Paradigms in Physics: A new upper-division curriculum

Corinne A. Manogue,

Philip J. Siemens,

Janet Tate, and Kerry Browne

Department of Physics, Oregon State University, Corvallis, Oregon 97331

Margaret L. Niess and Adam J. Wolfer

Department of Science and Mathematics Education, Oregon State University, Corvallis, Oregon 97331

共

Received 28 December 2000; accepted 21 January 2001

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We describe a new curriculum for the final two years of a B.S. program in Physics. Case studies in
the junior year provide concrete examples or Paradigms as pillars to support systematic Capstone
lectures in the senior year. In each of nine three-week Paradigms, the junior progresses from a
descriptive lower-division understanding to an advanced analysis of a topic defined by phenomenon
rather than discipline. Students generally view the new format with favor. They are better at
visualization and make important connections among physics disciplines. Independent assessment is
ongoing. ©

2001 American Association of Physics Teachers.

关

DOI: 10.1119/1.1374248

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I. INTRODUCTION

In the fall of 1999 two dozen Physics and Engineering

Physics majors at Oregon State University

共

OSU

兲

plunged

enthusiastically into their junior year, which was also the
third year for faculty teaching a new curriculum of upper-
division studies in physics. They participated in a series of
nine three-week intensive case studies taught with a variety
of classroom methods and topics that bear scant resemblance
to the courses followed by OSU juniors a few years ago.
Senior-year students, who became the second class to gradu-
ate from the new program, began a set of survey courses
which rounded out and knit together the junior-year ex-
amples from several viewpoints. These senior courses corre-
spond more closely to the traditional disciplines and meth-
odology of upper-division physics courses, as do the
laboratory-lecture courses in electronics and optics that run
alongside through both years. A senior thesis or engineering
project completes the undergraduate training of these aspir-
ing scientists.

The experimental curriculum is structured to help students

organize their own knowledge in ways that parallel the pro-
fessional’s organizing strategies. It is intended to remedy nu-
merous drawbacks of the conventional approach by using a
variety of pedagogical techniques, applying insights into the
cognitive structures that are being constructed by advanced
students. While some of these techniques are inspired by
those which have been successful in lower-division and pre-
college physics instruction, many are new. Upper-division
students must deal with problems of far greater complexity
and must learn to see patterns which cross the boundaries of
traditional physics subdivisions.

This narrative is primarily an account of the intentions,

experiences, and observations of the faculty who planned
and implemented the new curriculum. Many of our impres-
sions are anecdotal, no doubt deserving the skepticism of a
critical reader. However, our students’ progress has been
monitored by independent experts in the teaching of science,
eager to observe how the newly adapted methods play out at
a level of instruction for which little documented experience
is available. A summary of the evaluation by these education
researchers

共

MN and AW

兲

is included as Sec. IV of this

report.

A. Challenges for a new curriculum

The old upper-division physics curriculum at OSU was

typical of most similar institutions. Each of several subdisci-
plines was taught separately as a sequence of courses two to
three quarters in length. Two sequences

共

Electronics, Optics

兲

were laboratory based, the others theoretical, applying ab-
stract principles to deduce concrete examples. Some theoret-
ical sequences

共

Electromagnetism, Classical Mechanics, and

Mathematical Methods

兲

were taken in the junior year and

some

共

Quantum Mechanics and Thermal Physics

兲

in the se-

nior year. Students had to master each topic as it arose, since
it arose only once. Individual faculty members typically
taught an entire sequence independently, and there was little
opportunity to bring out the underlying unity of the various
subdisciplines. Because students had to take several se-
quences in parallel, they frequently struggled when they en-
countered difficult material simultaneously in several differ-
ent sequences. The level of difficulty in the junior-year
courses was similar to the level in the senior year, making
the junior year a significant barrier; locally, it was referred to
as the ‘‘brick wall.’’ We suspect that this basic scenario de-
picts a national problem.

B. Response to the challenges

Our solution has been to introduce a two-tiered upper-

division course of study involving a nonstandard division of
topics compared to the traditional subject areas. This allows
students to consider the main topics twice: first emphasizing
analytical skills and a multi-faceted approach to problems,
then emphasizing deductive skills and disciplinary integra-
tion. The junior-year curriculum involves a sequence of case
studies of paradigmatic physical situations and conceptual
examples, some involving two or more subdisciplines. We
thus equip students with concrete examples on which to base
an abstract deductive framework. The senior year consists of
more advanced courses, each of which consolidates an indi-
vidual physics subdiscipline, in addition to electives offering
introductions to some major areas of current research.

We aim to improve students’ comprehension by cultivating

their analytical and problem-solving skills, to provide
bridges between the content of different subdisciplines, and
to offer a more varied and flexible learning experience. Since
we see our solution as rooted in fundamental aspects of the

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Am. J. Phys. 69

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, September 2001

http://ojps.aip.org/ajp/

learning experience, we may hope that our results and meth-
ods may also prove to be useful in other allied disciplines,
e.g., mathematics or chemistry.

Our new curriculum for junior-year physics majors con-

sists of a sequence of nine courses, each lasting about three
weeks and meeting for seven hours per week. Each course is
a case study involving a single physical situation or simple,
conceptual principle. We call these case studies Paradigms.

The Paradigms serve a dual function. The topics, shown in

Table I, were chosen to span many of the principal examples
usually developed in the deductive subdisciplines, but with-
out restriction to the ideas and strategies of a single subdis-
cipline. In addition, they emphasize the development of ana-
lytical and problem-solving skills, often involving integrated
observational and/or computer laboratories. For example, in
the unit on Waves in One Dimension, the students study
traveling and standing waves in a coaxial wave guide. They
make experimental observations and analyze them math-
ematically, testing the limits of an ideal model. After study-
ing pulses and their resolution into normal modes in this
nondispersive context, they compare the propagation of
quantum Schro¨dinger waves in computer simulations.

The Paradigms are followed by six single-term Capstone

courses that systematically present the usual deductive sys-
tems of physics. The topics and sequences are shown in
Table II. The format is condensed compared to our previous
year-long sequences in these disciplines, since the students
are already familiar with many of the central examples. For
example, the Capstone in Classical Mechanics uses topics
from the Paradigms such as harmonic and anharmonic oscil-
lations and central forces as illustrative examples when dis-
cussing the Lagrangian and Hamiltonian formulations. Dur-
ing the senior year we also offer a selection of specialty
courses surveying the phenomena and methodology of mod-
ern research areas, such as solid state physics, nuclear and
particle physics, advanced optics, and computational physics.
These are topics for which there was insufficient time in our
old curriculum.

The inherent flexibility of our curriculum is a significant

asset. Students pursuing variations on the basic Physics de-

gree, degrees in related fields, or interdisciplinary degrees
can pick appropriate topics without being locked into year-
long commitments. For example, our Engineering Physics
majors choose a subset of the Paradigms and Capstones ap-
propriate to their engineering specialization. In addition, a
number of nonphysics majors and graduate students

共

chem-

ists, mathematicians, geophysicists, oceanographers, and en-
gineers

兲

take some of our upper-division courses; the Para-

digms can assist them by addressing specific needs they may
have, or specific deficiencies in the background they need for
a senior Capstone course. Students who have difficulty with
a particular topic may be able to revisit or retake that Para-
digm the following year without getting out of step with the
whole program. And the one-quarter senior-year deductive
Capstone courses make good entry-level courses for graduate
students with isolated weaknesses in their background.

