Investigating Teaching Practice: Setting the Stage

The first session used videotapes of excerpts from an elementary classroom to establish a framework for thinking about creating a classroom climate where mathematical reasoning is the norm and how this is reflected in the mathematical content teachers need to know.

PRESENTERS

What Mathematical Knowledge Is Entailed in Teaching Children to Reason Mathematically?

Deborah Loewenberg Ball, University of Michigan

Hyman Bass, Columbia University

Reaction to the Presentation by Deborah Ball and Hyman Bass

James Lewis, University of Nebraska



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Knowing and Learning Mathematics for Teaching Investigating Teaching Practice: Setting the Stage The first session used videotapes of excerpts from an elementary classroom to establish a framework for thinking about creating a classroom climate where mathematical reasoning is the norm and how this is reflected in the mathematical content teachers need to know. PRESENTERS What Mathematical Knowledge Is Entailed in Teaching Children to Reason Mathematically? Deborah Loewenberg Ball, University of Michigan Hyman Bass, Columbia University Reaction to the Presentation by Deborah Ball and Hyman Bass James Lewis, University of Nebraska

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Knowing and Learning Mathematics for Teaching What Mathematical Knowledge Is Entailed in Teaching Children to Reason Mathematically? Deborah Loewenberg Ball, Hyman Bass INTRODUCTORY COMMENTS BY BALL Consider the task of establishing a classroom culture in which mathematical reasoning is both possible and called for. The wording of this task is part of the popular rhetoric these days, but our goal is to think about the task from a mathematical perspective. The idea of classroom culture is ubiquitous in elementary school teachers' talk. Everybody talks about creating classroom climates or cultures in which students feel comfortable and safe and talk to each other. What does it mean to think about the mathematical resources involved in creating a mathematical classroom culture, not just the generic idea of classroom culture? Mathematical reasoning is also something that everybody talks about. It is in the NCTM Standards (National Council of Teachers of Mathematics [NCTM], 1989). It is in every text you look at, but people mean different things by these words. Here, too, we want to probe what it means to look at children 's mathematical reasoning, thinking about it mathematically. There are many other perspectives that bear on mathematical reasoning: cognitive perspective, socio-cultural perspectives, perspectives that look from the bottom up at how children reason. Those are all valuable, but we want to focus on what it means to look at children 's mathematical reasoning as emergent mathematical justification. What lenses do elementary teachers need to understand that what they as teachers are involved with is the development of children's capacity to construct proofs, to understand and follow mathematical proofs, to understand the need for justification, and to be able to distinguish valid justifications from invalid justifications? This is not discordant with discussions about classroom culture or mathematical reasoning but means reflecting about these phrases in combination with each other and specifically thinking about them from a mathematical perspective. We're going to show you two different pieces of videotape—one very basic and one more complicated. Both are from my third-grade classroom. The purpose of looking at this tape is to give a concrete example of young children talking with each other in a group about a mathematical idea and analyzing how these ideas are crucial to the resources required to teach. What do the videos cause you to see about the classroom and

