force and acceleration as vector
quantities, our pretest results seem to be consistent with results reported elsewhere
and suggest that without intervention only a small fraction of students will be able to
determine the direction of the acceleration for general two-dimensional motion.
There is generally improvement in student performance from pre- to post-test.
However, most students do not seem to be applying the procedural knowledge that we
are attempting to encourage with these labs.
Results are also poor for questions requiring reasoning about forces based on
knowledge about acceleration direction. Our post-testing indicates that most students
in our lab do not develop a functional understanding of Newton's second law as an
equation relating two vector quantities.
51
Conclusion
The labs described in this chapter were designed to try to improve our
introductory physics students' understanding of the vector nature of acceleration and
force as well as to improve their conceptual understanding of conservation of energy
and the relationships among quantities characterizing rotational motion. After taking
the traditional labs and even after TUP we found our students' understanding in these
areas in particular to still be deficient and sought to use digital photography of various
moving objects fitted with blinking LED's to deepen student understanding in some
of these content areas. We hoped that students practice with vector manipulation in
real world examples of objects moving along curved paths would help develop the
necessary tools for them to appreciate the utility of Newton's Second Law in many
situations and thus seek to use it in understanding a broad range of circumstances.
Previous research at other universities indicated that most students cannot readily
determine the direction of an acceleration after standard instruction. Our labs seem to
also have been marginally successful in improving student performance in this regard.
Students do not seem to have significant difficulties with the lab procedures or
with the equipment, and they generally like making the photographs and are able to
perform the required analysis correctly in the lab. However, post-testing of student
understanding of the underlying concepts has yielded disappointing results. Some
students, for example our physics majors, do quite well with the post-test questions.
On the other hand, many students in the algebra-based course — our intended
52
audience — do not. There are also different success rates among the various topics
presented, with students improving performance on concepts related to energy and
rotations more than they improved on concepts related to Newton's Second law.
The differences in performance among our different introductory physics
populations, as well as differences in the successful application of curricular
modifications at the University of Maine and the University of Washington
heightened our interest in attempting to predict which among our students were more
likely to grasp the concepts presented in lab. We sought through the use of pretests to
identify potential factors that may contribute to student understanding of the topics
presented in the labs. Our attempt to identify some of these factors is detailed in
chapter four of this dissertation.
Before these labs were developed we attempted to address the same
instructional goals through extensive use of motion maps as part of the lecture
instruction.24 This also met with limited success. As a stand-alone intervention, the
labs seem to be only marginally successful, at least as measured by our multiplechoice lab final. There is some encouragement however in the fact that there is some
improvement in student performance when the labs are closely aligned with lecture.
We hope to teach a lecture section that emphasizes the vector nature of kinematic and
dynamic quantities in the near future with these labs in place as reinforcement of this
approach to further improve student understanding. We believe that the goals that
have motivated the design of these labs are important ones and we will continue to
53
attempt to explore factors which may assist in helping us achieve these goals, as well
as potential barriers to students' understanding.
54
CHAPTER 3: PROPORTIONAL REASONING AS A PREDICTOR
OF S U C C E S S IN P H Y S I C S
Introduction
The lab curriculum development project described in chapter 2 was an attempt
to improve our students' understanding of physics concepts through the application of
physics education research that was primarily done at other universities with different
student populations. We had less success using the research-based materials
developed at the University of Washington than the developers of those materials
had. One obvious possibility was that the implementation of these materials was
deficient in some way. However, our collaborators at California State University
Fullerton and Arizona State University had also seen poorer student performance
when implementing the Tutorials than they had observed at the University of
Washington. A second possibility was that the difference in student populations
made a significant difference in what we might expect. The Tutorials were developed
through research with students who were in the calculus-based introductory physics
course at the University of Washington. The majority of these students have taken
high school physics, and admission to the UW is reasonably selective. These students
are pre-engineering majors, and must pass these physics courses to be admitted into
the engineering program. The engineering program is competitive: for aeronautics
engineering, "admitted students generally have grade point averages well above 3.0;"
for electrical engineering "the average admission prerequisite GPA varies between
55
3.4 and 3.6."
In contrast, NMSU is less selective, and fewer than half of our
students have had high school physics. Students are admitted directly into the
engineering program, and often do not take physics until after they have taken some
engineering courses. In general, our students are not as well prepared for physics as
the students who participated in the research at the University of Washington.
One of the motivations that we had for focusing on students in the algebrabased introductory mechanics course was that we had observed that the students in
this course seemed to struggle with the ideas presented in the lab more than students
in the calculus-based course. While this might not be surprising, this difference in
performance was apparent even on questions that required very little math.
Anecdotally, physics instructors often comment on the differences between these two
student populations in terms of both ability and attitude toward physics.
For these reasons, we wanted to have some measure of the level of our
student's preparation for physics courses. Are there predictors for success in physics
that would allow us to compare how well one student population is prepared
compared to some other student population? Is it possible to characterize what makes
students in the algebra-based course different than students in the calculus-based
course? In this chapter we describe previous attempts to measure preparation for
physics, and our own experiments with using proportional reasoning pretests as a
predictor for success. We do not believe that one can have a complete understanding
of how a factor, such as proportional reasoning, influences success in a physics
course without also considering other closely linked factors. To this end, we also
56
attempt to give a brief perspective on related influential cognitive theories that have
guided previous attempts to identify factors predictive of success in physics courses.
Description of Statistics Used
Throughout this dissertation we will use standard (frequentist) statistical
hypothesis testing when measuring the significance of differences in 'treatment' and
'control' groups. Central to this technique is the idea of the null hypothesis. Under the
null hypothesis, it is assumed that the two groups being compared are not inherently
different. In general, data is collected and a test statistic is calculated that describes
how far the data fall from the null hypothesis value.42
Test Statistics
One of the oldest test statistics still in use today is the chi-squared statistic, x2It establishes the contingency between two variables by comparing the observed
frequency of data values (f0) against the frequency one would expect to observe for
these data values under the null hypothesis (i.e. no association between the two
variables), fe. Its value is calculated from x2 = E
f
'e
where the summation is
taken over all cells in a contingency table. Larger values of this statistic indicate that
the null hypothesis of independence of variables may be false.
Another test statistic used extensively in chapter four of this dissertation is the
t-statistic. The t-value measures the number of standard errors (dy) that the sample
57
mean (Y) falls from the null hypothesis value (uo) - t = -—•. For a random sample of
size n, the standard error in 7, is related to the population standard deviation a by:
dy = -j=. Again, using a t-test, larger t-values indicate increasing un-likeliness of the
null hypothesis.
Associated with each test statistic is the P-value. This value tells us the
probability of exceeding the value of a test statistic under the null hypothesis
condition49 (that there was no inherent difference between experimental and control
groups to begin with). We will keep with what has become convention since
Fisher's 1925 work43 in considering p<0.05 as being statistically significant for test
statistics - i.e. there is sufficient evidence to reject the null hypothesis. It should be
kept in mind that non-significantly different sample means do not imply equal
population means, but may imply that there is not a large enough sample size to
detect any difference.44
Often throughout chapter four, our sample sizes from the two populations will
be of different sizes and will possibly have different variances. The WelchSatterthwaite t-test is used when the two population variances are assumed to be
different (the two sample sizes may or may not be equal)45 and hence must be
estimated, and therefore in this dissertation it is often also quoted separately.
The effect size46 of a treatment, also called Cohen's d, is a measure of the
difference between the treatment and control means (ma - nib) as a fraction of their
common standard deviation (a) - d = ™a
mb
58
. Researchers have kept with Cohen's
convention in referring to effect sizes of about 0.2 as small, effect sizes around 0.5 as
medium, and effect sizes above 0.8 as large.
Correlation Coefficients
The correlation coefficient is a well defined numerical index which describes
the degree to which a relationship between two random variables can be modeled
linearly. It is commonly represented by the letter 'r' or the greek letter 'p' and
provides a number to associate with the correspondence between variables observed
on a scatterplot diagram. It ranges in value from negative 1.00 to positive 1.00.
The magnitude of the correlation coefficient describes the degree of association of
the two variables, while the sign tells us the nature of the relationship: a positive
correlation coefficient tells us that a large positive value of one variable coincides
with large positive values of the correlated variable, while a negative coefficient tells
us that large positive values of one variable coincide with large negative values of its
correlated variable (see Fig. 3.1).
Researchers are usually interested in predicting some measure of student
success such as students' scores on a final exam or students' final grades. In a
scatterplot diagram, these variables will usually appear on the y-axis. Factors which
may possibly be predictive of our chosen success measure will usually appear on the
x-axis. In general factors which are good predictors will be well correlated with our
response (success) measure.
59
r=-0.9
# m«
• * v
y
r=-0.8
•*
•
a
• •• • »
» «»
• ••
• •
• ••
*
»
•^
x
e
»
. .
r=0.0
' * .
•
•
»
« •
• •.» *«
»
r=0.8
*«
r=0.3
y
«»
r=1.0
>
• ••»
• • » •
** *
x
Figure 3.1. Sample scatterplot diagrams for two variables x andy and their
associated correlation coefficients. Adapted from Kachigan. 47
Cohen46 has popularized a rough, qualitative scale applicable to correlation
coefficients found in the social sciences. He referred to correlations of magnitude
above 0.5 as large, correlations between 0.3 and 0.5 moderate, correlations 0.3 to 0.1
small, and correlations less than 0.1 insignificant.
It is often instructive to researchers using correlation coefficients to focus on
their relationship to the coefficient of determination - commonly called R . As its
name implies the value of R is equivalent to the square of the value of the correlation
coefficient, r. Where multiple predictors are used the coefficient of multiple
60
correlation R, and the coefficient of multiple determination still called R , are the
multivariate analogs and satisfy the same relationship.
Usually the variable that we would like to predict, e.g. student scores on a
final exam, will have some distribution (it is unlikely that all students will have the
same score on a final exam). Nearly all reasonable distributions will have an
associated mean and (sample) variance. The variance of a distribution is a measure of
the average spread of the distribution's values from its mean. It is found by: s2 =
— Xi=i(yi — y)2- Whe n two variables are correlated, the average spread of values
from the 'line of best fit' is less than the average spread of values from the mean.
Using SStotaias a measure of the spread of values from the mean, and SSerroras a
measure of spread from the line of best fit, we can see that the ratio of ^ ^ plays an
ss
Total
important role in describing how well the line of best fit (linear regression model48)
'describes' some of the variance in the observed y-values (Fig. 3.2).
Statisticians talk about 'partitioning' the variance in y-values into that which
is attributable to regression (and by extension to the predictor or x-variable) and that
which is attributable to error.49 The value of/? 2 is calculated from these measures
by: R2 = 1 - f ^
= 55f/re"i0" Taking, for example, students' final exams as our
chosen measure of physics success, R2 tells us what fraction of total variation in
students' final exam scores (measured by SSjotai) is accounted for by our predictor
variable(s). Many researchers quote the correlation coefficient, with the knowledge
61
3.2(b)
r=0.3
SS
Totai = d x 2 + d 2
SSrotai = d i 2 + d 2 2 + d 3 2 + ••• + d 1 2 2
SSurror = ^ 2 + e 2 2 + e 3 2 + - + e 1 2 2 = 0
2
SSError
K = l-f^=l--p— = 1
scrotal
i? 2 =
S-'Total
=
l-
e
l
+ e2
:
Scrotal
+ d32 +
+ e3
+ '
• + d27
• + e 27 2
= (0-3)2
Figure 3.2. Scatterplots ofy vs x with mean y-value (dashed line) and line of best fit
(red solid line) superimposed. Circles below show enlarged view for various data
points' (counting from left for point # n) distances from the mean - d„, and distances
from line of best fit - en.
that squaring this number gives us an estimate of the fraction of total variation that is
accounted for.
Review of Previous Studies
Predictors of Physics Grade, Mathematics and Education Reform
In an early study of factors correlated to success in a physics course,
Blumenthal50 reported in 1961 on the correlations between grades on a mathematics
62
pretest and a final exam of an introductory physics course. His study involved 159
students over three semesters at City College of New York. The final exam for the
course consisted of two exams: Final exam I was a twenty-five question, multiple
choice exam 'concerned with physics knowledge' and final exam II was an 8question 'physics problem' exam. In his own words: "The term 'physics problemsolving abilities' as used herein refers to success in solving problems in physics
involving the use of mathematics."
Blumenthal found no significant correlation between students' scores on the
mathematics pretest with their scores on final exam I. However, there was a 0.4
correlation between final exam II and mathematics pretest. Both exams are
undoubtedly measures of success in a physics course, however, the difference in
correlation serves as a useful reminder of an inherent problem associated with
predicting success in physics - that various styles of exams used as measures of
success in physics courses may produce different correlations with predictors. This
phenomenon of different correlations based on qualitative vs quantitative physics
problems has been noticed by others.77In this dissertation, we will refer to exams that
have a focus on numerical solutions or on symbolic manipulation of equations as a
'traditional exam'. Exams that have significant focus on physics concepts and on
qualitative reasoning we will refer to as 'conceptual exams'. In recent years, this
distinction has been of particular importance to physics education researchers since
Hake54 has shown that courses structured where there is interaction among students
and between students and instructors, referred to as 'interactive engagement' courses,
63
tend to produce better gains in students conceptual understanding than traditional or
passive courses.
Mathematics as a Predictor
Mathematics is a natural first candidate as a predictor of success in physics.
The results obtained using mathematics as a predictor have been varied across
institutions and also seems to depend on the type of exam (whether traditional or
conceptual) used as a measure of success.
Adams and Garrett51 in their 1954 publication of their study conducted at
Louisianna State University, reported a correlation coefficient of 0.435 between
grades in a physics course and grades in a mathematics course. Only students who
successfully completed the sequence: mathematics course, first physics course,
second physics course, in consecutive semesters were included in the study. As
described later in this section, Rottmann et al. claim that this selection may have had
some effect on the measured correlations. The study was conducted over 3 years
from 1947-1949, and included 877 students in total. Physics grades in the first
physics course (taken in the Fall semesters) were used as the measure of success in
physics.
In contrast, Hudson and Mclntire used an eighteen-question mathematics
pretest with topics including parametric equations, linear equations, graphical
analysis, and trigonometry as a potential predictor of success. They conducted their
64
study on 200 algebra-based physics students at the University of Houston and found
only weak correlations1" between math pretest score and physics final score.
Hudson and Rottmann53 over a 3 year time-span, reported in a study involving
1403 students in an introductory physics course, 913 of whom completed the course
and 490 of whom subsequently dropped the course. They found a 0.42 correlation
with final grade and a mathematics diagnostic pretest for students that completed the
course and a 0.23 correlation coefficient between math pretest score and 'projected
final grade' for students who subsequently dropped the course.
The studies reported in this section, have found mathematics to be a useful
predictor of success in physics courses. Most of the studies cited also sought to use
other predictors but have found mathematics to be most useful for their course. Even
today, with emphasis on non-traditional modes of instruction, and with significant
gains being reported by instructors using these methods54'55 it still holds true that
'mathematical skill' is a good predictor of success in physics courses regardless of
instruction method.55 It is important to note however, that none of these researchers
found correlations above 0.5 in any of their studies even given the traditional nature
of some of their courses.
111
They did not calculate a value, however using their tabulated results, using their
binning of 8 pretest bins and 5 final grade bins, a correlation coefficient of 0.3 can be
found. This coefficient is not terribly different from results reported by other
researchers.
65
Piagetian Ability as a Predictor
As a part of his study mentioned in the first section of this chapter,
Blumenthal50 also included 'ability to reason' and 'general scientific knowledge'
based on scores on college entrance examslv as predictors of final exam scores. The
only significant predictor for exam I was general scientific knowledge. For exam II,
however, he found coefficient of multiple correlation (essentially the square root of
R-squared where more than one predictor variable is used) of r=0.55 for prediction of
this final exam score. Final exam II correlation with general scientific knowledge
(r=0.2) and mathematics ability (r=0.4) were the two primary variables contributing
to this multiple correlation coefficient.
In the decades that were to follow, formal operational capacity (sometimes
referred to as Piagetian ability) became a second popular variable used by many
researchers as a predictor of success in physics courses.
Jean Piaget was a professor of experimental psychology and genetic
epistemology at the University of Geneva who for many years served as the director
of the International Bureau of Education and of the Institut J.J. Rousseau. He sought
to form a theoretical framework to explain how the human mind advances from a
lesser state of knowledge to a state of 'higher knowledge'. He posited that all humans
progress through the logical formalization of scientific knowledge by a process of
1V
This entire entrance exam was given across 4 municipal colleges in the City of New
York. The author breaks up scores on the exam into three sections: 1) Ability to
reason 2) general scientific knowledge 3) mathematics ability.
66
equilibration of thought structures, and in some cases transformation from one
cognitive level to another.56
Piaget described human intellectual development as a progression through
four stages: sensory-motor, pre-operational, concrete operational, and formal
operational stages (see Fig 3.3).
Piagetian Stages of Intellectual Development (1958)
Formal Operational
11-15 yrs old
/
Concrete Operational
7-11 yrs old
~7
Pre-Operational
2-7 yrs old
~1
r
Sensory-Motor
0-2 yrs old ~1
Figure 3.3. Representation of human intellectual development (from left to right)
according to Piaget's 1958 model.
He called the first stage the sensory-motor stage, characterized by an infant's
developing muscles used in performing reflex actions, and on framing the
67
permanence of an object's existence. He referred to the second stage of development
(observed from about two to seven years of age) as the pre-operational stage of
development. A major feature of this stage is the development of a child's framing of
the world in a less 'ego centric' fashion and the development of moral characteristics
such as empathy. He believed that in this stage that children observe how objects as
well as people 'behave'. The third stage of intellectual development occurs between
the ages of seven and eleven years old, and is called the concrete operational stage.
During this stage children are able to organize, seriate, and match objects, and can
manipulate data. However at this stage, while they are able to determine how their
own action influences the world around them, they experience difficulty in generating
hypotheses. The fourth and final stage of intellectual development, called the formal
operational stage, is marked by the ability to perform abstract manipulation on classes
of objects, with 'concrete reality' taking a less significant role, and with 'possibility'
playing a more prominent role. At around the age of fifteen, humans undergo the last
transition of intellectual development from the concrete operational to the formal
operational stage.
In the late 1960's and throughout the 1970's, researchers in the US began to
take note of the work done by Piaget.57 They began testing their students for the
presence of certain elements of mathematical and logical structure called 'operational
capacities' that signal the last transition in human intellectual growth from concrete
operational to 'formal' reasoning, even though Piaget's model assumes that a collegeaged population would be largely at the formal operational stage.57 They argued that
68
the main logical operations which characterize formal reasoners can be summarized
by: 1) The ability to draw (verbal) analogies; 2) The ability to identify correlations; 3)
The ability to deduce the generating principle of, and sequentially follow, verbal and
numerical chains; 4) The ability to isolate a variable; 5) The ability to use
proportional reasoning.
Piaget himself proposed that age was overwhelmingly the dominant factor in
determining one's transition through these levels of development. The evidence is
strong that it does indeed have a large role to play in performance on 'Piagetian
tasks'.58,59'60 It is not, however, the only pertinent factor: many researchers have
reported that a large fraction of college students were not at the formal level as
measured by Piagetian guidelines.61'62'63'64'65'66
Piaget's is not the only theory on intellectual development. Other very
influential cognitive psychologists like Bruner, Sternberg or Gardner might argue that
Piaget only measured one aspect of intelligence (for instance Gardner might call the
intelligence measured by Piaget 'logico-scientific intelligence' as opposed to six other
'types of intelligences'). Because Piaget used experiments closely related to physics
and he focused specifically on scientific reasoning, Piaget's theory of development
has held particular attraction for physics education.
Paper-and-Pencil Tests of Piagetian Ability
Physicists, and especially physics education researchers, have been at the
forefront of investigating how Piaget's theory of genetic epistemology applies to
69
physics classrooms. ' '
Two researchers in particular - Robert Karplus and Anton
Lawson - devoted significant effort towards trying to develop pencil and paper tests
that measure the same abilities as the original Piagetian interview tasks. The Lawson
Test of Scientific Reasoning Ability (see appendix D) resulted from some of this
7ft
research.
Lawson was also very instrumental in summarizing the research
demonstrating the unity of 'formal thought'71 and in showing its relationship to a
variety of other measures of cognitive ability.
In the past researchers used other
pencil-and-paper tests of scientific reasoning ability such as the Tomlinson-Keasey
Campbell test73 and the Science Logic test.74 The Lawson Test, however, appears to
be the most widely used of these tests currently.
Various Other Factors Used as Predictors
With the rise in acceptance of the Piagetian model of development, many
physicists sought to modify the focus of their courses to promote the development of
'reasoning ability'.
As a result, when reporting on prediction studies, researchers
tended to be a bit more descriptive of the pedagogical content of their courses.
Piagetian ability, mathematics ability as well as other factors were explored as
possible predictors of success in physics courses.
Cohen et al. used a battery of tests at the beginning of their physics courses,
as well as SAT verbal and math scores as predictors of physics success for a group of
195 students enrolled at the University of Vermont. Included in each student's battery
of tests was at least one Piagetian task. Students in this study (reported inl978) were
70
randomly chosen from 4 different intro physics courses (with grades assigned by
different instructors). Choosing a combined subset of 53 students drawn from two of
the courses - one course with the highest average math SAT score and the other
course with the lowest average SAT math score - they found that verbal SAT score
had no correlation with student's final course grade. They also found that Piagetian
level only just significantly correlated with final course grade and that SAT math
score was the best predictor of course grade with a correlation coefficient of 0.43.
77
Griffith,
working with small class sizes of between 20-30 students over
multiple semesters at Pacific University, used the Science Logic Test (a test of
Piagetian ability developed by Griffith and Weiner) as a predictor of success. He
found correlations of between 0.2 and 0.5 (from one semester to another) with
students' scores on a conceptual final.
Liberman
used the Tomlinson-Keasey / Campbell test as a measure of
Piagetian ability and found a correlation of 0.49 with scores on this test and final
exam score for a group of 67 undergrads enrolled in a physics course for non-majors.
7R
In 1981, Halloun and Hestenes as part of an effort to develop separate tests
to measure incoming physics knowledge and mathematics ability, reported on the
70 sn
predictive ability of these tests. ' The mechanics test was designed to highlight
differences between common-sense and Newtonian concepts. The questions on this
test went on to form the basis for two of the most commonly used assessments of
o i
conceptual understanding in mechanics - The Mechanics Baseline Test and the
R7
Force Concept Inventory.
The math test used as a predictor consisted of 33
71
questions containing (a) ten algebra and arithmetic items (b) six reasoning items (c)
four items on graphs (d) six reasoning items (e) five calculus items. This math test
was obtained by choosing from a long list of math questions that had the highest
correlation with the physics test. In a study of more than 1000 students over 3 years
enrolled in both calculus-based and algebra-based introductory physics course at
Arizona State U, as well as 50 high school students, Halloun et al. found that factors
such as gender, age, major and high school mathematics showed no correlation with
physics performance. However, they found a 0.48 correlation coefficient between the
mathematics pretest and students' final score for the calculus-based physics students.
They also found a 0.43 correlation coefficient between mathematics pretest scores
and final scores for algebra-based physics students. They further reported a 0.56
correlation between students' pretest scores and post test scores on the same
mechanics test described above (asked at the beginning and ending of the course
respectively).
In 1979, in a study involving 60 undergraduate physics students in a course
for non- physics majors, Liberman and Hudson83 used the Tomlinson-Keasey /
Campbell test battery (a composite of pre and post tests), as a pretest measure of
formal reasoning ability. The final course score, a composite of 4 exams and
homework scores, was used as the predicted variable. They found a 0.49 correlation
coefficient between pretest and final score. It is interesting to note that because
students scored very well on the proportional reasoning aspects of the TKC test, the
scores from this section were left out of the analysis.
72
Whether or not students took high school physics is another variable used to
attempt to predict success in introductory physics courses. In the Fall semesters of
1990, 1991 and 1992, Hart and Cottle84 collected a total of 508 questionnaires at
Florida State University with students reporting on their: 1) grade in last math course;
2) major; 3) high school physics background; 4) whether or not they completed a
degree at a community college or not. Using the Mann-Whitney test, they found a
statistically significant difference in physics course grades of students who: 1) Took a
physics in high school compared to those who did not (Z=5.4, p<10"4); or who 2)
had a math grade above B- in their last math course compared to those who did not
(Z=5.0, p<10"V 5 This study was later replicated by Alters86 with over 200
introductory algebra-based physics students at the University of Southern California
during the academic year 1993 through 1994. An identical questionnaire was given
on the first day of class. He reported a similar significant difference in course grade
between students who took high school physics and those who did not.
