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Peter Appelbaum
appelbap(at)arcadia.edu
Arcadia University, USA
The Manifesto of our conference, Childrens Mathematical Education, calls our attention to the ways that our efforts are tools for both the development of the child and for solving critical problems in a global society. Indeed, these two ways of thinking about our work are always interwoven, since individual children are always present and at the same time future members of our society. That is, our global society is nothing other than ourselves. Education in general, and mathematics education in particular, is central to the basic existence and aims of social life. I start today by reminding us that this manifesto is, if anything, gentle in its call for principals, education officials, the general community, and, most importantly, parents and teachers, to consider how and why fundamental mathematical concepts are at the heart of both personal and social development. Toward this aim, I ask us to think about two concepts that undergird most of the intellectual work of teachers and curriculum workers in mathematics education, and to think critically about the ways in which they impact on our beliefs about what we should do and what might be changeable: the support of sense-making by pupils, and the overarching aim of facility with representations.
By sense-making, I am referring to the common assumption that our task as mathematics educators is to help young people to make sense of mathematics. We receive mathematics as a reasonable and logical world within which one should be made to feel comfortable and secure. We often agree that mathematics is the one place where we can be certain about what we know and whether or not we are correct. My comments today have to do with the power of these assumptions to enable specific kinds of educational experiences while also perhaps failing to allow pupils to fully appreciate the wonders and powers of mathematical modes of inquiry and understanding.
With representations, I ask us to consider also the power of our typical pedagogies, which tend to lead students from the concrete to the abstract, and also to move students away from specific instances of mathematics in the world toward general representations of these instances. We might think of the representations (of the ideas) as the actual material and content of the mathematics itself. I mean here simple things like numerals to represent numbers of things; drawings of shapes to represent ideal geometric relationships; fractions to represent parts of wholes, proportions, and ratios; equations to represent functional relationships, letters to represent variables that may take on different values, and so on. Other representations model mathematical concepts and relationships, such as base-ten blocks for arithmetic operations, drawings of rectangles or circles for fractions and ratios, or graphs which visually represent algebraic equations. In my experience, much of mathematics education aims to help students to develop artistic virtuosity with mathematical representations for communicating their ideas. However, if we take this artistic virtuosity seriously, then critics of artistic practice sometimes suggest that representation is not always the aim of art, and in fact, representation often violates art itself. What could young children, as mathematical artists, do, then, if they would not primarily be practicing forms of mathematical representation? I will return to this question, because it is connected most directly with our Manifestos call to consider the broad, social contexts even as we focus on the individual mathematicians in our kindergartens, primary school classrooms, and on the adolescent mathematicians with whom we work day-to-day.
Stop Making Sense
Brent Davis (2008) recently wrote, In the desire to pull learners along a smooth path of concept development, weve planed off the bumpy parts that were once the precise locations of meaning and elaboration. We have, he says, created obstacles in the effort to avoid them. Davis describes huh moments, when it is possible to enter authentic mathematical conversations. For example, we might ask someone to describe what we mean when we write 2/3 = 14/21. Responses vary from pictures of objects to vectors on a number-line, but all share a conceptual quality of relative change so that increasing one thing leads to a proportional increase in another thing or group of things. However, when we ask the same person to describe what is happening in the expression -1/1 = 1/-1, we usually get a kind of huh, which communicates a moment where the mathematics has lost its sense, but which also potentially begins an important (mathematical) conversation. In my own work on what Davis calls the huh moments when mathematics stops making sense to us, and we grope for models apparently not available (Appelbaum, 2008) I, too, have noted the potential for the non-sense-making characteristics of mathematics to generate different kinds of teacher-student relationships, and most significantly, different kinds of relations with mathematics within associated critical mathematical action (Appelbaum, 2003). Mathematics curriculum materials too often hide the messiness of mathematics where sense dissolves into paradox and perplexity, but more importantly they construct a false fantasy of coherence and consistency. As most professional mathematicians understand, mathematics at its core is grounded in indefinable terms (set? point?), inconsistencies (Gdels proof? Cantors continuum hypothesis?) and incoherence (the limit paradox in calculus?). At a more basic level, multiplying fractions ends up making things smaller even though multiplying conjures images of increasing to many people; two cylinders made out of the same piece of paper (one rolled length-wise, on width-wise) have the same surface area but hold different volumes; were taught to add multiple columns of numbers from right to left with re-grouping, when it is so much easier to think left to right starting with the bigger numbers. In some cases, it is impossible, speaking epistemologically, for mathematics as a discipline to make sense; in others, it might be more valuable pedagogically to treat mathematics as if it does not make sense. To do so would celebrate the position of the pupil, for whom much of the mathematics is new and possibly confusing anyway.