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Knowing and Learning Mathematics for Teaching what do they make you think about the mathematics that teachers might need to know to establish and maintain a classroom culture in which mathematical reasoning is central? We invite you to think about the tape, to make comments, and to talk to each other. INTRODUCTORY COMMENTS BY BASS I came into the work as a mathematician, and my role is essentially the same as the assignment for the workshop, to look for the mathematics entailed in the core tasks of teaching, the kinds of problems a teacher has to solve, design of lessons, interpretations of student thinking and work, and how to assess whether to pursue an idea or return to the lesson plan. Many of these decisions entail mathematical knowledge and considerations, and the idea is to try to discern what that mathematics was, to place it in a larger mathematical context. The problem is how to look at these core tasks of teaching. The idea of doing this is compatible with the work of a university mathematician. The courses that we teach—for example, our calculus courses—are, in fact, designed to serve professional communities, like engineers, or economists or biologists. We do, in fact, look at how, in practice, they use mathematics, and we design curricula to meet those needs. Somehow, however, in teaching teachers about mathematics, we don't seem to have looked carefully at how the mathematics is actually used in practice. We treat it as a disembodied subject matter to be imported by the teacher, and the very complicated and difficult process of importation, which entails considerable knowledge in its own right, is not part of the picture. So, the method used here was to start with a site of practice, analyze mathematics in use, and pull out curricular and epistemological ideas from that use. As a mathematician, this was new for me, because I'm accustomed to analytical reasoning where the truth of things ultimately derives from the meanings of the words. Here I had to pay attention to data and try to justify claims not by deduction but by evidence from those data. I took that quite seriously; and, at the outset, I fully expected to come up with a list of mathematical topics which I would embed in somewhat larger recognizable domains of mathematics as interesting packets of knowledge that would be helpful for teachers. What I want to emphasize is that we're paying attention to something that is not a mathematical topic in the typical sense. It was not drawn from rhetoric of the standards or from my own inclination to pay close attention to the importance of proofs. Rather, it was something that was driven by observation of the data, by looking at what was actually happening in this particular class and giving more specification to what these words mean. We have two objectives. One is to explain what it means to create a community of reasoning among a class of third-graders and, secondly, to find out what mathematical resources a teacher might bring into play to create such a community. Now, that may be a characteristic of this particular class, but the presumption, I think, in any research of this kind is that close observation of a complex enactment of teaching suggests principles that are applicable much more broadly. In a way, one of the surprising and, to me, very satisfying outcomes was that the description of mathematical reasoning we found appropriate to third-graders—and I emphasize that this was based on observa-

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Knowing and Learning Mathematics for Teaching tion—was completely consonant with the way you would describe mathematical reasoning among professional mathematicians. Let me now turn to some of the ideas we found. This is a conceptual framing, and again I want to emphasize that it was drawn from the data. What is reasoning? It's one of the principal instruments for developing mathematical understanding and for the creation of new mathematical knowledge. We're thinking here of the reasoning of justification, warrants for claims, as opposed, for example, to the reasoning of the inquiry in discovery. Such reasoning, in our view, requires two foundations. One is a body of prior knowledge, either inherited or assumed, on which to stand as a point of departure. You can't get something for nothing. You have to buy it with some sort of currency. And this foundation furnishes the elementary bonds in chains of reasoning. The other is the linguistic medium and the elementary and often unarticulated rules of logic with which to formulate claims and the networks of relationships that are offered to justify them. A classroom dedicated to creating a culture of shared mathematical reasoning has both of these foundations: the basic common knowledge and the linguistic structure and conventions for mathematical communication. Both of these must be made part of the community consciousness of students, as well as the teachers. First, let's talk about the basic common knowledge. This is defined relative to the community of reasoners. If these were professional mathematicians, this base might consist of an axiom system for some mathematical structure like Euclidean geometry or group theory, simply admitted as given, plus a body of previously developed and publicly accepted mathematical knowledge derived from these axioms. In another setting, a university instructor who lists prerequisites for a course is defining part of the presumed common knowledge in the environment of that course. In a third-grade classroom of children with diverse backgrounds, this base of common knowledge is at first not wholly known or formed. It is determined and shaped through ongoing empirical inquiry, observation, and orchestration by the teacher. The children's knowledge comes from a mix of prior experience and mathematical learning, and it grows through the learning in the classroom. The crucial issue is the following: How can one justify a mathematical claim? One way is to simply assert it dogmatically with whatever force of authority its advocates can evoke; and, in fact, this is a course open to and often taken by teachers of young children. But that route is the antithesis of what we mean by mathematical reasoning. On the other hand, the process of reasoning itself typically consists of a sequence of steps, each of which has the form of justifying one claim by invocation of another, to which the first claim is logically reduced. This process, which merely transforms one claim into another is not a vicious circle because the reduced claim is typically of a more accessible and elementary nature, and in a finite number of steps, one arrives at a claim which requires no further warrant. Why? It's universally persuasive in the reasoning community because it's part of the base of common knowledge. Thus, the base of common knowledge defines the primordial steps requiring no further warrant, which form the kind of stepping stones of an argument. As we like to say, it defines the granularity of acceptable mathematical reasoning in that particular environment. The base of common knowledge consists of knowledge of

