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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
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.
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).
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,
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.
Presidentís 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: