A research-based approach to improving student understanding
of the vector nature of kinematical concepts
Peter S. Shaffer and Lillian C. McDermott
Department of Physics, University of Washington, Seattle, Washington 98195-1560
共Received 6 April 2005; accepted 26 June 2005兲
In this paper we describe a long-term, large-scale investigation of the ability of university students
to treat velocity and acceleration as vectors in one and two dimensions. Some serious conceptual
and reasoning difficulties identified among introductory students also were common among
pre-college teachers and physics graduate students. Insights gained from this research guided the
development of instructional materials that help improve student learning at the introductory level
and beyond. The results have strong implications for the teaching of undergraduate physics, the
professional development of teachers, and the preparation of teaching assistants. © 2005 American
Association of Physics Teachers.
关DOI: 10.1119/1.2000976兴
I. INTRODUCTION
The Physics Education Group at the University of Washington 共UW兲 has been engaged in a long-term, large-scale
investigation of student understanding of motion.1 In this
project, we examined the ability of students to determine
qualitatively the magnitude and direction of the instantaneous velocity and acceleration of an object from knowledge
of its trajectory. Analysis of the results provided some of the
underpinnings for the treatment of kinematics in our two
curricula, Tutorials in Introductory Physics (TiIP)2 and Physics by Inquiry (PbI).3
Studies on student understanding of motion have been
conducted at all levels of instruction.4,5 The findings have
informed the treatment of kinematics in some innovative
curricula.6 Of particular relevance to the research and
research-based curriculum development discussed in this paper has been the work of Reif and his colleagues, who used
kinematics as a context for studies on cognition.7 In an intensive small-scale study, university students were asked to
find the acceleration of objects moving along various trajectories under different conditions. Drawing on insights gained
from observations and in-depth interviews, Reif formulated
some specific procedures to help students solve kinematics
problems. From this experience, he proposed a set of instructional guidelines for teaching problem solving more generally.
Like Reif, our group views teaching as a science. Student
learning is our primary criterion for determining teaching
effectiveness. We think of research, curriculum development,
and instruction as an iterative cycle. In this paper, we describe how we used information obtained through this process to design and assess instructional strategies to improve
student learning in kinematics.
II. MOTIVATION FOR THE EMPHASIS ON
VECTORS AND OPERATIONAL DEFINITIONS
A series of events that occurred early in the present investigation greatly influenced the direction of our research. Between 1988 and 1994, we administered problems on the motion of a simple pendulum to four populations. One group
consisted of about 125 undergraduates in introductory
calculus-based physics at UW, in which lectures on kinematics had been completed. A second group of 18 students was
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Am. J. Phys. 73 共10兲, October 2005
http://aapt.org/ajp
enrolled in a UW physics course for preservice high school
teachers, most of whom had studied kinematics in a previous
course. There were 22 UW Teaching Assistants 共TAs兲 in the
third group. Most were graduate students. In the fourth group
were graduate students taking the Ph.D. qualifying examination at UW and Montana State University.
The first three populations were asked for the direction of
the velocity and the approximate direction of the acceleration
of a pendulum bob at various points after it is released 关see
Fig. 1共a兲兴.8 A similar problem on the Ph.D. qualifying examination was based on a child on a swing 关see Fig. 1共b兲兴. In
addition to the questions on velocity and acceleration, the
graduate students were asked to draw free-body diagrams for
the child and seat at the bottom of the trajectory if the child
was not trying to swing higher.
Only about 30% of the introductory students gave a correct response for the velocity at all points and none did for
the acceleration. The performance of the preservice teachers
was better for the velocity 共70% correct兲, but similar for the
acceleration 共⬍5% correct兲. Almost all of the TAs in the
third group and the graduate students in the fourth group
drew correct velocity vectors, but only 15% did so for the
acceleration. In all four groups, many students tried to determine the acceleration from the forces. However, in these
problems, the relative magnitudes of the forces are not given.
To find the approximate direction of the acceleration, a kinematical analysis is necessary.9
On the Ph.D. qualifying examination, fewer than 15% of
the students gave correct free-body diagrams at the bottom of
the trajectory and also ranked the forces correctly. Most
stated that the string tension is equal to the weight of the
child and seat. Some explicitly stated that the net force is
zero and thus the acceleration is zero, not realizing that neither quantity can be zero if the velocity is changing. Such
errors indicate that difficulties with kinematics persist beyond introductory physics and are not adequately addressed
in later courses. The results from the graduate students were
a major motivation for focusing our research on the ability to
apply the operational definitions of velocity and acceleration
as vectors 共v = lim ⌬x / ⌬t and a = lim ⌬v / ⌬t兲 to the analysis
⌬t→0
⌬t→0
of real-world motion.10 We wanted to deepen our understanding of the nature of student difficulties and also to determine their prevalence.
© 2005 American Association of Physics Teachers
921
Fig. 1. Questions used early in the investigation. In each case, students were
asked to draw velocity vectors and to indicate the approximate direction of
the acceleration for various points in the motion.
III. OVERVIEW OF THE INVESTIGATION
A major goal of the project was to develop a set of tutorials in one- and two-dimensional kinematics for TiIP and to
expand the treatment of kinematics in PbI. The primary environment for research and curriculum development was the
introductory calculus-based mechanics course at UW, in
which tutorials from TiIP2 supplement instruction by lectures, textbook, and the laboratory. Each tutorial is part of a
sequence that consists of a pretest, worksheet, homework,
and post-test. The pretests establish a baseline for student
understanding before tutorial instruction 共usually after the
relevant lectures兲 and provide insights into the nature of specific difficulties. The pretests also help students recognize
what they are expected to understand. The 50-min tutorial
sessions typically have about 20–25 students. Structure is
provided by worksheets with sequenced questions designed
to guide students, working collaboratively in groups of 3–4,
through the reasoning required to develop a functional understanding. Teaching assistants help students by asking additional questions. The TAs prepare for this role in a required
weekly seminar in which they take the pretests and work
through the tutorials in the same way as the introductory
students. Tutorial homework reinforces and extends what
students are expected to learn in the group sessions. The
effectiveness of the tutorials is assessed by post-test questions on course examinations.
