ࡱ> [ bjbj 7ΐΐ  /////CCC8{<DCI(####WWWzI|I|I|I|I|I|I$MOI/WWWWWI//##4I W/#/#zI WzI V7R9#6C-6"8fII0I<8Pc 4P4R9R9P/R:WW WWWWWII WWWIWWWWPWWWWWWWWW : ROY WAGNER ON MATHEMATICAL ABSTRACTION: UNFILTERED REACTIONS Brian Greer brian1060ne@yahoo.com Recently, following a trail on Scholar Google, I came across a paper in this journal by Wagner (2019) that I found extremely illuminating. In the paper, he offers a new definition of abstraction in mathematics. Experiencing multiple resonances, the following is a spontaneous and unpolished response. And it is a response to that paper only, as I have not read his other work yet. On the assumption that readers have read, or will read, Wagners paper, I do not summarize it apart from pointing to connexions that I see with many of my current preoccupations. Wagners proposed recharacterization of abstraction, as I read it, lies squarely within a confluence of recent reflections on mathematics that seek enlightenment, not through the traditional ruminations of philosophers, but in observations of mathematical practices how those humans designated as mathematicians behave when they do what is consensually recognized as doing mathematics. This change reflects, it seems to me, a revolutionary shift, from trying to define mathematics as some kind of entity that can exist to trying to describe doing mathematics as a family (or family of families?) of evolving practices embedded in historical, cultural, social, political in short, human contexts. Reuben Hersh was an outstanding leader in this movement, as exemplified in his 2014 compilation Experiencing mathematics: What do we do when we do mathematics?. Of course, within doing mathematics there are distinctions to be made in particular, I suggest, between initial discoveries/creations and later systematizations. Wagners paper questions the typically unexamined assumption that primary ideals to aim for when doing mathematics, with the ultimate goal of achieving absolute certainty, are precision, consistency, avoidance of ambiguity, and the extended family of such aspirations. For example, in a very short abstract he includes the words intangibility, imperfect, unstable, dynamic, inconsistent. He thereby aligns himself a position exemplified by the forthright statement that any project that would eliminate ambiguity from mathematics would destroy mathematics (Byers, 2010, p. 24 and see the review reproduced in Hersh (2014)). Thus, after briefly surveying prevalent characterizations of mathematical abstraction, Wagner invites us to consider it as the practice of incomplete, underdetermined, intermittent and open-ended translations between systems of presentations. He recommends do[ing] away with the suggestion that such translations [between available representations] generate a set of invariants that serve as the goal or content of abstraction. (Note: I do not follow Wagner and others in preferring presentations on the grounds that the (Oxford) dictionary defines the prefix re- in this case as meaning expressing intensive force). I discuss below why I find that position so relevant to school mathematics. Another suggestion is to characterize abstraction as additive transforming actions, processes, or relations into objects that can be manipulated rather than substractive, stripping down to some essence. And he suggests redrawing the boundaries of the concrete. I wonder if he deliberated whether to propose a new definition of abstraction (what is mathematical abstraction, really) rather than describe an important interpretation of certain aspects of observable practices and give it a new name. Enough preamble. Let me, perhaps arrogantly, confess to the feeling that having extremely limited knowledge of the standard canon of philosophy may be an advantage, since I have to think these issues through for myself. In any case, recent thinking has led me to a number of positions that I see reflected in Wagners paper, of which the following are the ones that first leap to mind. I make no claims to systematic coverage of his very rich analysis, and as befits one diagnosed with chronic tangentivitis, there are digressions. Philosophical issues Bad questions, problematic words There are several reasons why I am prepared to argue that What is mathematics?, despite its popularity (very notably, the title chosen by Hersh (1997) or was he teasing?) is a bad question. In a court of law, it could be ruled out as a leading question, since its very phrasing implies that mathematics is both singular and timeless, and some kind of entity rather than something done by people within families of practices. Channeling the fable of the blind men and the elephant, I am sceptical of any statement that begins Mathematics is ; an improvement would be: One important aspect of mathematics is . I have developed an aversion