In one of the landmark studies that introduced gender as a contributing factor
into predicting success in introductory physics courses, McCammon87 studied 206
students in various introductory physics courses at East Carolina University. They
purposefully elected not to use Piagetian ability as a predictor variable, citing the low
correlations observed by Griffith in his 1985 study. The final exam in McCammon's
study would most likely be considered traditional, as she describes questions
appearing on the exams as "problems which require translation between english and
mathematics." The predictor variables used in this study were 1) critical thinking
11
appraisal using the Watson & Glaser Thinking Appraisal Test, which tests students
on assumption making, deduction, and the interpretation and evaluation of arguments;
2) primary mental ability, an assessment of reasoning, perceptual speed, spatial
relations and tests of verbal meaning; 3) a measure of math anxiety; 4) Arithmetic
Skills Test - a test developed by the college entrance exam board which typically
contains algebra problems including problems involving inequalities, solving
equations etc. and 5) Elementary Algebra Skills Test. No clear indication was given
on how variation in assessment instruments used by different instructors was
controlled for in her study. In this study, these multiple predictor variables held a
multiple correlation coefficient with grade of 0.4. Of the individual factors
investigated, the most significant predictors of final grade were the algebra test with a
correlation coefficient of 0.32 and the arithmetic test with a correlation coefficient of
0.3. The other predictors had lower correlation coefficients and were therefore
considered below the commonly accepted level of significance for this number of
students. In a separate analysis of the 91 women in the study, however, they found a
correlation coefficient of 0.56 between algebra test score and final grade. This result
suggests that gender might be an important factor in attempting to account for
differences in student performance in physics. We will explore this in further detail
in Chapter 4 of this dissertation.
74
Summary of Previously Used Predictors
In the large majority of studies conducted so far, attempting to find predictors
of success in physics courses, there appears to be a pattern that the most predictive of
variables tend to be centered around mathematics, with some sensitivity to which
mathematics questions are included on the mathematics pretest. There also appears to
be some sensitivity to whether the measure of success on the course was traditional or
conceptual. There have also been some potential problems in the studies conducted.
Some of the previous studies may have placed insufficient emphasis on specifying
when
their 'predictor' assessment was administered (in many cases predictors and
achievement tests were all used in the same battery of tests). In some studies, course
grade, itself composed of multiple midterm (and sometimes even homework) scores,
was used as a measure of success in physics. This complicates our understanding of
which aspects of success the predictors are tied to. In some studies, results from
only subsections of researchers' physics populations were included, which may not
have been representative of the entire introductory physics population.
In general the correlations found, even those using many factors, have not
consistently yielded correlations above r =0.6. A large fraction of the variance in
predicting student success in physics remains unexplained, with even the most
generous estimates leaving more than 50% of the variance unaccounted for. This
estimate agrees with the findings of Sadler and Tai (in their analysis placing this
figure at 64%) as well as Kost et al.90 The large fraction of unexplained variance
75
points to the need for either improving our prediction instruments, or for the
identification of yet to be discovered predictive factors.
In the subsequent sections of this chapter, we will detail some of our attempts
to use pretests as a predictor of score on the final exam in our laboratory course
(described in chapter two). Our goal was to have a pretest that would help us to
reasonably identify underprepared students, so that in the future we would be able to
suggest an appropriate intervention for these students. A further criterion influencing
the design of our pretests was that they should be short, so as not to significantly
infringe on the time allocated to covering physics material in the lab course.
Lawson Test Questions as Predictors of Success
The Lawson Test was found by Coletta to be a good predictor of FCI gain.91
In our earliest attempts to use diagnostic pretests as a predictor of final performance
at NMSU, we used a subset of fourteen questions taken from the Lawson Test of
Scientific Reasoning (see appendix D) which we deemed contextually applicable to a
physics course. We added one additional question (question 15) which asked students
to compare the densities of pieces of a broken block. The final form of the fifteen
questions given to the students is shown in appendix E.
Lawson suggested various subcomponent ability measures associated with
each question appearing on the Lawson test.70 The questions appearing on the fifteen
question pretest included questions testing proportional reasoning ability,
76
conservation of mass, conservation of volume, control of variables and probabilistic
reasoning.
These pretests were given during the lecture portion of two introductory
algebra-based physics courses in the Fall of 2005. Seventy-seven students from the
Physics 211 lecture course turned in answers to the pretest, while from the Physics
221 lecture course, 48 responses were turned in. For the Physics 211 lecture course 65
students' pretest scores were compared to the final exam of the lab course for which
they were simultaneously enrolled. Of the 48 students taking the 15 question
scientific reasoning diagnostic in Physics 221 however, few were simultaneously
enrolled in the lab course, so instead their pretest scores were compared to their
lecture course final exam. In Physics 221, 33 students completed the course and 15
students dropped the course. The average pretest score of the students who
completed the course was 10.7 out of 15, while for those students who ended up
dropping the course, the average score on the pretest was 8.3.
Correlation of students' pretest scores with scores on the final grade shows
that about 30% of the variation in students' final scores is accounted for by their
performance on the 15-question diagnostic. These results are on the order of the best
predictive pretests used in physics courses that we have seen. Our students generally
had no problem with the conservation of mass problems. However, they seemed to
struggle the most with the proportional reasoning problems. Since the 15-question
pretest was still a bit time consuming, we decided to construct a shorter pretest
consisting only of proportional reasoning items.
77
Results of Using Questions From the Lawson Test as a Pretest
Physics 211
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physics courses. Top Graph: For students completing the Physics 211 course (N=65)
the correlation coefficient was 0.54. Bottom Graph: For the Physics 221 course
(N=33) the correlation coefficient 0.58.
78
Table 3.1. For Physics 211 students, this matrix shows the correlation between score
on the final exam and various subcomponent ability measures of the Lawson test
(according to Lawson10) including proportional reasoning ability, conservation of
mass, conservation of volume, control of variables and probabilistic reasoning.
Correlation Matrix - Physics 211
Final exam
Final exam
15 0 pre
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Correlation Matrix - Physics 221
Final exam
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79
1
Proportional Reasoning Pretests
Each semester since Fall 2007 we have used two of three different, short
proportional reasoning pretests, assigned to students within the first few weeks of the
lab course. Each pretest was composed of five multiple-choice questions. Version
one of the pretest consisted of the four proportional reasoning items on the Lawson
test, plus one additional density question described in the section above. We refer to
this version as the original pretest version (see appendix F).
In this chapter, we are primarily concerned with the use of proportional
reasoning ability as a predictor of success in our physics course. An additional goal
involving use of our pretest, however, was to determine the effect of context
variations on performance on our proportional reasoning pretests and in our course.
As a consequence of this goal (explained further in chapter four), we also present data
for two pretests which were context-variations of our original proportional reasoning
pretest. These two pretest versions are referred to as the modified and discrete
modified pretest versions respectively.
The modified and discrete modified pretest versions were interleaved with the
original pretest version during different semesters (the discrete modified pretest
version was only used during the Fall 2007 semester). For the Spring semester of
2008, the pretests were only asked of Physics 211 students, and were administered in
the latter half of the course. We present data on student performance on these pretest
versions in the next section.
80
Performance on Pretests
There is a difference in student performance on the three versions of the
pretests given to our algebra-based physics population, with Physics 211 students
having a mean score of 2.31 out of five on the original pretest version (Std. Error
0.11); of 2.55 (Std. Error 0.11) on the modified pretest version; and of 2.77 (Std.
Error 0.20) on the discrete modified pretest version. For the calculus-based course,
Physics 215, there was a significant difference in performance on the three versions
of the pretest (explored in further detail in the next chapter), with students achieving
mean scores out of five of 3.07 (Std. Error 0.13) on the original version; mean 3.47
(Std. Error 0.13) on the modified version; and 3.18 (Std. Error 0.25) on the discrete
modified version of the pretest. For reference we also include the results from a
course comprised of a small number of students who are mostly physics majors,
Physics 213: The mean score on the original pretest version was 3.87 (Std. Error
0.36); on the modified pretest version the mean was 4.17 (Std. Error 0.48); and on the
discrete modified pretest version the mean was 3.43 (Std. Error 0.43) for this group of
students.
81
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Figure 3.5. The left column graphs show the distribution of scores on the 5 question
pretest for Physics 211 students. The right column corresponds to Physics 215
students. Each row represents one of the 3 pretest versions. Superimposed in the
background of each graph is the mean score for each pretest population (a single
vertical line) and the corresponding standard error of the mean (a narrow band in
red on either side of the mean).
82
Performance on Final Exam
As described in Chapter one of this dissertation, students enrolled in the lab
course at NMSU are given a lab final that is composed of about 36 multiple-choice
conceptual questions (about three questions per topic covered on each meeting during
the semester). The test is administered on the last day of the lab course and accounts
for about 20% of their lab grade. The labs covered in the lab course for our algebrabased, calculus-based and physics majors courses are exactly the same except for one
lab each semester (the algebra-based students are generally not required to do the
Torque lab and in its stead often do the Buoyancy lab). For ease of comparison
among the introductory physics courses, the lab final results presented from here
onwards are of the 33 questions that were common for all lab sections. The labs
covered from semester to semester were the same and the conceptual questions asked
on the lab finals each semester were very similar in nature. Class averages and
distributions are virtually the same every semester for each course. We show the
distribution of class scores over the period Fall 2007 to Spring 2010 in Figure 3.6
below.
For the Physics 211 course, the mean score on final was 15.60, with a
standard error of 0.26, out of a possible 33 points. For the calculus-based physics
group, Physics 215, the mean score was 17.02 with as standard error of 0.33. For
reference the Physics 213 course had a mean score of 21.77 with a standard error of
1.02. There is much overlap in performance on the final exam between the algebra83
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Figure 3.6. Shows the distribution of final exam scores for the three introductory
mechanics courses. The red hollow bar in the background is centered on the mean
score for each course and its width is indicative of the standard error of the mean.
based and calculus-based introductory physics populations. This speaks favorably to
our goal of designing labs that were accessible to the algebra-based students,
however, we are somewhat disappointed that our calculus-based students in
particular, do not perform better.
84
Pretest Version and Correlation
The overall correlation of the original pretest version with the final exam for
N=211 students over four semesters was 0.38 (significant at the p<10"4 level) for our
algebra-based physics students, and 0.54 for N=140 calculus-based physics students
(significant at the same level). For the modified pretest version there was generally a
lower correlation with N=168 algebra-based physics students, having a 0.29
correlation (significant at the p<10"3 level) with score on final, while for N=109
calculus-based physics students, there was a correlation coefficient of 0.34 (same pvalue). Due to the low numbers of students taking the discrete modified pretest
version, we only include these results for the sake of completeness. There was a
correlation of 0.49 between student scores on this pretest with the final for 50 Physics
211 students and a 0.42 correlation coefficient with 39 Physics 215 students.
These results show that our proportional reasoning pretests account for about
1/5 the total variation in students' final grades. This is slightly better than the
mathematics pretests used in the past, although it is not as predictive as some of the
other studies that have included multiple factors. Given that the pretests were only 5
questions long and asked at the beginning of the semester, however, they exhibit a
high efficiency in predictive ability, indicating that proportional reasoning ability may
be a central skill required for student success in conceptually driven introductory
physics courses such as ours. It is interesting to note that each pretest version had
different predictive ability for our two main introductory mechanics populations, with
85
Physics 211 Original Pretest Version
Physics 2 1 1 Modified Pretest Version
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Physics 215. For each group of students with a particular pretest score (e.g. 1) there
is a corresponding distribution of scores received by these students on the final exam.
86
a generally higher correlation between final score and pretest score for our calculusbased populations. The fact that there were different correlations for different types
of proportional reasoning questions will be explored further in chapter four. This
result seems to show that 'proportional reasoning ability' may not be the only factor
at work when using our pretests as predictors.
Fall 2009 Pretest Versions
Fall 2009 was a unique semester in our efforts to optimize our use of
proportional reasoning pretests as predictors of final grade. In this semester, we used
a pair of pretest versions (described in further detail in chapter 4) that were different
from the questions we used on our proportional reasoning diagnostics in previous
semesters. On the two (Fall 09) versions, we retained the corresponding versions of
questions two and three from our 'original' and 'modified' pretests. The other
questions used this semester were new or significantly modified from previous
versions.
We also had an opportunity to use the entire Lawson Test as one of our
weekly administered pretests on the last lab meeting of the semester - one week prior
to the final exam. The temporal proximity of these two tests may have had some
effect on the correlations observed. For comparison we have included in appendix G
the correlations with various components of the Lawson test with the final exam to
87
give a further picture of how the correlations compare with various subsections of the
Lawson test.
For 128 algebra-based physics students there was a 0.47 correlation
coefficient between the proportional reasoning questions on the Lawson test and the
final exam score. This was the highest correlation among all of the subsections of the
Lawson test. The Total Lawson test score had a 0.65 correlation with the final.
Considering the fact that the Lawson test has no questions that are based on physics
content presented in the course, we were surprised to learn that it accounted for nearly
40% of the variance in students' final exam scores. Our proportional reasoning
pretest, asked at the beginning of this semester, had a 0.42 correlation with the final
for these students.
For 63 Physics 215 students taking pretests, final exam and Lawson test, there
was 0.64 correlation with the final exam score (a value similar to Physics 211).
Again the proportional reasoning items on the Lawson test correlated best with the
final with a coefficient of 0.63 (a value about as high as the total score on the Lawson
test). Interestingly also for this group, the pretest at the beginning of the semester had
a correlation of 0.62 with the final score.
These findings suggest that proportional reasoning may stand on its own as a
valid predictor of success in a physics course such as our own that emphasizes
conceptual understanding. It is also interesting to note that the correlations found
with proportional reasoning items were about the same whether these questions were
asked at the beginning or at the end of the semester.
88
It should be noted that our labs do not attempt to directly improve our
students' proportional reasoning abilities. In a lab course at Rutgers University for
engineering students, the same pretest used in Fall 2009 at NMSU (unmodified
version) was given to students near the end of Fall 09 in the lab section of their
course. (Since this was given some time after relevant instruction it might be more
accurately considered a proportional reasoning post-test). In that course, students' lab
and lecture grades are combined into an overall physics grade and there is greater
synchronization of the material presented. Students' ratio reasoning is explicitly
elicited as part of invention tasks designed to improve students' conceptual
understanding of kinematics. Those students performed markedly better on this
pretest than our mechanics students, averaging 3.6 out of 5. Also, the correlation
between students' final exam score and pretest score for this group of students was
0.22 (N=101; p-0.023). While this correlation was significant at the 0.05 level, it was
not nearly as strong as the correlation that we have observed at NMSU with our of
Phys 211 and Physics 215 populations.
In an effort to improve the predictive ability of our pretests, we have
experimented with adding measures of students' working memory capacity into the
prediction of students' final exam score at NMSU. We were led to this construct by
anecdotal instructors' characterization of algebra-based physics students as
'memorizers', while this characterization was not made of calculus-based students.
As described in subsequent sections, some psychologists have found significant
correlations between measures of working memory capacity and higher level
89
cognitive functions. We hoped that measuring our students' working memory would
highlight any differences in the efficient use of memory between our introductory
physics students, and how these differences affected student success in our physics
course. A description of our efforts in this regard, along with some preliminary
results, is presented in the next section.
Working Memory Capacity
In an attempt to explore potential measurable factors that would help us
characterize some of the differences between the algebra-based and calculus-based
introductory physics populations, we conducted preliminary experiments to measure
our students' working memory capacities. Further, we determined how predictive
these measures were of students' final exam scores. As far as we are aware, there
have been no attempts to use the construct of working memory capacity as a predictor
of success in physics, although as described in subsequent sections, a few researchers
have used similar constructs in predicting physics grades.
Description of Working Memory Capacity
Working memory as a construct falls within the domain of informationprocessing theories, which generally view all cognitive processes in terms of a
relatively small number of mental sub-processes and representations which interact
with stimuli to produce definite responses.93 Lautrey94 notes the shift in dominance
90
from Piaget's theory to information-processing models of cognition towards the end
of the 80's, with the new emphasis on exploring the limits imposed by processing
capacities to explain individual differences.
Early neo-Piagetian researchers have used measures of 'processing capacities'
as a means of explaining the persistent problem of Piagetian horizontal decalage the inconsistent solution of problems supposedly requiring the same cognitive
processes according to Piaget's stage theory.95'96 Among the capacities used by these
early researchers are M-Capacity (central computing space) and field dependence /
independence - a measure of cognitive style closely linked to the ability to extract
useful information hidden among useless information (one common task to measure
this is the popular hidden figures task). Other researchers abandoned the Piagetian
model altogether.
ENVIRONMENT (INPUT)
MtRCEPTUAL
PROCESSING
CYCLE
'1
PROCEDURAL
MEMORY
COGNmVE
PROCESS!!*!
CYCLE
womcwa
MEMORY
COGKTT1V6
PROCESSING
CYCLE
DECLARATIVE
MEMORY
MOTOR
SCESSWO
ro<
CYCLE
'
RESPONSE (OUTPUT)
Figure 3.8. Information-processing model of memories and processing cycles.
Individual differences occur in either the capacity of the memories or the speed of the
processing cycles.
91
Figure 3.9. Baddeley and Hitchl974 model of the inner components of Working
Memory
Baddeley and Hitch describe Working memory (see Figures 3.8 and 3.9) as a
highly dynamic form of memory that operates over periods of seconds and
temporarily stores selected information for detailed analysis." Working memory
span tasks typically involve some complex activity (such as reading sentences aloud)
performed concurrently with an item retention task (such as remembering the last
word of each sentence read).100 The span or capacity of a span task is usually quoted
as the maximum number of items one can remember while efficiently performing the
'complex activity' associated with the task. Working memory has been linked to
cognitive roles such as: memory, focus, attention101, and inhibiting irrelevant
information.102'103 Researchers also find the working memory construct useful in
explaining why processing speed varies with age (with adults typically exhibiting
greater processing speed than children),93 and in offing insights into reduced
performance when experiencing math anxiety (Ashcraft and Kirk10 ) and general
'choking under pressure' (Beilock and Carr 105 ). We believed WMC would make a
92
good candidate to account for variance in individual students' lab final exam scores
independent of that which is accounted for by the coarser, Piagetian four-stage
measure.
Previous Use of Working Memory as a Predictor
In one of the most cited experimental results in support of the success of
executive function (shown in Fig 3.9 as a component of WMC) variables as
predictors of cognitive success, Bull and Scerif,106 tested 93 third grade students from
six Scottish elementary schools on a variety of executive function measures. The
mathematics test consisted of single and multi-digit addition and subtraction as well
as the Group Mathematics Test.
It was found that a number of executive function
measures correlated significantly with achievement on their mathematics test, the
highest of which were 0.44 (p<.01) with counting span (a working memory span task
that involves counting aloud while remembering specified numbers) and -0.46
(p<.01) with a measure of field dependence/ independence . More recently, Lepine et
al.100used various span measures as potential predictors of scores on nationally
administered tests of reading and mathematics for 93 French 6 graders. They found
correlations of about 0.35 with test scores when compared to scores on span tasks as
described by Barrouillet et al (2004). Aside from the results of studies involving
children's reading and mathematics,108 Hambrick and Engle109 noted the lack of
research on the role of working memory in problem solving or performance on real
93
complex cognitive tasks (as opposed to 'toy model' cognitive tasks), echoing the
sentiments of Hutchins (1995).
In one of the few studies to use elements of the working memory capacity
construct as a predictor of success in a physics course, Cilliers et al110 measured 2D
and 3D spatial rotation ability, memory tests of meaningless words and symbols
buried within a paragraph, as well as visual perception speed on a group of 75
students at the University of South Africa at Unisa. They did not find significant
correlation with any of these psychometric measures with exam or final score.
Whereas these processing measures have been shown to be highly correlated with one
another, their relation to achievement in physics has been disappointing in
general.95'111
Working Memory Capacity as a predictor at NMSU
As a preliminary investigation into whether or not working memory capacity
bore any relationship with student success in our introductory physics labs, we
solicited volunteers from among students enrolled in our physics labs between Spring
2008 and Fall 2009. The study was strictly voluntary and students were not offered
any extra-credit or other incentive to participate in our study - as a result of this, the
number of students participating in this study was relatively low.
We used the Blackboard online learning management system's quiz tool to
host and time our span tasks. In order to prevent interviewees from anticipating which
94
spans they were to be tested on, the ordering of the individual screens was such that
the spans tested were random. Engle et al.112 described a similar random ordering of
set sizes in their experiment. Our span tasks were also self paced working memory
span tasks. The time savings automated span tasks affords over individualized,
experimenter-run task measures have been also noted by Engle et al.112 The ease and
time savings of computer run span tasks have been found to translate into larger
potential experimental populations, a problem with many of the studies involving
working memory thus far. Our span tasks were available to students from the lab
course's online course management web page throughout the duration of the course this meant that students could participate whenever they found some free time. In
spite of this only about 120 students participated in the study.
Description of Our N-Back Test
The n-back test is a simple span task where the student is shown a series of
cards (or slides) after each of which the student is asked to recall the card that was
shown to her "n" cards previously. In our version of the task, in order to minimize
mental processing power used in choosing students' responses (leaving nearly all of
their mental processing power to be devoted to the memory task), we narrowed our
available responses to two choices: same if students thought that the slide in the
current question item was the same as the slide in the question item 'n' slides back; or
different if they thought they were different. It is precisely because of the relatively
low processing component (complex activity usually associated with working
95
memory span tasks) associated with the n-back task that some researchers have
questioned the validity of using this task as a measure of working memory
1 1 1
capacity.
We will however accept the n-back task as a valid measure of working
memory capacity.
After an introduction into the rules of the 'game,' including information on
what is meant by a slide (see Figures 3.10 and 3.11.) and which slide we are asking
them to compare the present slide to, students are led into a practice-run quiz, where
they are allowed to play the game with feedback given at the end of a short 'practice
run' quiz. The release of the scored version of the quiz is conditional upon volunteers
successfully completing at least two items correctly on the 5-question practice run,
including the final question which asks whether or not they understand the rules of
the game.
3 . (Points: 5,0)
n-back=2
O a.
same
G b.
different
1
Save and View NextJ jj^ext Question J
Finish j
' Help •
Figure 3.10. Question 3. The large number one with the box around it is referred to
as a slide. In this question students are asked to compare this slide (number 1) with
the slide encountered two slides previously (n-back=2).
96
5 . (Points: 5.0)
n-back=4
',; a,
same
..' b.
different
Save and View Next
| Next Question
F i n i s h ]>•.«•
••! H e l p j .
Figure 3.11. Question 5 (shown to students two screens later). Asks students to
compare this slide (number 2) to the slide shown 4 slides ago.
Scoring of Our N-back Test
There are differences in the scoring criteria used by researchers in determining
memory spans. For instance, Bull106 considered the span to be one less than the value
where there was a double failure on recall of both cards used to determine the span
length (i.e. a success rate greater than or equal to 50%), while Leather114 considered
success rates of more than two out of three trials to be indicative of successful span
capacity at that level. Further, some of the original researchers into the construct such
as Daneman and Carpenter115 define (word) span as the maximum number of
sentences the subject could recall while maintaining perfect recall of final words (i.e.
a perfect success rate). Conway et al.116 have shown, however, that there still exists
high construct validity even when different scoring schemes are used. The results
quoted in the remainder of this chapter are for a scoring schema where spans are not
97
calculated directly, instead the percentage of correct responses is recorded for each
student. We wish to emphasize, however, that we have also calculated these results
using scoring schema similar to the researchers cited above without much difference
in the correlations quoted.
Results of Our N-back test
In our preliminary study to use the working memory capacity construct as a
predictor of success in our laboratory physics course, we used an n-back test as a
measure of our students' working memory capacity. Over four semesters at NMSU
about 60 Physics 211 and about the same number of Physics 215 students volunteered
to take our standard working memory test (WMT). The mean final exam scores of
these students were slightly higher than the average score of the general introductory
physics population reported earlier in this chapter.
There was a correlation slightly above 0.2 for students enrolled in both
courses, when comparing their score on the working memory test ('Grade' on the
scatterplot) with student's score on the final exam (see Fig 3.12). When brought in as
a second predictive variable of final exam score into a regression equation already
including our proportional reasoning pretest scores, however, the combination of low
correlation and low student numbers leads to the result that as a predictor variable,
score on the WMT may not be a significant factor (p>0.4).
98
Scatterplot of Final Score vs WMT score
Final33
40
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30
' • • i > i
40
i
i i I i
5 0
i
i
i
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Figure 3.12. Scatterplot of students' final exam score vs percentage of working
memory capacity item correctly answered.
The time students took to take the test was also not a good predictor of
success on the lab final exam for our students (see Fig 3.13 in appendix H).
Neither was the efficiency of working memory score, which was calculated by
dividing the score on the working memory test by the time taken to make this score
(see Fig 3.14 in appendix H).
In summary, even though our experiment was preliminary, we feel that the
results from our n-back test were not encouraging. It is possible that relying on results
from a subset of students who volunteer to take a test biases our results to show weak
to no correlation with final exams. It is also possible that some students may have
99
played unfairly to inflate their scores. While the aforementioned are both possible
causes of the low correlations observed between n-back scores and final exam scores,
we believe the lack of correlation to be indicative of a weak connection between
student ability to perform well on the n-back task, and student understanding of
physics as measured by our lab final exam. Further testing may be necessary to show
otherwise.