More than 20 000 students took part in the present investigation. Of these, about 11 000 were students in more than
60 different sections of introductory calculus-based mechanics at UW. Additional data came from calculus- and algebrabased courses at eight colleges and universities that were
pilot-sites for our curricula: Georgetown University, Grand
Valley State University, Harvard University, Pierce College
共CA兲, Purdue University, Syracuse University, the University
of Cincinnati, and the University of Colorado.11 TAs 共mostly
graduate students兲 at UW, Montana State, and Purdue also
participated. Another 200 participants were enrolled in special UW physics courses for K–12 teachers. Information obtained from the analysis of pretests and post-tests was
supplemented by our observations and by our interactions
with students as they worked through the curriculum. In this
way we became acutely aware of their many struggles with
the concepts, vector representations, and limiting procedures
of one- and two-dimensional 共1D and 2D兲 kinematics.
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Fig. 2. Questions used to examine the relation between vector skills and
conceptual understanding: 共a兲 1D pretest on collision of two carts, 共b兲 1D
matched pretests on manipulation of vectors, and 共c兲 2D post-test on manipulation of vectors.
IV. PROBLEMS USED FOR DISTINGUISHING
VECTOR SKILLS FROM CONCEPTUAL
UNDERSTANDING
Most previous studies on kinematics have involved the
interpretation of observed or simulated motions and their algebraic or graphical representations.12 The ability to use and
interpret vectors has been examined in several studies and
found to be weaker than many instructors realize.13 To gauge
the extent to which difficulties with the kinematical concepts
could be attributed to lack of skill with vectors, we gave
three tests at various times during the calculus-based mechanics course and one at the end. All four tests required the
same vector operations, but two involved a physical context
and two did not.
A. Test on 1D problem: Colliding carts
The velocity vectors of two carts are shown before and
after a collision 关see Fig. 2共a兲兴. Students are asked to determine the direction of the average acceleration of each cart
and to compare the magnitudes for the indicated time interval. No knowledge of dynamics is required, nor is formal
P. S. Shaffer and L. C. McDermott
922
Table I. Results from 1D pretest on the colliding carts 关Fig. 2共a兲兴 and 1D post-test on the colliding pucks 关Fig.
4共a兲兴.
Pretest
a
Post-test
Undergraduates
N ⬃ 5040
TAs
N ⬃ 170
Undergraduates
N ⬃ 1845
50%
90%
70%
40%
…
5%
…
25%
30%
65%
75%
40%
25%
5%
20%
10%
10%
20%
60%
55%
Directions of accelerations
Correct 共aA to left and aB to right兲
Most common incorrect
aA to right 共pretest兲
aB to left 共post-test兲
Relative magnitudes of accelerations
Correct 共aA ⬎ aB兲
Incorrect
aA = aB
aA ⬍ aB
Directions and relative magnitudes
Correct on both
a
Includes results from most of the Ph.D. granting universities in the study, which had very similar results. About
35% of the students 关N ⬃ 200兴 at Harvard University and the UW honors section of the calculus-based course
answered both pretest questions correctly. About 5% of the students 关N ⬃ 95兴 from the two-year college and the
four-year, non-Ph.D. granting university answered both pretests questions correctly. These data are not included.
knowledge of vectors necessary. The changes in the velocities can be found by subtracting the initial from the final
velocity for each cart. Because the time interval is the same,
the ratio of the magnitudes of the average accelerations is the
same as for the changes in velocity. Thus, cart A has the
larger acceleration, which is opposite to the direction of the
acceleration of cart B.
This problem was given to more than 5000 students in
introductory calculus-based physics at several universities at
various times during the first three weeks of class. The timing did not matter. Overall, about 20% of the students gave
both correct directions and relative magnitudes for the accelerations 共see Table I兲. The success rate for TAs on this problem was 60%. It seemed unlikely that this result and the
poorer performance of the graduate students on the pendulum and swing problems were primarily due to lack of basic
vector skills. We wondered to what extent difficulties with
vectors were responsible for the trouble the introductory students had with the colliding carts problem.
B. Tests on 1D matched problems: Concept application
and vector manipulation
We gave two matched, multiple-choice questions in several lecture sections 关see Fig. 2共b兲兴. In one version of the
question, a cart strikes and rebounds from a wall. The velocity vectors are shown and the students are asked to find the
direction of the average acceleration 关N ⬃ 360兴. In the other
version of this question, students are asked to find the difference between the same two vectors with no physical context
关N ⬃ 115兴. The problems were similar, but students did better
共65%兲 on the one involving only vector subtraction than on
the other 共45%兲. On other questions from the same test that
were identical, performance in the various lecture sections
was the same.
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C. Test on 2D problem: Vector manipulation
We gave a question, previously used by Nguyen and
Meltzer,14 on a final examination in a section of the calculusbased course 关N = 100兴 关see Fig. 2共c兲兴. Only the manipulation
of vectors is required. About 95% of the students gave a
correct response.
D. Commentary
A comparison of the results from the four tests indicates
that many more students were able to subtract vectors when
there was no physical context than when a collision was
involved. After completing introductory mechanics, students
were much more able to manipulate vectors than to apply
them in real-world problems. Clearly, the difficulties that the
introductory students had with kinematics extended beyond
vector formalism.
V. PROBLEMS USED FOR PROBING CONCEPTUAL
UNDERSTANDING
To extend our knowledge of student difficulties in kinematics, we administered qualitative problems 共similar to
those used by Reif7兲 to many students. Explanations were
required. We gave the pretests described in this section to
both the introductory students and TAs before they worked
through the tutorials described later in the paper. Before taking the 2D pretests, the students had worked through the 1D
tutorial. Within equivalent groups, responses on similar pretests were similar and have been combined.
