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Part I: Recommendations for departments, programs and all courses in the mathematical sciences

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Illustrative Resources


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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 problems;
  • 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.

All proceedings 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 applets, video clips, slide shows, JavaScript-based tutorials, and more, in the areas of pre-calculus, calculus, and engineering mathematics.

Alexander Bogomolny’s Interactive 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 mathematics teaching.

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 from International 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 numbers.

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 software.

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 Sketchpad, Cabri Geometry, Cinderella, and GeoGebra (which is free), are available on the Math Forum Geometry Software website.

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.

Bernhard Kutzler, 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

Elementary Service Courses

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 Exemplary 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.

Elementary Statistics

Elementary 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.

Calculus

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 & Mathematica (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 website maintained 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.

Elementary Discrete Mathematics

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.

Linear Algebra

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.

Differential Equations

ODE 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 University.

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."

History of Mathematics

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 Resources

Additional examples about using technology in upper-level mathematics courses are in Part 2, Section C.2.

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