I. Introduction

I don't know the situation in Norway, but in the United States cries for reform from educational crisis managers have produced new mandates for curriculum change. We have been characterized as "a nation at risk" because of the low quality of our educational system, and as a result we have been deluged with proposals for improvement. We have been urged to go "back to basics", though often it is not clear what is basic. Our former Secretary of Education campaigned for a return to the classic works of western civilization. Appalled by the results of international studies of academic achievement, almost every state has developed competency tests in mathematics, English language, and social studies. Most every professional association of teachers has developed new standards and proposals for the improvement of education.

Whatever their merits many of these recommendations share a common tendency: they treat subject matters as fixed entities with unambiguous meanings, well known methods, and clearly bounded content. Although central concepts and principles for organizing instruction may be drawn from academic disciplines (such as biology, geography, history, or physics), once chosen they typically are assumed to be the only reasonable choices that could have been made. From this perspective, the teacher's job -- though it may be difficult to perform -- is simple to describe: communicate the meaning, develop skill in the methods, and cover the material. Missing from the debate is a sense of the complexity of subject matter, a complexity that generates conflicting meanings, alternative methods, fuzzy boundaries, and thus complementary possibilities for teaching.

Many curriculum developers, textbook authors, and especially teachers share the same tendencies. In the first paper I described the difficulties teachers have when grappling with curriculum problems. Invariably they have trouble seeing that the subjects they teach offer a range of possibilities. The geography teacher cannot get his students to grasp "the basics", but what is basic in one conception of geography may be less essential detail in another. The mathematics teacher wants her students to learn to prove theorems; but deduction and proof, though important are not the only kinds of mathematical methods. These limited views of subject matter are a severe handicap for teachers. The central curricular issue always involves bringing a subject matter and a learner together. If we see the meanings and methods of the subject matter as static -- embalmed, as it were, between the covers of the textbook -- our options for promoting this interaction are severely limited. Teachers need a wide range of views of what they teach so that they have alternatives from which to choose.

This paper assumes that subject matter as a curricular commonplace is a variable, not a fixed, term and, therefore, that detailed analyses of particular subject matters are curricular resources in ways not typically imagined in schools. If, for example, we say that children should learn "literature", we must ask ourselves which set of literary concepts and principles we mean and in what senses exposure to this version of "literature" is educational. Among literary critics, there is some variety of views as to what constitutes a reading of a text; and these different views implicate contrasting sets of reading skills, sensitivities, and strategies for teaching. As a result, educators have choices about the educational goods, which could be achieved. Such choices are obscured when "literature", or any subject matter, has imposed upon it a fixed image of its shape and educational possibilities. If, instead of viewing subject matters as fixed, we see them as variable, then we can use them as curricular resources from which choices can be made to serve particular educational needs.

II. Controversies in the Philosophy of Mathematics

In this section of the paper, I shall use a traditional set of questions to locate four pairs of contrasting views on the nature of mathematics. I have picked these views from a variety of sources: philosophy, history, and discussions by mathematicians of curriculum issues. Given the generating questions, my choice of theoretical formulations has been somewhat arbitrary. I could have chosen differently, but the selections I have made seem to me to offer an adequate variety to start us thinking more flexibly about mathematics curriculums in general and geometry in particular. Of course, the test of their adequacy comes later when we apply them to curricular problems in mathematics. Should readers be stimulated to propose a different selection, my purpose will have been served.

What are the Origins of Mathematical Activity?

A long-standing answer to this question has been that all mathematics stems from the need to understand and master the physical world. Mathematics may be abstract and often remote from its origins, but from the time of the Greeks until the nineteenth century it was generally accepted that the axioms of mathematics were self-evident truths about the world around us. Confidence in this view reached a peak in the eighteenth century. Euler, for example, in a paper on the design of ships' masts said that he felt no need to check his conclusions with experiments because they had been deduced from unimpeachable postulates. Yet only a century later, the mathematical community was shaken by the development of non-Euclidean geometries, which clearly showed that it was possible to develop different models of the physical world. Worse, these models were as useful for helping us to understand the world as the Euclidean model. Perhaps the axioms were not so self-evident after all. A further blow to the traditional view was struck by the Wright brothers who were flying even when classical dynamics had proven it to be impossible.

