* 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

Student Responses

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