Mathematics software

Summary: Mathematics software has long been used as a teaching aid and has become an important tool in applied mathematics.

Mathematics software refers to a wide variety of computer programs designed to manipulate, graph, or calculate numeric, symbolic, or geometrical data. Along with the development of computer technology and wider access to personal computers, these types of programs gained popularity at end of the twentieth century. Within the mathematics community, it has influenced instruction, applications, and research. Instruction has changed so that mathematics is more accessible to larger numbers of students; it is more engaging, more visual, and more focused on conceptual understanding rather than on computational facility.

Mathematics software has changed research and the nature of mathematical proof so that computers are now tools for exploration, for applications, and for performing repetitive tasks. There are numerous journals devoted to the development, use, or implementation of software in research and teaching, such as the Transactions on Mathematical Software journal. Computer software has influenced what mathematics is being taught, how it is being taught, the nature of its applications, and the way mathematics is explored. Computer software provides society a different modality for learning, understanding and applying mathematics.

What is Computer Software?

“Computer software” is a general term reserved for a collection of computer programs that provide step-by-step instructions for a computer to perform specific tasks. There are four major types of software:

• • Operating systems: System software, often called the computer “platform” (for example, Microsoft Windows, Mac OS, and Linux);
• • Computer languages: The code and syntax used for developing software (for example, Java, C/C++, Visual BASIC, Pascal, and Fortran)
• • General applications software: Software designed for general purposes (for example, word processors, database systems, spreadsheets, and communications software)
• • Specific applications software: Software designed for performing content-based tasks (for example, MATLAB, Mathematica, Geometer’s Sketchpad, SPSS, and MINITAB)

Software for the subject of mathematics falls in the category of Specific Applications Software.

Mathematics Software

The term “mathematics software” refers to computer programs designed to manipulate, graph, or calculate numeric, symbolic, or geometrical data. The journal Transactions on Mathematical Software (TOMS), produced by the Association of Computing Machinery (ACM), provides current information on available mathematics software. Through TOMS, the reader can gain access to large indexed mathematics software repositories. The majority of the software is written in Fortran or C++ for solving mathematics problems occurring in the sciences and engineering. Research scientists are invited to use these modules in developing their own software.

Software for Applied Mathematics

According to the National Research Council (NRC), computer software has had a major impact on applied mathematics and has illuminated new areas for mathematical research. The use of computer software in research in applied mathematics is prevalent, especially when repetitive computations are necessary. The most prominent computer software packages used in college-level instruction in the early twenty-first century are MATLAB, Mathematica, and Maple. These are computer algebra systems (CAS) that perform both symbolic and numeric computations. Software is used for statistics applications by professionals in mathematics, sciences, education, and social sciences, such as SPSS, SAS, BMDP and SPlus. These allow users to easily explore and visualize data and automate the computational aspects of many commonly used statistical procedures, which can be significantly difficult for larger data sets. It also facilitates more complex modeling and computer-intensive methods like exact tests, resampling techniques like bootstrapping, and many types of Bayesian statistical procedures, which are named for mathematician Thomas Bayes.

Software for Mathematics Research

Mathematics software is also gaining prominence in the fields of pure mathematics such as number theory, abstract algebra, and topology. An outstanding example of the impact of software on topological research occurred in 1976 when a computer program was used to check all of the possible cases in the Four-Color Map conjecture.

To understand the Four-Color Map conjecture, consider a map of the United States. Suppose the task is to color the individual states so that no two contiguous states are the same color. How many different colors are necessary to complete the task? Such a question arose in 1852. The Four-Color Map conjecture states that, at most, four colors are needed to color the map. In 1976, Kenneth Appel and Wolfgang Haken finally proved this conjecture (thus establishing it as a theorem) by using a computer program, representing the first time computer software was used in the proof of a mathematics theorem.

This computer-based proof led to considerable controversy within the mathematics community. The controversy centered on the nontraditional nature of the proof, which required a computer program for testing all of the possible cases, namely 1936 maps. Some mathematicians argued that this procedure did not constitute a formal mathematical proof, which is typically based on deductive logic and mathematical principles (such as definitions, axioms, and theorems). Instead, it was an exhaustive test of all possible cases, made possible by a computer program. Thus, neither deductive logic nor mathematical structure was required. Regardless of the controversies surrounding the proof of the Four-Color Map theorem, the result was to alter the attitudes of mathematicians toward the role of computer software in formal mathematical proof. Consequently, since the 1970s, computer software has become a major research tool for both pure and applied mathematicians.

