JUDAH L. SCHWARTZ

CAN TECHNOLOGY HELP US MAKE THE MATHEMATICS CURRICULUM INTELLECTUALLY STIMULATING AND SOCIALLY RESPONSIBLE?

ABSTRACT. In order to answer the question posed in the title of this paper, we must take a wide perspective and explore the goal societies have for maintaining educational systems, how curriculum contributes to the attainment of these goals, how mathematics in the curriculum contributes to effectiveness in attaining these goals and finally some of the ways in which appropriately crafted technology can help to make mathematics a more effective part of the curriculum.

1. THE EDUCATIONAL GOALS OF SOCIETY AND THE CONTRIBUTIONS OF MATHEMATICS TO THEM Societies maintain educational systems for a variety of purposes. While societies strike different balances among these purposes, traditionally (in the spirit of Dewey’s observations, 1959) such purposes include aiding the personal growth and development of citizens, preparing people for the world of work, and transmitting the culture and values of the society. • Assisting the personal growth and development of individuals Most societies claim to want their educational systems to help young people develop as distinct individuals with fulfilling lives and relationships. To be sure, the balance between the development of the individual qua individual and the development of the individual qua member of the collective varies from one society to the next. • Preparation of people for the world of work By far the dominant expectation of education in most societies, at least as articulated by political leaders, the media and often the public at large, is to prepare people for productive contribution to the world of work. • Transmission of the culture Over the long term probably the most important role that a society expects its various educative agencies to fulfill is that of transmitting the culture and the values of the society. How do modern societies transmit their culture and heritage from one generation to the next? It International Journal of Computers for Mathematical Learning 4: 99–119, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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is clear that school, home, family, religious institutions and the media all play important roles in addressing this goal of education. One may analyze the behavior of any aspect of the educational system through the prism of these three stated purposes. In particular, we may ask how school curricula do and/or do not further the attainment of these aims in a society. In order to understand how mathematics contributes to the furthering of a society’s aims for its educational system, it is particularly important examine school curricula. For the most part, educative agencies of the society other than school such as the family, the church and the media play only a minimal role in the mathematics education of the society. Transmitting the Culture Inclusion of language and literature in the school curriculum is meant to transmit the culture and values of the society. To be sure, other societal institutions play an important role in this regard – perhaps even a dominant one. The inclusion of music and the visual arts into the school curriculum is often justified on the grounds of contributing to the school’s role in transmitting the culture and values of the society. In contrast, people rarely justify the presence of mathematics in the curriculum on these grounds (e.g., the 1989 NCTM Standards make only passing mention of such issues). Perhaps one reason for this difference in perception lies in the nature of the experience of people with these different fields. In the case of literature or music, most people have, at some point in the course of their formal education, and often for many years beyond, some direct personal experience with literature and/or music. People watch drama on television or the stage. Some people act in amateur theater groups. People listen to music. Some people sing in choirs. People read books. Some people write them. Most people deliberately undertake to partake of literature and music, sometimes more actively, sometimes less so. In contrast, most formal school experience never gives students the opportunity to do anything with mathematics except lean back and let it wash over them. In short, it would seem that the elements of cultural heritage are those things that most everybody engages in, however pale that engagement may be compared to the achievements of the Mozarts and the Shakespeares of the society. Might the role of mathematics in this respect be different? I do not believe that the current situation will change until there are opportunities for young people to participate in the making of mathematics in the ways in which they participate in the making of language and music. I do not mean simply rehearsing the theorems and experiments others have devised. I mean that they must also have the tools to devise and use quantitative

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measures of their own, to make and analyze models of their own, and to fashion and explore conjectures of their own. I will argue later that newly emerging technologies can provide an unprecedented set of tools with which to equip our students to do precisely this. Preparing People for the World of Work The presence of the natural and social sciences in the school curriculum of all of our societies is largely attributed to the need to prepare the next generation of people to function in the global economy. People who function in very diverse sorts of economic positions need to have a sense of place and time as well as some understanding of that which is physically possible and that which is not. Of course, the specialized needs for education in the natural and social sciences will vary depending on what role a person plays in the world of work. The engineer is likely to have greater need for a background in physics than the social worker, who in turn is likely to have a greater need for a background in anthropology than the engineer. Because modern societies are committed to not asking people to make career choices before puberty at the earliest, there is a period of general education which must educate broadly, if not deeply, in the natural and social sciences. While I believe that mathematics is neither a natural or social science, it is closely linked to them because of the role that mathematical tools play in these disciplines. As a result, mathematics is clearly recognized as having a central role in preparing people for the world of work. This recognition is sometimes informed, thoughtful and deep. Often however, this recognition is the result of tradition that is not reinspected or reconsidered for decades, and sometimes longer than that. Roughly speaking, the mathematics of a general primary and secondary education consists of a variety of topics in arithmetic, geometry, algebra, and calculus. In addition, many have argued that a general primary and secondary mathematical education should also include topics in logic, discrete mathematics and mathematical structures (indeed such recommendations are central to the 1989 NCTM Standards). The rhetoric of education also calls for students learning these subjects with understanding and insight. Unfortunately, it is the rare student who attains more than a mechanical, manipulative mastery of the mathematics that he or she learns at school. Might it be different? Certainly, recent advances in technology make it possible for us to address nontraditional and exciting new content in powerful ways.1 Technology even makes possible the addressing of