to binary questions about the nature of doing mathematics: Is it universal or culture-bound? Is it discovered or invented? I can admire the neatness of the dual statement (of doubtful provenance) that mathematics is the art of giving the same name to different things whereas poetry is the art of giving different names to the same thing, but the very neatness is questionable. When Hamlet says to die, to sleep, no more is he giving different names to the same thing, or vice versa? Wagners paper pressed many sensitive buttons about particular words. I shudder when I see the phrase the concept of number. Or the concept of a triangle (people doing experiments with rats will tend to use it when the rat can be trained to consistently find food behind the door with a triangle drawn on it). A major improvement is Vergnauds (2009) formulation of conceptual field. Another refinement is the distinction between concept definition and concept image (e.g. Tall & Vinner, 1981). And then there is all the work a few decades ago in cognitive psychology about prototypes: All birds are birds but some (e.g., robins) are more birds than others (e.g., penguins) (and see fuzzy logic but thats another story). What are people trying to do with the phrase a mathematical object?. Is it some variety of metaphor? Calling on a small piece of knowledge of philosophy, my concept image of object is the stone kicked by Johnson in refuting Berkeley. Closely related uses of the word exist and its cognates create perhaps the most serious source of confusion. I will just say that I find a question such as Do imaginary numbers exist? unhelpful. Then there are real and concrete. Wilensky (1991) quotes with approval the definition of a mathematician as someone for whom imaginary numbers are just as real as real numbers. Well, that blackballs a lot of people who consensually would be regarded as having been mathematicians (and it also illustrates how mathematicians, while often thinking of mathematics as timeless, find it hard to escape the image of mathematics prevalent at this point in history). Wagners suggested redrawing of the boundaries of concrete in relation to representations I find attractive. Wilensky (1991) hints at a Johnson-Wilensky Test for concreteness, namely Can it be kicked? (as opposed to the metaphor of kicking ideas around), but surely a less stringent variation is needed. Ontological Vagueness This phrase pinpoints a current concern of mine. First, to go back to concept. If I had the power, I would ban the use of the phrase the child has the concept of X, most especially in the case X = number, together with the idea of a touchstone, such as the conservation of number task, that decides when a child has crossed the threshold. For close on 40 years I have been puzzled about the ontological status of cognitive structures (Jeeves & Greer, 1983, pp. 65-69). In those pages, we used a lengthy quotation from Feldman and Toulmin (1975, p. 426) and it still seems apposite: Nowhere, it seems, are the differences between the problems involved in formally representing a theory and the problems in empirically testing it so difficult to keep separate as in the area of cognition. Just because the theoretical system in question can plausibly be represented as corresponding to some mental system in the mind of an actual child, we may be led to conclude that the formalism of the theoretical system must be directly represented by an isomorphic formalism in the mind of the child In this way, ontological reality is assigned to the hypothetical mental structures of the theory simply on the basis of the formal expressions by which they are represented in the theory. As I read it, this goes to the heart of what I would call the Piaget-Bourbaki fallacy (see below). I cannot find citations of Feldman and Toulmins paper or elaborations of their discussion; if any reader can enlighten me, I would be very grateful. Likewise, I have misgivings when I find one of my heroes, Hans Freudenthal, writing: The final goal of teaching and learning is mental objects. I particularly like this term because it can be extrapolated to a term that describes how these objects are handled, namely by mental operations (Freudenthal, 1991, p. 19, emphasis added). Contrast that with a material object, for example a representation of a graph of a function on a screen and software that allows it to be operated upon (e.g., by varying a parameter) using a mouse (extension of a hand). B: Representations and Doing Mathematics Representational flexibility As a dilettante, I enjoy solving (not difficult) mathematical problems (puzzles). Here is one (Greer and Harel, 1998, p. 18): Given 16 logic blocks differing in size (large, small), thickness (thin, thick), shape (square, triangle) and colour (red, blue), can you arrange them in a circle so