To Understand Post-Counting Numbers and ... - Pat Thompson

3 downloads 0 Views 157KB Size Report
One way is that the situation is about changes in James' numbers of marbles. .... Thompson and Dreyfus [75] noted likewise that the 6th-graders in their study ...
To Understand Post-Counting Numbers and Operations† Patrick W. Thompson Luis A. Saldanha Vanderbilt University

White paper prepared for the National Council of Teachers of Mathematics.

Running head: To understand numbers



Preparation of this chapter was supported by National Science Foundation Grant No. REC-9811879. All opinions expressed are those of the authors and do not reflect official positions of the National Science Foundation.

Thompson & Saldanha

To understand numbers

Writers of the Standards 2000 have attempted to redress imbalances that became evident after the first version of the Curriculum and Evaluation Standards [46] and Professional Standards for Teaching Mathematics [47]. Critics of the original Standards sometimes forget, however, that they were written amidst a prevailing predisposition in American curricula toward procedures and computations [44] and a strong predisposition among American teachers toward calculational discourse [66]. Evidence that students were learning far less than they should was overwhelming, especially in regard to understanding situations in which mathematics could be applied [7, 44, 57]. Those first Standards, unfortunately, were interpreted far more literally than anyone intended. It seems the perceived imbalances were more in readers’ interpretations than in writers’ intentions. To some extent, these earlier documents invited the interpretations they got. The Curriculum and Evaluation Standards often appeared to address pedagogy at the expense of addressing issues of what should be taught and what students should learn. The Professional Standards for Teaching Mathematics addressed actions of teaching, but did not address the substance of teaching — the ideas and methods students should build, and how reformed instruction might aid their building them. The Standards 2000 might have a similar fate, but for a different reason. In its first draft, Standards 2000 writers strove to avoid perceptions of imbalance. Thus, it was replete with calls for a balance between concepts and skills. But the draft Standards 2000 failed in the same way as its predecessors to address concepts students should learn and understandings students should have when they understand those concepts. Thus, though the draft Standards portrayed the appearance of balance between concepts and skills, it actually gave very little attention to students’ conceptual understanding. Consequently, attempts to recruit people to a reform movement that is centered around “student understanding” could have the ironic outcome of producing a cadre of reformers having incompatible meanings of “to understand” and incompatible images of what would be acceptable evidence that students understand specific ideas. We are not saying that that everyone should agree on one view of the understandings DRAFT - 1

Thompson & Saldanha

To understand numbers

students should have. Rather, it would be helpful if the Standards were to offer one stance, even as a sacrificial example. The importance of explicitness Writers are often vague about what they have in mind when saying “understand x” or “have a conceptual understanding of y.”1 Being vague about ideas we want students to learn misses opportunities to make them explicit, and the community’s ability to debate the value and intricacies of learning them is hindered severely. As Herscovics [31] noted, many students experience difficulties learning an idea not because it is inherently difficult, but because they have developed understandings of ideas it builds on that thwart coherent understandings of the idea they are trying to learn. By being vague in what we intend that students learn, we fail to discuss potential obstacles that are due to students having developed understandings other than ones we intend. We also fail to make explicit the long-term benefits we envision being gained by students who do develop the understandings we intend. We appreciate the negative reaction this position engenders in some people. It may seem anti-postmodern and anti-constructivist to say that we should decide upon understandings students should have. Our position, however, is that, as professional mathematics educators, we are remiss not to do so. It is our job to develop thoughtful, informed opinions on what is important that students understand. Moreover, it is misguided merely to define what we hope students understand in terms of what teachers should do. That would be anti-constructivist, for it would amount to saying, “We want students to understand what they are shown.” Put another way, learning objectives should be stated independently of specific instruction. To mix the two invites confusion about when a learning objective is achieved.2

1

We use phrases like “concept of x” reluctantly. To say “concept of x” suggests we are comparing a person’s concepts and a correct view of x. We do not mean this at all. Rather, we mean “conceptual structures that express themselves in ways people would conventionally associate with what they understand as x.” But saying that is too cumbersome, so we continue to use “concept of x” and “understanding of x”. 2 When learning objectives are stated in terms of what teachers should do, then we invite people to judge

DRAFT - 2

Thompson & Saldanha

To understand numbers

Discussions of understandings we think students should gain from instruction have a practical side. They focus debate on building consensus on what students should learn from instruction and how various forms of instruction might support that learning. We note that this debate is about what students should know and not about skills students should have. The two debates are related, but they are not identical. Evidence from many sources points to a lack of substance and lack of coherence in mathematics instruction in the United States. The TIMMS report of 8th-grade mathematics instruction in the United States, Germany, and Japan states this clearly. Finally, as part of the video study, an independent group of U.S. college mathematics teachers evaluated the quality of mathematical content in a sample of the video lessons. They based their judgments on a detailed written description of the content that was altered for each lesson to disguise the country of origin (deleting, for example, references to currency). They completed a number of in-depth analyses, the simplest of which involved making global judgments of the quality of each lesson’s content on a three-point scale (Low, Medium, High). (Quality was judged according to several criteria, including the coherence of the mathematical concepts across different parts of the lesson, and the degree to which deductive reasoning was included.) Whereas 39 percent of the Japanese lessons and 28 percent of the German ones received the highest rating, none of the U.S. lessons received the highest rating. Eighty-nine percent of U.S. lessons received the lowest rating, compared with 11 percent of Japanese lessons. [64, p. iv] The teachers in the TIMSS video study knew they were going to be videotaped, so it is reasonable to expect that they “gave it their best.” We also assume that teachers had intentions about what they hoped students learn, so the preponderance of low ratings suggests that whatever the U.S. teachers in this study intended, it evidently did not entail coherent mathematical ideas, students’ generalizations, or student-drawn inferences. That no U. S. lesson’s content received the highest quality rating and that 89% of the U.S. lessons’ content received the lowest quality rating points to an incoherence, among this sample’s U.S. teachers, regarding what students

that a learning objective has been achieved once the teacher has done what was prescribed — regardless of whether students learned what was intended.

DRAFT - 3

Thompson & Saldanha

To understand numbers

should learn. This makes it even more imperative that mathematics educators engage in open discussions of the ideas and modes of reasoning we intend students’ schooling to foster. One major source of personal reform is when teachers realize that what they are teaching does not support what students should be learning. We hope no one interprets us as “teacher bashers”. We are mathematics teacher educators as well as researchers; we work daily with teachers and prospective teachers. We are mathematics teachers ourselves. However, we also have a “public health” perspective on problems of mathematics teaching. A community resolves problems most effectively when it discusses their scope, severity, and sources openly and objectively. A new discourse about what students should understand Our concern for inadequate explications of meanings is not merely a matter of rhetorical preference. Rather, without explication, many readers will trivialize statements about students’ understandings of number and operations or interpret them inappropriately. We feel justified making this statement on the basis of a number of research studies. We cited one result of the TIMSS video study, that 89% of U.S. lessons received the lowest quality rating. We doubt that any of the teachers in this sample would claim that they deliberately chose to teach low-quality content. Moreover, this result is especially powerful because the TIMSS sampling technique was to draw nationally-representative samples from each of its participating countries [64]. Two additional studies are Post, Harel, Behr, and Lesh [53] and Ma [43]. Post et al. [53] gave several versions of a ratio and rate test to 218 intermediate mathematics teachers in Illinois and Minnesota. The test reflected concepts covered in the intermediate (4-6) mathematics curriculum. Teachers scored between the13-year-old NAEP average and the17-year-old average on items drawn from the 1979 NAEP; overall performance among teachers varied widely across test versions, but average performance scores ranged from 60% to 69% across test versions, and more than 20% of the teaches scored less than 50%. An important result in the Post et al. study is teachers’ performance on problems like this: “Melissa bought 0.46 of a pound of wheat flour for which she paid $0.83. How many pounds of DRAFT - 4

Thompson & Saldanha

To understand numbers

flour could she buy for one dollar?” [53, p. 193]. Only 45% of the teachers answered this question correctly and 28% left the page blank or said “I don’t know.” Teachers were also asked to explain their solutions as if to a student in their class. In the case of the “Melissa” problem (above), only 10% of those who answered correctly could give a sensible explanation of their solution. The authors concluded: Our results indicate that a multilevel problem exists. The first and primary one is that fact that many teachers simply do not know enough mathematics. The second is that only a minority of those teachers who are able to solve these problems correctly were able to explain their solutions in a pedagogically acceptable manner. [53, p. 195] Ma [43] compared elementary school teachers’ understandings of mathematical topics they commonly taught. She found that despite the U.S. teachers’ advantage in educational backgrounds3, Chinese teachers were far more likely than U.S. teachers to exhibit richly connected and pedagogically powerful understandings of what they expected students to learn. Thus, both the Post et al. and Ma studies point to the distinct possibility that phrases like “deep understanding” and “with meaning” will not communicate well without careful, detailed explication. Numbers and Operations Before discussing what it means to understand post-counting numbers and operations, we will relate our method of conceptual analysis to techniques in mathematics for constructing number systems formally, and we will discuss integers as a number system from mathematical and psychological perspectives. Mathematical foundations and mathematics education Mathematicians rely heavily on symbol systems to aid their reasoning. Symbol systems are tools for them. Mathematicians strive to develop symbol systems (inscriptions and conventions for

3

Chinese elementary school teachers enter normal school after ninth grade, graduating two to three years later.

DRAFT - 5

Thompson & Saldanha

To understand numbers

using them) that capture essential aspects of their intuitive understandings and means of operating, so that they need not rely explicitly on conceptual imagery and operations as they move their reasoning forward or generate further insight. This has been a hallmark of mathematical activity as long as there has been mathematics. In the 18th and 19th centuries, mathematicians found that their symbol systems, used according to established conventions, led to contradictory results. Many mathematicians saw that the problem was in their intuitions of number, not in the symbol systems they had devised. The idea of “mathematical development of number systems” arose from the need to make commonly held meanings for number more precise. This development was influenced also by historical accounts of the emergence of new numeric understandings, such as eventual acceptance that “nothing” is a numeric amount, that deficits can be measured, and that number systems could be extended even if no one understood what they were.4 Number systems from two perspectives In discussing the development of number systems we will adopt a useful yet completely arbitrary distinction. It is between “mathematical numbers” and “psychological numbers”. This is not a factual distinction. Rather, it is a rhetorical ploy that gives greater ease to speaking of numbers from two points of view. The first is the perspective of mathematicians who work with numbers at advanced levels; the second is from the perspective of psychologists who are interested in cognitive processes that are active when people reason numerically. Mathematical numbers are mathematicians’ attempts to explicate psychological numbers. By “mathematical numbers” we mean number systems as developed within a method for expressing structural characteristics of systems that they understand intuitively but imprecisely. The primary aspect of this method is that of “extension by embedding”. By “psychological numbers” we mean understandings people have, in mature or formative stages, that correspond

4

As was the case for complex numbers, and later with numbers systems like hyperreal numbers which are used in non-standard analysis [30, 65], surreal numbers, and quaternions, which have no known use

DRAFT - 6

Thompson & Saldanha

To understand numbers

roughly to what mathematicians attempted to capture formally. Mathematically, a number system is a set of objects called together with operations defined on elements of that system which satisfy certain properties. Numbers do not exist apart from the operations defined on them. New number systems are defined in terms of already-defined ones. One mathematical motivation for extending a number system to a larger sphere is to define a system that is closed under operations that weren’t closed before. The whole numbers are not closed under subtraction (3 - 7 is not a whole number), integers are closed under subtraction but not under division (7 ÷ 3 is not an integer), rational numbers are closed under division by non-zero divisors but not under root extraction ( 3 5 is not a rational number). Integers According to our rhetorical ploy of distinguishing between mathematical and psychological numbers, we will give a brief overview of integers as developed mathematically and a brief overview of integers as psychological constructs. In doing so we hope to draw a parallel between them to show that the mathematical development does have psychological implications. Integers as mathematical numbers Constructing the integers mathematically from whole numbers means to develop a new number system which has in it a subsystem identical to the whole numbers, which preserves addition and subtraction of whole numbers, and which also extends the whole numbers so that it (the new system) is closed under subtraction. A common technique for extending a number system so that the extended system is closed under an operation is to create pairs from the original system and then partition those pairs into classes. The classes are defined so that every pair in a class would produce the same result under the operation were the old system actually closed. This defines the elements of the new system.

except to challenge prevailing intuitions of number systems [14].

DRAFT - 7

Thompson & Saldanha

To understand numbers

For example, even though 2-5 is undefined in the whole numbers, 2-5, 3-6, 4-7, and etc. all would be the same as 2-5 if it were defined. In each case the subtrahend exceeds the minuend by 3. Each pair of whole numbers in this list represents the result of subtracting a subtrahend that is 3 larger than the minuend. Put another way, an integer is determined by a pair of whole numbers. Because many different pairs have the same difference, whatever an integer is, it can be determined by an infinite number of whole-number pairs. The new system still needs a formal criterion for when two whole-number pairs determine the same difference. Moreover, the criterion cannot use subtraction to define it, because you can’t say “two pairs are equivalent if they have the same difference (result of subtracting)” when we are not guaranteed that either has a difference at all. Figure 1 shows two whole-number pairs — (a,b) and (c,d) — that have the same difference. Since the two pairs have the same difference, we can “rotate” the bottom pair in Figure 1 and fit the two pairs together, as in Figure 2. This gives a rule of equivalence — two pairs of whole numbers (a,b) and (c,d) are equivalent (have the same difference) if and only if (a+d) = (b+c). a b

c d

Figure 1. Two pairs of whole numbers that have the same difference. a b

c d

Figure 2. (a,b) and (c,d) have the same difference if and only if (a + d) = (b + c). Since integers are somewhat like differences of whole numbers (not the actual result of subtracting, but the pair of whole numbers subtracted), adding integers must be like adding DRAFT - 8

Thompson & Saldanha

To understand numbers

differences. To continue the present example, if (a,b) and (c,d) are two whole-number pairs that each represent a difference, then we must define their sum so that this sum represents a difference that is the sum of (a,b)’s and (c,d)’s differences. The pair (a+c,b+d) represents a comparison that has as its difference the sum of (a,b)’s and (c,d)’s differences (Figure 3). a

c b

d

Figure 3. The pair (a+c,b+d) has the sum of differences in (a,b) and (c,d) as its difference. We hasten to point out that in Figures 1-3 the pairs (a,b), (c,d) etc. are not integers. The integer normally represented as “+5” is defined to be the class of whole-number pairs equivalent to (5,0) — the class of pairs in which the first entry exceeds the second by 5. This is represented as [(5,0)]. The integer normally represented as “-3” is defined to be the class of whole-number pairs equivalent to (0,3) — the class of pairs in which the second entry exceeds the first by 3, which is represented as [(0,3)]. That is, any specific comparison of whole numbers is but one instance of a class of equivalent comparisons. This is similar to the relationship between a specific person and the species Homo sapiens. A person is a Homo sapiens, and is equivalent in species to any other member of Homo sapiens, but Homo sapiens is not a specific person. This approach has a number of immediate consequences. First, every integer has an additive opposite. [(a,b)] and [(b,a)] are additive opposites, for the pair (a+b,b+a) has a difference of 0. So, -[(a,b)], the negative of [(a,b)], is defined to be [(b,a)].Put another way, the additive opposite of any integer x is the integer that is comprised by the class of pairs having differences of the same size as those making x, but the comparisons are in the opposite direction. Another immediate consequence of this approach is that subtraction of integers can now be defined in terms of addition of integers, for now the concept of “opposite” is well defined.5

5

We strongly criticize textbooks that define integers as “whole numbers together with their opposites.” The concept of opposite is founded upon the concept of integer, so defining integers in terms of opposites is tantamount to relying on students’ thoroughly understanding what you intend to define for them.

DRAFT - 9

Thompson & Saldanha

To understand numbers

Integers as psychological numbers The prior section provided a brief mathematical development of integers. But it did not address what people understand as they develop a concept of integers that is consistent with the mathematical development. Equivalence classes of whole-number pairs are a convenient definition of integers, but they do not convey what people might have in mind that an equivalence class formalizes. Explicating that requires a conceptual analysis of integers, specifically an analysis of what constitutes a difference of two whole numbers. Psychological integers emerge as people develop general images of quantitative comparisons and changes [8, 29, 75, 78]. We try to capture this in Figure 4 and Figure 5. Figure 4 shows that, early on, children can conceive specific comparisons in specific settings. Figure 5 shows that, later, children can conceive specific comparisons as being gotten from arbitrary quantities whose only salient characteristic is that they differ by the amount in question.

Specific comparison determines a specific excess

Specific comparison determines a specific deficit

Figure 4. Early conceptions of difference.

DRAFT - 10

Thompson & Saldanha

To understand numbers

Generic comparison determines a specific excess

Generic comparison determines a specific deficit

Figure 5. Later conceptions of difference. Vergnaud [78] and Thompson [71, 75] point out the great difficulty students have reasoning relationally about situations typically identified with integer reasoning. They argue that it is students who develop the ability to reason about transformations of quantities who also grasp integers as numbers in themselves. For example, 1. James played two games of marbles. In the first, he won 7 marbles. In the second, he lost 10. What has happened altogether? Upon reading Problem 1 primary and middle-school students commonly think that James has 7 marbles after the first game, and therefore he cannot have lost 10 marbles in the second – he could only have lost 7. That is, they interpret both “7” and “10” as referring to numbers of marbles that James has at different times, and they are all the marbles he has at those times. This is entirely natural, and is a productive basis for learning elementary mathematics [8, 9]. To conceive the entire situation coherently, however, they need to think of “7” and “10” in one of two ways. One way is that the situation is about changes in James’ numbers of marbles. Another is that the situation is about excesses or deficit in his number of marbles at one moment in time when compared with his number of marbles at another moment in time. Either way it becomes evident that, altogether, James lost 3 marbles. Whatever number of marbles he had before the first game, he had 3 fewer after the second. The need to have a different conception of numbers is even more evident in the case of Problem 2, a follow-up to Problem 1.

DRAFT - 11

Thompson & Saldanha

To understand numbers

2. Is it possible that James ended the first game owing 3 marbles? In the case of Problem 2, if we think of James’ initial number of marbles as a whole number of marbles then are unable to understand that he might have started Game 1 in debt. In this case, numerals do not refer to numbers of marbles. They refer to relationships. To end the first game having -3 marbles, James must have begun it having -10 marbles. This points out the need for students to rethink even the basic notion of “having” a certain number of marbles. “Having” -10 marbles makes sense only if we understand “having” as meaning net worth. A net worth of -10 marbles refers to the deficit that results from comparing the number of marbles James possesses (or could possess if he called in all his debts) with the number of marbles he owes other people. Similarly, “having” 10 marbles would not mean to possess 10 marbles. Rather, it would mean that you possess (or could possess if you collect debts owed you) 10 marbles more than you owe. You could actually possess any number of marbles and have a net worth of 10 marbles. Figure 6 emphasizes that, psychologically, whole numbers are not integers. Whole numbers, as psychological numbers, are understood as sizes of collections. Integers, as psychological numbers, are not sizes of collections. They are amounts of excess, deficit, or change determined by a comparison of amounts. The integer “+5” does not refer to five things. Rather, it refers to the result of comparing two quantities of which one has five more than the other, or to changing one amount into another by adding 5 to it. As such, to understand “+5” as an integer, one must assimilate it to a scheme of operations that entails the potential transformation of a class of situations [78]. Put another way, whereas understanding “5” as referring to a number of things means to assimilate it to a scheme of operations that creates a cardinality, understanding “+5” as an integer means to assimilate it to a scheme that creates relationships between pairs of cardinalities. A scheme for integers would allow students to see immediately that “net worth of $5000” could refer to an infinite number of possibilities — such as a person having $1,000,000 in assets and $995,000 in liabilities or a person having $6000 in assets and $1000 in liabilities.

