Can We Solve the Problem Solving Problem Without Posing the Problem Posing Problem? [reprinted from a chapter that I wrote in J. P. Ponte et al. (eds.), Mathematical Problem Solving and New Information Technologies © Springer-Verlag Berlin Heidelberg 1992] Abstract: The first section of this paper deals with the inter-related nature of problem solving and problem posing. Although some time and attention is devoted in schools to problem solving activities in mathematical domains. problem posing as an intellectual activity is almost totally neglected. This is particularly unfortunate because the intellectual progress of mankind in mathematical and scientific domains depends on our being able to make and explore conjectures. i.e. problems that we pose for ourselves. We are thus confronted with a two-fold problem. First how might we change the way we educate people so as to help them make and explore conjectures .. Second. not all conjectures are interesting or productive. How can we help people to develop and exercise taste and judgement with respect to the conjectures that they make and explore? How does one decide what problems are worth working on? Information technologies offer the possibility of constructing "intellectual mirror" software environments that can scaffold the posing of powerful problems. The properties of such environments will be described. Finally some of the implications and consequences of the use of such environments in classrooms will be discussed. On the Inter-related Nature of Problem-Solving and Problem-Posing The dominant paradigm in most schools around the world is, understandably, the transmission of the culture of the society to a next generation of youngsters. By and large this means that a considerable fraction of time in school is devoted to the teaching and learning of factual information or what is sometimes called "declarative knowledge". If all schools had to do was to help youngsters to learn what is already known, it could be argued that the process could be accelerated and its efficiency improved by the use of Computer Assisted Instruction (CAl) technology. Certainly in those areas of society in which we feel it is important to train people to know the already known, such as training soldiers to clean rifles and automobile sales people to sell new and used model cars, this technology has more than amply proved its worth. However, we have broader ambitions for schools in democratic societies. While it is still the case that we expect schools to be places for the transmission of the culture and for teaching the already known, it is also the case that we expect schools to teach our youngsters to think critically about their societies and the world around them. In the rhetoric of the education community, we would like them to become good "problem-solvers". In the United States in recent years there has been a growing amount of attention paid to the teaching and learning of problem-solving. Most of the efforts of researchers and teachers in this area fall clearly into one of two, largely non-overlapping camps. The ftrst camp consists of people who want to teach general problem-solving strategies that can be used broadly across intellectual domains. The second camp is largely filled with those who think that learning to solve problems is best done in the context of particular domains. I belong to the second camp. Aside from certain simple meta-strategies I do not think that there are problem solving heuristics that are realistically useful across domains. Getting from Boston in the United States to Oporto in Portugal, conjugating irregular French verbs, solving a differential equation and finding a decent restaurant in Moscow are all problems. It is hard for me to believe that there are any powerful commonalities among the strategies that a reasonable person would pursue in attempting to solve this diversity of problems. For those of my persuasion who are interested in the teaching and learning of science and mathematics, the pedagogic problem becomes one of finding ways of getting students to appreciate the power of some of the heuristic strategies that have proven to be valuable in science and mathematics. Here I refer to such strategies as shifting one's frame of reference, or invoking symmetry and invariance, or approaching a complex situation through a successive series of approximations, each modeling the problem situation a bit better than the previous one. Perhaps the most seminal of these problem solving strategies is one that Polya captures with his dictum, "find a similar but easier problem and solve it! ". I find it interesting that this most powerful of strategies for solving problems asks people to pose a problem as a step on the road to solving a problem. Is there something to be learned from this about the way we build curricula? Can we make the posing of problems an important part of the way we teach and learn mathematics and science? At the moment, it is certainly the case that our curricula in these subjects do not attach great importance to problem-posing. The Role of Conjecture in Intellectual Progress Students have odd notions about the nature of mathematics and science as school subjects and as intellectual disciplines. As subjects in school that they are asked to learn, many students, and teachers, believe that their tasks are, respectively, to learn and to teach the science and mathematics already made by other people. It is not hard to see the probable reason for this attitude. The problems that students solve and that teachers grade are already there printed neatly in the text. Moreover, the problems are carefully fashioned to have relatively clean solutions using methods that have just been explained in the preceding section. With this image of science and mathematics as school subjects, it is not surprising that the attitude that carries over to science and mathematics as intellectual disciplines is one of a body of knowledge with clean, right and wrong answers and with little, if any, uncertainty. Moreover, the disciplines are believed to be complete, with no further intellectual development required or possible except at the "frontiers of science". Needless to say, these frontiers are believed to be necessarily abstruse and complex and certainly beyond the comprehension of normal mortals. But the essence of good science and mathematics does not lie only at the cutting edge. It is entirely possible to do first rate science and mathematics in those parts of the subject that are believed to be sufficiently well understood to be taught in schools. Changing Attitudes and Expectations Clearly this is not the case