Technology in the Mathematics Classroom: Guidelines from the Field
By
Shelley Goldman
The promise of using technology in mathematics classrooms to improve teaching and learning is especially bright. What a natural connection. Gathering data, analyzing it, looking for patterns, transacting among representations, developing algorithms, and thinking spatially are all greatly facilitated by computer use. These are some of the very activities that have been endorsed by the national mathematics standards (NCTM, 2000). Unfortunately, the promise of technology in school mathematics is largely unrealized. Many math classrooms have no computers, and in math classrooms where computers are present, they are used mainly for skill practice or as a supplement to the official math work.
Simply loading classrooms with technological devices won’t improve learning; principled use of computers, software, multimedia, and Internet resources is needed. Through a decade of research and curriculum development with the Middleschool Mathematics through Applications Project (MMAP)^{1}, we delineated a set of guidelines for technology in the mathematics classroom. They are shared with the hope that they might contribute to realizing the promise that technology holds for enhancing math learning. MMAP was one of the first projects in math education to simultaneously unite reforms in comprehensive curriculum and technology development. We experimented with the ways middle school math could be made more powerful by coupling content activities with computer technology. The results thus far have yielded a comprehensive middle school curriculum as well as products such as published research papers, CDs and web resources for teachers.
The overarching goal of the project was the development of the technologyintegrated, comprehensive curriculum. The centerpieces of the curriculum are designbased projects that draw on the integrated use of technology and provide plenty of opportunities for middle school mathematics to be made visible and used. The projects––designing a research center for Antarctic scientists, creating and breaking codes, helping to save guppies while their natural habitat is restored––offer opportunities for a set of standardsbased math topics such as proportional reasoning and function to be introduced, explored, developed, and practiced. Traditional paper materials and computer technologies are bundled with guides for teachers to sequence lessons and activities. MMAP technology environments range from friendly, gamelike simulations to more businesslike spreadsheets, but they all support activities that meet middle school standards and are mathematically and pedagogically significant (Goldman & Moschkovich 1998a, 1998b; Greeno et. al. 1997,1998; Knudsen & Briskman in press).
In the Antarctica Project, for example, students design an Antarctic research station for a reasonable cost that can adequately house four scientists as they work and live for two years. The application, ArchiTech, was developed to provide students with a computer aided design (CAD) system for working on and analyzing features of the station design. Students use the software to create many different designs that meet the project constraints and to experiment with and improve their designs by making changes in available design parameters as well as by changing and analyzing the shape, size, and features of their plans. The project takes up the topics of proportional reasoning and direct and inverse variation as well as providing review and practice opportunities in calculation, scale and measurement.
One of our goals was to develop technologies and research the roles they played in keeping more middle school students engaged in, and grappling with, mathematical processes and mathematical concepts. We saw an emerging body of complementary technology development addressing similar goals (e.g. JASPER, Function Probe, Geometer’s Sketchpad, Earth Lab, AT&T Learning Network). Unfortunately, classroom practice with technology outside of those instances was not yet a sustained, integrated aspect of classroom practice. We wanted to learn if it was feasible for technology to become a longterm partner in making the core math curriculum concepts and skills accessible to students. If it was feasible, we wanted to identify what issues stood in the way of universal adoption.
Early in the development process we knew that technology was motivating to students and thought we could capitalize on that fact to keep them interested in school mathematics. Frankly, we understood little about that motivating aspect, but relied on it while expecting to learn more. After seven years of development and classroom research, we moved beyond that real, yet simplistic depiction of how technology was powerful for students because we had documented many instances when technology leveraged learning in significant ways. Technology not only brought students to the table to do their classroom work, it also engaged them deeply and over time with mathematical content and activities that they found sustaining and relevant.
As it turns out, technology is neither panacea nor glitzy new classroom toy. It should and, not surprisingly, does get used only when and where it is helpful and, indeed, helps achieve learning goals. We documented many manifestations of the power of technology. Simulation environments, for example, provided sophisticated tools and manipulators for students, allowing them to gain handson experiences that took them beyond blocks and tiles, and promoted experiences that anchored explorations and conversations that were steeped in key mathematical concepts. Students moved easily among mathematical representations such as visual maps, tables, charts, graphs and standard mathematical expressions. Technology facilitated students’ efforts to generate, explore, discuss, and analyze data, which are all important goals in current mathematics teaching and learning (NCTM 2000).
