Contents | Committee | Preface | Ch 1 | Ch 2 | Intro to Ch 3-9 | Ch 3 | Ch 4 | Ch 5 | Ch 6 | Refs for Part 1 | Other Reports | Ch 7 | Ch 8 | Ch 9

Chapter 7
The Preparation of Elementary Teachers

The power to reason mathematically is a natural human capacity. Young children enter school already curious about number and size, and with ideas about how to join, remove, and split quantities. Mathematics instruction in the elementary years can---should---be designed to cultivate this curiosity. Encouraged to solve problems, children become aware of their ideas; and as they learn to analyze their own, their classmates, and their teachers thinking, these ideas become more refined and many-sided. It is during these early years that young students lay down those habits of reasoning upon which later achievement in mathematics will crucially depend.

Teaching elementary mathematics requires both considerable mathematical knowledge and a wide range of pedagogical skills. For example, teachers must have the patience to listen for, as well as the ability to hear, the sense---the logic---in children's mathematical ideas. They need to see the topics they teach as embedded in rich networks of interrelated concepts, know where, within those networks, to situate the tasks they set their students and the ideas these tasks elicit. In preparing a lesson, they must be able to appraise and select appropriate activities, and choose representations that will bring into focus the mathematics on the agenda. Then, in the flow of the lesson, they must instantly decide which among the alternative courses of action open to them will best sustain productive discussion.

It is by now widely acknowledged that many practicing teachers were not adequately prepared by the mathematics instruction they received to meet these challenges. As K-12 students---often even in the primary grades---they lost their curiosity about mathematics. When the rules and procedures one is taught are not conceptually anchored, memorization must pass for understanding, and mathematics becomes an endless, senseless parade of disparate facts, definitions, and procedures.

College students who today choose to become teachers are by and large still products of such K-12 instruction. Even if those who opt to teach middle- or high-school-level mathematics have experienced their mathematics education positively, many who choose elementary teaching have not. Intimidated by mathematics, the latter generally avoid mathematics courses wherever possible.

It seems, then, that we are caught in a vicious cycle: poor K-12 mathematics instruction produces ill-prepared college students, and undergraduate education often does little to correct the problem. Indeed, some universities mandate next to no mathematics coursework for the prospective elementary teacher. However, simply increasing the number of required credit hours is no solution---courses that allow students to get by using the same stratagems that got them through K-12 just perpetuate the problem.

In order to break this cycle, college students with weak mathematics backgrounds must have opportunities to reconnect with their own capacities for mathematical thought. Those among them who decide to enter the classroom and are willing to engage the conceptual riches of the elementary curriculum can become effective mathematics teachers. But they, just like the children they will someday teach, must have classroom experiences in which they become reasoners, conjecturers, and problem solvers.

Future teachers will need to connect fundamental concepts to a variety of situations, models, and representations. They will have to learn to notice patterns and think about why those patterns hold; pose their own questions and know what sorts of answers make sense; look for connections among different methods for solving the same problem or different ways of representing the same quantity. In short, developing these new mathematical habits means learning how to continue learning.

This is a daunting agenda. But if teachers are to help their students become strong mathematical thinkers, it must be met. And the medium through which this agenda can be realized is the very mathematics they are charged with teaching in the realms of number and operations, geometry, early algebraic thinking, and data.

Conventional belief has it that elementary school mathematics is simple and to teach it requires only learning prescribed facts and computational algorithms. However, recent work has revealed the conceptual richness of this early content, demonstrating that teaching elementary school mathematics can be intellectually challenging. Though each of us once inhabited the mathematical world of the young child, that world is lost to most of us. To re-enter it, Deborah Ball and Hyman Bass argue in a recent paper,

one needs to be able to deconstruct one's own mathematical knowledge into less polished and final form, where elemental components are accessible and visible. We refer to this as decompression. Paradoxically, most personal knowledge of subject matter knowledge, which is desirably and usefully compressed, can be ironically inadequate for teaching. In fact, mathematics is a discipline in which compression is central. Indeed, its polished, compressed form can obscure ones ability to discern how learners are thinking at the roots of that knowledge. Because teachers must be able to work with content for students in its growing, not finished state, they must be able to do something perverse: work backward from mature and compressed understanding of the content to unpack its constituent elements. (2000a, p. 98)

Ball and Bass's description of the challenge the elementary teacher faces in connecting to the mathematical world of the child holds for the instructor of the mathematically naïve adult. In a college course, prospective teachers' ingenuous questions will require instructors to "decompress'' their mathematical knowledge to find responses satisfying to both mathematician and teacher.

Although some questions elementary-school teachers pose may be stimulating, others are certain to be very disturbing. Instructors teaching teachers for the first time will occasionally feel dismay, or even shock. How can such basic notions not be understood? What is there to think about? But the gaps in these teachers' mathematical backgrounds are consequences of systemic rather than personal failings, and it is essential that, recalling this, instructors work to maintain the necessary stance of interest, generosity, and respect.

To repeat, the challenge is to work from what teachers do know---the mathematical ideas they hold, the skills they possess, and the contexts in which these are understood---so they can move from where they are to where they need to go. For their instructors, as we have seen, this means learning to understand how their students think. The habits of abstraction---of compression---and deductive demonstration, characteristic of the way mathematicians present their work, have little to do with the ways children build their mathematical world, experientially, modeling concepts on actions---counting out, dividing up, comparing heights or ages. . . . Mathematics courses for teachers must aim, first of all, at helping them develop ways of giving meaning to the mathematical objects under study, only then moving on to higher orders of generality and rigor.