Spatial Working Memory Test
117
Daneman and Carpenter
suspected that the reason that traditional memory
span tasks failed to correlate with comprehension ability was because the source of
individual differences in memory capacity rested with the brain's functional capacity
- the amount of capacity left over for storage after the requirements for processing
had been met. They presumed that less skilled cognitive performers process
information less efficiently, resulting in reduced storage capacity. At NMSU, we also
attempted to measure students' working memory capacity through use of a spatial
working memory task, also commonly referred to as counting span task.106These
tasks involved a higher level of processing power as students had to count the number
of squares, located at different screen positions, while remembering the counts from
previous slides. The release of this test was conditional upon successful completion of
the practice test and the n-back task. This was done because we expected low
participation rates and wanted to have the highest number of students volunteering for
the same task (the n-back task) as possible. The number of students completing the
100
spatial working memory task was 25, lower than the number of students taking the nback task.
The task we used was identical to the one described by Towse et al.,118except
that the time for which each slide was displayed was not computer driven, but was
instead user defined (the user clicked 'next question' when she was ready to move on
to the next question).
3 . (Points: 5.0)
•
•
• • • •
• • • •
•
•
n-back=l
C
a.
6 blue squares
O b.
7 blue squares
O c.
8 blue squares
O d.
9 blue squares
Save and View Next !
Next Question
[Finish J,..:":
>;;•.:•:] H e l p i
Figure 3.15. Spatial working memory test slide.
101
Results Using Spatial Working Memory Capacity. Neither the spatial working
memory capacity score (r= 0.03) nor the time taken to complete this task (r=0.14)
correlated well with students' final exam grade for the 25 students who volunteered to
take this test. It also failed to significantly improve the overall pretest predictive
ability when added as a second variable when trying to predict final grade. The low
numbers of students taking this test makes it difficult to say whether or not this test
was more or less predictive of physics grade than the n-back test (see Figures 3.16
and 3.17 in appendix H).
Summary of Working Memory Results
The results of our working memory capacity experiments did not lead us to
finding a secondary predictor of success in physics as we had hoped. In our
estimation, neither n-back nor spatial tasks, upon preliminary investigation, held out
much hope of being more predictive of success in physics than the predictors already
identified by other PER researchers. In addition, we believe the working memory
construct may not hold out much promise of adding to the predictive ability of other
pretests. We do wish to emphasize, however, that our investigation into this construct
was preliminary, and the number of students participating was low, leaving open the
possibility that other researchers may find this construct predictive of success in
physics.
102
Conclusion
One common theme in many of the studies conducted so far is that there are
many problems associated with the variables used in attempting to predict success in
physics courses. Using tests in a battery may introduce an unwanted time factor119 as
does using final grade which is composed of a collection of many midterm grades.
Equating success measures across courses taught by different instructors using
different exams is also problematic.
Tests of scientific reasoning ability such as the Lawson test seemed to be at
the heart of many of the most predictive pretests used. Our results using at first the
15 item subset of these questions, then with the smaller subset of proportional
reasoning items, resulted in pretests which were shorter than many of the traditionally
used tests of Piagetian ability, without the loss of much predictive power. In chapter
four, we explore factors which affect students' performance on these proportional
reasoning pretests. We try to determine what effect surface-features such as question
context have on student performance, and whether any differences in performance
have deeper implications for some student populations over others.
Finally, whereas other researchers have found the working memory capacity
construct to be a useful predictor of higher cognitive ability97so far we have not found
this to be the case for our physics courses. A study with larger student numbers may
be necessary in order to observe if working memory span accounts for unique
variance that is unaccounted for by proportional reasoning ability. Such a test in
103
conjunction with our proportional reasoning test, however, will lose the feature of
brevity and may not be a better predictor than the Lawson test.
104
C H A P T E R 4: C O N T E X T D E P E N D E N C E OF P E R F O R M A N C E
ON PROPORTIONAL REASONING TASKS
Two Perspectives on Student Difficulties
As described in the previous chapter, Piaget saw learning as a process of
acquisition of abilities, resulting in progression through various stages of
development. In an example of a description of such a process, he states:120 "Once
the child possesses the operation of seriation he has opened himself to a whole range
of new behaviors.. .he becomes able to construct, understand and cope with new
relationships among objects not possible before. Indeed the acquisition of seriation is
a Copernican revolution for the child."In Piaget's view of intellectual development
(as described in the previous chapter), learning is achieved through the equilibration
of thought structures in a process of adaptation through assimilation and
accommodation.
Many physics education researchers viewing the repeated and replicable
student difficulties encountered by students when asked certain physics problems122
believe that students come into their physics classrooms with certain relatively
strongly held beliefs which are usually referred to as misconceptions.
Researchers
framing student difficulties in this manner tend to adopt a model for the development
of physics knowledge similar to Piaget's adaptation and often employ strategies such
as elicit, confront and resolve,
in attempting to correct these student difficulties.
105
McCloskey et al.
for instance, asked students to consider the horizontal
motion of a ball tied to a string whirled in a circle above a person's head. He asked
students to draw the horizontal path of the ball if the string were suddenly cut. He
found that about one-third of the students interviewed drew curved paths upon
release. He went on to use apparatus similar to that shown in Figure 4.2c to ask
students to predict the path of a ball bearing initially constrained to move in a circular
path. He found that most students were surprised that upon leaving the circular
segment, the ball bearing did not continue to move in a circular trajectory.
McCloskey attributed these findings to students possessing a strongly held impetus
theory of motion, where any object set in motion was given an embedded 'force'
called impetus, which would keep it moving in this way until this impetus died out.
This theory of motion was prevalent in the 14th-16th Centuries, and as such is
sometimes referred to as a 'medieval' impetus misconception when used to explain
students' wrong answers on similar questions.
An alternate view on repeatedly observed student difficulties is held by
researchers holding to the knowledge in pieces model espoused by DiSessa126 and
others.
In this view of student thinking, often called fine-grained constructivism,
DiSessa states that in answering physics problems students are heavily guided by
their sense of 'intuition'. He claims that it is only through the reorganization of the
elements of intuitive knowledge that scientific knowledge can be attained. He refers
to the smallest elements of intuitive knowledge: things that 'are' because "that's the
way things are," as phenomenological primitives (p-prims). He believes that
106
students' cognition is a manifestation of the activation of a certain set of p-prims
which are called and ranked based on their reliability (i.e., past experiences in
'similar' situations determine which p-prims are called and which among these are
called first). One example of a p-prim is the Ohm's p-prim, where an agent acts
against a resistance to produce some result. This p-prim often leads students into
framing many physics situations as whether or not an agent or the resistance is
'winning' or 'overcoming'.
Many fine-grained constructivists share DiSessa's view that much of students'
intuitive knowledge is loosely connected and inarticulate. Researchers such as
Hammer and Elby
refer to these pieces as 'resources' while others such as
Minstrell129 call these units of thought 'facets'. These researchers believe that when
students answer questions, they activate and assemble these elements 'on the fly', and
their answers are strongly sensitive to context.130'131 Experiments showing that
students can be cued into answering virtually the same problem in markedly different
ways support a knowledge-in-pieces view.
The instructional implications of viewing knowledge in such a fine-grained
manner are immediate. Rather than viewing student learning as a process of
replacing misconceptions with more productive beliefs, these researchers view
teaching as a process of utilizing student 'schema', or activating student 'resources'.
In attempting to help students construct physics knowledge from their own sense of
intuitions, programs such as the Maryland Tutorials in Physics,
107
encourage students
to activate the correct set of resources when solving physics problems. They often
refer to this activation of correct resources as sense-making.
In one example where the cuing of different resources elicited different
student responses, Frank et al.132asked students to consider the time of flight for three
balls rolled off of a table (see Figure 4.1). Half of the students were asked to compare
the time of flight for the three balls given that each ball had a different initial
(horizontal) velocity when sent rolling off of a table (they referred to this as a speedcue). The other half of the students were asked to compare the time of flight for the
three balls given that each of the three balls travelled different horizontal distances
before they hit the floor (they referred to this as a distance-cue).
SO.
(b)
Experiment 1:
Ejperiitifint2:
_2*_
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-3*-
Projectile question: distance-cuing.
Figure 4.1. Students are asked to rank the time it takes for the ball to travel from the
table to the floor in three different situations: a) speed-cuing: students are given that
the three balls have different initial speeds b) distance-cuing: students are given that
the balls travel different horizontal distances before hitting the floor.
McCloskey asked similar trajectory-based questions in his earlier mentioned
study, and attributed students' incorrect responses to students misconceptions of each
108
ball containing different 'impetuses' which would 'resist' the pull of gravity
differently. Viewed from this misconceptions standpoint, one would expect both
speed-cuing and distance-cuing situations to equally lead students to the incorrect
conclusion that the fastest (or furthest-travelling) ball would have the most impetus,
enabling it to resist the pull of gravity longest, and therefore having the longest time
of flight. Frank et al. found however, that for the speed-cued situation, students were
more likely to answer that the fastest-travelling ball would travel for the shortest time,
than for similarly distance-cued students to answer that the furthest-travelling ball
would travel for the shortest time. Students choosing this answer on the speed-cued
questions based their answers on the fact that faster travelling objects take a shorter
amount of time to travel, a reasoning pattern inconsistent with the 'medieval impetus
misconception' mentioned earlier. This result led researchers such as Frank to
conclude that depending on the context of questions asked different resources (more
distance equals more time etc.) were activated.
Example of Context-Dependent Performance
Figure 4.2c shows a question appearing on the FCI40 in which students are
asked to predict the trajectory of a ball after it exits a circular channel. Over half of all
students predict that the ball continues in a circular path (answer A). McCloskey,134
who introduced this question, was an early proponent of the misconceptions
interpretation of student difficulties on physics problems. As mentioned earlier, he
109
saw this result as part of students' naive impetus theory of motion developed through
their every-day experience with moving objects. Kaiser et al. however,
in a study
conducted at the University of Michigan, sought to challenge whether or not students
answered this question incorrectly because of a strongly-held 'circular impetus'
misconception of motion or whether students' responses were more context
dependent, and therefore more likely to be explained by a resources model of student
thinking.
In Kaiser et al.'s study 80 students recruited from a hallway were asked two
versions of this circular channel question (see Figure 4.2). Half of the students had
taken physics either at the high school or college level. There were equal numbers of
males and females in the sample. The researchers asked half of the students to predict
the path of a ball that rolls through a coiled tube (shown in Fig 4.2a and described in
their paper). These students were then asked to predict the path that water coming out
of a hose would take if the hose was coiled as in Figure 4.2a. These two questions
were posed in reverse order to the other half of students. The researchers chose the
water problem since "water shooting from a curved garden hose is a closely related
event that is familiar to most people."
Students performed significantly better on the water question than on the ball
question, x2 (1) = 12.13, p<.005. They also found a smaller gender effect on
performance for the water problem than there was for the ball problem (x2 (1) = 4.59,
p<.05 compared to x2 (1) = 9.45, p<.005). Notably, they found no evidence of
transfer, as students answered at about the same rate on both problems whether asked
110
Figure 4.2. Students are asked to predict the trajectory of: a) water exiting a curved
hose b) a bullet exiting the barrel of a curved gun and c) a marble as is leaves a
circular tube (appears on FCI).
first or last x = 2.26, p>.10.
In a modification of this experiment involving 81 female students a second
intuitive context of the same problem was added (Figure 4.2b). Here students were
asked to draw the trajectory of a bullet as it leaves the curved barrel of a gun. In a
similar experimental format, the two intuitive examples were asked before the ball
problem for half the students and after for the other half. Again there was no evidence
of transfer of correct solutions from the familiar problems to the less familiar
problem, x 2 (1) = 1-29, P<.005; and there was also similarly better performance on the
intuitive problems.
Ill
The findings of Kaiser et al.'s study brought to light two important results that
are pertinent to physics education research. That students could solve this problem in
the familiar contexts and not in less familiar contexts leads one to think that students
do not hold rigid beliefs that all objects constrained to move in circular paths inherit a
property of 'circleness' of their motion (a 'misconceptions' view posited by
McCloskey). Researchers such as Elby would argue, on the contrary, that this is an
example of different contexts activating different resources, and that the significantly
different response rates support a resources view. Another important finding
highlighted in this study was that the difference in performance between males and
females was less marked for the contexts that were 'familiar'.
We wanted to explore further whether the context of our proportional
reasoning pretest would have some effect on student performance. In consultation
with Andy Elby we discussed the possibility that familiarity of contexts used in
physics problems may matter to some students and have some effect on their
performance. Elby suggested that familiar contexts may activate more appropriate
resources, which would enable students to solve our proportional reasoning problems.
It is against this backdrop that we decided to modify the versions of the pretest used
to test proportional reasoning ability.
112
Background to study at NMSU
As part of our efforts to explore whether or not some part, or all of what is
claimed to be measures of proportional reasoning ability by our pretests is really a
manifestation of a conglomeration of non-ability factors such as student familiarity
with scientific contexts and vocabulary, for 6 semesters we have been using two
versions of a pretest designed to measure proportional reasoning ability. One version
of the pretest (referred to as the original version throughout this chapter), has been
administered every semester (except Fall 2009) in our lab course since Fall 2007.
This version consists of four proportional reasoning questions drawn from the
Lawson Test of Scientific Reasoning136 in addition to one density item, as explained
in Chapter 3. Each semester since Fall 2007 we have used an alternate pretest within
our lab course, interleaved among pretests of the original version. Our goal was to
see if there was: 1) A difference in student performance on the two versions of the
pretests by controlling for other factors on the two pretests except for the contexts of
these pretests and 2) a difference in predictive ability for these two pretests.
There does seem to be some predictive power in questions about 'scientific
reasoning' in terms of student performance in physics classes, as has been described
in the previous chapter of this dissertation. We wanted to see, however, if a pretest
based on a more familiar situation might trigger students to use 'sense-making' in
their solution of physics problems, as described by researchers such as Elby and
113
Redish,
'
and whether this might have an effect on their performance on such
questions or on the predictive ability of these pretests.
Method
In the Fall of 2007, the second version used as an alternate to the original
pretest version was a pretest designed to highlight common-sense reasoning in
proportional reasoning items. In this example, referred to as the 'discrete modified'
version of the pretest in the previous chapter [see appendix F], the Lawson questions
were re-written to include the same numeric quantities, but with a context that we
thought would be more familiar. The mathematical steps required to solve the ratio
problems were exactly the same. The context however, was changed to one based on
serving rice in a soup kitchen, a more everyday context in our estimation. The two
versions of the tests were alternated, so that every consecutive pretest was of a
different version. Students arriving at the lab were then given a pretest from the top
of the stack, thus randomizing the version of the test each student received. As
shown in Figure 3.5 of the previous chapter, there was a significant difference in
performance on the alternate version of the pretest for our algebra-based physics
population. This version (discrete modified) was used as an alternate version for only
one semester however, because of its failure to account for an additional difference
between the two sets of problems: The discrete modified pretest version was
composed of proportional reasoning problems involving materials that were discrete
114
(a countable number of scoops of rice) while the original Lawson questions involved
materials that were continuous (water poured from one container to another). The
difference in student performance on discrete versus continuous proportional
reasoning questions has been shown by Lawton139 and cited by Pulos and
Tournaire as a context variable that affects student performance.
From Spring 2008 to Spring 2010 (except Fall 2009), the second version that
we used as a more commonsense alternative to the original pretest version was in the
context of a mother using two different sized dosage cylinders to pour medicine for
her children (see appendix F). For the alternate version of the broken block question
(question one on the 5 item pretest), we used 'cheesiness' as a commonsense
alternative to mass density, except in Spring 2010 where we asked about the density
of a broken brownie (appendix F).
Results for all students
Because the discrete modified version of the pretest was only asked for one
semester (Fall 2007) and the total numbers of students taking this pretest was low, we
will ignore the results obtained in the data we present here. Of the over 700 students
taking either the original pretest version or the modified pretest version during the
period Fall 2007 to Spring 2010, there was a 0.27 difference in score out of 5 (by
students taking the different versions of the pretests, with students scoring marginally
(10%) better on the modified version of the test version. This difference meets the
115
commonly-used threshold level of significance of 5% (see Table 4.1 and Table 4.2)
using the appropriate Satterwhite t-test for unequal variances (as described in chapter
3). The effect size for this difference is, however, small (d~0.2). Table 4.1 and Table
4.12 show summary statistics of student performance on the pretests.
Table 4.1. Descriptive statistics for 2 pretest versions Fall 2007 to Spring 2010
Pretest Version
N
Mean Score
Std.
Std.
(out of 5)
Dev
Error
Original
431
2.654
1.760
0.085
2.924
Modified
315
1.588
0.090
1.690
0.125
Difference (2-1)
0.270
Table 4.2 Summary oft-test analysis for difference in Table 4.1
Method
Variances
df
t-Value
Pr > |t|
Pooled
Satterthwaite
Equal
Unequal
744
711.8
2AS
2.19
0.034
0.031
This result implies that for a significant portion of our introductory mechanics
population, the original pretest version is more difficult to answer than the modified
pretest version, while to an expert, these pretests require the same proportional
reasoning steps and abilities. We can then attribute this small effect to the difference
in contexts, controlling for proportional reasoning ability (this difference is
significant at the p<0.05 level which is commonly considered significant140).
116
Difference in Student Performance by Course
A primary motivating force behind the development of lab exercises described
in chapter two of this dissertation was to address student difficulties faced by our
algebra-based physics population. Many instructors recognize a difference in the
algebra-based and calculus-based physics students beyond that which can be
attributed to a difference in mathematical ability. Instructors often complain that the
engineers are obsessed with 'asking for the right formula', while the algebra-based
students are sometimes known as 'grade grubbers' with little to no interest in actually
understanding the material. Despite these apparent differences, the overwhelming
majority of research in PER has been done with calculus-based students141 with the
(perhaps incorrect) expectation that difficulties revealed by these students are also
present in the algebra-based population.
As a first attempt to explore whether or not the pretest versions were 'more
different' for one group of students over another, we looked for differences in the
results for our calculus-based physics students compared to those for our algebrabased physics students. A look at differences in overall score on each pretest version
can be gleaned from Figures 4.3 and 3.5. In this chapter we will look at the
differences in performance of the two introductory physics populations on individual
questions as well as differences in performance on the two pretest versions within
each course.
117
The calculus-based physics students outperformed the algebra-based physics
students on every question on both versions of the pretests. On the original pretest
version, Physics 215 students scored about 0.8 points higher on average than Physics
211 students, and on the modified version of the pretest, the difference in the mean
Physics 215 score was about one full point higher than the score for their Physics 211
counterparts. The effect size was medium for both pretests, with course having a
slightly larger effect on the modified version (see Tables 4.3 and 4.4 in appendix I).
On every question on the standard pretest version, the average Physics 215
student performed significantly better than the average Physics 211 student. This
difference in performance between Physics 211 students and Physics 215 students
was at the (p<0.05) level for question number five and (p<0.01) for question number
four. For all other questions, the probability the observed difference in performance
was due to random sampling was less than a 0.1% as can be seen in Tables 4.3 and
4.4 in appendix I. The effect sizes of these differences were all small-medium. In
total, based on performance on this test, these two introductory physics populations
are significantly different.
On the modified version of the pretest however, there was not a significant
difference in performance for students enrolled in the two mechanics courses on
question number five. There was however a statistically significant difference in
performance between the two groups on all of the other questions of the pretest. The
difference in average total score on this pretest between the Physics 211 and Physics
215 students was also statistically significant [see Tables 4.5 and 4.6 in appendix I].
118
The effect sizes of these differences were medium (larger than those for the
corresponding questions on the original pretest), for all questions except number five
- which virtually showed no effective difference in the two populations' scores.
Using this pretest version as a guide (except probably for question five), there
remains a significant difference in performance between these two introductory
physics populations.
By looking into achievement on each of the pretests within the two main
courses, we were able to explore each population's sensitivity to version of the pretest
taken. Students enrolled in the algebra-based course performed about the same on the
two versions of these pretests. For our Physics 215 (calculus-based mechanics)
population however, students who answered the modified pretest version scored
about 0.4 points out of 5 better on average than students who answered the original
version. This difference was significant at the 0.05 level [see Figure 4.3 and Tables
4.7 and 4.8 below], however the effect size was small. When comparing question-byquestion performance within each course, for Physics 211 there was a significant
difference in performance on the two versions of question number two and question
number three (p<0.05 andp<0.01 respectively), but there was not a significant
difference in performance on the other questions of the pretest. For the Physics 215
students however, there was a statistically significant difference in performance on
the two versions of the pretest only for question three (p<0.01).
119
Table 4.7. Descriptive statistics for difference in performance on 2 pretest versions
within each course.
Course
(Physics)
Pretest
Version
N
Mean
Score
(out of 5)
Std.
Dev
Std.
Error
211
I
2
Diff(2-1)
254
191
2.307
2.550
0.243
1.749
1.572
1.675
0.110
0.114
0.160
215
1
2
Diff(2-1)
161
118
3.075
3.466
0.392
1.664
1.448
1.577
0.131
0.133
0.191
Cohen's d: Physics 211 d=0.15, Physics 215 d=0.25.
Table 4.8. Summary oft-test analysis for difference described in Table 4.7.
Course
Method
Variances
df
t-Value
(Physics)
Pooled
Equal
443
1.51
211
429.2
1.54
Satterthwaite
Unequal
Pr>|t|
0.131
0.125
Pooled
Equal
277
2.05
0.041
Satterthwaite
Unequal
268.9
2.09
0.037
215
120
Algebra-based physics
Pretest: 'Standard' version
40%
30%
20%
10%
Algebra-based physics
Pretest: Modified version
40%
30%
20%
10%
0
1 2
3 4
N = 245
5
0
40%
30%
20%
10%
0
1 2
3
4
5
Calculus-based physics
Pretest Modified version
Calculus-based physics
Pretest 'Standard'version
40%
30%
20%
10%
1 2
3 4
N=191
-
l_l
0 1 2
3 4
N=118
5
N=161
5
Figure 4.3. Distribution of student scores on two versions of the 5-question pretest.
Top graphs shows results for algebra-based physics course - Physics 211, while the
bottom graphs shows Physics 215 results.
121
Physics 21 1
80
o
<D 6 0
O 40
O
sP 20
I
Q1
PH
Q2
Q3
I
Q4
Q5
Original Version
Modified Version
Physics 21 5
80
id
60
40
o
u
20
iI i
[—1
Q1
Q2
Q3
Q4
I
Q5
Figure 4.4 Comparisons of student performance (including error bars) on each test
question of the original version and the modified version of our proportional
reasoning pretests. Separate results are given for Physics 211 and Physics 215
students.
Discussion of Performance by Course
The result of differing overall performance between the algebra-based and
calculus-based introductory mechanics populations on both proportional reasoning
pretests may not be surprising to many researchers. Barnes,69 in one of the earliest
reports of Piagetian-type tests used as a predictor of success in introductory physics
courses, also observed that mean scores on his pretest varied as he considered
introductory physics courses with different prerequisites and consisting of different
introductory physics populations (e.g. biology majors etc). Variation in performance
122
on these tasks across institutions, has also been noted by Loverude et al.
Viewed
from a Piagetian standpoint one might attribute the difference in performance by our
two introductory physics populations (and as in Loverude's case: by students at
different institutions) on these pretests as indicative of these students' differing
proportional reasoning abilities.
Generally speaking, when glancing at student performance on the two pretest
versions within each of our introductory physics courses, the results on these two
versions appear similar. This result lends credence to the view that regardless of
context, both tests are largely measures of proportional reasoning ability and that this
ability is an issue that is separate from context in which the question is posed. One
may say that both of these proportional reasoning tests show that calculus-based
physics students are better proportional reasoners than their algebra-based physics
counterparts. However, context does play a significant role in student performance on
these tests, especially within our calculus-based physics population. For the algebrabased students, even though we do not observe statistically significant differences in
the mean performance on the two pretests, we do observe subtle shifts in the
distribution of scores on the two versions (see Fig 4.2). We believe these differences
in our students' overall performance indicate that familiar contexts might be more
successful at eliciting and activating the right set of resources, necessary for some
students to tap into their proportional reasoning abilities. The implications of this
result to our curriculum development goals are then immediate, as familiar contexts
can be used as a bridge to tap into students' hidden abilities, while building up to
123
broader understanding of many physics concepts. As part of our efforts to further
investigate which groups among our student were more sensitive to context
dependence of our proportional reasoning pretest, we looked for possible gender
effects as explained in the next section.
Gender Effects on Performance
We further wished to investigate whether males and females performed
differently on our proportional reasoning pretests, within each of our courses. In a
study very similar to our own, McCollough143 constructed a 'stereotypically female'
version of the FCI which she called the Revised FCI (RFCI). The contexts appearing
on the RFCI were "shopping, cooking, jewelry and stuffed animals," as opposed to
(what she saw as) the male-dominated contexts appearing on the regular FCI of
cannonballs and hockey, as well as the all-male illustrations. She interleaved the
versions of the FCI and RFCI given to students. She asked the questions to over 300
students enrolled in general education classes (such as English) because at her
university she saw that the physics classes were male-dominated. She found that
whereas there was no difference in performance in the female students' scores, the
male students scored significantly worse on the RFCI. We looked at whether or not
student performance on the two versions of the proportional reasoning diagnostic here
at NMSU exhibited any gender-related effects.