that each neighbouring pair differs on only one attribute? I am pleased that I quickly realized that this problem could be solved by sketching a 4-dimensional hypercube and reading off a path from one vertex passing once through each of the other vertices and ending back at the starting point. This example illustrates the long familiar point that a well-chosen representation often makes a problem easy to solve. In general, it seems obvious that representational flexibility is an essential component of mathematical expertise. A related aspect is that different solutions for a problem afford, with differences in accessibility, multiple extensions and variations. Thus, my favourite proof of the theorem misattributed to Pythagoras is that based on the observation that the altitude to the hypoteneuse of a right-angled triangle divides it into two similar triangles each of which is similar to the initial triangle, which extends naturally to a broader result based on a powerful general idea about dimensionality. Opening new representational windows The phrase is from Kaput (1998). Historically, a great deal of the development of mathematics has been made possible through the long-drawn-out development of structured ideas interactively with representational systems for them. Kaput (1984) provided a detailed historical analysis of these processes in the case of calculus. A good example for me of a new representational window is provided by software such as Geometers Sketchpad within which one constructs not diagrams but procedures for constructing diagrams. Thus, a procedure can be defined that will draw a quadrilateral and join the midpoints of adjacent sides. Appreciation of the invariance whereby the resulting figure is a parallelogram is ramped up by grabbing a vertex of the original parallelogram with the mouse, moving it around, and hence making the whole configuration dance. One final point. As elaborated in the theory of speech acts, an utterance cannot be interpreted merely on the basis of the string of words it contains (many jokes rely on this kind of ambiguity). By analogy, Goldin and Kaput (1996) analyse representational acts. But thats another another story. Issues of Mathematics Education Premature formalization If pressed to nominate the most fundamental problem in school mathematics education, the above would be my response. The first general comment is that the pace is much too fast. An indelible memory is of sitting beside an 8th grade student trying to solve a linear equation in x. By analogy with Freires description of an impoverished conception of reading as barking at text, he was pawing at symbols. Nowadays in many countries, there is pressure for all students to learn algebra before leaving school. That can be challenged (Greer, 2008). Freudenthal (1991) points to the grave error that he terms didactical inversion. Specific instantiations of group structures were known in multiple mathematical contexts numerical, symmetries, permutations and so on. These were what Freudenthal (1991, p. 20) calls rich structures, replete with context. Eventually the definition of the generalized group with a few simple axioms emerged. Starting with that definition, or anything similar, when teaching mathematics Freudenthal considers a grotesque didactical error. When the New Math was laid to rest, he delivered its epitaph: New Maths wrong perspective was to replace the learners insight with the adult mathematicians (Freudenthal, 1991, p. 112). I remember when Sputnik went up and the pure mathematicians told us that children must learn set theory, bizarrely confusing old-style foundations of mathematics and foundations of mathematics education. Essentialism The mathematician Wu (1999) provides me with the perfect target. The reader is invited to access his paper to appreciate why I characterize it that way (and, in passing, to note his crude insulting of Freudenthal). The following is a richly flavoured quotation in relation to initial teaching of fractions: [] instead of offering mathematical explanations to children of why the usual algorithms are logically valid a simple task if one starts from the precise definiton of a fraction, algorithms are justified through connections among real-world experiences, concrete models and diagrams, oral language and symbols [] It is almost at if one makes the concession from the start: We will offer anything but the real thing). (Wu, 1999, p. 2, emphases in original if I had been adding emphases, I would have chosen simple and real). Later (p. 6) we learn that Wu (1998) summarizes a complete, self-contained mathematical treatment of fractions that explains every step logically that starts with the definition of a fraction as a number (a point on the number line, to be exact). The summary concludes with the statement that The