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traditional content in profoundly more interesting and challenging ways. Moreover, it permits both students and teachers to ask and explore a richer and more realistic range of issues, and to develop intellectual “habits of mind” that may serve them in good stead well beyond the immediate setting in which they are developed. (For an example of the way in which habits of mind are developed into a curricular structure, see Goldenberg [here].)

Aiding Personal Growth and Development There are a variety of elements of the school curriculum that are directly addressed to this purpose of education. School programs in the visual arts, music and crafts of various sorts are largely addressed to helping youngsters develop aesthetic sensitivity and provide opportunity for personal fulfillment. In addition to these curriculum content areas, since the close of the second World War, and in many instances preceding that, the rhetoric of educators has often called for the school to play an important role in the social and emotional development of students (Powell, Farrar and Cohen, 1985; Cuban, 1993), all too often at the expense of the acquisition of important substantive knowledge. With the growth of the conservative perspective in the past several decades the role of the school curriculum in aiding personal growth and development has become a hard-fought political issue (see for example the critiques of Hirsch (1996)). Increasingly we find pressures on school and school curricula that emphasize developing academic knowledge, at the expense of helping youngsters develop intellectual independence and thoughtful, critical inquiring modes of thinking. These opposing pressures can have important consequences for curriculum content. The presence of mathematics in the school curriculum is rarely justified in terms of the enhancement of the personal growth and development of students or as arena in which young people can grow and develop selfconfidence and self-esteem. Nor do people tend to think of mathematics as an arena in which students come to develop any sense of the aesthetic. To the extent that school can play this sort of role, it is going to be in subject areas other than mathematics. I believe that, given the nature of mathematics instruction and curriculum available to most students in most societies, this is an accurate perception. This recognition is painful to me as a mathematics educator and I wish it were otherwise. In particular it is painful because it means that most people never experience mathematics as a source of beauty and delight.

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Might it be different? I will argue that the judicious and thoughtful use of technology can affect both the practice and the perception of mathematics in schools. In what way(s) could and should the mathematics curriculum be affected by the rapidly developing technologies that surround us? I believe that thoughtful use of technology can effect the role of mathematics in schools, with respect to the educative purposes of aiding personal growth and development, preparing people for the world of work, as well as for transmitting the culture and values of the society. This paper will address the following question – given the reasons that societies maintain educational systems whose curricula universally include at least some mathematics, and the ways in which mathematics curricula could be changed in light of new technologies, what changes might, in fact, be considered desirable by those holding different educational goals?

2. THE STRUCTURE OF MATHEMATICS CURRICULA In order to analyze how mathematics curricula might contribute more effectively to the attainment of society’s educational goals, we need to understand how curricula are currently structured and some alternatives. There is a tendency among those who try to delineate the content of mathematical curricula to confound the mathematical content of the curriculum with the kinds of behaviors they would like to see students exercise both in and out of mathematics classes. For example, the New Standards Project,2 an education reform effort in the United States, lists the following eight elements in a single, undifferentiated list that it regards as appropriate to secondary school. • • • • • • • •

number and operation concepts geometry and measurement concepts function and algebra concepts statistics and probability concepts problem-solving and mathematical reasoning mathematical skills and tools mathematical communication putting mathematics to work

Within this list, there are some indications of a stress on mathematics as part of the preparation of youngsters for the workplace (the choice of topics, the focus on reasoning and communication, and the final bullet’s explicit reference to work). On the other hand, it is not clear what importance is attached in this curricular structure to such goals as the

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transmission of the culture or the personal growth and development of students. The list, however, is a puzzling one since the last four entries on the list could refer to any body of mathematical content. These elements of the list tell us something about what they would like students to be able to do mathematically, but nothing about the mathematical content that they are to do it with. On the other hand, each of the first four entries on the list refers to a particular kind of mathematical object, its attributes, and the actions that can be done with and on such objects. Similarly, the curriculum standards of the National Council of Teachers of Mathematics (NCTM, 1989) lists the following thirteen elements in a single list that it regards as appropriate for youngsters ages 11 to 14. • • • • • • • • • • • • •

Mathematics as Problem Solving Mathematics as Communication Mathematics as Reasoning Mathematical Connections Number and Number Relationships Number Systems and Number Theory Computation and Estimation Patterns and Functions Algebra Statistics Probability Geometry Measurement