124
Results of NMSU Study Separated by Gender
Within the algebra-based physics course two very important differences were
observed when data collected from the students enrolled in this course were separated
by gender. For the male Physics 211 students, there was almost no difference in
performance on the two versions of the proportional reasoning pretests. However,
there was a significant difference in performance on the two versions of the pretests
for female students, who scored on average about 0.6 points higher out of five (d~0.3)
on the version of the pretest which contained the more familiar context (see Tables
4.9 and 4.10 in appendix J and Figure 4.5 below).
Another very important feature of the gender difference in performance on the
two versions of this pretest taken by the Physics 211 students is that the difference
between mean male students' scores versus mean female students' scores on the
original (standard) version of the pretest (mean 2.53 vs 2.09 respectively - d~0.3) is
significant, with a p-value less than 0.05 (see Table 4.11 and 4.12 in appendix J);
whereas the difference in the two genders' mean scores on the modified version of the
pretest is not significantly different (p>0.55). This result carries with it the
implication that while the context of the original pretest version, carries with it the
commonly observed gender difference in performance on tests like the FCI and in
physics courses,144 there is no evidence that the context of the modified pretest
version carries with it a bias in either direction.
There was a similar pattern for our Physics 215 students. For the male
students in this course, there was no significant difference in performance on the two
125
versions of the pretests. Even though there is a relatively low number of female
calculus-based physics students, the difference in performance on the two versions of
the pretest was statistically significant at the (p<) 0.05 level as was the case for
Physics 211 female students [see Tables 4.13 and 4.14 in appendix J and Figure 4.6
below]. The effect size for this difference was also medium (d~0.5)
Algebra-based physics
Pretest: 'Standard' version
Algebra-based physics
Pretest: Modified version
40%
30%
40%
30%
20%
10%
20%
10%
0
1 2
3 4
Males: N = 122
5
0
Algebra-based physics
Pretest: 'Standard' version
1 2
3 4
Males: N = 93
5
Algebra-based physics
Pretest: Modified version
40%
30%
20%
10%
40%
30%
20%
10%
0 1 2
3 4 5
Females: N = 88
0 1 2
3 4 5
Females: N = 128
Figure 4.5. Distribution of Physics 211 students' total scores on two versions of 5item pretest. Results for male students in top graphs andfemale students in bottom
graphs.
126
Again, in our calculus-based physics course, for the original pretest version
there was a significant male bias in terms of performance: Males scored on average
about 0.7 points higher out of 5 than their female counterparts on this test version
(d~0.4). In contrast, on the modified pretest version the two genders' scores were not
significantly different [see Tables 4.15 and 4.16].
Calculus-based physics
Pretest: 'Standard'version
Calculus-based physics
Pretest: Modified version
40%
30%
20%
10%.
40%
30%
°0%
10%
0
1 1
1 2
3 4 5
Klale s:N [ = 1 22
0
Calculus-based physics
Pretest: 'Standard' version
11
1 2
3 4
Males: N = 87
Calculus-based physics
Pretest: Modified version
40%
30% —
20% —
10% —
40%
30%
20%
10%
J
L
0 1 2
3 4 5
Females: N = 37
0 1 2
3 4
Females: N = 28
5
Figure 4.6. Distribution of Physics 215 students' total scores on two versions of 5item pretest. Results for male students in top graphs andfemale students in bottom
graphs.
127
In short, altering the context of our proportional reasoning test served to
improve the proportional reasoning performance of our female introductory physics
students. We did not observe the same significant decrease in male performance that
McCullough did.
Our findings are consistent with observations made by Trimmer145 in
reviewing the literature on gender differences pertaining to physics: that females tend
to perform better than males on questions that are contextualized and on questions
that involve a human presence. We posit that these contextual characteristics are
primarily responsible of our modified pretest version having significantly improved
female performance, while leaving male performance virtually unaffected.
It is instructive to consider some further observations made by Trimmer on
contextual characteristics of questions where females outperform males. She
observed that questions that contain open instructions or that require extended
writing, with words such as 'explain', 'evaluate', 'transform' and 'analyze' and
questions that require 'routine use of mathematical knowledge and manipulation',
generally led to females outperforming males. In contrast she found that males
outperformed females in questions requiring very short answers or where only one
correct answer was sought (such as multiple choice questions); questions involving
calculations; questions beginning with the words 'identify', determine', 'locate', or
'calculate'; questions that involve the use of addenda such as diagrams, tables and
graphs; complex information-dense questions; and questions framed in the negative.
128
Our study provides further evidence that by carefully choosing the contexts
and wording of our physics problems, we can reduce the male gender bias prevalent
in our classrooms today. Steele145 has showed that, by cuing people in genderdiscriminatory and race-discriminatory situations in math and science testing, one can
cause women and minority populations to do more poorly than otherwise. We believe
that context factors play a larger role in standardized test scores than it is currently
given credit for, and that highlighting these factors is a potentially valuable
contribution to be made by physics education researchers.
Fall 2009 Pretest Versions
As explained in chapter three, the pretests used in Fall of 2009 were slightly
different. In addition we also had an opportunity to ask the entire Lawson Test during
the last lab of this semester. For students who were present for both pretests, we had
matching pre-course and (virtually) post-course testing data for two questions which
appeared on the pretests given during the first lab as well as on the Lawson test. For
half of these students the pre-test and post-test questions were the same, while
(depending on which pretest they took at the beginning of the semester) the other half
of these students had matching 'modified pre-' and 'original post-' testing results.
We thought these results would be useful in showing how individual students
responded to the two versions of questions - Whether or not there would be many
129
frequency count of algebra-based physics students'paired preand post-test scores
Score on
original
pretest
Score on
original
post-test
Score on
original
post-test
1
1
17
1
7
0
2
2
3
3
26
Score on
modified
pretest
10
2
2
2
2
5
2
2
33
N=60
N=si
Jbrequency count ofcalculus -based physics students'paired preand post-test scores
Score on
original
pretest
Score on
original
post-test
Score on
original
post-test
1
1
5
0
A
1
1
1
1
1
13
Score on
modified
pretest
N=27
6
1
2
0
1
0
1
2
16
N=29
Figure 4.7 Frequency counts of students' corresponding pre- and post- test scores
(out of two) on two questions - Questions two and three on 'original' and 'modified'
pretests shown in appendix F. For instance, in the algebra-based course, of the 61
students taking the original pretest and original post-test, seven of them scored zero
on the pretest but scored two on the post test on these two questions. Also, of the 29
students taking the modified pretest version in the calculus-based course, 16 students
got both these questions correct (had a score of two) on both the pre-test and posttest.
130
students who could answer the modified question versions at the beginning of the
semester yet fail to answer the original question versions at the end of the semester.
Due to the low numbers of students taking both pretests however, we cannot make
conclusive statements based on our results about whether or not more students who
took the modified version of questions two and three at the beginning of the semester,
performed worse later on in the semester when taking the original version of these
questions. Theoretically one could use the paired results of those students who took
the original pretest version before and after as a baseline measure of proportional
reasoning improvement through the semester, to assist in making this case.
The large number of students receiving the same pre- and post-test scores on
these two proportional reasoning questions, as can be seen by the frequency counts
along the diagonals on Figure 4.7 above, implies that there was little change in
students' proportional reasoning throughout the semester.
A sixth question, placed at the end of both pretest versions used at the
beginning of the Fall 2009 semester, was a question on the revolution of two
interlocked gears [see appendix K]. This question was suggested to us during an
earlier (Spring 2004) student interview as a question that might be more recognizable
as a 'proportional reasoning problem.' As mentioned in chapter three, this pretest was
also given towards the end of the same semester to a group of underprepared
engineering students at Rutgers University. These students, as part of a special course
on introductory mechanics that spans two semesters, worked explicitly throughout the
131
Fall of 2009 (the first semester of their 2-semester course), on invention tasks
designed to build scientific reasoning ability and to help students decide "when to use
Figure 4.8. Two interlocked gears. Students were asked to determine the number of
revolutions made by the smaller gear, given revolutions of the larger gear.
Performance^ on Gear Qyestion
Phys211
Phys215
Rutg Eng
Figure 4.9. Student Performance on gear question with error bars. For NMSU
students performance on the two pretest versions is shown.
132
math and what math to use."
This pretest was administered to these students near
the end of their first semester in this course. As predicted by our interviewee, students
performed very well on the gear question. Over 75 percent of students in all groups
answered this question correctly, which cannot be said of any other proportional
reasoning question we have asked. This finding brings home the fact that students'
poor performance on the proportional reasoning questions appearing on the Lawson
test cannot be taken on its own as evidence that students don't 'have proportional
reasoning ability'. It also brings up the important question of why many students
recognize that ratio reasoning is required in problems such as this one, but do not
recognize that it is to be used in other situations.
The fact that this same question was the last question appearing on two
different versions of proportional reasoning pretests allowed us to test for the effect of
context cuing on performance on this question. For 40 calculus-based physics
students, about 98% of students correctly answered this gear question appearing on
the modified version of the pretest, whereas only 78% of 41 students enrolled in the
same course answered this question correctly on the original version of the pretest.
This was a very significant difference (t=2.75, P<10"4).
The difference in performance on this question appearing at the end of the two
randomly assigned pretest versions was not significant for 145 Physics 211 students
(t=0.58, P=0.56). About 80% of students got this question correct regardless of
pretest version chosen.
133
On the entire 5-question pretests used in Fall 09, students performed
comparably to previous semesters. During this semester, however, there was a
statistically significant difference in performance on the two versions by Physics 211
students (df 143, t=2.17, P=0.03) while this difference was not observed for the
Physics 215 group (df 79, t=0.19, 0.85). We believe this to be an artifact of the
relatively low numbers of students taking each pretest and place much higher
significance on the results collected over 6 semesters (mentioned in previous
sections).
Mean score on 5 Question
Fall09Pretest
3 _.__
_
m
w.
1
0
Fall 09 Original
_h.
2
W>.
Phys211
W
V//A
Phys215
Fall 09 Modified
Rutg Eng
Figure 4.10. Student Performance five-question pretests (excluding gear question).
For NMSU students, performance on the two pretest versions is shown.
134
Pretest Correlation with Final Exam
As stated in chapter three, the modified pretest version had a generally lower
correlation with success in our physics lab course as measured by our lab final exam.
For the original pretest version the correlation with the final was between 0.4 and 0.5,
whereas for the modified pretest version the correlation was about 0.3. Here we
describe the correlations with final when the different populations are divided by
gender.
Results from the discrete pretest version as well as from our female Physics
215 students' are included for completeness of the presentation However, as these
numbers are low, these correlations are not as reliable as the other coefficients
presented.
There is some small gender difference between the two course populations in
terms of how predictive the pretests are [see Table 4.19]. It should also be noted that
again it appears as though the original pretest version is more predictive of success on
our final exam than the modified pretest version regardless of gender. This holds true
in both the algebra-based and calculus-based courses.
In the Fall semester of 2008 we used one additional measure of student
success in our lab course. During the last week of lab we asked the Teaching
Assistants (TAs) to rate their students based on factors such as how well students
understood what was going on in lab and the quality of the questions they asked in
lab.
135
Table 4.17. Correlation coefficients of score on different pretest versions with final
exam score.
pretest
Version
Physics 211
Males
Physics 215
Females
Males
Females
Original
0.29(72)*
0.46(82)***
0.50 (91 )*** 0.40 (25)*
Modified
0.28(61)*
0.39(64)**
0.36(63)**
0.26(19)
Discrete
0.68(18)**
0.27(14)
0.37(33)*
0.20(6)
Modified
* p < 0.5
* * P< ICr2
*** P< 10"4
Table 4.20 in appendix L shows the correlations among TAs' ratings of
students, the students' scores on the proportional reasoning pretest (taken at the
beginning of the semester), and the students' scores on the final exam. The TAs'
ratings of students were fairly well correlated with students' scores on the final exam
(correlation coefficients generally about 0.6). The TAs' ratings of students were also
generally better correlated with the original pretest version than the modified pretest
version. Again, if one were to choose between these two pretests based solely on their
predictive ability of success in physics as defined by TAs' ratings one would be
inclined to choose pretest version one.
In designing our final exams, we generally try to choose 'real life' contexts in
which to frame our questions. This fact does not seem to translate into higher
correlations between our students' scores on our modified version (everyday context)
proportional reasoning pretest with their final exam scores (compared to correlation
between the original proportional reasoning pretest with final). In Karplus'
136
interpretation of Piaget's framework of accounting for different student thinking
patterns,148 it can be said that the original pretest, being more abstract, probes a higher
stage of development (the formal stage) than the modified pretest which probes the
concrete stage. In Karplus' view, abstract reasoning is preferable to concrete and as a
consequence the more abstract of the two pretests is preferable. The fact that an
independent measure of accounting for student differences - rankings by TA's correlated better with the more abstract of the two pretests, lends credence to the view
that the abstract measure may be a better predictor of success in physics regardless of
measure of success used.
It may be argued however, that this lower correlation is an indication that
despite our best efforts, the labs we have designed were unsuccessful in promoting
our students' reconciliation between mathematical and everyday reasoning.137'149 We
have not done expectations surveys150 within our labs to test for changes in student
views on what the purpose of the labs were. One way to test this would be to use both
pretest versions in a course designed around sense-making ideas and see if we get a
different result in the pretests' correlations with the success measures used in such a
course.
Discussion and Conclusion
We have observed that our different introductory physics populations perform
differently on our proportional reasoning pretests. We saw that our calculus-based
137
students performed better than our algebra-based students on these pretests regardless
of context. Using a strict Piagetian interpretation of these results one would conclude
that there are more formal reasoners among the calculus-based physics population
than among the algebra-based physics population. That contexts affects performance
on our proportional reasoning tests (at least among our calculus-based physics
population), leads us to lean towards a less strict reasoning ability interpretation of
these results towards including more of a resource-based model of student thinking.
We believe that tests such as the original proportional reasoning diagnostic
incorrectly lump students who have the ability to reason proportionally in some
contexts along with students who cannot reason proportionally at all - both of these
groups receiving the wrong answer on these tests.
In the past a few other researchers have observed that context plays some role
in student performance on Piagetian tasks. Tschopp and Kurdek151 for example
concluded that Piagetian tasks often differ in levels of difficulty oftentimes based on
the students' level of familiarity with the experimental materials. Tournaire,95 in
studying elementary school students in grades 3, 4 and 5 on various proportional
reasoning problems across various contexts, concluded that "proportional reasoning
begins as an essentially fragmented ability. It is likely that solving proportional
problems in different contexts, or with different number structures, involves different
abilities."
We observed that when we shifted contexts in proportional reasoning tasks
that our female students tended to perform better, while our male students'
138
performance remained the same. This finding was surprising to us, and alerted us to
the fact that we may have paid insufficient attention to the contexts used in our
curriculum development materials and in our courses. One application of these
findings may be in the promotion of 'human' context-based problems and text-books
within our courses, and (of particular importance to us) in the curriculum we develop,
as a means of tapping the abilities of female students.
One potential problem of using this approach, however, is observed where we
noted that the abstract problems may be more closely related to success in physics.
This would necessitate that we design curriculum that initially probes students' everyday thinking, but eventually providing a clear path from everyday thinking to full
scientific reasoning.
139
CHAPTER 5: CONCLUSION
Introduction
In this dissertation we have described curricular modifications that we have
made to our introductory mechanics laboratories in an attempt to improve students'
conceptual understanding. Although these modifications were made with a research
basis, our post-testing suggests that the conceptual gains that our students made were
modest. For the four laboratories that we described in Chapter 2, we remain satisfied
with the laboratory procedures we have developed in terms of their efficacy in
connecting the motion of actual physical objects to a representation of this motion
that is amenable to analysis. Students like the procedure, and seem to benefit from the
procedural practice that is offered in the labs. Moreover, for our Physics 213 students
(physics majors), there was reasonable success in students' understanding of
Newton's second law as a vector equation as measured by our final exam questions.
For our Physics 211 students (the intended audience for our lab development efforts)
and our Physics 215 students, however, we did not observe the conceptual gains that
we had hoped for. The labs appeared to be least successful with the population that
we had hoped to have the most impact on - the students in the algebra-based course.
As part of attempting to tailor our laboratories to the students in the algebrabased course, we tried to characterize the differences in preparation between these
students and students in the calculus-based course. A brief summary of the results of
140
our efforts to characterize differences in student preparation between these two
students groups is presented in the next section.
Summary of Findings
In a review of literature relevant to predicting success in introductory physics
courses highlighted in Chapter 3, we saw that many of the most successful predictors
in physics were based upon tasks used to determine Piagetian stages of intellectual
development. In our efforts to characterize differences that were relevant to the
successful implementation of the conceptual labs that we designed in our NMSU
introductory physics populations, we also found that questions designed to measure
students' scientific reasoning ability were quite useful predictors of success.
Questions excerpted from the widely used Lawson Test of Scientific Reasoning plus
one additional item accounted for about one-third of the total variance in students'
final exam scores. This fraction of total variance is cited by other researchers as
among the largest accounted for in studies in physics, even using multiple
predictors.89'90
Upon further investigation of results from the Lawson test, we found that our
students struggled most on proportional reasoning items, and in fact, pretests based on
these items alone also bore good predictive power of student success in our physics
lab courses: We found the predictive ability of a short 5-question pretest on
proportional reasoning ability - four proportional reasoning items from the Lawson
141
test, plus one additional density item - to be of comparable predictive power to other
longer pretests used as predictors in physics in the past. That so many of our students
do so poorly on this pretest shows that a significant portion of our incoming students
lack this fundamental skill.
In a later study, we attempted to use our students' working memory capacities
to improve on the predictive ability of our pretests, as well as to potentially provide a
further measurable characterization of differences between our introductory physics
populations. Our results from this investigation were largely inconclusive, with
measures of our students' working memory capacities using the n-back test only
weakly correlating with their final exam scores. In addition, there was virtually no
correlation between final exam scores and: (1) A spatial working memory test; (2) the
time taken do the working memory task; or (3) a measure of student efficiency in
answering our working memory questions.
Results from our introduction of contextual modifications to the proportional
reasoning pretests used as predictors were guided by a desire to explore whether
contextual changes influenced students' epistemic dispositions
as indicated by their
success rates on proportional reasoning problems. Initially, we introduced an alternate
proportional reasoning diagnostic which was set in the context of serving rice in a
soup kitchen. This pretest, however, failed to control for the difference in reasoning
required to solve for proportional reasoning questions involving discrete items, as has
been noted to influence success on proportional reasoning problems by other
researchers.96 Later, as an alternate to the original proportional reasoning pretest, we
142
used a proportional reasoning pretest that (similar to the original) involved the use of
a liquid (modified pretest).
We found that these contextual modifications had significant consequences on
our students' success rates on proportional reasoning questions. In particular, we
observed that on a pretest version modified to include a more every-day context (in
our estimation) - one pouring medicine into two different sized dosage containers that students were generally more successful at answering our proportional reasoning
problems. When broken up by course, this difference in performance was statistically
significant for our calculus-based students, but fell short of achieving statistical
significance for our algebra-based population. We concluded from this that not all of
what we measured using the original proportional reasoning pretest can be
attributable to proportional reasoning ability, and that some of our students' poor
performance on these questions can probably be attributed to a lack of familiarity
with the context with which the problems are presented .
In what was perhaps the most surprising result of our study, we found that for
students in both algebra-based and calculus-based introductory physics courses, the
effect of context modification was significant for female students, whereas for male
students there was not a noticeable difference in performance on the two pretest
versions. Furthermore, within each course the original proportional reasoning pretest
version showed males significantly outperforming females, whereas this was not the
case on the modified pretest version.
143
Finally, in measuring the correlations between students' scores on both
versions of the pretest with their final exams scores, we generally found lower
correlations between the modified pretest version with final exams than the original
pretest version with final exam. This was true for male and female students alike, and
was also true for calculus-based and algebra-based introductory physics courses.
Implications for instruction
At NMSU we have seen that whereas there is general improvement from
pretest to final exam for many of our students across different introductory physics
course populations, our majors seem to benefit the most from our reform attempts
while our algebra-based population seems to benefit least. In spite of this
disappointment, some hope is presented in the fact that students in lectures closely
aligned with the labs' content, generally exhibit above average performance, as was
highlighted in chapter two. We believe that reinforcement may be key towards further
improving results from our underperforming student populations. It is possible that
through increasing awareness of the methods utilized in the laboratory to our lecture
instructors, that the coherence between physics presented in lecture and lab might
lead to improved performance.
Results reported by the University of Maine seem to suggest that their
students have benefited from breaking the acceleration along a curved path into
tangential and radial components in addition to the method of vector subtraction we
144
have presented in this dissertation. Our past experience at NMSU leaves us hesitant
to attempt such an approach, as our algebra-based students (the intended audience for
this lab development) have traditionally struggled with understanding vector
components. This fact, however, does not preclude the possibility of attempting a
dual vector-component approach with our calculus-based population.
We believe that different success rates of the same curricula at various
institutions implies a need for careful consideration of the target population when
designing curricula. Hake,54 in popularizing the use of <g>, made a very important
first step in accounting for different incoming states of students. It is our opinion that
a further measure needs to be introduced that gives some indication of the maximum
possible improvement in performance that can be reasonably expected for different
populations, in order to be of further guidance to institutions, instructors etc. We
think that such a measure will be useful for instructors who hope to implement
curriculum in populations that are significantly different from the populations in
which the research was conducted. Meltzer supposes the existence of a 'hidden
variable'55 related to identifying successful physics students. Factors such as selfefficacy show promise in predicting success,
and are among factors that most
educators across various institutions would agree should contribute towards overall
student success.
That our short proportional reasoning pretests play such a prominent role, on
their own, in accounting for which among our students will be successful in our
courses at NMSU, highlights the strong connection between this skill and
145
understanding physics. This result gives some merit to Piaget's view - that
proportional reasoning ability is one central factor (among others), necessary for
students' to reason scientifically. It can be surmised that in any physics course,
whether conceptual or mathematical, students need to represent and manipulate very
abstract ideas (such as momentum, energy, and torque) and mathematics is an
indispensable tool in doing the book-keeping of all of these abstractions. That some
mathematics pretests used in the past have not been highly successful as predictors,
may indicate that raw mathematical ability is insufficient to guarantee success in
physics. In this regard, it may be enlightening for instructors to pay close attention to
the abilities identified by Piaget as having particular bearing on scientific reasoning in
general.
It is very important to note that the context in which questions are asked has a
strong effect on which subsets of students enrolled in our courses do better and which
do worse. We are encouraged by the finding that in choosing contexts that are
familiar to students, we can potentially help some students overcome what at first
seems to be deficit in ability. We hope that by guiding students to work with the
concepts we are trying to convey in a more familiar context, and then linking these
skills directly to formalism necessary for physics, that we will be able to better utilize
students' incoming abilities toward physics understanding.
The influence of context on our female students' performance suggests a need
to pay particular attention to the contexts in which the material that we develop, is
presented. Careful selection of contexts may play a significant role in reducing
146
potential gender bias and encouraging more gender-equitable participation in physics.
In order to design our physics courses in ways that promote diversity,154 it is
important that we pay particular attention to the contexts in which our questions are
asked. These contexts influence which among our students will go on and be
considered 'successful' and which will not.155 Rennie and Parker156 in research
supporting gender-equitable teaching and assessment in physics found that teachers
can create gender-neutral or gender-inclusive assessment tools by careful portrayal of
stereotypes in particular contexts. They advocate concrete problems over abstract
problems and find that appropriate contexts make problems easier to visualize and
more interesting. We hope to be guided by these principles in any curriculum
development we may undertake in the future.
147
APPENDICES
148
APPENDIX A
MOTION IN TWO DIMENSIONS TUTORIAL AT THE UNIVERSITY OF
WASHINGTON
Mt h
MOTION m TWO DIMENSIONS
*
L Vdtwily
Anwtojeci i* rowing around an oval track. Sfceleh itie tr»j««;K»cy <sf rte *>bjeet tin a large sheet of
paper. (Make your diagram lai%e.%
A.
Choose a point to serve as an origin for your «*irdina,!e system. Label Shal point O (for
origin). Select twt» locations »F the object ihw are about nnc-icighfh of the oval apart and
label them A and B.
f.
Draw the paoacion
vector* f«r isiwh *:*!'
„
Co y y l w r
f
IWP
sd
,
, > »,
,
„.
.
