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In this section we list the programs and software that teachers have found most useful in teaching discrete mathematics. Overall, general-purpose programs, such as spreadsheets and drawing programs, or programs that allow open-ended explorations are much more useful than the many limited-use or ``drill-and-kill'' programs that dominate the market. As one can see in the articles [9,36], there is essentially no general-purpose software designed for discrete mathematics that is appropriate in grades K-12; nevertheless software intended for other purposes has proven useful.

Teachers interested in using software may be interested in contacting CLIME, the Council for Logo and Technology In Mathematics Education (see the Appendix). The organization, which is affiliated with the NCTM, publishes a newsletter; issues have included a ``top-ten mathematics-education software list'' (based on a survey of teachers in a mentorship program at Stevens Institute of Technology), as well as a complementary ``top-ten lessons-with-software list''. In our recommendations below, we refer to the software survey as the ``CLIME survey.''

Spreadsheets  (Excel, ClarisWorks, Lotus 123, etc.) (4-up)

Spreadsheets contain rows and columns of cells in which one can enter either numbers or algebraic formulas. Although one doesn't normally think of spreadsheets as educational software, they are in fact one of the most useful tools to have in the middle- or high-school classroom, especially for Algebra or Pre-Algebra students. Spreadsheets were rated ``No. 2'' in the CLIME Survey (``No. 1'' was the Geometer's Sketchpad, described below).

Spreadsheets are ideal for explorations involving iteration and recursion, such as experimentation with population growth and other models of change, such as described in [11,15]. Of course, calculators can perform many of the same functions, but spreadsheets can do many calculations in parallel, and so can be much faster and easier to use. Also, data from spreadsheets can often be exported to graphics programs to be displayed in different formats.

Classroom Use. The following are some examples of projects in which students used spreadsheets.

Geometer's Sketchpad   (7-up)
Key Curriculum Press; Windows or Macintosh, $170 (plus shipping); site licenses are available.

Sketchpad is a drawing program that allows one to do precise geometric constructions. Teachers can use it to create demonstrations and examples; students can experiment with examples to make conjectures or verify theorems. It is one of the all-time favorites of mathematics teachers everywhere. It was rated ``No. 1'' in the CLIME survey. In addition to being a great experimental tool for traditional geometry, it is also excellent for demonstrating tessellation concepts, or creating tessellations. One can create animations easily by saving ``scripts''. Sketchpad can also be used to generate fractal patterns, such as the Sierpinski Triangle or Koch Snowflake (see Fractals for the Classroom, discussed in section 3.2). It is also possible to create activities that illustrate graph (network) concepts, such as those described for Netpad in [9].

A similar program, which uses slightly different conventions, but is not as well known, is Cabri (Texas Instuments), a limited version of which is installed on the new TI-92 graphing calculator. Another recommended though less-popular program is the Geometric Supposer (8-up) (Sunburst Communications; Mac, $129; Apple/Windows, $99), developed by Dr. Judah Schwartz of the Educational Development Corporation. For lower grades (5-up), there is the Geometric PreSupposer (Sunburst; Apple/Windows, $99). Standard drawing programs such as in ClarisWorks may also be used effectively (e.g., for tessellations and fractals).

Logo   (3-up)
Logo Computer Systems Inc. (LCSI); $70-$200 (Windows, Mac, Apple IIe); (see also the references in [22] for other versions).

The best description of Logo is perhaps the one given by Hal Abelson of MIT: ``Logo is the name of a philosophy of education and a family of computer languages that aid in its realization.'' The idea behind Logo is that we learn by constucting our own knowledge. Logo, as conceived by Seymour Papert, was intended to be the educational clay that would facilitate the building of mathematical knowledge.

A Logo program is a set of commands to a ``turtle'' that moves and draws on the computer screen in response to those commands. Students can construct new commands, and in turn use them as building blocks for even more elaborate creations. For example, students can write Logo programs to generate fractal curves, such as the Koch Snowflake. Logo's structure makes it especially easy to implement iteration and recursion.

Classroom Use. I (Kowalczyk) have extensive experience using and teaching with Logo. For example, I used it to simulate ``gnomon growth'', which leads to Fibonacci spirals, as described in [20]. (The text Excursions in Modern Mathematics described in Section 2 has good material on this topic.) Charlie Hennessy LP `95, who teaches 6-8th grades, reports that it is a good tool for getting students comfortable with the concept of a variable, since students can change parameters and see the effects immediately. Philip Lewis, a high school teacher, describes success using Logo to teach vector algebra and elementary algebra with an algorithmic approach [22].

Further Information. One of the most popular versions of Logo over the last 10 years has been Logowriter (LCSI; Mac/Apple IIe, $199). This version is slowly being replaced by its more sophisticated relative, Microworlds (LCSI; Mac, $99), which features an unlimited number of turtles, parallel processing, a melody editor, drawing tools, and the ability to create hypermedia links. MathLinks (LCSI; $79) (rated ``No. 4'' in the CLIME ``top-ten lessons with software'' list) is an interactive set of activities designed to help students develop mathematical thinking and become mathematical problem solvers. Activities are organized around the topics of polygons, repeating patterns, permutations and combinations and transformations. Turtle Math (LCSI; Mac, $69),[*] another version of Logo, was developed to address the needs of the elementary classroom. It is accompanied by 36 classroom-tested activities and materials geared to grades 3-6, which involve students in such discrete math topics as geometry, similarity, patterns, and probability.

Tesselmania!   (3-12)
Dale Seymour; $48 (Mac), $69 (Windows), $79 (CD-ROM).

This program is designed for demonstrating and creating tessellations. Tesselmania is easy to use and good for sparking interest. Judy Nesbit LP `94 reports using the program for a student tessellation contest: ``Students have been very enthusiastic about this project. They learn a lot and many of them are quite creative!'' The program includes animation that is good for illustrating the different types of polygons that tile the plane and the transformations (reflections, translations, and rotations) involved in creating the tiling. The program comes with a resource guide, which has a series of lesson plans for teaching tessellations and using the program. Judy Brown LP `92 comments: ``Tesselmania is the perfect tool for teaching teachers about tessellations. It gives teachers the ability to experiment with a number of different types of transformations and tessellations in a quick and accurate manner.''

Many, including myself (Franzblau), have found that in teaching tessellations it's much better to start by having students create tessellations by hand; but once they understand the concepts, the program may be useful as a demonstration, or to inspire student creativity. Introducing any software too early may give students the misconception that a computer is necessary to create the tessellations.

I find that paper and pencil, drawing programs, or Geometer's Sketchpad are better tools for creating interesting tessellation designs. Although Tesselmania allows one to add decorations easily, and does the transformations automatically, its drawing capabilities are limited.

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