In the twentieth century, the focus has shifted from the physical world to the human mind. If mathematics does not uncover the inherent design in Nature, then at least mathematicians can impose an order on the natural world. In this view, mathematics results from the human need to create pattern and order out of chaos. Whether or not the real world is orderly, we see it in the light of the patterns we create. In return for giving up the notion of absolute truth, we gain freedom to invent abstract axioms and investigate the consequences of our choices. But there is a paradox here because these new axiom systems have been spectacularly successful. As mathematicians have become less concerned with discovering truths about the real world, their inventions have been increasingly useful in mastering it.

What Position should we take in the face of this dilemma? Should we side with Kronecker who said that "God created the integers and the rest is the work of man"? Or should we accept Thom's view: "No one can reasonably escape the impression that the most important mathematical structures ... appear as fundamental data imposed by the exterior world."? (1971: 699) Perhaps we should adopt the middle view that mathematics originates in the physical world but the human mind can transcend the origins of mathematics. In this case, Morris Kline warns us of danger:

Ideas concocted solely by the human mind and bearing no relation to the meaningful and weighty phenomena of the physical world may become thin, impoverished, and insignificant however facile the reasoning. ... Mathematicians may like to rise into the clouds of abstract thought, but they should, and indeed they must, return to earth for nourishing food or else die of mental starvation. (1959: 475)

This is, of course, one of the most vexing philosophical questions which dates back at least to the Pythagoreans. It contains an ontological problem ("Do mathematical entities exist?"), an epistemological problem ("What is mathematical truth?"), and a logical problem ("Is mathematics a giant tautology?"). (See Benacerraf and Putnam, 1964.) Any answer to it also could involve us in disputes between various "isms" (e.g. empiricism, nominalism, intuitionism, ...) which, for our purposes, would not be productive. I shall simply describe two contrasting answers to an apparently simpler question, "What do mathematicians study?"

One possible view is that mathematicians study formal systems, sets of propositions that have been shown to be true by certain agreed upon rules. A formal system consists of a primitive frame (i.e. its basic terms, its conventions for combining terms into propositions, and its agreements about which propositions will be called true), its rules of formation for constructing less elementary terms and propositions, and its rules of procedure for deriving theorems (i.e. additional propositions which are true). (Curry, 1964) Notice that it does not matter what the primitive terms represent. They could be numbers, points, variables, symbols, or even the propositions of another formal system. For the mathematician, the system exists independently of its possible representations and is itself the object of study. Its primitive frame determines whatever meaning or truth value we give to the propositions in the system. It was in this spirit that Bertrand Russell said, "Mathematics is the subject in which we never know what we are talking about nor whether what we are saying is true."

Notice also the emphasis on the entire system, not just on its parts. A proposition, or set of propositions, has meaning because of its place in the system. It does not stand on its own, so we cannot give meaning to it or evaluate it on its own merits. Instead we try to evaluate the acceptability of the whole system, and so we concern ourselves with its usefulness, its economy, its consistency, its completeness, or its decidability. One of the interesting and powerful developments in mathematics has been the discovery that questions about formal systems can themselves be analyzed by formal systems. (See Hofstadter, 1979.)

The contrasting view of mathematics, exemplified by Courant and Robbins in their classic text What is Mathematics?, sees mathematics as the abstract study of patterns. Not just number patterns or geometric patterns but patterns wherever they can be found: in nature, in human activity, in symbols, in the human mind. To think mathematically is to abstract pattern from whatever phenomena one is considering and then to study it with mathematical methods. Queing theory, an important branch of mathematics now used in the design of communications networks, is said to have been invented by von Neumann as he was standing in line in a grocery store. Why, he might have asked, did the express checkout line say "6 items or less?" Is this the most efficient policy for getting customers out the fastest with the least dissatisfaction while leaving the most dollars behind? Assumptions were made, consequences deduced, patterns emerged, and thus queuing theory was born.