A. 1D pretest: Ball moving up and down an inclined
ramp
One of the pretests on 1D motion involves a ball rolling up
and then down a ramp. Students are given two strobe diaP. S. Shaffer and L. C. McDermott
923
for the velocity, but only about 20% did so for the
acceleration15 共see Table II兲. When a demonstration or simulation accompanied the pretest, the results were the same.
B. 2D pretest: Object moving along a closed, horizontal
track
Fig. 3. Examples of pretests administered to large numbers of students.
Students were asked to draw velocity and acceleration vectors at various
points during each motion. 共a兲 1D pretest on ball moving up and down a
ramp. 共b兲 2D pretest on object moving at constant speed along a closed,
horizontal track. Some students were also asked about the case that the
object speeds up from rest.
grams, each showing half of the motion. The turnaround
point is on both diagrams 关see Fig. 3共a兲兴. Students are asked
to draw vectors at specified points to indicate the magnitude
and direction of the velocity and acceleration of the ball.
共Sometimes only acceleration vectors were required.兲
The questions about the direction of the velocity and acceleration can be answered from knowledge of the motion.
Dynamics is not required. The displacement and hence the
velocity are up the ramp as the ball moves upward. The
reverse holds for the downward motion. The velocity decreases in magnitude as the ball rolls up, is zero instantaneously at the turnaround, and increases as the ball rolls
down. The change in velocity 共and hence the acceleration兲 is
always directed down the ramp. Students usually know from
the lectures, textbook, or laboratory experiments that the
magnitude of the acceleration is constant.
Versions of this pretest were given to more than 20 000
undergraduates, at times ranging from the first to the third
week of class. About 80% of the students answered correctly
We decided to probe student understanding of 2D motion
in physical contexts that are conceptually easier than the pendulum. We used curved, closed and open trajectories with
shapes chosen to elicit specific difficulties. The students were
asked to draw velocity and acceleration vectors at several
points. Various versions were given to almost 7000 students.
One version involves an object moving along a closed, horizontal oval track with constant speed and, in some cases,
also with increasing speed 关see Fig. 3共b兲兴.
To find the direction of the velocity, students could draw
the displacement for a small time interval about each point
and let the interval approach zero. They could then reason
that the velocity is tangent to the trajectory at each point and
that the lengths represent the relative speeds. They also could
simply state that the velocity is tangent to the trajectory.
The question about the acceleration also could be answered in various ways. Students might state that the acceleration has a tangential component if the speed is changing
and a perpendicular component if the direction is changing.
Thus, the acceleration has only a perpendicular component
for constant speed. If the speed is increasing, the acceleration
has both tangential and perpendicular components except at
the instant at which the object starts from rest, when the
acceleration is purely tangential. Students also could draw
velocity vectors at two instants near a point and estimate the
change in velocity.
Correct diagrams, with or without explanations, were
Table II. Results from 1D pretest on the ball on ramp 关Fig. 3共a兲兴 and 1D post-test on the motion of two blocks
关Fig. 4共b兲兴. Not all students were asked about both the velocity and the acceleration.
Pretest
Velocity
Correct 共up along ramp, zero, down along ramp兲
Incorrect
Nonzero vector drawn at point where v = 0
Acceleration
Correct 共down along ramp at all points兲
Incorrectb
acceleration mimics velocity
acceleration straight down 共at one or more
points兲
acceleration zero at top
Post-test
Undergraduatesa
TAs
Undergraduatesa
N ⬃ 715
…
…
80%
…
…
15%
…
…
N ⬃ 20 110
N ⬃ 285
N ⬃ 575
20%
75%
75% 共top only兲
20%
20%
5%
10%
…
10%
50%
15%
10%
a
Includes results from most of the Ph.D. granting universities, which had very similar results. About 35% of the
students 关N ⬃ 500兴 at Harvard University and in the UW honors section of calculus-based physics answered the
question about acceleration correctly. These data are not included.
b
Categories not mutually exclusive.
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P. S. Shaffer and L. C. McDermott
924
Table III. Results from 2D pretests 关Fig. 3共b兲兴 and post-tests 共Fig. 5兲 involving motion at constant speed in a
horizontal plane.
Pretest
a
Velocity
Undergraduates
TAs
Undergraduatesa
N ⬃ 450
…
N ⬃ 390
90%
…
95%
5%
…
⬍5%
N ⬃ 6900
N ⬃ 75
N ⬃ 390
20%
65%
80%
15%
10%
20%
25%
…
5%
5%
⬍5%
⬍5%
Correct 共tangent to trajectory, equal length兲
Incorrect
Vectors curved or extend from point to point
Acceleration 共direction only兲
Post-test
Correct 共a perpendicular to trajectory at all points兲
Incorrect
acceleration toward “foci” or “center” of track
acceleration tangent to trajectory
acceleration zero everywhere
a
The results are from undergraduates at Ph.D. granting universities.
treated as correct answers. As can be seen in Table III, when
the speed was constant, about 90% of the students gave correct responses for the velocity, but only 20% did so for the
acceleration. The results in Table IV for increasing speed are
much poorer. Before taking the 2D pretest, some students
had attended lectures on 2D motion, including uniform circular motion. Because results were similar with and without
lectures, the data have been combined.
VI. ANALYSIS OF INCORRECT RESPONSES ON 1D
AND 2D PRETESTS
Most students were able to draw vectors representing the
velocity of the ball rolling up and down the ramp, as well as
along the oval when the speed was constant. On both pretests, performance was much weaker on questions about the
acceleration. There were many errors at points at which the
velocity changes direction. The instant at which motion begins caused much difficulty for students at all levels. It was
the most challenging point on the 2D pretest and the pendulum problem.