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Knowing and Learning Mathematics for Teaching certain facts and concepts, of the meanings of mathematical terms and expressions, of procedures and resources for calculations and problem solving. It is always present in either latent or active forms, and it may be passive and only implicit in either the students ' or the teachers' talk. It plays a role for both teachers and students in a classroom where mathematical reasoning is expected. First, consider the teacher. The teacher must both uncover and build students' common mathematical knowledge. Being attuned to the class 's base of common knowledge, it is crucial to understand where the students are and where they're prepared to head. The teacher's explanations, for example, depend on a close coordination with students' current base of accepted knowledge. At the same time that such coordination is important, teachers also must work to establish and extend this base of common knowledge. Decisions about when to introduce a term or an idea, when to make a distinction, and when to raise a challenge—all of these are fundamental in helping students to build and extend what they know. Working to extend not only individual students' knowledge but also what is commonly accepted among them is central to the teacher's work. Next, what about the students? What importance does it have for students? We see two dimensions in this. First, when a child or the teacher reasons before the class, the elements on which the argument is built are presumed to be part of the base of common knowledge. Whether this knowledge is, in fact, common is an empirical question and one of active concern to an investigation by the teacher. The plausible presumption of its commonality is a working premise of the reasoning, and this very process of reasoning, with suitable interventions by the teacher, can, in fact, help plant such presumed common knowledge into the common base when it's not already solidly there. Independently of this empirical question, an important objective of the teacher is to encourage children to build mathematical arguments on the basis of presumed common knowledge, for this is an important part of teaching them what it means to reason mathematically. In the course of this, the teacher also wants to make the evolving base of common knowledge a part of the class' consciousness—that is to say, to make it common. So, now let me turn to the other component of this construct, mathematical language. We understand language in a very broad and inclusive sense, comprising all of the linguistic infrastructure that supports mathematical communication, with its requirements for precision, clarity, and the economy of expression. It includes the nature and role of definitions in mathematics, the nature and rules for manipulation of symbolic notation, and the compression of concepts afforded by their uses. It is important to recognize disagreement that stems from divergent or unreconciled uses of terminology from disagreement that is rooted in substantive and conflicting mathematical claims. The ability to do this requires a sensitivity to the nature and role of language in mathematics. We're concerned, as well, with the transformation of mathematics embedded in experiential settings, described informally with common language, into more formal mathematical expression susceptible to efficient mathematical manipulation. We emphasize that mathematical language is not simply an inert canon inherited and learned from a distant past. It is, as well, a medium in which learners, as mathematicians, act and create. Notations are introduced to reduce computation and manipulation to