«'*«'m'« * i * *P«*« * * r df«w»«w-
«l»e t w o JctCatKNMt <4
and # and draw tlis
vector that
represents the
displLncemcinl fawn
A la-B,
2,
\
IJCK+IH* how U» us* the *t:is|rfiK«ncni vector lo determine She «liwcti«i of i!t« av'Miige
velocity s»f the object between A and i<. Dww it veet«r t» rejwcNeiM itu; iivwige velocity,
3. Chewsc m ptsmx %m the ovalfeelween ptsittte 4 *«d W. ami label that jwini fl*.
A* poiw H * i* ehf**en 1*> tic th}<*r aiul elcwer la j»in« A, ckte* the direction of (tic awrrage
velocity «v«r the interval ^ H ' change? i f so, haw?
4, ftesMihe she direction ttf the i«*.ttriia»w*»«s> velocity €»i'th« mbjwt at pniwl A.
How w©tth( yuti t'tiaisw-ieriw IIK? diwsettan «f the wsMi»rrt«ie«i:* vcluciiv m tm$ %mM <«
ills* waicostff ?
P«e* y«t« anwer defwnd mt wteiltw tte «Jtoje£t is speeding rtp. stewing itown, m
moving; wish «M*isl«it \jxsert"? Hxphun.
B, M ym wefts u> dwwse a different origin (tor tji<? ettwrtltotite syMerit, wbk'h of tin* «**tow that
yam have (!*!»»» in pan A. w«dd dw*»g$ tuwl wbk*h wtitihl mit cluing? ?
fttft.vittf*ill
Meftermaif, Staffer,« M i * i , t ; , ' W i » l i .
fr»iri«,tef/-v>"Fkyik'i
149
CClraski? Halt, IIK.
FSIM lalidmi-JtJOI
16
m^-—————^mmiBe=SBBB
11, Amterathxn for motion -wltb (.cinyfunt spet-d
Suppfls? that the ufojset in s*elioti I is moving around (he track at amstam spend. ITratw vectors te»
represent the velocity at two points on she track t t a an? relatively dose together, fDra* ywii
vectors tor^*',) Latiel (fee two points Carad J3>.
A, On a $tjmm» pm of your paper, copy tte velocity *«et«f* pc and #^ Pnwn these vectors.
tietetmifie the cfaiti$i in tneheity m*<w-f M*.
I,. Is the i«igte formed by the '"head*" of i% and the "tail" ef Ar greater than, less than, m
cqmimW'f
As point D is chosen to lie etasc? aiKl closer to point C, dots* Ae above angle
incfmvtt deertase,« remain the Mime? E*pljMli h»* yen «M tell,
D»e«< (he shove anjgfc upproiKh a Umtimg value"? If so, what is te limiting value?
2.= Describe how tw H « the change in velochy vcdo* ?»<*ete«wii«c the mmrgs mmtemkm
irf: the ©fe^t het%«M C and /X Draw SI v%ttMf w *epf*»*nt the avwap aaceknttion
Between point* C and 0 ,
What hippo!* Mi the; rm»piiSiMfc «f Ai» sw» point P isrfwwettSo lie cloxcr aiwl etasesr tit
pmituCr1 &«iittesicwl«itJiwt«tanpttMhc:W)*w wty? Ikptaifi,
Cansidef the dirertkift «1 the iie«teraii«:i»t «i jMsiiii C, lb HK aitfte kstwcew dhe
»t«|ffljlici« m l o r swl the welaelty w&mgtmm than, km $hm, wmjmt t» W f
(l¥«ir.;; CMvetrtitwtiHjvtte « $ k tetweett » **«*» Is tfcftned «i (te Mikgle liHimi*t
when tltey ««frfwrf '*iill»t»-ttil,"l
fuimmk mfaMMte-terj'Ptopics
MriDwrooll, Staffer, *>.E«J„ V. Wwh.
0f"wnik» HUf I,, to£>
I%si tetiita, W 6
150
Mpt-ifm
w IWG
dlfttfttmioni
mmmm
B. StiRpcse she object started/horn resl at pottst £ and mewed towards point F whit iitcrcastiif
speed. H»w WCKIM yew find Mie aewteratioii at paint E?
Demwibe-ttmdirccliosi of (he m^tarcwtofJ of (lit abjeti m paint E.
ۥ
AS j*everat point*on each*rfUse sHaft'siMi*bchw,straw it v*cu*thai rejWEsaHs the
acceleration sf she object.
AeeeltfEatmn vectors for
speeding up (TOHI rest iil piiim 4
'\
; >
Tpjt view (iiafrrajti
TV:f- vtew diagram
CttafartefKg the direct i<Kt «f Hw irecderMion i t « d » pntat DO site trajectory for « « h >e*se.
Is the KJceteratrort directed toward the ''center*' «f the «w»! »t (Wety paint «t» ibc trajectory
fyt i*jtrwr of these cases?
Sketch arrows to show ihc disreeiiwrii of the ii««lersHofi fur the following tra|cc?txwte<<:
Cwtstaitt *|tg#i|
•
$ p « t l i l l j t Uft-
CO
ffcSiMMMfrMpNl
• eiwwiiir
Tutorials in huradmrmri,- IRISH'S
© I t e r t i w I W I , fee.
Hist Edition, 2002
151
APPENDIX B
SELECTED LABS AND ASSOCIATED HOMEWORK EXERCISES
Motion in 2 Dimensions Lab
Introduction
In this laboratory, you will examine the acceleration of objects that are moving in two
dimensions, or 'in a plane'. You can imagine this motion as motion that is confined to lie on a
giant, flat sheet of paper. The orientation of this 'flat sheet of paper' will change in different
instances, but the motion of our objects can always be traced upon this virtual sheet of paper.
- Av
By definition, (average) acceleration is a = — . This is a vector equation, so the direction of
A?
the acceleration must be the same as the direction of the change in velocity (since the change
in time is a scalar). We will use this fact to help us find the direction of acceleration for
motion in two dimensions.
Lab Objectives
After completing this lab and the associated homework, you should be able to:
1.
Subtract two vectors, and identify (Av) as the difference of 2 vectors.
2.
Use a change in velocity vector to determine the direction of the acceleration
of an object for a small time interval.
3.
Describe the direction of acceleration for objects in parabolic, oval and
other types of two-dimensional motion.
Outline of Laboratory
Approximate sequence of the lab and homework:
1.
Practice interpreting time-exposure photographs.
2.
Learn how to graphically subtract two vectors.
3.
Relate displacement to velocity.
4.
Deduce the acceleration of an object from velocity vectors for straight-line
motion.
5.
Find the acceleration vector for an object experiencing parabolic motion.
6.
Find the acceleration vectors for an object moving in an oval path.
7.
Derive the acceleration vectors for complex paths of motion.
152
8.
Draw acceleration vectors corresponding to oval-shaped motion.
Recording motion and interpretation of pictures
In this laboratory we will be using images similar
to the strobe photographs that were used in the
Descriptions of Motion lab. We will attach
blinking lights ('blinkies') to objects in twodimensional motion and take long-exposure digital
photographs of these objects, which will help us to
analyze the motion of our objects. These
photographs will be similar to the one shown at right.
Our blinkie emits light for a period of time (we will refer to this condition as being 'on') and
immediately following this period undergoes a period where it does not emit light ('off). The
blinkie goes through this sequence of being 'on' then 'off then 'on' continuously. If the room
is dark enough, the resulting picture only shows the light that entered the shutter, while the
blinkie was on, at the position from which the light was emitted. If the shutter was open long
enough to record many cycles of light and darkness (from the blinkie), the resulting image
will show lines of light (and spaces of darkness) emanating from the position of the blinkie.
We can then analyze this image (of lines of
light amid a backdrop of darkness) to
determine the direction of the velocity and
acceleration of the moving object to which the
blinkie was attached.
2.1: The drawing at right represents a
photograph taken of the motion of a toy
hovercraft initially travelling along a straight
line, then suddenly being kicked. It is known
that our attached blinkie emits light for 0.1
seconds (one-tenth of a second), and that the
floor tiles that can be seen in the background
are 10 cm on a side. Based on this drawing:
A. Is the speed of the hovercraft at point A:
greater than, less than, or equal to the
speed of the hovercraft at point B? Explain
B. Use a ruler to determine the ratio of the speed of the hovercraft at point A to the speed of
the hovercraft at point B.
C. Estimate the speed of the hovercraft at point A.
D. Estimate the speed of the hovercraft at point B.
153
E. Before the hovercraft was kicked, was it speeding up, slowing down, or moving at
constant speed? Explain how you can tell.
2.2: In physics, vectors are often represented by lines whose length is
proportional to the magnitude of the vector, and whose direction is depicted
by an arrow-head (at the tip of our line) pointing in the direction of our
vector. For example, a 200 km per hr wind from the north may be
represented by the vector shown at right.
200km/hr
A. Measure the length of the vector shown at right above. What speed
does 1 cm represent?
The relationship between the length of a drawn vector and the magnitude of the quantity that
vector represents is known as
the scale of the drawing.
i
B. For the motion of the toy
hovercraft shown in section
2.1, draw a vector that
represents the velocity at point
A. Use a protractor to ensure
that you have not changed the
directions of the original
vectors. Use a scale of 1 mm
= 2 cm per second.
C. At the same scale, draw
|
another vector for the
velocity of the hovercraft at point B.
D. How does the length of the velocity vector at point B (which represents the magnitude of
the velocity at B) compare to the length at point A? Does the ratio of the lengths of the
vectors you have drawn accurately reflect the ratio between the lengths of lines of light at
points A and B in your diagram?
154
=>
Check your answers above with your lab instructor before continuing.
Subtracting two vectors
Given a vector^, the vector -A is the vector having the same magnitude as A, but pointing
in the opposite direction.
_
_
A
3.1: Given the vectors A, B and C shown at right, draw the
vectors -A, -B and —C in the space below.
^
^
Vectors are translation invariant, which means
that you can slide the vector A across or down or
wherever, as long as it points in the same
direction and has the same magnitude as the
original vector, then it is the same vector.
All of these vectors
are equivalent
3.2: Two vectors can be added
graphically by placing the tail of
one vector against the tip of the
second vector. The result of this
vector addition, called the
resultant vector (R) is the vector that has its tail at the tail of the first vector and has its tip at
the tip of the second vector.
In the space below, add vectors B and C shown in 3.1 and at right.
155
3.3: Vector subtraction is just a special case of vector addition. One can view A -B as
The procedure we have used for adding two vectors can therefore be used for
vector subtraction, where in this case the two vectors to be
added are A and -B.
_
A. In the space below, subtract vector C from vector A.
B. Now subtract vector A from vector C. How does A -Ccompare to
C-A?
Finding change in velocity vectors and acceleration direction.
The average acceleration of an object for a time interval defined as the ratio of the change in
the velocity of that object divided by the length of the time interval: a = ^At • This is a
vector equation - the quantities on either side of the equal sign must have the same direction
as well as the same magnitude. Therefore, in order to find the direction of the acceleration
we need to be able to determine the direction of the change in velocity vector. (Since time is
not a vector quantity, the change in time Af does not influence the direction.) The direction
of the acceleration is the same as the direction of the change in velocity.
4.1: To find the change in any quantity, we need to subtract its initial value from its final
value. To find the change in velocity, we subtract the initial velocity vector from the final
velocity vector.
A. Consider the velocity vectors A
and B that you drew in section 2.2.
In the space at right, redraw these
vectors to scale, labeling the
velocity at point A asvjmtiaiand the
velocity at point B as fmal. The
direction of these vectors must be
the same as in the original diagram
in section 2.1.
156
The change in velocity vector Av is equal to the final velocity
final
minus the initial velocity
v
initial •
Av -
v
final
~
v
initial = v'final
+ (~v
initial)
B. In the space to the right above, find the change in velocity vector for the hovercraft as it
moves from point A to point B.
C. Draw a small arrow to indicate the direction of the average acceleration for the toy
hovercraft between point A and point B.
4.2: It is probably a good idea at this point to comment on why we are doing all this. One of
the most important fundamental ideas in introductory mechanics is Newton's second law.
This law is a vector equation (again, the direction as well as the magnitude must be the same
on either side of the equal sign) that relates the net force acting on the object to the
acceleration of that object: I.F -ma .
Since we found the direction of the acceleration the hovercraft above, we can now use
this to learn something about the direction of the net force. In this case that net force
direction is just the direction of the kick. In what direction was the hovercraft kicked?
4.3: The drawing at
right was created by
drawing the blinkie
lines from a photograph
similar to the one at the
beginning of section 2.
We will assume that the
object was moving
clockwise (That is, the
object reached point A
before it reached point
B.)
To find the direction of the acceleration for a portion of the motion, we can determine the
change in velocity vector for that region. For example, to find the change in velocity vector
for a small region around point A, we use the light stripe just before point A to determine the
velocity at the beginning of a small time interval, and we use the stripe just after points to
determine the velocity at the end of this small time interval. Often, the change in velocity
vectors are small, so it is useful to draw the vectors larger than the light stripes as shown
below. In this case we have drawn the vectors five times as large as the light stripes.
157
A/
— Vfiml - Vinitial
'final
A. In the space at left above, perform a similar vector subtraction to find the direction of the
change in velocity vector for point B.
B. For each of the following cases determine whether it is possible for the change in velocity
vector to be zero. For each case, include a graphical vector subtraction to support your
contention.
Case 1: Neither the speed nor the direction of the velocity vector changes in the interval
between two adjacent stripes.
Case 2: The magnitude of the velocity vector remains the same, but the direction of the
velocity vector changes in the interval between two adjacent stripes.
158
Case 3: The magnitude of the velocity vector changes, but the direction of the velocity
vector remains the same in the interval between two adjacent stripes.
For the remainder of this lab you will be using 'blinkies' attached to objects to create longexposure photographs, and then you will analyze these photographs to study acceleration in
two dimensions. You will complete three exercises: (1) motion of a toy hovercraft moving
along a ramp, (2) motion of a pendulum moving in an oval, and (3) motion of a toy roller
coaster cart moving along a track. You do not need to perform these exercises in any
sequence, and you should ask your lab instructor at this point which experiment to perform
first. You can do either the hovercraft or the pendulum experiments first. You will use data
from the roller coaster experiment as a basis for some of your homework.
General instructions for camera use
You will be using a digital camera to take these pictures. The camera is set up in a 'shutter
priority' setting that allows the user to control the amount of time that the camera shutter is
open. In this setting, the ambient light level is measured and the camera adjusts the aperture
to control the amount of light.
These digital cameras take about one-half of a second for the shutter to open after the shutter
control button is pushed. It may take a few tries before you get a photograph that captures the
motion you want to analyze.
Once you have a picture your group is happy with, use the USB cable to download the picture
to the computer at your lab table. Follow the instructions on the handout on the table to
invert the picture. (This is similar to creating a negative.) Then print a copy of the picture for
each member of your group, as well as a picture for the whole group.
As with the exercise on the previous page, we will need to draw fairly large vectors to
represent the initial and final speeds compared to the size of the stripes. As long as we are
careful to preserve the relative sizes of the vectors as well as their angles, we will be able to
determine the direction of the change in velocity vector.
159
Exercise I: Parabolic Motion
The diagram at right shows our
hovercraft about to move along an
inclined plane.
^ — ^ ^ ^
^^JS"""""""'"^
^ ^ ^ ^
^^T^^
"""" *4 ^ ^ ^ - ^
^^*^^
/'
^ ^ \ ^
5.1: In a darkened room, take a
^S*S%.
/'
^s^S^fj
picture of the motion of the hovercraft
^'*5!!!!S!^
'
^a^^^^^^^
as it travels in a parabolic arc as
^ s *S ftsS as* iiS ^^
shown. To get the best possible
picture, we should aim for our hovercraft to begin its motion in one of the lower corners, and
end its motion on the opposite lower corner of the inclined plane. You may want to repeat
this procedure a few times, and only use the 'best run' for your analysis.
You will find that the parallel stripes on the ramp do not look parallel in the photograph. But
you can use the grid painted on the ramp to tell which direction is 'down the ramp' and which
direction is 'across the ramp' at each point in the photograph.
5.2: After printing this photograph:
A. Find the change in velocity vector for the hovercraft for a portion of the path when it is
on its way up the ramp (somewhere near point A in the diagram). What is the direction of
the acceleration vector for this portion of the path?
B. Find the change in velocity vector for the hovercraft for a portion of the path when it is at
the turnaround point for this path (somewhere near point B in the diagram). What is the
direction of the acceleration vector for this portion of the path?
Find the change in velocity vector for the hovercraft for a portion of the path when it is
on its way down the ramp (somewhere near point C in the diagram). What is the
direction of the acceleration vector for this portion of the path?
D. For this parabolic motion, summarize your answers about the direction of the
acceleration for the hovercraft. That is, how can you describe the direction of
acceleration for all points of the parabolic motion?
Check your answers above with your lab instructor before continuing.
160
Exercise II: Motion in an oval
The diagram at right shows a pendulum
swinging in an oval.
6.1: In a darkened room, Place the camera
on the floor pointing up at the pendulum
bob. Start the pendulum swinging in an
oval as shown, and take a picture of the
motion of the pendulum.
6.2: After printing this photograph:
A-i'
A.
Find the change in velocity vector for
~~ ~
_
•*--"'
B
the pendulum bob when it is at its
maximum distance from the center of the oval (somewhere near point A in the diagram).
What is the direction of the acceleration vector for this portion of the path?
B.
Find the change in velocity vector for the pendulum bob for a portion of the path when it
is at a point closest to the center of the oval (somewhere near point B in the diagram).
What is the direction of the acceleration vector for this portion of the path?
C.
Find the change in velocity vector for the pendulum bob for a portion of the path when it
is at an intermediate distance from the center of the oval (somewhere near point C in the
diagram). What is the direction of the acceleration vector for this portion of the path?
D.
For this motion, summarize your answers about the direction of the acceleration for the
pendulum bob. That is, how can you describe the direction of acceleration for all points
of the oval?
Check your answers above with your lab instructor before continuing.
161
Exercise III Motion along a track and parabolic motion
The diagram above shows the track that the toy roller coaster will travel along. Unlike a real
roller coaster, our toy will be launched so that it travels in a parabolic arc.
7.1:
In a darkened room, take a picture of the motion of the coaster as it travels along the
path shown. Be sure that your picture includes a clear blinkie signal from the cart for all of
the labeled points above, because you will be using the photograph you make as the basis for
answering some homework questions. You may want to repeat this procedure a few times,
and only use 'the best run' in your final analysis.
7.2:
Print one copy of the photograph for each group member so that you can use it to
answer homework questions. (If you have time, you may want to complete question 1 of the
homework while you are in the lab room.)
162
Motion in 2 Dimensions Homework
1. For each pair of vectors shown below, use a ruler (and protractor if necessary) to perform
the indicated subtraction as shown in the example.
Example
C-
C=B-A
oD=E-F
11.
H
c>G=H-I
in.
J=K-L
K
2. Suppose that two vectors A and B have the same magnitude (length). For each part
below show an example of possible directions of A and B if subtracting one from the
other gives:
I.
zero.
ii. a third vector that has twice the magnitude of one of the original vectors.
iii. a third vector that has the same magnitude as one of the original vectors.
163
In each case i. through v. below, find the change in velocity vector Av from the initial point
to the final point for an object moving along the path shown. IndicateAv with a dashed line
vector, and graphically determine its magnitude. Use a scale 1 centimeter = 1 meter per
second. In the space to the right of each case, draw an arrow that shows the direction of the
average acceleration for the time that the object travels from the initial to the final point.
{Recall that the average acceleration is defined by the vector equation a* = —- Since At is a
scalar, the direction of a must be the same as the direction of A v.}
Example
Initial:
/ v = 4 m/s
H =0
Final:
4 m/s
Acceleration direction:
None.
\4
Acceleration direction:
\M-
Acceleration direction:
\M
Acceleration direction:
\M
Acceleration direction:
\M-
Acceleration direction:
Final:
v"= 5 m/s
Final:
v"= 3 m/s
• Initial:
v = 4 m/s
Final:
v*= 4 m/s
• Initial:
v"=4m/s
Finab
v = 3 m/s
• Initial:
7 = 4 m/s
164
4. A toy car with a blinkie attached moves in a clockwise direction around a racetrack. A
drawing of the trail made by the blinkie is shown. The car starts at rest from points. By the
time it reaches point D it is traveling at a constant speed, and continues at this speed until it
reaches point G. It then slows down to
,B
a stop.
y^
s'
/ ^
i. On the diagram at right, draw
velocity vectors for each of the
points A-G
Be sure that the
relative magnitudes of your vectors
are consistent.
/
/
ii. On the same diagram at right,
draw the acceleration vectors for
each of the points A-G. If the
acceleration is zero at any point(s)
indicate that explicitly.
iii. How does the magnitude of the
acceleration at point E compare to
that at G? Explain.
5.
A car is slowing down (but not
turning left or right) as it passes over the
crest of a hill as shown. Indicate the
approximate direction of the acceleration of
the car for the time that it travels between
points A and B. Show how you determined
your answer.
165
sF
Newton's Second Law Lab
Introduction
In this laboratory we look at Newton's second law. In the addition of forces lab, we looked at
situations where the net force acting on an object was zero. Since Newton's second law
states that the net force acting on a body is equal to its mass times its acceleration: S F = ma,
as long as there is no acceleration, there is no net force. That is, when an object is not
moving, or is moving with constant speed and in a constant direction, then the acceleration is
zero and the vector sum of all the forces acting on the body is also zero. In this lab, we look
at Newton's second law more generally, including cases where the net force is not zero. In a
sense, the purpose of this lab is to connect what you did in the last lab (drawing free-body
diagrams to find the net force) to what you did in the lab before that (finding change-invelocity vectors based on photographs of blinkies attached to objects). This connection is the
essence of Newton's second law: The net force acting on an object, based on a vector sum of
these forces, is proportional to the acceleration of that object, based on the vector difference
between final and initial velocities for that object for a small time interval. Moreover, the
direction of the net force must be the same as the direction of the acceleration, because
Newton's second law is a vector equation.
Sometimes we can use what we know about the forces acting on an object to make inferences
about the change in velocity that we expect for that object. In other cases, we can use what
we know about the change in velocity for an object to make inferences about the forces acting
on the object.
Lab Objectives
After completing this lab and the associated homework, you should be able to:
1.
2.
3.
4.
Find the direction of the acceleration by vector subtraction.
Determine the direction of the net force based on the vector sum of the
forces involved in a free-body diagram.
Compare the direction of the net force with the direction of the acceleration.
Make inferences about the magnitudes and directions of forces based on
Newton's Second Law.
Outline of Laboratory
Approximate sequence of the lab and homework:
5.
6.
Practice subtracting vectors to find the direction of the acceleration.
Practice adding the forces from a free-body diagram to find the direction of
the net force.
166
7.
Based on photographs taken of objects with blinkies, determine acceleration
direction, draw free-body diagram, and construct a vector sum diagram that is
consistent with Newton's second law.
2.
Roller coaster again
In the Motion in Two Dimensions lab, you
took a photograph of a roller coaster and
studied the acceleration of the coaster at
various points. An example of a photograph of
a coaster from that lab is reproduced here, and
is cropped to show the path of the coaster as it
moves from left to right at the bottom of the
ramp.
For this example, the streaks of light are
curved because the coaster changed direction during the time interval At that the light was
on. But a vector drawn from the beginning of a streak to the end of a streak represents the
displacement Ax of the cart during the time interval, and the average velocity during this
interval can be found from the definition of average velocity:
ave
_ Ax
~~At
T7'
2.1
On the diagram above, draw vectors to represent the velocity of the cart for light
streaks 1 and 2.
2.2
We will use these velocity vectors to give us an indication of the change in velocity
for the time interval At' that the blinkie is off between these streaks. Find the change in
velocity vector for this time interval.
2.3
The (average) acceleration for the coaster for the time interval between blinks can be
Av
found from the definition of acceleration aave - — . What is the direction of the acceleration
At
for the time that the cart is at the bottom of the ramp? Explain.
2.4
Because it is a vector equation, Newton's second law, ~LF = ma, requires that the
direction of the net force be the same as the direction of the acceleration.
What is the direction of the net force acting on the coaster in the time interval At' ?
2.5
What forces are acting on the cart during the time interval At' ? (Ignore air resistance
and friction, because these forces are very small for the coasters.)
167
2.6
Draw a free body diagram
for the coaster at the bottom of its
path, using the forces that you listed.
Be sure that the directions of the
forces you drew are consistent with
what you learned about these forces
in lecture and in the Forces lab.
2.7
Construct a vector sum
diagram by adding the forces from
your free-body diagram. The
resultant is the net force. When
adding these vectors, be sure not to
change the direction of the vectors!
However, you will need to adjust
the lengths of these vectors so that
the direction of the net force is the
same as the direction of the
acceleration.
•=!>
Do not continue past this point before checking your vector sum diagram with your
instructor.
2.8
Based on your vector sum diagram, which of the forces acting on the coaster while it
is at the bottom of the track is the largest?
3.
Pendulum
For the blinkie attached to a pendulum bob:
3.1
Hold the pendulum to the side, and release it.
Take a picture of the motion. A reasonable time
setting for this photograph is 1.3 seconds. It might
take a few tries before you have a picture that has
most of one swing from left to right, but not part of a
return swing. Print out a copy of your inverted
photograph for your group to analyze.
