Chapter included in The Post-Formal Reader. Joe Kincheloe and Shirley Steinberg, editors, Garland (1999).
 

1. The Incident
 

Devin takes five cubes and puts them on his mat. Josh tosses two into the "pot" in the middle and grabs a "ten", thinks again, picks up one of his two cubes, and places the ten in the middle of his mat, the cube on the right with some others. Caresse carefully counts three cubes, puts them on her mat, rearranges them to fit in a nice, neat row with five others already there, and counts how many she has all together: "1, 2, ..., uh, 3, 4, 5, 6, ... 7, 8;" she continues counting the "cubic" portions of a "ten" rod in the middle of her mat: "9, 10, 11, 12, 13, 14, 15, 16, 17, eighteen."
 

We're playing "Target Number Up and Down." In this game, before we start, each child picks a "target number" between fifty and one hundred. Rolling a conventional six-sided die, the child takes the number of base-ten blocks from the "pot" corresponding to the number rolled on the die. Each child does this at the same time with their own die. When they get to their "target number," adding on with each die roll, they continue the game, from now on subtracting the number rolled from the cubes on their mat and putting them into the "pot" until they have no more left on their mat. The children are glad to play this game with me on the floor in the hall outside their classroom. Their teacher taught them how to play yesterday. They start right away as soon as I tell them what we're doing. "Oh, O.K., we played that yesterday," they say pleasantly, writing down target numbers on their mats, such as "89," "93," and "97." The atmosphere is a gentle contentment as the play continues.
 

The teacher asked me to encourage the children to look for "short cuts." For example, if they have 17 and roll a 4: instead of taking four cubes, then putting the four with the seven, counting out ten little unit cubes, and trading them for a "ten" rod, a student might realize that they could take a ten to begin with, if they put six cubes into the pot (because three of the four new ones, together with the seven on the mat, would make ten, leaving one more cube for a total of 21). So I observe the five children in the hall with me, ready to pounce on an appropriate moment for suggesting the idea of looking for shortcuts. I quickly notice Tyrone in this very situation: with two tens and three ones on his mat, he rolls a six, which gives him 29. Then he rolls a six again. He takes six cubes from the pot, drops them on his mat, adds four from the pile already there, puts the ten cubes back in the pot, takes a new ten, and ends up with three tens and five ones, 35. I say, "Tyrone, Ms. Taggen suggested that we should look for shortcuts today. Like, just now, instead of taking the six new cubes, then making a ten and trading, could you have thought ahead, and, instead of taking the six and trading for ten, could you think of taking the ten and putting some cubes into the pot from your mat?" He instantly understands: "Sure," he says with a smile, "I could just take the ten and put in four of the nine I had here." How did you figure that out? "It's easy -- I added four to the six I rolled to make the ten when I traded." "Oh," I say. "See if you can plan that way for a shortcut as you keep going." He nods, rolls a 1, and takes one unit cube from the pot. It's amazing to me how compelling this activity is for the children. They all just keep on rolling, taking cubes, and trading, with no pause or break.
 

Josh, who had been listening to our conversation even as he kept on rolling, smiles, rolls a 3, puts seven into the pot, and takes a ten. "Explain what you just did," I ask Josh. "I put in seven and took a ten." "Why?" "Because I had 48 and 3 more makes 51, so I made the one and took a ten for the fifty." "How did you know to leave just one cube on your mat?" "To make fifty- one," he says, as if I am pretending to be stupid. Apparently, the cubes are not modeling the operation of addition for Josh; he is just making the cubes match the sum he gets when he adds the numbers in his head.
 

I glance at Devin. He has about thirty ones on his mat, and a ten. "Devin, are you trading for tens?" "Oh," he says, counting out ten blocks and trading, counting out another ten and trading, and so on. He has been pleasantly rolling the die, and taking the number rolled from the pot -- no trading for tens, just mindless collection of cubes. He has been perfectly happy to enjoy the activity this way.
 

Caresse continues to carefully count out a block for each dot on her die, and slowly place the cubes in rows on her mat, next to a ten rod. When the cubes match a ten rod, she trades for a new ten for her collection on her mat, and continues counting individual cubes. She has four tens on her mat, three in a cluster and one with seven cubes lined up against it. She rolls a 5. Caresse counts one at a time: 1, 2, 3; stops and trades for a ten; continues to count: 4 ... 5 ... I ask, "Caresse, instead of counting until you get ten little cubes here, could you think ahead and find a short cut, so that you can just take a ten and put some cubes back into the pot?" "What do you mean?" she asks, annoyed that I have interrupted her well-organized routine. I show her. "Look, " this is what you just had before..." Reproducing her mat, I put the die down with a 5 on top. "You rolled a five. Now instead of counting out five, and trading for a ten, could you tell me how many cubes will be here after you trade?" Lining up the cubes with two sticking out beyond her ten, she says, "two." "So," I suggest, "if you know that, can you save yourself energy and not take all the cubes first?" "I don't know," she falters. "Think about it as you keep playing." "O.K.," she says quickly, anxious to get back to her rolling and taking cubes.
 

It is hard to keep track of each student, even though I only have five in the group. I haven't said anything yet to Pag -- What is she doing? "Pag, tell me what you're thinking about as you play." She has 8 tens and 2 ones on her mat. She rolls a 1. She takes a cube, saying, "I rolled a one so I take a one." She rolls a one again, saying, "I rolled a one again." Now she has 8 tens and 4 ones. She rolls a 6. "So I take six," she says. She places the six cubes on her mat and rolls again. "Pag, could you trade now?" "Yes," she says, "there's ten there but I like to wait until I have more to trade." "Oh." She rolls a 4. She counts out six from her mat, puts them in the pot, and takes a ten. "How did you do that?" "Well, I took four, I mean, if I took four, I would trade the ten here already, so I put the six in plus the four I could take makes the ten." "Great, Pag, Ms. Taggen wanted me to ask you to look for that kind of short cut today." She smiles and continues to roll.
 