The two-tiered approach to the upper-division curriculum

addresses the needs of physics students from the most basic
to the most applied curricula. Because students experience
the broad sweep of physics earlier, they can begin to formu-
late realistic career goals in time to apply for relevant sum-
mer internships or other jobs between their junior and senior
years. In addition, they can tailor their experiences during the
senior year to their particular career goals. Our graduate-
school-bound students encounter basic quantum mechanics
and thermal physics early enough to help on their Graduate
Record Examinations. Our applied students are able to par-
ticipate in the co-op program of off-campus internships
while still maintaining a coherent academic program. Our
courses may be particularly helpful for students who aim to
use their B.S. in Physics as part of their pre-service training
for careers as high school physics teachers. We believe our
integrative, paradigmatic approach will improve the training
of high school teachers and offer them an up-to-date model
for instruction.

C. Context for implementation

Our institution, Oregon State University, is a typical

medium-sized

research

university.

Our

introductory

calculus-based physics sequence is primarily a service course
for engineers and students from other sciences, but also pro-
vides the entrance to our undergraduate major programs in
Physics and Engineering Physics. Many of our Physics ma-
jors transfer from community colleges as juniors with basic
math courses and only a single year-long introductory phys-
ics sequence. As a result of these circumstances, most of the
material in our curriculum for majors is crammed into the
last two years. Few of our Physics majors have adequate
opportunities to develop their analytical and problem-solving
skills before they enter the upper division. Efforts to reform

Table I. Case studies offered in the junior year. Detailed syllabi for the nine
Paradigms are available at our web site: http://www.physics.orst.edu/
paradigms.

Fall quarter

Winter quarter

Spring quarter

Static Vector Fields

Waves in One Dimension

Periodic Systems

Oscillations

Quantum Measurements

and Spin

Rotational Motion

Energy and Entropy

Central Forces

Reference Frames

Table II. Survey courses offered in 1999–2000. Detailed syllabi are available at our web site.

Fall quarter

Winter quarter

Spring quarter

Capstone

共

junior year

兲

Quantum Mechanics

Electromagnetic Theory

Classical Mechanics

Capstone

Specialty

Mathematical Methods in Physics

Thermal and Statistical Physics

Nuclear and Particle Physics

Capstone

Specialty

Optics

共

with laboratory

兲

Optics 2

共

with laboratory

兲

Optics 3

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with laboratory

兲

Specialty

共

graduate level

兲

Specialty

Advanced Mechanics including Chaos

Computational Physics

Solid State Physics

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Am. J. Phys., Vol. 69, No. 9, September 2001

Manogue et al.

the introductory curriculum are under way here, as at other
institutions including community colleges, but even under
the best of circumstances it will be some time before such
changes can be implemented by all the community colleges
that prepare students for our program. Therefore, we have
decided to focus on what we can do now with the upper-
division curriculum.

The single most important requisite for success in a

change of this magnitude was the unanimous support of the
Physics Department. Those who were not directly involved
in the project helped by providing release time, advice and
suggestions, temporary postponement of some other depart-
ment priorities, and a wide variety of other support for the
project. We found an effective mechanism for obtaining pro-
ductive input from the entire department in the early plan-
ning stages which helped to build consensus. As part of the
process of determining an appropriate rearrangement of con-
tent for our new courses, experienced faculty recorded the
subject matter of our old curriculum in small natural chunks
on index cards color-coded by discipline. After these cards
had been rearranged by the committee into a tentative plan,
each faculty member in the department was invited, indi-
vidually, to consider the proposed curriculum. Their sugges-
tions for change were instantly converted into a rearrange-
ment of the cards. Some rearrangements resulted in apparent
improvement; others revealed disadvantages of the sugges-
tion. Eventually this departmental game of Solitaire con-
verged on an optimized curriculum that was acceptable to
all.

Financial support has been essential to develop the new

curriculum. External funding from the National Science
Foundation supported the external evaluation activities as
well as summer salary for faculty preparing instructional ma-
terials and experimenting with new pedagogical strategies.
Such funding may not be necessary for institutions adopting
our program after its development is completed.

Internal funding is another matter. Initially, we could see

no way to phase in our new curriculum more slowly than we
did—all the junior courses changed in the first year, the se-
nior courses in the following year. Internal funding from the
department and higher levels of the Oregon State administra-
tion have supported release time and acquisition of curricular
materials. In particular, it was critical that faculty teaching
the Paradigms for the first time did not have other simulta-
neous teaching duties.

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All of the faculty directly involved in

teaching are regular research faculty with the substantial load
of commitments which that entails. One of us

共

CAM

兲

is also

a coordinator for the project.

兴

It is our view that this amount

of support will turn out to be a comfortable minimum for
institutions that opt to make the change rapidly, as we did.
Meanwhile we are looking for ways in which other institu-
tions might be able to make the change more gradually and
with fewer resources.

Our external budget also has included funding for a single

teaching assistant for the Paradigms classes. During the ini-
tial years, the efforts of outstanding and dedicated physics
graduate students working with us

共

on part-time appoint-

ments

兲

have been critical to the program’s function. While

our own efforts were focused on the development and imple-
mentation of the new program, their attention centered on the
students. One of the challenges we need to address is how to
run the laboratory and small-group activities described below
without support from teaching assistants.

II. CONTENT OF THE CURRICULUM

A course of studies can help a student become a physicist

in many ways. Some of the student’s needs are specific to the
discipline of physics; others are common needs shared by all
beginning scientific professionals.

A. Specific physics content

The basic principles guiding the choice of Paradigms top-

ics are important to emphasize. First, we chose simple ex-
amples, with only enough complexity to adequately develop
the needed concepts. Second, we chose central concepts and
examples which lie at the heart of physics—concepts which
professional physicists use often. Third, we chose examples
and concepts common to more than one of the traditional
subdisciplines of physics or to an important area of applica-
tion or research. Finally, we sought to include enough sub-
ject matter from each subdiscipline to provide a sufficient
basis for the senior courses.

All of the Paradigms build on a basic knowledge of clas-

sical physics acquired in a traditional introductory calculus-
based physics course; some also presume an introductory
course in modern physics, including elements of quantum
physics and special relativity. Mathematics prerequisites in-
clude calculus through vector analysis and an introduction to
ordinary differential equations. However, we have found that
the majority of our students benefit when we revisit a num-
ber of these mathematics topics as part of the Paradigms
sequence.

1. Content by discipline

An illuminating way of viewing our reorganization of the

curriculum is to see how the main topics of the traditional
courses are distributed among the Paradigms and Capstones,
shown in Table III. First, compare the last row, topics not
included in the new curriculum, with the last column, topics
not included in the old curriculum. We see that nearly all the
content of the previous curriculum is included in the new.
Although there are certainly differences in relative weight
assigned to individual topics, we might well have made these
changes within the traditional curriculum to reflect the needs
and prospects of our students. For example, we now ap-
proach

Coriolis

forces

with

extensive

computer

visualization;

spin is covered more thoroughly and colli-

sions receive less emphasis. The additional professional
skills and interdisciplinary training of the new curriculum
have been accommodated without loss of traditional topics.

Table III does not show the additional specialty courses

which now enrich the senior year. This year, for example, we
offered ten-week surveys of subatomic and solid state phys-
ics, in addition to the courses in computational physics, la-
sers, and wave guides carried over from our previous cata-
log. Nor does it show the junior-year electronics laboratory
and senior thesis, both continuations of previous successful
components of our majors’ curriculum.

2. Order of presentation

A second look at Table III shows how substantial material

from each discipline appears in the Paradigms. By compari-
son, under the old system, our students had to wait until their
senior year for basic quantum and statistical concepts. An-
other such change, not shown in the table, shifted most cir-
cuit theory from spring electromagnetism lectures to the pre-
ceding fall’s electronics lab. Conversely, advanced topics

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Manogue et al.

such as Lagrangian formalism, radiation, and Bessel func-
tions no longer scourge the juniors, but now are reserved for
the better prepared seniors.

