What strategy shall we choose for thinking about and selecting among cognitive technologies in mathematics education? I argue for the need to move beyond the familiar cookbooks of 1,001 things, in near random order, that one can do with a computer. Such lists are usually so vast as to be unusable in guiding the current choice and the future developments of mathematics educational technologies. Instead, we should seek out high leverage aspects of information technologies that promote the development of mathematical thinking skills. I thus propose a list of "transcendent functions" for cognitive technologies in mathematics education. What is the status of such a list of functions? Incorporating them into a piece of software would certainly not be sufficient to promote mathematical thinking. The strategy is more probabilistic-other things being equal, more students are likely to think mathematically more frequently when technologies incorporate these functions. Some few students will become prodigious mathematical thinkers, whatever obstacles must be overcome in the mathematics education they face. Others will not thrive without a richer environment for fostering mathematical thinking. This taxonomy is designed to serve as a heuristic, or guide. Assessments of whether it is useful will emerge from empirical research programs, not from intuitive conjecture. Indeed, until tighter connections can be drawn between theory and practice, the list can only build on what we know from research in the cognitive sciences; it should not be limited by that research.
Finally, why should we focus on transcendent functions? There are two major reasons. We would like to know what functions can be common to all mathematical cognitive technologies, so that each technology need not be created from the ground up, mathematical domain by mathematical domain. We would like the functions to be transcendent in the sense that they apply not only to arithmetic, or algebra, or calculus, but potentially across a wide array, if not all, of the disciplines of mathematical education, past, present, and future. The transcendent functions of mathematical cognitive technologies should thus survive changes in the K-12 math curricula, since they exploit general features of what it means to think mathematically-features that are at the core of the psychology of mathematics cognition and learning. These functions should be central regardless of the career emphasis of the students and regardless of their academic future. Lessons learned about these functions from research and practice should allow productive generalizations. The transcendent functions to be highlighted are those presumed to have great impact on mathematical thinking. They neither begin nor end with the computer but arise in the course of teaching, as part of human interaction. Educational technologies thus only have a role within the contexts of human action and purpose. Nonetheless, interactive media may offer extensions of these critical functions. Let us consider what these extensions are and how they make the nature or variety of mathematical experience qualitatively different and more likely to precipitate mathematics learning and development. These functions are by no means independent, nor is it possible to make them so. They define central tendencies with fuzzy boundaries, like concepts in general (Rosch & Mervis, 1975). They are also not presented in order of relative importance.
I will illustrate by examples how many outstanding, recently developed mathematical educational technologies incorporate many of the functions. But very few of these programs reflect all of the functions. And only rare examples in classical computer assisted instruction, where electronic versions of drill and practice activities have predominated, incorporate any of the functions. One could approach the question of technologies for math education in quite different ways than the one proposed. One might imagine approaches that assume the dominant role for technology to be amplifier: to give students more practice, more quickly, in applying algorithms that can be carried out faster by computers than otherwise. One could discuss the best ways of using computers for teacher record-keeping, preparing problems for tests, or grading tests. In none of these approaches, however, can computers be considered cognitive technologies.
I will illustrate by examples how many outstanding, recently developed mathematical educational technologies incorporate many of the functions. But very few of these programs reflect all of the functions. And only rare examples in classical computer assisted instruction, where electronic versions of drill and practice activities have predominated, incorporate any of the functions. One could approach the question of technologies for math education in quite different ways than the one proposed. One might imagine approaches that assume the dominant role for technology to be amplifier: to give students more practice, more quickly, in applying algorithms that can be carried out faster by computers than otherwise. One could discuss the best ways of using computers for teacher record-keeping, preparing problems for tests, or grading tests. In none of these approaches, however, can computers be considered cognitive technologies.
A different perspective on the roles of computer technologies in mathematics education is taken by Kelman et al. (1981) in their book, Computers in Teaching Mathematics. They describe various ways software can help create an effective environment for student problem solving in mathematics. Their comprehensive book is organized according to traditional software categories and curriculum objectives: computer assisted instruction, problem solving, computer graphics, applied mathematics, computer science. programming and programming languages. The spirit of their recommendations is in harmony with the sketch I propose in this chapter, although their orientation is predominantly curricular rather than cognitive. My stress on transcendent functions is thus a complementary approach, taking as a starting point the root or foundational psychological processes embodied in software that engages mathematical thinking.
In my choice of software illustrations I have leaned heavily toward cases that manifest most clearly the specific loci supporting the seven Purpose or Process functions. Although programming languages, spreadsheets, simulation modeling languages such as MicroDynamo (Addison-Wesley), and symbolic calculators such as muMath (Microsoft) and TK!Solver (Software Arts) can be central to thinking mathematically in an information age (e.g., Elgarten, Posamentier, and Moresh, 1983), I have seldom chosen them as examples. Although I take for granted the utility and power of these types of tools in the hands of a person committed to problem solving, their usefulness stems in part from the extent to which they incorporate the purpose and process functions. For example, Logo graphics programming provides the different mathematical representations of procedural text instructions and the graphics drawing it creates (Process Function 3); and simulation modeling languages and spreadsheets are excellent environments for mathematical exploration (Process Function 2), since hypothesis-testing and model development and refinement are central uses of these interactive software tools. But other environments in which these tools are used-for example, drill and practice on programming language syntax or abstract exercises to write programs to create fibonnaci number series need not offer much encouragement for mathematical thinking. In other words, the intrinsic value of such tools in helping students think mathematically is not a given. The stress on Functions remains central.
In my choice of software illustrations I have leaned heavily toward cases that manifest most clearly the specific loci supporting the seven Purpose or Process functions. Although programming languages, spreadsheets, simulation modeling languages such as MicroDynamo (Addison-Wesley), and symbolic calculators such as muMath (Microsoft) and TK!Solver (Software Arts) can be central to thinking mathematically in an information age (e.g., Elgarten, Posamentier, and Moresh, 1983), I have seldom chosen them as examples. Although I take for granted the utility and power of these types of tools in the hands of a person committed to problem solving, their usefulness stems in part from the extent to which they incorporate the purpose and process functions. For example, Logo graphics programming provides the different mathematical representations of procedural text instructions and the graphics drawing it creates (Process Function 3); and simulation modeling languages and spreadsheets are excellent environments for mathematical exploration (Process Function 2), since hypothesis-testing and model development and refinement are central uses of these interactive software tools. But other environments in which these tools are used-for example, drill and practice on programming language syntax or abstract exercises to write programs to create fibonnaci number series need not offer much encouragement for mathematical thinking. In other words, the intrinsic value of such tools in helping students think mathematically is not a given. The stress on Functions remains central.
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