DRAFT - 12

Thompson & Saldanha

To understand numbers Whole numbers: Numbers of things 0 1 2 3 4 5 6 7 8 9

… -9 -8 -7 -6 -5 -4 -3 -2 -1 ± 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 … Integers: Amounts of change or amounts of difference Figure 6. The whole numbers as “embedded” within the integers. A scheme for integers is at the heart of understanding how to balance a checkbook. Each transaction is a transformation of your bank balance, and your bank balance is being tracked at two locations — at your home and at your bank. Sometimes you have entries in your records that the bank doesn’t have in its, which means that you have transformed your balance by those transactions, but the bank has not. You have two avenues to see whether you and the bank agree on what is your balance. You may transform the bank’s balance as if it had received the items it hadn’t, by subtracting from its balance the amounts of uncleared checks and by adding the uncleared deposits. Or, you may “undo” those transactions in your records that are still on their way to the bank, thus adding to your balance the amounts of uncleared checks and by subtracting from your balance any uncleared deposits. In either case the underlying idea is of keeping track of a balance’s transformations, not just of check or deposit amounts. Dep Hayward 48.32 2015 Billiard Co. 22.41

My Checkbook Register

Bank’s Version of My Checkbook Register

Num 2012 2013 2014 Dep

Item Elect Co. Gas co. Albertsons Fullers

Amt -113.24 -94.23 -193.48 +782.82

Bal 2234.81 2140.58 1947.10 2729.92

Num 2012 2013 2014 Dep 2015 Dep 2016

Item Elect Co. Gas co. Albertsons Fullers Billiard Co. Hayward Babysitter

Amt -113.24 -94.23 -193.48 +782.82 -22.41 +48.32 -17.50

Bal 2234.81 2140.58 1947.10 2729.92 2707.51 2755.83 2738.33

2016 Babysitter 17.50

}

Figure 7. A bank balance as recorded in two different registers. Reasoning relationally is in general more difficult than reasoning about the items related. Pallermo [50] noted this in his study of children’s difficulties understanding “less” and “fewer”. Thompson and Dreyfus [75] noted likewise that the 6th-graders in their study repeatedly DRAFT - 13

Thompson & Saldanha

To understand numbers

interpreted descriptions of changes in a turtle’s location as if describing the turtle’s ending location. Vergnaud [78] found in a series of studies that children often confused the states related by a transformation with the transformation itself.6 As noted in the prior section, rules for processing sign-augmented numerals do not make operations on integers. Learning objectives regarding operations on integers must be coherent with learning objectives regarding integers themselves.7 If we expect students to understand integers as amounts of excess or deficit, or as amounts by which quantities change, then any learning objectives for operations on integers must take this into account. As such, expressions like -(-2 - -4), to be sensible to students, cannot be about whole number amounts. They must be about excesses, deficits, or transformations of amounts. Of course, students can attempt to master rules for processing sign-augmented numerals without any understanding of what they are about. But such attempts are short sighted and produce results like those in national evaluations of mathematical performance and international studies of mathematical performance.8 Multiplicative numbers The 1996 National Assessment of Educational Progress [54] gave the following items to 8th and 12th graders. 3. Luis mixed 6 ounces of cherry syrup with 53 ounces of water to make a cherryflavored drink. Martin mixed 5 ounces of the same cherry syrup with 42 ounces of water. Who made the drink with the stronger cherry flavor? Give mathematical evidence to justify your answer. (NAEP M070401)

6

This is very similar to Piaget’s [51] observation that young children see “being ahead” as “being faster”. However, one car being ahead of another is a comparison of their locations relative to some common point, whereas one car being faster than another is a comparison of transformations of their locations in a given amount of time. 7 To some people integers are no more than marks on paper. We exclude them from this remark, intending to include those people whose schemes allow them to see equivalence among comparisons of excess and deficit. 8 The 1996 Nation’s Report Card [54] reported that over 50% of 8th graders in its national sample answered this question incorrectly: The lowest part of the St. Lawrence River is 294 feet below sea level. The top of Mt. Jacques Cartier is 1277 feet above sea level. How many feet higher is the top of Mt. Jacques Cartier than the lowest point of the St. Lawrence river? (NAEP M069301)

DRAFT - 14

Thompson & Saldanha

To understand numbers 1980 Population

4.

1990 Population

Town A Town B = 1000 people

= 1000 people

In 1980 the populations of Towns A and B were 5000 and 6000, respectively. In 1990 the populations of Towns A and B were 8000 and 9000, respectively. Brian claims that from 1980 to 1990 the two towns’ populations grew by the same amount. Use mathematics to explain how Brian might have justified his answer. Darlene claims that from 1980 to 1990 the population of Town A had grown more. Use mathematics to explain how Darlene might have justified her answer. (NAEP M069601 Question 3’s intent was to see the extent to which students could quantify intensity of flavor (higher ratio of cherry juice to water means more intense cherry taste). Question 4’s intent was to see whether students could compare quantities additively (by difference) as well as multiplicatively (by ratio). Compared additively, both towns grew the same amount (1000 people). Compared multiplicatively, Town A’s 1990 population was 8/5 (160%) as large as its 1980 population, whereas Town B’s 1990 population was 9/6 (150%) as large as its 1980 population. While these problems might seem straightforward, they challenged 8th and 12th grade students who took part in the1996 National Assessment of Educational Progress [54]. Less than half the 12th graders gave even partially acceptable answers to Problem 3; about one-fifth of 8th and one-fourth of 12th graders gave partially-correct responses to Problem 4 (Table 1). Though NAEP students’ performance was poor, it was consistent with a long line of studies that examined students’ abilities to reason about ratios and relative amounts [25, 26, 28, 33, 34, 48, 49, 76]. Nevertheless, U.S. students have done poorly on such items even in comparison to other countries [16, 44].

DRAFT - 15

Thompson & Saldanha

To understand numbers

Table 1. Student performances in 1996 NAEP on Questions 3 and 4 (* data not released) Grade 8

12

Problem 3 Correct:.......................... * Partially correct ............. * Incorrect ........................ * Omitted.......................... * Off task.......................... * Correct:.....................23% Partially correct ........26% Incorrect ...................42% Omitted.......................9% Off task.......................1%

Problem 4 Correct: ......................... 1% Partially correct........... 21% Incorrect...................... 60% Omitted ....................... 16% Off task ......................... 2% Correct: ......................... 3% Partially correct........... 24% Incorrect...................... 56% Omitted ....................... 16% Off task ......................... 1%

Explanations of US students’ poor performance on questions like these are complicated. One reason is that the understandings tapped by these questions can “go wrong” at many developmental junctures during students’ schooling. As such, we will focus on describing one path to successful development of schemes that would support success on these and even more sophisticated problems. We will continue the approach used for integers by discussing rational and irrational numbers from mathematical and psychological perspectives. The mathematical treatment will be quite brief. A full treatment is beyond the scope of this chapter. Fractions and more as mathematical numbers (rational and real numbers) The mathematical construction of rational numbers is tremendously general and abstract for the same reasons that the mathematical definition of function is general and abstract. The construction in its present form addresses a history of paradoxes and contradictions, and the present definition is the result of a long line of accommodations to eliminate them. For example, one motivation for the modern development of rational numbers is a spin-off from the period in which the calculus was arithmetized (grounded in the analysis of real-valued functions; see [18, 83]). The original notion of derivative as a ratio of differentials (first-order changes in two quantities’ values) fit with the notion of rate of change as a relationship between two varying quantities. But contradictions that traced back to notions of derivative-as-ratio led d’Lambert, for one, to wonder whether “ dy dx ” should be thought of as merely a symbol that represents one number instead of as a pair of symbols representing a ratio of two numbers [17]. Questions about DRAFT - 16

Thompson & Saldanha

To understand numbers

rates-as-numbers and about continuity of functions threaded themselves into formal constructions of rational and real number systems. Rational numbers were constructed as equivalence classes of integer pairs — a rational number being the thing represented by any of a class of equivalent fractions. Real numbers were constructed as above-bounded classes of rational numbers (by Dedikind’s approach) or as equivalence classes of convergent sequences of rational numbers (by Cauchy’s approach). The method of extension from the integers to a system closed under non-zero division creates the rational number system. The method of extension from the rationals to a system closed under convergence of rational-number sequences creates the real number system.9 We hasten to add two points. First, the mathematical developments of rational and real number systems interconnect so many issues that they are typically not treated until advanced undergraduate or introductory graduate mathematics courses. As such, we have absolutely no idea what school mathematics textbook authors or other writers intend when they say they want middle-school students to “understand” rational and irrational numbers. To us, understanding either number system, as a system, means to understand how it is created, why it was created and what problems were solved by its creation, and the issues are hardly within reach of very bright high school students after studying pre-calculus functional analysis. So, to us, using the terms “rational numbers” and “real numbers” with school students introduces a false rigor and cannot be productive. What are they supposed to understand about numbers when they see a demonstration that

