**PHS 542: Integrated Mathematics and Physics (3
semester hours)**

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**Catalog
description:** Mathematical
models and modeling as an integrating theme for secondary mathematics and
physics. Enrollment by teams of
mathematics and physics teachers encouraged.

**Course learning goals:**

**(1) **learn techniques for coordinating algebra and
pre-calculus courses with physics by exploring models that are common to both
disciplines.

**(2)** develop
student activities that support math-physics coordination.

**(3) **initiate and promote an ongoing dialogue between
math and physics teachers that illuminates mutual interests and concerns and
fosters collaboration.

**(4) **collaborate to establish a common language and set
of representational tools that math and physics teachers can both use with
their students within their respective disciplines.

**(5)** develop
classroom research techniques whereby teachers can develop an understanding of
the modeling methodology.

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**Course
content:**

Utilizing the Modeling Method of instruction, abstract
mathematical concepts such as *variable*, *function* and *rate*
will be used in development of mathematical models of physical situations.

Emphasis will be placed on use of technology, which allows
for less emphasis on the process of data-gathering, and shifts the focus to
data interpretation, model identification and generalization.

Math and science standards will be examined to identify and
extend possible alignments.
Discovering the commonalities these standards have can point to ways
that science instruction can improve student performance on mathematics
standard-based assessments, and vice versa.

An emphasis will be placed on the underlying structure of
student learning as a means of guiding modeling instruction.

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**Suggested prior course:**

For physics teachers, a Modeling Workshop is recommended.

**Basic Classes of Models:**

**(**Notice that these
mathematical models are called* functions* by mathematicians.)

**1) **Linear model: Rate of
change = constant ().
Common representations are graphs and equations for straight lines (e.g.,
velocity, acceleration, force, momentum, energy).

**(2) **Quadratic model: Change
(in rate of change) = constant () .
Usual representations are graphs and equations for parabolas (e.g., accelerated
motion, kinetic and elastic potential energy).

**(3) **Exponential model: Rate
of change is proportional to amount ().
Representations include graphs and equations of exponential growth and decay
(e.g., population growth, radioactive decay).

**(4) **Harmonic model:**
**Change (in rate of change) is proportional
to amount (, ).
Usual representations are graphs and equations of trigonometric functions
(e.g., waves and vibrations, harmonic oscillators, situations in electricity
and magnetism such as simple AC circuits and LC circuits).

If we wish to consider two-dimensional situations, we can
combine one or more of the above model classes to form complex models:

**(5) **vector valued models, represented by graphs and parametric equations
(e.g., vectors in 2 dimensions, uniform circular motion, the unit circle,
projections from a vector point of view).

Punctuality and active participation in class and group
activities are crucial to the learning experience. Homework will be assigned, but the emphasis is on
cooperative learning experiences and collaborative curriculum development
activities.

All participants are encouraged to subscribe to the modeling
listserv so that the discourse established during the course can continue once
the school year begins. In this
way, teachers will be supported in their efforts to disseminate what they
acquire as a result of their participation in this course. Their efforts will form a foundation
for future participants in this course.

**ÒTop Ten ReasonsÓ to enroll in PHS 542: Integrated
Mathematics and Physics (with apologies to David Letterman)**

10. This course
aims to establish a community of high school teacher - leaders to support the
integration of mathematics and physics.
We will develop a forum within which physics and math teachers can talk
with each other, and find ways to mutually reinforce one anotherÕs efforts.

9. Finding a common language and set of
representational tools for use with our students is crucial to this enterprise,
and will be an ongoing focus of discussion.

8. Physics courses are typically weak in
the mathematical analysis of the models they develop. There is very little time to analyze the functional
properties that are identified, and rarely are they generalized to non-physics
contexts. This collaboration is a
golden opportunity for mathematics classes, who are crying out for non-trivial
applications, to step in and make use of the groundwork (grunt-work?) that is
already being done in the physics laboratory.

7. Physics students learn to look at rate
of change in a narrow kinematic context.
With a little coordination, this concept of change of rate will be
generalized and applied to a broader range of processes.

6. Science is about discerning and
representing structure.
Mathematics has been called the Òlanguage of structureÓ. Such a coincidence of interests should
be exploited whenever possible, as there are never enough instructional minutes
to do the job.

5. Technology tools will allow us to place
less emphasis on the gathering of data, and focus our efforts on analysis and
deployment of the resulting models.

4. Math and science standards will be
examined for areas of alignment that can be mutually exploited and reinforced.

3. Assessment of integrated learning (both
formative and summative) will be discussed, critiqued and extended as often as
possible throughout the course, and in listserv discussions during the ensuing
school year.

2. The listserv will facilitate ongoing
dialogue between participants and instructors, and will support and encourage
the efforts of teachers who disseminate what they learn in this course at their
school site or district.

1. Refreshments will be solicited to fuel
the intellectual engagement and dialectic that will inevitably result from
consorting with a group of motivated educators.

This course is a golden opportunity for math and physics
teachers to multiply their effectiveness by cooperating with one another. It will be a win-win proposition for
all comers. DonÕt miss out on a
groundbreaking opportunity to be at the cutting edge of educational reform in
your discipline.