The sequence in which the Paradigms are offered is influ-

enced by constraints on our students’ background and par-
ticipation which may not apply to other institutions. First,
our university accommodates many transfer students from
local two-year community colleges, who have studied rea-
sonable courses in classical physics but have little or no
background in modern physics. Their mathematics back-
ground may also be weak. These late arrivals take our Intro-
ductory Modern Physics course and sometimes vector calcu-
lus and/or differential equations alongside their first term of
Paradigms; we accommodate them by appropriate schedul-
ing of the order in which the Paradigms courses are offered.
Also, some of our Engineering Physics students participate

in a five-year co-op program which takes them off campus in
the spring of their third and fourth years; these students can-
not take the spring courses until their fifth year, so we have
scheduled advanced topics in that term. Most universities
and colleges will have constraints of their own; the flexibility
inherent in the Paradigms approach should allow other insti-
tutions to find a suitable sequence should they elect to adopt
our approach. This flexibility should also make it easy to
deploy our curriculum in a semester-based setting. We are
exploring sequences which may be appropriate to the small-
est schools which alternate upper-division courses on a two-
year cycle.

Each Paradigm is offered as a separate course for two

quarter-hours of credit

共

versus three quarter-hours for our

traditional lecture classes

兲

, in order to give students flexibil-

ity in arranging their schedules and choice of experiences.

Table III. Principal topics of previous traditional curriculum

共

columns

兲

and new curriculum

共

rows

兲

Course unit

Mathematical

Methods

Classical Mechanics

Electromagnetism

Quantum Mechanics

Thermal

Physics

Not included

in old courses

Vector Fields

Vector calculus
Visualization

Statics, 3D geometry
Vector theorems

Delta functions

Computer
Visualization

Oscillations

Fourier series,
integrals
Complex
exponentials

Small oscillations
Anharmonic
pendulum

LRC circuit
Resonance

Orthogonal expansions
State space

State space

Lab component

Energy & Entropy

Probability
Thermodynamic
potentials

Statistical inference

Waves in 1
Dimension

Normal mode
expansions

Vibrating string

Standing and
traveling waves
Coax cable

Eigenmodes
Wave packets

Lab component

Quantum
Measurements
and Spin

Matrix algebra
Representations
Basis transforms

Hamiltonian

Eigenvalues, probabilities
Repeated measurements
Spinors, spin 1/2

Measuring
probability

Bell inequalities

Central Forces

Legendre functions
Separability

Angular momentum
conservation
Kepler, others

Angular momentum
conservation
Spherical harmonics

Periodic Systems

Coupled oscillations

Band structure

Distribution
functions

Phonons
Bloch waves

Rotational Motion

Tensor notation

Rigid rotation
Inertial tensors

Tensor notation

Basis rotations

Lab component

Reference Frames

Rotating frames
Relativity

Relativity
Lorentz transf.

Lab component

Math Methods
Capstone

Partial differential
equations
Complex analysis

Green functions

Mechanics
Capstone

Formal Lagrange
and Hamilton
methods

Electromagnetism
Capstone

Dynamics, media
Waves, radiation

Optics Capstone

Boundary conditions

3D waves
Coherence

Quantum Capstone

Atoms, fine structure
Angular momentum
coupling
Scattering

Thermal Capstone

Statistical
mechanics and
applications

Not included
in new courses

Time dependent
perturbation
theory

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Manogue et al.

However, the order in which the courses are taken cannot be
entirely arbitrary, since some of the units build on knowl-
edge or skills acquired in others. For example, the Paradigm
on Oscillations develops methods of Fourier analysis that are
an essential background for the Paradigm on Waves in One
Dimension.

We begin each of the first two quarters with a week-long

Preface, discussed below. The last week of the spring term is
the Postscript, a finale to the junior-year series involving
presentations from senior scientists

共

in the first year, for ex-

ample, we included an astrophysicist, a geophysicist, and a
materials scientist

兲

on how they use some of the Paradigms

concepts in their own research.

B. Professional preparation

Progress from student to professional is marked by a series

of changes in mindset regarding the individual’s role in ac-
quiring knowledge. The student must ultimately learn to ad-
dress new and old knowledge directly, free of mediation by
the teacher. The nascent scientist must learn how to record
new results to preserve them and make them available to
other researchers. And each inquirer must learn how to
progress, not only by following experienced guidance, but by
pooling insights with peers, and eventually by following
one’s own counsel.

Our Physics majors need to acquire several skills that are

common to a range of related scientific professions. They
need to know how to approach a problem and solve it. They
need to access knowledge resources knowledgeably, to em-
ploy computers confidently, to analyze data quantitatively, to
model and approximate appropriately. The Paradigms ad-
dress all these needs.

1. Role definition and modeling

The student whose goal is to satisfy the teachers must

become the scientist whose goal is to acquire knowledge.
The intense involvement required by the Paradigms courses
is meant to facilitate this transition. Drawing from multiple
text sources—textbooks,

MAPLE

scripts, notes—the Para-

digms direct the students’ attention to comparable content in
divergent notation, so that they quickly learn to adapt. One
very successful outcome is that the Paradigms students take
the multitude of different notations they encounter in stride,
unlike their old-curriculum counterparts. This outcome is ex-
pected as the students turn their attention from the bearers of
the information they are learning, to the information itself. It
may indicate that they are internalizing their knowledge as
they acquire it.

Another way the Paradigms help students take charge of

their own learning is by providing enhanced opportunities to
confront natural phenomena. In the laboratory exercises and
other concretely visualized examples, each student gains a
repertoire of immediate experiences. As these are analyzed,
they become useful objects of analogy for future reasoning
about unfamiliar or inaccessible phenomena. By insisting
that each student draws conclusions from the experiences
provided in each Paradigm, we begin a habit of exercising
judgment in a professional context. Many students welcome
this opportunity but, at least initially, some are hesitant to
advocate their own interpretations. They often express anxi-
ety about revealing their opinions, which evidently has not
been encouraged in some previous situations.

2. Problem-solving approaches

The Paradigms are meant to form a link in an evolutionary

chain as the students’ way of solving problems adapts to
their changing role. By the end of their introductory courses,
students are accustomed to guided discovery: they follow a
path indicated by an instructor to gain the prescribed per-
spective.

They

are

beginning

learn

peer-assisted

discovery:

they discuss common problems with other stu-

dents, for example by working together on homework.
Lower-division students at a nearby university who are fol-
lowing the Inquiry Method have a head start on this process.

In the Paradigms, we encourage the further development of
peer-assisted discovery with frequent group activities, in-
cluding collaborating in the laboratory, sharing a computer
screen in a visualization exercise, and gathering around a
whiteboard in a classroom discussion of a theoretical prob-
lem. Facility in the peer-assisted mode of discovery, which
dominates most scientific workplaces, is essential for profes-
sional success.

After the Paradigms, our Physics majors also have an op-

portunity to sample independent discovery as they do re-
search for their senior theses. This last transition is often
completed at the Ph.D. or postdoctoral level.

3. Scientific skills

By drawing information from a variety of sources in a

single course, a Paradigm helps the students develop the
practical tools of scientific literacy. They also learn to use
computers, both for numerical and symbolic manipulation as
well as for accessing and transferring information via com-
puter networks.

In addition to accessing external resources, Paradigms stu-

dents also develop reasoning skills they need in their profes-
sions. They learn to carry out quantitative confrontation of
observational data with theoretical expectations. They are
encouraged to analyze the conditions under which a model or
approximation is appropriate, and to draw conclusions from
their analysis. These abilities are a necessary part of every
scientist’s repertoire.

4. Documentation and communication

Another habit cultivated in the laboratory-based Para-

digms is recording the students’ experiences. They learn to
document their observations, both quantitatively and qualita-
tively. They learn the value of this documentation when they
return for more advanced analysis of earlier observations.
They also learn to record their analyses, creating a paper trail
of both intermediate numerical results and intermediate steps
in the reasoning process. Finally they learn to sort out their
results into an organized set of conclusions, which they
record in analytical reports. These habits are needed by every
professional.