Here too, the reform places emphasis on mathematics as part of the preparation of youngsters for the workplace. Here too, it is not clear what importance is attached in this curricular structure to such goals as the transmission of the culture or the personal growth and development of students. This list, too, is puzzling. The first four entries could refer to any body of mathematical content. These elements of the list tell us something about the insights the NCTM would like students to have and what they would like students to be able to do mathematically, but nothing about the mathematical content that they are to do it with. At the same time the remaining entries on the list refer to a set of mathematical subfields, each of which deal with particular kinds of mathematical objects, their attributes, and the actions that can be done with and on such objects. Both of these approaches suffer from the category error of confounding mathematical objects and their attributes with both mathematical and general cognitive actions. Nonetheless, it is the case that we would like

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students to learn about a variety of mathematical objects as well as develop the ability to do certain kinds activities with them. Thus, I suggest that a list may not be the appropriate logical structure to use to describe the skeletal structure of a curriculum. Recognizing that mathematical objects and mathematical actions are both important and are different from one another leads me to conclude that a more sensible structure to describe the skeleton of a curriculum is a matrix, one of whose dimensions describes objects and the other actions. One benefit of such a structure is that it, more than a list, lends itself to a nonlinear perspective on curriculum. What sorts of mathematical objects do we want people to be able to use with some agility, and what kinds of actions would we like people to be able to do with these objects? To illustrate the nature of such a matrix, below is a list of objects and actions which is similar in spirit to the recommendations outlined earlier. My candidate list of mathematical objects3 includes • number and quantity – integers, rationals, reals – measures (length, area, volume, time, weight, etc.) • shape and space – shapes defined by topological properties such as connectedness and enclosure – shapes defined by metric properties (on planes and other manifolds) • pattern and function – on the domain of numbers and quantities – on the domain of shapes and spaces • data – collections of counted/measured numbers/quantities) – structured data sets, distributions, moments, etc. • arrangements – permutations, combinations, networks, trees, etc. My candidate list of mathematical actions4 with these objects includes • representing, formulating and modeling – observe and gather data, both qualitative and quantitative (counted and measured) – make informed and reasonable judgments about that which should be considered and that which could be ignored in a given situation

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– make reasonable estimates of lengths, weights, times, areas, volumes, costs, etc. – represent relationships among mathematical objects (quantities, shapes, patterns, etc.) graphically and symbolically • manipulating and transforming – manipulate mathematical objects (quantities, shapes, patterns, etc.) using appropriate operations – transform relationships among mathematical objects using appropriate symbolic and graphical operations • inferring and drawing conclusions – make inferences about assumptions and revise models as needed – understand and utilize the distinction between necessary and sufficient conditions – compare and contrast – make inferences about invariance, symmetry, extreme cases, point of view • communicating5 – communicate clearly in both oral and written (verbal and graphical) form Here too, as with the earlier lists, it is not clear how a mathematics curriculum with such a structure might contribute to the attainment of society’s educational goals. I will, however argue in the next section, that the OBJECT × ACTION framework for the organization of the curriculum lends itself particularly well to the design of technological tools for the enhancement of the mathematical curriculum and its ability to play a larger role in the attainment of all three of society’s educational goals. I will argue that technological tools designed from such a perspective have this potential because they support student exploration of mathematical phenomena. 3. HOW MIGHT OBJECT × ACTION BASED TECHNOLOGY ENHANCE THE CONTRIBUTION OF MATHEMATICS CURRICULA TO THE ATTAINMENT OF SOCIETY’S EDUCATIONAL GOALS? Increasingly, over the last decade, teachers and researchers have found that thoughtfully crafted technology-based expository curricular materials and exploratory tools can have profound effects on both the content of the

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mathematics curriculum and what happens in the mathematics classroom (for two extensive reviews use Olive, 1996; Kaput, 1992). These effects, can influence the role of the mathematics curriculum in the way society seeks to address the purposes of education. I believe that the positive effects of appropriately crafted software on mathematics learning stem, in the main, from the fact that the users of such software, be they students or teachers, control either the intellectual content of their efforts, or the mode of interacting with the content, or both. In my view enabling a person to have such idiosyncratic, self-driven, creative mathematical experiences is the key to changing the way the subject is perceived and thus the ways in which it can help us, as a society, to address our educational goals. As a way of making this discussion more specific, let us consider five aspects of mathematical activity. They are: (a) (b) (c) (d) (e)

conjecturing and exploring acquiring, evaluating and analyzing data modeling one’s world conceptually grounding manipulative skills deepening and broadening understanding

