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Why is Amp` eres law so hard? A look at middle-division physics
Why is Amp`
eres law so hard? A look at middle-division physics
Corinne A. Manogue Department of Physics, Oregon State University, Corvallis, Oregon 97331
Kerry Browne Department of Physics, Dickinson College, Carlisle, Pennsylvania 17013
Tevian Dray Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Barbara Edwards
§
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
(Dated: February 4, 2006)
Because mathematicians and physicists think dierently about mathematics, they have dierent
goals for their courses and teach dierent ways of thinking about the material. As a consequence,
there are a number of capabilities that physics majors need in order to be successful that might
not be addressed by any traditional course. The result is that the total cognitive load is too high
for many students at the transition from the calculus and introductory physics sequences to upper-
division courses for physics majors. We illustrate typical student diculties in the context of an
Amp`eres law problem.
I.
INTRODUCTION
We have all seen it happen. A student who got straight As in lower-division math and physics classes starts the
post-introductory courses for physics majors and is totally bewildered to be suddenly getting Bs and Cs. The middle
of the pack become angry or frustrated with the level of diculty in our courses. I just dont know how to get
started! echoes in the hallways. And too many of the weakest students give up or quietly disappear. Students who
come to our oce hours for help seem able to do the homework problems with a few hints, but freeze completely on
exams. What is happening? To build more eective curricula we need to develop a better understanding of what
makes the transition to upper-division physics so hard for some of our majors.
At some schools, as is the case at Oregon State University (OSU), the transition occurs in middle-division courses
whose content is electrostatics and magnetostatics. The middle-division consists of those courses taken immediately
after introductory calculus, introductory physics, and modern physics, and which serve to introduce the major. At
other schools the middle-division courses cover topics such as waves, mathematical methods, or classical mechanics.
For the past nine years, we have been focusing on this transition in two NSF-funded projects at OSU. In this paper,
we share the insights we have gained that are relevant to the teaching of these courses.
The Paradigms in Physics program
1
comprises a complete reorganization and revision of upper-division theory
courses to cultivate students analytical and problem-solving skills. The nature and goals of the program as a whole
have been discussed in detail.
2,3
One of the goals in the rst few courses is to ease the problems that students have
transitioning from lower-division to upper-division courses. Group activities require students to employ geometric
reasoning and build mathematical skills in the context of strongly focused physical examples. We encourage movement
away from routine problem-solving following well-dened templates and toward the use of multiple representations
and synthesis.
The purpose of the Vector Calculus Bridge Project
4
is to understand the dierences in perspective between math-
ematicians and physicists and why these dierences cause transition problems for students. Informed by these under-
standings we designed and classroom-tested curricular materials at OSU. We also developed resources for mathematics
faculty to help them appreciate the needs of their physical science and engineering students. These resources include
a series of papers
58
that emphasize the importance of the vector dierential dr in both rectangular and curvilinear
coordinates, group activities and an instructors guide focused on student development of geometric reasoning, and
an ongoing series of faculty development workshops.
9
The Bridge Project has now evolved to the point that we are
using what we have learned to address the educational needs of students in middle-division courses.
The Paradigms and Bridge projects are perhaps unique in terms of the sheer scope of the curriculum that they
address. From this broad perspective we have learned that there are overarching expectations that we implicitly hold
for our students: students at this level are required to solve problems involving many steps and to engage in complex
logical arguments; they must generalize their nascent conceptual understanding to examples that involve unexpected
additional structure; and they must pull together resources from many previous experiences, recognizing that what 2
they learn today is not simply related to what they learned yesterday, but may involve a web of information from
many previous courses learning is not linear.
The expectations on this abstract list should come as no surprise. How do they impact our students in practice? To
make our discussion concrete, we include a detailed task analysis of an Amp`eres law problem, highlighting common
student diculties. None of the individual diculties will sound overwhelming; once students have had a chance to
address them, they nd the solutions straightforward. Nevertheless, so many ideas come together that, even under
the best of circumstances, many students need to scramble to keep up. Our task analysis suggests the question, Is
the total cognitive load in middle-division courses too high? Synthesis has become so automatic to us that we may
fail to recognize how new it is for our students. Are we giving them sucient resources to be able to do everything
we ask of them?
In Sec. II we give a broad discussion of two major dierences between the way mathematicians and physicists
use mathematics. In the rest of the paper we explore the consequences for students as they try to bridge this gap
by applying what they have learned in mathematics courses to physics courses beyond the introductory level. In
Sec. III we introduce a standard Amp`eres law problem and discuss typical textbook solutions. Section IV discusses
a detailed task analysis of this problem. In Sec. V we return to the broader theme of the capabilities that we want
our middle-division students to be constructing, and suggest that designing curricula that pay explicit attention to
the transition students need to make may help more students be successful. Section VI briey links our work to the
work of others.
This paper does not pretend to report on education research (but see Ref. 10). We have not done careful studies to
learn how prevalent particular student problems are. Nor have we systematically compared the results of educational
interventions that we suggest here to either traditional methods or those based on education research. It would
be impossibly cumbersome for us to write, or the reader to read, properly qualied sentences; we ask the readers
indulgence. When we write, Students think . . . , we really mean, In our many years of working with students,
faculty, and TAs from a diverse set of institutions, we suspect that at least some, and probably a signicant number of
students may think . . . , and that regardless of what they are actually thinking, if we tailor our educational interactions
with them as if they think . . . , then apparently, it seems to help them learn more and/or they at least appear to be
more satised with their learning experience, without our actually assessing that.
In all seriousness, we hope that what we suggest will not only provide numerous fruitful questions for education
research but also inspire traditional educators to look more closely at what is happening in their classrooms.
II.
MATHEMATICS IS NOT PHYSICS
Mathematicians are responsible for much of the lower-division education of our students, and yet mathematicians
and physicists view mathematics in inherently dierent ways. This contrast in perspective has dramatic repercussions
when our students try to apply the mathematics they have learned in the physics classroom, as illustrated in Sec. IV.
We have found that many of the dierences between the problem solving strategies of mathematicians and physicists
t under two main headings.
A.
Physics is about things
In our conversations with physics and mathematics faculty the most striking dierences arise from the fact that
physics is about describing fundamental relationships between physical quantities whereas mathematics is about rig-
orously pursuing the consequences of sets of basic assumptions. Conventional lower division mathematics is primarily
about teaching students to manipulate mathematical symbols according to well-dened rules without asking about
the interpretation of these symbols. Calculus reform has helped somewhat, but even application-based curricula that
are designed to stress multiple representations have limited time to focus on the interpretation of results. Rightly,
interpretation is the realm of science. As professionals who have spent our careers interpreting equations and nding
ways of representing information, the rst question that we ask about a new formula is, What physical quantities do
the