In one of the later Paradigms, students are expected to

report on at least one journal article related to the course
material from the American Journal of Physics. Most stu-
dents choose to make an oral presentation as well as the
required written report. The oral presentation is a first step
toward professional communication in science, and the re-
search activity opens up a new resource for them. Most stu-
dents are not readers of AJP, and they learn that it is acces-
sible to them. They may also observe that it is very
affordable for students.

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Manogue et al.

III. METHODS OF TEACHING

Our curriculum incorporates new developments in peda-

gogy in many ways. We reorganize the order of presentation
of topics and the way they are grouped. We incorporate a
wide variety of activities both in the classroom and in the
students’ preparation. And we employ an array of devices for
evaluating the students’ performance.

A. Instructional organization

The most striking way in which our new curriculum dif-

fers from the previous one is that it fundamentally reorga-
nizes how the content is presented. Two sweeps cover the
upper-division material instead of one. The new, extra layer
of case studies groups topics by affinity of the phenomena
observed and concepts employed rather than the equations
invoked. Mathematical sophistication is developed in a con-
text of Physics applications. And many of the Paradigms are
organized into learning cycles of hypothesis and observation.

1. Case study format

The Paradigms replace previous parallel-track lecture

courses, each meeting three hours per week for a ten-week
term.

Instead the Paradigms run serially, each Paradigm

lasting three weeks at seven class hours per week. They meet
for one hour on Monday, Wednesday, and Friday and two
hours on Tuesday and Thursday. The students have the ad-
vantage of concentrating on one theory course at a time,
instead of dividing their attention among the traditional two
or three. Several have commented that, just as they tire of a
topic, it changes. On the other hand, students must develop
the flexibility to change topics frequently, with little elapsed
time to become accustomed to new facts, methods, and con-
cepts.

Each Paradigm typically draws subject matter from sev-

eral traditional disciplines, as illustrated in Table III. The
resulting cross-disciplinary connections, a characteristic
strength of the case-study method, make it especially valu-
able in advanced studies to counter the fragmentation that so
often accompanies specialization. In addition, we hope that
the repeated example of assembling knowledge from varied
sources can help students develop habits of resourceful prob-
lem solving.

2. Redistributed mathematical content

One of the perennial problems in designing any upper-

division physics curriculum is the appropriate placement of
mathematical methods

共

beyond the calculus sequence ordi-

narily taken from a mathematics department

兲

. If the math is

taught separately, then students have trouble envisioning
how they will use it. They focus on extraneous aspects and
find the techniques difficult to remember when they need
them, sometimes a year or more later. In contrast, when the
math is offered in context, the primary focus on physics
makes it difficult for students also to see the underlying pat-
terns of the mathematics. They have trouble generalizing to
similar mathematical situations when the physical contexts
may appear radically different. Weaker students are often
overwhelmed by having to learn the mathematics simulta-
neously with the physics.

Our double-tiered structure allows us to use both place-

ments strategically. During the junior year, the math is taught
primarily in context by incorporating it directly into relevant
Paradigms; while at the beginning of the senior year, we

teach a separate course in mathematical methods. By then,
students have seen eigenfunction expansions in the context
of classical oscillations as well as in the quantum hydrogen
atom and they are eager to see what these two problems have
to do with each other. They have considerable experience
with simple examples of the methods, learned in context, and
are ready to ask questions like: How do I know when I can
use this technique and how do I recognize when it will fail?

There are a few basic mathematical methods common to

many of the Paradigms which need to be highlighted before
the senior year. To accommodate this need, we use the Pref-
aces, a week at the beginning of the term, before the Para-
digms begin in earnest. In the fall term, the Preface is used to
ensure that all of the students have accounts in the computer
lab and to get them started using

MAPLE

, a computer algebra

system that is used as an instructional tool in many of the
classes.

MAPLE

labs are then used to help students visualize

some basics of complex functions and power series needed
for the first term’s Paradigms. In the winter term, the Preface
is used to explore rotations as preparation for the Paradigms
on Spin and Central Forces.

3. Incorporating modern viewpoints

We have taken advantage of the restructuring of our cur-

riculum to present some traditional topics from a modern
viewpoint.

The winter quarter Paradigms begin our first formal pre-

sentation of quantum mechanics. In the Preface, we use the
rotation matrices as a simple, finite-dimensional exercise for
examining concepts such as eigenfunctions and eigenvalues;
Dirac bra-ket notation is introduced. In the Paradigm on
One-Dimensional Waves students solve explicitly for the
eigenstates of the one-dimensional particle in a box. Then, in
the Paradigm on Quantum Measurements, the students
plunge immediately into a detailed study of the Stern–
Gerlach experiment, where spin is used as a vehicle to teach
the quantum postulates. The only mathematical manipula-
tions required are small-dimensional matrix calculations, so
that students can focus on basic concepts. Finally, in the
Central Forces Paradigm, students engage in an investigation
of the hydrogen atom, deriving and then solving problems
with the eigenstates. The fact that students turn spontane-
ously to bra-ket notation in their problem solving shows us
that our strategy of alternating between the matrix and wave
function representations of quantum mechanics is a powerful
one.

The Energy and Entropy Paradigm

共

thermal physics

兲

be-

gins

with

explicit

discussion—with

numerous

examples—of macroscopic thermodynamics, so that the
meaning and usage of these time-honored quantities is
clearly laid out. In this process thermodynamics is presented
as the quantum mechanics of macroscopic systems in which
thermodynamic state functions are defined as quantum aver-
age values and the required probabilities are quantum prob-
abilities. But wave functions are not measurables and quan-
tum probabilities are not immediately accessible. In lieu of
this, a minimum bias

共

maximum entropy function

兲

postulate

is used to invert relevant macroscopic knowledge to infer the
unknown probabilities. This Baysean process yields prob-
abilities consistent with the few macroscopic constraints
known about a given system. Partition functions are an im-
mediate by-product. The circle is then closed when students
are shown that the quantities and thermodynamic laws ob-
tained in the inferential approach are identical to those of

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Am. J. Phys., Vol. 69, No. 9, September 2001

Manogue et al.

macroscopic thermodynamics introduced at the beginning of
the course. In this way students see that statistical physics is
the quantum mechanics of macroscopic systems. In a few
examples, partition functions for simple microscopic models
are constructed and the observable thermodynamics implied
by them are studied in qualitative and quantitative lab experi-
ments.

Standard approaches to electrostatics typically introduce

the electric field first, use line integrals to obtain the potential
at a particular point, and lastly employ gradients to arrive
back at the vector field, completing the circle. We complete
the same circle, but begin instead with the scalar potential
familiar from voltmeters and oscilloscopes. Explicit attention
must be paid to helping students visualize the scalar fields by
exploiting the power of three-dimensional computer graphics

共

we employ color to represent the value of the field

兲

. The

extra attention pays off as students come to view both the
electrostatic potential and the electric field as fundamental
properties of space.

4. New juxtapositions

In several other Paradigms, we have found that the un-

usual juxtaposition of topics has introduced a new synergy
into our courses. For example, the Paradigm on Central
Forces begins with a treatment of the classical case of orbits
around a gravitational point source and then goes on to ex-
amine the hydrogen atom in the context of quantum mechan-
ics. It is far easier to highlight the similarities and differences
between the classical and quantum concepts of angular mo-
mentum when they follow each other by days rather than
months. Students who have just seen the value of an effec-
tive potential in finding classical turning points are primed
also to see its role in the radial equation for the hydrogen
atom.

Inertial frames are at the heart of special relativity. Yet the

use of Einstein-like thought experiments involving rocket
ships leads to an intuitive notion of inertial frame which is
really local—perfect for the extension to general relativity,
but subtly different from the Newtonian concept taught in
introductory physics. The juxtaposition of noninertial

共

rotat-

ing

兲

frames and inertial frames

共

special relativity

兲

in the last

Paradigm forces students to confront these subtleties. Is the
surface of a nonrotating earth an inertial frame? The answer
depends on one’s point of view. In many of the Paradigms,
puzzles like these have invigorated our own conversations as
well as those with students.