Yet, so much of contemporary mathematics education practice is devoted to helping students make sense of mathematics! What if, instead, we stopped trying to make sense, and instead worked together with students to study the ways in which mathematics does and does not make sense? Instead of school experiences full of memorization and drill on techniques, we would imagine classroom scenarios full of conversation about the implications of one interpretation over another, or of explorations that compare and contrast models and metaphors for the wisdom they provide.
Elizabeth de Freitas (2008) describes our desire to make mathematics fit a false sense of certainty as mathematical agency interfering with an abstract realm. She encourages teachers to intentionally trouble the authority of the discipline, in order to belie the reasonableness of mathematics. In this way, we and our pupils can better understand how mathematics is sometimes used in social contexts like policy documents and arguments, business transactions, and philosophical debates, to obscure reason rather than to support it. Stephen Brown called this kind of pedagogy, balance[ing] a commitment to truth as expressed within a body of knowledge or emerging knowledge, with an attitude of concern for how that knowledge sheds light in an idiosyncratic way on the emergence of a self" (Brown, 1973, p. 214)
So, you may wonder, what does this mean about curriculum materials and textbooks? Obviously somebody somewhere with a lot of authority has actually sat down and written this Numeracy Strategy, says one teacher with whom Tony Brown (2008) spoke. its not like they dont know what they are talking about. Tony Brown blames the administrative performances that have shaped mathematics for masking what Brent Davis calls the huh moments, and what de Freitas describes as the self-denial that accompanies rule and rhythm. Teaching in this senseless world of mathematical practice need not abandon science and the rational. It merely shifts teaching away from method and technique toward what Nathalie Sinclair calls the craft of the practitioner, as she evokes the metaphor of teaching as midwifery from Platos Theaetetus (see also Appelbaum 2000). As midwives, teachers assist in the birth of knowledge; students experience not only the pain and unpredictability of the creative process, but also the responsibility for the life of this knowledge once it leaves the womb. One must care for and nurture ones knowledge, whether it acts rationally or not. Can we be confident that the ways we have raised our knowledge will prepare it for when it is let loose upon the world? Will our knowledge be embodied with its own self-awareness and ethical stance?
A Dubious Theory
A demand that everything make sense, and that this sense be so simple that it is virtually instantaneous if at all possible, dominates the way we work with mathematics in school. We design a curriculum that introduces a tiny bit of new thought once per week or even less often, because we worry that a pupil will feel lost or confused, and not be able to move on to the next tiny new step that follows. I imagine instead a curriculum where children beg for new challenges, and where these children delight in the confusion that promises new worlds of thinking and acting, of children we do not just get by in mathematics class, but who love mathematics as part of their sense of self and their engagement with their world. The French philosopher and social theorist Michel de Certeau (1984) blamed the social sciences for reducing people to passive receivers of knowledge. And indeed, educational research and practice has been dominated by the social sciences for the past century, so we have been living the successes and failures of these approaches to education and now need to look at them critically as we reassess our work in mathematics. de Certeau suggested that the social sciences cannot conceive of people as actors who invent new worlds and new forms of meaning, because they study the traditions, language, symbols, art and articles of exchange that make up a culture, but lack a formal means by which to examine the ways in which people re-appropriate them in everyday situations. This is a dangerous omission, he maintained, because it is in the activity of re-use that we would be able to understand the abundance of opportunities for ordinary people to subvert the rituals and representations that institutions seek to impose upon them. With no clear understanding of such activity, the social sciences are bound to create little more than a picture of people who are non-artists (meaning non-creators and non-producers), passive and heavily subjected to receiving culture. Social sciences thus typically understand people as passive receivers or consumers rather than as makers or inventors of culture, ideas, and social possibilities. Indeed, I believe this is exactly the situation we find ourselves in as we seek ways to make mathematics meaningful for young people and for young people to take advantage of mathematical skills and ideas as they participate in their local and global communities.