2 cannot be represented as a fraction? This does not prove that

irrational, for it is not an existence proof. It doe not prove that

2 is

2 actually exists. Rather, it

proves that there is no rational number that, squared, gives 2. Even if it were an existence proof (as in, “all roots exist and any root that is not rational is irrational”), the logic of this proof

9

The reason that “convergence of sequences” is an important criterion for closure is that there needs to be a number to approximate that corresponds to what we think we are approximating with ever more precise rational approximations. That is, we are assured, within this new system, that there is a square

DRAFT - 17

Thompson & Saldanha

To understand numbers

implies that complex numbers are irrational. Even more, in order to say that the standard proof of 2 ’s non-rationality proves that

2 is irrational, we must presume that students understand the

real numbers (as rational and non-rational real numbers) in order to introduce irrational numbers as a subset of them. That is, we must presume they already understand what we intend they learn.10 Second, the fact that mathematical developments of rational numbers and real numbers interconnect a myriad of issues at the foundations of mathematics makes a direct mapping between these number systems and psychological numbers impossible. We cannot look to the mathematical development of rational and real numbers for guidance on how to think about students’ thinking and its development. Rather, the mathematical development of these number systems provides strong background constraints on how we envision the end products of successful conceptual development. Fractions and more as psychological numbers Our discussion of psychological numbers in this section will take a somewhat circuitous route from examples of mildly sophisticated reasoning, to discussing factors in its development, to a discussion of more general characteristics of advanced multiplicative reasoning and some relationships with mathematical rational and real numbers. In our discussion we will draw heavily from foundational research by Gerard Vergnaud [79-81], Les Steffe and Ron Tzur [59, 62, 77], Tom Kieren [35, 36, 38], the Rational Number Project [2, 3, 40, 41, 52, 53], Jere Confrey [11, 12] and our own work in quantitative reasoning [55, 69, 71-74] As noted already, Post et al. [53, p. 193] gave this problem to a sample of intermediateschool mathematics teachers: Melissa bought 0.46 of a pound of wheat flour for which she paid

root of 2, even if we know we cannot calculate it exactly. 10 We suspect that most students will think that irrational numbers are relatively rare, since they know of so few. There are infinitely many irrational numbers, and the order of infinity is larger than that of the rationals. One way to imagine the vast difference in numbers is this: If stars in the universe represent rational numbers, then the irrational numbers fill the empty space between them.

DRAFT - 18

Thompson & Saldanha

To understand numbers

$0.83. How many pounds of flour could she buy for one dollar? A standard solution would be to set up an equation, as in

.46 x = , and then solve for x. However, as Post et al. found, setting up .83 1

this equation correctly and understanding what it represents are not the same. A solution to this problem that relies on reasoning instead of on equations might go like this: If .46 lb. costs $.83, then 1/83rd of .46 lb. costs $.01. Thus 100/83 of .46 lb. costs $1.00. So, about 5/4 of.44 lb. costs $1.00, or about .55 lb. costs $1.00.11 A more sophisticated expression of the same reasoning would be $1.00 is 100/83 as large as $.83, so you can buy 100/83 of .46 lb. for $1.00. The latter reasoning, is, in effect, what one would do when solving the equation

.46 x = by multiplying .83 1

each side by 1 and dividing both sides by .83. What conceptual development leads to such reasoning? A variety of sources suggest it is through the development of a scheme of conceptualizations of measurement, multiplication, and division, conceptualizations that build on each other and complement one another. Note that we speak of conceptualizations of measurement, fractions, multiplication, and division. This is not the same as measuring, multiplying, and dividing. Measurement schemes To conceive of a measured quantity is to imagine the measured attribute as segmented [45, 61] or in terms of a coordination of segmented quantities [51, 56, 72]. The idea of ratio is at the heart of measurement. To conceive of an object as measured means to conceive of some attribute of it as segmented, and that segmentation is in comparison to some standard amount of that attribute. A mile, measured in yards, is 1760 yards because the length of one mile is 1760 times as long as the length of one yard. That same distance is 5280 feet because it is 5280 times as long as the length of one foot. It does not mean just that a mile contains 5280 feet, in the sense that a fielded

11

Of course, this is mathematically equivalent to x = 46 ÷ 83, but if students think the point is to go straight to an equation, they probably will not develop the reasoning that will express itself in writing such equations from a basis of understanding what is going on.

DRAFT - 19

Thompson & Saldanha

To understand numbers

soccer team contains 11 players. The distinction is between an additive, part-whole relationship between a set and its elements and a multiplicative comparison in which the unit is held apart from the thing measured. The ratio nature of measurement is trivial to people who have a quantitative scheme of measurement, but it is nontrivial to children who are building one. A conceptualized measure entails an image of a ratio relationship (A is some number of times as large as B) that is invariant across changes in measurement units. For example, Figure 8 illustrates that if m is the measure of quantity B in units of quantity A (i.e., B is m times as large as A), then nm is the measure of quantity B in units of 1/nth of A. Conversely, if q is the measure of quantity B in units of quantity A, then q/n is the measure of B in units of nA. Put another way, the measure of a quantity is m times as large when you dilate the unit by a factor of 1/m, and its measure is 1/mth as large when you dilate the unit by a factor of m. B

B

A A Figure 8. Change of measurement unit entails a constant ratio. A conceptual breakthrough behind the example of Figure 8 is that the quantity of interest and its comparison with the original unit’s magnitude do not change even with a substitution of unit.12 Wilki [84] emphasizes this point by distinguishing between numerical equations and quantity equations. A numerical equation, like W=fd, says how to calculate a particular quantity’s measure and the result depend on the particular units used. Quantity equations explicate a

12

The first author observed a 4th-grade lesson on the metric system in which children measured their heights. He asked one boy who had measured his height with a yard stick what he got. “140 centimeters.” How tall are you? “Four feet seven.” Do you know how many centimeters make 4 feet 7 inches? (Pause.) “No.”

DRAFT - 20

Thompson & Saldanha

To understand numbers

quantity’s construction. The equation [W] = [f][d] says that accomplished work, as a quantity, is created by applying a force through some distance, regardless of your measurement units, and that the magnitudes remain invariant regardless of unit substitution. We saw this distinction, between numerical and quantitative formulas, in a 5th-grade teaching experiment on area and volume. The first author presented the question in Figure 9 during a follow-up interview. Portions of two children’s interviews follow the diagram.

17 in

2

Figure 9. What is the volume of this box? PT: (Discusses with BJ how the diagram represents a hollow box and what each number in the diagram represents about it.) BJ: I don’t know. There’s not enough information. PT: What information do you need? BJ: I need to know how long the other sides are. PT: What would you do if you knew those numbers? BJ: Multiply them. PT: Any idea what you would get when you multiply them? BJ: No. It would depend on the numbers. PT: Does 17 have anything to do with these numbers? BJ: No. It’s just the area of that face. … PT: (Discusses with JA how the diagram represents a hollow box and what each number in the diagram represents about it.) JA: Oh. Somebody’s already done part of it for us. PT: What do you mean? JA: All we have to do now is multiply 17 and 6. PT: Some children think that you have to know the other two dimensions before you can answer this question. Do you need to know them. JA: No, not really. PT: What would you do if you knew them? JA: I’d just multiply them. PT: What would you get when you multiplied them? JA: 17. To BJ, the formula V=LWD was a numerical formula. It told him what to do with numbers once he had them. However, it had no relation to evaluating quantities’ magnitudes. To DRAFT - 21

Thompson & Saldanha

To understand numbers

the second child, JA, the formula V=LWD was a quantity formula. To him, it was V = (LW)D, where (LW) produced a measure of a face’s area, and (LW)D produced a volume. JA recognized that being provided one face’s area was as if “somebody’s done part of it for us,” that part being the quantification of one face’s area. Proportional reasoning is important in conceptualizing measured quantities. Vergnaud [79, 80] emphasized this when he placed single and multiple proportions at the foundation of what he called the multiplicative conceptual field.13 A single proportion is a relationship between two quantities such that if you increase the size of one by a factor a, then the other’s measure must increases by the same factor in order to maintain the relationship. More formally, if x is the measure of one quantity, and if f (x) is the measure of the other, then f (ax) = af ( x). A double proportion is a relationship among three quantities, one thought of as a function of the other two, in which the dependent quantity is proportional to each of the others. More formally, if x and y are measures of two quantities, and if f(x,y) is a third quantity’s measure, then: f (ax,by) = af (x,by) = bf (ax, y) = abf (x,y). The concept of torque concerns measuring the twisting force produced by applying a force at some distance from a resistant hinge or fulcrum. It provides one example of how being able to conceptualize a double proportion provides a way to think of quantifying a new quantity.

Hinge, or fulcrum

Figure 10. Torque as amount of twisting force Torque, in its simplest form, can be thought of as “amount of twisting force”, as in “twist

13

Confrey [11] takes a counter position, claiming that multiplicative reasoning emanates from a primitive operation called splitting, which does not emanate from proportionality but instead underlies it. We cannot here discuss the two in contrast, as that would take us astray of our present purpose. We will say only that we agree with Steffe [63] that when we consider the developmental origins of splitting we see that

DRAFT - 22

Thompson & Saldanha

To understand numbers

my arm.” It is well known that physics students have difficulty understanding its quantification. However, as a multiplicative concept its quantification can be thought of as this: (T1) Suppose that applying a force of y force-units (f-units) at a distance of x distance units (d-units) from a fulcrum produces T twist units. (T2) Increasing the applied force by a factor of a while applying it at the same distance increases the twisting force by a factor of a. That is, applying a force of ay f-units at a distance of x d-units produces a twisting force that measures aT twist units (Figure 11).

