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

* *

** Instructor: **Robert Rowley, M.S. in Physics. Bob has 60 graduate credits in physics
at ASU. He taught physics,
calculus, programming, and digital electronics for 25 years. He developed the

This class will meet at
ASU-Tempe campus in the Physical Sciences Center, room H-355, from 4:00pm to
6:30pm MTWR for 5 weeks from July 2 to Aug. 2, 2018. No class on Wed. July 4.

** **

**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 purpose: **

Science is about modeling
the natural world (or, modeling our observations of the natural world). Math is
a set of formal tools for making our models clear and precise, and for working
out the consequences with rigor. We all make internal mental models of the
natural world, but without formalizing them (with the use of math) they tend to
be ad hoc, vague, unreliable stories.

Math, on the other hand, has
precision and clarity of structure, but without being applied to meaningful
models it tends toward being a meaningless proliferation of empty rules.
(Caveat: those tendencies are just that—tendencies, not absolutes.)

Science and math both become
more vibrant, fecund, and rewarding when joined together, and thatÕs true of
education as well as professional practice. Thus, we want to integrate them all
throughout the instructional process.

**Course objectives:**

- learn ways to coordinate algebra and
pre-calculus with physics by exploring models that are common to math and
physics.
- develop student activities that support
math-physics coordination.
- establish a common language and set of
representational tools that math and physics teachers can use with their
students.

** **

**Course content:**

Abstract mathematical
concepts such as *variable*, *function* and *rate*
will be used to develop mathematical models of physical situations. Emphasis
will be placed on use of technology, which de-emphasizes the process of
data-gathering, and shifts the focus to data interpretation, model
identification and generalization.

Geometric algebra will be
introduced. Geometric algebra (GA) is a unified mathematical language for the
whole of physics. GA is suitable for a first year physics course (at advanced
high school and post-secondary levels). Participants will learn how to
formulate, analyze and apply basic math models for Newtonian mechanics without
introducing coordinates. Teachers will learn GA using **the Primer on Geometric Algebra for introductory mathematics
and physics**.
Downloadable at http://geocalc.clas.asu.edu/html/IntroPrimerGeometricAlgebra.html

**Expectations:** Punctuality and active participation in class and
group activities are crucial to learning.
Homework will be assigned, but the emphasis will be on cooperative
learning.

**Suggested prior course: **For physics teachers, a Modeling Workshop is
recommended.

**Basic mathematical
models:**

** **

**(1) ** Linear --
rate of change = constant ()

graphs and equations for
straight lines (i.e., velocity, acceleration, force, momentum, energy, etc.)

**(2) **Quadratic –** **change (in rate of change) = constant ()

graphs and equations for
parabolas (accelerated motion, kinetic and elastic potential energy, etc.)

** **

**(3) **Exponential - rate of change = proportional to amount
()

graphs and equations of
exponential growth and decay (population growth, radioactive decay, etc.)

**(4) **Harmonic --** **change (in rate of change) = proportional to amount (, )

graphs and equations of
trigonometric functions (waves and vibrations, harmonic oscillators,
electricity and magnetism, etc.)

**(5) **Parametric and vector valued functions

graphs and parametric
equations (vectors in 2 dimensions, uniform circular motion, the unit circle,
projections from a vector point of view)

The theme of this course is
to investigate these elementary functions as models of various physical and
geometric phenomena, in one and two dimensions, with some emphasis on the idea
that they are the simplest ways a change in one quantity relates to the change
in another.

**Why enroll in this
course: **

* Physics courses are typically weak in mathematical analysis
of the models they develop. There
is little time to analyze the functional properties that are identified, and
rarely are they generalized to non-physics contexts. This course remedies that
deficiency.

* Physics students learn to look at rate of change in a narrow
kinematic context. This concept will be generalized and applied to a broader
range of processes.

* 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.

* This course is a unique
opportunity for math and physics teachers to amplify their effectiveness by
cooperating to find a common language and set of representational tools for use
with students. **Physics teachers: please alert your math teachers to this
course, and invite them.**

WHY GEOMETRIC ALGEBRA? David Hestenes wrote:

--------------------------

Physics teachers are universally
dismayed by the paltry understanding of mathematics that students bring from
their mathematics courses. Blame is usually laid on faulty teaching. But I hold
that the crux of the problem is deeply embedded in the curriculum. From the
perspective of a practicing scientist,* the mathematics taught in high school
and college is fragmented, out of date and inefficient*!

The central problem is found in
high school geometry. Many schools are dropping the course as irrelevant. But
that would be a terrible mistake for reasons already clear to Galileo at the
dawn of science.

* Geometry is the starting place for physical science, the
foundation for mathematical modeling in physics and engineering and for the
science of measurement in the real world.

* Synthetic methods employed in the standard geometry course
are centuries out of date; they are computationally and conceptually inferior
to modern methods of analytic geometry, so they are only of marginal interest
in real world applications.

* A reformulation of Euclidean geometry with modern vector
methods centered on kinematics of particle and rigid body motions will simplify
theorems and proofs, and vastly increase applicability to physics and engineering.

** **

**A basic pedagogical principle:*** The depth and extent of student learning is critically
dependent on the quality of the available mathematical tools.*

Therefore, we can expect a
well-designed curriculum based on vector methods to produce significant
improvements in the depth, breadth, and usefulness of student learning. É

Downloadable at http://geocalc.clas.asu.edu/html/IntroPrimerGeometricAlgebra.html

------------------------------

**References for further study of
GA:**

D. Hestenes, ÒOersted Medal
Lecture 2002: Reforming the mathematical language of physics, Ò Am. J. Phys.**
71:** 104-121 (2003). Get print copies from
Jane Jackson, ASU Dept. of Physics.

D. Hestenes,* New Foundations
for Classical Physics* (Kluwer, Dordrecht,
1986, 2nd ed. 1999)

Both are at David Hestenes'
website: http://geocalc.clas.asu.edu