with the teaching and learning of science and mathematics in our schools now. What must change so that we might move our schools in this direction? I believe that to do this we must expand dramatically the time and attention we devote to the posing of problems. By posing of problems, I mean both the formal posing of new problems by teachers as well as, and perhaps more importantly, the development and exploration of conjectures on the part of the students. A major aspect of the problem confronting us is the need to change the attitudes of people in and out of schools about the importance of problem posing and the making and exploring of conjectures. Doing this amounts to a strong redirection of the culture of schools and the expectations of various publics about what schools can reasonably be expected to do. This redirection will not occur easily or readily. It is unlikely that preaching and forecasting of educational doom will help, despite the fact that current practice fails dismally to educate more than the merest sliver of our youngsters to anything like scientific and mathematical competence. Lest it not be clear, my sense of competence includes the development of students' ability and willingness to make and explore conjectures. It seems to me that the only viable strategy is to adopt the advice of the late Jerrold R. Zacharias. He suggested that the way to bring about viable education reform was to "wheel in a Trojan mouse". By this he meant, to introduce a change in materials and practice that is perceived to be contiguous with and consistent with what is current in the schools, but which contains the germ of long-term, deep and systemic change. Considering the complexity of the educational system this is a major problem. Somehow, we must act in such a way that attitudes of students and teachers and the public all begin to change. Further, we must do this in a way that does not manifestly threaten what students and teachers and the public think ought to happen in schools. Thus, simple minded cries for new curricular content will not do. Curricular content is probably the most salient aspect of what goes on in schools, and any call to change the manifest content is bound to arouse suspicion and distrust. This is not to say that the curriculum should not change, but rather it is a word of caution about pinning extensive hopes on the effectiveness of a strategy that is based on changing curricular content. By the same token, it will not do to simply call for a longer school day, or a longer school year or more homework. While these changes in themselves are no doubt desirable, it is not clear that they will have the kind of qualitative effect we seek to have on the teaching and learning of science and mathematics. These subjects are not taught adequately anywhere in the world. The most likely outcome of American youngsters spending as much time in school and doing as much homework as Japanese youngsters is that American students will do as well as Japanese youngsters in solving the problems that are used on the cross national tests of scientific and mathematical competence. As desirable as that may be, that is not the kind of change that is needed in the US. The kind of change that is needed in the US is precisely the kind of change that is needed in Japan and indeed, everywhere. We need to begin to educate youngsters to conjecture and to pose problems and to question evidence and authority. We need to have succeeding generations ask naturally and spontaneously about everything they see in the world around them, "What is this a case of ?". Where can we turn, if not to completely new curricular content and not to dramatic expansions of time in school? I would like to suggest that there is a kind of answer provided by technology that may help in just the ways we have outlined. I hasten to add that I am not referring to traditional CAl uses of microcomputers but rather to a use which is conceived of within and consistent with an entirely different pedagogical framework, i.e. guided inquiry. It is to this form of computer use in education that I now turn my attention to. A New Role for Information Technology Information technologies offer the possibility of constructing "intellectual mirror" software environments [1] that can scaffold the posing of powerful problems. In this section I will describe some of properties of such environments, as well as discuss some of the implications and consequences of the use of such environments in classrooms. What is "intellectual mirror" software and why do I believe that software of this sort offers a reasonable promise of helping to move us in the direction discussed earlier in this paper? Reflecting the Consequences of Users' Actions The first, and perhaps most important property of "intellectual mirror" software environments is based on the fact that humans are not very good at making intricate and concatenated chains of inferences. To the extent that we wish them to be able to do so, it is important that we provide them with intellectual prosthetic devices that will help them. In science and mathematics it is often the case that there is a substantial logical distance between the starting points offered by nature and our conjectures about nature and the detailed implications of our models. It is precisely in this arena that appropriately crafted software environments can aid dramatically in extending our ability to explore our formal models. Briefly stated, therefore, intellectual mirrors are software environments that allow users to explore the entailments and logical consequences of the formal descriptions and models that they make of natural phenomena or of rule-governed systems. These software environments have no built-in pedagogic agendas. They do not ask the user questions, nor do they evaluate the "quality" of the user's efforts. A Short Discourse on the Nature of Primitives More needs to be said about "intellectual mirrors" because what has been described so far is a reasonable description of almost all programming languages. As wonderful as I think programming languages are, I do not think of them as practical "intellectual mirrors" because of their generality. Most programming languages can be used to in truly wondrous varieties of ways. This flexibility and diversity is a direct consequence of being able to build explicitly increasingly complex procedures from very simple primitives. One way to think about an "intellectual mirror" environment is as a special purpose programming language with a severely limited domain of application. Because of this limited domain, it is possible to imagine a different sort of primitive