Jeffery M. Saul

Michael C. Wittmann

Edward F. Redish

** sponsored in part by NSF grants RED-9355849
and DUE-9455561*

Context for this research

- Second semester introductory calculus-based physics (3 semester sequence), University of Maryland
- Corequisite: Calculus II

Methods of investigation

- classroom observations
- "pretests"
- examination questions
- student interviews

How do students interpret and apply mathematics in introductory physics?

**Students need facility with many mathematical
representations in learning introductory physics:**

Graphs

- interpret physical phenomena based on graphical representation
- construct a graph from experiment
- appreciate related quantities from a given graph

Equations

- recognize the relationship between physical situation and the associated equation
- understand the idea of a function

Vectors

- understand what a vector represents
- tie different representations to a well defined coordinate system

**Example from electrostatics Midterm
Examination (N=95)**

Consider a region of space where there is an electric field given by:

where:

Part A: Determine the value of E at the point labeled A.

- 44% gave correct response:

- 14% used the correct procedure but made a computational mistake
- 41% could not apply vector ideas in this problem

- were unable to find on diagram**r**

- were unable to find**r - ro**

- were unable to find the magnitude of**r - ro**- mixed scalers and vectors

- 1% did not answer the question

Part B: Draw electric field lines for this region of space.

- 47% gave correct response
- 28% drew field lines corresponding to a point charge at A
- 4% drew field lines corresponding to point charges at A and
- 5% drew field lines corresponding to a point charge at the origin
- 16% other

**Example from mechanical waves
Pretest (N=57) Individual interviews (N=9)**

Consider the pulse below at t = 0 moving in the x direction with velocity vo.

The displacement of the spring from its equilibrium position at t = 0 is given by:

Part A: Sketch the shape of the spring after it has traveled a distance xo.a correct response (pulse displaced, amplitude unchanged)

pretest: 56%; interview: 44%

correct response? (pulse displaced, amplitude decreased)

pretest: 35%; interview: 56%

Part B: Write an equation for y as a function of x when the pulse reaches xo.

- correct response:

pretest: 7%; interview: 0%

- "non-function"

pretest: 44%; interview: 67%

- sinusoidal:
**e.g.**

pretest: 2%; interview: 22%

- other:

pretest: 44%; interview: 11%

**Summary**

Students have difficulty interpreting and applying mathematical ideas when learning introductory physics.

In the context of equations:

- students fail to demonstrate a functional understanding of vectors when interpreting vector equation
- students fail to recognize the relationship between the physical situation and the associated equation
- students fail to understand the idea of a function

**Research-based curriculum**

Students, working independently, make predictions about a propagating pulse and its mathematical form.

Students, working in groups:

- at a given time, relate the shape of a pulse to an equation describing that shape
- at a later time, relate the shape of a pulse to an equation describing that shape
- construct a single equation that describes the shape of the pulse as a function of both position and time
- describe different shape pulses physically and mathematically
- use video software to mathematically model the shape of an actual pulse

*This page prepared by:
Richard
Steinberg*

*University of Maryland
Physics Department
College Park, MD 20742-4111
(301) 405-6184*