Chapter 3 outlines the mathematics content teachers need to know for the K-4 classroom. What follows in this chapter expands upon that discussion. Precisely because what goes on in the elementary classroom will seem alien to many readers of this document, vignettes drawn from actual lessons are used to elucidate the issues. In these scenes, children articulate their mathematical thinking, showing their teachers what they understand and where they are confused. The scenes do not exhaust the territory, but they are representative of the mathematical issues that arise in a typical classroom when mathematics teaching is organized to elicit and build upon children's thinking. (Some scenes are taken from grade 5 classrooms, but are included in the belief that what comes up in a fifth grade class is likely to come up for the fourth grade teacher, too.) The vignettes are followed by discussion of the mathematics the teachers will need in order to identify the sense in their students thinking, know when a key mathematical idea is being missed, or anticipate when significant mathematical territory is being broached. What mathematical knowledge will help teachers navigate these situations in ways that support building stronger mathematical conceptions?

These scenes taken from classrooms are not intended as models of exemplary teaching. They have been chosen, not for emulation, but to illustrate the kinds of knowledge and skills required of elementary teachers. Because lessons in which children practice routine procedures typically do not present mathematical challenges to teachers, they are not included in this document. Their absence is not meant to imply that such activities have no place in the elementary mathematics classroom.

For readers who have little contact with aspiring or practicing elementary-level teachers, excerpts from teachers learning journals and episodes from teacher education courses are also included. These are intended to communicate what teachers themselves report as new mathematical insights.

All of the vignettes are drawn from actual classrooms. Many of the scenes of elementary-level lessons are paraphrases of cases written by the teachers themselves. Others are taken from videotape or records of classroom observations. Scenes from courses for teachers and excerpts from teachers journals are based on published literature, unpublished field notes, and personal communications.

The recommendations for course content draw on research about teacher and student knowledge. They also follow from the assumption that most teachers (though certainly there are exceptions) have had few, if any, opportunities to learn content that is just now entering the elementary curriculum, particularly topics in early algebra, geometry, and statistics.

This chapter discusses teaching and learning at both elementary-school and teacher-preparation levels. To minimize confusion, the term "children'' is used to refer to elementary school students, and "teachers'' refers to both practicing and aspiring teachers.

Number and Operations

Understanding number and operations and developing proficiency in computation have been and continue to be the core concerns of the elementary mathematics curriculum. Although almost all teachers remember traditional computation algorithms, their mathematical knowledge in this domain generally does not extend much further. Indeed, many equate the arithmetic operations with the algorithms and their associated notation. They have little inkling of how much more there is to know. In fact, in order to interpret and assess the reasoning of children learning to perform arithmetic operations, teachers must be able to call upon a richly integrated understanding of operations, place value, and computation in the domains of whole numbers, integers, and rationals.

Summary of number and operations content

In order to begin to explore the mathematics content knowledge required for teaching at the elementary level, consider the classroom scene below, which captures children at work on subtraction.

Scene 1, from a second grade classroom: The children have been finding the difference between Jorge's height, 62'', and the height of Cinthia's little brother, Paulo, 37''. (Currently they are using inches so that the heights will be two-digit numbers. Later they will use centimeters to get three-digit numbers.) Many of the children use dots and ten-sticks to represent two-digit numbers.

(based on Hiebert et al., 1997, pp. 153-155)

Formed by an education equating mathematical strength with computational proficiency, too many teachers have been left with an impoverished understanding of the number system. To orchestrate a classroom discussion like the one above, or those presented below, teachers must be able to do more than demonstrate remembered procedures---for example, they must be able to select problems that anticipate the issues their students will next need to confront, and then assess whether what the children make of those problems advances the mathematical agenda. Such skills require much deeper understanding of number and operations than most teachers now hold. The bulleted items discussed below identify key points of entry into the mathematics of the elementary grades.

Each operation can model a variety of actions or situations.

For years, elementary textbooks have suggested that teachers teach their students to solve word problems by finding "key words'': "altogether'' means add, "left'' means subtract, etc. Beyond such superficial clues, many teachers associate each operation with just one possible action: joining with addition; taking away with subtraction; repeated addition with multiplication; and either finding the number of groups of a given size or, given the number of groups, finding the size of each group (usually not both) with division (Graeber & Tanenhaus, 1993). Hampered by their own limited understanding of the operations, teachers have had little more to offer their students. What is required is a sense of the different kinds of situations that can be modeled by each of the four basic operations.

In the classroom scene above, the teacher has selected a problem involving comparison of heights. By thinking through the situation, the children develop different solution strategies: Gabriella considers how much Paulo has to grow; Roberto thinks about "shrinking the big guy down.''

A teacher who is aware of the range of situations that can be modeled by subtraction can consciously choose problems that promote a variety of ways of thinking about the operation. The following is an example of a common "take-away'' problem: Sue Ellen had 62 cents and then bought an item for 37 cents. How much money did she then have? A second type of subtraction problem involves joining, with the starting quantity unknown: Manny went to the beach yesterday and picked up 37 shells for his shell collection. He now has 62 shells. How many shells did he have before his last visit to the beach?

Teachers must come to recognize the variety of situations---of joining, separating, and comparing, with an unknown in various positions---that can be represented by addition and subtraction. Similarly, multiplication and division can be associated with a rich store of interpretations: multiple groups, splitting, shrinking and stretching, counting rectangular arrays, counting combinations. In many interpretations of multiplication (and in contrast to addition and subtraction), the numbers are associated with different units: e.g., 34 might model 3 bags, each with 4 donuts. Such multiplication problems have two types of division problem as analogs: partitioning into groups of a given size and partitioning into a given number of groups.

In developing more broadly based conceptions of the kinds of situations modeled by the operations, it is also important to become familiar with such other modes of representation as the number line or arrangements of blocks. Area representations of multiplication are particularly useful.

Keeping in mind Scene 1, consider a second, in which, at the start of a new unit on division, the teacher has given the class a set of what she considers division problems.

Scene 2, from a combined third/fourth grade classroom:

Jesse has 24 shirts. If he puts eight of them in each drawer, how many drawers does he use?

