Back to Table of Contents
5: Use computer technology to support problem solving
and to promote understanding
At every level of the curriculum, some courses should incorporate activities
that will help all students progress in learning to use technology:
- Appropriately and effectively as a tool for solving
- As an aid to understanding mathematical ideas.
Tools for Visualization and
for Promoting Understanding
A number of sites
provide links to software tools that can be used for classroom demonstrations
or that students can explore on their own to obtain a concrete feeling for
mathematical concepts. Many are part of the National Science Foundation’s
digital library initiative, NSDL.
MathDL, the MAA Mathematical Sciences Digital Library contains
a number of useful resources: The Journal
of Online Mathematics and its Applications (JOMA) publishes
peer-reviewed web materials containing dynamic, full-color graphics for
learning mathematics. MathDL's Digital
Classroom Resources section offers further free peer-reviewed and
classroom-tested learning materials, Convergence
provides an extensive set of resources for teaching the history of mathematics,
Classroom Capsules and Notes
brings together the best of 12 years of short classroom materials from the
MAA print publications, MAA
Reviews is a large database of books and book reviews, and MAA Writing Awards, currently
under construction, will contain pdf copies of the articles that have won MAA
journal writing awards.
since 1994 of the annual meetings of the International Conference on
Technology in Collegiate Mathematics (ICTCM) are available on the
Internet. These include both abstracts and full texts of talks given at the conferences.
The Center for Technology in Teaching
Mathematics (CTTM) at the University of Rhode Island promotes the
exchange of ideas, materials, information, and experiences among the three
public institutions of higher education in Rhode Island. The site offers Java
more, in the areas of pre-calculus, calculus, and engineering mathematics.
Mathematics Miscellany and Puzzles website has perhaps the largest
collection of Java and Flash applets for mathematics that exists on the
Internet and contains much additional information about mathematics and
The Connected Curriculum Project,
with support from the NSF, has created interactive learning environments for
a wide range of mathematics and mathematically based applications. Materials
combine the flexibility and connectivity of the Internet with the power of
computer algebra systems. They can be used by groups of learners as an
integrated part of a course, or by individuals as independent projects or
supplements to classroom discussions. Most of the CCP learning materials fall
into one of three categories: modules, projects, or textbooks. All use at least
some of the following: hypertext links, Java applets, sophisticated graphics,
a computer algebra system, realistic scenarios, thought-provoking questions
that require written answers, and summary questions designed to lead students
to see the forest as well as the trees. The CCP products are aimed at
precalculus, linear algebra, single-variable and multi-variable calculus,
differential equations, and mathematics for engineering.
MERLOT, Multimedia Educational Resources for
Learning and Online Teaching,
is a collection of free peer-reviewed material, which includes
mathematics. It was founded by the California State University Center
for Distributed Learning and rates collections as well as individual
software tools. ’MERLOT's strategic goal is to improve the
effectiveness of teaching and learning by increasing the quantity and
quality of peer-reviewed online learning materials that can be easily
incorporated into faculty-designed courses.â?
The website Manipula Math with Java
Education Software contains 279 mathematics applets on geometry (located
under the ’Middle Schoolâ? heading), trigonometry, calculus, vectors, complex
numbers, conics, and miscellaneous. Walter Fendt’s website contains Java
applets for arithmetic, elementary algebra, plane geometry, stereometry,
spherical geometry, trigonometry, vector analysis, analysis, and complex
David Hill, Temple University, and Lila Roberts, Georgia Southern University, have developed a website
whose goal is "to connect mathematics instructors with effective
teaching tools" that engage the learner on a level beyond dialogue. The DEMOS with POSITIVE IMPACT project
collects and posts demonstrations that can be incorporated into a lecture and
that use some form of instructional technology, where technology is
interpreted very broadly, from physical equipment to calculator and computer
Math Tools is "a community
library of technology tools, lessons, activities, and support materials for
teaching and learning mathematics." It is a Math Forum project, funded
in part under the NSF’s NSDL initiative, and currently includes over 3000
software tools and supporting materials, ranging In level from calculus down
to pre-kindergarten. Under the assumption that an imperfect but useful tool
is better than no tool at all, the material is not peer -reviewed, but
Amazon-style user -rated, and discussions about using the tools are an
important part of the forum. Consequently, user complaints have sometimes
caused software developers to improve their products.