This kind of misinterpretation is critical to our "consumer culture," in which people are assigned to market niches and sold products, concepts, modes of life, and predictable desires. In curriculum as in advertising, such social science persists, so that we see students as consumers of knowledge whose desires are shaped by the curriculum via the teacher, teachers as consumers of pedagogical training programs, and so on. de Certeau employs the word "user" for consumers; he expands the concept of "consumption" to encompass procedures of consumption and then builds on this notion to invent his idea of "tactics of consumption". School curriculum tries to sell students on the value of mathematical knowledge; we sometimes call this motivation. New curriculum materials are published and sold as part of a global economic system that demands new and improved products in a cycle of perpetual obsolescence and innovation.
What would it mean for youth who are learning stuff that many adults already know to be artists creators and producers when we seem to want them to consume and use instead? The critical notion turns out to be how we make sense of the art. Susan Sontag (1966) wrote about what she named a dubious theory that art contains content, an approach that she claimed violates art itself. When we take art as containing content, we are led to assume that art represents and interprets stuff, and that these acts of representation and interpretation are the essence of art itself. Likewise in school curriculum, we often imagine the curriculum as content, and move quickly to the assumption that this curriculum represents and interprets. This makes art and curriculum into articles of use, for arrangement into a mental scheme of categories. What else could art or curriculum do? Well, Sontag suggests several things: To avoid interpretation, art may become parody. Or it may become abstract. Or it may become (merely) decorative. Or it may become non-art. (Sontag 1966: 10)
New Worlds of Mathematics Education
Parody, abstraction, decoration, and/or non-art are three types of tactics for art and curriculum. I think, too, that they can be used to stop making sense of mathematics for young children, and instead, in the words of our conference manifesto, they can help us not only to pose questions, but also to look for solutions. Common work of our conference is focused around four main issues: Mathematics as a school subject; Teacher-training; Teachers work; and Learning Mathematics. I conclude with a brief outline for applying the de-Certeauian-Sontagian tactics in each of these four realms. With my suggestions, I am encouraging each of us to consider how school mathematics could be experienced as something other than a representation of content, or something other than an abstract representation of ideas. This does not mean that I want us to abandon representations or the representation of ideas, but that our methods of teaching would not stress this as our primary purpose.
Mathematics as a school subject: Normally, we emphasize two kinds of experiences in school mathematics, and through these we create an implicit story about what mathematics is. We either develop ideas out of concrete experiences, or we model real-life events with mathematical language. An example of the first would be to work with numerals to represent numbers of objects, in order to stress for young children the differences between cardinality and ordinality, or to develop arithmetical algorithms for adding, subtracting, multiplying or dividing numbers. We might work with base-ten blocks, number lines, collections of objects, drawings of objects, and so on. An example of the second might be to create a story problem out of a real-life situation, such as to ask how many tables we need for a party if each table can seat six people, and we expect fifteen people to attend our party; or, to ask, given eighty meters of fencing material, what shape we should use to have the most area for our enclosed playground. Now, suppose we wanted to transform our pedagogy so that the work in our classroom were one of parody, abstraction, decoration, or non-art, rather than representational art. Children might parody routine questions by acting out seemingly absurd situations where the reckoning leads to ludicrous results, or they might ask and answer questions that shed humorous or critical light on typical uses of the mathematics. For example, 4-year-olds who have counted the number of steps from their classroom to the door of the building, in ones, threes, and fives, might then count the number of drops of water to fill a bucket in ones, threes and fives, even though it seems to make no sense to do so this would only be a parody, though, if the children themselves suggested it as a silly thing to do that they wish to do nevertheless. Similarly, ten-year-olds might design alternative arrangements of their classroom that make use of unusual shaped desks, such as asymmetric trapezoids, circles, etc. Mathematics might be abstract if children did more comparing and contrasting of questions, methods, and types of mathematical situations, rather than focusing on the particular questions or on practicing specific methods. For example, 8-year-olds might first organize a collections of mathematics problems first into three categories, and then the same problems into four new categories, rather than solving the problems themselves; the classification of the problems into types would constitute the mathematical work, rather than the solution of the mathematical problems. Mathematics as decoration might be accomplished through a classroom project where students experiment with different representations of a mathematical idea for communicating with various audiences. After working with ratio and proportion, for example, a class of 11-year-olds might form small groups, one of which creates a puppet show for younger children, one of which composes a book of poems for older children, and another of which prepares a presentation for adults at their neighborhood senior citizens community center, all on the same subject of applying ratio and proportion to understand the ways that a recent election unfolded. In this sense of considering the appropriate way to describe ratio and proportion for a particular audience, the mathematics is more of a decorative from of rhetoric than a collection of skills or concepts; the important concepts have more to do with democratic participation in elections than with the mathematics per se. Mathematics as non-art uses artistic work that is not considered art as its model we could ask, when is creative mathematical work not mathematics? One answer is, when it is something else other than mathematics per se for example when it is an argument for social action presented at a meeting; when it is an example used to demonstrate a philosophical point; when it is a recreational past-time; etc. In other words, mathematics as non-art would be mathematics not done for its own sake; mathematics as non-art would be mathematics for the purposes of philosophy, anthropology, literature, poetry, archaeology, history, science, religion, and so on. As long as the activity has purpose other than the mathematics itself.