3.2
For the pendulum near position A:
Graphically determine the change in velocity vector.
168
3.3
What is the direction of the acceleration of the pendulum? What is the direction of
the net force?
3.4
A free body diagram for the pendulum bob is shown at right for a time
when it is near point A. The force along the wooden pendulum is a tension
force on the bob by the wooden stick.
/ T BW
XBV
W
BE
Note that a free-body diagram tells us about the directions of the forces acting
on an object, but it does not tell us about the magnitudes (sizes) of these forces.
Y
However, the net force and the acceleration must point in the same direction.
When we add the vectors together (without changing their directions!) we can adjust the
relative lengths of the vectors in order to ensure that the net force points in the same direction
as the acceleration.
3.5
Construct a vector sum
diagram based on this free body
diagram. Again, you will need to
adjust the lengths of the vectors
without changing their directions so
that the net force points in a
direction that is consistent with
Newton's second law.
3.6
Repeat the steps of exercises 3.2 to 3.5 for the pendulum blinkie when it is near point
B, near the bottom of its swing. (This time you will have to draw your own free-body
diagram.)
3.7
For which of the two positions shown above does the acceleration (and the net force)
point toward the center of the circle that the pendulum blinkie is moving in? What must be
true about the speed of an object moving in a circle in order for the acceleration of that object
to point toward the center of the circle?
169
3.8
Compare the vector sum diagram for the pendulum bob near the bottom of its swing
with the vector sum diagram you obtained for the coaster near the bottom of its path. What
similarities and differences do you observe?
4.
Hovercraft on a level
surface
For the blinkie mounted on the
hovercraft:
4.1
Push the hovercraft
across the board so that its
motion is roughly parallel to the
stripes and so the hovercraft motion is horizontal in the photograph. Take a picture of the
motion. A reasonable time setting for this photograph is 2 seconds. Print out a copy of your
inverted photograph for your group to analyze.
4.2
Choose a position somewhere close to the center of your photograph, and graphically
determine the change in velocity vector.
4.3
What is the acceleration of the hovercraft? What is the net force?
4.4
Draw a free body
diagram for the hovercraft
on the board.
4.5
Construct a vector sum
diagram based on your free body
diagram. Again, you will need to
adjust the lengths of the vectors
without changing their directions so
that the net force is consistent with
Newton's second law.
170
4.6
In your own words, describe the special case of Newton's second law that is
illustrated by the hovercraft's motion.
5.
Block on track - Speeding up
For the blinkie attached to the wood
block:
5.1
Place the block at the top of the
track with the wood side against the
track (cork side up). Make sure that the blinkie is facing the camera. Release the block from
rest, and take a picture of its motion as it slides to the bottom. A reasonable time setting for
this photograph is 2 seconds. Print out your inverted photograph for your group to analyze.
5.2
For the block on the ramp:
Graphically determine the change in velocity vector.
5.3
What is the direction of the acceleration of the block? What is the direction of the net
force?
5.4
Draw a free body diagram
for the block when it is on the ramp.
5.5
Construct a vector sum diagram based on your free body diagram. Again, you will
need to adjust the lengths of the vectors without changing their directions so that the net force
points in a direction that is consistent with Newton's second law.
171
•%>
Check your vector sum diagram with your instructor..
6.
Block on track - Slowing down
For the blinkie attached to the wood
block:
6.1
Place the block at the top of the
track with the cork side against the track
(wood side up). Make sure that the
blinkie is facing the camera. Give the
block a firm shove toward the bottom of the ramp, and take a picture of its motion as it slides
downward. You do not need the block to reach the bottom of the track -just be sure that you
have a photograph that includes the block slowing down. A reasonable time setting for this
photograph is 2 seconds. Print out a copy of your inverted photograph for your group to
analyze.
6.2
For the block on the ramp:
Graphically determine the change in velocity vector.
6.3
What is the direction of the acceleration of the block? What is the direction of the net
force?
6.4
Draw a free body diagram
for the block when it is on the ramp.
6.5
Construct a vector sum diagram based on your free body diagram. Again, you will
need to adjust the lengths of the vectors without changing their directions so that the net force
points in a direction that is consistent with Newton's second law.
172
6.6
Compare your free-body diagram in this case with your free-body diagram for
ifHS.
jX) '•kslKft
5l
^,fc-'?¥#,^'*«fc'
1-&
/
*
L A roller coaster is set up as shown above. An Inversion of the bliiriMe
„^-
"=v
X
st after
ii. What is the direction of the acceleration at point A? Of the net force?
Explain.
iii. Draw a free-body diagram for the coaster at point A, and use this to make a
vector sum diagram. Be sure to label the net force on the vector sum diagram.
iv. Of the forces you have drawn on your free-body diagram, which is the
largest? Explain how you can tell.
2.
Repeat parts i., ii., and iii. of question 1 for point B shown in the coaster
photographic inversion
3.
A skateboarder falls as she travels down the right side
of the bump. At the instant shown, she is slowing down. Use
velocity vectors to find the direction of the acceleration of the
skater. Then draw a free-body diagram of the skater, and
show that the net force on the skater is in the same direction as the acceleration.
4.
In each case shown on the next page:
174
a.
Draw and label velocity vectors for times just before and just after
the instant shown.
b.
Use these velocity vectors to find the direction of the acceleration
of the object at the instant shown
c.
Draw a free-body diagram for the instant shown.
d.
Add force vectors to show that the net force is in the same
direction as the acceleration.
Example: A skier speeding up on a ski slope.
•&1
A skier slowing down on a ski slope.
Free-body diagram
Acceleration direction
Force addition
A car coasting (i.e., no friction) uphill.
175
iii.
A football at the highest point of its trajectory. (Ignore air
resistance.)
Free-body diagram
Acceleration direction
^>-<•>•--^^
Force addition
iv. A bouncing ball for the time interval during which it is in contact with the
floor.
lSi,fcsaB-^i:.^iM- ^iitafc-jiaiMMiBB^BMBi
v. A roller coaster slowing down as it travels upwards on a loop.
•
176
Changes in Energy Lab
Introduction
With this lab, we introduce a different way of analyzing motion, by considering the changes
in the energy for moving objects. In this lab we will introduce two kinds of energy - kinetic
energy and potential energy - and observe how these quantities change along an object's
path.
Kinetic energy is the energy associated with the motion of an object. When the center of
mass of an object moves, we say that the object has translational kinetic energy, and when an
object rotates about its center of mass, we say that it has rotational kinetic energy. Potential
energy is energy associated with the position of interacting objects. For example, when an
object moves further from the center of the earth, we say that the gravitational potential
energy (PEG) has increased. (Strictly speaking, it is the gravitational potential energy of the
system of earth + object that increases.) When a mass attached to a spring stretches that
spring, we say that the elastic or spring potential energy of the spring-mass system has
increased. In this lab we will only consider translational kinetic energy and gravitational
potential energy.)
In some situations we can use either Newton's laws or energy ideas to solve a problem, and
deciding which approach to use requires practice. A few general rules of thumb: First, in
cases where forces change as a function of position (for example, the normal force for an
object moving on a curved surface), Newton's laws become difficult to use. Energy is not a
vector quantity, and it may be more easily used in these cases. Second, when we apply rules
about energy, we do not learn anything about how much time a motion takes. So if a problem
does not require or contain information about time, considering the energy of the system
under study may be useful.
Lab Objectives
After completing this lab and the associated homework, you should be able to:
177
5.
Relate the gravitational potential energy (PEG) of an object to its
position, and determine whether the PEQ increases, decreases, or stays the
same in a given motion.
6.
Relate changes in potential energy of a simple system to changes in the
object's speed.
7.
Calculate gravitational potential and translational kinetic energies for
simple situations.
Outline ofLaboratory
Approximate sequence of the lab and homework:
8.
Analyze a blinkie photograph of a moving object in order to relate the
position of the object to its speed.
9.
Make qualitative predictions about the speed of a roller coaster as its
position changes.
10.
Evaluate whether the shape of a track makes a difference in changes in
potential energy.
11.
Evaluate the extent to which differences in mass affect the resulting
motion of a body.
Roller Coaster and Moving Coaster
In this lab you will be using the roller coaster and track. The roller coaster has a
blinkie on it that flashes at a constant rate. You will take a digital photograph of the
coaster as it moves along the track, and the streaks in the photograph created by the
blinkie will allow you to make inferences about the speed of the coaster at each
position.
A long-exposure photograph of the type you will be taking is shown below. Some
points along the coaster's path are labeled A through E. (Your group will also be
given a larger (inverted) copy of a portion of this photograph for analysis.)
178
\
nz^w^%r-yt .^<m&ifi*r'T-'&.'
s? (At this poJEt, you do mc
of the §peeds so omly qualitative
W-M-^
: to dletennffiui© the auisenca!!
what point on the track would you expect the speed to be greatest? Explain.
Expl
Examine the enlarged photograph given to your group, in which the coaster moves
down a ramp, starting from rest. On the diagram, the release point is labeled with the
letter A, and the release height is marked by a line across the photograph.
Choose a point that is near the top of the ramp where the light stripe is several
millimeters long in the photograph. On the picture, label this location as Point B.
Prediction:
Let's call the speed of the coaster at point B: vB. and the vertical distance between A
and B: AyAB. Imagine finding the point where the coaster achieves a speed that is
double its speed at point B, or a speed of 2vB. Call that point C. Will the vertical
distance between point C and point B be greater than, less than, or equal to the
distance between point B and the release point? Explain why you think so.
Now examine your diagram. Look for a streak on the photograph that is about twice
as long as the streak at point B. Call this location Point C.
Is the vertical distance between point C and point A greater than, less than, or equal
to double the vertical distance between point B and the point A? (It doesn't matter to
which point on the streak you measure the height, as long as you are consistent. The
start or end of the streaks are probably easiest.)
You should have found that the vertical distance between point C and point A is
greater than two times the vertical distance between point B and the release point.
Approximately how many times greater is this distance?
Now examine your diagram again. Look for a streak on the photograph that is about
3 times as long as the streak at point B. Call this location Point D.
Compared to point B, approximately how many times below the release height is
point D?
In the absence of friction and air resistance (as well as any distortions due to the
photograph), the spacing of points B, C, and D should be such that C is four times as
far below the release point as B, and D is nine times as far. In practice, your ratios
will probably be a bit smaller than that. We find in general that the increase in speed
is proportional to the square root of the change in vertical height. Alternatively, the
change in height is proportional to the square of the speed. This observation leads to
180
the following definitions of gravitational potential energy (PEG) and kinetic energy
(KE):.
A PEQ = mg A h
KE = lA mv2.
In SI units, with mass in kg, height in m, and velocity in m/s, both PEQ and ICE will
be measured in Joules, the SI unit for energy.
For most of this lab, we talk about the change in PEG between two positions. There
are two important points to make about this definition. First, the PEG is, strictly
speaking, defined for the system of earth and object together, and depends on the
distance between them. Secondly, for most problems, the change in PEG is more
relevant than the value of PEQ itself, so PEG is typically defined to be zero at a fixed
reference height that is convenient. This height can be chosen at will, but it is
important to stick with a single reference height. For the measurements you have
just taken, we measured all of the heights from the height of the release point, and we
might have chosen it as zero height. For that choice of reference, the potential
energies at points is zero, and the potential energy of the coaster at points B, C, and
D are negative.
Reexamine your results from above. Compared to point B, the coaster at point C has
twice the speed. How much greater is the kinetic energy of the coaster at point C
compared to point B? Explain.
Is your result consistent with the difference in height that you observed in the two
cases?
Compared to point B, the coaster at point D has three times the speed. How much
greater is the kinetic energy of the coaster at point D compared to point B?
Is your result consistent with the difference in height that you observed in the two
cases?
Based on your results, at approximately what height would you expect a speed 2.45
times the speed of point B?
You should check your prediction using your diagram, though you will probably find
that the distance doesn't quite match your prediction. (Why not?)
181
Factors influencing the speed of the coaster
In this section we will examine how the shape of the track and the mass of the cart
affect the speed of the coaster. We start by making some predictions.
Prediction:
A. Imagine that two identical coasters are released from rest from the same
height on tracks attached to the same board. One track is steeper, but both reach
the same lower level. Will the speed of the coaster at the bottom be greater on the
jCoaster
Shallower track
Steeper track
steeper track, greater on the shallower track, or the same on both tracks?
B. Now imagine that we add some mass to the coaster, and again release it from
rest on the steep track. Will the speed at the bottom of the track be greater with
the added mass, greater without the added mass, or the same in both cases?
Explain the basis for your prediction.
Your group will now take three photographs to test your predictions. Since you will
be comparing these photographs, you should take the photographs one after another
without moving the camera. In addition, it is important to remember the sequence in
which you took the photographs. First, release the coaster from rest on the steep track
without added mass. Second, release the coaster from rest on the shallow track
without added mass. Finally, release the coaster from rest on the steep track with 60
grams of mass added to the coaster.
For the first two photographs, compare points at the same height about halfway
down the two tracks. What do you notice (approximately) about the speeds of the
two coasters?
182
Compare the speeds of the coasters at the bottom of their respective tracks? What
do you notice (approximately) about the speeds of the two coasters?
Based on your diagrams, what can you conclude about the change in potential
energy for the two coasters? Did the change in potential energy depend on the
shape of the track?
Note that the change in gravitational potential energy between two points depends
only on the difference in vertical height between the points. The shape of the track
does not matter, only the change in height.
4.3
Now compare the photographs showing the coaster on the steep track both
with and without the added mass: At the bottom of the track, is the speed of the
coaster with greater mass greater than, less than, or equal to that of the coaster with
the smaller mass?
It may seem counterintuitive that the speed of the coaster does not depend on the
mass. Note that the formulas for both potential energy and kinetic energy are
proportional to the mass of the object. Imagine that the heavier coaster had twice the
mass. Its change in potential energy would be twice as much, and it would therefore
have twice the kinetic energy at the bottom of the track, but the change in speed
would still be the same! Indeed, the change in speed over this interval depends only
on the height difference between top and bottom.
Conservation of energy
In the examples you have examined in this lab, as the coaster travels down the track
its gravitational potential energy decreases, and its kinetic energy increases. At any
point along the path of the coaster, the potential energy plus the kinetic energy add up
to the same value. In the absence of friction and air resistance, we find that the total
energy (kinetic plus potential) stays the same. For this reason, physicists call the total
energy a conserved quantity,
We can choose a reference height and define the potential energy of the earth-coaster
system at that point to be zero. We can then define the potential energy at every point
183
at height h as PEG = mgh, where h is the height of the coaster above our refere
.1
The coaster picture from section 2 is reproduced below.
nergy as zero at the release height.
— • • "
#,,
~~ }ms«o <* tS. nfcU tin o ^ J*'
For the coaster at the instant of release at points, the potential energy is zero (since
stgy:
sr w i l l K
It is helpful to represent the two kinds of energies as bars on a bar graph. For
units, and we can re
Points
K^ram
PE
Grav
Point £
^rans
PE
Qav
Point C
K^T™*
PE
&av
Point D
^ans
^Oav
Point £
^
^
PE
(
1
A. Draw the bar to represent the kinetic energy of the coaster at point C. Explain
how you knew to draw it as you did.
B. Fill in the bar chart diagram for points B, D, and E.
C. Is it possible to have a negative potential energy? Is it possible to have a
negative kinetic energy? Explain.
D. Compare the dash in the photograph that is just above point C to the one just
above point F. Based on your comparison, has mechanical energy (i.e., potential
energy plus kinetic energy) been conserved from C to F? Explain how you can
tell.
5.3
Suppose instead that we had chosen point E as our reference potential. Redo
the bar chart for this case.
185
Points
KETram P E ^
Point B
KE^ PE^
Point C
KE^ PE^
Please check your results above with your instructor.
186
Point D
KE^ PE^
Point £
KETrans P E ^
Changes in Energy Homework
1. Three toy roller coasters are set up on different tracks as shown. Roller
coaster A has a mass of 30 grams, roller coaster B has a mass of 50 grams, and
roller coaster C has a mass of 70 grams. The coasters are released from rest at the
top dashed line, and there is no friction or air resistance.
Coaster^
Coaster B
Coaster C
Rank, from greatest to least, the speeds of the carts when they reach the dashed line.
Explain how you determined your ranking.
2. A toy hovercraft is given a quick
shove at the bottom of a ramp. The
ramp makes an angle of 30° with the
horizontal. There is no friction
between the hovercraft and the ramp. The hovercraft has a mass of 0.4 kg. At the
187
instant shown, the hovercraft is at point A and has a speed of 2 m/s. The
hovercraft travels up the ramp, comes to rest at point B, and then travels down the
ramp.
i. What is the kinetic energy of the hovercraft at point A? Show your work, and
express your answer in Joules.
ii. What is the potential energy of the hovercraft at point B? Explain how you
can tell.
iii. How high above point A is point B? (Give the vertical distance, not the
distance along the ramp.) Show how you obtained your answer.
iv. What is the speed of the hovercraft when it reaches point A again? Explain
your reasoning.
188
3. A roller coaster is set up as shown above. The coaster is released from rest at
the location of the coaster on the track in the photograph. There is no friction or
air resistance. The white horizontal lines that are superimposed on the
photograph are evenly spaced.
i. If we define the gravitational potential energy to be zero at the release point A,
what is the total energy (potential plus kinetic) of the cart when it reaches point
C? Explain.
ii. If the speed of the cart is 12 cm/s at point B, what is the speed at point D?
Explain how you determined your answer.
iii. Fill in the bar chart graph below, showing the potential and kinetic energies at
the points labeled A - E. The kinetic energy of the coaster at point C is shown.
Points
Points
Point C
l
189
Point D
Point £
^Trans
^Grav
iv. The bar chart graph below is for the same five points, but with a different
height chosen as zero potential energy. The scale of the chart is the same as for
part ii. Fill in the rest of this chart showing the kinetic and potential energies at
points A - E.
Point/i
Kenans
PEQav
PointB
^^Trans
™Om
Point C
^^Trans
190
^Gnv
Point D
^^Tans
^Grav
Point £
^Trans
^ G H V
Rotational Motion Lab
Introduction
All of the previous labs have dealt with systems whose centers of mass were moving from
one place to another - systems undergoing translational motion. In this lab, we take a first
look at systems in rotation - systems where the object is rotating about an axis. When we
studied systems in translation we first focused on the kinematics of translation, i.e., the study
of the motion of the systems without worrying about the causes and influences of that
rotation. Later we looked at dynamics, or the study of how the forces acting on a system and
the mass of the system affected the motion. Similarly, for the study of rotation we will first
investigate rotational kinematics, paying attention to the motion, and then we will look at
some of the factors that influence rotation.
For rotating objects, we can define an angular displacement, the change in angle that a line
connecting a point on the object to the center of the object makes in some time interval; an
angular velocity, the rate of rotation; and an angular acceleration, the time rate of change of
the angular velocity. Mathematically, the relationship between these quantities is similar to
the relationships between their translational analogues of displacement, velocity, and
acceleration. In addition, we can connect the translational and rotational variables for rigid
bodies. In this lab, we explore the relationships between these kinematic quantities. In
addition, we introduce the idea of moment of inertia, which is a rotational analogue to mass.
Lab Objectives.
After completing this lab and the associated homework, you should be able to:
8.
Determine angular displacement, angular velocity and angular
acceleration based on measurements of a rotating object.
9.
Relate rotational kinematic quantities to translational kinematic
quantities for points on a rotating object.
10.
Reason qualitatively about the effect of moment of inertia on angular
acceleration.
Outline of Laboratory
Approximate sequence of the lab and homework:
191
12.
Analyze a blinkie photograph of a disc moving with roughly constant
angular velocity.
13.
Calculate angular velocity and angular displacement.
14.
Analyze a blinkie photograph of a disc moving with roughly constant
angular acceleration.
15.
Calculate angular acceleration.
16.
Change the moment of inertia of a rotating disc, and again calculate
angular acceleration.
Angular displacement
A plywood disc is set up with a set of 4 blinkies attached to it.
The blinkies are all attached to the same circuit, and they all turn
on and off at the same time.
2.1:
The distance from the center of the disc to each blinkie
is shown here.
Measure the flash rate of the blinkies Count the number of
times one of the leds flashes in thirty seconds. (It is probably
best to do this as a class and write the count on the blackboard.)
Blinkie
A
B
C
D
Radius
0
4 cm
8 cm
11 cm
How many times does it flash per minute? How many times
does it flash per second?
2.2:
Set the camera set for a one-half second exposure. Turn on both switches on
the blinkie circuit, start the disc spinning, and take a picture of the disc as it rotates.
Make sure that each blinkie makes several light streaks, but that they don't go around
the disc completely. Print out an inversion of the picture for study.
2.3:
Does blinkie B travel the same distance as blinkie D during each time
interval? Explain how you can tell.
2.4: Do the four blinkies rotate through the same angle
during each flash? On your printed picture, measure the angle
that each blinkie travels through during a single on/off cycle,
and fill in the table. (You can measure the angle from when a
blinkie turns on until it again turns on.)
192
Blinkie
A
B
C
D
Angle
When an object is translating (when its center of mass is moving), the displacement
of that object in some time interval is the distance Ax from the initial position to the
final position. For rotation, we can define the angular displacement as the change in
angle from the final position to the initial position A9 during some time interval. It
is useful to measure this angle in radians.
A6
Blinkie
In one complete revolution, the angular displacement of an
object is 360° or 2K radians. Convert the angular
displacements that you measured above to radians, and record
your result here.
A
B
C
D
Arc length
Radius r
Blinkie
A6
2.5:
Using your data above, find the
A
0
distance (arc length) traveled by each
B
4 cm
blinkie during one on/off cycle. You can
C
8 cm
do this by determining what fraction or
D
11 cm
percent of a circle the blinkie travels in one
on/off cycle, and then calculating what fraction this is of the distance traveled in a
complete revolution (i.e., one circumference).
2.6:
For a point on a rotating object, write down an algebraic relationship between
the radius r, the angular displacement A0, and the arc length As.
Angular velocity
3.1:
From the number of flashes per second found in section 2.1, find the time At
for one complete on/off cycle.
3.2
Use the information from parts 2.5 and 3.1 to find the
speed v of each blinkie, measured in cm/sec.
Blinkie
A
B
C
D
Speed
When an object is translating in one dimension, the velocity of that object in some
time interval is the distance Ax from the initial position to the final position divided
Ax
by the time interval At:v- — .
At
For rotation in a plane, we can define the angular velocity oo as the angular
displacement A6 (again measured in radians) from the initial position to the final
position 9f - 9i divided by the time interval At:
Radius r
CD
Speed v
Blinkie
co =A0_
A
0
At '
193
B
C
D
4 cm
8 cm
11 cm
3.3:
Find the angular velocity for each blinkie on the disc, and fill in the table at
right.
3.4: For a point on a rotating object, write down an algebraic relationship between
the radius r, the angular velocity co, and the speed v.
For rotating objects, the use of radian measure for angles simplifies the relationships
between translation and rotation. A radian is defined as the ratio of the arc length to
the radius, and since this is a length divided by a length, a radian is dimensionless.
The angular displacement A0 is therefore also dimensionless, and the angular
velocity co has units of 1/seconds.
4.
Angular acceleration
Since the bearing on the disc has very little friction, the motion you studied in
sections 2 and 3 was roughly constant. That is, the angular velocity of the disc did
not change very much from one blinkie flash to another. As with the translational
kinematics that you studied at the beginning of the semester, a more general treatment
of rotation includes cases where the angular velocity of an object is changing with
time. Just as acceleration in translation is defined as the change
in velocity per unit time, we can define angular acceleration (a)
as the change in angular velocity per unit time.
The disc you will be using for this experiment has a blinkie to
mark the center of the disc, and another blinkie mounted near the
edge of the disc. The flash rate for this outer blinkie is
approximately 3 flashes per second, so we will use this value in
the calculations we make.
A third blinkie circuit with (but with no led!) is mounted across the disc and is only
used to balance the disc. A small weight attached to a string wrapped around the
edge of the disc will be used to change the angular velocity of the disc.
4.1:
Place the four 100-gram brass cylinders into the holes that are closest to the
center of the disc as shown. Turn on both blinkies. Wind the string around the edge
of the disc until the small hanging weight is close to the disc. Release the disc, and
take a 1-second exposure photograph of the disc as it rotates. Print out an inverted
photograph of your picture for analysis.
4.2: Is the angular velocity w of the disc constant? Explain how you can tell from
your photograph.
194
4.3:
Measure the angular displacement of each on/off cycle in your photograph.
(You can do this by measuring the angle from when the blinkie turns on to the next
time it turns on.) You should find that the change in A0 between adjacent cycles is
approximately constant - the angle made by each flash increases from the previous
one by about the same amount. What is this change in A0 measured in degrees? In
radians?
4.4: Pick two adjacent on/off cycles, and use them to calculate the change in
angular velocity Aco. Show how you calculated this value below.