The children are startlingly on task, it seems. They don't pause, but roll take, roll take, roll, take trade. Soon they are hitting their target numbers. They start subtracting what they roll from their mats. I suggest to the group that they keep looking for shortcuts when they roll.
 

Josh continues to "miss the point." He subtracts in his head, then makes the cubes match his result. The cubes are a waste of his time, I think. Tyrone continues to "get it completely." With each roll, he thinks about taking or putting cubes, thinks about trading, then figures out a shortcut. Devin continues to move the number of cubes he rolls, usually collecting so many cubes that they are falling off of his mat before he trades them in in bunches of tens. On the way down to zero from his target number, he sometimes puts the cubes into the pot from his mat, sometimes forgets and takes the number of cubes from the pot and puts them on his mat. He enjoys the task, even as he is not engaged in any way with the concepts that the blocks and the activity are intended to model. Pag, on the way down, gets confused about whether she should be taking cubes or putting them into the pot. She soon stops worrying and alternates. Brief queries to Devin and Pag cause them to be more careful on their next roll, but they quickly jump back into their established routines. Caresse, however, has understood what Ms. Taggen wants. She is slower than the others, counting out cubes ever so carefully each time she rolls; but by the end she is using a shortcut each time she can, and accurately.
 

Is this activity a "success"? It looks like math and sounds like math, and it is "hands-on", so it must be good, right? The searing realization that the activity is lacking in a link between the concepts of place value and procedural forms of knowledge for at least three of the five children gnaws at me as I drive back to campus. I decide to tell this story of my morning in my N-8 Mathematics Methods course. The students have been espousing an uncritical love of manipulatives and hands-on activities for the last few weeks. Perhaps this will interest them. But I also want my students to note that such an activity helped me very rapidly get to know an enormous amount about the five students I worked with. As a performance assessment task it might have merit. Can they appreciate this?
 
 

2. The Lesson.

An hour later I am relating my story to my class. I demonstrate Devin randomly collecting cubes with overhead base-ten blocks. I explain Pag sometimes subtracting from her mat, sometimes subtracting from the pot, usually but not always using a reasonable shortcut procedure. I mimic Josh, adding or subtracting in his head and then making the cubes match his calculation. "What do you think?" I ask. My students quickly suggest the activity does not link conceptual and procedural knowledge. They have been primed for this. They note a number of problems with the activity: Since each child is playing by her or him self, there is no interaction among them, no need to communicate ideas or explain their result or process. The activity itself is somewhat meaningless, with little purpose. The target number choice is random and does not matter -- it is a false choice. Otherwise, there is no decision-making and no reason to care about accuracy. Indeed, the willingness of the children to continue the activity surprises my students. They want to know why the children did not invent their own game. In fact, they suggest this should be the activity: the group should design a better game. Short of this, my students offer attempts to turn the activity into one with some semblance of purpose.
 

Variations Offered.
 

Group students in pairs working toward a commonly agreed-upon target number. They take turns. The first person back to zero wins. Here students might care about each others' adding and subtracting.
 

Class members predict the number of rolls it might take for a given target number. Each person tries it and the class records the data. For various targets, students are chided into trying to become more accurate in their predictions. Prediction strategies are discussed at length in-between repeated data collection.
 

Pairs work with a common target. Players take turns rolling the die. On each roll, they may choose to add or subtract the number to or from either their own mat or their competitor's. First person back to zero wins.
 

Same as above except players may either take the number of new ones from the pot or give the other player the same number from their own mat. They then discuss how to make the game better.
 

Players start so that one person has 100, the other has 0. They take turns rolling the die. On each roll they may add that number to their own or subtract that number from their partner's mat. The goal is to work cooperatively to get each person to 50, and then back to the original start-up situation. They play several times and try to use fewer rolls each time, trying to figure out a reasonable "par" for the game. Students are again asked to improve upon the game.
 
 
 

I am pleased. But I wonder: Will my students think this way when they are the teacher? Why don't more teachers think this way?
 

I ask my students what they know about the five children I talked about. Very little, they say, because the children did not have a chance to explain their reasoning for their actions. I share my own thoughts: that Josh basically doesn't need the cubes to model the operations; that Devin and Caresse have been "taught" that they can do this, and the lesson has provided a model for them, but that I wish they had been given an activity that allowed them to construct this idea on their own in the context of useful borrowing and trading; that Pag has the procedure down pat, but needs more opportunities to explain the procedure in terms of the concepts; that Devin understands place value but needs a meaningful activity that puts adding and subtracting with place value in a context that has a purpose for him. We talk on a bit, but in the end my students reaffirm their naive worship of hands-on manipulative activities as more crucial than the construction of purposeful projects in the classroom. We talk about ideas that would encourage the students talking to each other, and listening to each others' ideas about place value and operations. The need for relationships between number facts or their application to the problems of the "real world" does not yet enter this conversation, constructing in Joe Kincheloe's terms "cognitive illness." This makes me think.
 

3. One Month Later.

I have been hired by a suburban school district to run a workshop on "Meaningful Mathematics with Manipulatives." My assignment is to offer teachers an advanced discussion on problems that arise with manipulatives and to suggest inexpensive home-made materials. After an introductory conversation in which the 20 in attendance mainly critique manipulatives as messy and taking too much time, I offer the Target Number Game as something to think about. I ask them to think about two things: (a) the quality of homemade material as a model of place value (I have distributed around the room graph paper, cardboard, and plastic strips and squares, sticks and nubs); and (b) the potential of the game. They play the game. I ask them what they think. One person says it offers practice in adding and subtracting. Another says it is hands-on. A third says she doesn't like the graph paper strips because her students would eat them. I ask for concerns or criticisms of the activity. No one has any, until after a long pause one teacher suggests that, while this game might be "good" for more affluent districts, her "kids" need to first practice the basics before they can be applied to this sort of activity. I hand out a sheet with my students' concerns and their suggestions. The teachers complain, "If it isn't any good, why did you make us play it?" "Because I want you to think about how you might do the same thing with the activities suggested in your teachers' manual. They're not always as good as your ideas would be." They are bewildered.
 