We organized the incorrect responses on the pretests into
four overlapping categories based on the incorrect reasoning
used by students. Many of the errors on the pendulum and
swing problems also fall into the same categories. The first
deals with difficulties at arbitrary points along the trajectory;
the second and third, with special points. The difficulties in
the fourth category stem from dynamics.
Table IV. Results from 2D pretests 关Fig. 3共b兲兴 and post-tests 共Fig. 5兲 involving motion with increasing speed in
a horizontal plane.
Pretest
Velocity
Correct 共v = 0 at start, then tangent with increasing
magnitude兲
Incorrect
Nonzero velocity at starting point
Acceleration
Correct 共a tangent at start; perpendicular and tangential
components elsewhere兲
Incorrect
acceleration tangent to trajectory everywhere
共excluding point A兲
acceleration incorrect for point A 共starting
point兲
acceleration zero at start
Post-test
Undergraduatesa
TAs
Undergraduatesa
N ⬃ 3375
N ⬃ 75
N ⬃ 685
30%
70%
65%
55%
30%
30%
共N ⬃ 3375兲
共N ⬃ 50兲
共N ⬃ 685兲
⬍5%
20%
35% 共60%兲b
45%
10%
15%
85%
70%
45%
20%
20%
30%
a
The results are from undergraduates at Ph.D. granting universities.
The second number represents the percentage of students who drew correct acceleration vectors if the point at
which the object starts from rest is ignored.
b
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P. S. Shaffer and L. C. McDermott
925
A. Incorrect reasoning about kinematics at arbitrary
points along a trajectory
1. Not recognizing that instantaneous velocity is tangent
to the trajectory
Some students drew velocity vectors that were not tangent
to the curve on the 2D pretest. Most followed the curvature.
Other vectors were straight with both tangential and radial
components, possibly indicating the direction in which the
object was going to move. Often they started at the point of
interest and ended at the next labeled point. Such responses
could represent confusion between average and instantaneous velocity and, in some cases, confusion between the
concepts of displacement and velocity.
In some cases, students commented that the velocity is the
slope of the curve, indicating possible confusion between a
trajectory and a position versus time graph. Such statements
were more common on pretests involving motion on open
curves.
2. Not distinguishing between velocity and acceleration
and sometimes using identical vectors for both
Many of the errors made by the students suggested confusion between the concepts of velocity and acceleration.16 On
the 1D pretest, some students drew acceleration vectors that
seemed to mimic the velocity. They often stated that “the
acceleration would be in the direction that the ball is moving.” Sometimes they showed the magnitude of the acceleration as decreasing on the way up and increasing on the way
down. Even students who drew acceleration vectors in the
correct direction often made this error. Overall, more than
half of the students indicated that the magnitude of the acceleration changed as the ball moved up or down the ramp.
Other errors also suggested confusion between velocity and
acceleration. A common response on the 2D pretest was to
draw both velocity and acceleration vectors tangent to the
oval.
Confusion between velocity and acceleration also was evident on the colliding carts pretest 关see Fig. 2共a兲兴. Many students claimed that the acceleration of cart A is to the right.
Some seemed to be thinking of an average or “overall” velocity and reasoned that because the car’s initial velocity 共to
the right兲 is larger than the final velocity 共to the left兲, the
acceleration is to the right. Even some students who answered correctly seemed to relate the acceleration to the direction of the final velocity, not to the change in velocity.
3. Mistakenly assuming that the acceleration is zero
because the speed is constant
On the 2D pretest, about 20% of the students stated that
the acceleration is zero for an object moving with constant
speed along the oval track. They treated the motion as if it
were one-dimensional, not realizing that a change in direction of the velocity means a change in the velocity vector and
so corresponds to a nonzero acceleration. 共This was the most
common error made by the graduate students on the pendulum problem.兲
4. Mistakenly assuming that the acceleration is directed
toward special points
For motion with constant speed around the oval, with constant speed, about 15% of the students drew acceleration
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vectors that were clearly directed toward the center of the
oval or toward points that some students labeled as “foci.”
Some students seemed to be over-generalizing what they had
learned about uniform motion along a circle or motion along
an elliptical orbit.17
B. Incorrect reasoning for a turnaround point
1. Mistakenly using a nonzero vector for the velocity at
the turnaround point
The most common error 共15%兲 on the velocity portion of
the 1D pretest was to draw a velocity vector for the ball at
the top of the ramp. Some did so even though they stated that
the velocity is zero. Often they showed a velocity vector up
the ramp at the turnaround on the top diagram 共which shows
the ball during its ascent兲 and/or a velocity vector down the
ramp on the bottom diagram 共which shows the ball during its
descent兲. Students also incorrectly drew velocity vectors at
the turnaround point for the pendulum.
2. Mistakenly assuming that the acceleration is zero at a
turnaround point
About half of the students stated that the acceleration is
zero at the turnaround point for the 1D pretest involving the
ball on the ramp. Similar errors were made at the end points
on the pendulum problem. Often students reasoned that because the velocity is zero, the acceleration is zero. This
widely-recognized conceptual error is closely related to the
tendency to confuse velocity and acceleration.
C. Incorrect reasoning for the point at which an object
starts from rest
1. Not treating the instantaneous velocity as zero for an
object starting from rest
Some students seemed to believe that an object about to
move must have nonzero velocity. Many drew nonzero vectors at the starting point for the oval and pendulum.
2. Mistakenly assuming that the instantaneous acceleration
is zero for an object starting from rest
On the 2D pretest, about 85% of the students gave incorrect answers for the acceleration at the starting point. About
20% claimed that it is zero.18 The undergraduate and graduate students made a similar error on the pendulum and swing
problems.