Vanessa writes: 24 -- 8 = 16,   16 -- 8 = 8,   8 -- 8 = 0, and then writes 3 for the answer.

If Jeremy needs to buy 36 cans of seltzer water for his family and they come in packs of six, how many packs should he buy?

This time Vanessa writes: 6 + 6 = 12,   12 + 12 = 24,   24 + 6 = 30,   30 + 6 = 36.  (She doesn't identify her answer.)

You go into a pet store that sells mice. There are 48 mouse legs. How many mice are there?

Matthew organizes his work in a chart of two columns:

Then in a neat box he writes,

12 m 4 l = 48 l

Above the box, he writes the number 12.

The teacher wonders, what does this say about kids' understanding of division if they use all the operations except division? (based on Schifter et al., 1999a, pp. 55-57)

Scenes 1 and 2 illustrate another issue, the interrelationships of the operations, that many teachers need to work on.

A given situation can be modeled by different operations.

The "key words'' mindset leads many teachers to believe that, for any given word problem, there is just one operation that can be used to solve it correctly. However, as Scene 2 shows, a "division problem'' can be solved by adding, subtracting, or multiplying. And in Scene 1, where some children readily see their way to the solution as a process of finding a missing addend, others subtract. Instead of ruling out any of these methods as incorrect or problematic, a teacher who understands the ways operations are interrelated can seize the opportunity to explore such connections more deeply.

The children whose classroom gave rise to Scene 2 did eventually learn their division facts. With a richer understanding of what division means and how it is related to the other operations, they were able to see how particular facts can be derived from other facts, making the process of recall easier.  (Later in the lesson from which Scene 2 derives, Matthew is given another "division problem.'' He responds, "It's 63 9. What number times 9 is 63? Seven. . . . [I]t is [division], but my thinking is multiplication.'')

Understanding how the operations are related and how these relationships can be called upon in solving problems is critical for teachers if they are to interpret and advance their students thinking. Just such a revelation is recorded in this excerpt from the journal of one participant in an inservice course.

In my group we did it [15913] the "regular way'' [using the division algorithm], then by equally distributing base-ten rods, [and then] by going around and counting out by ones. Then [the instructor] came to our group and 15913 suddenly became 159 -- 13 = 146,  146 -- 13 = 133,   133 -- 13 = 120,   . . .  and so on. I had never before thought of division as directly related to subtraction. . . . As simple as it sounds that one interaction really made an impact on me, as and were just something I did by a rote method, with not much thought as to how , , , andare all related. (Schifter et al., 1999b, p. 177)

This teacher was now resolved to bring these ideas to her fourth grade students.

Returning to Scene 1, consider that the children's methods rely on decomposing numbers into tens and ones. This highlights another set of ideas teachers must understand.

The principles of place value involve significant conceptual issues for young children and for teachers.

Most teachers are readily able to identify the ones place, the tens place, etc., and can represent multidigit numbers in expanded notation. Nonetheless, they often lack understanding of core ideas: how place value permits efficient representation of numbers; that the value of each place is ten times larger than the value of the next place to the right; how a number can be decomposed into tens and ones in a variety of ways (53 can be viewed as 5 tens and 3 ones, or 4 tens and 13 ones, etc.); how the operations of addition, multiplication, and exponentiation are used in representing numbers as "polynomials in 10''; and how decimal notation allows one to determine quickly which of two numbers is larger. Teachers should be familiar with the notion of "order of magnitude'' and should have a sense of the relative magnitudes of numbers.

Not only must teachers be able to state these ideas, they must be able to recognize and apply them flexibly. One activity that has been used successfully to help teachers develop such facility asks them to create a number system using the symbols A, B, C, D, and 0 (Schifter & Fosnot, 1993). The letters allow the possibility of assigning different values to the symbols. Working in small groups and offered a set of base-five blocks as a thinking tool, teachers are asked to show how to represent large numbers in their system and to calculate with multidigit numbers. If they should get that far, they are also asked to explore divisibility and give an account of numbers smaller than 1.

Multidigit calculation provides opportunities to both deepen understanding of place value and build meanings for operations.

The following errors are commonly seen in elementary classrooms:

In these examples, children are applying their single-digit math facts but are mis-remembering their computational procedures. Because they are not thinking about the size of the numbers they start with or about what the operations do, they form no reasonable estimate of the outcome. If neither the children, nor their teachers, have learned to approach such problems with the expectation that they should make sense, it is difficult to correct the misconceptions underlying these errors.

Teachers with richly developed meanings for the operations (a sense of the variety of situations and representations associated with the operations) and a flexible understanding of place value (for example, knowing how to decompose numbers into convenient parts and operate on them) are positioned to help such children. They can recognize the strong thinking of children like Gabriella, Roberto, and their classmates in Scene 1, and they can help children who make such errors as those shown above go back to what they do understand about numbers and operations in order to help them recognize their errors.

Solving multidigit problems in their heads---"mental arithmetic''---and then sharing the strategies they employ, is an especially useful exercise. Teacher educators have found (Schifter et al., 1999b) that many students come to courses believing that conventional algorithms offer the only valid methods of computation. Those who invent their own strategies often feel sheepish, as if they are relying on "crutches,'' or are embarrassed by their lack of "sophistication.'' Once the hold of these prejudices is loosened and they begin to maneuver about the number system more fluently, they begin to see how the base-ten structure can be used flexibly and efficiently.

As various methods of calculation are encountered, teachers must consider the logic behind each: Does this method always work? Some need to consider why very basic procedures are justified, say, 58 + 24:   50 + 20 = 70,   8 + 4 = 12,   70 + 12 = 82. In one seminar, a teacher watching a video of second graders solving problems such as this, blurted out, "I can imitate this method to apply to other numbers, but I don't see why it works. It's just another meaningless algorithm to me!'' In this situation, representations such as blocks or number lines, which help teachers think about what the operations do, can help them see a justification for the procedures. Some teachers must think through the general principle that addends can be decomposed and the parts recombine in any order, yet conserve the sum; or that when subtracting, if the same amount is added to or subtracted from both quantities, the difference remains constant.