The Mathwright Library is a collection of
interactive, electronic mathematics and science "WorkBooks" and
"Microworlds." Members of the Library may download materials freely
during their subscription period, and year-long subscriptions are available
for a nominal fee, either by individuals or by institutions. The aim of the
Library is to invite students into the world of mathematics and science
through structured Microworlds and WorkBooks that allow them to ask their own
questions, read at their own pace, and experiment and play with various
mathematical topics. The WorkBooks were developed by college and secondary
school mathematics and science teachers, with initial funding by the National
Science Foundation and support by the IBM Corporation.
Classroom resources for using dynamic geometry software
packages, such as Geometer’s
and GeoGebra (which is free), are
available on the Math Forum Geometry
A highly technological approach to mathematics
instruction, using Java resources, Excel files, and a variety of dynamic software
sources, is described in ’The
Dynamic College Classroomâ? by Douglas Butler, Oundle School, UK.
managing director of bk teachware, wrote an article in which
he considers which mathematical problems students should solve with
technology and which problems should be solved without it. Kutzler's website contains an up-to-date, searchable
listing of conferences on the use of technology in mathematics education.
OpenOffice has a
free office suite that includes a spreadsheet application ("Calc")
which can read, edit, and save in Excel format. The suite also includes
a word processor compatible with MSWord, a presentation program compatible
with PowerPoint, and a database program.
Technology Throughout The Curriculum
Most college algebra
and precalculus textbooks and many courses now incorporate the use of
graphing calculators. Some also include spreadsheet programs, usually Excel,
computer algebra systems, such as Derive or Maple, and dynamic geometry
programs, such as the Geometer's Sketchpad or Cabri Geometry. For example,
the syllabus for Math
Modeling & Problem Solving I
at Francis Marion University states that ’The new TI-83 Plus graphing
calculator will be used extensively in this courseâ?¦We will also use other
technology such as the Excel program, Microsoft Word, Graph-Link, and Maple
on the computers for computations and graphs.â? The syllabus for College Algebra
at Dakota State University includes the sentence ’Students will use The
Geometer’s Sketchpad, Excel, and Maple in this course.â?
Publishers of books
for elementary service courses have developed tutorial software packages to
accompany many of their lower-level texts. These packages include the ALEKS system from McGraw-Hill, The
Learning Equation from Thomson Learning, MyMathLab and MathXL from Pearson
Addison-Wesley and Pearson Prentice Hall.
In 1995/1996, the
INPUT (Innovative Programs Using Technology) Project solicited entries for
the first INPUT competition. The result of the project's work is the handbook
Introductory College Mathematics Programs, which was designed to
provide concrete, detailed descriptions of what some innovative instructors
are doing, how they are doing it, and what technologies they are using.
statistics textbooks, especially those for business statistics, now typically
include electronic data sets to be analyzed using statistical and spreadsheet
software, such as Excel, SPSS, or Minitab. A survey of course syllabi on the
Internet indicates that this software is incorporated into a significant
fraction of these courses.
John C. Pezzullo
maintains a collection of webpages that together
make up a freely accessible, multi-platform statistical software package. It
also contains links to online statistics books, tutorials, downloadable
software, and related resources.
Robin Lock, St. Lawrence University, has created a listing of Internet
websites with Java applets that illustrate statistical concepts.
The book Teaching Statistics:
Resources for Undergraduate Instructors, edited by Thomas Moore (MAA,
2000) contains a section on the use of technology, which begins with a series
of tasks designed to help teachers evaluate and compare statistical
software packages. There are also sections on the use of graphing calculators
in teaching statistics and information about Internet resources and
innovative use of technology (including videos) in the classroom.
With support from
the National Science Foundation, several calculus reform projects in the
1990s led the way toward the use of calculator and computer technology in
calculus instruction. Now exercises making use of graphing technology are
incorporated into virtually all calculus textbooks, and offerings at a number
of schools include computer laboratory exercises using Maple, Mathematica,
Mathcad, MATLAB, or Derive. While some instructors make use of laboratories
they have devised themselves or obtained from colleagues, others use one of a
number of published collections of calculus laboratories. An alternative
approach is found in the Calculus
(C&M) course, started at the University of Illinois,
Urbana-Champaign, in 1989 by Jerry Uhl and Horacio Porta. In this
course most of the instruction is conducted in a laboratory setting,
and the course text is a Mathematica notebook, which students access
and interact with online.
The International Education
Software website contains 279 Java applets that dynamically illustrate
mathematical concepts and are freely available for viewing. The majority are
relevant to single and multi-variable calculus. An annotated listing of
additional information about technology in calculus instruction is given in a
by Martin Flashman, Humboldt State University.