Notice that the patterns that we find can be more or less abstract. They could be firmly rooted in the phenomena that initiated our inquiry; or they could be so abstract that their origins were unrecognizable. Out of smaller patterns bigger structures grow. Here too the emphasis is on a whole -- the abstract pattern that emerges from our thought -- but the scale need not be as grandiose as a formal system. We can study local phenomena and find small scale patterns or we can search for global patterns.

What are the Fundamental Mathematical Methods?

It is clear that Descartes hoped to develop a universal method for mathematics. His posthumously published Rules for the Direction of the Mind contain fragments of his plan, but the scheme that he started was never finished. It is likely that it never would have been completed because, as Polya tells us. "The more you know, the more gaps you can see in this project." (1981: 22) This is probably the last such attempt in the history of mathematics, for the idea of a single mathematical method has been discarded. It is now recognized that there are a variety of approaches to mathematical questions and that one's methods should be chosen in a way which is appropriate to the problem being attacked. Nevertheless, it is possible to discern broad groupings of methods: numerical, algorithmic, dialectic, computational, analytic and so on. Instead of elaborating on these, I shall focus on the contrast between mathematical proof and mathematical invention.

The first school holds that deductive methods are the hallmark of mathematics. It has been one of the achievements of Bourbaki -- actually a group of mostly French mathematicians -- to systematize the enormous quantity of new and startling mathematics which had been developed in the nineteenth and early twentieth century. This work has been continued by his successors so that now all important branches of mathematics have been axiomatized. This means that it is now possible to present ideas which once rested only on intuitive and informal arguments in a rational and ordered way. Symbols are used with care, definitions are precise, theorems are rigorously proved, and the primitive terms and postulates upon which the structure rests are kept to the barest minimum.

This is the pattern set by Euclid in the Elements over 2000 years ago. It turns out, however, that the axiomatics of Euclid are, by modern standards, flawed and lacking in rigor. For example, by his methods it is not hard to prove that all triangles are isosceles, a most undesirable and inconsistent result. Other similar and distressing examples can be found. (Maxwell, 1959) Though it is easy enough to uncover the flaws in each particular proof, it is not so easy to fix up Euclid's axiomatics so that such proofs cannot be made. Among other things, a careful treatment of "betweeness" and "interior" is required, and these concepts are deeper than they seem. Elaboration of them and a more rigorous axiomatization of Euclidean geometry were accomplished by Hilbert. Attempts to present this axiomatic structure in a simple way have been "defeated by the dreadful complexity of his work." (Thom, 1971: 696)

Though no one denies the tremendous advances that have been made by axiomatics, there are many mathematicians who say that this is not the way in which mathematics is discovered. Deductive methods are necessary to make intuitively understood ideas more precise and to communicate results to others, but new ideas and new theorems are developed by intuitive and less formal methods. "It is characteristic that no new theorem of any importance came out of the immense effort at systematization of Nicolas Bourbaki." (Thom, 1971: 697-98) Kline who calls the dedication to mathematical rigor exemplified in some "new math" texts rigor mortis has repeated this theme. (1974) But the most persuasive argument for non-deductive methods has been created by George Polya who, in numerous books and articles, has tried to specify and exemplify the less formal methods which mathematicians use. Ideas are developed by complex processes involving -- in addition to generalization and deduction -- specialization, analogy, observation, plausible reasoning, and even guessing. "In many classrooms 'guessing' is taboo, whereas in mathematical research 'First guess then prove is almost the rule.'" (1981, vol. 2: 157) His arguments are too detailed and too specific to be reproduced here, but his writings could be a source of inspiration for teachers tired of purely deductive methods. They ought to know that there is a debate in the mathematical community about the appropriateness of single-minded use of deductive methods.

What are the Eventual Outcomes of Mathematical Activity?

Mathematicians no longer believe that the outcome of mathematical activity is knowledge of the truth. The development of non-Euclidean geometry in particular and formal axiomatics in general have permanently demolished that notion. Since our axiomatic systems must start with undefined terms and postulates, our deductions can be no more true than our primitive frame. But we cannot say that this frame is either true or false because terms and postulates are not interpreted and so have no inherent meaning. Any particular representation of them might be true or false, but this would be an empirical question and so beyond the scope of mathematics. Mathematics leads to knowledge but not to knowledge of the truth. It is hypothetical knowledge of logical and formal relationships, and, at a more mundane level, it is instrumental knowledge of procedures and techniques for solving problems.