5. Exploiting learning cycles

A variety of studies of learning at the secondary and

lower-division post-secondary levels have recognized a se-
quential pattern, the learning cycle.

5–7

It seems natural to

inquire whether this pattern persists as the learner’s under-
standing approaches the state of the art.

We utilize a schematic learning cycle in planning several

of the Paradigms. In particular, two of the Paradigms have
been designed to conform to a formal pattern based on a
five-stage learning cycle: interest, experience, analysis, ex-
periment, integration. In the Paradigm on Oscillations, the
cycles form a nested structure which will be described else-
where. In the Paradigm on Rotational Motion of Rigid Bod-
ies, overlapping learning cycles are employed. Two parallel
cycles span the three-week experience: a study of rotational
dynamics, and a study of tensors. The first experience of the

dynamics cycle is itself a cycle involving the construction
and characterization of a rigid body and its small-amplitude
oscillations about fixed axes; the second subcycle begins
with the free rotations of the same body and concludes with
gravity-driven precession of the rotor, introducing the Euler
equations of motion in the analysis. The parallel cycle begins
with a subcycle introducing inertial tensors and rotation ma-
trices, and concludes with a subcycle treating their alterna-
tive characterizations and eigenrepresentations. This struc-
ture seems quite effective, with the math component
providing tools for understanding the physical systems, and
the laboratories providing motivation and examples for the
math. The students show a surprising level of interest and
enthusiasm for subject matter traditionally considered dry
and arcane. An ongoing challenge is to improve the primitive
experimental techniques.

6. Advanced courses

After the Paradigms, our curriculum returns to the deduc-

tive didactic with the Capstones’ overview of the traditional
disciplines. Each of these analytical courses gives its own
interpretation of the examples forming the Paradigms, re-
flecting a way of thinking which is as characteristic of the
discipline as it is of the teacher. The varying ‘‘takes’’ on the
experiences of the junior year give the students a variety of
patterns to incorporate into their own conceptual structures.

The alternation between the introductory and advanced

survey courses, versus the case studies of the Paradigms and
the specialized senior thesis, gives the student’s undergradu-
ate experience a rhythm reminiscent of a long-term learning
cycle. The final interpretation of the experience takes place
as each student chooses and plans a post-graduate career.

B. Instructional activities

Extensive research at the lower-division level has shown

that, by and large, students are not learning what faculty
think they are teaching.

2,8–11

Lower-division curricula that

incorporate interactive experiments

or adopt the experimen-

tal method to the exclusion of the lecture

have demon-

strated success in avoiding or clearing up misconceptions.

1. Classroom methods

To help address this issue at the upper-division level, the

weekly schedule of the Paradigms has deliberately included
multi-hour blocks of time to allow us to employ a variety of
teaching methods. These methods include integrating labora-
tory investigations into the instruction—both computer simu-
lation laboratories

共

e.g.,

CUPS

and

SPINS

兲

in situations for

which actual experiments are not possible, and also real ex-
periments that allow students to discover new information or
gain first-hand experience with concepts encountered in the
classroom. Influenced by successes in the lower division,

2,15

we have tried several strategies for collaborative small group
activities. Our main efforts involve both guided

MAPLE

worksheets

and interactive problem-solving in small

groups.

Direct laboratory observations are a mainstay integrated

into most introductory physics courses. Expert-level instruc-
tion in advanced laboratory techniques often features an in-
tegrated theoretical component; for example our electronics
course includes circuit theory and a phenomenological de-
scription of semiconductors, and our optics course reviews
the propagation of electromagnetic waves at boundaries. But

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Am. J. Phys., Vol. 69, No. 9, September 2001

Manogue et al.

advanced theoretical courses usually rely on descriptions of
observations with occasional lecture demonstrations. In sev-
eral of the Paradigms, we have found it useful for the stu-
dents to observe and interact with simple systems which ex-
hibit the advanced physics concepts as well as the simple
ones. For example, loaded dice give a starting point for a
discussion of statistical inference and entropy. The anhar-
monic motion of a pendulum provides a concrete application
of Fourier series, while a simple LRC circuit illustrates the
treatment of damped motion by Fourier integrals. One-
dimensional waves are palpably illustrated on an elastic rope,
then electronically observed in a coaxial cable; these experi-
ences prepare the students to appreciate computer simula-
tions of quantum wave packets. And rigid rotors tacked to-
gether from simple materials allow students to gain a
kinesthetic experience of an inertial tensor while marveling
at the counterintuitive aspects of rotational motion.

An example from the Paradigm on One-Dimensional

Waves illustrates our approach. Here the students encounter
the classic problem of transverse waves propagating without
dissipation in a rope under tension. An interactive lecture
demonstration of standing waves in a rope is introduced, and
the students locate the frequencies of the standing waves,
measure the tension in the rope, and then predict the mass
per unit length of the rope, which they later measure with the
help of a scale and a ruler. This is a vehicle for a discussion
of boundary conditions and superposition and the ideas of
reflection and transmission. The students then work together
in groups of three or four in the laboratory to measure the
speed of propagation of an electromagnetic wave down a
coaxial cable. During the course of this exercise, they natu-
rally encounter the concept of attenuation from an experi-
mental point of view, and the entire laboratory then focuses
on measuring transmission and reflection coefficients with
the added complication of damping. The students seek out
the appropriate equation of motion that correctly describes
the observed damping. They consider what ‘‘weak’’ damp-
ing means and investigate attenuation length. In this context,
they must define ‘‘short’’ and ‘‘long’’ and are confronted
with the fact that any physical quantity must be compared
with another of the same dimension. Finally, they come full
circle: they set up standing waves in this damped system and
must model the expected behavior in

MAPLE

In a Physics Education Research Master’s project,

Katherine Meyer found that effective small group activities
at this upper-division level shared the following characteris-
tics:

they are short, containing approximately three
questions,

they require groups to apply the same techniques
to different examples, allowing students to com-
pare and contrast several cases expeditiously, and

they

are

followed

summary

lecture/

discussion with the instructor.

For example, in the activity which she ranked highest, Linear
Transformations, each group is asked to calculate and then
report to the class the effect of a two-dimensional linear
transformation on a group of representative vectors. As the
class discussion proceeds, someone inevitably asks if the
vectors that are unchanged by the transformation have any-
thing to do with eigenvectors. The class as a whole is aston-
ished that the answer is yes. They have learned how to solve

eigenvalue/eigenvector equations in mathematics classes, but
the geometric meaning has never registered. After this short
experience, it is much easier to convey the role of eigenfunc-
tions in quantum mechanics.

Because collaborative activities require lots of classroom

time, we have been obliged to limit their use to carefully
chosen, critical topics. Interestingly, it is becoming apparent
that the most valuable time for a collaborative activity may
be when a new topic is being introduced, to ensure that the
topic is set in context. If students fail to understand a simple
idea

共

such as the physical meaning of an eigenvector in the

example above

兲

then their learning can come to a complete

halt and subsequent activities are lost to them. A short activ-
ity

共

such as the worksheet on Linear Transformations

兲

can

make it possible for high-content presentations such as tra-
ditional lectures to carry meaning for more students.

Our experience has also uncovered a remarkable synergy

obtained by juxtaposing the ideas of three-dimensional phys-
ics, especially electrostatics; the mathematical skills of vec-
tor calculus; and the visualization capabilities of modern
technology. Using the impressive graphical and algebraic-
manipulation capabilities which are available on

MAPLE

共

and

other similar computer algebra systems

兲

, we have written a

number of guided worksheets which allow the students to
explore the connection between spatial visualization and for-
mulas. These worksheets are incorporated directly into lec-
tures and class discussion sessions which take place in the
computer lab. The approach is different from that of most
current physics texts incorporating computer algebra which
teach students to use technology to solve entire problems, in
that we still rely heavily on solving equations by hand, re-
serving

MAPLE

to enhance students’ visualization skills.

