A Comprehensive Mathematics Curriculum with Mathematica

Joanne Mason, Director of Math Studies

The Harker School, San Jose CA

Copyright Joanne Mason, 2004. All Rights Reserved. Contact joannem@harker.org.

Session Overview

Introduction - The mathematics program at Harker: What

Principles and beliefs about mathematics, curriculum, Mathematica, & technology: Why

Advanced Math and Calculus: A comprehensive, technology-intensive, applications-oriented introductory survey course

Examples of problems and projects - Examples of student work with Mathematica

Some Observations

Q&A

Based in part on...

"A Comprehensive Mathematics Curriculum with Mathematica", at Joint Mathematics Meetings, The Effective Use of Computer Algebra Systems in Teaching Mathematics, AMS/MAA, Phoenix, January 7, 2004

Email:  joannem@harker.org
To receive a copy of the paper...

Web:  http://faculty.harker.org/joannem/
To get copies of Mathematica notebooks and other materials...

The Harker School, San Jose, California

Harker's Mathematics Curriculum

Some Significant Features of Mathematics Program

Principles of Mathematics, Curriculum, Mathematica, & Technology

I.e., Why do we use Mathematica so extensively in the mathematics program

Two new paradigms for mathematics and for mathematics learning

The "Interactive Document"

Electronic Virtual Library

The tools that perform computation are also the tools with which mathematical knowledge is stored, organized, indexed, retreived, created, and learned.

Advanced Math and Calculus

Adv Math & Calculus is an introductory comprehensive, 11th-13th grade level, applications-oriented course in Calculus, data analysis, mathematical modeling, and other applied topics.

An Outline of Mathematica in Advanced Math: A Comprehensive Introduction/Survey of Elementary Mathematica

Projects and Student Notebooks

Traffic Flow - Advanced Mathematics. Students must determine workable traffic configurations for a street network with given flow parameters and constraints. They must also determine traffic flow with additional constraints such as a street closure. The problem requires solving systems of 5 or 6 eaquations with 7 or more unknowns as well as combining system solutions with systems on inequalities. Presentation format.

The Vase - Calculus.  Students must devise a data gathering system to obtain sufficiently accurate measurements of a large pottery vase on the teacher's desk. Using calculus, students determine the volume of the vase and the volume and mass of the clay used in the vase. [Immersion not possible; material is absorbent and unstable. Cameras and scanners are broken :-) ]  Students then devise a method to enter the data and use Mathematica to create an accurate graphic rendition of the vase.

Ant on a Helix - Multivariate Calculus.  [CalcLabs with Mathematica, Selwyn Hollis. Accompanying Stewart's Calculus].  An ant is travelling up a helix. A light source is positioned at given point on the z-axis. Find the equation of the plane curve of the shadow traced as the ant travels around and up. Students additionally added the step of animating the action.

Hydro Power Plant - Multivariate Calculus.  Students represent a hydropowerplant mathematically. There are three turbines, each with different power curves. Students use optimization methods [in this example: Lagrange Multipliers] to determine the recommended distribution of flow to the three turbines under different flow conditions and determine the power outputs.

Gravitation - Multivariate Calculus [final project]. An open ended project, students were to develop a Mathematica-based exploration of classical gravitational theory. Images describing sungle body, two body, and many-body cases were developed. Some students opted to explore largescale systems. Others considered the development of such a system in time. This example shows the incorporation of energy considerations to show the orbit of a body travelling around two bodies.

Earthquake [3 examples] - Differential Equations [based on a project in Edwards and Penney]. Students explore the resonance effects of vibration effects of a multistory building in an earthquake of varying intensities. The building is modeled as a compound spring system. Graphs are developed showing "safe" and "catastrophic" frequencies of vibration.

Forced Spring Systems - Differential Equations [based on Duke Univ Connected Curriculum Project project  http://www.math.duke.edu/education/ccp/materials/]. Students explore vibration and resonance in simple spring systems under different combinations of forcing and damping.

Leslie Matrices - Linear Algebra [final project]. Comprehensive paper on Leslie Matrices. Incudes both theory and proofs of relevant theorems and numerical solution of given problems.

Math Art [Bradley Stoll, bradleys@harker.org]. Examples of recreational graphics and animations produced by students in Calculus.

Valentines Day Plots [Danae Romrell, danae@harker.org]. Precalculus project requiring students to use polar plots to design colorful valentines day card designs.

Some final Observations

June [tba], 2005

Institute on Project Learning in Math and Science using Mathematica, at the Harker School, San Jose, CA

A week-long institute/seminar, in the San Francisco Bay Area of California [South Bay - Silicon Valley], for advanced high school teachers and college teachers...

Training in the educational use of Mathematica [Beginning, Intermediate, Advanced...]

Project Learning - Curriculum design, planning, development, and evaluation; coaching and seminars developing and preparing actual science or math projects using Mathematica

Thanks!

Joanne Mason,  408-345-9290
joannem@harker.org

Danae Romrell
danaer@harker.org

Bradley Stoll
bradleys@harker.org

Misael Fisico
misaelf@harker.org

The Harker School
500 Saratoga Ave 
San Jose, CA 95129 USA