 
 

4. Pomo Pugnaciousness
 

In the above constellation of stories, is there a hidden story about race and economic inequality? There must be since neither are raised. Is there another story about the role of mathematics in the school curriculum? There must be because this is not mentioned. Is there a story about power and knowledge? There must be since they were not mentioned. Of course we can critique the encounter, the reporting of the encounter, or my own participation and choices. We can offer alternative curricula which establish "place value" and "addition of two-digit numbers with regrouping" within thematic units, project-based inquiries, or more efficient skill drill exercises. What strikes me, however, is the disjunction between my own abhorence of the repetitive rolling and adding, rolling and subtracting, and the students' contented embracement of this same experience. I have constructed a dichotomy, a dualism, which I find viscerally frightening, but which my methods students accept and adapt, and which the teachers in my workshop do not judge or notice as relevant to the decisions they make in their practice as teachers.
 

I see the mindless participation of the second-graders as an enactment of ironic pleasure. In his introduction, Joe Kincheloe refers to Terkel's (1970) description of workers withdrawing emotionally from their labor, and students learning early in their school lives, "moving through the day without affect, staring straight ahead at nothing in particular," that school has no larger purpose. Kincheloe notes how quickly children learn that school has nothing to do with their passions -- indeed, he writes, their emotional health is irrelevant. What activities like "Target Number" do is establish motivation as a problem of a scientistic pedagogy in search of technical and scripted solutions: how do we motivate children to attend to the regrouping and place value? These activities turn each of the five children into isolated problems to diagnose, leading to prescriptions: keep Josh off of the base-ten fix; give Tyrone and Caresse an extra dose; retest Pag for the right level of dosage; and Devin, he needs special clinical attention. Kincheloe is on target when he raises the specter of modern positivists chopping up learning into chunks of data to be chewed in decontextualized fragments and stale morsels of chalkdust. More to the point, the dismembered mathematics these children are asked to consume feeds them as much a message of mathematics as politically neutral and aesthetically inert as it lays out a dish of pabulum. Absent are attributes of intuition, imagination, surprise, anger, and curiosity. But the reconstruction of mathematics as including these and other attributes is not so clearly established. This is partly due to the role mathematics is often assigned in the common sense construction of formal, rational thought. The legacy makes it severely threatening to challenge such a view of mathematics as ironic or deceitful because, in challenging such pervasive presumptions, one risks the danger of being misunderstood as attacking the accuracy and coherence of someone's rationality rather than the notion of rationality itself.
 

Brian Rotman (1993) has helped us to see that the threat is even more severe. "No doubt," he writes, "the idealized imaginings of mathematics answer, as a familiar, unproblematic, innocuous part of everyday wishing and thinking, to the desire for order, regularity, repeatability, form, pattern, and harmony." (p.156) Socially constructed as much as any other cultural artifact, mathematics appears "true" because our construction of "true" is imbricated in Rotman's list of the "everyday wishing and thinking," but also because of a tautological definition of consistent and persistent truth and reliability that dovetails with a mathematics built upon a tradition of picking those regular, repeatable, formed, patterned, harmonious concepts and procedures that have the properties of regularity, repeatability, form, pattern and harmony! "But," as Rotman writes, "poised behind such desires is an absolute desire, introduced into the meaning of number and so into the imaginings themselves ... the desire is for no less than that the grandeur and imprimatur of eternity be stamped on the objects of mathematics and the truth one discovers. In this way one can identify with a transcendent being, can move, outside history in His [sic] dominion ..."
 

"The fantasy of a transcendental origin, an ultimate guarantor of truth unsituated in time, space, or history, for whom or out of whom the infinity of numbers is/was/will be always there, has proved difficult to resist," continues Rotman. (p.157) Yet why are we led from a seemingly harmless "game" (although I would have to claim "Target Number" does not conform to a genuine definition of "game" in a mathematical or philosophical sense) to the claim that young children are being forced to live outside of time and history in a meaning-drained fantasy of "truth"? Only in a form of scientistic research that examines the incident under a microscope without its links and web-like connections to innumerable other like incidents and social contexts would allow us to marvel at such a wildly "extreme" claim. It is in the day-to-day repetition of similar "mathematical" encounters that the mathematics is reconstructed perpetually in a way that supports such an absurdist framework. Such practices work to reconstruct the active learner as an example of Piaget's assimilator, thinking outside of reality; the accomodator, immersed in relationships and exchanges among the thinker and a world of objects, is perceived as "slow" in developing appropriate skills and language facility.
 

Similarly, it is in the racially-charged community, in which African American parents tend to persist in condemning the public school curriculum as not academic enough, lacking in the teaching of basic skills, and deficient in discipline, that the scene takes place. The school is 50% white, 50% black. The white parents tend to persist in demanding ever-more thematic, integrated teaching and non-competitive, cooperative projects. The African-American principal tends to persist in calling for more rigor as "preparation for middle school," where the children are combined with students from other schools in the district. The white parents like to say they have a black principal, but work around her to accomplish their goals. In this context, Tyrone and Caresse need the pedagogy offered; the two African American children in the group of five "need" this pedagogy, much like recent "special" programs for "urban youth" (a code for race) provide the "self-esteem" and "culturally-based curricula" that "these" children "need" (Appelbaum 1994). Josh, a white child, possibly does not "need" this pedagogy; he already adds and subtracts well and would "flourish" in a thematic unit project that challenged him to apply his skills in "meaningful ways." Devin, another white child, does not learn well in such formats; his teacher tells me he thrives on personal, individualized attention away from the group, and she expects him to be "up to grade level" by the end of the year if she continues to set aside time every week for a short conference session. Here we can see a racially coded unfoldment of pedagogy and assessment that might be compared with the conclusions of others that find a pattern of white children being treated as "apprentices" who already have knowledge, and African-American children who are treated as if they do not have knowledge and experience instruction as "teaching." (Gee, 1987; Ladson-Billings, 1995) There are also hints toward gender-influenced interpretations. Pag, a white girl, can adapt to what the teacher wants, but she and Caresse, who competently demonstrates comprehension of the task but also persists in her own (less "efficient"?) style of performance, are easily compared "unfavorably" with the "male standard" set by Josh and Tyrone. Thus are gendered interactions with mathematics set in motion well before this second grade incident.
 