I contend that our ability to enhance the ability of students and teachers in each of these areas can be made substantially greater by the thoughtful use of thoughtfully-crafted software, and that moreover, by so doing, we are likely to enhance the ability of mathematics curricula to contribute to the attainment of society’s educational goals. a. Conjecturing and Exploring The act of making and exploring conjectures about mathematical objects is the heart-and-soul of making mathematics. A feature of many of the technological environments for mathematics education that have been developed recently is that of making it easy for users to make and explore conjectures. How do these exploratory environments make it easy for users to make and explore conjectures? Perhaps the easiest way to answer the question is to invoke once again OBJECT × ACTION metaphor. These environments track the user’s interaction with the environment and parse that interaction into the objects that the user is acting on and the actions that the user is carrying out on those objects. This parsing of the interaction makes it possible for the environment to formulate the set of actions as a procedural entity, with a name if desired. This procedure may then be repeated on other objects. Interesting properties discovered to be true in the case of acting on one object may then be searched for when the

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procedure is carried out on other objects. Thus the exploratory environment substantially lowers the difficulty of exploring conjectures. For example, consider the following two illustrations from the Geometric superSupposer (Schwartz and Yerushalmy, 1992) – the first being a standard theorem in Euclidean Geometry, the second an insufficiently explored conjecture with an associated challenge. (i) Join in order the mid-points of the segments AB, BC, CD, and DA that form the four sides of a quadrilateral. This construction is an ACTION carried out on a quadrilateral OBJECT. The resulting figure is seen to be a parallelogram. The environment allows the user to explore the generality of this result by repeating the construction in two rather different ways; (1) by constructing or choosing a new quadrilateral on which the construction procedure might be carried out once again, or (2) by dynamically dragging any of the original four points A, B, C or D that define the original quadrilateral. Each of these modes builds belief in the plausibility of the conjecture in quite different psychological ways.6 (ii) Consider an arbitrary triangle ABC (OBJECT). Reflect each vertex over its opposite side. Join the image points to one another thus forming a new triangle (ACTION). This procedure is clearly a function from the domain of triangles to the range of triangles. Conjecture: If this function is composed with itself repeatedly, the image triangle either collapses to a line or approaches an equilateral triangle. A related challenge: Given an arbitrary triangle, can one find its pre-image under this operation? Is it unique? i.e., does this function have an inverse? These two examples illustrate the power of such environments to engage people at a variety of levels of mathematical sophistication. The first example is an exploration of a standard, but surprising, theorem in Euclidean geometry studied in classrooms the world over. The second illustrates a conjecture that has engaged research mathematicians, and to the best of my knowledge has not been proven or disproven at the time of this writing. The reader ought not think that such technological possibilities are somehow the special province of geometry. Similar technological environments that allow users to perform actions on functions as mathematical objects have also been written (Schwartz and Yerushalmy, 1995). This kind of environment leads quite naturally to a deeper understanding of the solving of both linear and nonlinear differential equations. This is because a differential equation can be regarded as manipulating with a differential operator (ACTION) on an as yet undetermined function (OBJECT) resulting either in the appropriate inhomogeneous function (or zero in the case of a homogeneous equation). Taylor expansions are naturally thought

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of as procedures that take two arguments – the function being expanded and point about which the expansion is carried out. In such environments, once the procedure is written, it can be carried out in rapid succession on many different values of its arguments. It has been found, in many parts of the world, that if students are given tools that make it easy for them to explore conjectures, they will both make and explore conjectures. This means that they will have, in some measure, something of the experience of making mathematics. This, in turn means that mathematics can begin to play a role in the development of self-confidence and self-esteem of the student. If people emerge from the school experience with mathematics as an engaging element of school life that they found satisfying and perhaps even rewarding, then over time, it is likely that mathematics will come to be seen as part of the culture in fact as well as in name.

b. Acquiring, Evaluating and Analyzing Data We can now offer students tools for acquiring and analyzing data that differ dramatically from those available to the student of several decades ago. In the natural sciences data acquisition is often carried out with the aid of automated instrumentation that can be easily reconfigured and that is not subject to the fatigue of observers and errors of recording and transcribing data. In the social sciences the connectivity of the internet makes it possible to collect data easily and with much less regard to geography than was hitherto possible and with frequency and comprehensiveness that would not have been imaginable even ten years ago. New data tools do more than simply make the acquisition of data easier and more extensive than it has ever been before. They also make possible much richer, deeper and broader analyses of data than was hitherto possible. An example to illustrate the point – until the advent of some of the dynamically manipulable visual data analysis tools on the computer, the application of a wide variety of statistical analytic techniques were performed mechanically and with little understanding of, or appreciation of, the appropriateness of any specific technique in a given situation. The new visual statistical tools (such as the electronic spread sheet or tools designed for explorations of data such as DataDesk (1989) or Statistic Workshop (Rubin, 1991)) have allowed for the development of intuitive understanding of the nature of the data sets that are being analyzed and a sense of when a particular analytic approach is or is not appropriate. Further, these new graphic analytic tools allow for the development of intuitions about the nature of the geometry of spaces of more than