The technology also provided a scaffold for moving students from concrete ways of thinking, operating and talking to abstract mathematical concepts, practices and language. It set the stage for students to make and test conjectures, find and analyze patterns and data, and draw conclusions. Finally, the technology made it easy to process large amounts of data, which freed teachers to use complex projects in the classroom without getting bogged down in excessive or inaccurate computation.
Guidelines for Technology in the Math Classroom
The intense, longterm research and development process we completed resulted in the identification and adoption of working guidelines for technology integration in mathematics classrooms. The MMAP design process was cyclical and comprised of generating ideas, prototyping, testing the application project ideas and software with small groups of middle school students, redesigning and then extensively field testing in classrooms. The research and development that led to the guidelines spanned thousands of hours of partnership with, and observations of, teachers and students at work with materials in their classrooms (Greeno, et.al. 1999). It also included six years of summer institutes and monthly meetings with a group of thirtyplus teachers where we debated the feasibility of, and tested, various combinations of application projects, classroom mathematics activities, and software functionality. What follows is a set of guidelines for successful integration of technologies in schools, particularly within mathematics. Although not exhaustive, these are the foundation from which we continue to explore, experiment with, and evaluate technology work.
1. Technology should be an integral part of learning and teaching and should not be used as an addon to the curriculum or simply for its own sake.
The use of computers and other pieces of technology in isolation from the core curriculum creates the needless separation of technology and important content. Computers need to be yet another tool that students have available for learning, as appropriate. Students and teachers will learn to use computers for solving problems when they are useful and integral parts of the curriculum, not separate and exotic addons.
By integrated, we mean that there should be a proper balance between the use of computer technologies and other resources and tools, and work with technology should bring you closer to the learning goals, not distract from them. It should not have an elevated status or dominate in terms of curriculum time. In MMAP, we designed the use of computers to weave tightly with learning goals and activities, and we estimated that students used computers only onethird of project time. In other classrooms, technologies should be used when they help students better understand or manage particular skills or concepts.
School learning is a social process, where teachers and students work together to explore, shape, refine, and make use of ideas. Thus, technologies should not be designed or chosen solely for the purposes of transferring information or practicing operations and skills. Instead, computer environments should foster easy representation, exploration, and public inspection, as well as easy modification of ideas, concepts and disciplinary practices. They should facilitate access by many to both individual and group activities and work.
2. Technologies should be designed, chosen, and used to support content area learning goals.
Important as it is, designing technologies with regard to social and cognitive aspects of learning is not enough; the special demand of teaching and learning a particular subject matter also needs to be considered. In mathematics education, we see several appropriate roles for computers:
 As simulation environments, in which students can test conjectures or generate related cases to make conjectures and look for patterns.
 As analytic tools, with which students can begin to describe relationships and patterns, arrange and manipulate tables and graphs, create formulas, and explore the relationships of these mathematical representations.
 As communication tools, with which teachers and students can engage in questioning, reasoning, record keeping, sharing, discussing, and explaining the work they are doing.
 As computational tools, with which students and teachers can operate on large or complex numbers, quantities, and values.
Computer environments for middle school should help students to focus their attention on math topics, such as scale and proportion, functions, geometric thinking, and logic. Within and across these topic areas, the technology should allow teachers and students to collect, organize, and operate on data; create and use mathematically meaningful representations; and create and test hypotheses. Computer environments should also make it easy to provide students with multiple representations of their activities. They should also make it easy and efficient for children to create and use visual representations, charts and spreadsheets, graphs, tables, and formulas.
3. Technologies as friends, not foes, of teachers.
Computers must make very obvious contributions to more successful learning if they are to be worth teachers’ time and efforts to learn and to manage. Even when there is achievement gain to be realized, bringing computers into the classroom can be seen as, and can actually be, just another problem for already overworked and overburdened classroom teachers. Taking on teaching with technology is a huge job, and it takes quite a bit of restructuring of the classroom experience.