5. Teaching as Story-Telling.

Curriculum is more than a pipeline through which facts and skills get injected into students. As Keiran Egan has emphasized in many contexts, teaching is a form of storytelling about the content, and about what it means to know and learn (Egan, 1988, 1990, 1992). Yet his poignant example for mathematics illustrates well the sort of story that is often told about mathematics and its central purpose in the school curriculum. Egan encourages teachers to compare their curriculum organization with the story-telling qualities of fairy tales, one attribute of which is the dramatization of binary opposites. For mathematics, and for place-value in particular, Egan suggests the magical drama of power versus powerlessness. In a unit coordinated with Colonial American social studies, students are introduced to the theme: they hear about how pioneer families would help each other out when a crow needed to be removed from a barn. Crows are about as good as people at counting -- they can recognize around five objects in a cluster. So, if a farmer went into a barn and waited for the crow, the crow would know that one person went in and would not fly in himself until that farmer went out. Again, if two, three, four or five farmers went in, and one or two walked out, in an attempt to fool the crow, the crow would still know that a couple of farmers were waiting with shotguns to shoot him dead. Now, if a bunch of farmers help each other out, and a group of seven or eight go into the barn, one farmer can hide out while the rest go back out. The crow will lose count, and fly in to find his nest. BLAM: the hiding farmer no longer has to worry about the crow eating his seed stock. Extensions of this story can move children into the study of different animals and their relative ability to count. The story would be consistent with the drama: whales and dolphins, who can count up to 12, can outsmart people and save themselves in various ways; counting is placed in the life cycle in terms of the power or lack of it that the counting ability enables the animals have. Back to arithmetic: The myth of the origin of troops in the military is conveyed and acted out by the children. A general calls his advisors and says, we need a better system for keeping track of our soldiers, or we will never beat our enemy. One advisor after the next fails to come up with a scheme, until one genius suggests having each soldier drop a stone into a vase as they walk past the general: each time a vase is filled with ten stones, that group of soldiers is clustered into a team of ten men; ten vases are grouped into ten tens of men, and so on. The general then is able to plan the movement of his troops with such precision and creativity that the battles are won with great finesse. Thus is the notion of counting and place value intimately linked with the normalization of counting as a tool of power and the ability to take another living being's life. Even the Calculus, often the keystone of school mathematics, is implicated through its etymological origin in the meaning of its name -- the word is Greek for the stone or pebble used for counting. Taking the story a step further, Davis and Hersh (1981) string along Archimedes, scribes and astrologers, and the mathematicians of Napoleon, continuing on to the development of operations research techniques during World War II, the marriage of mathematics and physics in the atomic bomb projects, Norbert Weiner's controversial work in prediction and feedback that led to his later work against the "nonhuman use of human beings," the origins of the computer industry in the intensified cold war space race, and the futurist notion that while World War I was the chemist's war, and World War II was the physicist's war, World War III will be the mathematician's war.
 

Christine Keitel (1989) has written another story of mathematics as power, in terms of its role as a "technology" or tool that people use to accomplish newly possible tasks. Ledger systems of accounting, for example, made it possible for a whole culture of mercantilism and a merchant class to emerge in Medieval Europe. As Keitel notes, however, this form of accounting and the use of columns for adding and subtracting numbers of and costs of objects, also structured a form of culture that previously did not exist, to which people adapted in the slowly emerging assumptions of trade systems within capitalism. More specifically, Cline-Cohen (1982) writes a history of mathematics education as one of "calculation," pointing out that calculating the population to be governed and the idea of a calculating population (assisting the running of government and the emerging capitalist system) were intimately linked. Students in schools today are smoothly enculturated to this notion of being calculated and studied as objects of pedagogy and administration, and are similarly "prepared" to both calculate and be calculated. Valerie Walkerdine (1988, 1990) continues the story one step beyond, by recording and analyzing the ways in which bourgeois democracy was to be upheld, not by a coercive pedagogy, but by a "natural" pedagogy of love, in which reason would unfold. "Reason was to become the goal of a technology designed to provide reasoners who could govern, and those who might, at least, be hoped to be reasonable, not pushed to rebellion by repressive and coercive pedagogy." (Walkerdine, 1990, pp. unnumbered)
 

In Egan's storytelling, the teacher constructs a resolution of the story: mathematics should be learned and practiced for the power it yields. You, too, can count beyond stupid animals and uneducated people, so you too can get the jobs such skills promise. Of course, the story might also be resolved in other ways -- a less optimistic moral that focuses not on the power promised but the objectification of people through number: a libertarian fear of big government; a distrust for the numbers claimed by military and government statistics; a distaste for the anonymity and subsequent loss of community fostered by contemporary studies of "average" people and the loss of attention to particular individuals (Greene, 1973). We might study how the school has turned each of us into objects of study, through calculations of various projective data about our "ability" or "personality," twisting each of us into a particular projective future. We might further examine the role of numbers in mystifying the public rather than communicating information: "termination units" in discussion of new weapons in Pentagon budgets, hard-to-understand units of radiation leakage that spring forth in local debates about the location of a new toxic waste dump or faltering power plant, the manipulation of testing data to satisfy local taxpayers that the schools are accomplishing their stated goals; data used in arguments about "raced ability" (Bachman, 1996; Kincheloe et al., 1996).
 