Other, more complicated procedures are often challenging to teachers. Consider the following steps, commonly devised by primary grade children, for solving  35 -- 16:

This procedure raises such questions for teachers as: Why is the 1 subtracted rather than added? Will this work for any subtraction problem, even one with numbers larger than two digits? When does this method apply and what comparable method can be used when it doesn't? Can the steps of this procedure be articulated as an algorithm? A college student writing in her journal, excerpted below, touches on these issues.

Furthermore, I am able to apply this same type of thinking to subtraction problems. I have more difficulties with subtraction problems, but I am working on increasing my comfort level. What I discovered to be very interesting were the many ways a subtraction problem could be broken down. . . . The example of  35 -- 16 was a great one. This led to many different discussions. I was able to look at this problem and say:

(Student journal, spring, 1999)

This aspiring teacher, her classmates, and others enrolled in comparable classes certainly know the computation algorithms before beginning the course. What they learn is flexibility in decomposing numbers, figuring out how to recombine them to perform the operations, and thinking about the operations in terms of the actions they model. The numbers they are operating on remain in view and do not get lost in a thicket of disconnected digits. (Thus, "3 -- 1''; in the computation above is correctly recognized as a representation of 30 -- 10, which equals 20.)

Similarly, with multidigit multiplication and division, learners (both children and teachers) first think in terms of groups in order to sort out calculation procedures. Again, teachers can begin an exploration of multiplication through practicing mental calculations. Then they can analyze their own, their classmates', and children's methods of calculation. Once the idea of rectangular array is introduced into thinking about groups, the area model, in particular, brings to light the partial products of two-digit multiplication. Later, these ideas can be formalized as the distributive property.

Comparing procedures can make the reasoning behind algorithms transparent

In Scene 1, Ms. Lo Cicero asks the class how Roberto's and María's methods are alike. In this way, she highlights particular steps in the procedures and draws students' attention to analogous lines of reasoning in the different representations. For another example of comparing procedures, consider the following scene.

Scene 3, from a fifth grade classroom: The class has been given the homework problem 728 34. One child, Henry, presents this solution method:

3410 = 340
3420 = 680

Henry explains to the class, "Twenty 34s plus one more is 21. I knew I was pretty close. I didn't think I could add any more 34s, so I subtracted 714 from 728 and got 14. Then I had 21 remainder 14.''

Michaela describes the steps of the conventional division algorithm: "34 goes into 72 two times and that's 68. You gotta minus that, bring down the 8, then 34 goes into 48 one time.''

Apparently, their teacher has not shown the conventional division algorithm to her students, and Michaela's classmates say they don't understand her solution. Asked to explain, Michaela takes the class through the steps again, but with the same response. Then the teacher asks the class to compare the two procedures to identify similar parts, assisting them by inserting a "0'' next to Michaela's "68'' so that the children could more easily see where Henry's 680 shows up in Michaela's process. Through the discussion that ensues, using Henry's solution as a point of reference, some of Michaela's classmates can begin to see the justification for the steps she had taken.

(based on NCTM, 2000, pp. 153-154)

The reasoning behind Henry's method is clear to him and his classmates. But when Michaela presents the conventional long division algorithm with its more efficient notation, the rationale for her procedure eludes both her and her classmates. The teacher recognizes the parallel reasoning behind the two methods and draws her students' attention to it, thus giving them access to what was an initially opaque process.

The steps of the conventional algorithms, particularly for multidigit multiplication and division, are often every bit as mystifying to teachers as they are to children. The former, too, can compare procedures, devised by themselves or by students, or by other cultures, to bring to light their conceptual bases.

It is also useful to examine commonly applied incorrect procedures for solving multidigit multiplication problems, such as those instantiated in the following strategies observed in the work of teachers and children.

Each of these incorrect methods derives from misapplying additive procedures. After all, when adding 16 + 28, one can operate on the tens, operate on the ones, and add the results, etc. By analyzing these procedures, teachers have opportunities to deepen their understanding of multiplication and the distributive property, and to become sensitive to the tendency to extend additive procedures to multiplicative situations.

Presenting and exploring these various methods of calculation highlights for teachers the differences among the operations. In addition and subtraction, the units are the same; in multiplication and division, more than one unit is involved. In addition or subtraction, one can decompose the addends or both the minuend and subtrahend, respectively. In multiplication, additively decomposing both factors frequently makes the calculation more complex. And in division, additively decomposing the divisor is not useful. Up to now, this discussion has been confined to the arithmetic of whole numbers. And children do begin to learn about numbers through counting. Soon, though, the world of number expands to include integers and rationals. In order to support children through this transition, teachers, too, must have explored these new concepts.

As with whole numbers, teachers must learn to give meaning to operations with integers.

Many young children are exposed to numbers less than 0 outside of school, through discussions of weather in the wintertime, say, or by keeping track of scores in some of their games. In an example cited above, a second grader says that  5 -  6  equals "one in the hole" and knows how to use that idea to compute  35 -- 16.  However, in general, operating with integers presents new issues.

However, the teacher is concerned about the limitations of the elevator representation. For example, it allows the children to think about subtracting a positive integer as "going down" or about subtraction as the distance between floors, but the representation does not help the children develop a sense of "taking away" numbers less than zero. Nor could they make sense of certain addition expressions, e.g. 6 + (-- 6) .
(based on Ball, 1993)

Knowing the rules for computing with integers is insufficient for understanding operations with numbers less than zero. As with whole-number operations, teachers and children must learn to think of the variety of situations that can be modeled by the operations. Now, however, as the numbers represent both magnitude and direction, the situations increase in complexity.

Fractions introduce a new kind of number.