Teachers Work: So, mathematics as a school subject can and should take on the character of parody, abstraction, decoration or non-art. If this is to occur, there are important implications for the teachers work. For one thing, the teacher would not be providing clear presentation or explanations of mathematical concepts or procedural skill. Instead, we can learn from current work at the University of Amsterdam on the types of teacher help that support mathematical level-raising (Dekker & Elshout-Mohr 2004, 2005; Pijls 2007; Pijls, Dekker & Van Hout-Wolters 2007). In their studies, they have found that teacher help directed at mathematical content explanations and demonstrations, is rarely more valuable than teacher help directed at collaborative learning and groups processes, and in fact sometimes teacher help focused only on the group processes leads to more significant conceptual level-raising. In other words, the nature of useful teacher work involves making it possible for pupils to participate as creators and consumers of mathematical art that is not representational, and which does not aim at simplifying the path to sense-making. Instead, teacher work essentially makes it possible for pupils to experience together the authentic practices of sense and non-sense through events such as parody, abstraction, decoration, and non-art. In the group processes that are supported by teacher-help, the mathematics is secondary to the group process in the teachers mind. The teacher is helping the pupils to use mathematics in order to accomplish the group process, rather than using organization of the group in order to accomplish representation or sense-making of mathematics. This seems backward, given that our job is to teacher mathematics! It is almost counterintuitive! But, indeed, when we think this way, perhaps, there is a new sense to be made of mathematics teaching.
Teacher-training: What, then, are the implications for teacher training? I believe the key things to think about are the differences between preparation for representation and sense-making, which has been the primary direction of mathematics education for the last century, and preparation for the support of artistic practice. We have inherited a technology of teaching methods steeped in cognitive psychology which direct the teachers attention to individual cognitive development. This has certainly been useful, and will continue to be useful to all of us in our work. However, I am suggesting today that we foreground another orientation to our work, which Eliot Eisner (1991) called criticism and connoisseurship. Ordinarily, the teacher training that I am most familiar with involves extensive practice in the application of methods, diagnosis and remediation. Eisners ideas suggest instead that teachers-in-training spend more time immersed in experiences that are not directly focused on the representation of teaching and learning, or on making sense of what pupils can and cannot do, but instead on criticism and connoisseurship in the context of schools.
Connoisseurship is the art of appreciation. It can be displayed in any realm in which the character, import, or value of objects, situations, and performances are distributed and variable, including educational practice. The word connoisseurship comes from the Latin cognoscere, to know. It involves the ability to see, not merely to look. To do this we have to develop the ability to name and appreciate the different dimensions of situations and experiences, and the way they relate one to another. We have to be able to draw upon, and make use of, a wide array of information. We also have to be able to place our experiences and understandings in a wider context, and connect them with our values and commitments. Connoisseurship is something that needs to be worked at but it is not a technical exercise. The bringing together of the different elements into a whole involves artistry.