4.4:
Pick two different on/off cycles that are also adjacent and again use them to
calculate the change in angular velocity Aco. You should find that that you get about
the same value.
4.5:
The angular acceleration a is defined as the change in angular velocity Ac;
divided by the change in time (i.e., the time interval over which this change takes
place):
Aco
a =
At
Since each on/off cycle for the blinkie at the edge of the disc takes about one-third of
a second, we can use this time interval to calculate angular acceleration. What is the
angular acceleration of the disc? What are the units of angular acceleration?
195
The acceleration of a point on the disc in the direction of its motion is called
the tangential acceleration. For a point that is rotating and changing speed such as the
blinkie that we have measured here, the tangential acceleration is related to the
angular acceleration by atan = ocr.
(Note that this point also has an acceleration that points toward the center of the circle
- the radial acceleration - that is equal to the velocity squared divided by the radius.)
4.7
If the distance from the center of the disc to the blinkie is 11 cm, what is the
tangential acceleration of the blinkie?
5.
Moment of inertia
So far in this lab we have focused on the kinematics of rotation, paying attention to
the relationships between angular displacement, angular velocity, and angular
acceleration, and connecting these quantities to their translational counterparts. We
now begin to pay attention to the dynamics of rotation. - a study of the factors that
affect rotation. In translational motion, the acceleration of an object is related to the
net force acting on that object through Newton's second law, HF = ma. In rotational
motion, the moment of inertia I is analogous to mass m in translation. If we think of
the mass of an object as a measure of how hard it is to change the velocity of that
object, then the moment of inertia is a measure of how hard it is to change the angular
velocity of a rotating object.
5.1:
Take the four brass cylinders from the inner holes and place them in the holes
closest to the edge of the disc as shown. Note that you have not
changed the mass of the rotating system, and you will use the
same hanging weight. Predict what will happen to the angular
acceleration of the disc when you repeat the exercise from section
4.1.
6
5.2: Repeat the exercise from section 4.1, again taking a 1 -second photograph.
Print out an inverted photograph for analysis.
5.3: Repeat the procedure from sections 4.3 - 4.5 to find the angular acceleration
of the disc over two different intervals. Is the angular acceleration greater than, less
196
than, or equal to the angular acceleration of the rotating system with the brass
cylinders closer to the axis of rotation?
5.4: If we think of the moment of inertia as a measure of how hard it is to change
the angular velocity of a rotating object, does the rotating system have a greater
moment of inertia with the cylinders closer to the axis of rotation or further from the
axis of rotation?
As we have seen, the moment of inertia of a rotating system depends on the position
of the mass of that system - the mass distribution. When mass is further from the
axis of rotation, the moment of inertia increases, as it becomes more difficult to
change the angular velocity of the system. In addition, the moment of inertia depends
on the amount of mass in the system - if we had put a brass cylinder in all eight holes,
we would have found that the angular acceleration was smaller than with only four
cylinders.
Rotational Motion Homework
1.
Three children, Aricelia, Bao, and Chuck, are
playing on a merry-go-round. Their positions on
the merry-go-round are shown in the top-view
picture at right. The merry-go-round is rotating
clockwise and is neither speeding up nor slowing
down.
A. Rank the speeds of the children. If allI
of the children are moving at the same
speed, state that explicitly. Explain your
reasoning.
197
B. Rank the angular velocities of the children. If all of the children have the
same angular velocity, state that explicitly. Explain your reasoning.
C. Rank the times that it takes each child to complete one revolution. If all of the
times are the same, state that explicitly. Explain your reasoning.
D. With Aricelia, Bao, and Chuck on the merry-go-round, a fourth child, David,
wants to bring the merry-go-round from rest to a rotation rate of 1 revolution
every 4 seconds in as little time as possible. That is, he wants to be able to speed
the merry-go-round up from rest as quickly as possible.
Does it matter where the other three children sit on the merry-go-round? If you
were David, where would you ask the other three children to sit? Explain based
on the idea of moment of inertia.
2.
In lab you released a mass tied to a string that was wrapped
around a disc, and took a picture of the blinkie as the mass fell and
the disc rotated. Suppose that instead you had unwound the string
until the mass was just above the floor, and then someone in your
group had pulled quickly down on the edge of the disc as shown.
If you took a picture of the blinkie after the quick pull but while
the mass rose was still rising, what would the picture look like?
Explain.
198
3.
What is the angular velocity of the earth as it spins on its axis? Express your
answer in radians per second. Show how you determined your answer.
4.
A blinkie photograph made using the same procedure
as in section 4 of the lab is shown at right. The blinkie is
flashing five times per second.
i. What is the duration of a single on/off cycle for the
blinkie?
ii. What is the approximate angular velocity (in radians per second) of the blinkie
at point P? Show how you determined your answer.
iii. If the speed of the blinkie at point P was 32 cm/sec, what is the distance from
the center of the disc to the blinkie? Show how you determined your answer.
iv. What is the approximate angular acceleration of the blinkie (in radians per
second squared) of the blinkie at point P? Show how you determined your
answer?
199
APPENDIX C
SELECTED FINAL EXAM QUESTIONS
A football is thrown so that it
follows the path indicated by the
dashed line (from left to right).
Air resistance can be neglected.
1. When the ball is at point A on
this path, what is the direction of
the net force on the ball?
A
A heavy steel ball is attached to a light rod
to make a pendulum. The pendulum begins
at rest at instant^, as shown. The ball is
then pulled to one side and held at rest. At
instant B, the pendulum is released from
rest. It swings down and passes its original
height at instant C, as shown. Ignore air
resistance.
2. The sum of gravitational potential
Ball
energy (PEG) + kinetic energy (KE) of the
A
pendulum is:
a) Greater atB.
b) Greater at C.
c) Equal at both instants.
d) It is impossible to determine without more information.
A roller coaster starts at rest at the top of a track. The coaster is released and rolls down the
shallower track as
shown in the
diagram. At the
~"~£-£™1'5tpr
bottom of the track,
the speed of the
.Shallower track
coaster is recorded.
• » • •
Steeper track ^ ^
The experiment is
repeated with the
1.... ,.
coaster rolling
down the steeper
track shown in the diagram.
6
6
6'
3. Compared to the original experiment down the shallower track, the speed of the coaster at
the bottom of the steeper track is:
a) The same as at the bottom of the shallower track.
b) Greater than at the bottom of the shallower track.
200
c) Less than at the bottom of the shallower track.
d) Impossible to tell without more information.
A new roller coaster is designed that has twice the mass as the original coaster. The
experiment was again repeated with the heavier coaster placed on the top of the shallower
track and released from rest. At the bottom of the track, the speed of this coaster was
recorded.
4. Compared to the original coaster rolling down the shallower track, the speed of the heavier
coaster at the bottom of the track is:
a) The speed of the heavier coaster will be 2x the original speed.
b) The speed of the heavier coaster will be greater than the original speed;, but without more
information, it is impossible to know by how much.
c) The speed of the heavier coaster will be 0.5x the original speed.
d) The speed of the heavier coaster will be less than the original speed, but without more
information, it is impossible to know by how much.
e) The speed of the heavier coaster will be the same
as the original speed.
Three children, Aricelia, Bao, and Chuck, are
playing on a merry-go-round. Their positions on the
merry-go-round are shown in the top-view picture at
right. The merry-go-round is rotating clockwise and
is neither speeding up nor slowing down. Each of
the children will move in a circle as the merry-goround rotates: The radius of the circle for Aricelia is
one meter; for Bao it is two meters; and for Chuck it
is 3 meters.
5. A correct ranking of the speeds of the children
would be:
a) Aricelia > Bao > Chuck.
b) Chuck > Bao > Aricelia.
c) They are all the same.
d) It is impossible to tell.
6.
a)
b)
c)
d)
A correct ranking of the angular velocities of the children would be:
Aricelia > Bao > Chuck.
Chuck > Bao > Aricelia.
They are all the same.
It is impossible to tell.
7. A skateboarder goes over a circular bump. Velocity vectors
are shown for the skateboarder just before and just after the time
he is at the top of the bump.
201
^
Which of the following statements is true about the skateboarder at the instant he is at the top
of the bump as shown in the diagram?
a) His acceleration is zero, and the normal force on him is equal to his weight.
b) His acceleration is straight down, and the normal force on him is equal to his weight.
c) His acceleration is straight down, and the normal force on him is less than his weight.
d) His acceleration is straight down, and the normal force on him is greater than his weight.
e) His acceleration is straight up, and the normal force is equal to his weight.
f) None of the above statements is correct.
8. Velocity vectors are shown for two positions, A and B,
for a child on a swing. The average acceleration
between points A and B is
a) zero.
b) upward.
c) downward, in the direction of the gravitational force.
d) to the left, in the direction of motion.
e) to the right, to balance the forces acting on the child.
A skateboarder slows down as she coasts up a ramp.
Velocity vectors are shown for the skateboarder just
before and just after the time she is at the position shown
in the diagram.
9. Which of the following statements about the skateboarder at the instant shown in the
diagram is true?
a) Her acceleration is down the ramp, and the net force acting on her is toward the center of
the earth in the direction of gravity.
b) Her acceleration is up the ramp in the direction she is moving, and the net force acting on
her is toward the center of the earth in the direction of gravity.
c) Her acceleration is down the ramp, and the net force acting on her is up the ramp in the
direction she is moving
d) Her acceleration is down the ramp, and the net force acting on her is also down the ramp.
e) None of the above statements is correct.
202
APPENDIX D
LAWSON TEST OF SCIENTIFIC REASONING
Version
Directions: This is a test of your ability to apply aspects of scientific and mathematical
reasoning to analyze a situation to make a prediction or solve a problem. Make a dark
mark on the answer sheet for the best answer for each item. If you do not fully understan
what is being asked in an item, please ask the test administrator for clarification.
1. Suppose you are given two clay balls of equal size and shape. The two clay balls alsc
weigh the same. One ball is flattened into a pancake-shaped piece. Which of these
statements is correct?
a. The pancake-shaped piece weighs more than the ball
b. The two pieces still weigh the same
c. The ball weighs more than the pancake-shaped piece
2. because
a. the flattened piece covers a larger area.
b. the ball pushes down more on one spot.
c. when something is flattened it loses weight.
d. clay has not been added or taken away.
e. when something is flattened it gains weight.
To the right are drawings of two cylinders filled to
the same level with water. The cylinders are
identical in size and shape.
Also shown at the right are two marbles, one glass
and one steel. The marbles are the same size but
the steel one is much heavier than the glass one.
When the glass marble is put into Cylinder 1 it
sinks to the bottom and the water level rises to the
6th mark. If we put the steel marble into Cylinder
2, the water will rise
a. to the same level as it did in Cylinder 1
b. to a higher level than it did in Cylinder 1
c. to a lower level than it did in Cylinder 1
because
a. the steel marble will sink faster.
b. the marbles are made of different materials.
c. the steel marble is heavier than the glass marble.
d. the glass marble creates less pressure.
e. the marbles are the same size.
203
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o
frl
STEB.MAR8UE
CYUNOBU
5. To the right are drawings of a wide and a
narrow cylinder. The cylinders have equally
spaced marks on them. Water is poured into
the wide cylinder up to the 4th mark (see A).
This water rises to the 6th mark when poured
into the narrow cylinder (see B).
Both cylinders are emptied (not shown) and
water is poured into the wide cylinder up to the
6th mark. How high would this water rise if it
were poured into the empty narrow cylinder?
a. to about 8
b. to about 9
c. to about 10
d. to about 12
e. none of these answers is correct
6. because
a. the answer can not be determined with the information given,
b. it went up 2 more before, so it will go up 2 more again.
c. it goes up 3 in the narrow for every 2 in the wide.
d. the second cylinder is narrower.
e. one must actually pour the water and observe to find out,
7. Water is now poured into the narrow cylinder (described in Item 5 above) up to the
11 th mark. How high would this water rise if it were poured into the empty wide
cylinder?
a. to about 7 1/2
b. to about 9
c. to about 8
d. to about 7 1/3
e. none of these answers is correct
8. because
a. the ratios must stay the same.
b. one must actually pour the water and observe to find out.
c. the answer can not be determined with the information given.
d. it was 2 less before so it will be 2 less again.
e. you subtract 2 from the wide for every 3 from the narrow.
9. At the right are drawings of three strings hanging from a bar. The three strings have
metal weights attached to their ends. String 1 and String 3 are the same length. String
2 is shorter. A 10 unit weight is attached to
the end of String 1. A 10 unit weight is also
1
?L
* „A
attached to the end of String 2. A 5 unit
M
!
weight is attached to the end of String 3.
I
1
I
The strings (and attached weights) can be
swung back and forth and the time it takes
to make a swing can be timed.
Suppose you want to find out whether the
length of the string has an effect on the
time it takes to swing back and forth.
Which strings would you use to find out?
©
a. only one string
b. all three strings
c. 2 and 3
d. 1 and 3
(*)
e. 1 and 2
204
10. because
a. you must use the longest strings.
b. you must compare strings with both light and heavy weights.
c. only the lengths differ.
d. to make all possible comparisons.
e. the weights differ.
11. Twenty fruit flies are placed in each of four glass tubes. The tubes are sealed. Tubes 1
and 11 are partially covered with black paper; Cubes 111 and IV are not covered. The
tubes arc placed as shown. Then they arc exposed to red light for five minutes. The
number of flics in the uncovered part of each tube is shown in the drawing.
REDUOHT
1
T
J
i
i
in
9
f
IV
•
t
t
0 10 .: 10
u
*
t
1
t
)
t
REDUGHT
IJiis experiment shows that flies respond to (respond means move to or away from)
a. red light but not gravity
b. gravity but not red light
e. both red tight and gravity
d. neither red light nor gravity
12. because
a. most flies are in the upper end of Tube 111 but spread about evenly in Tube n .
b. most flies did not gotothe bottom of Tubes I and III.
c. the flies need light to sec and must fly against gravity.
d. the majority of flies are in the upper ends and in the lighted ends of the tubes.
c. some flics arc in both ends of each tube.
13. In a second experiment, a different kind of fly and blue light was used. The results are
shown in the drawing.
BLUE UGKT
I
\
I
r
1
1
i
!
*
1
t
e
§
t
I
i
(J w
t
t
t
t
• io
t
)
t
These data show that these flies respond to (respond means move to or away from):
a. blue light but not gravity
b. gravity but not blue light
c. both blue light and gravity
d. neither blue light nor gravity
14. because
a. some flies are in both ends of each tube.
b. the flies need light to see and must fly against gravity.
c. the flies are spread about evenly in Tube IV and in the upper end of
Tube III.
d. most flies are in the lighted end of Tube II but do not go down in Tubes I ar
III.
e. most flies are in the upper end of Tube 1 and the lighted end of Tube II.
205
000
15. Six square pieces of wood are put into a cloth bag and
mixed about. The six pieces arc identical in size and
shape, however, three pieces are red and three are
yellow. Suppose someone reaches into the bag
Iv I I
I I
1
(without looking) and pulls out one piece. What are
\ Y | | Y | I YI
the chances that the piece is red?
a. 1 chance out of 6
b. I chance out of 3
c. 1 chance out of 2
d. 1 chance out of I
c. can not be determined
16. because
a. 3 out of 6 pieces are red.
h. there is no way to tell which piece will be picked.
c. only 1 piece of the 6 in the bag is picked.
d. all 6 pieces arc identical in size and shape.
e. only 1 red piece can be picked out of the 3 red pieces.
17. Three red square pieces of wood, four yellow square pieces, and five blue square
pieces are put into a cloth hag. Four red round pieces, two yellow round pieces, and
three blue round pieces are also put into the bag. All the pieces are then mixed about.
Suppose someone reaches into the bag (without looking and without feeling for a
particular shape piece) and pulls out one piece.
000
0000
00 0 0 0
000®
0©
000
What are the chances thai the piece is a red round or blue roundpiece?
a. can not be determined
b. 1 chance out of 3
c. 1 chance out of 21
d. IS chances out of 21
e. 1 chance out of 2
18. because
a. 1 of the 2 shapes is round.
b. 15ufthe21 pieces are red or blue.
c. there is no way to tell which piece will be picked,
d- only 1 of the 21 pieces is picked out of the bag.
e. I of every 3 pieces is a red or blue round piece.
19. Parmer Brown was observing the mice that live in his field. He discovered that all of
them were either fat or thin. Also, all of Ihem had either black tails or white tails. This
made him wonder if there might be a link between the size of the mice and the color
of their tails. So he captured all of the mice in one part of bis field and observed them.
Below arc the mice that he captured.
Do you tltirtk there is a link between the size of the mice and the color of their tails?
a. appears to be a link
b. appears not to be a link
c. can not make a reasonable guess
20. because
a. there are some of each kind of mouse.
b. there may be a genetic link between mouse size and tail color.
c. there were not enough mice captured.
d. most of the tat mice have black tails while most of the thin mice have white
tails.
e. as the mice grew falter, their tails became darker.
206
21. The figure below at the left shows a drinking glass and a burning birthday candle
stuck in a small piece of clay standing in a pan of water. When the glass is turned
upside down, put over trie candle, and placed in the water, the candle quickly goes out
and water rushes up into the glass (as shown at the right).
This observation raises an interesting question: Why does the water rush up into the
glass?
Here is a possible explanation. The flame converts oxygen into carbon dioxide.
Because oxygen does not dissolverapidlyinto water hut carbon dioxide does, the
newly-formed carbon dioxide dissolvesrapidlyinto the water, lowering the air
pressure inside the glass.
Suppose you have the materials mentioned above plus some matches and some dry
ice (dry ice is frozen carbon dioxide). Using some or all of the materials, how could
you test this possible explanation?
a. Saturate the water with carbon dioxide and redo the experiment noting the
amount of water rise.
b. The water rises because oxygen is consumed, so redo the experiment in exactly
the same way to show water rise due to oxygen loss.
c. Conduct a controlled experiment varying only the number of candles to see if
that makes a difference.
d. Suction is responsible for the water rise, so put a balloon over the top of aji
open-ended cylinder and place the cylinder over the burning candle.
e. Redo the experiment, but make sure it is controlled by holding all independent
variables constant; then measure the amount of water rise.
22. What result of your test (mentioned in #21 above) would show that your explanation
is probably wrong?
a. The water rises the same as it did before.
b. The water rises less than it did before.
c. The balloon expands out.
d. The balloon is sucked in.
23. A student put a drop of blond on a microscope slide and then looked at the blood
under a microscope. As you can see in the diagram below, the magnified red blood
cells look like little round balls. After adding a few drops of salt water to the drop of
blood, the student noticed that the cells appeared to become smaller.
Magnified Red Blood Cells
Afusr Adding Salt Water
This observation raises an interesting question: Why do the red blood cells appear
smaller?
Here are two possible explanations: I. Salt ions (Na+ and C1-) push on the cell
membranes and make the cells appear smaller. II. Water molecules are attracted to the
salt ions so the water molecules move out of the cells and leave the cells smaller.
To test these explanations, the student used some salt water, a very accurate weighing
device, and some water-filled plastic bags, and assumed the plastic behaves just like
red-blood-cell membranes. The experiment involved carefully weighing a water-tilled
bag in a salt solution for ten minutes and then reweighing the bag.
What result of the experiment would best show that explanation I is probably wrong?
a. the bag loses weight
h. the bag weighs the same
c. the bag appears smaller
24. What result of the experiment would best show that explanation II is probably wrong?
a. the bag loses weight
b. the bag weighs the same
c. the bug appears smaller
207
APPENDIX E
15 QUESTION SUBSET OF LAWSON PRE-TEST
1. Suppose you are given two clay balls of equal size and shape. The two clay balls also weigh
the same. One ball is flattened into a pancake-shaped piece. Which of these statements is
correct?
a. The pancake-shaped piece weighs more than the ball
b. The two pieces still weigh the same
c. The ball weighs more than the pancake-shaped piece
2. because
a. the flattened piece covers a larger area.
b. the ball pushes down more on one spot.
c. when something is flattened it loses weight.
d. clay has not been added or taken away.
e. when something is flattened it gains weight.
GUSSUARBUE
. 3. To the right are drawings of two cylinders filled to the
same level with water. The cylinders are identical in
size and shape.
o
Also shown at the right are two marbles, one glass and
one steel. The marbles are the same size but the steel
one is much heavier than the glass one.
When the glass marble is put into Cylinder 1 it sinks to
the bottom and the water level rises to the 6th mark. If
we put the steel marble into Cylinder 2, the water will
rise
a. to the same level as it did in Cylinder 1
b. to a higher level than it did in Cylinder 1
c. to a lower level than it did in Cylinder 1
4. because
a. the steel marble will sink faster.
b. the marbles are made of different materials.
c. the steel marble is heavier than the glass marble.
d. the glass marble creates less pressure.
e. the marbles are the same size.
208
STEEL U*MLE
Kx-&
^
iilfll
5.) To the right are drawings of a wide and a narrow
•^ cylinder. The cylinders have equally spaced marks
on them. Water is poured into the wide cylinder up
to the 4th mark (see A). This water rises to the 6th
mark when poured into the narrow cylinder (see B).
Both cylinders are emptied (not shown) and water
is poured into the wide cylinder up to the 6th mark.
How high would this water rise if it were poured
into the empty narrow cylinder?
a. to about 8
b. to about 9
c. to about 10
d. to about 12
e. none of these answers is correct
6. because
a. the answer can not be determined with the information given.
b. it went up 2 more before, so it will go up 2 more again.
c. it goes up 3 in the narrow for every 2 in the wide.
d. the second cylinder is narrower.
e. one must actually pour the water and observe to find out.
7. Water is now poured into the narrow cylinder (described in Item 5 above) up to the 11th
mark. How high would this water rise if it were poured into the empty wide cylinder?
a. to about 7 1/2
b. to about 9
c. to about 8
d. to about 7 1/3
e. none of these answers is correct
8. because
a. the ratios must stay the same.
b. one must actually pour the water and observe to find out.
c. the answer can not be determined with the information given.
d. it was 2 less before so it will be 2 less again.
e. you subtract 2 from the wide for every 3 from the narrow.
2
209
At the right are drawings of three strings hanging from a
bar. The three strings have metal weights attached to
their ends. String 1 and String 3 are the same length.
String 2 is shorter. A 10-unit weight is attached to the
end of String 1. A 10-unit weight is also attached to the
end of String 2. A 5-unit weight is attached to the end of
String 3. The strings (and attached weights) can be
swung back and forth and the time it takes to make a
swing can be timed.
d
®
Suppose you want to find out whether the lengdi of the
string has an effect on the time it takes to swing back
and forth. Which strings would you use tofindout?
a. only one string
b. all three strings
c. 2 and 3
d. 1 and 3
e. 1 and 2
®
<J1Q! because
a. you must use the longest strings.
b. you must compare strings with both light and heavy weights.
c. only the lengths differ.
d. to make all possible comparisons.
e. the weights differ.
(ly
Six square pieces of wood are put into a clodi bag and
mixed about. The six pieces are identical in size and shape,
however, three pieces are red and three are yellow. Suppose
someone reaches into the bag (without looking) and pulls
out one piece. What are the chances that the piece is red?
a. 1 chance out of 6
b. 1 chance out of 3
. c. 1 chance out of 2
d. 1 chance out of 1
e. can not be determined
//uj because
a. 3 out of 6 pieces are red.
b. there is no way to tell which piece will be picked.
c. only 1 piece of the 6 in the bag is picked.
d. all 6 pieces are identical in size and shape.
e. only 1 red piece can be picked out of the 3 red pieces.
210
0 00
0 00
13. Three red square pieces of wood, four yellow square pieces, and five blue square pieces are
put into a cloth bag. Four red round pieces, two yellow round pieces, and three blue round
pieces are also put into the bag. All the pieces are then mixed about. Suppose someone
reaches into the bag (without looking and without feeling for a particular shape piece) and
pulls out one piece.
000
0000
00000
®©0®
00
0.0 0
Wliat are the chances that the piece is a red round or blue round piece ?
a. can not be determined
b. 1 chance out of 3
c. 1 chance out of 21
d. 15 chances out of 21
e. 1 chance out of 2
14. because
a. 1 of the 2 shapes is round.
b. 15 of the 21 pieces are red or blue.
c. there is no way to tell which piece will be picked.
d. only 1 of the 21 pieces is picked out of the bag.
e. 1 of every 3 pieces is a red or blue round piece.
15yA uniform block of cheese is cut into two unequal pieces, labeled A and
B. The mass density of an object is defined as the mass of that object
divided by its volume .
A correct ranking of the mass densities (from largest to smallest) of the
original block, the largest piece (A), and the smallest piece (B) is:
a. Original block, largest piece, smallest piece.
b. Smallest piece, largest piece, original block.
c. All mass densities are the same.
d. Not possible to determine without additional information.
211
y
x
Original block • >^
/ A/VA
,*y
APPENDIX F
THREE VERSIONS OF PROPORTIONAL REASONING PRETEST
Original Pretest Version
1. A uniform block of trinitramine is cut into two unequal
pieces, labeled piece A (larger piece) and piece B (smaller
piece). The mass density of an object is defined as the mass
of that object divided by its volume.
