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.

- (
*6th grade*) Students acted as landscape consultants to students from other countries (communicating over the Internet), doing their planning, estimates, and budgeting with the spreadsheet. (Sr. Diane Mollica LP `95, private communication.) - (
*High school*) As preparation for finding probabilities mathematically, students rolled dice and entered their data on a spreadsheet; they then estimated the probabilities of various outcomes, such as rolling ``7'' with two dice. (Br. Patrick Carney LP `91, private communication.) - (
*High school*) In a unit on understanding the electoral college, students used a spreadsheet to compute the number of electoral votes each state would get using various methods of apportionment. Using their data, they showed how Hayes won over Tilden in 1876 in the electoral college--even though Tilden won the popular vote. (William Bowdish LP `92 and David Fogle LP `93, private communications.)

**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.

**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.