Children are introduced to rational numbers through their work with fractions. Although most young children are familiar with the numbers ¹/2,  ¹/4, and perhaps ³/4, the idea of fraction is challenging. To many, these numbers represent a quantity less than one, or, perhaps, part of a whole, but they might also talk about how "your half is bigger than my half" or be unable to interpret the meaning of, say, ²/3. And even when children seem to understand the meaning of fraction in some situations, that understanding often proves fragile and context dependent. For example, in a class of third graders who had been working on fractions for some weeks, the question arose, Which is larger, ⁴/4 or ⁵/5? Some argued for ⁴/4 because the parts are larger; others, for ⁵/5, because there are more parts. No one argued they were equal (Ball & Wilson, 1996).

Adults are generally unable to recall a time when their concept of number was exhaustively defined by the experience of counting whole numbers. Yet, listening to children being introduced to the idea of fraction and realizing how this challenges their very notion of number, offers adults an opportunity to think through how the concept of number expands as one moves from the system of integers to rationals. It is no longer merely a matter of counting units. Instead, one must now count the number of units in one quantity, count the number of units in a second quantity, and derive a third number---a new kind of number---that places the first quantity in relation to the second (Behr & Hiebert, 1988; Carpenter, Fennema, & Romberg, 1993).

Many children, and older students as well, see fractions only as pairs of natural numbers plugged into arithmetic procedures. So, for example, in the second National Assessment of Educational Progress, when students were asked to pick an estimate for ¹²/13 + ⁷/8  from the choices, 1, 2, 19, and 21, most chose the latter two, presumably having combined either their numerators or their denominators. They failed to recognize that ¹²/13 and ⁷/8 are each quantities close to 1 and, thus, their sum is close to 2 (Carpenter et al., 1981).

For teachers to be able to perceive the mathematical ideas children must put together in order to develop the idea of fraction, their own understanding of the concept must be expanded. The same number created by finding the part of a whole can also be seen as an expression of division (further discussed below), as a point on the number line, as a rate, or as an operator.

Most teachers know what a fraction is under at least one of its interpretations, but they often lack a sense of relative size. Having memorized a method for finding a common denominator and comparing numerators, they cannot determine, say, which of a pair is larger--- ⁵/7 or ⁷/9? or ⁵/8 or ⁷/12?---without applying that procedure. Working with area diagrams, teachers can explore such fraction pairs and learn to use other strategies, e.g., given common numerators, comparing the denominators; or considering how much smaller each fraction is than 1, or how much larger than ¹/2. Such observations can, in turn, be applied to more cumbersome pairs of fractions.

Developing meaning for calculating with fractions enriches understanding of both fractions and operations.

The following vignette illustrates two major mathematical ideas---unit and fraction as quotient---that need to be investigated by teachers and children, too.

Scene 5, from a fifth grade classroom: To work on a representation of "3 divided by 15," children produce the scenario "three pizzas need to be shared among 15 people," and solve the problem by dividing each of the pizzas into five slices. When the teacher changes the problem to 3 divided by 16, there is not such a neat solution.

One child suggests cutting each circle into 16 pieces. Another, whose scenario is "share 3 pieces of cheese among 16 mice," says that if you divide each piece into five slices, one mouse won't get any. Another girl says she should have divided each of the three pieces of cheese into six slices. A debate ensues about what to do with the extra slices. Some of the students think it would work to divide each of the leftover slices into eight pieces, and they are challenged to explain to their classmates how they have figured out that it will be eight. Another issue for debate is what each of the little pieces should be called: one sixteenth? one eighth? one forty-eighth? (Each of the slices, it has been agreed, is one sixth of a piece of cheese.)

(based on Lampert and Ball, 1998, pp. 139-141) 

The first major mathematical idea, "unit," is central to work with fractions. In Scene 5, the children correctly suggest that the small bits of cheese could be represented as ¹/16,  ¹/8,  or  ¹/48 ---depending on the unit chosen. The second major idea is the relationship between fraction and division. Many teachers have considered fractions only as parts of a single-unit whole---e.g.,  ³/16  involves dividing a single whole into 16 equal pieces and taking three of those. The idea that ³/16  might mean three wholes divided equally into 16 portions is new to them. And they would have difficulty explaining how the quantity ³/16  is the same as the quantity ¹/6  ^{+ 1}/48.
(Lampert & Ball, 1998, p. 142)

In order to help her students work out why the correct answer is ⁵/4, first the teacher needed to see why ⁵/8 made sense to some of her students, what would be a question in this context whose answer is ⁵/8 (e.g., If you start with 2 cups of flour, how much of the flour do you need for the two batches?). Understanding how ⁵/8 could make sense to some of her students, she was in a better position to help them see why ⁵/4  is the sum of ³/4 + ²/4.

Addition and subtraction require working with single units, but multiplication and division involve more than one. This is precisely what is so difficult about devising word problems or diagrams for, say, ³/4¹/2: What is the unit for ³/4? for ¹/2? for the quotient? And how does that shift when multiplying ³/4¹/2? On the other hand, sorting out these issues can bring to light the reasoning, so elusive to both children and adults, behind the invert-and-multiply algorithm. (These ideas are further elaborated in the chapter on mathematics for middle grades teachers.)

Decimal fractions extend the ideas of place value to numbers less than 1; as with calculations with common fractions, decimal computation can enrich understanding of the operations.

Scene 6, from a fifth grade classroom: In September, the teacher had given the class a set of word problems, among them, "Rob wants to read one hundred pages of his book before his next conference in seven days. How many pages should he read each day?" Now, two months later, she asks her students to look at the problem again, but to find the answer on a calculator. The children all report, 14.285714. After reviewing an earlier discussion about the interpretation of their original answer, 14 remainder 2, the teacher asks "What is the `.285714' in the calculator's answer?"