Hinge, or fulcrum

Hinge, or fulcrum

x d-units

x d-units

Figure 11. Apply a force a times as great at the same distance produces aT twist units (T3) Increasing the distance at which the original force is applied by a factor of b increases the amount of twist by a factor of b. That is, applying a force of y f-units at a distance of bx d-units produces a twisting force of measure bT twist units (Figure 12).

Hinge, or fulcrum

Hinge, or fulcrum

x d-units

bx d-units

Figure 12. Apply a force at b times the distance from the hinge produces bT twist units. (T4) Combining (T2) and (T3), we get that applying a force of ay f-units at a distance of bx d-units produces a twisting force of measure abT twist units.14 If we adopt the convention that a force of 1 f-unit applied at a distance of 1 d-unit produces 1 twist unit, then a force of u ⋅1 funits applied at a distance of v ⋅1 d-units produces u ⋅ v twist units. This is the conceptual

Confrey and Vergnaud are not in opposition. 14 We stress once again that it is this step, combining T2 and T3, that is enabled by the ability to conceive

DRAFT - 23

Thompson & Saldanha

To understand numbers

foundation for quantifying torque by the quantity formula “force times distance”. Multiplication schemes Contrary to most textbooks, and contrary to Fishbein’s [19] well-known treatise, multiplication is not the same as repeated addition. To be fair, we should say conceptualized multiplication is not the same as repeated addition. The conceptual foundation of multiplication of whole numbers is quite like the Biblical “multitudes” – creating many from one. That is to say, multiplication of whole number is the systematic creation of units of units. At an intuitive level this is repeated addition; at an operational level it could be either a transformation brought about by composing [the mental operation of extending one number by another] with itself a number of times …, or, equivalently, as a hierarchy of integrations. [68, p. 316] The difference between conceptualized multiplication and repeated addition is between envisioning the result of having multiplied and determining that result’s value. Envisioning the result of having multiplied is to anticipate a multiplicity. One may engage in repeated addition to evaluate the result of multiplying, but envisioning adding some amount repeatedly cannot support conceptualizations of multiplication. Suppose we have three quantities, one of whose measure we think of as being a function of the other two. By convention the standard unit of the third, dependent, quantity is defined as that amount made by one unit of the independent quantities. That is, suppose x and y are the independent quantities’ measures, and suppose f(x,y) is the third quantity’s measure expressed as a function of the of the two. Then the third quantity’s standard unit is meas(StandardUnit) = f (1,1) = 1. This says that if a double proportion relates the three quantities, then the third quantity’s measure will be the product of the two other’s measures. We can express this formally as:

a double proportion.

DRAFT - 24

Thompson & Saldanha

To understand numbers f (x, y) = f (x ⋅1, y ⋅1) = x ⋅ y ⋅ f (1,1)

= x ⋅ y ⋅meas(StandardUnit) = x ⋅ y ⋅1 = x ⋅ y. The very first step — the understanding reflected in the statement that f(x,y) = f (x ⋅1,y⋅1) — is the key to moving from multiple proportion to multiplication as the operation to evaluate a quantity’s measure. This step reflects the cognitive move from thinking of the values of x and y as numbers of things to thinking them as entailing a ratio comparison between two quantities’ measures.15 x inches is a number of inches. (x ⋅1) inches specifies that the thing having measure x inches is x times as large as the same kind of thing having measure 1 inch. This highlights the move from thinking of a length’s measure as a collection of inches to thinking of its measure as a ratio comparison between the attribute in question and a standard length (Figure 13.). 12 little lengths, called “inches” versus The total length of this

The length of this

Figure 13. Measure as a number of things vs. measure as a ratio comparison. 16

15

A ratio comparison between two quantities is to compare them in terms of relative size — the size of one is so many times as big as the size of the other. 16 The difference between 12 inches as a collection of inches and 12 inches as a measure that is 12 times as big as one inch is the difference between seeing a collection additively and seeing it multiplicatively. Seen additively, comparing 1 inch to the collection entails removing that inch and noticing what is left — 11 inches. Seen multiplicatively, comparing 1 inch to the collection entails removing a copy of 1 inch. The collection still contains as many inches as before, and yet you’ve pulled out one of its members to compare against the group.

DRAFT - 25

Thompson & Saldanha

To understand numbers

This last observation, that understanding objects as entailing a single or multiple proportion relies on understanding measures as reflecting a ratio comparison, also highlights the special way in which multiplication must be conceived -- as entailing a multiple proportion. Generally, most students’ understanding of multiplication does not support proportionality. As directed by a great deal of curriculum and teaching, they understand multiplication as a process of repeated addition. But an extensive research literature has documented how quickly this conception becomes limiting and problematic [15, 19, 24, 27, 42]. Fischbein [19] and Burns (get ref) propose repeated addition as the fundamental initial idea of multiplication. Because of children’s prior numerical and quantitative experience (which is fundamentally additive), this conceptualization makes multiplication easy to teach and (in this limited calculational sense) easy to learn. But thinking about multiplication as repeated addition leads to severe difficulties in later grades. Specifically, if 5 × 4 means 4+4+4+4+4 -- “add 4 five times”, then it is difficult to say what 5 32 × 4 means. It can’t mean “add 4 five and two-thirds times”. But we need not start where Fischbein, Burns, and most textbooks start. If we start with the basic meaning of 5 × 4 as “five fours”, then 5 32 × 4 means “(five and two-thirds) fours, or five fours and 2/3 of four”. The principle difference between “add 4 five times” and “five fours” is that the former tells us a calculation to perform while the latter suggests something to imagine.17 In exactly the same way, when we read 5 32 × 4 35 we can understand that we are being

Another way to distinguish between the two is to characterize the nature of question being answered by the comparison. A collection seen additively entails questions about how much more or less this collection has in comparison to another. A collection seen multiplicatively entails questions about the measure of the collection in units of a subcollection. 17 This example carries a larger significance. It is that how students read mathematical statements (i.e., the meanings attached to their verbal decoding) can have a large impact on the connections they can eventually form. If they read “5 times 4” as a command to calculate, then “5 times x” is highly problematic. But if they read “5 times 4” as “five fours”, this supports the image that we are speaking of a product — (5 times 4) is a number that is 5 times as large as four. “5 times 4” becomes a noun phrase — it points to something. Thus, understanding “5 times x” as “a number, and it is a number that is 5 times as large as the number x” can be a natural extension of their understanding of numerical multiplication, if only their

DRAFT - 26

Thompson & Saldanha

To understand numbers

asked to think about some number of (four and three-fifths). We are being asked to think of 5 (four and three-fifths) and 2/3 of (four and three-fifths). ADD A PARAGRAPH OR TWO TO DO TWO THINGS: CONNECT THIS IDEA OF “SOME NUMBER OF SOME THINGS” TO STEFFE’S INTERIORIZED NUMBER SCHEMES, AND THEREBY SEPARATE THIS FROM CONFREY’S NOTION OF SPLITTING. SECOND, TALK ABOUT THE ROLE OF IMAGERY AND OPERATIONS, THEIR DEVELOPMENT, AND THE FACT THAT THIS CAPABILITY EMERGES THROUGH CHILDREN’S REFLECTION ON ITERATIVE ACTIVITY. It is important to understand that when we start with the idea that the expression “__×__” means “some number of (or fractions of) some amount,” we are not starting with the idea that “times” means to calculate. We are starting with the idea that “times” means to envision something in a particular way -- to think of copies (or parts of copies) of some amount. This is not to suggest that multiplication should not be about calculating. Rather, calculating is just one thing one might do when thinking of a product. Non-calculational ways to think of products will be important in comprehending situations in which multiplicative calculations might be useful. The comprehension will enable students to decide on appropriate calculations. When people write “5 × 4 = __ ” they are suggesting that five fours make some number of ones, and by convention others understand them as a request to fill in the blank with an appropriate number. They are asking, “How many ones do we have if we have 5 fours?” The question, “How much does that make in all?” requests a calculation or some other determination of total amount. Though we will discuss fraction schemes in a later section, it is worthwhile to point out here that to understand multiplication multiplicatively is also to understand fractions as entailing a proportion. The phrase “quantity X is 1/n of a quantity Y” means that Y is n times as large as X.

understanding of numerical multiplication is appropriately general in the first place.

DRAFT - 27

Thompson & Saldanha

To understand numbers

It does not mean merely that X is one out of n parts of Y.18 Thinking of 1/n as “one out of n parts” is to think of fractions additively -- that Y is cut into two parts, one being X and the other being the rest. It is merely a way to indicate one part of a collection. When students’ encompassing image of fractions is “so many out of so many”, it possesses a sense of inclusion — that the first “so many” must be included in the other “so many”. As a result, they will not accept the idea that we can speak of one quantity’s size as being a fraction of another’s size when they have nothing physically in common. They will accept “The number of boys is what fraction of the number of children?”, but they will be puzzled by “The number of boys is what fraction of the number of girls?” To think of multiplication as producing some number of some amount, and to think at the same time of the product (what you make by taking some number of some amount) in relation to the individual factors, entails proportional reasoning. To understand 5 × 4 multiplicatively, as 5 fours, children need to understand that 4 in 5 × 4 is not just 4 ones, as in 20 = 4 + 9 + 7. Rather, this 4 is special — it is 1/5 of the product.19 In general, when children understand multiplication mulitiplicatively, they understand the product (nm) being in a special relationship to n and to m: m is 1/n as large as (nm) and n is 1/m as large as (nm). The use of “ugly numbers” sometimes helps clarify the idea that multiplication entails a

18

One could object that nothing is gained by defining other people’s meanings out of existence, even if they are somehow flawed. This poses an interesting question — to what extent do we want to respect meanings that don't work and end up hurting children if a particular meaning becomes what they *really* mean. We understand that a person can say "one out of n,” really mean it additively at the moment of saying it (like, one of those six boys has a blue shirt), and yet flip effortlessly and unawarely to "the number of boys over there is six times as large as the number of boys with blue shirts". Our objection is not to the use of the words "one out of n". Our objection is that mathematics education’s past mistake was to think that the former automatically conveys the latter. What we want to say is that it takes an incredible, systematic, programmatic effort over multiple years to ensure that the majority of children internalize the multiplicative point of view, so that it becomes "the way they see things." It is so important that we are willing to say, "Blind adherence to the phrase 'one out of n' as conveying the meaning of "1/n" allows incredible problems to build undetected. It is so problematic that we need to mount a campaign advocating that teachers and mathematics educators express themselves consistently so that there is no question what they mean." 19 We are indebted to Jose Cortina for this observation.