Every teacher who used MMAP made substantive changes in the ways he or she approached classroom and curricular activities. Teachers planned differently, set up new physical spaces in the classroom, and innovated ways to move students through activities, space, and time. They thought about new approaches to mathematics content and new ways to assess mathematics learning. Restructuring their classrooms became a neverending job. Integrating technology was not a trivial task to the teachers with respect to their workload, so it needed to be worth it on the learning results side of the equation. It had to make a difference in the quality of the mathematics experience for it to come to the top of teachers’ priority lists.
4. The issue of inequitable access to computers can be addressed by proving the need for computers in the curricula.
The inequities associated with computer use in school are well documented. Some schools and classrooms have computers; many don’t. If mathematics learning and teaching is enhanced by welldeployed and thoughtful use of computers, how can we start bringing these experiences to more students?
There are no simple answers. Yet, as curriculum developers and researchers, we feel a responsibility for responding to inequities. It is not enough to say we cannot use technology because of access issues. Instead we must demonstrate a compelling curricular need for computers and dispel the myth that computers are a luxurious, frivolous, or incidental addon. When the value of computers was proven for improving workplaces, computers, software, and staff development followed. The same will be true of schools. When computers can be seen as contributing to improved success in core content areas, schools and teachers will make it a priority. With MMAP, teachers saw a greater number of students engaged, exploring, making conjectures and testing them out, debating how to proceed or how to analyze data, and generally becoming better problemsolvers. The impact potential was clear and they committed eagerly to new kinds of work.
In conclusion, to realize the promise of technology in the mathematics classroom, teachers need to see compelling demonstrations and results that make obvious a positive and indisputable impact on teaching and learning. The technologies developed must be targeted at meeting the social and learning needs of students. Technologies must not only be there, but they must be capable of engaging the students with meaningful content and in ways that support their connections to it. They must be in direct service of curricular and learning goals.
There is a price to pay financially and professionally as schools explore the potential of technology. Teachers must see that it advances their work if they are to make technology work a priority. Finally, and perhaps most importantly, technology integration will play a role in the school mathematics equity and access equation.
We have been able to document the ways in which technology has positive impact on the quality of the students’ mathematical experiences. Knowing this, we need to find ways to create more access to, and staying power for, technology in standardsbased classrooms. MMAP technology has helped us better understand the power of technology in the math learning enterprise. In that sense, it has provided a case study for what is possible.
Notes
1. MMAP was funded by grants from the National Science Foundation. This article reflects the opinions of its authors and does not necessarily reflect those of the National Science Foundation. I’d like to thank our colleagues at the Institute for Research on Learning for their work in defining these guidelines.
Bibliography
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Goldman, S., & Moschkovich, J.N., (1998a). Environments for collaborating mathematically. In J. Schnase and E. Cunniues (Eds.), Proceedings of the First International Conference on Computer Support for Collaborative Learning (pp. 143146). Bloomington, IN.
Goldman, S & Moschkovich, J. (1998b). Technology environments for middle school: Embedding mathematical activity in design projects. In J. Bruckman, et. al, Proceedings of the ICLS 98: International Conference of the Learning Sciences (pp. 112117). Atlanta: Georgia Tech University.
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Greeno, J.G., & Goldman, S. (Eds.) (1998). Thinking practices in mathematics and science learning. Mahwah, NJ: Lawrence Erlbaum Associates.
Greeno, J. G., McDermott, R., Cole, K., Engle, R. A., Goldman, S., Knudsen, J., Lauman, B., & Linde, C. (1999). Research, reform, and aims in education: Modes of action in search of each other. In E. Lagemann & L. Shulman (Eds.), Issues in education research: Problems and possibilities (pp. 299335). San Francisco: JosseyBass.
Greeno, J. G., & the Middleschool Mathematics through Applications Project Group (1997a). Theories and practices of thinking and learning to think. American Journal of Education, 106, 85126.
Greeno, J. G. & the Middleschool Mathematics through Applications Group (1997b). Participation as fundamental in mathematics education. In J. A. Dossey, J. O Swafford, M. Parmantie, & A. E. Dossey (Eds.), Proceedings of the Nineteenth Annual Meeting, North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 115). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Knudsen, J. & Briskman, P. (In press). Problem contexts and representations in middle schoolers learning functions. Pennsylvania Mathematics Teacher Council Yearbook.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.
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