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three-dimensions – such spaces being the mathematical setting for the understanding of multivariate data of any sort. Here are two examples of interesting possibilities involving mathematical actions carried out on data sets as mathematical objects. (i) It is a commonplace in the social sciences to explore the correlation of two variables by plotting them on a two-dimensional Cartesian plane and “regressing” one on the other. Assuming the usual notation, regressing Y on X means minimizing the sum of the squares of the vertical distances from the data points to the regression line. Clearly, the regression line so obtained will, in general, differ from the regression line obtained by regressing X on Y. Since one is looking for correlation (rather than causation), it is not clear why this asymmetry should exist. However, one can easily generate other criteria of “good fit” including minimizing the sums of the squares of minimum distances to the regression line, the sums of the absolute minimum distances to the regression line, or any of a wide variety of other measures that treat the variables symmetrically. (ii) Looking at a scatter diagram in a plane, it is clear enough what one might mean by a cluster of data points and what one might mean by an outlier. Similarly it is clear enough that a unique straight line can be passed through any two (noncoincident) points, or that a unique plane might be passed through any three (noncollinear) points in three dimensions. How can these understandings and intuitions that work so readily in two and three dimensions be extended to help deal with multi-variate data with the same sort of intuitive understanding? We can use suitably crafted software to enhance in quite dramatic ways the kind of understanding that until quite recently has been attempted only on the printed page. (For more such examples see The Fourth Dimension: Toward a Geometry of Higher Reality Rucker, 1984.) The dramatic amplification of the ease of acquiring and analyzing data has heightened our sensitivity to an issue that should have been prominent in the study of the sciences all along – i.e., the evaluation of the validity of the data. It is easy to close one’s eyes to this problem if there are too few data. However, as more and more data become available, it becomes clear that one must devote more attention to the question of how to design a data collection protocol and how apposite to the analysis one wishes to perform are the data that are subsequently collected. c. Modeling One’s World There is a sense in which one can legitimately say that the study of the sciences is not the study of nature, but rather the study of the models that humans fashion in order to describe and explain natural phenomena.

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The building and testing of models is the central tenet of scientific epistemology. Unfortunately, the teaching and learning of science in the schools does not emphasize this notion adequately, if at all. New computer-based tools make it possible to change the place of model-building in the school curriculum. There are now a variety of environments that allow students to build and explore models of both natural and social science phenomena.7 Many of these are specially designed software packages with iconic interfaces that make the semiqualitative formulation of models easier to do. They take over the specification of the computational procedure implied by the mathematical model the user has formulated by linking icons representing the levels and rates of change of levels of the elements of the situation being modeled. Similarly, the generic productivity tool, the spreadsheet, is an important modeling tool in both the social and natural sciences. Its proper use helps the user develop a richer and deeper sense of the phenomenon being modeled and a better appreciation of the limits of the model.8 Students who learn to understand both the power and the limits of their mathematical models will be better able to resist the growing popular infatuation for pseudo-science and false advertising and the all-too-frequent resultant misleading of the public. d. Conceptually Grounding Manipulative Skills Another potential consequence of the use of appropriately crafted technology is a shift in the nature of manipulative skills, away from rote ceremonies and in the direction of procedures that while being exercised mechanically and with some degree of automaticity, are, at a deep level, understood. Given that, at present, many people in the workplace use bits of mathematics mechanically and without understanding, this would seem to be a major improvement. A simple example of this sort of environment serves to illustrate the point (Schwartz, 1989). Imagine a four-function calculator in which you can selectively disable keys. Suppose you disable the following keys 2, 3, 4, 5, 6, 7, 8, 9, *, / and ask the student to enter the number 34562 into the calculator. Children in the very lowest grades will quickly develop place value schemes to avoid having to add 1 many thousands of times. In this instance we are challenging users to deny themselves the use of particular mathematical objects as they try to accomplish a given objective. Alternatively, consider what might be done to accomplish the multiplication of two numbers if the * key is disabled or the addition of two numbers if the + key is disabled. In contrast to the previous instance, in

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this case we are challenging users to deny themselves access to particular mathematical actions as they try to accomplish a given objective. Imagine further that the program has a “one significant digit” mode. In this mode, all the keys initially are functional, but as soon a a single significant digit is entered then the numeral keys 1 through 9 are all disabled. Thus, only numbers such as 20, 3000, 0.07, 0.005, etc. can be entered. Doing standard calculations with the calculator in this mode is an excellent way of developing an understanding of the structure of the computational algorithms that are most often taught and learned by rote with little conceptual understanding. This sort of challenge can be readily made in geometric domains as well. For example, while the primitive figures available for use in the Geometric superSupposer include equilateral triangles, quadrilaterals that can be inscribed in circles, etc., users of the this environment can be challenged to build such relatively complex objects by defining procedures on simpler ones such a points and line segments. These examples serve to illustrate a further pedagogical point. It is quite useful, as a way of understanding the depth of students’ understanding, to ask the students to do a task without the full complement of tools (both OBJECTS & ACTIONS) ordinarily available to them for doing that task. This forces them to devise alternative ways of doing what the missing tools would ordinarily do for them. This, in turn, leads to a deeper understanding of the procedures that we ask students to develop some automaticity with.9 e. Deepening and Broadening Understanding By virtue of the fact that software environments can make the intangible interactively manipulable, and do so in several different representations simultaneously, the depth and breadth of understanding that can be reached by students and teachers is substantially enlarged. This expanded understanding comes about by virtue of the ability of the user of the technology to manipulate an abstraction in one representation and see the consequences of his or her actions in several different representations. To the degree to which understanding is built on an ability to move nimbly across representations, the contribution of the technology here is clear. Here is an example. Both symbolic and graphical representations of functions of a single variable are important for students of algebra to master. Graphing calculators and software graphing environments all make it possible for users to modify the symbolic representation of a function and see the consequences of their actions on both the symbolic and the graphical representation of that function. Some software environments10 allow for the converse as well, i.e., allowing users to modify the graphical