DRAFT - 28

Thompson & Saldanha

To understand numbers

proportional relationship: In 4 73 × 7 23 (read as [four and three-sevenths] seven-and-two-thirds), 1 1 of the product. Now, what does “ 3 “ mean? Recall what we said it 3 47 47

7 23 is special. It is

means for a to be 1/n of b. The statement “a is one-nth of b” means that b is n times as large as a. Let a=7 23 , b= (4 73 × 7 23 ) , and n=4 73 . Then, to say that 7 23 is

(4 73 × 7 23 ) is 4 73

1 of (4 73 × 7 23 ) means that 4 73

times as large as 7 23 (see Figure 14).

7 23

7 23

7 23

7 23

7 23

Figure 14. (Four and three-sevenths) seven and two-thirds. One shaded region ( 7 32 ) is

1 of the total, because the total is 4 73 times as large as 7 32 .20 3 47

Division schemes A long list of research studies point to students’ difficulties in conceptualizing division and its relationships with multiplication and fractions. Greer [22, 23] and Harel [27] noted the great frequency with which students will decide upon the use of multiplication or division depending on the size or type of numbers involved, not on the underlying situation. Graeber [20] found the same to be true among pre-service elementary teachers. Ball [1] discovered that even college mathematics majors had difficulty proposing situations in which it would be appropriate to divide by a fraction. Simon [58] found that pre-service elementary teachers’ knowledge of division was tightly bound up with their whole-number schemes, meaning, which resulted in weak connections among various meanings of division and in their inability to identify the unit of the result. In short, it is unquestionable that division is a problematic area of mathematics teaching and learning.

20

Of course, the argument is symmetric in regard to both numbers. But to understand the symmetry

DRAFT - 29

Thompson & Saldanha

To understand numbers

The fact that division is a problematic area does not clarify what is problematic or what constitutes an understanding of it. In addressing this issue we will begin with a widely used distinction between two “types” of division settings (sharing and partitioning), and demonstrate that , among other things, operational understanding of division entails a conceptual isomorphism between them. Sharing is the action of distributing an amount of something among a number of recipients so that each recipient receives the same amount.21 Partitioning is the action of cutting up an amount in units of some standard amount. “13 people will share 39 pizza’s. How many pizza’s will each person receive?” usually is classified as a sharing situation, because of the request to distribute the pizzas into a pre-established number of groups. “You are to give 3 pizzas to each person, and you have 39 pizzas. How many people can you serve?” usually is classified as a partitioning situation, because of the request to partition the number of pizzas into preestablished group sizes. It is well known that students often see these situations as being very different [4, 5, 22, 24], yet both can be resolved by the numerical operation of division.22 How is it possible that the results of sharing and partitioning are evaluated by the same numerical operation? To most students (and many teachers) this is a mystery, or the issue is not raised at all. To see why the numerical operation is the same in either situation entails the development of operative imagery — the ability to envision the result of acting prior to acting — and to suppress attention to the process by which one obtains those results. Suppose we have some number of candy bars. Consider this as one mass of chocolate. •

The process of sharing ends up with each recipient having the same amount of chocolate. So,

presumes an understanding of commutativity. 21 We will speak as if all sharing situations are equi-sharing, and that all partitioning situations are equipartitioning. 22 It is also well known, by these same studies, that if you change the numbers so that in either situation each person receives 2/3 pizza, students will see these problems even differently yet. We agree with Confrey [11] that this result is entirely an artifact of persistent experience with stereotypical problems in which products are always larger than either factor and quotients are always smaller than either minuend or divisor.

DRAFT - 30

Thompson & Saldanha

To understand numbers

if we put the chocolate into 7 pieces, each piece contains 1/7 of the chocolate. So, the number of candy bars in each piece is 1/7 as large as the number of candy bars. •

The process of partitioning cuts up the mass of chocolate into a number of pieces of a given size (perhaps with an additional piece that is a fraction of the given size). Suppose we cut up this mass into pieces, each piece the size of 7 candy bars, and consider any remaining bars to make some fraction of a piece (i.e., of 7 candy bars). We can determine the number of pieces by removing one candy bar from each piece and a proportionate part of a candy bar from any fraction of a piece. Each piece (or fraction of a piece) is 7 times as large as what we remove from it, so the total is 7 times as large as what we remove in all. The mass of chocolate is 7 times as large as what we remove, what we removed counts the number of pieces, therefore the number of pieces is 1/7 as large as the number of candy bars. The two analyses above show that, in either case, the result of sharing or the result of

partitioning produces a number that is 1/7 as large as the number of candy bars. The above analysis also reveals that fractional reasoning and proportional reasoning are integral to understanding that partitioning and sharing are equivalent. Finally, students can see that measuring and partitioning, and therefore measurement and division, are equivalent even though at the level of activity they appear to be very different. When they understand the numerical equivalence of measuring and partitioning they understand that any measure of a quantity induces a partition of it and that any partition of a quantity induces a measure of it. Fraction schemes A common approach to teaching fractions is to have students consider a collection of objects, some of which are distinct from the rest, as depicted in Figure 15. But, what might Figure 15 illustrate to students? Three circles out of five? If so, they see a part and a whole, but not a fraction. Three-fifths of one? Perhaps. But depending on how they think of the circles and collections, they could also see three-fifths of five, five-thirds of one, five-thirds of three, or fivethirds of three-fifths (see Figure 16). DRAFT - 31

Thompson & Saldanha

To understand numbers

Figure 15. What does this collection illustrate? They could also see Figure 15 as illustrating that 1 ÷ 3 5 = 12 3 —that within one whole there is one three-fifths and two-thirds of another three-fifths, or that 5 ÷ 3 = 1 2 3 —that within 5 is one 3 and two-thirds of another 3. Finally, they could see Figure 15 as illustrating 5

3

× 3 5 = 1—that five-thirds of (three-fifths of 1) is 1.

If we see as one collection, then is one-fifth of one, so is three-fifths of one. If we see as one collection, then is one-third of one, so is five-thirds of one. If we see as one circle, then circles, so is one-fifth of five, and three-fifths of five.

is five is

If we see as one circle, then is three circles, so is one-third of three, and is five-thirds of three. Figure 16. Various ways to think about the circles and collections in Figure 15. Figure 15 is customarily offered by texts and by teachers to illustrate 3 5 as three out of five. We rarely find texts or teachers discussing the difference between thinking of 3 5 as “three out of five” and thinking of it as “three one-fifths.” How a student understands Figure 15 in relation to the fraction 3 5 can have tremendous consequences. When students think of fractions as “so many out of so many” they are puzzled by fractions like 6 5 . How do you take six things

DRAFT - 32

Thompson & Saldanha

To understand numbers

out of five?23 Kieren [35, 37, 38] and the Rational Number Project [2, 3, 40] have given the most extensive analyses of rational number meanings. Their approach was to break the concept of rational number into subconstructs,—part-whole, quotient, ratio number, operator, and measure—and then describe rational number as an integration of those subconstructs. Our feeling is that their attempt to map their systems of complementary meanings into the formal mathematical system of rational numbers will necessarily fall short. As we already noted, many of the mathematical motivations for developing the rational numbers as a mathematical system did not emerge from meanings of subconstructs. Rather, it emerged from the larger endeavor of arithmetizing the calculus (see prior discussion). Moreover, our analysis places the idea of psychological rational numbers squarely within multiplicative reasoning as a core set of conceptual operations, and prior analyses were less focused on conceptual operations per se. A fraction scheme is based on conceiving a reciprocal relationship between two quantities in terms of relative size. Amount A is 1/n the size of amount B means that amount B is n times as large as amount A. Amount A being n times as large as amount B means that amount B is 1/n the size of amount A. We worded these sentences very carefully, avoiding saying “1/n of”. This is to emphasize the slippery connection, already mentioned earlier, between comparison and inclusion. Fraction sophisticates can say “A is some fraction of B” and get away with it because they imagine that A and B are separate amounts (see Figure 13). Children are often instructed, and therefore learn, that the fractional part is contained within the whole, so “A is some fraction of B” connotes a sense of inclusion to them, that A is a subset of B. As a result statements like “A is 6/5 of B” makes no sense to them.

23

We cannot count the number of times teachers, in-service and pre-service, have said “change

and they’ll understand.” This misses the point that it is problematic if we must change 65 it to means that students cannot understand fractions as entailing a proportional relationship.