3. Mistakenly assuming that the instantaneous
acceleration has a radial component for an object starting
from rest
A very common error on the pendulum and 2D pretests
was to indicate a radial component to the acceleration at the
starting point. In some cases, students drew only a radial
component; in others, they drew both radial and tangential
components. In the former case, this answer often seemed to
be a memorized response based on uniform circular motion.
In other cases, they seemed to be thinking that the acceleration should have a radial component because the object was
starting to move along a curve.
P. S. Shaffer and L. C. McDermott
926
D. Incorrect (or incomplete) reasoning about application
of dynamics to kinematics
Many students tried to use incorrect 共or incomplete兲 dynamical arguments to find the acceleration from insufficient
information about the net force. They did not recognize that
the direction of the net force cannot be determined from
dynamics when one of the forces is “passive,” 共for example,
the normal force by a sloping surface or tension in a rope兲.
The direction must be inferred from kinematics.
1. Not associating the direction of the acceleration with
that of the net force
On the pretest involving the ball on the ramp, about 20%
of the introductory students drew acceleration vectors
straight downward. They often stated that the acceleration is
down because gravity is the “cause” of the motion or that
gravity is the force “responsible.”19 Some said explicitly that
the magnitude is 9.8 m / s2, as in free-fall. They did not associate the difference in direction of the motion 共along the
incline versus straight down兲 with a corresponding difference
in the acceleration. They neglected to consider the normal
force exerted on the ball by the ramp. Instead of relating the
direction of the acceleration to that of the change in velocity,
some students simply assumed that the force was down the
ramp, and thus so was the acceleration. On the oval pretest, a
number of students justified the direction of their acceleration vectors by statements about the direction of the force
“causing the motion.” A similar error was made by graduate
students on the pendulum problem, for which about 20%
said the acceleration is straight downward.
2. Confusing net force and acceleration
About 40% of the introductory students stated that the
accelerations of the two carts on the colliding carts pretest
are equal. Often they based their explanation on Newton’s
third law. These students did not seem to distinguish between
acceleration and net force. Although the magnitude of the
force exerted by each cart on the other is the same, the accelerations depend on the relative masses, which are not
given. Confusion between the net force and acceleration also
might have contributed to poor performance on the problem
in which the cart strikes a wall and rebounds.
E. Commentary on results from 1D and 2D pretests
On the 1D pretest, very few students gave a correct answer
and explanation for their choice of direction of either the
velocity or the acceleration vectors. The errors and explanations often indicated that students did not have a functional
understanding of 共1兲 the operational definitions of velocity
and acceleration as rates of change of displacement and velocity, respectively, or 共2兲 the vector nature of these quantities. Many students seemed to be recalling memorized answers, apparently not understanding how these had been
obtained or the conditions under which they held. Reif has
made similar observations.7
The difficulties that students encountered on the 2D oval
pretest indicated a general failure to apply the operational
definitions of velocity and acceleration. Students generally
failed to transfer what they had learned about motion in one
dimension to two dimensions. They often drew on their
memorized knowledge of specific cases.7 Some tried to use
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Am. J. Phys., Vol. 73, No. 10, October 2005
their knowledge of the radial acceleration for an object moving with uniform speed in a circle, the derivation of which
appears in almost every introductory course and textbook.
Because in this case the radial acceleration is constant and
can be expressed by a simple formula, the reasoning involved in the limiting process often is ignored by students.
Therefore, they cannot transfer to situations in which the
velocity is not constant in magnitude and the motion is not
circular.
VII. DEVELOPMENT OF TUTORIAL CURRICULUM
The results discussed in the previous sections suggest that
students need targeted help to develop an understanding of
the kinematical concepts that extends beyond the ability to
apply formulas. We found that an effective instructional approach is to help them develop facility in applying the operational definitions for velocity and acceleration as vectors.
In this section, we illustrate the implementation of this strategy in the design of a curriculum. We hope the discussion of
how we have used research to guide instruction will be helpful to others in their teaching of kinematics.
To help students construct a coherent conceptual framework for a given topic, we begin with the identification and
analysis of the intellectual hurdles that they commonly encounter. This detailed knowledge helps guide the design and
assessment of instructional strategies to address specific difficulties. However, a list of errors cannot drive effective instruction. Many difficulties are interrelated and interdependent and therefore must be treated together. Moreover, as we
have noted, the context is critically important. A brief description of the development and assessment of the 1D and
2D tutorials follows. The design of both embodies certain
generalizations that we have drawn from our experience and
an instructional strategy that we have often found effective.20
A. Kinematics in one dimension
It seemed likely that students would benefit from direct
experience in determining a direction for the change in velocity and in relating this quantity to the direction of the
acceleration. We therefore designed a tutorial to guide students through the process of applying the operational definition of acceleration to a motion in one dimension.
The context for the tutorial worksheet is essentially that of
the ball on the ramp 1D pretest. Students are asked a sequence of questions that not only help them identify the steps
needed to determine the direction of the acceleration, but
also serve to elicit incorrect ideas. The students first consider
the motion of the ball up the incline and draw vectors that
represent the velocity at several instants separated by equal
time intervals. They then are asked to find the change in the
velocity between two specific instants. On the basis of a
velocity versus time graph that they are given for the motion,
they decide how the change in velocity would differ for a
time interval that is half as large. The students are led to
recognize that the change in velocity for each unit of time is
the same throughout the motion. They then consider how the
result of dividing the change in velocity by the corresponding time interval would change as the interval becomes
smaller and are led to associate this quantity with the instantaneous acceleration. Our observations of students as they
work through the tutorial indicate that many have not
thought about a limiting process before and are surprised that
P. S. Shaffer and L. C. McDermott
927
the acceleration is not zero. The students repeat the same
steps for motion down the ramp. Even after this experience,
many still think that the acceleration is zero at the top. This
error is a common conceptual difficulty that we elicit through
the pretest. By guiding students in applying the operational
definition for acceleration at the turnaround point the tutorial
helps them confront and resolve this error.