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representation of the function by translation, dilation and reflection and to see the consequences of their actions on both the graphical as well as the symbolic representations of the function. An anecdote from Chazan (in press) serves to capture some of the power of students learning algebra with environments such as this at their disposal. . . . one of the students asked me if expressions like 4x2 have a slope. Since the graph was not a straight line, he was not sure whether it had a slope or not. The question warmed my heart. He was taking slope as a property of a class of expressions and, now that he was investigating a new class of expressions, he wondered whether this new class would have a similar property . . . I decided that the students could benefit by thinking about this question of what the slope of a curved line is or could be. I suggested they give it some thought. Some of the students thought maybe the “slope” of 4x2 should be 4, after all the 4 seems to be in the same sort of place as the 2 had been in the linear expressions; it is in front of the “x” . The next day, I came back to this question and used x2 , “Take a number and multiply it times itself,” as the example for discussion. A student whom I’ll call Alexander said he had an answer. He explained that he thought that the slope of x2 was x. After trying to explain his reasoning verbally, he came to the board and argued that he was going to put x2 into the same form as a linear expression, some number times “x” plus some other number; for x2 he argued this was x∗x + 0, with the first x as the coefficient of the second x. So, x was now the slope. He said this made sense to him because in a curved line, the slope is always changing . . . For Alexander, the form of an expression had some meaning; he used this form to make an argument.

4. CURRICULUM IS NECESSARY – NOT SUFFICIENT While the focus of this paper is on mathematics curriculum and the opportunities offered by technology to improve and enhance the curriculum, it is important to recognize that other dimensions of the problem of improving mathematics education are also directly impacted by the growing presence of technology. I mention some of these here very briefly to underscore my belief that improving curriculum is necessary, but not sufficient. Consider, for example, the possible effects on teachers. If exploratory environments of the sort discussed above become commonplace, then the roles of teachers, particularly at primary and secondary level, are likely to be affected greatly. If students are equipped with tools that allow them to make and explore conjectures, then it is likely that they will rapidly press the edges of the teacher’s knowledge of the subject (Lampert, 1995). If this happens, then teachers themselves will be pressed to actively seek to expand their mathematical knowledge on an ongoing basis. Perhaps

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an even more important likely consequence is that teachers will learn to work in open, interactive environments. Such environments often confront teachers and students with unexpected developments and answers are not always known in advance. In my view, these changes would clearly be a welcome improvement over the status quo in most schools. There is also likely to be a serious impact on the publishers of curricular materials. We are already seeing a dramatic change in the way information is distributed around the world. The four-color, two kilogram mathematics text is likely to wane in importance as more and more curricular materials become available on the World Wide Web. The stranglehold that a small number of publishers or a Ministry of Education currently may have on a country’s mathematics curriculum may well be loosened as teachers find themselves increasingly able to gain access to a rich source11 of instructional materials as well as instructional recommendations for the use of those materials in classrooms Finally, one should consider the potential effect of this kind of mathematics curriculum development on assessment of the mathematical attainment and competence of students. Although it is not universally the case, it is now common for students whose mathematical knowledge and achievement is being assessed to be allowed to use hand calculators while taking their examinations. In many instances students are permitted to use graphing calculators. In those situations in which such tools are made available, it is clear that the nature of the problems we pose to the students in order to assess their achievement must and does change. As these tools become more and more commonplace, and as they grow in sophistication so that mechanical mathematical procedures can be automated and carried out at the user’s request by the tool, it is inevitable that the nature of the problems we pose will become both more challenging and more conceptual. 5. SHOULD THE MATHEMATICAL CONTENT OF THE CURRICULUM BE CHANGED? If the new technologies offer such promising opportunities for improving mathematics education, it behooves us to ask whether we should continue with the same mathematical content, or, whether we can take advantage of the opportunity to rethink what content might be in the curriculum in the light of the new kinds of tools available to us. The relative importance of any particular piece of mathematical content cannot be decided upon without taking into account who is doing the deciding, what relative priority they assign to the different societal aims