The class begins to talk about this and, after a few minutes, Jeremy raises his hand. "I think I get it. In the 14.285714 it's like the 2 is a paragraph and the 8 is a sentence and the 5 is a word and the 7 is a letter and the 1 is part of a letter. I don't know what the 4 is. Only the 14 [the two left-most digits] really counts anyway. The other pieces are really small, especially after you get beyond the sentences." "That's pretty interesting," the teacher says. But recognizing the limitations of Jeremy's metaphor, she offers him an opportunity to qualify it: "Does it make sense to you that it could work that way?"

"Well, it doesn't really make sense," he answers. "I mean, you don't have pieces of words to be read and things like that. It does make sense in some ways though; like how I said, it's really only the first few numbers that make a difference. The rest are too small to matter." (based on Schifter et al., 1999c, pp. 108-110)

Many children and teachers, too, believe decimals bring with them a new set of rules to remember, but the main principles that underlie decimal fractions are the same as those that govern whole numbers. Precisely because they are tacit with respect to whole numbers, formerly unproblematic concepts now need to be considered, thus providing an opportunity to return to whole numbers with new insights.

For example, in Scene 6, a fifth grade class is presented with an eight-digit number between 14 and 15 and the children are challenged to interpret the digits to the right of the decimal point. Although Jeremy's metaphor is limited (a paragraph is not necessarily ²/10 of a page; a sentence is rarely ⁸/10 of a paragraph), it does bring him and his classmates to an important observation: "It's really only the first few numbers that make a difference. The rest are too small to matter." Indeed, this can be said of any multidigit number, whole ones included. The digits to the left are more significant than those to the right in determining the magnitude of the number. (This is the basic idea behind scientific notation; and the same principle is highlighted in stem-and-leaf representations of data.)

Like the fifth graders in Scene 6, teachers, too, need to develop meanings for the digits to the right of the decimal point. Having them create their own representations of decimals (with blocks or diagrams) provides opportunities to bring out misunderstandings and highlight issues that need to be explicit for teachers---what is the value of each place and how do the digits combine to represent a single quantity?

Teachers must come to see that any number can be approximated arbitrarily closely by finite decimals. The study of repeating decimals invites work with the calculator and, especially given the calculator's finite capacity, exploration of its limitations. Why is it that a calculator cannot exactly represent ¹/3? How can you characterize all fractions that have finite representations?

To summarize this discussion of decimals, consider the following journal excerpt, by an inservice instructor who asked her class to compare whole number and decimal addition.

People at first posited many differences, but by the end of the discussion came to see none! For example, the first thing stated as a difference was that you line whole numbers up from right to left and then you line decimals up from the decimal point. Then someone suggested that if you have two decimals with the same number of places, you also line them up from the right and that, in fact, you are always (no matter what kind of numbers you are dealing with) lining up like places with like places and that this was a similarity, not a difference. . . .

Another difference mentioned was that when you add decimals, the quantities got smaller, but when you add whole numbers, the quantities got bigger. People actually thought about this for a moment. I suggested we add some decimals, so they proposed  .15 + .16.  Everyone then stated that no, in fact, even decimal addition makes numbers bigger [when only positive addends are under consideration]. Someone stated authoritatively that when you multiply, the result ends up being smaller. I asked disingenuously how this could be since just last session people were telling me that multiplication was nothing more nor less than repeated addition. If addition of decimals produces larger quantities, how could multiplication of decimals produce smaller quantities? Some people laughed. . . .

The third difference was proposed by Nancy, who said that the places got smaller as you go to the right of the decimal and larger as you go to the left. I asked her to come up and show us what she meant and she approached the board, pulled up sharply, and said, "Oops, never mind!" as she figured out that even to the right of the decimal, the farther to the left you go, the bigger the number.

Then someone suggested that regrouping changes as you move across the decimal. I wondered how, and several people said it worked the same way regrouping works in whole number addition, but that the places had different names.

(Yaffee, personal communication, 1997)

This recap of a discussion among teachers working to understand number and operations highlights several themes. First is the habit of noticing superficial characteristics of calculation procedures. When dealing with rule making at this level, one must remember different rules for the different kinds of numbers. Moreover, when relying solely on memory, one is likely to come up with such misremembered "facts" as "adding decimals produces a sum smaller than the addends," but with no resources to challenge them. Returning to basic principles, however, different rules may merge into one. The arithmetic for decimals is essentially the same as for whole numbers.

Second, shallow or mistaken ideas, such as those offered in this recap, can sit alongside correct understandings. When the classroom environment is safe enough to bring such notions out into the open, they can be challenged, corrected, replaced, or modified.

And third, this journal entry illustrates how the basic themes of place value and operations recur in the context of work with rational numbers.

One might have noticed that, in this discussion of number and operations, the formal statements of properties (some prefer to call them "the laws of algebra") have not played a central role. To many teachers at the elementary level, algebraic notation obscures rather than illuminates. Introducing formal axioms in the expectation that teachers will be struck by the beauty and logical economy of our number system is naïve. Only after they have done the kind of work described above---have come to know the various kinds of situations modeled by the operations, developed a variety of representations for them, and worked with these representations to explore calculation with whole numbers, integers, and rationals---can they make sense of algebraic notation. Without it, mechanical application of rules (commutativity, associativity, etc.) is likely to leave them with those familiar feelings of disconnection.

Work with algebra is discussed in the following section.

Algebra and Functions

Although the study of algebra and functions generally begins at the upper-middle- or high-school grades, some core concepts and practices are accessible at a much earlier age. If teachers are to cultivate the development of these ideas in the elementary grades, they must understand those concepts and practices and recognize how they are manifested in the mathematical thinking of young children.

Summary of Algebra and Functions content

When children begin their study of algebra in middle or high school, they learn a new language, an efficient way of representing properties of operations and relationships among them. Now they are expected to make meaning for such sentences as,

If, in earlier grades, students lose their ability to make sense of mathematics and, as a consequence, can attach no meaning to arithmetic expressions, they have nothing on which to build their algebra. On the other hand, to those already familiar with those properties and relationships, the challenge is learning the conventional system of notation.