than the kind of primitive one normally finds in programming languages. Let us consider this difference. To make the discussion concrete, let us digress for a few paragraphs to describe one such environment The Geometric Supposer is an "intellectual mirror" environment meant to encourage and invite the making and exploring of conjectures in Euclidean geometry. Users' may make elaborate Euclidean constructions on a geometric shape of their choice, e.g. isosceles triangle, a pair of tangent circles or a user constructed quadrilateral whose diagonals form particular pairs of vertical angles and whose lengths are in a specified ratio. The "construction tool-kit" consists of such primitives as the drawing of line segments, perpendiculars, parallels, angles bisectors, inscribed and circumscribed circles etc. Measurement tools are available for the user to inspect the properties of the construction. It will often be the case that seemingly "interesting" properties of the construction will be suggested by such measurements. In order to inspect the generality of such "interesting" properties, the environment allows users to repeat the construction and measurements on arbitrarily many instances. Although I shall continue describing the Geometric Supposer later, the description to this point is sufficient to allow us to return to the discussion of logical and psychological and/or pedagogical primitives. To be workable at all, a programming language must have a manageable number of different primitives. Therefore, there may not be too many of them. Because of generality of the language, however, it is necessary for these primitives to be truly primitive symbol manipulating operations. One can think of these primitive as the logical primitive objects and operations of the symbol system. The usual way this problem is addressed in a computer-literate culture is through the building of "macros" and procedures that can be used to extend the language. Thus one can fashion complex "primitives" out of simpler ones for special purposes, limited only by the constraints of talent and imagination. In an "intellectual mirror" environment, the primitives are of a different sort. They can be thought of as the psychological or pedagogical primitive objects and operations of the domain. Let us consider how this distinction applies in the case of the Geometric Supposer. The logical primitive objects of Euclidean geometry are the point and the line and the logical primitive operations of the subject are those operations that can be performed with idealized straight edges and compasses. In the Supposer the primitive objects are clearly more complex than the logical primitive objects of the subject. Similarly, the primitive operations are more complex than the logical primitive operations of the subject. I believe that the objects and operations of the Supposer are better pedagogical primitives in geometry for most people than are the point, the line, and the straight edge and compass operations. It is, of course true, that the Supposer objects and operations can be understood in terms of the logical objects and operations of the subject. We have the experience of centuries of geometry teaching and learning to support this assertion. A Pedagogy of Intellectual Depth in Bounded Domains What I am suggesting is that we begin the subject with these pedagogically primitive objects and operations and work in two directions to develop the subject. One direction is the classical one of establishing how each of these primitives can be thought of as concatenations of the logically simpler objects and operations of the subject. The other direction is rather different Because the primitives of the Supposer environment are moderately complex and because the environment makes it easy for users to concatenate these primitives richly while still being able to inspect easily the consequences of such concatenation, it is an environment that entices users into "supposing", i.e. making conjectures and exploring them quickly and in some depth. The reader is entitled to ask why I recommend starting "in the middle" and working in two directions when it is possible to start "at the beginning" with the logical primitives and build "macros" that can bring one to the starting point I am recommending. I hold the position I do for two reasons. The first is that the explicit, but reasonably abstract, notion of procedure has still not permeated the culture deeply enough for us to be able to build on it. Evidence to support this observation comes from as widely diverse sources as the difficulties youngsters have in learning to program computers to the almost universal avoidance by adults in the business world who use spreadsheets and word processors of the facility to write "macros". The second reason has to do with the nature of what motivates many, if not most, students. It seems that only a very small number of them share the a priori enthusiasm that many scientists and mathematicians have for the reductionist aesthetic that celebrates the building of intellectual edifices starting from the sparsest of ingredients. A large fraction of our students, and the larger population that they eventually join, have a more pragmatic set of criteria for intellectual worth. To be sure this often means, can the knowledge of a particular piece of subject matter be turned to advantage in the outside world in which one lives. But people are engaged by more than just the pragmatic. They are often engaged by interesting complexity, particularly if it is complexity of their own making. We can think of "Intellectual mirror" environments as offering people the opportunity to fashion and explore complex situations in domains that our culture has come to regard as important. It is evident that there are substantial implications of the use of such environments for enhancing the role of problem-posing in our schools. Capturing Particularity and Inferring Generality I wish now to describe a second feature of "intellectual mirror" environments and examine its consequences. Before doing so, I shall return, for the sake of concreteness, to the description of the Geometric Supposer. There is a focal problem in the teaching of geometry. The human cognitive apparatus seems to require diagrams and images in order to aid thinking about spatial matters. The diagrams we construct, however, are of necessity, specific. For example, although we can construct a regular 8-gon and a regular 12-gon, we cannot construct a regular N-gon. We posses no visual notational scheme for shapes that approaches the generality of our notation schemes for algebraic constructs. On the