It may sound like I am advocating an elitist notion here, but I do not mean this; indeed, I want us to think mainly about the depth of knowledge that all people have in their everyday lives as connoisseurs of those things they taste deeply, and to imagine how we could help young people to take those ways of learning and thinking and making meaning, and see that they are relevant in school (Gustavson & Appelbaum 2005; Appelbaum 2007). Now, what Eisner makes clear in his writing, is that educators need to be more than connoisseurs. They need to become critics. Our models for ourselves need to be those reviewers of films, albums, music videos, and video-games that we read and listen to for pleasure, and that help us to know which artistic works we will enjoy and find valuable, even those critics with whom we love to disagree. Criticism is the art of disclosure, of revealing more than the obvious; as John Dewey pointed out in his book Art as Experience, criticism has as its aim the re-education of perception. The task of the critic is to help us to see.
Thus connoisseurship provides criticism with its subject matter. Connoisseurship is private, but criticism is public. Connoisseurs simply need to appreciate what they encounter. Critics, however, must render these qualities vivid by the artful use of critical disclosure. (Eisner 1985: 92-93)
I see direct connections with our conference manifesto, which describes teachers as crucial to the evolution of mathematics education:
The quality of teaching and learning mathematics depends on many elements, affected and determined by each other. While many factors, such as social structures of inequity and diversity, are seemingly beyond the purview of the individual teacher, the teacher in the classroom remains a central element, responsible for what is going on during lessons in the immediate context. Teachers must understand their role, both within the classroom, and as a part of larger social and political structures. They must blend their interactions with pupils and their understanding of mathematical content objectives with their own ethical and moral commitments as a change agent in society.
So, in my own work in teacher education, I strive to work as a connoisseur and critic, in order to support the artistry of my students who wish to be teachers. And I welcome conversations with you over coffee, tea, a beer, wine, and so on, to share such stories. But back to the main theme of this presentation: what sort of mathematics learning is enabled by a teacher with extensive background in connoisseurship and criticism?
Learning Mathematics: Well, we could simply say, pupils of mathematics would be succeeding when they are demonstrating abilities to use mathematics in order to achieve a parody, to communicate an abstraction, as a decorative element in other contexts, or as non-mathematics across the curriculum. But more directly, I offer the following: Young people learning mathematics are artists whose tactics of parody, abstraction, decoration and non-art are forms of consumption that re-appropriate school mathematics as a tool of connoisseurship, and thus, of remaking their world anew in each act of mathematics they commit. Here is a very active and vibrant way to imagine mathematics learning: as artistry, as doing, as alive, and as transforming the world in every tiny moment. Mathematics in this sense is a collection of tactics for doing this. And learning mathematics is an apprenticeship in the artistry of social participation. Their mathematical actions, as art, are not aimed at a purpose that involves curricular illustration, but instead become the embodiment of critical pedagogy that engages both the mathematical artist and the artistic mathematician in critical citizenship (Springgay and Freedman 2007). I end, then, with a challenge to you: are you ready to allow the children in your life and work to become connoisseurs of mathematics? That is, to become more than knowers, to become critics of mathematics? Mathematics as criticism is an art of disclosure, of revealing more than what is obvious on the surface. Here is the magic recipe for achieving this: think more about coordinating activities where the children are active artists of mathematics than about how to represent or explain clearly a mathematical concept. I know, it goes against so much of our desires to make things easier for the child. In the end, though, if we stick to this plan, we will be lucky enough to spend time with current and future crafters of beautiful worlds, young people who use mathematics to shed insight on contemporary society, to ironically critique common sense practices, as tools for appreciating and interpreting culture and societal problems, as the medium of decoration and entertainment, and simply as so valuable as to be part of all things not usually named mathematics.
This paper was presented as a Plenary address at Childrens Mathematical Education, Iwonicz-Zdrj, Poland, August 17-22, 2008.
Contact: HYPERLINK "mailto:appelbaum@arcadia.edu"appelbaum@arcadia.edu Webpage: HYPERLINK "http://gargoyle.arcadia.edu/appelbaum/"http://gargoyle.arcadia.edu/appelbaum/
References
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Sontag, Susan. (1966). Against Interpretation 2 5 E M Q o u v w S X {
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Springgay, Stephanie, and Debra Freedman (eds.). (2007). Curriculum and the Cultural Body. NY: Peter Lang.
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