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Knowing and Learning Mathematics for Teaching manageable proportions. How and when to do this is an important skill and one needing and deserving to be taught. Definitions are not simply delivered names to be memorized. They are seeded or conceived in concepts; they gestate through active investigation and reflection, and when they come to term, they are born out of a need to describe a rich or important idea in need of easy reference to facilitate its entry into common discourse. These are what we call emergent definitions. The decision about what to name and when and how to name it calls for a developed mathematical sensibility and discrimination, one that teachers of mathematics at entry level would be well served to acquire. Another persuasive role of definitions, not always recognized as such, arises when a notion given meaning in one context is then given expanded meaning in an enlarged context, a process that we refer to as expanding definitions. This raises important questions about the criteria, usually left only implicit, by which such extensions are made. Now, all of this may seem fairly esoteric and a bit simplistic to mathematicians, but let me illustrate how these notions can play out in the early grades. Take, for example, the study of number— basic number systems. The children typically first encounter whole numbers, and they may have some informally developed sense, say, of even and odd, and they probably could quickly decide correctly whether a given small whole number is even or odd. At first, they usually have no formalized definition of these notions and typically lack, for example, the unit-digit criteria for recognizing evenness or oddness. Eventually, they may be led to some working definition, say, of even numbers as those that can be evenly divided into two groups with none left over or, for example, that the numbers alternate: even, odd, even, odd, etc. Perhaps they can be made to see that the unit digit controls this property. Later, when first introduced to positive and negative integers, they may be asked the question of whether zero or negative 3 is even or odd. Now, is this a question of finding mathematical truth? Mathematically, the answer is no, because at the moment, these notions don't have meaning in that environment. For them, at this moment, evenness or oddness of these numbers has yet to be defined. The issue is, rather, how should these notions be defined for such numbers and by what criteria. One can try to apply a working definition of even in this larger number domain, but then various questions arise. Does the working definition of even make sense? What does it mean to divide negative 6 into two equal parts? Or there may be several equivalent definitions of even for whole numbers. In fact, there are. You can divide numbers into two equal parts, or you can separate them into pairs, with none left over, or you can apply the alternating definition. These are all quite distinct but in significant ways mathematically equivalent. So, if there are several definitions for even numbers, do these generalize to give a common notion for integers, and if so, should one prove this, and if not, how should one prioritize them? Do evenness and oddness have reasonable meaning for fractions? Is one-and-three-quarters even or odd? The same questions are posed. We don't assign a meaning there and for exactly these kinds of reasons. What would the alternating definition mean for fractions? What would dividing into two equal parts mean in that setting? We don't encounter that. The decisions have already been made for us. We don't enter that domain, but if we did,

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Knowing and Learning Mathematics for Teaching we would see that these issues of expanding definitions do present themselves. So, these notions have, at the moment, not yet been defined, and how should they be defined? To derive definitions, one course is to derive them from some model used to represent integers—for example, money and debt or an elevator building with many basements or a frog leaping back and forth on the number line. Alternatively, one might try to extend the operation to consider another mathematical notion, not only evenness or oddness but the very operations of arithmetic. Once you've defined what negative numbers are, what should addition and multiplication mean for them? At the moment, the issue is not what they are and how to discover what they are but, rather, how should one define them, and the criteria by which that's done involves significant mathematical considerations. It's not simply a matter of arbitrary convention. Implicitly, the sort of criteria we use are those of preserving certain patterns of regularity. For example, the alternating definition extends in a very natural way to the negative numbers, but the notion of dividing something into equal parts or an equal number of pairs, or a whole into number of pairs with nothing left over, presents a different way of prolongation to the other domain. For example, when extending the notion of addition to negative numbers, you might want to preserve certain mathematical properties like the distributive law with respect to multiplication and addition, but arguments made about arithmetic with negative numbers, which invoke those properties, have tacitly made the presumption that the definition must be made in such a way as to preserve those properties, just as using exponential notation when the exponent is no longer an integer or a rational number raises the question of what meaning should be assigned to it. You want to do this in such a way that you preserve the usual laws of exponentiation. So, these are simply some illustrations of why questions of mathematical language, which at first may seem fairly sterile of content and seem to be merely a matter of convention, in fact involve, in the way one evolves and constructs such language, substantial mathematical issues. We're not trying to suggest in any sense that we think of this as something that happens without consideration for the mathematics and the tasks in which children are apt to engage. I wanted to set the context about the sort of work students have been doing before we watch the tape, because you'll see their work on one of these problems. There are two clips from the same lesson, both taught early in the school year. So, you can see the beginning of efforts for the students to make mathematical justifications to one another. This is not a case of looking at a classroom that has a well-established culture yet, but you can see it beginning to emerge. One of the problems the students worked on the first or second day of class was the coin problem, which is taken from the Curriculum and Evaluation Standards (NCTM, 1989, p. 24): “I have some pennies, nickels, and dimes in my pocket. I put three of the coins in my hand. How much money could I have?” Another problem students did subsequent to that was a permutations problem involving taking the digits of the date, September 12 and the questions, how many 3-digit numbers can you make with the digits 9, 1, 2. They also worked with 2-digit and 4-digit numbers, and there, too, they were beginning in both of these problems to consider the question not just of answers but of how many solutions