2. Out of class

Much of the time students devote to learning is spent out-

side the classroom. During this time they use many re-
sources: pencils and paper, computers, books, notes, on-line
information, and consultations with instructors and with each
other.

Many students are eager to take advantage of the comput-

er’s facility not only at numerical computations and graphics,
but also at algebra, calculus, and other symbolic manipula-
tions. These early adopters soon begin turning in homework
problems in computer format, with

MAPLE

doubling as a

word processor! However many other students warm more
slowly to computerization, often frustrated by the unintelli-
gent machine’s demands for arcane technical trivia. We en-
courage each student to develop an individual style integrat-
ing computer usage as part of a flexible problem-solving
strategy. We try to insist that all students exercise basic skills
of verbal reasoning and of pencil-and-paper computation as
well as of key computer applications.

The fact that traditional textbooks are not structured along

the same lines as our new courses posed an obvious problem.
The syllabus of each Paradigm refers students to written ex-
positions of the course content. A standard set of textbooks,
adopted to serve both Paradigms and Capstones, is supple-
mented with varying quantities of notes prepared specifically
for the Paradigms. We have chosen traditional textbooks
which have a reasonably modular format, so that sections
can be studied out of sequence. By the middle of the junior
year, the students adapt to blending the contents of sources
with varying notation. We consider it an advantage that stu-
dents cultivate this indispensable skill at an early stage of

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Am. J. Phys., Vol. 69, No. 9, September 2001

Manogue et al.

their careers. An additional advantage is that the same books
are familiar to the students when they reappear in the senior
Capstone courses.

Up-to-date collections of course materials are available

around the clock on the Web, although we are not making
much use of the interactivity of that medium. We do take
advantage of its speedy dynamics by posting solutions of
assigned problems quickly after they have been collected

共

hard copy

兲

for grading.

The peer interactions modeled in the classroom continue

intensely among the students as they prepare their assign-
ments. The Physics Department encourages and facilitates
these interactions by providing the students with a couple of
large rooms where they can study together. Operated by the
local chapter of the Society of Physics Students, this area
houses a collection of useful textbooks, old course notes,
problems and exams, computer terminals, whiteboards,
tables, and sofas where the students can work together. In-
structional faculty and graduate teaching assistants often stop
by to join the discussion; equally often, a strongly interacting
cluster of students erupts from the study area in search of an
instructor who can help with their questions. During the first
year of the Paradigms, the graduate teaching assistants found
their time monopolized by the demanding curiosity of the
undergrads; within a year they learned to share the chores—
and fun—with the faculty instructors.

It has been a challenge to steer a balanced course between

collaborative learning and individual development. We en-
courage working together because of its many advantages in
the learning process and in later workplace situations. How-
ever some students experience the group effort as a tempta-
tion to become passive.

共

A cautionary note: students copy

each others’

MAPLE

worksheets indiscriminately without un-

derstanding the implications.

兲

One way to compensate for

this tendency is to require individual essays interpreting the
collaborative projects. This ensures that each student spends
time reflecting on the experiences and integrating them into
an organized view of the physical world.

C. Evaluation of student performance

Evaluation serves two main purposes: to give the partici-

pants feedback to use in managing activities during the
course, and to record the students’ achievements for their
credit and the instructors’ analysis. The distinction between
these two roles of the evaluation process becomes especially
prominent due to the brief duration of the Paradigms. Only
the quickest students can consistently demonstrate mastery
of early material before the end of the course. Ironically the
early feedback is needed most by the students who are not
yet ready to document their achievements.

1. Self-assessment

The frequency of class meetings makes it important for

students to keep up, while making it hard to provide grading
services in time for the next application of the learned mate-
rial. We provide opportunities for our students to practice
recurring mathematical manipulations on Web-posted ex-
amples with posted solutions. In this way we can reserve
individual grading for more substantial homework exercises.

More importantly, an active classroom provides students

with myriad opportunities to check their own understanding.
Small group activities which require groups to report to the
class as a whole can serve as valuable checkpoints for stu-
dents’ self-assessment. We have found that some of our most

effective small group activities share a common structure:

Each group works with a slightly different example and the
informal follow-up presentations help the students them-
selves to highlight the similarities and differences. Students
are more responsive to Socratic teaching styles in the Para-
digms environment. After just a few weeks, the simple de-
mand ‘‘You tell me what will be on the exam’’ elicits a far
more interesting discussion, and more accurate suggestions
from the students, than in the past.

In the Quantum Measurements Paradigm, students are told

from the first day of class that they will be expected to solve
four basic types of problems: time-independent problems in-
volving spin 1/2, generalizations of these

共

typically to spin

兲

, time-dependent problems involving spin 1/2, and generic

time-dependent problems. Students have many opportunities
to assess their progress toward these goals as these four types
of problems are discussed in class, modeled in computer labs
using a specialized program called

SPINS

and practiced on

homework.

2. Formal feedback during courses

To encourage students with a variety of learning styles, we

use a number of evaluative tools to mark progress in the
Paradigms, ranging from homework problems, through re-
ports on laboratory-related activities, to a cumulative
method: the Inventory of Achievement. Based on the overall
learning goals and strategies of each Paradigm, different
courses use different methods.

Three of the Paradigms utilize laboratory reports as a way

to evaluate the students’ progress. A typical report would
include a description of measurements together with a quan-
titative analysis. The students are asked to test hypotheses by
confronting expectations with experience, and to draw con-
clusions from this comparison. Most are able to do so, when
appropriately prompted. Students found overly prescriptive
lab manuals as unsettling as those which were too open-
ended; we are learning to find the right mixture.

Two of the Paradigms provide students with a list of about

a dozen announced goals to be documented during the
course. These goals are used to provide a running evaluation:
an Inventory of Achievement. Corrected work is returned to
the student with annotations, along with a sheet evaluating
progress toward the goals. Eventual documentation of full
accomplishment of all goals gives a top grade; goals not or
only partially met translate to a lower course grade. In this
evaluation scheme students are not penalized for being slow
to catch on, since only their ultimate achievement is re-
corded. But they still get feedback that relates directly to
their grades, which seems to be important to many of our
students.

3. Rhythm of feedback

As faculty, we have years of experience

共

including our

own schooling

兲

with the traditional schedule, so that we

spontaneously encourage a familiar rhythm of weekly home-
work, review, midterms, and exams. This is not so with the
current mode of the Paradigms. A significant problem in the
first year was to find and establish a natural rhythm for these
intensive courses. Experience in succeeding years has been
more favorable.

Originally, each faculty member set his/her own schedule

for homework and integrated lab project due dates, often
differing week by week within a single course. The results

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Am. J. Phys., Vol. 69, No. 9, September 2001

Manogue et al.

were devastating to the students, particularly to the large
population of our students with outside work and/or family
responsibilities. An important recommendation to anyone at-
tempting a similar curriculum change is to develop a consis-
tent weekly pattern which enables the students to be respon-
sible about coordinating their academic lives with their other
commitments.

We experimented with homework frequency within the

context of our daily class meetings. Daily homework, even
when assignments were short, was too incessant; weekly
homework did not allow enough practice. In particular, some
students came up short, by the end of the year, in their fa-
miliarity with simple algebra and calculus manipulations.
Twice-weekly assignments proved a good compromise for
most Paradigms the second time we taught them.