DRAFT - 33

6

5

to

1 51 , for it

1 51

Thompson & Saldanha

To understand numbers

To say “A is m n as large as B” means that A is m times as large as 1 n of B. So, “A is 6/5 as large as B” does not mean that there are six of something in A. Rather, it means that A is 6 times as large as (1/5 of B). Our conceptual analysis of fraction knowledge resembles the approach taken by Lesh, Behr, Post, and Harel [2, 3, 41]. They characterized m/n of B as m one-nths of B. The difference between their approach and ours is that we point out the need for a direct link to students’ conceptualizing reciprocal relationships of relative size. Without our insistence on students having that link they could appear to be speaking of fractions in ways we intend while in fact be thinking of additive inclusion, as we described earlier. Students gain considerable mathematical power by coming to understand fractions through a scheme of operations that express themselves in reciprocal relationships of relative size. For example, “related rate” problems are notoriously difficult for U.S. algebra students. It takes Jack 20 seconds to run as far as Bill does in 15 seconds. Bill’s top speed is 18 ft/sec. About how fast does Jack run? One way to reason through this using basic knowledge of fractions is as below. The chain of reasoning is based on Figure 17. In it we see that Jack and Bill ran the same distance but in different amounts of time. It is important to note that 5 seconds is 1/4 of Jack’s time and 1/3 of Bill’s time.

20 secs 15 secs

1/4 of Jack’s time

1/3 of Bill’s time

Figure 17. Jack and Bill took different amounts of time to run the same distance. •

Bill runs 1/3 of the distance in 1/3 of his time.



Jack runs 1/4 of the distance in 1/4 of his time. DRAFT - 34

Thompson & Saldanha

To understand numbers



1/3 of Bill’s time is as large as 1/4 of Jack’s time.



If Bill were to continue running another 1/4 of Jack’s time, he and Jack will have run for the same amount of time.



If Bill were to continue running another 1/4 of Jack’s time, he would run an additional 1/3 of the distance



Jack runs 3/4 as far as Bill in the same amount of time



Therefore, Jack runs 3/4 as fast as Bill. So he runs at about 13.5 ft/sec.24

Fractions as reciprocal relationships of relative size also draw on relationships among measure, multiplication, and division. For example, suppose we interpret “a ÷ b” as “the number of b’s in a”. Then whatever number is a ÷ m n (i.e., whatever number is the number of m n in a) it is n times as large as a ÷ m, because m n is 1 n as large as m. (see the discussion following Figure 8). When we recall that a÷m is 1 m as large as a (see discussion of equivalence between partitioning and sharing), we can conclude that whatever number is a ÷ m n , it is n times as large as 1 m of a. Put more briefly, a ÷ m n = n m ⋅ a . “The number of m n ’s in a is n times a large as

1

m of a.” This is a conceptual derivation of the “invert and multiply” rule for division by

fractions, and its interpretation is straightforward when thinking in terms of relative sizes. Similarly, we can use fraction relationships to reason about algebraic statements like x = y/32. “x=y/32 means that x is 1/32 as large as y. That means that y is 32 times as large as x. So, y=32x. The evolution of students’ understandings of reciprocal relationships of relative size is still being researched, especially by Steffe and his colleagues [32, 60, 62, 63, 77]. The fact that

24

In the past, some people have understood examples like this to imply that we advocate solving all problems non-symbolically. Nothing could be further from the truth. What we advocate is that rules and shortcuts for operating symbolically should be generalizations from conceptual operations instead of being taught in place of conceptual operations.

DRAFT - 35

Thompson & Saldanha

To understand numbers

this understanding happens so rarely among U.S. students makes it quite hard to research its development. But the fact that these understandings happen so rarely is a significant problem for U.S. mathematics education, for there is some evidence that it is expected more routinely elsewhere [16, 43, 44]. It should be a dominant topic of discussion that U.S. instruction fails to support its development, and the reasons for that failure should be understood in detail. At present, we can say little more than “it doesn’t happen because few teachers and teacher educators expect it to happen.” The matter of students’ and teachers’ inattention to what numbers are and what it means to operate on them is reminiscent of a discussion between Samuel Cutler [39] and John Conway [13] in which Kutler wondered about Conway and Guy’s [13]’s treatment of fractions. Kutler asked: … they [Conway and Guy] give this example: 2/3 x 1/4 = 4/6 x 1/4 = 1/6. What is the logic of this? Do [they] think that the readers have learned that the definition of the product of a/b and c/d is ac/bd and that they have forgotten it? Do they think that the reader knows or will investigate the justification for compounding the ratios of numbers, or what? [39] Conway responded: Guy and Conway DO think, unfortunately, that readers have learned that that is the definition of the product, which it really isn't, except in formal axiomatic contexts. Suppose one stick is three-quarters as long as another, whose length is two-and-a-half inches. Then do you really think that the reason the shorter stick's length is 15/8 inches is because the DEFINITION of a/b times c/d is ac/bd? It ISN'T! It's because what actually happens is that if you cut the longer stick into 4 equal quarters and throw one of them away, the total length of what's left will actually be 15/8 inches. [13], Kutler spoke from a presumed image that the result of multiplying fractions is determined by definition, by an algorithm. Conway pointed out that the result of multiplying fractions is determined by what multiplication of fractions means. In his example, 3 4 × 2 12 refers to a length that is 3 times as long as is 1 4 of 2 21 . That is the way he turned the multiplication expression into a thing to which it referred. It happens that the result is 15/8 not by definition, but by

DRAFT - 36

Thompson & Saldanha

To understand numbers

coincidence. The rule “a/b × c/d = ac/bd” is a generalization that derives from the meanings of multiplication and of fractions in conjunction with a particular notational system for expressing those meanings. We agree with Conway that it is misguided to portray multiplication of fractions as being defined by a generalization, and we support the larger implication that it is misguided to portray numbers and operations as being defined by what could be (under a more enlightened pedagogy) generalizations from fundamental meanings. Multiplicative numbers — a synthesis The title of this section does not point to what we will do in it. We will not synthesize what we’ve said about measurement, multiplication, division, and fraction. Rather, it points to what multiplicative numbers are. Multiplicative numbers are a synthesis. We do not mean this in a realist sense. We mean it in a cognitive sense. Multiplicative numbers become “real” when people understand them through complementary schemes of conceptual operations that are each grounded in a deep understanding of proportionality. At the same time, a focus on developing these schemes will enrich students understanding of proportionality. Most evidence points to the firm conclusion that it is rare for multiplicative numbers to be real to U.S. students. The extent to which other countries’ educational systems and broader culture support students’ creation of that reality is an open question, but an important one. Nevertheless, our system would be improved were it to support students’ creation of multiplicative numbers. We need not compare ourselves to other countries in setting goals for ourselves. Conclusion We have attempted to make explicit one image of what “to understand” means in regard to numbers and numeric operations beyond counting and whole numbers. In doing so we described conceptual operations that might underpin those understandings. As we said in an opening paragraph, we do not consider our work in this chapter to be definitive. Rather, we offer it as a “sacrificial example” that, we hope, contributes to focusing scholarly discussion on what we DRAFT - 37

Thompson & Saldanha

To understand numbers

intend students learn from instruction. We have not discussed issues of symbolization, representation, symbolic skill, problem solving, and so on. The issues we’ve raised are largely independent of symbolism and symbolic procedure. Conceptual operations are about what one sees, not what one does, and about reasoning that is grounded in meaning. That is not to say that “doing” is unimportant. Rather, it addresses the common-sense issue that you should conceive of what you are trying to accomplish before trying to do it. As we noted in Footnote 24, rules and shortcuts for operating symbolically should be generalizations from conceptual operations instead of being taught in place of conceptual operations. Once done, symbolic operations can become the focus of instruction. But as a number of people have stressed, it is important that students have a rich web of meanings to fall back on when they become confused or cannot recall a remembered procedure[6, 10, 21, 67, 70, 82]. There is a practical drawback to a scheme-based characterization of learning objectives. It is that assessment of whether students have achieved a learning objective is more complicated when expressing objectives in terms of schemes of conceptual operations than when expressing them in terms of behavioral skills. When learning objectives are stated in terms of skills, determining whether a student has achieved the objective is straightforward. When learning objectives are stated as schemes of operations, students’ behavior must be interpreted to decide whether it reflects reasoning that is consistent with the objectives’ development. This complication is unavoidable. However, coming to see reasoning in students’ behavior is part of coming to understand the scheme-based approach we’ve used, so attempts at making teachers sensitive to students’ reasoning may have the side-effect of preparing them to think of learning objectives in terms of conceptual operations. In closing, we re-emphasize our motive for this chapter. We believe that conditions for improved instruction entail an enduring discussion of what the community intends students learn. When teachers and teacher educators lack clarity and conceptual coherence in what they intend students learn, they introduce a systematic disconnection between instruction and learning. This DRAFT - 38

Thompson & Saldanha

To understand numbers

becomes is a major obstacle to creating instruction that empowers students to think mathematically. Note that we said that the problem is a lack of clarity and coherence in what teachers and teacher educators intend that students learn; we did not say that there is any one correct intention. Nevertheless, not all intentions are equally coherent or equally powerful. We hope this chapter helps initiate prolonged discussions of what those intentions should be.

DRAFT - 39

Thompson & Saldanha

To understand numbers

References 1.

D. L. Ball, Prospective elementary and secondary teachers’ understanding of division, Journal for Research in Mathematics Education 21 (1990) 132–144.

2.

M. Behr, G. Harel, et al., Rational number, ratio, and proportion, in Handbook for research on mathematics teaching and learning, D. Grouws, Editor, Macmillan, New York, 1992, p. 296–333.

3.