whole treatment is elementary and, in particular, is appropropriate for grades 5-8. In other words, it eschews any gratuitous abstractions. (p. 7, emphasis most decidedly added). I have no idea what he means by abstractions and doubt that he means it in the sense of Wagner, though he comes close to the latters definition by referring to connections among real-world experiences, concrete models and diagrams, oral language and symbols as already quoted (except that he used that phrase dismissively as obviously (to him) wrong). All of this could be ignored, perhaps, as an example of what mathematicians does in their spare time, if it were not for the fact that his opinions carry political weight. A broadly similar approach is presented in Common Core State Standards for Mathematics (2010), the curricular framework in the United States. Thus, the first mention of fractions, in grade 3 (p. 24), sets as the headline goal understand a faction as a number on the number line. The fraction a/b is introduced thus: Represent a fraction a/b on a number line diagram by marking off [how?] a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. In Grade 5, the interpretation of a fraction a/b as division of a by b is introduced as a theorem (with a special notation to avoid the statement of the theorem being a/b = a/b). Mathematics educators (e.g. Lamon, 1999) generally take the view that rational numbers have multiple aspects: 3/5 can occur in representing 3 of 5 equal parts, the result of dividing a quantity of measure 3 by 5, the ratio of 3 to 5, and an operator such as 3/5 of and so on. Wu (1999, p. 2) is concerned that children will find it difficult to cope with such polysemy. Well, thats what teachers are for (whether elementary teachers are appropriately prepared for such a difficult task is another story). Conceptual restructuring 3 8 is an impossibility, it requires you to take from 3 more than there is in 3, which is absurd (De Morgan, 1810, pp. 103-4, emphasis added) Augustus De Morgan, no mathematical slouch, continued by setting out complex rules for solving linear equations in one variable, which children these days would be expected to solve using a unitary method, by classifying them into six different variants to avoid having to confront negative numbers. This example is my favourite for illustrating how difficult, historically, have been the processes of moving from the unthinkable to the thinkable, from the ineffable to the effable, from seeing to believing (echo of Cantor). Another angle is that it has been shown that unschooled children cope well with addition and subtraction of negative numbers in context (Mukhopahyay, Resnick, and Schauble, 1990), but thats yet another story. In the 80s and 90s, I researched the considerable difficulties of conceptual restructuring required to move from multiplication and division of natural numbers (beginning, naturally, as repeated addition) to multiplication and division of positive rationals. The most striking phenomenon is how difficult it is to take on board that multiplication makes bigger, division makes smaller which has been very starkly demonstrated (e.g., Mangan, 1989). I suggested the term nonconservation of operations (Greer, 1987) to describe the phenomenon that it is far from obvious that the required operation in a situation related to, say, constant speed, time, and distance, is the same regardless of the numbers attached to those variables. Such observations motivate suggestions for how to help children negotiate the necessary conceptual restructurings. Ironically, what I am trying to say was beautifully expressed by De Morgan (1831, p. 33): If we could at once take the most general view of numbers, and give the beginner the extended notions which he may afterwards attain, the mathematics would present comparatively few impediments. But the constitution of our minds will not permit this In the limited view which we first take of the operations which we are performing, the names which we give are necessarily confined and partial; but when, after additional study and reflection, we recur to our former notions, we soon discover processes so resembling one another, and different rules so linked together, that we feel it would destroy the symmetry of our language if we were to call them by different names. We are then induced to extend the meaning of our terms Mappings between representations dont come naturally One theme in Jeeves and Greer (1983) is whether children can map between representations. While I am not attempting to deal with the notion of internal representations (Goldin & Kaput, 2013) here, there are a number of experiments summarized in Jeeves and Greer (1983) that suggest how the ontology of an internal representation of a structure may be studied. One, in particular, involved the presentation of three learning tasks, each embodying the structure of the 2-group, but with the representations of the two elements being of (a) different shapes (b) different shades, (c) different sizes. The idea was to see if subjects would perform at the same level on all tasks, suggesting that the learning of the four associations (e.g. H*H = H, H*T = T, T*H = T, T*T = H) had to begin de novo each time, or whether performance would improve across the three tasks, suggesting exploitation of the common structure. For the four groups tested (16 in each group), the results were clearcut. 