Nevertheless, mathematics has made crucial contributions to our ability to understand and cope with the world. With its help, we discover truths about the world and make accurate predictions. This activity is often called 'applied mathematics' to distinguish it from the 'pure mathematics' just described. The goal of applied mathematics is to gain control over the complexity of physical, social, or even mental situations. We do this by reducing the complexity; i.e. by building descriptive models so that they can be studied with the aid of mathematics. When mathematical results have been obtained, the predictions they imply are tested in the originating situation. If the predictions are accurate, we have made a successful application of mathematics. If not, we adjust our model until we meet with more success. In some ways, applied mathematics is harder than pure mathematics because its practitioners must have a broad knowledge of mathematics as well as a thorough knowledge of the situations being studied. Although some pure mathematicians tend to denigrate applied mathematics or see it as a separate field, it remains an important discipline through which the outcomes of mathematics receive expression. (Sutton, 1959)

We can take the point further. In most civilizations -- with the possible exception of the Romans -- mathematics has been a major cultural force. It has certainly helped to mold our twentieth century culture and is a vital aspect of that culture. In this sense, one of the important outcomes of mathematics is what Morris Kline calls a spirit of rationality.

It is this spirit that seeks to influence decisively the physical, moral, and social life of man, that seeks to answer the problems posed by our very existence, that strives to understand and control nature, and that exerts itself to explore and establish the deepest and utmost implications of knowledge already obtained. (Kline, 1953: 10)

III. Two Views of Computer Science

Like all emerging disciplines, computer science has had its share of difficulty in convincing the academic community that it is, in fact, a separate discipline with distinct problems and methods. In most universities in the United States, departments of Computer Science emerged from departments of Mathematics, often after internal political struggles that left their scars. As the new departments grew, they attracted -- or co-opted, as their skeptical adversaries might say -- new faculty members from a variety of other disciplines, most notably philosophers trained in logic, engineers, and physicists. Due in large part to the explosive growth in the number, power, and usage of computers and therefore in the number of students and employers interested in computer science, departments of Computer Science have flourished. In most large universities, the divorce has been completed. Departments of computer science are well established; in fact, so well established that some predict a "surplus" of graduates trained by these departments. We now see the process repeating itself to some extent in secondary schools, many of which are establishing their own computer science departments with teachers recruited from mathematics, business education, and sometimes physics.

All this activity does not demonstrate, by itself, that computer science is a distinguishable discipline. Much as university administrators might like to equate academic programs with academic disciplines, the intellectual world is not so tidy. Yet computer scientists have not been slow to mimic the behavior of their colleagues in more well established academic disciplines. They have their own journals, conferences, and newsletters as well as their own informal networks for communicating ideas and sharing new developments (which, naturally, use computer technology). They have associations that recommend curricular guidelines and set standards for accreditation. Computer Science has developed sub-disciplines and sub-fields, many with their own vocabulary and research problems. Therefore, computer scientists must specialize because it is virtually impossible to be an expert in the field as a whole. So factions develop as one speciality sees itself as super-ordinate to the others. Computer Science even has its own heroes and villains, though, with the exception of a few legendary figures like Church or Turing who did their work before there were computer scientists, there may be little agreement as to which is which.

A surer sign of maturity as a discipline is serious debate about the nature of the field itself; not partisan debate which pits one sub-field against another or the whole field against all comers, but debate in a more irenic style aimed at uncovering those matters about which legitimate differences persevere. As far as I can tell, such debate is taking place within computer science. Kalin (1987), in an essay review of Abelson and Sussman's new text for beginning computer scientists, indicates the main lines of controversy. Brian Harvey (1985), an educator as well as a computer scientist, has outlined the fundamental controversy. In several speeches, and more or less explicitly in his recent book, he has contrasted, two views of computer science, one "traditional" and one the "MIT style". Whether or not the labels are accurate -- surely there are still a few "traditional" computer scientists at MIT and more than a few "MIT" disciples elsewhere -- the terms in which he characterizes the differences are instructive. So let me try to clarify the distinctions which he makes.

