• Reasons for Using Technology
• Implementation of Technology in Teaching
• Best and Worst Practices in Using Technology
• Evolution of Teaching Mathematics with Technology

VI. Technology Recommendations

Technology and the Learning of Mathematics

Professional teachers and mathematicians often differ about the exact extent of the use of technology in the teaching of mathematics. Included in the debate are whether or not to use it, when to use it, which technology to use, how to teach with it, and at which levels to use technology. Furthermore, technology is leading to developments in the teaching of mathematics which are still "works in progress" and are not ready to be fully evaluated. Nevertheless, there is a consensus in a large portion of the mathematical community regarding the use of technology and its position in mathematics education.

Reasons for Using Technology

In general, technology includes the various levels of available technology: calculators, CBL’s, computers, and laptops. The particular technology changes depending on resources and the level of coursework. There are a few guiding principles – like computers should not be used as fancy calculators; primarily, the determination of the technology depends upon the material being taught and the resources available. The strongest reason to use technology, of any sort, occurs upon those occasions when it can drive the introduction/discovery of new material.

The experience of mathematics educators lead us to believe that technology helps promote the following:

• Understanding the concepts of function and functional behavior;
• Support of the development of algebraic concepts;
• Visualization of solutions and functions;
• Understanding estimation and its applications;
• Development of intuition and pattern recognition;
• Confirmation of algebraic solutions
• Understanding technology’s failure to solve certain problems;
• Use of and appreciation of scientific notation;
• Evolution of a vision of mathematics as a laboratory science, which allows student group projects, written reports, and an interdisciplinary understanding of mathematics and other areas of study;
• Consideration of more realistic problems;
• Attention to applications even when complicated algebraic computations are involved
• Visual understanding of mathematics in general and functions in particular;
• Improvement in communication skills;
• Increased attention of students;
• Understanding of the reality of mathematics and the applications of mathematics.
• Use of a broader selection of functions, less meaningless memorization, and more emphasis on intellectual understanding.

Implementation of Technology in Teaching

Ultimately, technology is a tool for teaching. Accordingly, it can be used effectively or inappropriately. An unfortunate trend in mathematics education is that in individual cases, the technology is introduced without an understanding of the time needed to learn to make the process effective. The inclusion of technology into an existing course is the equivalent of starting an entirely new course preparation; nothing stays the same if the implementation is done correctly. The following aspects of including technology in a course need to be carefully considered by instructors and administrators alike:

• Making testing work with technology
• Determining the type of technology as well as the make of technology to be used.
• Determining the amount of technology to buy.
• Considering which topics should be taught using technology, and which should not.
• Adjusting for the change in time expenditure which occurs when technology is introduced.
• Constructing strategies designed to be course-specific.

There are courses where the technology is easily included, such as differential equations and differential calculus; other courses do not make the inclusion as obvious, for example algebra I and abstract algebra. In each case, there is a common body of considerations that can be used as initial guidelines.

Best and Worst Practices in Using Technology

In each course, it is helpful to consider the following actions when designing the introduction of technology:

1. Limits: When considering the goals of the course, include appropriate levels of technology and yet do not force the technology to appear in an "unnatural" or contrived manner. Also, technology has its limits. For example, different "makes" and even different models of technology include different hierarchies in the order of operations and in the handling of fractional exponents (as in graphing ).
2. Core Material: For each course, there is a core of material which may or may not be introduced with technology but which should always be known and understood independent of technology. For example, recognition of basic graphs and the production of basic graphs should be possible without technology. For example, in an algebra class, a student should be able to produce and recognize the graphs of the following functions without the use of technology: and
3. Algorithmic: Although technology is excellent at performing algorithms repeatedly, students should not be given work which induces in them the belief that mathematics is best undertaken with the "brain turned off and the calculator turned on".
4. Keystrokes: Keystrokes or other explanations of how to use a particular technology should be kept at a minimum, especially as the course level increases. Instead, the explanation of the steps in the process should be included.
5. Written Work: The use of technology does not eliminate the need to write mathematics and to communicate it. An emphasis on the communication of the construction of the answer is vital. Consequently, "handwork" must be included in all problems. This means the inclusion of the mathematical sense of what the student did to solve the problem and why; normally, it does not include lists of keystrokes.
6. Holistic approach: For technology use to be successful and to ensure a more successful transition to post-secondary mathematics work, the presentation of mathematical work, on the part of the student and the teacher, needs to include the use of problem-solving/mental analysis, written and oral communication, and the appropriate level of technology use.
7. A Tool: In all mathematics classes, it must be remembered that the technology is a tool, not the purpose of the course. Mathematical understanding is the sole guiding principle, not teaching to the technology's capabilities.

Examples of topics in precalculus or algebra which benefit greatly from the use of technology include:

• Higher degree polynomials
• Range
• Intersection of curves
• Non-standard angles in trigonometry
• Exponential functions
• Visualization of roots and factoring
• Solutions of inequalities

Other topics do not lend themselves to technology nearly as well. The challenge is to understand the difference. Ultimately, we must understand that there is a strong need for both the traditional and technological skills.

Evolution of Teaching Mathematics with Technology

The use of technology in the teaching of mathematics is an evolutionary event. While there are certain areas which are helped by the use of technology, there are other areas where the pedagogical benefit are not as clear. Layered upon this are the changes in curriculum, logistics and resources brought about by the use of technology in mathematics teaching. Finally and ultimately, we must answer the questions: Does technology devalue mathematics as a discipline? Is it possible that the beauty of mathematics and its logic become less appreciated?

With regard to the pedagogical questions, we must remember that technology must be introduced and developed with care. An interesting observation is that the really good mathematics students use technology the best and at the same time, the least often. If not developed properly, technology can hamper learning instead of enhancing it.

The changes in curriculum, logistics and resources brought about by the use of technology in mathematics teaching require both a more efficient use of resources – time, energy, and money – as well as more resources. Mathematics can now be viewed as a laboratory science with all the concomitant resource expenditures and lab preparations. This requires that teachers share information through in-house workshops and worksheet/activity sharing. At the same time, teachers must be allowed access to more external workshops, support technology persons, subscriptions, software purchase, planning time, released time for curriculum overhaul, and current technologies.

Finally, as a community, students, parents and teachers must continue to realistically evaluate and assess the benefits of technology. It would be a real shame to enable students to be more capable mathematically while at the same time decreasing their interest in and understanding of the beauty of mathematics.

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Last updated: May 11, 1999
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