9-year-olds, 11-year-olds and 15-year-olds performed at essentially the same level on the first task, but whereas the 9-year-olds showed no improvement across the subsequent tasks, the 15-year-olds did. The results were intermediate for the 11-year-olds, while undergraduate students performed better on the first task, and also showed marked improvement, approaching perfect performance on the third task. The results of this experiment, if they could be replicated, are very significant for any theory about how the ability to recognize and exploit structure develops. In conclusion, I suggest that if you want children to be able to map among representations you need to think hard about how to teach them to do it, and you need patience and time. Which takes us to a special class of manipulable, concrete, representations that are designed as teaching/learning aids. When silver bullets arent silver bullets A bunch of silver bullets (or any set of discrete, small hard things that afford rearranging and counting) can be quite useful for teaching many things about natural numbers, but there are reasons why such manipulatives have limited value for instruction. Wagner refers to Schoenfelds (1986) careful discussion of the arithmetic blocks designed by Dienes. Holt (1982) commented as follows: Children who already understood base and place value, even if only intuitively, could see the connections between written numerals and these blocks But children who could not do these problems without the blocks didnt have a clue about how to do them with the blocks They found the blocks as disconnected from realty, mysterious, arbitrary, and capricious as the numbers that these blocks were supposed to bring to life. A child once put it succinctly (Hart, 1993): Bricks is bricks, and sums is sums. The problem is that the mathematical understanding, which has to be achieved, is at the same time needed to gain this understanding (Van den Heuvel-Panhuizen, 2004, p. 285). Similar considerations may help to explain why it has been so hard to turn Polyas analyses into classroom gold in order to powerfully apply think of a similar problem you need to have a sophisticated awareness of what similar problem means. Abstraction in modelling A model is always a simplication, but you need to consider how appropriate that simplification is. Consider this story: It was a lesson under the heading of ratio and proportion and the teacher told me that she wanted to approach the mathematical concepts in a practical way. So she offered [] [a scenario involving mixing paints to reproduce a particular colour]. The problem seemed quite clear and pupils started to calculate using proportional relationships. But there was one boy who said: My father is a painter and so I know that, if we just do it by calculating, the colour of the room will not look like the sample. We cannot calculate as we did, it is a wrong method! In my imagination I foresaw a fascinating discussion starting about the use of simplified mathematical models in social practice and their limited value in more complex problems [] but the teacher answered: Sorry, my dear, we are doing ratio and proportion. (Keitel, 1989, p. 7). Of course, if all you want is to teach real mathematics, the teachers is right (see Toom (1999) for a clear exposition of this value system). The mathematician Charles Lutwidge Dodgson (better known by his pseudonym, Lewis Carroll) produced an analysis that seems way ahead of its time (Carroll, 1880). He proposed the following problem: If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill 100 cats in 50 minutes? There is a standard way of solving this using proportionality, but Carroll breaks the rules by insisting on thinking about the reality of fractional cats and rats and framing it as a modelling problem, for which the answer depends on the assumptions made (see Verschaffel, Greer, & De Corte, 2000, pp. 132-134 for further discussion). Phylogeny and Ontogeny The theory that ontogeny recapitulates phylogeny is long discredited but spawned many pernicious, including racist, offspring (Gould, 1985). To my mind, Piagets claim that the mother structures posited by Bourbaki correspond precisely to elements of his theory of cognitive development is