The controversy arises from different perspectives on the source of the problems which computer science should study. Computers have become so ubiquitous because they can be used as tools to help us perform tasks that are lengthy but repetitive. They calculate; they organize data; they follow rules; they remember what they have done and can reproduce it on demand. And they do these things at high speed. Consequently, they can be used by accountants, writers, lawyers, doctors, statisticians, engineers -- indeed, by all of us -- to do jobs which are dull, time-consuming, but repetitive. In Lady Lovelace's words, the computer "can do whatever we know how to order it to perform." This frees the user to concentrate on the more interesting tasks that require genuine creativity and intelligence. From this traditional point of view, the problems of computer science lie in those well defined tasks that we know how to perform. Given the tasks, how can we get computers to perform them correctly, efficiently, and simply so that the computer becomes a more effective tool for its users?

But what if we do not have a clear idea of the task that we would like the computer to perform? Must we wait until we have it clearly defined before we try to tell the computer to do it? The traditional computer scientist would say "Yes" or, at least, would tend to focus only on those tasks that were already clarified. The other camp would say "No" and would try to use the computer to help define the task more clearly. Indeed, they would tend to seek out situations that were not yet well understood, rather than ignore them. They would think new situations were more fun than tasks whose structures were well known; moreover, each new situation is an opportunity to test and extend the tools that have already been developed. From the "MIT" point of view, the problems of computer science lie in some more or less complex process of experience or thought. Given the process, how can we put together the tools at hand -- or developed for the purpose -- to build a computational model which adequately reflects the system's behavior?

Let us take an example from the field of artificial intelligence: the development of computer programs which "play" chess. Is this traditional computer science or the "MIT" version? It depends on how the activity is pursued, particularly on how the problem is defined. The traditional approach -- interpreting the task as "write a program which can beat a good player" -- developed specifications and algorithms for good play and incorporated these into a computer program. Although this approach was moderately successful, it turned out that the algorithms bore little resemblance to the way people actually thought through a chess game. Consequently, little was learned about chess playing or about intelligent behavior, though quite a bit may have been learned about algorithms. If, in contrast, one interpreted the task as "write a program to model the behavior of chess players", one would approach the task quite differently. First, one would study the thinking process of chess players. Then one would develop some preliminary theories about what was going on and try to incorporate these into a computational model. This model would then have to be tested on the computer, revised in the light of a better theory, and tested again. Each new model would capture a little more about the ways in which humans play chess and, by extension, the ways in which they solve problems in general.

Notice the effect each perspective has on what its proponents think the discipline should study. Those who think the outcome of their labors should be the performance of a defined task -- even one like chess playing whose detailed specifications have to be developed -- naturally are concerned with the optimal performance of that task, and, perhaps, of closely related tasks. So they will concentrate on the design and analysis of algorithms for doing specified tasks. They will be interested in how fast the algorithms execute, how accurate they are, and how much storage space or other resources they require. This leads them to questions about computer architecture, machine organization, and the characteristics of various operating systems. They also must consider the characteristics of the various programming languages available to implement their algorithms. And they must be concerned about software engineering, in particular the ease with which users, even non-technical ones, can make computers perform their tasks. In short, the traditional perspective directs us to focus on algorithms, computers, programming languages, and the "man-machine interface". Each of these foci can be pursued, and taught, as a separate specialty within computer science.

On the other hand, those who think the outcomes of computer science should be computational models of processes -- whose nature is, at best, only partially understood -- concern themselves with the structure and interpretation of computer programs. The emphasis shifts from the physical devices or the characteristics of particular languages and operating systems to the intellectual strategies that we can use to make computers do their work. More precisely, computer science becomes the study of computational processes; i.e. inventions of the human mind encoded in programs to be run on computers. In effect, we have moved to a higher level further away from the machine but closer to the processes we wish to describe. Our subject is more abstract, but in some ways easier because we no longer need be so concerned with the details of how the programs we develop actually get implemented in a particular language on a particular machine. Abstraction has a more important payoff: we can build computational models of the computer itself -- or of the various levels between our abstract computational processes and the machine. So each of the sub-fields of the traditional approach to computer science can be pursued using this more abstract approach. The consequences of this bootstrap operation are mind boggling. Alan Perlis puts it this way in his foreword to Abelson and Sussman's text:

