Mathematics

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Guidelines

We have devised a set of guidelines to shape our development of activities and materials. These guidelines reflect what we believe to be appropriate uses of technology in mathematics teaching. Many of our activities follow the guidelines below, as applicable:

Introduce technology in context

Address worthwhile mathematics with appropriate pedagogy

Take advantage of technology

Connect mathematics topics

Incorporate multiple representations

Introduce technology in context
Features of technology, whether mathematics-specific or more generic, should be introduced and illustrated in the context of meaningful content-based activities. Teaching a set of technology or software-based skills, and then trying to find mathematical topics for which they might be useful, is comparable to teaching a set of procedural mathematical skills and then giving a collection of "word problems" to solve using the procedures. Such an approach can obscure the purpose of learning and using technology, make mathematics appear as an afterthought, and lead to contrived activities. The use of technology in mathematics teaching is not for the purpose of teaching about technology, but for the purpose of enhancing mathematics teaching and learning with technology. Furthermore, in our experience, teachers who learn about technology while using it to explore mathematics topics are more likely to see its potential benefits and use it in their subsequent teaching. This guideline is in accord with the first recommendation of the President's Committee of Advisors on Science and Technology, Panel on Educational Technology (1997): "Focus on learning with technology, not about technology" (p. 7).

Address worthwhile mathematics with appropriate pedagogy
Content-based activities using technology should address worthwhile mathematics concepts, procedures and strategies, and should reflect the nature and spirit of mathematics. Activities should support sound mathematical curricular goals and should not be developed merely because technology makes them possible. Indeed, the use of technology in mathematics teaching should support and facilitate conceptual development, exploration, reasoning and problem solving, as described by the National Council of Teachers of Mathematics [NCTM] (1989, 2000).

Technology should not be used to carry out procedures without appropriate mathematical and technological understanding (e.g., inserting rote formulas into spreadsheets). Nor should it be used in ways that can distract from the underlying mathematics (e.g., adding too many bells and whistles in a PowerPoint presentation that the mathematics gets lost). In other words, mathematical content should not be compromised.

Another way to prevent technology use from compromising mathematics is to encourage users to connect their experiential findings to more formal aspects of mathematics. For example, students using software to explore geometric shapes and relationships should be asked to use previously proved theorems to validate their empirical results, or use their new findings to propose new conjectures. In other words, technology should not influence students to take things at face value or to become what Schoenfeld (1985) referred to as "naive empiricists." This guideline is in accord with the second recommendation of the President's Committee of Advisors on Science and Technology, Panel on Educational Technology (1997): "Emphasize content and pedagogy, and not just hardware" (p. 7).

Take advantage of technology
Activities should take advantage of the capabilities of technology, and hence should extend beyond or significantly enhance what could be done without technology. Technology enables users to explore topics in more depth (e.g., interconnect mathematics topics, write programs, devise multiple proofs and solutions) and in more interactive ways (e.g., simulations, data collection with probes). Technology also makes accessible the study of mathematics topics that were previously impractical, such as recursion and regression, by removing computational constraints.

Using technology to teach the same mathematical topics, in fundamentally the same ways, that could be taught without technology does not strengthen students' learning of mathematics and belies the usefulness of technology. Furthermore, using technology to perform tasks that are just as easily or even better carried out without technology may actually be a hindrance to learning. Such uses of technology may convince teachers and administrators that preparing teachers to use technology is not worth the considerable effort and expense necessary to do so.

This guideline supports the technology principle of NCTM Principles and Standards of School Mathematics: "Teachers should use technology to enhance their students learning opportunities by selecting or creating mathematical tasks that take advantage of what technology can do efficiently and well- graphing, visualizing and computing" (NCTM, 2000, p. 25).

Connect mathematics topics
Technology-augmented activities should facilitate mathematical connections in two ways: (a) interconnect mathematics topics and (b) connect mathematics to real-world phenomena. Technology "blurs some of the artificial separations among some topics in algebra, geometry and data analysis by allowing students to use ideas from one area of mathematics to better understand another area of mathematics" (NCTM, 2000, p. 26). Many school mathematics topics can be used to model and resolve situations arising in the physical, biological, environmental, social, and managerial sciences. Many topics can be connected to the arts and humanities as well. Appropriate use of technology can facilitate such applications by providing ready access to real data and information, by making the inclusion of mathematics topics useful for applications more practical (e.g., regression and recursion), and by making it easier for teachers and students to bring together multiple representations of mathematics topics. This guideline supports the curriculum standards of the NCTM (1989, 2000).

Incorporate multiple representations
Activities should incorporate multiple representations of mathematical topics. Research shows that many students have difficulty connecting the verbal, graphical, numerical and algebraic representations of mathematical functions (Goldenberg, 1988; Leinhardt, Zaslavsky & Stein, 1990). Appropriate use of technology can be effective in helping students make such connections (e.g., connecting tabulated data to graph and curves of best fit, generating sequences and series numerically, algebraically, and geometrically). "We, as mathematics educators, should make the best use of multiple representations, especially those enhances by the use of technology, encourage and help our students to apply multiple approaches to mathematical problem solving and engage them in creative thinking" (Jiang & McClintock, 2000, p. 19).

References

Goldenberg, E.P. (1988). Mathematics, metaphors, and human factors: Mathematical, technical, and pedagogical challenges in the educational use of graphical representations. Journal of Mathematical Behavior, 7, 135-173.

Jiang, Z., & McClintock, E. (2000). Multiple approaches to problem solving and the use of technology. Journal of Computers in Mathematics and Science Teaching, 19(1), 7-20.

Leinhardt, G., Zaslavsky, O., & Stein, M.K. (1990). Functions, graphs, and graphing: Tasks, learning and teaching. Review of Educational Research, 60(1), 1-64.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

-----. (2000). Principles and standards for school mathematics. Reston, VA: Author.

President’s Committee of Advisors on Science and Technology, Panel on Educational Technology. (1997). Report to the President on the use of technology to strengthen K-12 education in the United States.

Schoenfeld, A. (1985). Mathematical problem solving. New York: Academic Press.