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Knowing and Learning Mathematics for Teaching there were. How could you establish that you had all of the solutions? You can see some of the potential in these problems for creating the need for mathematical justification. We're going to watch the problem, “Write number sentences for 10.” One issue is how problems get framed: What language is used; why number sentences; what does this actually mean. When I show this problem to people who aren't familiar with elementary classrooms, they actually don't know what the problem means and they don't like the wording of it very much at all. They would prefer the problem to say something like, “write arithmetic expressions that equal 10.” The whole question about how you word things for students and when and how you make decisions to use different forms of mathematical language, less formal, more children-like is important. On the videotapes, you'll see cases both of following children's mathematical language and introducing more formal language. So, this issue about language has to do with all the judgments that go into deciding when to use what kind of language. We're not advocating a particular point of view about that. We're simply saying that's an example of a site in which teachers reason and make decisions about the precision of mathematical language. At first, students simply wrote two elementary problems, such as six plus four equals 10 or five plus five equals 10. They were only doing addition and only using two elements. After they've produced a few of these working by themselves, we collect some solutions, and the class is asked to think of some solutions to the problem that involve more than two numbers. As you view the video, consider the way in which the base of common knowledge and mathematical language are brought to bear. It may allow you to see something about a class discussion that you might not otherwise see. This classroom is in a school close to Michigan State University in which the children come from many different countries and speak many different languages. The question of language is interesting simply because it's also a very multicultural classroom. There are children who are second-language speakers, some of them quite limited English speakers. The class is the only third grade in the school. There are 19 students at this point in the year, which is just a function of how many third-graders there are in the school. Sometimes the classes were as large as 30 or 32. Sometimes they were as small as 17 or 19. The school was a very mobile environment. Children were coming in and out of the class, leaving the class, entering the class all throughout the year. It was not a stable kind of teaching environment. At this point in the session, participants watched the videos. See Appendix E for transcripts. I just want to make a couple of comments about this episode. It illustrates perfectly the early use of what we're calling the base of common knowledge of the children at that grade level to define, in this instance, the accepted granularity of reasoning and to have a child model, with the teacher's guidance, what it means to prove and communicate the proof of a mathematical claim. Kip, seizing the teacher's challenge to make sums with more than two terms, goes almost to the limit with his expression, one plus one plus one plus one plus one plus one plus one plus three equals 10. He probably got tired of repeating ones at the end and so finished off quickly with the three. Now,