In the tutorial homework, the students apply and reflect on
what they have learned. A major focus is on relating vector
representations for motion to verbal, graphical, and other
diagrammatical representations, as well as to real-world motions. The homework also provides practice in determining
the acceleration from initial and final velocities. The students
generalize their findings by articulating how an object moves
if its velocity and acceleration are in the same, or in opposite,
directions. One homework problem is based on the pretest
situation involving two colliding carts. We have given several post-test problems related to this tutorial after students
also have worked through the 2D tutorial.
B. Kinematics in two dimensions
The 2D tutorial helps students apply the operational definitions of the kinematical concepts to two-dimensional motion and also helps them develop skill in adding and subtracting vectors. The context is the horizontal oval track on which
the pretest is based. As is the case for 1D motion, the tutorial
includes questions to elicit student thinking and to help them
confront and resolve incorrect ideas. By engaging students in
the reasoning, we try to help them deepen their understanding.
The students are guided through the process of finding the
acceleration through the use of vectors for an object moving
along the oval at constant speed, increasing speed, and decreasing speed. They select an arbitrary origin, draw position
vectors for two points on the oval, draw the displacement
vector 共⌬s兲 between the two points, and note that it is independent of the choice of origin. The students then apply a
limiting procedure to find the direction of ⌬s as the second
point approaches the first. They observe that ⌬s becomes
tangent to the curve in the limit. They are led to associate the
limiting direction with that of the instantaneous velocity at
that point.
The students begin the process of determining the acceleration by drawing the instantaneous velocity vectors at two
nearby points. They find the change in the velocity vector
共⌬v兲 and consider how it changes as the second point approaches the first along the oval. They observe that the triangle formed by the velocity vectors and the change-invelocity vector is isosceles. They reason that, when the speed
is constant, the angle between each velocity vector and the
change-in-velocity vector approaches 90° in the limit of
small time intervals. They draw acceleration vectors at various points on the oval and consider how the curvature affects
the magnitude. They repeat the procedure for the case when
the speed of the object is increasing including the case for
which the first point is the starting point. The slowing-down
case is completed as homework.
In the tutorial homework, students generalize their results.
They note that the angle between the velocity and acceleration vectors 共drawn tail-to-tail兲 is 90° for constant speed, less
than 90° for speeding up, and greater than 90° when the
object is slowing down. They recognize that for the speed to
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Fig. 4. Examples of 1D post-tests. 共a兲 Two pucks collide on a frictionless
table. Students were asked to give the directions of the accelerations and the
relative magnitudes. 共b兲 Two blocks of different mass 共mA = mB / 2兲 move on
a frictionless incline. Students were asked to indicate the direction of the
acceleration of block B at the earlier instant and to draw the change in
velocity vector for block A.
increase 共decrease兲, there must be a component of the acceleration in the same 共opposite兲 direction as the velocity.
VIII. ASSESSMENT OF 1D AND 2D TUTORIALS
Both the 1D and 2D post-tests were given after students
had worked through the 2D tutorial. Tables I–IV indicate the
combined effect of both tutorials. There are fewer post-test
than pretest results because not all the tutorials were assessed
every quarter.
A. 1D post-tests
Two examples of post-tests on velocity and acceleration in
one dimension are shown in Fig. 4. Each has been given in
the form depicted and in several variations.
1. Collision of two pucks on frictionless table
This post-test 关see Fig. 4共a兲兴 is similar to the question on
the colliding carts discussed earlier, on which about 20% of
the students 关N ⬃ 5040兴 gave correct answers for the acceleration 共direction and magnitude兲. On the post-test, about
55% 关N ⬃ 1845兴 answered correctly 共see Table I兲.
2. Motion of two blocks up and down an inclined ramp
Two blocks 共A and B兲 of different mass move independently on a frictionless incline 关see Fig. 4共b兲兴. The initial
velocities of both blocks and the final velocity of block A are
shown. The first part of the post-test checks whether students
realize that the acceleration is not zero when an object turns
around 共block B兲. The second part asks students to find the
change in velocity for an object that turns around 共block A兲.
The first part of the post-test can be compared to the 1D
pretest on the ball and the ramp. On the post-test, which is
more difficult than the pretest, about 75% of the students
关N ⬃ 575兴 found the correct acceleration at the top 共see Table
II兲. Only 20% had done so on the pretest. On the second part
of the post-test, students were shown initial and final velocity
vectors and asked about either the change in velocity or acP. S. Shaffer and L. C. McDermott
928
2. Pendulum
Fig. 5. Example of 2D post-test. Students were asked to sketch velocity and
acceleration vectors at the labeled points for the case that the object moves
at constant speed and to draw acceleration vectors for the case that the
object moves with increasing speed.
celeration. This part can be compared to the colliding carts
question. On the post-test, 55% answered correctly, in contrast to 20% on the pretest.
3. Commentary
Although there were no formal assessments of the 1D tutorial alone, results from other problems showed that many
students did learn to distinguish the kinematical concepts
clearly from one another and to recognize that the acceleration is not zero at a starting or turnaround point. As we
discussed, the 1D and 2D tutorials together led to improved
performance on both 1D post-tests when compared with the
1D pretests. The percentage making errors that the tutorials
try to address decreased. As on the pretests, the most common errors involved inappropriate dynamical arguments.
The introductory students did about as well on both 1D
post-tests as the TAs on the 1D pretests. However, it is clear
from the 2D pretest that few of the introductory students
were able to transfer to two dimensions much of what they
had learned in the 1D tutorial.
B. 2D post-tests
A variety of post-tests on 2D motion were given after
students had worked through the 1D and 2D tutorials. Here,
we discuss the results from post-tests based on closed horizontal curves and on the pendulum.
1. Motion with changing speed along a closed horizontal
trajectory
Many of the post-test questions involved motion at changing speed along a closed horizontal trajectory 共see Fig. 5兲.