In the construction of our curriculum, we should note a common practice of using relatively "small" numbers because, as some psychologists would tell us, children need to learn about numbers that they can construct concretely. Big numbers are hard to see and feel and thus inappropriate for young children. The concept of place-value is introduced within this larger curricular context, in which big numbers are for the powerful big people, little numbers for the constructed powerless little people. The ageism within the curriculum is implicit but important. Walkerdine is relevant in this discussion as well (Walkderdine, 1989; Walkderdine and Lucy, 1989). In studies of girls and mathematics, working class girls were found to have intimate understandings of large numbers of abstract mathematical reasoning in their home life; middle class girls' lives were distant from such interactions with mathematics. School mathematics based on presumptions of middle class lives did not meet the needs of either female population. But the key point here was that we should not presume a certain universal model of development that psychologists could abstract from studies of children; in fact, such universal models are often flawed in terms of class, race and other categories by which children might be grouped and clustered. Place value might not even be a concept to be taught when various family and life experiences are taken into account -- like abstract concepts of good/bad, fair/unfair, tasty, fun, etc., place value might indeed be a concept that many children bring with them to school along with comprehension of large numbers.
 

By "teaching" place value, however, we tell a variety of stories about the world and the idea of numbers in that world. For most of us, numbers are a natural truth that we can see and use to understand reality. They are basic skills essential for successful life experiences and jobs. Rotman's recent work on a non-Euclidean arithmetic articulates how hard it really is to imagine that numbers are as socially constructed and context-specific as Euclidean geometry. The idea that there might be a more generalized notion of number and quantification for which linear series of counting is locally "reasonable" but totally absurd for other contexts is almost impossible to fathom. Indeed ethnomathematics has argued for years that "Western" mathematics is not as universal as we wish, and has told alternative stories of worlds incommensurable with this so-called universal truth of arithmetic (Pinxten et al., 1983, 1987; Fasheh 1989, 1990). By subjecting children to particular "models" of the concept of place value without first trying to understand what models they bring with them, we are denying even a range of cultural variants within this narrow cultural construct of "Western" mathematics (Lave, 1991; Carraher, 1989; Mellin-Olsen, 1987; Ladson-Billings, 1995). Instead of "teaching mathematics" we are teaching that in school one must understand what the adult tells you to do; a new layer of obfuscation is sometimes added, to neutral or negative affects, occasionally positive enrichment, on top of what might be brought tacitly with the child to school.
 

An understanding of the "popular culture of mathematics" that is indeed brought with the child to school might actually prove useful to both the teacher and the children; at times, however, we might find that popular culture resources for mathematics meaning might buttress regressive notions of mathematics as a disempowering form of alienation, yet for this information we should still be grateful (Appelbaum, 1995). In this respect, the role and nature of dice games in and out of school provides an interesting link to the Target Number activity. Because many children's games use dice, it is tempting to think that the use of dice in this context will speak to the problem of motivation as constructed by contemporary educational practice. Because dice are used in casinos for various gambling games, we might want to argue that school is teaching an interest in such games at the same time as preparing children for these adult activities. Because role-playing adventure games use dice, yet are marginal in their social acceptance and acknowledgment, student use of fantasy and role-playing adventure games for pleasure might be something to which teachers might want to attend. The particular construction of parallel play in the context of dice games might have interesting ramifications for students' placement of this activity within a perspective on games and competition. For example, the expectation that the interaction of the probability of dice rolling and the learning of something about place value, might be met with students imagining a way in which place value has something to do with chance and probability! Yet in all of the above, we must still be aware of the presumption within all dice games, gambling activities, and associated use of dice, that numbers and a linear series of infinite counting numbers constitute a truth constructed by a legacy of acceptance within a culture of military strategy, accounting systems, calculation as a governing practice, and other associated notions of "the way things are." When we "teach" place value, we are teaching this story about number and counting, and associated truths about these stories as stories of "reality."
 

Meanwhile, we should also understand that when we teach in this way, we are supporting the perpetuation of a certain notion of what teaching is, should be, and can be. If courses in methods of teaching mathematics critique such activities, and then place students in practicum classrooms where the "real" teachers condemn such critique and even are skeptical about the efficacy of the activity discussed in this article, then what we are teaching our future teachers is a lesson in avoiding acceptance of new ideas about teaching and learning. Preservice teachers experience conflict: by exposing them simultaneously to a range of strategies and examples of teachers not using these strategies, they are required to negotiate a terrain of crisis -- i.e., the structure of teacher preparation necessitates that teachers learn to disparage "professors' ideas" and embrace a disempowering strategy of non-engagement with curriculum in favor of a technical "neutral" psychologicization of practice. Played out as hostile non-engagement with professional development that reduces contributions to their professional practice to "what works" (Howley & Spatig, 1996), such a pedagogy is an apprenticeship in and mastery of the avoidance of post-formal thinking (Kincheloe, 1993). Teachers "must" learn how to see facts without seeing these facts in any social, cultural or political context. They "must" understand student behavior according to prescribed theories and facts, rather than through the filters of a range and variety of metaphors. They "must" learn to think of mathematics as a collection of neutral truths disconnected from particular contexts or purposes. Place value is then known and prepared as a collection of disassociated techniques and facts removed from meaning, purpose, flexible critique, etc. "Shortcuts" for exchanging cubes is then able to be held as a goal of instruction, within a prepackaged curriculum on place value devoid of social, cultural, or political purpose, indeed, far removed from any immediate purpose other than to exchange cubes for tens.
 