part of that story (and suffused with ontological vagueness). In this section, I restrict myself to two aspects of the differences between humankind creating mathematics and a child today learning mathematics. Many have pointed out that children in school are expected in a few years to come to grips with mathematics that took humankind millennia to develop; here I choose the phrasing of Hofstadter (2013, p. 391): [] school is seen as a magical shortcut that allows ideas arduously developed by humanity over thousands of years to be transmitted in a few years to a random human being. Under instruction, knowing where it is going Mathematics makes its road as it walks, with many wrong turns and blind alleys along the way but also revivals in places, such as treatments of infinitesimals, as carefully analysed by Wagner. Children learn in school under instruction, with their instructors, however imperfectly, being guided by some sense of the destination. Those who are inclined to write that the child actively constructs his/her world, and those who have been tempted to follow them, may have done mathematics education harm, in my opinion. Of course, such a bald statement demands elaboration, not always to my satisfaction. First, there is the solipsism of the phrase his/her world is it not our world? Do we not share foundational affordances: physical (gravity, cycles of seasons and days); biological (prenatal listening to mothers heartbeat, digits that can be counted on); social (language, for heavens sake!). Second, I think it is a fair question to ask how far the most brilliant of children would advance in mathematics without instruction. Here I call on Hamlyn: Teaching is one of the immense social influences that can affect a child, but its effects can be out of proportion to any other kind of social influence once the first beginnings of a childs life are past. In it once again knowledge builds on knowledge, but the form of experience that makes it possible is really quite unlike those forms of experience that come the individuals way when teaching is not involved. (Hamlyn, 1978, p. 144) In terms of having some sense (not teleological) of where we would like the child to be going, I invoke Freudenthals principle of guided reinvention (1991, p. 45): Children should repeat the learning process of mankind, not as it factually took place but rather as it would have done if people in the past had known a bit more of what we know now. (p. 48). Learning from History: Applied Phylogeny Perusal of history can help, at the most simple level by avoiding blind alleys. Even the most Eurocentric mathematics educator would not advocate teaching children Roman numerals. More substantively, we can identify where epistemological obstacles have proven to be exceptionally hard to overcome and try to learn from what finally worked, through what Kaput (1994, p. 83) in his survey of the development of calculus applied phylogeny. Looking at it again, that whole chapter could be seen as an exercise in undoing a didactical inversion, going back to the roots of calculus as the mathematical study of change, and using the new representational windows afforded by computer technology. In terms of school mathematics, he proposed (p. 78) that calculus might be regarded as a web of ideas that should be approached gradually, from elementary school onward in a longitudinally coherent school mathematics curriculum. He emphasized, with examples of his own innovative software, how it is possible to escape the limitations of functions defined by closed-form algebraic expressions in order to model dynamic systems (Fisher, 2021; Morrison, 1991) and to move from graphic displays to graphic action representations (pp. 148-151). Final Comment Much of my reading these days makes me optimistic that philosophers of mathematics, who Hacking (2014, p. 41) suggested tend to take mathematics for granted, are beginning to wake up, that historians of mathematics are doing the hard work, and that mathematics educators are addressing the sad reality that, for too many people, the experience of mathematics in school is personally alienating and intellectually stultifying. It doesnt have to be like that. References Byers, W. (2010). How mathematicians think. Princeton, NJ: Princeton University Press. Carroll, L. (1880). The cats and rats again. The Monthly Packet (February). Common Core State Standards for Mathematics (2010). Retrieved from: www.corestandards.org/Math/ De Morgan, A. (1910). Study and difficulties of mathematics. Chicago: University of Chicago Press (originally published 1831). Feldman, C. F., & Toulmin, S. (1975). Logic and the theory of mind. In W. J. Arnold (Ed.), Nebraska symposium on motivation: Conceptual foundations of Psychology. Lincoln, NE: University of Nebraska Press. Fisher, D. (2021). Global understanding of complex systems problems can start in pre-college education. In F. K. S. Leung, G. A. Stillman, G. Kaiser, & K. L. Wong (Eds.), Mathematical modeling education in East and West. Springer. Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht, The Netherlands: Kluwer. Goldin, G. A., & Kaput, J. J. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.),Theories of mathematical learning(pp. 397-430). Mahwah, NJ: Lawrence Erlbaum. Gould , S. J. (1985). Ontogeny and phylogeny. Cambridge, MA: Harvard University Press. Greer, B. (1987). Brief report: Nonconservation of multiplication and division involving decimals.Journal for Research in Mathematics Education,18(1), 37-45. Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of research on mathematics education (pp. 276-295). New York: Macmillan. Greer, B. (1994). Extending the meaning of multiplication and division. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 61-85). Albany, NY: SUNY Press. Greer, B. (2004). The growth of mathematics through conceptual restructuring. Learning and Instruction, 14, 541-548. Greer, B. (2005). Ambition, distraction, uglification, and derision. Scientiae Paedogogica Experimentalis, 42, 295-310. Greer, B. (2012). The USA Mathematics Advisory Panel: A case study. In O. Skovsmose & B. Greer (Eds.), Opening the cage: Critique and politics of mathematics education (pp. 107-124). Rotterdam, The Netherlands: Sense Publishers. Greer, B. (2021). Learning from history: Jens Hyrup on mathematics, education, and society. In D. Kollosche (Ed.), Exploring new ways to connect: Proceedings of the Eleventh International Mathematics Education and Society Conference (Vol. 2, pp. 487496). Tredition.  HYPERLINK "https://doi.org/10.5281/zenodo.5414119" https://doi.org/10.5281/zenodo.5414119 Hacking, I. (2014). Why is there philosophy of mathematics at all? Cambridge: Cambridge University Press. Hamlyn, D. W. (1978). Experience and the growth of understanding. London: Routledge and Kegan Paul. Hersh, R. (1997). What is mathematics, really? New York: Oxford University Press. Hersh, R. (2014). Experiencing mathematics: What do we do, when we do mathematics? American Mathematical Society. Hofstadter, D., & Sander, E. (2013). Surfaces and essences. New York: Basic Books. Holt, J. (1982). How children fail (2nd Edition). New York: Dover. Jeeves, M. A., & Greer, B. (1983). Analysis of structural learning. London: Academic Press. Kaput, J. J. (1992). Technology and mathematics education. In: D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515556). New York: Macmillan. Kaput, J. J. (1994). Democratizing access to calculus: New routes to old roots. In: A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 77156). Hillsdale, NJ: Lawrence Erlbaum Associates. Kaput, J. J. (1998) Representations, inscriptions, descriptions and learning: A kaleidoscope of windows. Journal of Mathematical Behavior, 17(2), 283-301. Keitel, C. (1989). Mathematics education and psychology. For the Learning of Mathematics, 9(1), 7-13. Mangan, C. (1980). Multiplication and division as models of situations. In B. Greer & G. A. Mulhern (Eds.), New directions in mathematics education (pp. 107-127). London: Routledge. Morrison, F. (1991). The art of monitoring dynamic systems. New York: Wiley Interscience. Mukhopadhyay, S., Resnick, L. B., & Schauble, L. (1990) Social sense-making in mathematics: Childrens ideas of negative numbers. Learning Research and Development Center, Pittsburgh. Retrieved from: files.eric.ed.gov/fulltext/ED342632.pdf Schoenfeld, A. H. (1986). On having and using geometric knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 225-264). Hillsdale, NJ: Lawrence Erlbaum. Tall, D., & Vinner, S. (1981).  HYPERLINK "https://link.springer.com/article/10.1007/BF00305619" Concept imageand concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169. Toom, A. (1999). Word problems: Applications or mental manipulatives. For the Learning of Mathematics, 19, 3638. Van den Heuvel-Panhuizen, M. (2004). Shifts in understanding: The didactical use of models in mathematics education (pp. 285-290). In H.-W. Henn & W. Blum (Eds), ICMI Study 14: Applications and modelling in mathematics education. Pre-conference volume (pp. 97-102). Dortmund: Universitt Dortmund. Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83-94. Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse, The Netherlands: Swets & Zeitlinger. Wagner, R. (2019). Mathematical abstraction as unstable translation between concrete representations. Philosophy of Mathematics Education Journal, 35. Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism: Research reports and essays, 1985-1990 (pp. 193-203. Norwood, NJ: Ablex. Wu, H. (1999). Some remarks on the teaching of fractions in elementary school. 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