Tom Leathrum, Jacksonwille State University, has developed a collection, Mathlets:
Java Applets for Math Explorations, which provides a set of interactive
learning tools for precalculus, calculus, and beyond.
While a graduate
student at the University of Illinois at Urbana-Champaign, Lisa Denise Murphy
wrote an evaluative article: Computer
Algebra Systems in Calculus Reform.
The Journal of Online Mathematics (JOMA)
has at least 81 items on calculus, which can be obtained from its website by
entering "calculus" as a keyword.
A supplement to
Kenneth Rosens's text Discrete
Mathematics and its Applications, 5th edition, uses Maple to focus on
the computational aspects of the subject. The website for Rosen's text also
contains links to several interactive
demonstrations for discrete mathematical topics.
Doug Ensley of Shippensburg University hosts a webpage that contains a variety of Flash
applications to accompany Introduction to Discrete Mathematics:
Mathematical Reasoning with Puzzles, Patterns and Games, by
Doug Ensley and J. Winston Crawley. Several of these applications
give assistance for proof development, while others illustrate additional concepts in discrete mathematics.
Susanna Epp, DePaul University, has assembled an annotated
list of Java applets and other instructional software for discrete
mathematics, including a set of Derive laboratories developed by Nancy
Hagelgans, Ursinus College.
The ATLAST project (Augmenting
the Teaching of Linear Algebra through the use of Software Tools), funded
by the National Science Foundation, was created to encourage and
facilitate the use of software in teaching linear algebra. The book ATLAST
Computer Exercises for Linear Algebra, 2nd edition, edited by Steven Leon, Eugene Herman, and
Richard Faulkenberry is an outgrowth of the project and uses MATLAB. Data files are downloadable from the
ATLAST website. Versions of most of the exercises are also available for
Mathematica and may be developed for Maple.
The linear algebra text by David Lay, University of Maryland, is designed to be used with MATLAB, Maple, Mathematica, or the
TI-83+, TI-86, TI-89, or HP48G calculator. Data sets keyed to exercises in
the text are downloadable from the Internet.
The book, Interactive
Linear Algebra: A Laboratory Course Using Mathcad, by Gerald Porter, University of Pennsylvania, and David Hill, Temple University, is intended to be used in a
course with a computer laboratory format and a focus on discovery learning.
The Math Forum
website Choosing a
Linear Algebra Text contains additional information about linear algebra
books published between 1985 and 2005 that use technology for instruction.
Architect Companion, created through the Consortium for Ordinary
Differential Equations Experiments with support from the National Science Foundation
and now available from Wiley Higher Education, is an interactive teaching,
learning and research environment for constructing and exploring mathematical
models of real-world phenomena. A guide
to using the ODE Architect was written by Michael Moody.
Large collections of
Java applets for multivariable calculus and differential equations have been
developed by a number of people: Richard
Williamson, Dartmouth College, with assistance from Scott Rankin
and Susan Schwarz, and a group consisting of Beverly West,
Cornell University, Steven Strogatz, Cornell University, Jean Marie McDill,
California Polytechnic University, San Luis Obispo, John Cantwell, St. Louis
According to the
review by Jan E. Holly, Revolutions
in Differential Equations: Exploring ODEs with Modern Technology,
Michael J. Kallaher (Ed.), "consists of articles presenting a host of
ideas for making use of technology in teaching differential equations.
Included are ideas for classroom examples, student exercises, and ways to
structure a course. Also included are descriptions of available software and
references to Web resources."
In Learning and Teaching
Ordinary Differential Equations 1 Chris Rasmussen and Karen Whitehead
discuss the use of technology, concluding that "we need to be deliberate
in how and why we decide to implement technology in the classroom." They
write: "Using a computer algebra system as a separate lab component or
only as a demonstration tool seems less likely to achieve the intended
learning goals" than "integrating technology into students'
experiences in the classroom."
The MAA magazine Convergence
provides a large number of online resources for teaching courses in the
history of mathematics. All of Euclid’s Elements together with
explanations, discussion, and Java applets are contained in the website
developed by David Joyce, Clark University. Historical Modules
for the Teaching and Learning of Mathematics, edited by Victor Katz
and Karen Dee Michalowicz, is CD containing a huge collection of pdf’s with
descriptions of individual lessons, organized by subject, for teaching topics
in the history of mathematics.
Additional examples about using technology in upper-level
mathematics courses are in Part 2, Section C.2.