other hand, the mathematics we seek to construct within the framework of geometry deals not with the properties of individual shapes but rather with properties of classes of objects. It is relatively common for students, and even teachers, to be deceived by the artifactually particular properties of particular diagrams and to reach inappropriate generalizations that are rooted in the particularity of the diagrams we use. By allowing the user to make constructions on particular shapes, and then enabling the construction as a procedure to be separated from the original shape and to be repeated on another exemplar, the Supposer scaffolds the transition from the particularity of the diagram to the generality of the mathematics. This property of the Supposer environment enables and encourages the making and exploring of conjectures. It is easy for a user to explore whether a property discovered to be true in a particular case is true in other cases. If there are other cases for which the discovered properties is true, then the characterization of the nature of the range of cases for which the property obtains becomes an apparent and important problem. A central goal of education is to get our students to internalize the need and desire to continually ask of everything around them, "What is this a case of ?" This continuing fugue between particularity and generality permeates every intellectual discipline and activity. The student of literature must come to understand that the greatness of the nineteenth century Russian novelists lies in their ability to explore the generality of the human condition through the particularity of three very different brothers or a fatuous landowner. The student of biology must come to understand that the genetics of drosophila can shed light on the ways in which all organisms transmit their characteristics to succeeding generations. It hardly needs to be pointed out that the making and exploring of conjectures is, in its very essence, a way of asking the question "What is this a case of ?" On the Need for Abstraction There is yet another property of "intellectual mirror" environments that needs to be examined. Here I refer to the need for the program to "know" with precision and accuracy what the user is talking about. Once again, it is useful to draw on a specific example from the Geometric Supposer. Many people who see the Supposer for the first time are somewhat surprised that it is not possible to enter a triangle, for example, by clicking' the mouse on three non-colinear points on the screen. Instead, the user is required to obtain a triangle on which to make a construction in one of the two ways. The first way allows the user to choose a randomly generated triangle from within a defined class, such as obtuse isosceles or scalene right. The second way allows the user to specify a triangle by a triplet of measures such as SIDE SIDE SIDE or ANGLE SIDE ANGLE. In either case the program "knows" what kind of triangle the user intends to work with. Suppose, on the other hand, that the user had to input a triangle by clicking on three points on the screen. Suppose further, that in an attempt to enter an equilateral triangle, the three points that the user clicked determined a triangle with angles of 59, 60 and 61 degrees. What should the program conclude? The power of an "intellectual mirror" lies in its neutrality with respect to the actions of the user. It must simply be a mirror that reflects, with as little distortion as possible, the consequences of his or her actions without attempting in any way to make inferences about the users' intentions. Imprecision in specifying the "object" of mutual concern to the user and the program is inimical to the achievement of this goal. I would like to argue that this need for an abstract description language that allows the user to communicate with the software environment is an asset rather than a disadvantage in terms of our long-term educational objectives. Despite all of the rhetoric surrounding the need for more "hands-on" experience in the education of our youngsters, we should not lose sight of the fact that "hands-on" experience is merely a stepping stone to a more important goal, namely "minds-on" experience. In a deep sense, getting youngsters to abstract generality from the particularity of their own experiences is among our most important ultimate aims as educators. Some Concluding Remarks In writing this paper, I have made a variety of assertions and analyses about the desired role of problem-posing and conjecture making in mathematical and scientific education. These assertions and analyses have been based on the experiences that my colleagues and I have had, both directly and indirectly, with large numbers of geometry students and teachers using the Geometric Supposer. There have been many reports [2] of the effects of this mix of new tools and habits of mind with traditional curriculum content on teachers and students. The settings from which these data are drawn vary widely as do the students and the teachers within those settings. The outcomes can be categorized in many ways. There are performance measures using traditional assessment instruments. Using these measures there are small differences in performance almost always in favor of the students who have become accustomed to making and exploring conjectures. More important, however, and probably more illuminating are different outcome indicators. These students come to school early in order to work on "their" mathematics on their own time. They discuss and argue mathematics with one another, and they end up taking further mathematics courses that they would otherwise not have taken. In addition, we have had a smaller and newer but equally encouraging collection of experiences with similar environments designed to encourage the making and exploring of conjectures in algebra at the secondary school level. I do not claim that expanding our conception of science and mathematics instruction so that it gives problem posing a central role will, by itself, repair the ills of our educational systems in these domains. But the insufficiency of problem posing as a prescription does not in any way detract from its necessity. Judah L. Schwartz  mathMINDhabits by Judah L. Schwartz is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. References 1. Schwartz, J.L.: Intellectual Mirrors - A Step in the Direction of Making Schools into Knowledge Making Places, Harvard Educational Review, 25(1), 51-61 (1989) 2. Schwartz, J.L., Yerushalmy, M., & Wilson, B. (eds.): What Is It A Case 0f? A Geometric Supposer Reader, Hillsdale NJ: Lawrence Erlbaum, 1993. |