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Knowing and Learning Mathematics for Teaching one objective of this task was to open up the children's knowledge of, and skills with, basic arithmetic. The immediate verification of Kip's formula was not something to be presumed within easy reach of the children—that is to say, not part of their basic common knowledge. Perhaps for a fifth-grade class, this judgment would have been different. A young child's initial sense of addition comes from counting, which is adding one term at a time. Adding many terms at once or adding two terms both larger than one is a more complex operation, not only for children but also mathematically. Thus, once Kip's dramatic formula was presented to the class, the culture of reasoning demanded that it be justified or proved in a manner appropriate to children at this level, and as a move toward creating the culture of reasoning, the teacher had the children publicly honor this mandate. When Ranya is called upon to explain why Kip's formula equals 10, she first just recites the equation. The teacher presses on with, “But how could you prove that to someone who wasn 't sure,” introducing in the process the mathematical term “prove,” which is a significant mathematical term that is later adopted as part of the children's vocabulary. Ranya replied that she counted, and she was then pushed further to make this counting public. “There 's one, and the next one is two, and the next one is three,” et cetera. In fact, it was presumably by this counting that Ranya had proved to herself that the formula was valid, and the teacher now required that this reasoning be made public to persuade the class, as well. The teacher then publicly validates Ranya's performance, and she later appeals to it as one of the early benchmarks for what it means to explain something in mathematics and not just to assert it. Do students have the common knowledge that one plus one plus one . . . plus three is 10? At the beginning of the school year, what the children know and don't know about basic arithmetic is not really known to the teacher. The question is what is a safe level of assumption about their knowledge, and certainly, everyone, I think, would agree that counting by one is something that could be presumed of everyone. In the second example, which is more complicated, the basis of the student argument is that you can add any number to the sum and subtract that same number from the sum and you get the same result. For example, any number plus a lot of zeros minus that same number plus 10 equals 10. There's a predictable result. To contextualize this segment a bit, this is in September, at the beginning of the school year, and this is the third of several tasks that had been presented to the class. The first one was the coin problem; the second one was the permutation problem. One thing to keep in mind is what was the purpose of these early tasks? One, since there was not a clear knowledge by the teacher about what the students knew and what their level of facility with the curriculum of the course was, the task had to be accessible to children with a variety of levels of knowledge and skill and, at the same time, challenging. There should be entry points with different levels of mathematical sophistication and challenge. Second, the tasks should involve some serious mathematics and engage the children in some of the material that would be part of the course. Third, the tasks should provide context in which to develop this culture of mathematical reasoning. Overall, the tasks were designed to expose some of the basic common knowledge which the teacher was in the process of trying to discover.

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Knowing and Learning Mathematics for Teaching Thus, the tasks provided not only work for the kids but also work for the teacher to discover what they knew. The first two problems, the coin problem and the permutation problem, were problems of a combinatorial character, and one important feature was that they admitted multiple solutions. In both of those problems, the number of solutions was finite, and the students could empirically generate lists of answers, until finally, they exhausted them all. When asked whether they had all the solutions, the students decided empirically that they did. When asked why, they would say, “Well, we keep coming up with the same answers over again,” or, “I looked at so-and-so's list and she didn't have anything that I didn't have already.” Essentially they were making an empirical argument, a kind of scientific or probabilistic claim, not a mathematical argument. It was the first encounter as a class with the challenge to try to mathematically prove a claim, “do you have a full set of possible solutions to this problem?” In that sense, the third problem was different in two important respects. First, this was a problem not only with multiple solutions but with infinitely many, and so, it presented an opportunity to see how the students encountered the notion of the infinite and what their disposition would be toward that. Second, unlike the combinatorial problems with coins and with permutations, the terrain of the problem was the central domain of arithmetic, which is what the mathematics curriculum is about at that level. Making up number sentences that equal 10 opens the doorway to essentially all the arithmetic operations and all the things the curriculum wants to do with them. In some sense, the students in this process are not only going to expose what they know and so define this sort of basic common knowledge but begin to show the edges of their common knowledge. They'll make constructions where they falter and, in some sense, identify the point where the teachers wants to then develop the curriculum to expand what this basically is all about. REFERENCE National Council of Teachers of Mathematics. ( 1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

