PHS 542: Integrated Mathematics and Physics (3 semester hours)
July 7 – 25, 2008
ASU Tempe Campus, Bateman Physical Sciences Center, room H-563
8:30 – 3:30pm, MTWTh. 8:30-12:30F

Instructor: Barry Walker, Science Department Chair, Briarwood Christian School, Birmingham, Alabama
Co-leader: Robyn Rosenthal, Mathematics Department Chair, Paradise Valley High School, Phoenix, Arizona
Guest lecturer: JoAnne Magden, Mathematics Instructor, Paradise Valley High School, Phoenix.

Description: Models and modeling as an integrating theme for 8th and 9th grade mathematics and physics.

Prerequisite: in-service teacher of middle school or high school mathematics or science. (Teams of two teachers from the same school
are especially encouraged to take the course.)

Vision: cognitive processes for understanding mathematics and physics are intimately linked and fundamentally the same. (Practical application: science instruction can improve student performance on mathematics assessments such as Arizona’s AIMS, and vice-versa.)

Course objectives are to develop:
1) a modeling approach to learning arithmetic, pre-algebra, algebra, and ninth grade physics.
2) student activities that support both math and physics (an approach to mathematics that makes it science-friendly).
3) structures that we impose on the physical world and conceptual tools used to represent them.
4) an understanding of learning and how model-based instruction incorporates theories of learning into student activities.
5) learning environments that encourage thinking, reasoning and understanding, with a special focus on proportional reasoning.
6) mathematical and graphical models to describe basic physics phenomena.

Course content:
The course vision states, “cognitive processes for understanding mathematics and physics are intimately linked and fundamentally the same.” Therefore, all content will be taught simultaneously as science and mathematics. Topics that previously have been thought of as “science” will be taught as mathematics/science and topics previously thought of as “mathematics” will also be taught as mathematics/science. Science cannot be fully understood without mathematics and conversely mathematics cannot be fully understood without science.
1) Grouping
2) Symbolization
3) Multiplicity
4) Part/Whole Relationships
5) Quantity/Number
6) Fractions
7) Systems
8) Proportional Reasoning
9) Graphing
10) Density, Constant Motion – linear models
11) Constant Acceleration - quadratic models
12) Newton’s Second Law - inverse models
13) Cognitive Science as it relates to mathematical/scientific understanding, thinking and reasoning.

Rationale: A solid understanding of quantity and number is essential to construct the concept of measurement, a necessary prerequisite to understanding science concepts. Proportional reasoning is crucial for physics and algebra.

Responsibilities of participants: Punctuality and active participation in class and group activities are crucial to the learning experience. Homework will be assigned, but the emphasis will be on cooperative learning experiences and collaborative curriculum development. No textbook.

Academic year follow-up: Participants will be encouraged to subscribe to a listserv (dcmod) so that the discourse established during the course can continue once the school year begins. Teachers will be supported in their efforts to disseminate what they have acquired as a result of their participation in this course. Their efforts will form a foundation for future participants in this course.

A glimpse into the progression of ideas in the course:

Both mathematics and science can be thought of in terms of systems.
  1. Initial activities will develop the ability to construct mathematical models that describe simple systems in such a way that it is obvious how the models are constructed and what the models are modeling.
  2. Students will utilize conceptual and representational tools that are necessary to think about and represent various models.
  3. These mental processes and representational systems will be used to construct scientific models of physical situations.
  4. Participants will learn when, why and how to use conceptual and representational tools to build multiple models of various phenomena.

A. Physical systems
1) Each physical system can be broken down into
a) parts
b) relationships among the parts
c) operations
d) structure
e) invariants.

2) Abstract concepts are developed in terms of either quotient relationships or product
relationships. (A common thread in Strands 1, 2, 3, 4 & 6 of the AZ Mathematics Standard)

B. Mental processes include describing and reasoning.

(1) Describing requires symbols to represent the complete range of concepts.
a) The process of constructing symbols to represent collections of objects requires mathematical operations. Constructing a mathematical model requires understanding number, not as a label, not as an abstraction of a quantity, but as a relationship, which means number is a constructed concept. (A common thread in Strands 1, 2 & 4 of the Arizona Mathematics Standard)
b) Measurement acts as a description of a whole in terms of its parts. The difference between number and quantity will be explored. (Aspects of strand 4 of the AZ Mathematics Standard)
c) Quantities can be converted into one another by equivalence relationships that are often referred to as ratios. (Aspects of Strand 4 of the Arizona Mathematics Standard)

(2) Reasoning about the system and changes in the system require a deep understanding of equivalence relationships. When we reason, we change a description. We impose equivalences on concepts and construct a new concept from them, which is the coordination of the two equivalences. We can reason transitively and proportionally.

Three basic classes of mathematical models in physics will be developed:
Linear models are descriptions of situations in which the quotient relationship between changing quantities is a constant. Common representations are graphs and equations for straight lines (e.g., constant motion, density).

Quadratic models are descriptions of situations in which the change in the quotient relationship with respect to a changing quantity is constant. These situations are represented by graphs and equations for parabolas (e.g., accelerated motion, kinetic energy).

Inverse models are descriptions of situations in which the product relationship between changing quantities is constant. These situations are represented by graphs and equations for hyperbolas (e.g., force, pressure-volume).