Typically, the object moves at constant speed or starts from
rest, speeds up, and eventually moves with constant speed.
Students are asked to indicate the approximate direction of
the acceleration at various points.
For points that correspond to constant speed, about 80% of
the students recognized that the acceleration is perpendicular
to the motion 共see Table III兲. Only 20% had done so on the
pretest. For points corresponding to increasing speed, about
35% answered correctly, whereas fewer than 5% did so on
the pretest 共see Table IV兲. The point corresponding to acceleration from rest was the most difficult for students. When
that point is ignored in the analysis, about 60% of the students gave correct answers.
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Students performed less well on post-test questions that
involved vertical motion, such as the pendulum. We have
posed a variety of questions in which students are asked to
draw velocity vectors at several points of the motion and to
indicate the approximate direction of the acceleration. The
success rate increases from essentially 0% on the pretest to
only about 15% on the post-test. Many of the same conceptual and reasoning difficulties appear on the post-test as on
the pretest. In the case of the pendulum, many students still
try to relate the acceleration of the bob to the forces exerted
on it. Early in the investigation, we recognized that the ability to determine the direction of the acceleration often is
adversely affected by a tendency to reason on the basis of
forces.21 Although such an analysis yields a range of possible
directions, the information is insufficient to determine the
direction. Kinematical analysis is necessary.
3. Commentary
As on the 1D post-tests, student performance on both
types of 2D post-tests approximately matches that of physics
graduate students on the corresponding questions. We generally consider a tutorial to be successful when the performance of introductory students on a post-test matches, or
surpasses, that of the TAs on the corresponding pretest. Although the kinematics tutorials meet this criterion, we are
continuing to strive for greater improvement by probing
more deeply into how difficulties with dynamics affect the
ability of students to determine acceleration.22
IX. APPLICATION TO PRECOLLEGE TEACHER
PROFESSIONAL DEVELOPMENT
Some of the participants in the study were enrolled in
special preservice and inservice physics courses that our
group conducts for the preparation of teachers in physics and
physical science.23 The curriculum, Physics by Inquiry, consists of a set of laboratory-based modules expressly designed
to help teachers do the reasoning required to construct a coherent conceptual framework. The published version of PbI
includes Kinematics, a module that treats motion in one dimension only. We have recently developed a module on 2D
kinematics that incorporates many of the strategies in the
tutorials but is more rigorous. The post-test performance of
teachers who have worked through these modules often has
surpassed that of students enrolled in tutorials in the introductory calculus-based course. We attribute this accomplishment to the treatment in PbI, which requires a greater degree
of intellectual engagement and more time than is available
for the tutorials.
X. CONCLUSION
Evidence from this study and other research by our group
documents the difficulties that kinematics presents not only
to introductory students, but also to many precollege teachers
and even graduate students. After standard instruction, students often cannot apply the formalism that they have been
taught to determine the velocity and acceleration of a motion
that they have observed or that has been described. In some
cases, inability to perform basic vector operations is the obstacle. However, in many instances, the difficulties are primarily conceptual, rather than mathematical. Certain conP. S. Shaffer and L. C. McDermott
929
texts seem to lead students to try to reason on the basis of
dynamics rather than kinematics, even when this approach
does not yield sufficient information about the acceleration.
The tendency to try to infer the magnitude and direction of
the acceleration from the forces involved, rather than from
the velocity, is especially common in the presence of gravitational interactions 共for example, a ball on an incline and a
swinging pendulum兲.
The findings from our research, as well as our experience
in teaching kinematics, have led to the emphasis in our curriculum on operational definitions of velocity and acceleration in terms of their vector representations. Because difficulties with kinematics in one dimension become even more
pronounced in two, we explicitly try to help students learn
how to apply vectors in the analysis of motion in one dimension and then help them transfer this skill to two dimensions.
Tutorials in Introductory Physics has not only helped the
introductory students for whom it was expressly designed,
but also has helped undergraduate and graduate TAs both
deepen their own understanding and recognize the difficulties inherent in this topic. The corresponding modules in
Physics by Inquiry have served the same purpose for precollege teachers. Thus, the results from this investigation have
implications for instruction from the introductory to the
graduate level and provide an incentive for ongoing research
and curriculum development in kinematics.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the collaboration of
many members of the Physics Education Group. Costas Constantinou, Gregory F. Francis, Mark D. Somers, and Stamatis
Vokos participated in the development of curriculum. Contributions to the research were made by Bradley S. Ambrose,
Sean M. Courtney, Paula R. L. Heron, MacKenzie R. Stetzer,
and John R. Thompson. Special thanks are due to the instructors whose classes were included in this investigation. We
also deeply appreciate support from the Division of Undergraduate Education and the Division of Physics of the National Science Foundation.
1
Results from these investigations are in D. E. Trowbridge and L. C.
McDermott, “Investigation of student understanding of the concept of
velocity in one dimension,” Am. J. Phys. 48 共12兲, 1020–1028 共1980兲;
“Investigation of student understanding of the concept of acceleration in
one dimension,” 49 共3兲, 242–253 共1981兲; and L. C. McDermott, M. L.
Rosenquist, and E. H. van Zee, “Student difficulties in connecting graphs
and physics: Examples from kinematics,” ibid. 55 共6兲, 503–515 共1987兲.
Documentation of similar findings in the honors section of the introductory course are in P. C. Peters, “Even honors students have conceptual
difficulties with physics,” Am. J. Phys. 50 共6兲, 501–508 共1982兲.
2
L. C. McDermott, P. S. Shaffer, and the Physics Education Group at the
University of Washington, Tutorials in Introductory Physics 共Prentice
Hall, Englewood Cliffs, NJ, 2002兲.
3
L. C. McDermott and the Physics Education Group at the University of
Washington, Physics by Inquiry 共Wiley, New York, 1996兲.