A correct ranking of the mass densities (from largest to
smallest) of the original block, piece A, and piece B is:
a.
b.
c.
d.
Original block
sT/
/
Original block, piece A, piece B.
Piece B, piece A, original block.
All mass densities are the same.
Not possible to determine without
additional information.
To the right are drawings of a wide and a narrow
cylinder. The cylinders have equally spaced marks on
them. Water is poured into the wide cylinder up to
the 4th mark (see A). This water rises to the 6th mark
when poured into the narrow cylinder (see B).
/ /
/
A
/
B
/
4-
2. Both cylinders are emptied (not shown) and
water is poured into the WIDE cylinder up to the
6th mark. How high would this water rise if it
were poured into the empty narrow cylinder?
a. to about 8
b. to about 9
c. to about 10
d. to about 12
e. none of these answers is correct
B'
3. because
a. the answer can not be determined with the information given.
b. it went up 2 more before, so it will go up 2 more again.
c. it goes up 3 in the narrow for every 2 in the wide.
d. the second cylinder is narrower.
e. one must actually pour the water and observe to find out.
4. Water is now poured into the narrow cylinder (described above) up to the 11th mark.
How high would this water rise if it were poured into the empty wide cylinder?
a. to about 7 1/2
b. to about 9
c. to about 8
d. to about 7 1/3
e. none of these answers is correct
212
5. because
a. the ratios must stay the same.
b. one must actually pour the water and observe to find out.
c. the answer can not be determined with the information given.
d. it was 2 less before so it will be 2 less again.
e. you subtract 2 from the wide for every 3 from the narrow.
Modified Pretest Version
1. A brownie is cut into two unequal pieces, labeled Annie's
piece (larger piece) and Russell's piece (smaller piece).
The mass density of an object is defined as the mass of
that object divided by its volume.
A correct ranking of the mass densities (from largest to
smallest) of the original brownie, Annie's piece, and
Russell's piece is:
a.
b.
c.
d.
Original brownie, Annie's piece, Russell's piece.
Russell's piece, Annie's piece, original brownie.
All mass densities are the same.
Not possible to determine without more
information.
The cold medicine you bought came with two cylindrical containers: a wider one marked off
in adult doses and a narrower one marked off in children's doses. When you fill the adult
(wide) cylinder to the fourth mark (see A), and then pour it into the children's (narrow)
cylinder, it rises to the 6th mark (see B).
By mistake you pour your child's medicine up
to the 6th mark in the ADULT dosage cylinder.
How high would this dose be if you pour it
into the children's cylinder?
a. To about 8
b. To about 9
c. To about 10
d. To about 12
e. None of these answers is correct
=
:
£3@l
6—1=
B<gi
because
a. the answer can not be determined with the information given.
b. it went up 2 more before, so it will go up 2 more again,
c it goes up 3 in the narrow for every 2 in the wide.
d. the second cylinder is narrower.
e. one must actually pour the water and observe to find out.
4. On another occasion you poured an adult dose into the children's (narrow) cylinder
(described above) up to the 11th mark. How high would this medicine rise if it were
213
poured into the empty adult (wide) cylinder?
a. To about 7 1/2
b. To about 9
c. To about 8
d. To about 7 1/3
e. none of these answers is correct
5. because
a. the ratios must stay the same.
b. one must actually pour the medicine and observe to find out.
c. the answer can not be determined with the information given.
d. it was 2 less before so it will be 2 less again.
e. you subtract 2 from the adult (wide) cylinder for every 3 from the children's
(narrow) cylinder.
Discrete Modified Pretest Version
1. A wedge is cut out of a wheel of cheese. Let's
define the "cheesiness" of any piece of cheese as
the weight of that piece divided by its volume.
Suppose you were to rank the "cheesiness" of the
original wheel, the wedge, and the remaining
wheel. From largest to smallestthis ranking would
be:
a. Original wheel, remaining wheel, wedge.
b. Wedge, remaining wheel, original wheel.
c. The cheesinesses are all the same.
d. We can't tell without additional information.
Suppose you've decided to volunteer at a homeless shelter, and you are asked to serve rice.
There are two scoops for the rice, one for an adult portion and one for a child's portion.
You notice that a bowl with enough rice for 4 adult portions is also enough for 6 child
portions.
2. A second bowl holds enough rice for 6 adult portions. This would be enough for:
a. about 8 child portions.
b. about 9 child portions.
c. about 10 child portions.
d. about 12 child portions.
e. none of these answers is correct
2. because
a. the answer can not be determined with the information given.
b. there were 2 more child portions before, so there will be 2 more again.
c. there are 3 child portions for every 2 adult portions.
d. the child portions are smaller.
e. you'd need to actually serve the rice and see what happens to find out.
214
4. A third serving bowl is now filled with enough rice for 11 child portions (described
above).
How many adult portions could you serve out of this serving bowl?
a. about 7 1/2
b. about 9
c. about 8
d. about 7 1/3
e. none of these answers is correct
5. because
a. the ratios must stay the same.
b. you'd need to actually serve the rice and see what happens to find out.
c. the answer can not be determined with the information given.
d. there were 2 less adult portions before so there will be 2 less again.
e. you subtract 2 from the adult portions for every 3 from the children's portions.
215
APPENDIX G
CORRELATION COEFFICIENTS AMONG VARIOUS COMPONENTS OF
LAWSON TEST (A-G), PRETEST (H) AND FINAL EXAM (I).
Phys 211 - Table of correlation coefficients
among various reasoning ability measures
A B C D E F G H I
1.00
0.07
-0.01 0.04
0.06
-0.07 0.11
-0.10 0.20
1.00
0.17
0.27
0.06
0.18
0.44
0.18
0.37
1.00
0.26
0.32
0.18
0.63
0.65
0.47
1.00
0.30
0.40
0.73
0.27
0.41
1.00
0.30
0.68
0.35
0.41
1.00
0.63
0.23
0.38
1.00
0.55
0.65
1.00
0.43
A: Conservation of M a s s (L1-L2).
B: Conservation of Volume (L3-L4).
C: Proportional R e a s o n i n g - L a w s on (L5-L8).
D : Control of Variables (L9-L14).
iL: rroDaDil lty r\easoning {LiLo-LilzU}.
F: Hypo-deductive Reasoning (L21-L24).
G: Total Law son Score.
H: Proportional Reasoning -pretest.
I : Final Exam.
* in parentheses are the question numbers corresponding to
reasoning abilities measured by Lawson test (appendix D).
216
1.00
Phys 215 - Table of correlation coefficients
among various reasoning ability measures
A B C D E F G H
I
1.00
0.14
0.26
0.19
0.03
0.13
0.32
0.07
0.19
1.00
0.37
0.07
0.34
0.11
0.48
0.40
0.38
1.00
0.53
0.58
0.34
0.89
0.69
0.63
1.00
0.08
0.24
0.65
0.44
0.51
D
1.00
0.20
0.66
0.49
0.39
E
1.00
0.56
0.24
0.13
A: Conservation of Mass (L1-L2).
1.00 0.69
B: Conservation of Volume (L3-L4).
C: Proportional Reasoning-Lawson (L5-L8).
1.00
D: Control of Variables (L9-L14).
E: Probability Reasoning (L15-L20).
F: Hypo-deductive Reasoning (L21-L24).
G:Total Lawson Score.
H: Proportional Reasoning -pretest.
I : Final Exam.
* In parentheses are the question numbers corresponding to
reasoning abilities measured by Lawson test (appendix D).
217
B
0.64
G
0.63
H
1.00
APPENDIX H
VARIOUS SCATTERPLOTS OF FINAL EXAM SCORE VS MEASURES OF
WORKING MEMORY CAPACITY
Scatterplot of Final Score vs WMT time
Final33
40
0
0
+
+
0
+
+
+
+
+
+ -*
<0>
-fl-
o
0
I
1
o
o
+
+
o
+
+ +
+
0
OD
+
0
+
0
0
+
o
0
+
o
0
0
-to
o
O
0
o
++ o
a-i
+
+
+ o o
4f
•
0 O
o
o
+t
+
o
+
o
o
o
o
+
I
2
0
o
I I I I I I I
3
i i i—i i i i i i i i—f-i
i I i i i
4
5
6
7
8
TimeWMT
Course
* * *
°OOphys211
+ + + Phys213
+-!- + Phys215
Figure 3.13. Scatterplot of students 'final exam score vs time spent to complete
working memory test.
218
Scatterplot of Final Score vs WMT ratio
Final33
40
oo
0+
«-
o
+
o
0 + 0
++
++
o+
o +o
+
o * o o o +
t 0- 0
+
o
+ 0
++ 0
+
o o
oo
o e>
o
o
+
o
+ o
o o
o
20
30
SpanScrRatio
Course
* * *
o o o Phys211
• Phys213
+ + + Phys215
Fig 3.14. Scatterplot of students 'final exam score vs time-based correct response
rate.
Scatterplot of final score vs Spacial WMT score
scoreSPWMT
Course
+ + +Cross-Li
+ + +Ptiys211
+ + + Phys213
+ + + Phys215
Fig 3.16. Scatterplot of students 'final exam score vs percentage of spatial working
memory capacity item correctly answered.
219
Scatterplot of final score vs Time of Spacial WMT
TimeMinsSPWMT
Course
+ + +
+ + + Crass-LI
+ + +Phys211
+ + + Pbys 213
+ + + Phys215
Fig 3.17. Scatterplot of students 'final exam score vs time spent to complete spatial
working memory test.
220
APPENDIX I
VARIOUS TABLES SHOWING DIFFERENCES IN PERFORMANCE ON
TWO PRETEST VERSIONS FOR TWO INTRODUCTORY PHYSICS
POPULATIONS
Table 4.3. Summary of difference in performance on each question on original pretest
version between Physics 211 students and Physics 215 students.
Question
Course
N
Mean
Std.
Std.
Score
Dev.
Error
(out of 1)
Phys211
254
0.461
0.500
0.031
Phys215
161
0.627
0.485
0.038
Qi
Diff(2-1)
0.494
0.167
0.050
Q2
Phys211
Phys215
Diff(2-1)
254
161
0.484
0.646
0.162
0.501
0.480
0.493
0.031
0.038
0.050
Q3
Phys211
Phys215
Diff(2-1)
254
161
0.453
0.640
0.187
0.499
0.482
0.492
0.031
0.038
0.050
Q4
Phys211
Phys215
Diff(2-1)
254
161
0.295
0.441
0.146
0.457
0.498
0.473
0.029
0.039
0.048
Q5
Phys211
Phys215
Diff(2-1)
254
161
0.614
0.721
0.106
0.488
0.450
0.474
0.031
0.036
0.048
Phys211
Phys215
254
161
2.318
3.086
1.754
1.666
0.110
0.131
0.769
1.720
0.173
Total
Score
Diff(2-1)
* Cohen's d: Ql d=0.34; Q2 d=0.33; Q3 d=0.38; Q4 d=0.31; Q5 d=0.23;
Total Score d=0.45.
221
Table 4.4. Summary oft-test analysis for difference described in Table 4.3.
Question
Method
Variances
df
t-Value
Pr>
Pooled
Equal
413
3.35
0.001
Unequal
347.9
3.37
0.001
Equal
413
3.26
0.001
Unequal
351.3
3.29
0.001
Equal
413
3.77
0.000
Unequal
349.4
3.80
0.000
Equal
413
3.06
0.002**
Unequal
318.9
3.00
0.003**
Equal
413
2.23
0.026*
Unequal
360.5
2.27
0.024*
Equal
413
4.44
0.000
Satterthwaite
Unequal 353.1
4.49
* Significant at the P < 0.05 level; **significant at the PO.01 level
0.000
Ql
Satterthwaite
Pooled
Q2
Satterthwaite
Pooled
Q3
Satterthwaite
Pooled
Q4
Satterthwaite
Pooled
Q5
Satterthwaite
Pooled
Total Score
222
Table 4.5. Summary of difference in performance on each question on the modified
pretest version between Physics 211 students and Physics 215 students.
Course
N
Mean
Std.
Std.
Dev.
Error
Score
(out of 1)
Phys211
191
0.036
0.429
0.496
118
0.627
0.486
0.045
Phys-215
Ql
0.492
Diff(2-1)
0.198
0.058
Q2
Phys211
Phys215
Diff(2-1)
190
118
0.592
0.754
0.163
0.493
0.432
0.471
0.036
0.040
0.055
Q3
Phys211
Phys215
Diff(2-1)
191
118
0.581
0.788
0.207
0.495
0.410
0.464
0.036
0.038
0.054
Q4
Phys211
Phys215
Diff(2-1)
191
118
0.304
0.576
0.273
0.461
0.496
0.475
0.033
0.046
0.056
Q5
Phys211
Phys215
Diff(2-1)
191
118
0.644
0.720
0.076
0.480
0.451
0.469
0.035
0.042
0.055
Phys211
Phys215
191
118
2.550
3.466
1.572
1.448
0.114
0.133
Total
Score
Diff(2-1)
0.916
1.526
0.179
* Cohen's d: Ql d=0.40; Q2 d=0.35; Q3 d=0.45; Q4 d=0.57; Q5 d-0.16;
Total Score d=0.60.
223
Table 4.6. Summary oft-test analysis for difference described in Table 4.5.
Question
Method
Variances
df
t-Value
Pr>
Pooled
Equal
307
3.43
0.001
252.1-
3.45
0.001
Equal
307
2.95
0.003
Unequal
272.2
3.04
0.003
Equal
307
3.81
0.000
Unequal
281.6
3.98
0.000
Equal
307
4.90
0.000
Unequal
234.1
4.82
0.000
Equal
307
1.39
0.165*
Unequal
259.9
1.41
0.159*
Equal
307
5.13
0.000
Unequal
263.4
5.23
0.000
Ql
—
Satterthwaite
Pooled
Unequal
Q2
Satterthwaite
Pooled
Q3
Satterthwaite
Pooled
Q4
Satterthwaite
Pooled
Q5
Satterthwaite
Pooled
Total Score
Satterthwaite
* NOT significant at the 0.05 level.
224
APPENDIX J
VARIOUS TABLES SHOWING GENDER SEPARATED DIFFERENCES IN
PERFORMANCE ON TWO PRETEST VERSIONS FOR TWO
INTRODUCTORY PHYSICS POPULATIONS
Table 4.09. Descriptive statistics for difference in performance between Physics 211
students on 2 pretest versions for male andfemale students, —
Pretest
Version
N
Male
Original
Modified
Diff(2-1)
122
93
Female
Original
Modified
Diff(2-1)
128
88
Gender
Mean
Score
(out of 5)
2.525
2.495
-0.030
Std.
Dev.
Std.
Error
1.726
1.592
1.670
0.156
0.165
0.230
2.086
2.636
0.550
1.748
1.548
1.669
0.155
0.165
0.231
* Cohen's d: males d=0.02; females d=0.33
Table 4.10. Summary oft-test analysis for difference described in Table 4.09.
Gender
Method
Variances
df
t-Value
Pr > |t|
Pooled
Equal
213
0.13
0.896
Unequal
205.4
0.13
0.895
Equal
214
2.38
0.018*
Unequal
200.8
2.44
0.016*
Male
Satterthwaite
Pooled
Female
Satterthwaite
* significant at the 0.05 level.
225
Table 4.11. Descriptive statistics for gender difference in performance by Physics 211
students on 2 pretest versions.
Pretest
version
Original
Gender
N
Male
Female
Diff
(MaleFemale)
122
128
Mean
Score
(out of 5)
2.525
2.086
0.439
Male
93
2.495
Female
88
2.636
Diff
-0.142
(MaleFemale)
* Cohen's d: original pretest d=0.25; modified d=0.09.
Modified
Std.
Dev.
Std.
Error
1.726
1.748
1.737
0.156
0.155
0.220
1.592
1.548
1.570
0.165
0.165
0.234
Table 4.12. Summary oft-test analysis for difference described in Table 4.11.
Pretest
Method
Variances
df
t-Value
Pr > |t|
Version
Original
Pooled
Equal
248
2.00
0.047*
Unequal 247.7
2.00
0.047*
Satterthwaite
Pooled
Equal
179
0.61
0.545
Unequal
178.9
0.61
0.544
Modified
Satterthwaite
* significant at the 0.05 level.
226
Table 4.13. Descriptive statistics for difference in performance between Physics 215
students on 2 pretest versions for male andfemale students.
Gender
Pretest
Version
N
Male
Original
Modified
Diff(2-1)
121
87
Female
Original
Modified
Diff(2-1)
37
28
Mean
Score
(out of 5)
3.223
3.575
0.352
Std.
Dev.
Std.
Error
1.615
1.467
1.555
0,147
0.157
0.219
2.541
3.357
0.817
1.709
1.254
1.531
0.281
0.237
0.383
* Cohen's d: males d=0.23; females d=0.54.
Table 4.14. Summary oft-test analysis for difference described in Table 4.13.
Gender
Variances
t-Value
Pr > |t|
Method
df
Pooled
Equal
206
1.61
0.109
Unequal
195
1.63
0.104
Equal
63
2.13
0.037*
Unequal
62.96
2.22
0.030*
Male
Satterthwaite
Pooled
Female
Satterthwaite
* significant at the 0.05 level.
227
Table 4.15. Descriptive statistics for gender difference in performance by Physics 215
students on 2 pretest versions.
Pretest
version
Original
Gender
N
Male
Female
Diff
(MaleFemale)
121
37
Mean
Score
(out of 5)
3.223
2.541
0.683
Male
87
3.575
Modified
Female
28
3.357
Diff
0.218
(MaleFemale)
* Cohen's d: original pretest d=0.42; modified d=0.15.
Std.
Dev.
Std.
Error
1.615
1.709
1.637
0,147
0.281
0.308
1.468
1.254
1.420
0.157
0.237
0.308
Table 4.16. Summary of t-test analysis for difference described in Table 4.15.
Pretest
t-Value
Pr>|t|
Method
Variances
df
Version
1
Pooled
Equal
2.22
0.028*
156
Satterthwaite
Unequal
57.1
2.15
0.036*
Pooled
Equal
113
0.71
0.482
Unequal
52.9
0.77
0.448
2
Satterthwaite
* significant at the 0.05 level.
228
APPENDIX K
FALL 2009 PRETEST VERSIONS
Fall 2009 Original Pretest Version
1. A uniform block of trinitramine is cut into two unequal
pieces, labeled piece A (larger piece) and piece B (smaller
piece). The mass density of an object is defined as the mass
of that object divided by its volume.
A correct ranking of the mass densities (from largest to
smallest) of the original block, piece A, and piece B is:
a.
b.
c.
d.
/
Original Wocfc
Original block, piece A, piece B.
Piece B, piece A, original block.
All mass densities are the same.
Not possible to determine without additional information.
To the right are drawings of a wide and a narrow
cylinder. The cylinders have equally spaced marks on
them. Water is poured into the wide cylinder up to the
4th mark (see A). This water rises to the 6th mark
when poured into the narrow cylinder (see B).
/T/
/
A
/
B /
*-
2. Both cylinders are emptied (not shown) and water
is poured into the WIDE cylinder up to the 6th
mark. How high would this water rise if it were
poured into the empty narrow cylinder?
a. to about 8
b. to about 9
c. to about 10
d. to about 12
e. none of these answers is correct
3. because
a. the answer can not be determined with the information given.
b. it went up 2 more before, so it will go up 2 more again.
c. it goes up 3 in the narrow for every 2 in the wide.
d. the second cylinder is narrower.
e. one must actually pour the water and observe to find out.
A ruler is suspended
at its center from a
frictionless pivot
shown at right. The
ruler balances when
the two masses: wy
and m2 are
120 mm
3fc
s ^
90 mm
CZJ m2
18 kg
229
suspended from the positions shown. The mass of mi is 18 kg while the mass of m2 is
unknown.
4.
A student moves the unknown mass (rri2) to a position that is 60 mm from the pivot. To
what position should the 18 kg mass (mi) be moved for the ruler to balance again?
a. 80 mm from the pivot
b. 90 mm from the pivot
c. 45 mm from the pivot
d. 120 mm from the pivot
e. Not possible to determine without additional information.
Three points: A, B and C on a rotating disk are shown in the top-view picture at right. The
disk rotates clockwise about a pivot at its center, and is neither speeding up nor slowing down.
Each point moves in a circle as the disk rotates: The radius of the circle for point A is one
meter; for point B it is two meters; and for point C it is 3 meters.
In one second, point B travels 3 meters along the circle it is moving in. What
distance does Point C travel through in one second?
a. 1.5m
b. 2m
c. 4m
d. 4.5m
e. none of these answers is correct
5. Three points: A, B and C on a rotating disk are shown in
the top-view picture at right. The disk rotates clockwise
about a pivot at its center, and is neither speeding up nor
slowing down. Each point moves in a circle as the disk
rotates: The radius of the circle for point A is one meter; for
point B it is two meters; and for point C it is 3 meters.
In one second, point B travels 3 meters along the
circle it is moving in. What distance does Point C
travel through in one second?
a. 1.5m
b. 2m
c. 4m
d. 4.5m
e. none of these answers is correct
230
6.
Two interlocked gears are shown at right. For every 40 revolutions made by the
smaller gear, it is noticed that the larger gear
only completes 16 revolutions. How many
revolutions does the larger gear complete,
given the smaller gear completes 30
revolutions.
a. 40 revolutions
b. 12 revolutions
c. 6 revolutions
d. 20 2/3 revolutions
e. none of these answers is correct
Fall 2009 Modified Pretest Version
1. A brownie is cut into two unequal pieces, labeled
Annie's piece (larger piece) and Russell's piece
(smaller piece). The mass density of an object is
defined as the mass of that object divided by its
volume.
A correct ranking of the mass densities (from
largest to smallest) of the original brownie, Annie's
piece, and Russell's piece is:
a. Original brownie, Annie's piece, Russell's
piece.
b. Russell's piece, Annie's piece, original
brownie.
c. All mass densities are the same.
d. Not possible to determine without more information.
The cold medicine you bought came with
two cylindrical containers: a wider one
marked off in adult doses and a narrower
one marked off in children's doses. When
you fill the adult (wide) cylinder to the
fourth mark (see A), and then pour it into
the children's (narrow) cylinder, it rises to
the 6th mark (see B).
2. By mistake you pour your child's
medicine up to the 6* mark in the
ADULT dosage cylinder. How high
231
would this dose be if you pour it into the children's cylinder?
a. To about 8
b. To about 9
c. To about 10
d. To about 12
e. None of these answers is correct
3. because
a. the answer can not be determined with the information given.
b. it went up 2 more before, so it will go up 2 more again,
c it goes up 3 in the narrow for every 2 in the wide.
d. the second cylinder is narrower.
e. one must actually pour the water and observe to find out.
A see-saw pivoted at its center is
shown in the diagram at right.
The see-saw balances when two
children: Ai and Bao, are seated
in the positions shown. Ai's mass
is known to be 18 kg but Bao's
mass is unknown.
240 cm-
180 cm
Ai
18 kg
Bao
X
4. Bao decides to slide himself to a position 60 cm from the pivot. To what distance from the
pivot must Ai position herself so they balance again?
a. 80 cm from the pivot
b. 90 cm from the pivot
c. 45 cm from the pivot
d. 120 cm from the pivot
e. none of these answers is correct
5. Three children, Aricelia, Bao, and Chuck, are
playing on a merry-go-round. Their positions on
the merry-go-round are shown in the top-view
picture at right. The merry-go-round is rotating
clockwise and is neither speeding up nor slowing
down. Each of the children will move in a circle
as the merry-go-round rotates: The radius of the
circle for Aricelia is one meter; for Bao it is two
meters; and for Chuck it is 3 meters.
In one second, Bao travels 3 meters along the
circle he is moving in. What distance does Chuck
travel through in one second?
a. 1.5m
b. 2m
c. 4m
d. 4.5m
232
e. none of these answers is correct
6. Two interlocked gears are shown at right. For
every 40 revolutions made by the smaller gear, it is
noticed that the larger gear only completes 16
revolutions. How many revolutions does the larger
gear complete, given the smaller gear completes 30
revolutions.
a. 40 revolutions
b. 12 revolutions
c. 6 revolutions
d. 20 2/3 revolutions
e. none of these answers is correct
233
APPENDIX L
CORRELATION MATRIX OF TA RATINGS OF STUDENTS AND OTHER
MEASURES OF STUDENT SUCCESSS.
Table 4.18. Table of correlation coefficients between TAs' ratings of students with
these students 'pretest scores (sorted by version of pretest taken) as well as with their
scores on the final exam.
Fall 2008 Correlation of TA ratings of students
N
Version
Total Prop Scr
Final Scr
26
25
1
2
0.26
0.05
0.64
0.61
24
25
1
2
0.66
0.46
0.68
0.68
18
21
1
2
0.54
-0.17
0.63
0.49
19
17
1
2
0.25
0.15
0.60
0.24
Version 1 = original pretest version; version2 = modified pretest version.
234
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Instead of P-values some researchers prefer to use effect size -Cohen's d- to
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