The first time we taught the Paradigms we experienced a

stage, around the end of the second week of each, when the
students were afraid they were not ‘‘getting it’’ and the fac-
ulty, in response, suffered a crisis of confidence. We now
make sure that students

共

and faculty

兲

know that this is a

normal and expected stage in the intensive format. By the
end of the three weeks, students and faculty generally report
being more comfortable with the level of understanding at-
tained.

By the end of the junior year, the format of the Paradigms

courses is generally viewed with favor by the students. They
frequently mention that classes every day assist their immer-
sion in physics thinking and their understanding of the con-
cepts. Many students feel that this immersion in physics
helps them build on the topics in lecture, instead of losing
concepts after a two-day layoff. The student dropout rate
appears to be decreasing, although this trend is not yet sta-
tistically significant. Some at-risk students are blossoming.

4. Summative assessments of individual achievements

The students are encouraged to work collaboratively dur-

ing the courses, so all instructors include homework and re-
ports that have a collaborative component in their assess-
ments. Students are also encouraged to understand that
individual contributions are ultimately very important. Thus
most of the Paradigms courses use an exam as part of the
evaluation of student performance. This is always a final
exam: no instructors deemed a midterm exam appropriate in
a three-week course.

Timing of the exams has been a thorny issue. The first

Paradigm’s final exam was given on a Wednesday evening,
with negative consequences for the Monday start-up of the
succeeding Paradigm. Subsequent exams have been admin-
istered on Monday evenings, with better success. Students
appear satisfied with only two days of integration between
the end of formal course work and the final exam. A related
problem is that there is no natural time to go over the exam
with the students after grading; we now provide an extra
session to do so. In the first year we lost out on this impor-
tant opportunity for consolidation.

An integrative experience can also be provided by requir-

ing the student to prepare a summary of conclusions for sub-
mission together with the other work at the end of the course,
in the format of a portfolio. Each of the two portfolio-based
courses included two major laboratory experiences. Each
student submitted a written report analyzing each experi-
ment, which was evaluated and returned, so that the student
could correct errors before drawing final conclusions. The
portfolio consisted of the laboratory reports, a few technical

exercises graded and returned during the course, and an es-
say summarizing the conclusions supported by the student’s
experiences. Due the Monday after the course ended, most
were handed in

共

almost

兲

on time. The limited scope of each

Paradigm helped keep the portfolios manageable. After the
portfolios were evaluated and the course grade assigned,
each student’s performance and experiences were debriefed
in a 20-min exit interview with the instructor. In the most
recent round of these courses, the portfolio was evaluated
using the Inventory of Achievement described above. The
Inventory requires about the same amount of faculty effort as
the usual grading system, but the exit interviews add a sig-
nificant effort, which might best be considered instructional
rather than evaluatory time.

IV. EVALUATION OF CURRICULUM,
INSTRUCTION, AND IMPACT ON STUDENTS

The process we have undertaken has been a complete re-

form of the structure, content, and instructional methodolo-
gies of our upper-division program. Of necessity, the reform
involved a number of components: assembly of necessary
resources, internal planning of flow of the content, and ex-
ternal review by an expert panel of advisors. Formative
evaluation procedures guided the development over the three
years while summative evaluation procedures provided a
comprehensive look at the effects of the new implementa-
tion. A variety of data collection techniques included peri-
odic term-by-term student feedback from e-mail questions,
classroom observations, quantitative measures of student
achievement including pre-upper division Grade Point Aver-
age

共

GPA

兲

, GPA during program, Graduate Record Exami-

nation in Physics

共

GRE

兲

scores, and feedback from instruc-

tors and graduate assistants about achievement of students.

Preliminary course syllabi and related information were

sent to a panel of eight faculty engaged in teaching upper-
division Physics at a variety of other institutions. The com-
ments of these reviewers were initially an important source
of information for us on potential problems both in the indi-
vidual courses and in the content and flow of the whole.
Their responses indicated that they believed that the new
curriculum would meet the needs of Physics majors; indeed,
a number of them expressed interest in considering the new
courses and structure for their own institutions.

In the Paradigms approach, students have exposure to sev-

eral faculty members, each with unique perspectives to con-
vey, and to a wide variety of textbooks and other instruc-
tional materials. These varied viewpoints can be significant
strengths of the new approach, but only if special attention is
paid to continuity. While we were first preparing the new
junior-year courses, the faculty who would be teaching them
held a number of meetings whose main focus was the flow of
the ideas and content through the Paradigms. We have ex-
plicitly addressed the need to have a number of physical
concepts and mathematical tools develop naturally over the
course of several Paradigms. An example is the concept of
basis states which builds gradually through all of the Para-
digms, beginning with the Fourier analysis of Oscillations
and Waves, is picked up again in Quantum Measurements
and Central Forces, and culminates in the Capstones. Flow
charts were developed to help us maintain this continuity.
During the first year of implementation, at the end of each
new course, we held meetings with faculty and teaching as-
sistants where we were informed by the work of the review-

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Am. J. Phys., Vol. 69, No. 9, September 2001

Manogue et al.

ers and the evaluation team. These meetings focused on the
effectiveness of our teaching strategies and the rhythm of the
intensive implementation. Yet, near the middle of the year,
we returned to the issue of content and continuity. These
discussions of pedagogy were novel in our department and
were one of the most valuable outcomes of our efforts. We
were energized by the interaction and the curriculum ben-
efited greatly from the coordination.

共

Of course, it should not

have been necessary to revise the entire curriculum in order
to get together to discuss pedagogy!

兲

As the program matured over three years of implementa-

tion, its impact has been verified with respect to student
learning and the alternative instructional modular approach.
For student learning, evidence from a comparison of the
GRE

共

Physics

兲

scores and pre-Physics GPA indicates that

the Paradigms have improved the support for the learning of
physics for average and below average students. In the pre-
vious program, students struggling early tended to withdraw,
changing to other majors. However, these students were
more often retained and supported in their continued work
with physics at no apparent expense to the above average
students. In the Paradigms students quickly recognized the
importance of working together, both the strong and the
weak students. And, their work was continuous over the term
with courses changing every three weeks. This extensive
group work appeared to contribute to a stronger support
mechanism for average and below average students, students
who typically need additional support to engage in the pro-
cesses.

Throughout the junior year of the program, students were

constantly involved in the application of mathematics to
physical phenomenon. And, with courses changing every
three weeks, they were involved in intense study of particu-
lar problems. The evidence of comparing the analytic
problem-solving abilities of the students prior to the Para-
digms program with those in the Paradigms indicates that the
students’ problem-solving skills and thinking skills were en-
hanced. When students were asked, ‘‘What was the most
important thing you have gained throughout the program?’’
they indicated:

‘‘I have more confidence to solve problems and
feel that I have a bigger tool box to start prob-
lems.’’

‘‘I have gained a fairly decent physical intuition.’’

‘‘... not get frightened of anything that is asked,
for example, find out how tall a tree will grow if it
behaves a certain way, I know how to do that
right now from a purely thermodynamic ap-
proach; or if I have to look at something in geo-
physics like seismic refraction, there are a lot of
principles I understand that help to understand
how the system will behave.’’

‘‘I have a bag of tricks, an arsenal for solving
problems or weaponry for solving problems. If a
problem comes your way of any sort, you know
how to tackle it.’’

Another important feature of the student growth in the

program has been a stronger integration of mathematics and
physics. Previously, the mathematics presented in the junior
year was perceived as separate from the physics program.
This finding suggests that students begin thinking as physi-
cists where mathematics and physics are considered inte-

grated. Students indicated that ‘‘unifying across disciplines’’
was a realization from the program. Students indicated an
improved comfort level with applications of mathematical
tools. In some students’ minds, they ‘‘gained experiences at
applying math to various problems where math provides a
different perspective in thinking about things.’’ The exten-
sion of the mathematics to the physical system was also
more clearly recognized by students in the Paradigms. In
their words they developed a ‘‘physical intuition ... to get
from the physical situation to the math expression.’’

