In an investigation that extended over several years and
included several colleges and universities, we examined stu-
dent understanding of kinematical concepts in one and two
dimensions.^5 In one problem used in this study, students
were shown a strobe diagram of a ball rolling up and down
an inclined ramp and were asked to draw acceleration vec-
tors at various points along the trajectorysee Fig. 2.We
examined the responses from about 15,000 students in intro-
ductory physics, 180 preservice and inservice teacherspri-
marily grades 9-12, and 300 physics graduate students who
were teaching assistants in the introductory course. The most
common incorrect answers were that the acceleration would
be zero at the turnaround point, or that it would be directed
vertically downward at all points. Only about 50% of the
teachers and 20% of the introductory students drew correct
sketches with acceleration vectors of constant magnitude al-
ways directed down the ramp. About 75% of the graduate
students gave correct responses.
C. Mismatch for K-12 teachers: Example from
electric circuits
The topic of electric circuits is part of many precollege
curricula, often in the context of batteries and bulbs. In our
research on student understanding of this material, we have
administered a wide variety of questions. One, which is
based on Fig. 3, has been given to several different popula-
tions, including introductory physics students and preservice
and inservice teachers of all grade levels.^6 The question asks
for a ranking of the brightness of the identical bulbs in the
three circuits, which have identical, ideal batteries. Explana-
tions are required. The correct ranking isA=D=EB=C.
The results from introductory students and K-12 teachers
have been approximately the same. Only about 15% in each
group have given a correct ranking. The preservice and in-
service teachers performed similarly, even though many of
the latter had previously taught this topic. Analysis of the
explanations by all the populations, including high school
physics teachers, revealed the widespread presence of two
apparent beliefs: the battery is a constant current source and
current is “used up” in a circuit.
The results from this question and from the one on balanc-
ing discussed earlier illustrate a general finding. Teaching a
topic does not necessarily deepen one’s own conceptual un-
derstanding. The following event, which occurred during a
professional development workshop, is illustrative. A high
school teacher with 12 years of classroom experience had
just completed experiments and exercises intended to help
students associate bulb brightness with current. When asked
to compare the brightness of a single bulb across a battery
with that of two bulbs in parallel across a second battery, she
observed that all three were equally bright. Surprised, she
exclaimed, “That would mean that the amount of current
from the battery is different in different cases, and that
doesn’t make any sense!” She suddenly realized that her as-
sumption that the current through a battery is always the
same was incorrect. Although she was likely adept at solving
textbook circuit problems, her understanding of the material
was far short of what it should have been.III. DEVELOPMENT AND ASSESSMENT
OF CURRICULUMThese examples illustrate some specific difficulties that
teachers often share with many university students. Because
of their responsibility to help their students learn, the situa-
tion for teachers is more serious and needs more attention.
They must know and be able to do more than is expected of
their students. We should therefore ask what we want young
students to know and be able to do and prepare teachers
accordingly. These questions have led to the development of
Physics by Inquiry (PbI), a laboratory-based curriculum pri-
marily intended for the preparation of preservice and inser-
vice teachers but also suitable for other populations.13,14
We begin instruction on all topics by drawing on research
that identifies where students are intellectually. We use this
information to design, test, and revise curriculum on the ba-
sis of experience in classes at UW and at pilot sites. Teaching
is by asking questions to help students construct a coherent
conceptual framework, rather than by telling. The emphasis
is not on solving standard problems, but on developing the
reasoning ability needed to apply relevant concepts to situa-
tions that have not been memorized. The curriculum explic-
itly addresses specific difficulties that research has shown
may preclude a functional understanding. Even when teach-
ers do not have these difficulties themselves, it is likely that
their students will.PbIhelps teachers develop the type of
knowledge necessary to be able to teach a given topic effec-Fig. 1. Question about balancing. Students are told that the bat, which has
uniform density, remains at rest when placed on a finger as shown. They are
asked whether the mass to the left of the balance pointPis greater than, less
than, or equal to the mass to the right of the balance point.
Fig. 2. Question about acceleration. Students are shown the diagram of a
ball rolling first up and then down the ramp. They are asked to draw vectors
for the velocity and the acceleration at each of the marked points.
Fig. 3. Question about electric circuits. Students are told the bulbs are iden-
tical and the batteries are identical and ideal. They are asked to rank the
bulbs from brightest to dimmest.764 Am. J. Phys., Vol. 74, No. 9, September 2006 McDermottet al. 764