tivelypedagogical content knowledge.^15 Ongoing assess-
ment that includes pretests and post-tests is an integral part
of the iterative process involved in our ongoing curriculum
development. We illustrate our design and assessment of cur-
riculum in the context of dynamics.
Design of curriculum. Student understanding of dynamics
has been the focus of much research by our group and
others.7,10The results have guided the development ofDy-
namics. This module builds directly onKinematics, in which
the concepts of velocity and acceleration are developed from
their operational definitions.Dynamicsbegins with the con-
cept of force as a push or a pull. As in all ofPbI, the equip-
ment is simple and inexpensive so that it is readily accessible
to teachers. Measurement procedures are as straightforward
as possible with no black boxes. We start with simple “pull
meters” made of rubber bands and meter sticks, rather than
with spring scales or force probes. Students build and cali-
brate the pull meters and explore how multiple pulls affect
the motion of a wheeled cart. They find that a cart subject to
a constant pull undergoes constant acceleration.
Experiments with wooden blocks on rough surfaces and
pieces of dry ice on level slate surfaces lead students to rec-
ognize that an interaction between surfaces can be thought of
in terms of a force. These experiments help students distin-
guish between a single applied force, for example, exerted by
a pull meter or a hand, and the net force that an object ex-
periences. The students build on their previous experience
with kinematics and explore cases in which the net force is
exerted in the direction of motion and in the opposite direc-
tion. They conclude that an object accelerates in the direction
of the net force. The well-known tendency to associate force
and velocity is explicitly addressed. For example, the stu-
dents consider hypothetical dialogues in which fictional stu-
dents express common incorrect ideas.
The students use spring scalescalibrated in newtonsto
conduct experiments on carts to which varying numbers of
identical objects have been added. They find that the net
force required to produce a given acceleration increases as
the number of objects increases. They are then led to develop
the concept of inertial mass and arrive at an algebraic expres-
sion of Newton’s second law. Subsequently, the students ex-
plore gravitational and frictional forces in more detail. They
also develop skill in drawing free-body diagrams. Newton’s
third law is introduced by experiments in which students find
that two magnets exert forces of equal magnitude and oppo-
site direction on each other, regardless of which magnet is
stronger. Subsequent experiments and exercises provide stu-
dents with experience in applying Newton’s laws to systems
of increasing complexity.
There is an emphasis on the development of scientific rea-
soning skills throughout Dynamics. The module stresses
graphing, proportional reasoning, and vectors. Ideas intro-
duced in theKinematicsmodule, for example, the interpre-
tation of the slopes and the areas under the curves for graphs
of position, velocity, and acceleration as functions of time,
are reinforced. Thus, mathematics and physics teachers are
given concrete ways to help students relate differentiation
and integration to real-world phenomena.
The process of scientific model building is made explicit.
In particular, the difference between observation and infer-
ence is stressed repeatedly. For example, students are ex-
pected to recognize that the extension of a spring scale from
which an object is hanging is not a direct measurement of the
gravitational force exerted on the object; rather, it can be
used in conjunction with Newton’s second law to deduce the
magnitude of the force.
Assessment of student learning. We have assessed student
learning by comparing results from pretests and post-tests.
The following post-test question, which requires multistep
reasoning, is an example.
A system of three incompressible blocks is pushed across
a frictionless table by a hand that exerts a constant horizontal
forcesee Fig. 4.^16 Students are asked how, if at all, the
acceleration of block A and the net force on block A changes
if block B is replaced by a block of greater mass while the
hand continues to exert the same constant force. To answer
correctly, students must recognize that the inertial mass of
the system has increased while the net force on the system
due to the handremains unchanged. Newton’s second law
may be applied to determine that the acceleration of the en-
tire system and thus that of block A has decreased. Using
similar reasoning, the students can then infer that the net
force on block A has also decreased.
When this question was administered in introductory
physics courses after standard instruction, fewer than 20% of
the studentsN 100 answered correctly.^7 About 90% of
the teachers N=45 who worked through theDynamics
module gave a correct response. We have also given this
question to introductory students after they had worked
throughTutorials in Introductory Physics.^17 This curriculum
addresses the intellectual issues discussed previously, but in
a form adapted to a large introductory course.About 55% of
the studentsN 720 answered correctly. Although this re-
sult represents a sizable gain over that obtained with standard
instruction, it is not good enough for prospective teachers.
Even when an introductory physics course is supplemented
with research-based materials, students are unlikely to de-
velop the depth of understanding that is possible with the
type of instruction provided byPbI. There is evidence from
other topics that not only is the resultant gain in conceptual
understanding greater, it is also persistent.^18
Commentary. To illustrate our instructional approach in
preparing teachers, we have used an example from dynamics.
A topic from earlier grades would have served equally well.
Elementary and middle school teachers need the same typeFig. 4. Question about Newton’s second law.aStudents are shown a figure
in which three incompressible blocks are pushed across a frictionless table
by a hand.bThey are then told that block B is replaced by a block of
greater mass. The question asks how, if at all, the acceleration of block A
and the net force on block A change if the hand exerts the same horizontal
force in both cases.765 Am. J. Phys., Vol. 74, No. 9, September 2006 McDermottet al. 765