When the elementary classroom is designed to encourage and build upon children's thinking---where students pursue their own questions---we find them interested in formulating and testing generalizations (Ball & Bass, 2000a, 2000b; Carpenter & Levi, 1999; Russell et al., 1999). This is particularly evident in their work on calculation and number theory topics: evens and odds, square numbers, factors. Children's interest in articulating these generalizations provides an opportunity for them to explore the ideas they will later learn to express in algebraic form. At the same time, it provides an opening to work on methods of justification.

Many elementary teachers have shared the situation of the child who enters algebra class without a sound background in arithmetic. They, too, struggled through their courses, memorizing rules for manipulating symbols. If, on the other hand, prospective teachers are offered a course that helps them make sense of number and operations (as described above), then they are prepared to learn to use algebraic notation to express relationships that have meaning for them.

However, to be able to support children in the classroom, teachers will need more than fluency in algebra. They must appreciate the power of generalization, be able to recognize when children are approaching this territory, and understand what counts as a justification.

To build on children's capacities to articulate their observations and to generalize requires teachers who understand the importance of generalization and who command a variety of methods of justification and forms of representation.

Scene 7, from a kindergarten classroom: The children are in pairs playing a version of the card game, War. For each round, they each put down two cards and whoever has the larger sum takes the four cards. Myra and Janie have just laid out their cards and Myra declares, "I get these." Janie protests, "But you didn't count yet! I might have more." Myra explains, "My two numbers are more than your two numbers, so when you put them together, mine is more."

(based on Seyferth, field notes, 1995)

(based on Bastable & Schifter, in press)

In Scene 7, Myra's strategy for determining which child has the larger sum subverts her teacher's goal for the lesson. Rather than practice counting or adding, Myra reasons about quantities in general (Thompson, 1993). A teacher who recognizes the power of her observations will sacrifice her immediate objectives in order to encourage such thinking.

Given a situation similar to that depicted in Scene 8, a teacher who registers the difference between the first and second pairs of statements can ask about the latter, will it always work? Indeed, the teacher who recognizes that the fourth statement is a corollary of the third (understanding that the children's meaning for "add two zeros to it" is "insert two zeros after the last digit") is positioned to assess whether any members of the class can take on that idea.

And the third and fourth graders in Scene 9 make inferences based on three different kinds of arguments: testing a conjecture on a set of specific numbers, reasoning by extending a pattern, and forming a deductive argument. Only teachers who themselves appreciate the differences among these different forms of justification can, in turn, help their students understand them.

Greater attention to algebraic thinking at the elementary level has encouraged teachers and researchers to look into young children's abilities to reason with variables (Ball, 1989; Carraher et al., 2000). For example, in one first and second grade combination class (Carpenter & Levi, 1999), students worked with such equations as,

Children working on the first equation concluded that the sentence is true when x = 5. They generated the second when asked to find "an open number sentence that is true for every number, no matter what you put in." For the third, they found a number of solutions and recognized that y must be twice as large as x. As this kind of material moves into elementary classrooms, teachers must be aware of different uses of variables: to express unknowns that can be solved, to express identities, and to express relationships between sets of numbers (Usiskin, 1988).

To work on these issues, teachers, too, must be given contexts---perhaps, like their students, to explore factors, divisibility, square numbers---in which they can come up with their own observations and assess the validity of their own and their classmates' claims. For elementary teachers, it is especially important that they learn to develop arguments using, in addition to algebraic notation, representations familiar to them. For example, the third and fourth claims in Scene 8 can be supported using an area model of multiplication. By identifying the same lines of reasoning in strategies employing different forms of notation, teachers can now learn to give meaning to once meaningless symbols. And as they become used to algebraic notation, they can learn to appreciate its power and flexibility, especially in comparison to geometric representation.

For elementary teachers, an understanding of commutativity, associativity, distributivity, identities, and inverses cannot be taken for granted. Lacking proper grounding, many will have the sense that these rules have simply been "pulled out of the air." Rather, they should be given tasks that help them make these generalizations from their experiences. For example, one might ask teachers to consider these properties for all four operations, devising representations or situations to demonstrate why they work when they apply, why they don't if they don't, and then to identify any patterns they discover. Although it may be obvious to teachers that addition and multiplication are commutative, it is not obvious that reversing the quantities when subtracting or dividing yields the inverse. Although associativity of addition may be clear, teachers need to think through why, for example,  (10 -- 7) -- 2  results in a smaller number than 10 -- (7 -- 2).

In one inservice class, teachers exploring these properties reported the following conclusion: "When you reverse the numbers for subtraction, you get the opposite---except when the numbers are the same." To understand what the teachers were saying, the instructor worked with them to represent their observation in conventional algebraic notation: If  a -- b =  c,  then b -- a =  -- c,  except when  a = b. Perplexed by this exception, the instructor challenged them. After a few minutes of confused discussion, the teachers explained that if  a = b,  then the answer is 0 both ways, not  -- 0, which they called "the opposite of 0." When asked for the opposite of 0, there was a long pause, until someone ventured, "Infinity---the opposite of nothing is everything." This opening provided an opportunity to think about additive and multiplicative inverses, additive and multiplicative identities, and how definition plays a role in creating a consistent system (Schifter, field notes, 1989).

With a deeper understanding of commutativity, associativity, distributivity, identities, and inverses, teachers can return to calculation algorithms---those conventionally taught in the United States, those taught in other countries, as well as those children frequently devise---to analyze them as applications of these properties.

Solving multistep problems in a variety of ways also provides opportunities for teachers to create meaning for algebraic notation.