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Knowing and Learning Mathematics for Teaching Reaction to the Presentation by Deborah Ball and Hyman Bass James Lewis We have all been charged with spending the day exploring the question, “What mathematical knowledge does it take to teach well?” The answer to that question depends, in part, on how a teacher approaches teaching. If the task at hand is learning to divide fractions, solving, for example, the problem 1¾ divided by ½, it takes far less mathematical knowledge to tell students to “invert and multiply” than it takes to make sure students understand why this procedure leads to a correct answer. In their presentation, Ball and Bass are interested in a slightly different question, namely “What mathematical knowledge does it take to teach children to reason mathematically?” One obvious answer is that it takes quite a bit more mathematical knowledge, and this is certainly underscored by the videotapes that were shown as part of the presentation. Watching the tape of Deborah Ball teaching a class of third-grade students, it becomes clear that she is attempting to establish a classroom culture in which mathematical reasoning is called for. In her classroom, she changes the core question from “What is the answer?” to “Why is the answer correct?” There is no doubt that this places a much greater demand on the teacher than would a more traditional approach of demonstrating a process for solving a problem and then having children practice very similar problems until they have mastered the technique. It seems to me the contrast is a bit like the difference between an outdoorsman walking through a familiar wooded area and a tourist hiking an unfamiliar trail. The tourist, and the elementary teacher with minimal knowledge of mathematics, is quickly lost once “off the trail.” In her presentation, Ball identified three responsibilities that a teacher must accept if the teacher is to help students reason mathematically. The teacher must uncover the students' current base of common knowledge; establish and extend the students' base of common knowledge; and model and guide the construction of acceptable mathematical arguments. An analysis of the videotape of Ball in her third-grade classroom shows her efforts to be sure that the students have a base of common knowledge and that the

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Knowing and Learning Mathematics for Teaching students have access to a common language. At the beginning of the videotape, Ball asks her students to “think of a number sentence that uses more than two numbers” and to “make a number sentence that equals 10, but has more than two numbers adding to 10” (emphasis mine). I saw that as two separate questions, but Ball's students seemed to have no trouble interpreting the charge to be a search for various expressions that equal 10. The answer offered by one of the students was to add one to itself a sufficient number of times to get 10. Ball immediately challenges the class to explain why the answer is 10. When one student essentially repeats the assertion that “one plus one plus one . . . equals 10, ” Ball points out that the student has only read the answer, not given a justification for the answer. Later, when one student says that “100 divided by 10 equals 10,” the class is again challenged to explain the answer. One answer offered is, “Lin said her mom taught her . . . about dividing by. ” Relying on what mom said is also found to be an insufficient explanation. At this point on the videotape a number of students offer ideas that appear to have some depth of understanding behind them (e.g., since 10 times 5 equals 50 then surely 50 divided by 5 equals 10). At this point, Ball records one of the assertions the students have been discussing (50 divided by 5 equals 10), but she stresses that the idea does not yet belong on the list of facts that everyone understands as part of the class's base of common knowledge. In a later class, Ball introduces the term “conjecture” and offers the class a definition of the term. She uses one of the student's number sentences as a conjecture and leads the class in a discussion of why the conjecture is correct. The basic idea under consideration by the class is that not only is (200 − 200) + 10 = 10 but that one can replace 200 by any other number in this sentence and get another number sentence that equals 10. (Note: I can't resist adding the parenthesis signs but the students seem comfortable with expressions like 200 − 200 + 10 and everyone seems to know what to do.) My concern with proper notation reinforces some of the challenges that the teacher faces on a daily basis. The teacher must be able to hear what a student is saying (or might be saying); the teacher must decide what is mathematically significant in the discussion; and the teacher must decide when to introduce new mathematical notation and when to couch the discussion in the language of the children. This particular videotape is rich in concepts such as the term conjecture, the discussion about what does or does not constitute a proof, and the basic language of mathematical reasoning used in the classroom. At the same time, there is a tolerance of imprecise language used by the children that would be unacceptable in my college classroom. At the end of the videotape showing in this session, Ball is clearly modeling and guiding the “construction of acceptable mathematical arguments” when she introduces the concept of a variable and the sentence (x − x) + 10 = 10 as being valid for any value x. I found the videotape of Deborah Ball teaching third-grade students to be fascinating. She clearly was establishing a classroom culture that called for mathematical reasoning, and her students demonstrated a sophistication regarding the need to prove statements rather than accept them on faith. As a mathematician teaching at the university level, I have very little knowledge of third-grade

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Knowing and Learning Mathematics for Teaching classrooms, but I suspect that similar classrooms are in short supply. I am also convinced that the mathematical knowledge required to teach in this manner is far superior to the mathematical knowledge that most of our students have when we certify them as ready to teach mathematics at the elementary school level.

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