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of education and what role they see mathematics playing in addressing those aims. For those who particularly value the importance of transmitting the culture and who see mathematics as an integral part of the culture, it will be important that the mathematical content in schools be such as to make manifest rich beauty of the subject in its own right. They are likely to place great importance at elementary level on such topics as set theory and representing numbers in nondecimal number bases. At secondary level, they may place importance on such topics as non-Euclidean geometries, algebraic structures and nonlinear dynamics. For those who particularly value the importance of preparing people for the world of work, the topics just mentioned will seem less important than a rich exposure to modeling the world around one mathematically. This group is likely to want the curriculum, from the earliest grades on, to place a good deal of emphasis on developing an ability to estimate the magnitudes of counted and measured quantities in the surround. At secondary level, this group is likely to attach great importance to the study of functions as mathematical representations of relationships among attributes of entities in the surround. For those who particularly value the importance of schools aiding the personal growth and development of students, it is likely that the specific mathematical content of the curriculum will be less important than the ways in which that content is engaged. It is my personal view that we are far from being able to reach a significant portion of our students with the argument that the subject of mathematics is beautiful and engaging. We are, I believe, substantially further away from being able to reach the general public with that argument. However, the public does believe that mathematics is an important element in preparing people for the world of work. I believe, therefore, that we can best effect significant change in mathematics education if we shape our curriculum so that it clearly maximizes the ability of students to function effectively in commerce, industry and the professions. Moreover, if we do this in a way that engages the students in the making of mathematics, it is likely that over time we will produce a generation of youngsters for whom mathematics has been rewarding and fulfilling. As such youngsters become the public-at-large we can and should allow ourselves the luxury or rethinking the question of curriculum content.

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6. WHO DECIDES WHAT SHOULD HAPPEN TO THE MATHEMATICS CURRICULUM? Finally, I turn briefly to the question of who will decide how the mathematics curriculum will evolve given the opportunities presented by the new technology. In my view, we must look to four different, and not quite disjoint, constituencies. These are the mathematics and mathematics education community, the community of mathematics-using disciplines, commerce and industry, and the public at large. They will each have views on the subject and in all likelihood these views will be strongly held. In the past the mathematics community, by and large, has acted as if the primary goal of a society’s educative institutions was the transmission of the culture and that mathematics was a central and integral part of that culture. The community of mathematicians and mathematics educators now has the option of seizing the opportunity offered by the new technologies to make mathematics a source of personal fulfillment and reward for individual students. If it does so, then it will go a long way toward its goal of making mathematics a real part of the cultural heritage of humanity. In contrast to the community of mathematics educators, in many ways, the community of mathematics-using disciplines has already embraced the opportunities offered by the new technologies. It has done this by changing the ways in mathematics is used within the various disciplines, often without waiting for the mathematics community to sanction the changes. The mathematics community needs to begin serious dialog with this community. The price of not doing so is high. There are already universities in the United States which have abolished their mathematics departments and where the responsibility for mathematics instruction is being assumed by engineers and economists, physicists and physiologists. Perhaps the greatest enthusiasm for change in mathematics education comes from commerce and industry. They see large numbers of youngsters arriving at the workplace ill-prepared mathematically for the careers that lie before them. Before we rush to capitalize on this enthusiasm for change, let us recognize that commerce and industry are obliged to maximize returns on investment to owners and shareholders and are not likely to have the greater good of the society in mind as they argue for change. As a result we might well expect that the changes sought by commerce and industry are likely to be instrumental and mechanical and short-term in nature and not particularly geared to promote the growth of the discipline or the growth of the young people who might develop an interest in it. The degree to which the public discourse influences the nature of the mathematics curriculum varies from country to country and often depends on the degree of centralization of the system. In highly decentralized

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systems, such as that in the United States, it is possible for the uninformed and out-of-date views of a mathematically and scientifically ill-educated public to influence regional and local curriculum decisions. There seems to be an inexorable pressure to polarize the discussion – as has recently been the case in California. There, public pressure has forced the State Board of Education to reject the efforts of the mathematics education reform movement of the past fifteen years in favor of a return to a mechanical, drill-based curriculum. I am not optimistic about the nature of the pressure on the mathematics curriculum that is likely to come from the public-atlarge. I expect the attitudes of the public-at-large toward mathematics will change slowly, at best. Perhaps if the mathematics community and the community of mathematics-using disciplines succeed in mathematically engaging their students in richer and more fulfilling ways, and in ways that are demonstrably more effective in preparing students for the world of work, public attitudes may shift noticeably in a mere generation or two.

7. A CONCLUDING REMARK I have tried to show how the mathematics curriculum, as shaped by various constituencies, plays a role in helping society address its aims for its educative institutions. The new technologies offer a range of opportunities for these various constituencies to shape the mathematics curriculum in new ways so as address these aims more effectively than in the past. It seems to me that we in the mathematics and mathematics education communities face a problem that is difficult, important and urgent.