A further consequence of the use of computer software in mathematical research is a trend for mathematicians to use open-source software, rather than proprietary software. Many commercial or proprietary software programs were originally developed and sometimes freely distributed as part of grant-funded projects or by individual mathematicians, computer scientists, and others to meet specific research or teaching needs. Some of these programs were also developed in conjunction with educators and students. With proprietary software, the user is denied access to the algorithms used in solving problems and thus cannot have complete confidence in the fidelity of the mathematical results obtained by the programs. On the other hand, open-source software provides the source code to its users so they can modify and apply it with confidence to their research endeavors. Sage is an important example of open-source software that contains one of the world’s largest collections of computational algorithms. For this reason, it is gaining in popularity among contemporary research mathematicians.

Software for Mathematics Education

Since the 1980s, computer software has been utilized regularly in the research of both pure and applied mathematics and it has made its way into mathematics classrooms. However, the adoption of mathematics software in teaching has not been without controversy. For instance, in 1993, students at the University of Pennsylvania complained about frustrations with Maple in calculus classes, citing a lack of support and faculty expertise. Some students even wore shirts printed with vulgarities about Maple, which attracted national attention. The use and implementation of software in classes has continued to generate debate regarding the balance between students exploring concepts and solving problems using traditional methods and computers. There are also questions regarding how much teaching time should be focused on instructing students in software use versus addressing concepts.

More recently, mathematics instruction in grades K–12 has benefited from computer software. This trend is due in part to the recommendations of major professional educational organizations and from federal programs and legislation. In 2000, the National Council of Teachers of Mathematics predicted that technology would enhance the learning of mathematics, support mathematics teaching, and influence the content that is taught. Educators have also praised the advantages of interactive software on student motivation and for providing a different modality for instruction—a modality that is visual, concrete, and interactive. Thus, anticipated impacts of computer technology on student achievement are encouraging. In 2002, the No Child Left Behind Act provided $15 million for research on the effects of computer technology on K–12 instruction.

Geometry Software

Computer software for teaching geometry is prevalent in American schools. The software of choice is dynamic software, which allows students to construct geometric shapes and actively explore their properties on the computer screen by (1) dragging vertices, (2) measuring component parts, (3) transforming them in the coordinate plan, (4) animating them, and (5) tracing points, and so on. Examples of dynamic geometry software are Cabri II Plus, The Geometer’s Sketchpad (GSP), and GeoGebra.

When using dynamic geometry software, high school students have been able to make new discoveries in Euclidean geometry. For example, in 1994, Ryan Morgan, a sophomore at Patapsco High School in Baltimore, used GSP to discover a generalization to Marion Walter’s theorem.

First, consider Marion Walter’s theorem: If the trisection points of the sides of any triangle are connected to the opposite vertices, the resulting hexagon has area one-tenth the area of the original triangle.

Based on the prior theorem, Morgan discovered the following: If the sides of the triangle are instead partitioned into n equal segments (for n= an odd integer) and each division point is connected to the opposite vertex, a central hexagon is still formed.

Morgan’s theorem states that this hexagon has an area

relative to the original triangle.

Discoveries by high school students, such as Morgan’s theorem, lend credence to using dynamic software for geometry instruction in the nation’s high schools.

Bibliography

American Mathematical Society. “Mathematics on the Web.” http://www.ams.org/mathweb/mi-software.html.

Bailey, David, and Borwein, Jonathan. “Experimental Mathematics: Examples, Methods and Implications.” Notices of the American Mathematical Society 52, no. 5 (2005).

DeLoughry, Thomas. “A Revolt Over Software: Penn Students Call Calculus Program Frustrating and Say Faculty Didn’t Know How to Use It.” The Chronicle of Higher Education 40, no. 14 (November 24, 1993).

Lutus, Paul. “Exploring Mathematics With Sage.” http://www.arachnoid.com/sage/.

Quesada, Antonio. “New Mathematical Findings by Secondary Students.” Universitas Scientiarum 6, no. 2 (2001). http://www.javeriana.edu.co/universitas‗scientiarum/universitas‗docs/vol6n2/ART1.htm.

Sarama, Julie, and Doug Clements. “Linking Research and Software Development.” Research on Technology and the Teaching and Learning of Mathematics 2 (2008).