Mathematica-Based Computer Laboratories for the

Enhancement Of a Calculus Sequence

Mike O’Leary and Raouf Boules Mathematics

Department Towson University

ICTCM, 2000

Background (1)

Towson is a comprehensive university located in the Baltimore metropolitan area

SAT mid range score of entering freshman class in Fall 2000 is 500-590 (Verbal), and 510-590 (Math)

Main textbook in use: Stewart’s Calculus: Early transcendental functions, Brooks/Cole 1999

Background (2)

Audience: Mainly math, math education, computer science, physics and chemistry (lesser degree)

Semester systemClass meets for 4 hours per week for

lectures and one hour in a computer laboratory

Philosophy (1)

Material supplements the mathematics of an otherwise traditional course in calculus, and is NOT a substitute for it (more on this later)

Only small set of built-in functions used in building small programs (Modules) for graphical demonstration or to bypass tedious repetitive calculations

Philosophy (2)

No prior knowledge of Mathematica assumed

Material is based on interactive text (A computer document from which symbolic, numerical, and graphic tools can be invoked. The results of these computations can be pasted into the document so that each learner has an individual record of his or her explorations)

Philosophy (3)

Students are encouraged to experiment and make discoveries

Students may work in small groups but submit a lab report individually

Students are expected to write answers, many comments, observations,…etc

Mathematica Role: Black Box or Teaching Tool (1)

History of Mathematica and its Use: Wolfram: Born in 1959 and received his

PH.D. in Physics from Caltec in 1979 Began in 1979 constructing a CAS to

solve his own problems in physics Mathematica released in 1988 by

Wolfram Research, Inc

Mathematica Role: Black Box or Teaching Tool (2)

History of Mathematica and its Use (cont’d): Became popular first in the research fields

of science, engineering, and economics where it was typically used as a black box to solve complicated problems

Later it entered the scene of education as people realized its potentials for demonstrations and as its front end became more user friendly

Mathematica Role: Black Box or Teaching Tool (3)

Example 1: Using Newton’s method in solving a nonlinear equation in a single variable Black box approach: use the command FindRoot (This also uses Newton’s method)

Approach adopted: see next demonstration

Mathematica Role: Black Box or Teaching Tool (4)

Example 2: Numerical integration Black box approach: Use the command NIntegrate (This uses a sophisticated numerical adaptive quadrature algorithm)

Approach adopted: build a module to show the graphical meaning of the tarpezoidal, and Simpson’s rules; another assesses the error in each as function of number of partitions and asks for conjectures