So the choice of perspective not only influences one's view of the phenomena to be studied, it has an even more profound effect on the methods that are considered appropriate. The traditional view emphasizes methods that are linear, algorithmic, and deductive; such things as systems analysis, flow charting, structured programming, and top-down design. The MIT view emphasizes methods that are more fluid, idiomatic, and interactive -- things indicated by somewhat less precise terms like tool building, local organization, modular programming, and bottom-up design. The contrast indicated in the last terms of each set probably best illustrates the difference between the two approaches. In a top-down approach, the structure of the final program is determined in advance before the work on the individual components begins and, most important, before one starts to work with the computer. Top-down programs should be planned and written at one's desk; then, when all the pieces have been developed -- or, at least, incorporated in the program -- one tests the program with a computer. In the bottom-up approach, the final structure evolves as we put the program together. Components are built, tested with a computer, then set aside while we work on other pieces. Bottom-up programs tend to be written at the computer so the bugs can be worked out as we go long. As we get larger and larger parts of the program working smoothly, our perception of the overall program structure becomes clearer. More important, our sense of what the program might do enlarges as we glimpse additional interesting and more general possibilities.

In summary, the traditional view of computer science assumes that the problems of computer science arise from the need to perform well-defined tasks and that its outcomes are the actual performance of such tasks. As a consequence, its proper subject matters are seen to include: computers and their organization, programming languages, the analysis of algorithms, and the interface between humans and machines. The methods of traditional computer science primarily involve systems analysis and top-down, structured programming. There is, however, an alternative, more inclusive view of computer science which assumes its problems originate from the need to develop models of physical or mental processes and that therefore its outcomes should be the development of such models (which, incidentally, become tools for further investigation). Its proper subject matter is 'computational processes', a molecular term involving a human mind writing programs to be implemented by machines. Its methods primarily involve modular programming and the invention of increasingly abstract tools.

Let me close this section with Perlis's metaphor for contrasting the differences between the cultures surrounding two competing programming languages, Pascal and Lisp. I think it also captures the differences between the two views of computer science.

I did not contrast the two views of computer science in order to provoke partisan debate about their relative merits. It is important to realize that they are both legitimate activities. Computer scientists go about their work in different ways, as do scholars in any discipline. This diversity should not discourage us. If we know how to tap it, it is potentially a rich resource for curricular invention. Instead of choosing between the two views, either or both can be used in appropriate ways to plan experiences for students. On the other hand, we run into trouble if we do not distinguish the two approaches; or worse, if do not even recognize that there are at least two ways to "do" computer science. This is one reason computer literacy courses, in the United States at least, seem such a mish-mash. Their developers have not discerned the alternative principles of organization which are available and made appropriate choices among them. Instead of mixing up our principles or trying to do a smattering of everything -- by, for example, including a little bit of three programming languages instead of a more in depth experience with one -- we should be clearer about why we are doing what we are doing. The important question, then, is not which view is best, but where should we start and how do we, eventually, get both views represented.

Similarly, we do not need to debate the merits of the different views of mathematics. The contrasts that I made certainly are not proposed as the only alternatives; nor are they necessarily the best. I think they are attractive possibilities, and in an earlier paper I outlined five modifications to a geometry course generated from these distinctions. However, a decision to pursue any of the modifications would have to rest on a detailed diagnosis of the symptoms presented by a particular group of students. In the absence of such a diagnosis, they remain abstract possibilities. Nevertheless I think they provide a useful set of alternatives which geometry teachers might want to consider.

Curricular deliberation is essentially a process of formulating an adequate variety of alternatives, weighing their various virtues and vices, and choosing among them. I have argued that teachers' perceptions about the subject matters they wish to convey seriously limit their choices. On the other hand, if their understanding of subject matter were less static and more variable, the curricular alternatives that they could envision would be enriched. Knowing we are limited by the perspectives we habitually bring to bear, consideration of the richness of variegated subject matters could help us to anticipate problems, uncover neglected opportunities, and discover new duties and possibilities for education.

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