NOTES 1. To mention a few examples of subject matter/software pairings that offer new content to the math curriculum: Logo as a way to include algorithms and computer programming to deliver big mathematical ideas, Modeling with Fractals with software such as Devaney (1989), and generalizing beyond 3D with geometry software (e.g. Get Out of the Plane, Schwartz, 1993). 2. The New Standards Project is a school reform effort founded about a decade ago by Lauren Resnick and Marc Tucker and run by the National Center on Education and the Economy. 3. There is a sense in which pattern and function, data and arrangement are composite objects, in that they are composed of collections of number/quantity objects (e.g. data), relationships among number/quantity objects (e.g. arrangement), or relationships among collections of number/quantity objects and or shape/space objects (e.g. pattern & function).

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4. This list of actions is an attempt to introduce some structure and specificity into the terms “reasoning, problem solving, mathematical connections, etc.” 5. One should not regard communication as taking place only after all the “mathematics” is done – but rather throughout the process of modeling, transforming and drawing conclusions. 6. If one thinks about the universe of quadrilaterals as spanning some sort of parameter space, then the first repetition mode corresponds to exploring the correctness of the conjecture in disjoint regions of the parameter space, while the second repetition mode allows one to explore a neighborhood in parameter space. 7. Stella, Model Builder, IQON, are just a few of such environments described in Mellar, Bliss, Boohan, Ogborn and Tompsett (1994). 8. A particular form of modeling environment deserves special attention here. I refer to the computer simulation. A simulation is an expression of a theory the author of the simulation holds of the nature of the elements and their inter-relationships in the phenomenon or system being simulated. As such it is important that a student, or indeed anyone, using a simulation, be able to inspect, and even possibly modify, the assumptions built into the simulation. Absent this ability, my enthusiasm for simulations wanes rapidly. I believe that simulations that do not permit the user to explore, or at least to know, the nature of their underlying assumptions can do more intellectual harm than good. 9. It is interesting that posing a problem of this sort is quite commonplace in other disciplines. For example, in computer science students are regularly asked to write a routine that does a particular computation without using a given command. Similarly, a chemistry student might be asked to design a synthesis, but with strong limitations placed on the reagents that may be used. 10. See, for example The Function Probe (Confrey, 1991), Calculus Unlimited (Schwartz and Yerushalmy, 1995). 11. See, for example, www.forum.swarthmore.edu and a wide range of websites devoted to mathematics education that are linked to it.

REFERENCES Chazan, D. (in press). Unreasonable Certainties? Predicaments of Teaching High School Mathematics. New York: Teachers College Press. Confrey, J. (1991). The Function Prob, Computer Software. Santa Barbara. Cuban, L. (1993). How Teachers Taught: Constancy and Change in American Classrooms 1890–1990, 2nd edn. New York: Teachers College. Devaney (1989). Chaos, Fractals and Dynamics: Computer Experiments in Mathematics. Menlo Park, CA: Addison-Wesley. Goldenberg, E. P. (1999). Principle, arts and craft in curriculum design: The case of connected geometry, International Journal of Computers in Mathematical Education 4(2–3): 191–224. Hirsch, E. D. (1996). The School We Need and Why We Don’t Have Them. New York: Doubleday. Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of Reserach on Mathematics Teaching and Learning (pp. 515–556). NY: Macmillan Reference.

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Lampert, M. (1995). Managing the tensions in connecting students’ inquiry with learning mathematics in school. In J. S. D. Perkins, M. West and M. Wiske (Eds), In Software Goes to School. NY: Oxford. Mellar, H., Bliss, J., Boohan, R., Ogborn, J. and Tompsett, C. (Eds) (1994). Learning with Artificial Worlds: Computer Based Modelling in the Curriculum. The Falmer Press. National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: author Olive, J. (1996). New information technology in mathematics education. In T. Plomp and D. Ely (Eds), International Encylopedia of Educational Technology (pp. 546–551), 2nd edn. Pergamon. Powell, A., Farrar, E. and Cohen, D. (1985). The Shopping Mall High School: Winners and Losers in the Educational Marketplace. Boston: Houghton Mifflin. Rubin, A. (1991). Statistics Workshop, Computer Software. Pleasantville, NY: Sunburst Communications. Rucker, R. (1984). The Fourth Dimension: Toward a Geometry of Higher Reality. Boston: Houghton Mifflin. Schwartz, Y. L. and Yerushalmy, M. (1995) Calculus Unlimited, Computer Software. www.visual-math.com. Schwartz, J. L. and Yerushalmy, M. (1992). Geometric superSupposer, Computer Software. Pleasantville, NY: Sunburst Communications Schwartz, J. L. (1989). What Do You Do With a Broken Calculator? Computer Software. Pleasantville, NY: Sunburst Communications. Schwartz, J. L. (1993). Get Out of the Plane, Computer Software. www.visual-math.com.

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