6. New Stories

It is important, however, to recognize that the failure of "Target Number," if indeed it fails in any way, is not due to its relationship to games in any sense. Games can be educational in the best sense of the word. What seems to be key is that participants have to make decisions and live with the consequences of their decisions (Goodman, 1995). When people criticize games as educational environments, they are usually worried that the practice in a theoretical environment that is part of the game is not going to help people make theoretical decisions in a "practical" environment. Fred Goodman writes that the issue really comes down to the quality of the metaphors produced by the game, as opposed to the validity of the model behind something that intends to "simulate" real life practical experience. The strength of an educational activity that has something to do with place value, then, would have to respond to Goodman's claim that the activity not be planned in accordance with how it helps a child practice "in theory" rather than "in reality" (due, in the construction of educational actions, to the nature of school as removing children from reality in favor of them practicing in theory ...), but planned instead in terms of how well it helps a child practice theorizing. It is here that possible conflicts arise out of dissonant notions of the purpose of schooling, parallel to the need for prospective teachers to learn an avoidance of theorizing about learning and schooling in support of a "what works" philosophy of practical application: We ask just what it is that the students are doing when they are playing "Target Number Up and Down." Are they practicing something in a theoretical fashion? Are they practicing theorizing about numbers? Are they practicing something that is in fact a practical skill? (Goodman, p. 189)
 

Indeed they might be doing any of those things. But the overwhelming sense I got by being there with them was that they were, in the best sense of the words, "passing time." For many people, passing time harmlessly, especially if it has some sort of social sanction to it, seems like a perfectly "reasonable" and attractive thing to do. If decision making can be minimized, then consequences can be avoided (or at least responsibility for negative consequences). There is a sense in which playing with dice is playing with the "theory" of probability, and that seems to be comforting to many. In this sense, the children who are not theorizing at the level of imagining short-cuts are, nevertheless, practicing theory. At the same time, they are not, as I might be, focusing on getting the "job done," but rather on keeping the time passing, on making time go by in an endless rhythm that avoids negative consequences. This is what Fred Goodman has elsewhere called "practicing the theory of existence." That this dovetails rather nicely with passing time in mindless routinized jobs is not all that surprising and supports traditional correspondence theories of social reproduction in overt ways (Bowles & Gintis). The cultural interpretation of such practice might be that students do not "understand" place value, but "get used to it." Perhaps this is what is meant in some circles by "mental habits," in others by "mathematical enculturation" (Bishop 1988).
 

"Target Number" raises for me a whole new set of questions about "assessment," including how to form a relationship with children versus when I need to carry out surveillance. The common reduction of learning to "accomodation" leads to a need to observe and analyze a student's progress in focuing on factual and procedural knowledge. "Critical equilibration," as Kincheloe describes it, would a teacher recognize the value of long-term accomodation experiences; the relationsdhip between teacher and student becomes a key feature in the stimulation of searching and research-based student activities that do more than enable a student to describe prescribed concepts in terms of facts and procedures. David Hawkins writing in the 1960's of children developing "mental habits" through a "vast and essential redundancy of ... practices" reminds me of the ways that "schemata" ("ways of going at a subject matter, strategies") need to be rehearsed, enjoyed, and reflected upon, first using the words and performing things "unreflectively," then slowly over time rehearsing them in activities that point to new aspects of comprehension and reflection (Hawkins 1980). At first we might decide that this supports the "getting used to it" approach to pedagogy. However, he writes of a "second level" which comes "when the schemata or strategies the learner has acquired are transformed, by the learner, into vehicles of a new kind of meaning and interest." (p.102). Hawkins raises yet another point about "abstraction" and "schemata." He asks us to think about an issue that often gets confounded with this notion of shifting from concrete particulars to the second level of attending to the schemata by which we deal with the particulars itself: this other issue comes up in the uses of "abstract" to denote formal rationalized schemes of operation in a manner that is "detached ... [or] ... looked at apart from all but a careful delimited context." (p.108) Manipulative materials designed to model concepts for school mathematics are carefully designed to avoid the first kind of abstraction initially, so that students may deal with concrete particulars of a sort that are not mere symbols but concrete examples of the concepts. The materials are highly stylized so that the intended concept is "pretty unsubtly there, if you already know the secret, while all the other inevitable properties of the concrete object are de-emphasized by standardization." "So it turns out," writes Hawkins, "that this material is abstract in both my senses, heavy with conceptual intent and cut off from nature's variety and interconnection." Are manipulatives inherently "bad" then? Rather, the pedagogical issue is parallel to a discussion of manipulatives; this pedagogy needs to be unraveled from the manipulatives and discussed on its own terms.
 

What Hawkins likes about manipulatives is that children are playful and "'eolithic,' and can find more unintended uses for the concrete materials than they can find for printed tokens, crayons and work books." By this he means that children can easily invent purposes for the materials, and use them in ways that then produce "meaning." Think of stories like the one in Parker's (1993) Mathematical Power, or Kohl's (1976/86) On Teaching: for the beginning of the year students are left to themselves to explore the materials that they will be using, and develop amazing, creative projects. These stories articulate the "eolithic-ness" Hawkins is noting.
 

I use the word eolithic in memory of our remoter ancestors who had to start life with objects not intended for any purpose, but who after picking up the stone, for example, invented uses for it. The first invention was not the object -- but the purpose. (p.108)

"By now," we see his answer.

When we speak of "abstract thinking" do we mean thinking that is in an insoluble capsule, unrelated to the wealth of experience that can make it come alive? That can be done with Cuisenaire rods and geoboards as surely with paper and pencil -- or almost as surely. There is a time for such thinking, of course, but it should be very late -- as late as a child's readiness to grasp the partial isomorphisms between the concrete reality and formalized systems of signs. Or do we mean the cultivation of intuition, of analogy, of the mind's power to order and schematize? No time could be too early, I think, for that. (p.109)

I suppose we need to ask that silly question, 'why do we have schools, and what should people do in them?' When we observe a typical classroom, we see a bunch of activities designed to train students as assimilators. When we critique what we observe, we suggest a focus on critical equilibration that emphasizes accomodation. When we reflect on people learning in out-of-school encounters, we begin to note, with Joe Kincheloe, how people do not make formalistic generalizations, but reshape cognitive structures to account for unique aspects of what is perceived in particulat contexts. The person learning in context thinks in terms of what she or he might encounter in similar situations, what strategies might work in such contexts. What might it look like to strive for such situated knowledge in the public school classroom? Deborah Loewenberg Ball (1992) raised the same issue in response to a third grade class' interaction with mathematics with and without manipulatives:

The context in which any vehicle -- concrete or pictorial -- is used is as important as the material itself. By context, I mean the ways in which students work with the material, toward what purposes, with what kinds of talk and interaction. (p.18)
 

"Target Number" encapsulates thinking in a way that requires surveillance of children's thoughts, as opposed to an activity that would elicit interest in children's intuitions and analogies. This is a subtle but important distinction. Assessment in the second case would be an attempt to consider how one might provide a rich environment for the child, a way that they might benefit from an idea that the adult has, a search for materials that enable the adult and the children together to think about something. In the first case it is reduced to tallying skill attainment. In the second case, assessment would tell the teacher whether or not he or she had provided an activity for which a purpose could be invented.
 

Because educational practice is often constructed by our perspective on it to conform to a problem of motivation and surveillance, we can sometimes misread an event in ways that do not call attention to the student's effort and related identification of interest and self. Stephen Brown's (1981) work helped me understand this by writing of mathematical problem solving and posing in terms of Dewey's (1913) Interest and Effort in Education. Dewey addressed the question about whether teachers should be responsible for getting children "interested" in the (possibly dull?) things they do in school. Another perspective might be that students would be assumed to provide the "effort", regardless or even perhaps because of the "uninterestingness" of the activities in school. Typical for Dewey, it turns out in the end that both of these contrasting points of view create a common basic fallacy -- they assume an externality from the self of the object, idea or end to be mastered. Interest, for Dewey, becomes the "principle of recognized identity of the fact to be learned or the action proposed with the growing self." Assessment would focus not on task-specific objectives with such a concern about "interest," but instead on "the predominating direction of [the student's] attention," ... the student's "feelings," ... what the student's "disposition has been while ... engaged in the task."
 

If the task appeals to him [sic] merely as a task, it is as certain psychologically as is the law of action and reaction physically, that the child is simply engaged in acquiring the habit of divided attention; that he is getting the ability to direct eye and ear, lips and mouth to what is present before him so as to impress those things upon his memory, while at the same time he is setting his thoughts free to work upon matters of real importance to him." (Dewey p.8,9; Brown, p. 33)

But let's return to preservice education. Here my dilemma is that preservice teachers typically have not experienced manipulative materials as students or teachers, and need to be introduced to manipulatives before they can think about any critique of them. In fact, my students are suspicious of newfangled techniques, and act as if I need to convince them of the efficacy of the manipulatives. Questioning their validity, function, purpose, etc. is often reduced to a "first level" of winning them over to a constructivist model of teaching based on manipulatives and "hands-on" learning, despite the fact that labeling these teaching strategies as "newfangled" misrepresents them and obscures their long history or successes. Absent entirely from this discussion are alternative pedagogies of arithmetic and number facts that connect them with social practice, cultural politics, and other branches of mathematics. The fact that reform-oriented literature and methods texts often characterize constructivist approaches that incorporate manipulatives and problem solving approaches as something new actually works against them for students who wish to get a job and keep it ("fit in"), rather than develop a visionary philosophy and transform the nature of education. How can we redesign certification programs so that students are able to contextualize, critique, and challenge current practice and still move beyond this initial critique to a point of subtle distinctions?
 

Aspiring teachers need experiences rich in critical equilibration followed by periods of anticipatory accomodation. One possibility that meets directly their own expectations for "practical experience" in school classrooms, but simulataneously challenges their presumptions about the purposes of a school, is to begin with the student teaching internship. This would prolong the period of critical equilibration. An extension of this certification program that continues to reverse the typical, commonsense professional paradigm would provide a series of less-intensive field-work placements with more careful analysis of these practicum encounters as "case studies" that form the basis for anticipatory accomodation. Final preparation for teaching would culminate in an extensive period away from field work devoted to "applied hermeneutics" -- use of meaning-making abilities to anticipate what may happen next, and what should be done to prepare for such eventualities. At first glance this organization could be misinterpreted as a reproduction of Piaget's move toward increased assimilation away from real-world immersion; the important features would be careful relationships among preservice teachers and their college-teachers who promote conversations toward the aim of critical equilibration and anticipatory accomodation.
 

Let's think now about inservice education. A school district recently responded to my request that inservice work be planned to avoid the entertainment "what works" philosophy in favor of long-term consideration of issues that grow out of the teachers' own experiences with manipulatives in their classrooms. Such work with teachers is consistent with reform efforts to provide support for the adoption of pedagogical practices promoted by the National Council of Teachers of Mathematics Standards documents (NCTM 1989,1991, 1995). In these projects, professors team up with individual teachers for extended periods of time, so that they can discuss the day-to-day nuances and dilemmas that arise in changing one's pedagogical strategies (Ohanian 1992, Parker 1994, Romagnano 1994, Davis 1996). In the context of this psychologicization of professional development, the professor becomes a clinical therapist as the teacher attempts to renegotiate meaning in their everyday practice. For researchers who have spent time trying to figure out how to get teachers to think about and talk about their teaching, the mathematics pedagogy becomes a topic for both people to take as an object of study. The asymmetry of power created by a pairing of professor and practitioner perpetuates theory-practice dualisms even as the research works to undermine them.
 