4
For a list of references to studies at the precollege level, see R. Duit,
“Students’ and teachers’ conceptions and science education, Bibliography,” Leibniz Institute for Science Education at the University of Kiel,
Olshausenstr. 62 24098 Kiel, Germany, or 具http://www.ipn.uni-kiel.de/
aktuell/stcse/stcse.html典.
5
For a list of references to studies at the university level, see L. C. McDermott and E. F. Redish, “Resource letter: PER-1: Physics education
research,” Am. J. Phys. 67 共9兲, 755–767 共1999兲.
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6
For example, Laws, Thornton, and Sokoloff have applied findings from
research to the development of real-time microcomputer-based laboratory
tools to help students make connections between real motions and their
graphical representations. A discussion of the instructional approach appears in P. W. Laws, “Millikan lecture 1996: Promoting active learning
based on physics education research in introductory physics courses,”
Am. J. Phys. 65 共1兲, 14–21 共1997兲; and R. K. Thornton and D. R.
Sokoloff, “Learning motion concepts using real-time microcomputerbased laboratory tools,” ibid. 58 共9兲, 858–867 共1990兲.
7
P. Labudde, F. Reif, and L. Quinn, “Facilitation of scientific concept
learning by interpretation procedures and analysis,” Int. J. Sci. Educ. 10
共1兲, 81–98 共1988兲; F. Reif and S. Allen, “Cognition for interpreting scientific concepts: A study of acceleration,” Cogn. Instruct. 9 共1兲, 1–44
共1992兲.
8
This problem was used by Reif in the studies described in Ref. 7.
9
There are many problems of this type that students should be able to
solve. Another example of the need to infer information about the acceleration from kinematical constraints is given in L. C. McDermott, P. S.
Shaffer, and M. D. Somers, “Research as a guide for teaching introductory mechanics: An illustration in the context of the Atwood’s machine,”
Am. J. Phys. 62 共1兲, 46–55 共1994兲. Students were presented with a problem involving two blocks of different mass connected by an inextensible
string passing over a pulley. About 15% failed to recognize that the accelerations of the blocks were equal.
10
For a discussion on the importance of emphasizing operational definitions
in teaching physics and physical science, see A. Arons, Teaching Introductory Physics 共Wiley, New York, 1997兲.
11
Most of the Ph.D. granting universities had similar results on the pretests
and post-tests. The results have been combined and are presented in the
tables and text. Footnotes to the tables give results from non-Ph.D. granting institutions, Harvard University, and a self-selected honors section of
UW calculus-based physics, which had a higher percentage of physics
and science majors than other sections.
12
For examples of research related to graphical interpretations of motion,
see R. J. Beichner, “Testing student understanding of motion graphs,”
Am. J. Phys. 62 共8兲, 750–762 共1994兲; F. M. Goldberg and J. H. Anderson, “Student difficulties with graphical representations of negative values of velocity,” Phys. Teach. 27 共4兲, 254–260 共1989兲; and the third
paper in Ref. 1.
13
See, for example, S. Flores, S. E. Kanim, and C. H. Kautz, “Student use
of vectors in introductory mechanics,” Am. J. Phys. 72 共4兲, 460–468
共2004兲; N. L. Nguyen and D. E. Meltzer, “Initial understanding of vector
concepts among students in introductory physics courses,” ibid. 71 共6兲,
630–638 共2003兲; R. D. Knight, “The vector knowledge of beginning
physics students,” Phys. Teach. 33 共2兲, 74–78 共1995兲; J. M. Aguirre,
“Student preconceptions about vector kinematics,” ibid. 26 共4兲, 212–216
共1988兲; and J. M. Aguirre and G. Erickson, “Students’ conceptions about
the vector characteristics of three physics concepts,” J. Res. Sci. Teach.
21, 437–457 共1984兲.
14
See the second paper in Ref. 13.
15
This result is similar to that obtained by Reif and Allen on a corresponding question in the small-scale study 共N = 5兲 described in the second article in Ref. 7.
16
Confusion between velocity and acceleration in one dimension has been
documented previously. See, for example, the first two papers in Ref. 1.
17
We also have administered pretest questions involving open, curved trajectories. Most students do not claim that the acceleration is directed
toward the center or the “foci”; however, the percentage of correct answers is similar to that on the oval pretest.
18
An object that starts to move from rest at t = t0 is usually treated as having
an acceleration that changes discontinuously at t0. Some students might
have had difficulty with this approximation.
19
There is a common tendency to associate causes of motion only with
active forces and not with passive forces. See, for example, J. Minstrell,
“Explaining the ‘at rest’ condition of an object,” Phys. Teach. 20 共1兲,
10–14 共1982兲; L. C. McDermott, “Research on conceptual understanding
in mechanics,” Phys. Today 37 共7兲, 24–32 共1984兲.
20
Some of the generalizations, their basis, and an instructional strategy that
can be summarized as “elicit, confront, and resolve” are discussed in
greater detail in L. C. McDermott, “Oersted medal lecture 2001: Physics
education research—The key to student learning,” Am. J. Phys. 69 共11兲,
P. S. Shaffer and L. C. McDermott
930
1127–1137 共2001兲 and L. C. McDermott, “Millikan lecture 1990: What
we teach and what is learned—Closing the gap,” Am. J. Phys. 59 共4兲,
301–315 共1991兲.
21
Reif had observed a similar tendency. See, for example, Ref. 7.
22
Research on student understanding of 2D vertical motion has been con-
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ducted at the University of Washington, at the University of Maine 共J. R.
Thompson兲, and at Grand Valley State University 共B. S. Ambrose兲.
23
For a description of these courses, see L. C. McDermott, “A perspective
on teacher preparation in physics and other sciences: The need for special
science courses for teachers,” Am. J. Phys. 58 共8兲, 734–752 共2000兲.
P. S. Shaffer and L. C. McDermott
931