Problems conventionally solved using algebra are, in fact, accessible to elementary grade children, as well as to teachers whose algebraic skills are under-developed. For example, consider the following problem (written quite a few years ago, when we paid less for our fruit): Ten apples and 5 bananas cost $1.65, and one apple and one banana together cost $.20. What is the cost of one apple? one banana? In one course for teachers, some teams got started by pulling out colored cubes: 10 red for apples, 5 yellow for bananas. Each cube stood for the cost of a single piece of fruit and, together, the 15 cubes represented $1.65. Pairing red and yellow cubes, then discarding them, the teachers subtracted $.20 for each pair. Left with 5 red cubes worth $.65, they concluded that each was worth $.13. Therefore, a banana cost $.07.

Having solved the problem in a way that called upon their own mathematical ideas, the teachers could then represent it algebraically. Identifying a as the cost of an apple and b as the cost of a banana, they could now see parallels between the steps they took with the blocks and the steps of the algebraic procedure.

As teachers come to see how algebraic representations correspond to representations of actions modeled by other solution methods, they become more confident and skillful with algebra. From here, they can appreciate the use of algebra in problems for which other representations are too cumbersome.

Work with patterns has been a part of the K-4 curriculum, but the concept of function is new.

Scene 10, from a kindergarten classroom: The class has been working on patterns. As the teacher presents today's pattern (green, orange, brown, green, . . . ), she also writes down the number associated with each element: "1" is written under the green square, "2" under the orange, "3" under brown, "4" under green. As has been routine, the children call out what comes next: orange, brown, green. Now the teacher points to the numbers she has written to show how she has identified the place of each square and poses a new question to the class: "What color square will go in the tenth place?" Different children call out different colors, as if this were a guessing game. But Roberto speaks forcefully. He stands up, stamps his foot, and declares, "It's green! I know it's green!"

(based on Cohen, personal communication, 1994)

Scene 11, from a third grade classroom: The teacher is working with a small group of children who are interested in the surface area of towers they make out of cubes. Starting with a  112  tower, it takes some time for the children to sort out what surface area is, but eventually they conclude that 10 faces of the unit cubes show on the surface. They backtrack to look at a    111   tower before they move on to a    113  . The teacher suggests that they organize their findings into a chart.

She now asks the children to think about the surface area of the next larger tower. in

Jeff holds a stack of five cubes. He then shows the group how he counts all the square units on each face of the tower--- 5 + 5 + 5 + 5 + 2.  He points to the 5 square units on each of the four vertical faces, then adds the 2 for the ends. But then he shows that it is also 4 + 4 + 4 + 4 + 4 + 2   if he counts around all the vertical square units of each cube in the tower and then adds the ends. He says it's (4 5) + 2 and (5 4) + 2. The others agree.

Two days later, the teacher gives the group cubes of different sizes---with edges of 1, 2, 3, and 4---and asks what they think the surface area of each is. The children record their findings: 6, 24, 54, 96---and Jeff declares the next one is 150, though he hasn't made one that size.

The teacher asks if the group could make any rules to describe how the areas change. First Jeff says the areas were all 3-numbers (multiples of 3). Ron says they are also counted by 6-numbers. Then he says the areas were always six times one face of the whole cube. Ron suggests that they added three 6s every time. He points out the distance between 6 and 24 is 18 and guesses the same is true for 24 to 54. When he sees the distance is 30, Jeff points out that's five 6s. They figure out it's seven 6s between 54 and 96 and observe: 3, 5, 7. So Jeff says it must be nine 6s to 150. (based on Schifter et al., in press)

Elementary teachers often create rich classroom experiences around patterns, usually beginning with a sequence of elements (sometimes in the form of sounds or movements), where a set of elements forms a unit that is repeated. Children learn to extend the pattern by determining and repeating the unit. Later, children have experiences with number patterns---finding the next number, then explaining how they did it. However, teachers rarely have a sense of how this is related to the mathematics their students will encounter in later years: specifically, that a function can be created by labeling the elements or units of a pattern by the natural numbers. For example, the kindergartners in Scene 10 have been presented with a function, and based on the regularity of the pattern, Roberto is able to identify the value in the range associated with a particular value in the domain.

Some of the patterns children find particularly intriguing point toward ideas that will prove significant in higher grades. For example, second graders working on square numbers notice that as they increase, a pattern forms. The numbers go up by 1, 3, 5, 7 . . . Third graders in Scene 11 notice a similar pattern for the surface area of cubes: As the length of each side increases by 1, the surface area increases by 3 sixes, 5 sixes, 7 sixes.

Those same third graders also notice a different pattern when they count the square faces on the surface of the 1 1 n  towers they build: The numbers go up by 4. Teachers who are familiar with the characteristics of particular classes of functions---in this case, linear and quadratic---would have a context for their students' observations.

The children's observation that "you add 4 to each one'' is consistent with a recursive representation of functions: f (n +1) = f (n) + 4.  Jeff points out how you can look at a 5-cube tower and see the surface area as 4 5 + 2,  approaching a representation in closed form:  f (n) = 4n + 2 . Teachers must be familiar with both.

Some of the new curricular materials for the elementary grades have children working with stories, graphs, and tables to describe situations that change over time, e.g., height, speed, distance, population. These include graphs not defined by elementary functions. The very ideas on which the children are working are content for teachers: What is the meaning of a horizontal straight line? a line tilted up? a line tilted down? What is the relationship between rate of change and accumulated change?

Those familiar with the usual college-level algebra course will recognize that it does not address the issues described in this section.

Geometry and Measurement

For many years, the geometry curriculum for the elementary grades mandated recognition of basic two-dimensional shapes, measurement of length with standard and non-standard units, and the ability to apply area and perimeter formulas for rectangles (and possibly a few other shapes).  Because so many students entered high-school geometry courses unprepared for it (Usiskin, 1987; van Hiele, 1986), topics in geometry have recently been given a more prominent role in the early grades. Their own experience with high-school geometry notwithstanding, for most elementary teachers, much of this material---some of it highlighted below---is new.

Summary of geometry and measurement content