Like my preservice students, teachers I work with (despite acquaintance with manipulatives through workshops and conferences) seem to need a multilayered learning experience parallel to what Hawkins refers to in regard to abstraction. An initial activity with manipulatives or problem solving is met with dismay and dismissal. Subsequent discussion and open-ended activities elicit complaints about a "lack of clear objectives," a "need for more structure," and a challenge to make the workshop or course more relevant to the realities of teaching in "today's schools." If a school has accepted me on faith (or, as I establish a "reputation," on personal knowledge of what I have to offer), the second or third meeting soon turns into a "conversion experience." "I see the light," one participant recently remarked, as he took center stage and began retelling his attempts to think about his classroom in terms of some of the metaphors we had discussed in our previous meeting. In a graduate course, particular problems are often secretly "tried out" in students' classrooms; by the third week, some "converts" speak positively about their experiments with these "new ideas." Like Hawkins, I find there needs to be a "playing around" period with these ideas about teaching and learning, a period of critical equilibration. With adults who come with expectations of a consumer society -- that they should get a clear bang for their buck, and early on too -- this presents a challenge for the facilitator's serenity and confidence in the approach, because the playing around appears to the participants as a disrespecttful lack of attention to their needs and desires. The complication comes with the determination of whether to push past the plateau of playing around with the materials of teaching and learning toward that other layer of abstraction Hawkins refers to, the one where the materials and strategies of teaching and learning can be theorized about and critiqued in a way that produces "meaning," a period that establishes anticipatory accomodation as a legitimate educational encounter. In my graduate courses, we view videotapes of our teaching. In these snapshots of classroom life, there are examples of creative teaching as well as clear evidence of malaise. We point to specific details as examples of how the teachers are already meeting NCTM Standards (1989, 1990, 1992, etc.). I find that some graduate students warm to the indirect "praise:" it helps them see that they have skills they can build on even as they still squirm with dissatisfaction. Others become alienated and wonder why we are "wasting time" with these tapes that cause so much anxiety and embarrassment. There is a point in the middle of the semester when people want to critique others' teaching, but do not believe they should. This is the critical moment in the course. Subsequent self-videotaping can elicit the same need for critique as the videos brought in by classmates and viewed in class; it is the second and third round of tapes that allows for a post-formal perspective on one's own teaching. Students become desperate for more courses to help them alleviate the tension. In these future courses, we begin to critique the "new" methods; instead of trying them out and dismissing them, we can like them and then challenge them to meet our new "standards" that have to do with social and cultural issues, political ramifications, and new-found crises of self-confidence in our teaching. Have I built a new cult of ME instead of the "transcendental truth" of pedagogy? What I work towards is the formation of a community and a network of professional contacts. The community diffuses this prophet-disciple danger when it is successful.
 

7. New Projects

I have tried in this analysis to demonstrate the ways that certain research questions and theoretical ground-work are understood only through a kind of shifting back and forth between micro and macro perspectives on social and epistemological terrains. It is within the cracks and crannies between and threads of interaction among classroom incidents, teacher education, and professional development that my own work as a mathematics educator can be interpreted as telling its own story; in this story, mathematics, research pronouncements, and pedagogy all become characters subject to interpretations of subjectivity and action, social structure and cultural change. What I think we "need to do" is ask for careful explications of the ways that a discussion of the epistemology of number, the presentation of pedagogical options in a perservice program, and the processes of lesson planning by practicing teachers inform each other. I also believe we "need" to ask when and how we might be able to inject at any articulation of these characters in the story new forms of linkage that will have effects that we can witness as transformational. With others (Goodman 1995; Joseph & Burnaford, 1994; Brown 1981) I can suggest the usefulness of metaphors and discussion of metaphors as having this kind of impact. Like Foucault (1980) I have had small successes as an "intellectual terrorist" by placing metaphors in locations in ways that cause the explosion of conceptual bridges. In the rebuilding of the bridges is a brief moment of hope that the new bridge can be different, that the construction will open up the possibility of an invention of purpose. The effect is one of new threads woven in new ways that emphasize "new interactions." Brown would call the purpose "problem generation." The trick is to do this in a way that wins the hearts and minds of those around you, in the style of the French Resistance, rather than to present yourself as the harbinger of chaos and destruction. Extending the metaphor of the French Resistance, must we then, those of us who work in education, resort to the image of restoring a former golden past, and can this be successfully played out? Can we think of recapturing the authority and means to determine who we are? We turn to Sartre, deBeauvior, and their theoretical progeny as we recognize along with Brown that the matters of education pertinent to mathematics educators are much broader than those of understanding mathematics.
 

We are in this sense searching for ways to generate questions rather than researching to provide answers, and it is in this sense that I can begin to interpret for myself and hopefully for the reader as well what this essay is "all about." Because we are discussing at least three sites of learning and knowing and because we are looking for implications of the articulation, I can benefit from an adaptation of thoughts on the pedagogy of mathematics as a metaphor for the learning and teaching of "education" as a discipline. As I have learned much from Stephen Brown on the art of problem posing for the learning of mathematics, I can apply his thoughts here as well and say that the learning and teaching of this research has as its goal the generation of questions to ponder rather than solutions to teaching/learning problems. The kinds of questions we ask and the problems we generate as mathematics educators are to a great extent the ways we are known to others. Efficacious dialogue in the setting of this essay, as in the teacher-student dialogue of a classroom, requires a sorting out of what each participant believes ("regarding the nature of mathematics, the nature of his or her own mind, and the personal significance of shared experiences," writes Brown). As Hawkins, Ball and Brown agree on, the "answer" is not some sort of "open math environment" unless it, like any number of other environmental options, honors dialogue as a genuine interchange in which the teacher as well as the student can hope to increase his or her awareness of self and the generation of problems, or purpose. What I read them as calling for is nothing other than a revolution in what mathematics education is all about. The pie in the sky quality of such a statement does not detract from the seriousness of its intent or need. Our "target number" for ourselves must become a vision of new "non-Euclidean" mathematics education. Locally it might look like mathematics education as we know it. Various practices would resemble those we have grown up with and understand in "Euclidean terms." But our comprehension of the purpose and practice of these techniques and metaphors would have undergone so profound a reconstruction that their presence could never have the same meaning for us again.
 
 

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¹ I want to thank Mildred Dougherty and Rochelle Kaplan for their responses to early drafts of this chapter.