One problem that plagued students in the traditional cur-

riculum was the use of varied notations. However, this prob-
lem seemed to disappear in the Paradigms. While the stu-
dents noted the ‘‘difference in notation’’ the fact that the
mathematics was more integrated with the physics helped
them make ‘‘sense of math formulas’’ such that they saw the
math as ‘‘words and not just symbols.’’ In some cases, stu-
dents noted ‘‘a thread running through all the classes; the
first time we saw a topic, half of us did not know what we
were looking at but when we saw it the second time we
could say, ‘Hey, we know what we are looking at.’ We could
learn what was going on ... it was just written a different way
and the more times you see something makes it less intimi-
dating and you can deal with the multiple notations. You can
read any book.’’ One student added: ‘‘I learned to use the
index in books to look things up in more than one book.’’

The Paradigms’ modular approach

共

with a three-week fo-

cus for each module

兲

required students to learn in a manner

different than their traditional mode of instruction. This
change was most problematic during the first term. Students
had difficulty learning how to learn in the new mode. They
had to adjust how they learned physics as well as how they
wove that learning among their other concurrent traditional
eleven-week courses. By the second term, students seemed
to adjust by developing strategies for dealing with the differ-
ences. As one student indicated, ‘‘I learned how to learn.’’

Students consistently pointed to pace and intensity

throughout the junior year as major obstacles for learning.
As they recognized a repetition of major concepts from Para-
digm to Paradigm, their stress over pace lessened, indicating
recognition that they had not missed major concepts

共

a fear

in the pacing issue

兲

. Reflecting over the year, however, stu-

dents recognized how the courses complemented each other.
‘‘They built on one another pretty well. Fourier analysis was
learned in one course and used in the next Paradigm and then
in others as well. They introduced a concept and then more
in depth for the next use.’’ It may be that this repetition, with
each level developing more depth, helped the average and
lower students remain in the Paradigms.

The modular approach with different instructors for each

module resulted in various important side benefits of the pro-
gram. Students were required to adjust to a new instructor
every three weeks. This adjustment required additional stu-
dent learning that was problematic for them until they had
the instructors more than once. As they indicated one of the
major obstacles in the program was ‘‘getting used to the new
system of three or four professors each quarter for each Para-
digm. We did not know what they wanted and what they
expected.’’ Another student indicated that ‘‘adjusting to the
new system, four or five professors in the first term and half,

关

meant I had to adjust

兴

the work load from previous years.’’

With the different instructors, however, a variety of learning
styles were met in one term. Where one student indicated ‘‘I
learn by lecture so the style is important,’’ another expressed

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Manogue et al.

a request for ‘‘more demonstrations and labs.’’ Some in-
structors used projects while others used examinations to as-
sess student understanding. ‘‘I like it being project-based not
final-based.’’

The expertise of the instructors each term was maximized

in the Paradigms. Students recognized the expertise in the
specific Paradigms. However, the use of modules did require
instructors to have an intense three-week assignment for one
course. Thus, their work did not have a consistency through-
out the term. Yet the students noted the availability of the
instructors for assistance as a positive of the program. ‘‘The
teachers, you can always find someone if you have a prob-
lem, even if he is not teaching the class; I asked Professor X
something the other day that was for something totally dif-
ferent and he helped... in terms of people resources we are
fine. We have awesome people, a really good department
where you can always find someone.’’

Graduate teaching assistants

共

TAs

兲

were assigned each

term providing students a consistency of support through this
learning process that was overwhelmingly indicated as im-
portant for student success. The students indicated that the
‘‘TA helped a lot in classes. It was great to have someone
else to bounce ideas off of.’’ During the first year of the
program, the TA was considered to be an essential feature
for possible success. As the program progressed to the third
year, the dependency on these assistants lessened, perhaps
because the instructors were no longer in the ‘‘constant de-
velopment’’ stage and thus had more time to work with stu-
dents.

At the completion of one year in the program, students

were asked to comment on the most important concepts
learned. While students would indicate particular physics
concepts, they also were able to reflect on the program as a
whole: ‘‘I doubled my intelligence in one year. I learned
more about physics and nature in the last year than in my
entire life. I feel a confidence when confronted by a physics
problem or situation that I can overcome it... the Paradigms
prepared me to solve hard problems.’’

V. PROSPECTUS

Teaching students through the Paradigms and Capstones is

a satisfying experience. The students’ response richly re-
wards the work. A graduate of the first class to complete the
Paradigms and Capstones

共

with below-average grades!

兲

wrote in an unsolicited e-mail from his job as a high-tech
designer, ‘‘I can’t thank you enough for teaching me how to
think. Your classes certainly did just that.’’

The strength of the curriculum derives not only from the

choice and arrangement of topics, but also from the many
different pedagogical strategies employed. Some Paradigms
are heavier on lecture content than others, some involve labs,
others are more focused on group problem solving. Students
comment time and again that they really appreciate the many
different experiences. Most students derive benefit from all
the approaches, but a few students do not respond well to
some methods. An important aspect of our approach is that it
is a different few students who have trouble in different Para-
digms! Some students do not like laboratory work, but they
do not encounter it in all the Paradigms. Some students are
uncomfortable with group problem-solving, but not all of the
courses rely heavily on this strategy. Some do not understand
the Inventory of Achievement, but this is adopted in just two
Paradigms.

We have just entered a new phase of teaching these

courses, in which the faculty members who developed them
hand them off to others. We are not surprised that this poses
new challenges. Although the courses appear modular at
first, they turn out to be extensively interconnected by hier-
archies of developing concepts, skills, and habits. We have
traced many such connections, but may be unaware of oth-
ers, which we may discover as we exchange duties. We in-
tend to keep extensive notes of the hand-off, with each ex-
perienced instructor providing support and documentation to
the new crew. We will also continue to improve and develop
student materials for the courses.

As we exchange assignments with each other and with

additional colleagues, we will gain experience that should
help us to assist other Physics Departments that may wish to
adopt our curriculum for their upper-division students. We
hope to conduct this dissemination in a research environ-
ment, documenting the experiences and achievements of the
students and teachers. We are now in the process of applying
for grant support for a collaboration to include several other
schools in early ‘‘technology transfer’’ and its concomitant
analysis and evaluation. We have already identified several
other institutions with interested faculty, and are eager to
hear from more.

ACKNOWLEDGMENTS

This work would not be possible without the dedicated

efforts of the faculty teaching in the Paradigms and Cap-
stones program: we thank Tevian Dray, William M. Hether-
ington, David H. McIntyre, William W. Warren, and Allen
L. Wasserman for excellent collaboration. We gratefully ac-
knowledge the important contributions of early teaching as-
sistants Jason Janesky, Cheryl Klipp, Steve Sahyun, and
Emily Townsend— their expertise, dedication, and enthusi-
asm were above and beyond the call of duty. CAM thanks
Katherine Meyer and Shannon Mayer for their important
contributions in developing and discussing small group ac-
tivities. We thank Albert Stetz for a constructively critical
reading of this manuscript. The external reviewers were also
an important and useful source of ideas and comments. We
are very grateful to the successive Chairs, Kenneth S. Krane
and Henri J. F. Jansen, and all of the members of the Oregon
State University Physics Department for their unanimous en-
dorsement of this project and for absorbing extra work to
make it possible. Finally, but not least, we thank the students
for their hard work and innumerable suggestions. This mate-
rial is based upon work supported by the National Science
Foundation under Grant No. DUE 96-53250. Any opinions,
findings, and conclusions or recommendations expressed in
this material are those of the authors and do not necessarily
reflect the views of the National Science Foundation.

兲

Electronic mail: corinne@physics.orst.edu

兲

Electronic mail: siemens@physics.orst.edu

兲

Present address: Chemistry Department, Lower Columbia College, Long-
view, WA 98632.

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