M. J. Behr, G. Harel, et al., Rational numbers: Toward a semantic analysis -- Emphasis on the operator construct, in Rational numbers: An integration of research, T. P. Carpenter, E. Fennema, and T. A. Romberg, Editor, Erlbaum, Hillsdale, NJ, 1993, p. 49–84.

4.

A. Bell, E. Fischbein, and B. Greer, Choice of operation in verbal arithmetic problems: The effects of number size, problem structure and context, Educational Studies in Mathematics 15 (1984) 129–147.

5.

A. Bell, M. Swan, and G. Taylor, Choice of operations in verbal problems with decimal numbers, Educational Studies in Mathematics 12 (1981) 399–420.

6.

W. Brownell, Psychological considerations in the learning and teaching of arithmetic, in The teaching of arithmetic (Tenth yearbook of the National Council of Teachers of Mathematics), Bureau of Publications, Teachers College, New York, 1935, p. 1–31.

7.

T. P. Carpenter, M. Lindquist, et al., Results of the third NAEP mathematics assessment: Secondary school, Mathematics Teacher 76(9) (1983) 652–659.

8.

T. P. Carpenter and J. M. Moser, The acquisition of addition and subtraction concepts, in Acquisition of mathematics concepts and processes, R. Lesh and M. Landau, Editor, Academic Press, New York, 1983,

9.

T. P. Carpenter and J. M. Moser, The acquisition of addition and subtraction concepts in grades one through three, Journal for Research in Mathematics Education 15(3) (1984) .

10. P. Cobb, K. P. E. Gravemeijer, et al., The emergence of chains of signification in one firstgrade classroom, in Situated cognition theory: Social, semiotic, and neurological perspectives, D. Kirshner and J. A. Whitson, Editor, Erlbaum, Hillsdale, NJ, 1997, p. 151233. 11. J. Confrey, Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions, in The development of multiplicative reasoning in the learning of mathematics, G. Harel and J. Confrey, Editor, SUNY Press, Albany, NY, 1994, p. 293–330. 12. J. Confrey and E. Smith, Splitting, covariation and their role in the development of exponential function, Journal for Research in Mathematics Education 26(1) (1995) 66–86. 13. J. H. Conway. (1998, September 17, 1998). Further fruitfullness of fractions, [Threaded discussion]. Historica Mathematica Discussion Group. Available: http://forum.swarthmore.edu/epigone/historia_matematica/smanprehdol/199809180107.WA [email protected] [1999, December 1]. DRAFT - 40

Thompson & Saldanha

To understand numbers

14. J. H. Conway and R. K. Guy, The book of numbers. Springer Verlag, New York, 1997. 15. E. de Corte, L. Verschaffel, and V. Van Coillie, Influence of number size, problem structure and response mode on children’s solutions of multiplication word problems, Journal of Mathematical Behavior 7 (1988) 197–216. 16. J. A. Dossey, L. Peak, and D. Nelson, Essential Skills in Mathematics: A Comparative Analysis of American and Japanese Assessments of Eighth-Gmders. NCES 97-885, Washington, DC, U.S. Department of Education. National Center for Education Statistics, 1997. 17. C. H. Edwards, The historical development of the calculus. Springer-Verlag, New York, 1979. 18. H. Eves, An introduction to the history of mathematics. 4th ed. Holt, Rinehart, and Winston, New York, 1976. 19. E. Fischbein, M. Deri, et al., The role of implicit models in solving verbal problems in multiplication and division, Journal for Research in Mathematics Education 16 (1985) 3–17. 20. A. Graeber and D. Tirosh, Multiplication and division involving decimals: Pre-service elementary teachers’ performance and beliefs, Journal of Mathematical Behavior 7 (1988) 263–280. 21. K. P. E. Gravemeijer, P. Cobb, et al., Symbolizing, modeling, and instructional design, in Symbolizing, modeling, and communicating in mathematics classrooms, P. Cobb and K. McClain, Editor, Erlbaum, Hillsdale, NJ, in press, 22. B. Greer, Non-conservation of multiplication and division involving decimals, Journal for Research in Mathematics Education 18 (1987) 37–45. 23. B. Greer, Non-conservation of multiplication and division involving decimals, Journal of Mathematical Behavior 7 (1988) 281–298. 24. B. Greer, Non-conservation of multiplication and division: Analysis of a symptom, Journal of Mathematical Behavior 7(3) (1988) 281–298. 25. G. Harel, M. Behr, et al., Constancy of quantity’s quality: the case of the quantity of taste, Journal for Research in Mathematics Education 25(4) (1994) 324–345. 26. G. Harel, M. Behr, et al., The blocks task and qualitative reasoning skills of 7th Grade children, Cognition and Instruction 9(1) (1992) 45–96. 27. G. Harel, M. Behr, et al., The impact of number type on the solution of multiplication and division problems: Further considerations, in The development of multiplicative reasoning in the learning of mathematics, G. Harel and J. Confrey, Editor, SUNY Press, Albany, NY, 1994, p. 365–386. 28. K. Hart, The understanding of ratios in secondary school, Mathematics in School 7 (1978) 4–6.

DRAFT - 41

Thompson & Saldanha

To understand numbers

29. N. Hativa and D. Cohen, Self learning of negative number concepts by lower division elementary students through solving computer-provided numerical problems, Educational Studies in Mathematics 25(4) (1995) 401-431. 30. J. M. Henle, An outline of set theory. Springer Verlag, New York, 1986. 31. N. Herscovics, Cognitive obstacles encountered in the learning of algebra, in Research issues in the learning and teaching of algebra, S. Wagner and C. Kieran, Editor, Erlbaum, Hillsdale, NJ, 1989, p. 60–86. 32. R. P. Hunting, G. Davis, and C. A. Pearn, Engaging whole-number knowledge for rational number learning using a computer-based tool, Journal for Research in Mathematics Education 27(3) (1996) 354-379. 33. R. Karplus, S. Pulos, and E. Stage, Proportional reasoning and the control of variables in seven countries, in Cognitive process instruction, J. Lochead and J. Clement, Editor, Franklin Institute Press, Philadelphia, 1979, p. 47–103. 34. R. Karplus, S. Pulos, and E. Stage, Early adolescents’ proportional reasoning on ‘rate’ problems, Educational Studies in Mathematics 14(3) (1983) 219–234. 35. T. E. Kieren, Personal knowledge of rational numbers: Its intuitive and formal development, in Number concepts and operations in the middle grades, J. Hiebert and M. Behr, Editor, National Council of Teachers of Mathematics, Reston, VA, 1988, p. 162–181. 36. T. E. Kieren, Rational and fractional numbers as mathematical and personal knowledge: Implications for curriculum and instruction, in Analysis of arithmetic for mathematics teaching, G. Leinhardt, R. Putnam, and R. A. Hattrup, Editor, Erlbaum, Hillsdale, NJ, 1992, p. 323–372. 37. T. E. Kieren, The learning of fractions: Maturing in a fraction world. Paper presented at the Conference on Learning and Teaching Fractions, Athens, GA, February, 1993. 38. T. E. Kieren, Rational and fractional numbers: From quotient fields to recursive understanding, in Rational numbers: An integration of research, T. P. Carpenter, E. Fennema, and T. A. Romberg, Editor, Erlbaum, Hillsdale, NJ, 1993, p. 49–84. 39. S. S. Kutler. (1998, September 17, 1998). Further fruitfullness of fractions, [Threaded discussion]. Historica Mathematica Discussion Group [1999, December 1]. 40. R. Lesh, M. Behr, and T. Post, Rational number relations and proportions, in Problems of representation in teaching and learning mathematics, C. Janvier, Editor, Erlbaum, Hillsdale, NJ, 1987, p. 41–58. 41. R. Lesh, T. Post, and M. Behr, Proportional reasoning, in Number concepts and operations in the middle grades, J. Hiebert and M. Behr, Editor, National Council of Teachers of Mathematics, Reston, VA, 1988, p. 93–118. 42. C. Luke, The repeated additon model of multiplication and children’s performance on mathematical word problems, Journal of Mathematical Behavior 7 (1988) 217–226. 43. L. Ma, Knowing and teaching elementary mathematics: Teachers' knowledge of fundamental mathematics in China and the United States. Erlbaum, Mahwah, NJ, 1999. DRAFT - 42

Thompson & Saldanha

To understand numbers

44. C. McKnight, J. Crosswhite, et al., The underachieving curriculum: Assessing U. S. school mathematics from an international perspective. Stipes, Urbana, 1987. 45. G. I. Minskaya, Developing the concept of number by means of the relationship of quantities, in Soviet studies in the psychology of learning and teaching mathematics, L. P. Steffe, Editor, School Mathematics Study Group and National Council of Teachers of Mathematics, Palo Alto, CA and Reston, VA, 1975, p. 207–261. 46. NCTM, Curriculum and evaluation standards for school mathematics. National Council of Teachers of Mathematics, Reston, VA, 1989. 47. NCTM, Professional standards for teaching mathematics. National Council of Teachers of Mathematics, Reston, VA, 1991. 48. G. Noelting, The development of proportional reasoning and the ratio concept. Part 2 Problem structure at successive stages: Problem-solving strategies and the mechanism of adaptive restructuring, Educational Studies in Mathematics 11(3) (1980) 331–363. 49. G. Noelting, The development of proportional reasoning and the ratio concept. Part I differentiation of stages, Educational Studies in Mathematics 11(2) (1980) 217–253. 50. D. Palermo, More about less: A study of language comprehension, Journal of Verbal Learning and Verbal Behavior 12 (1973) 211–221. 51. J. Piaget, The child’s conception of movement and speed. Basic Books, New York, 1970. 52. T. R. Post, K. A. Cramer, et al., Curriculum implications of research on the learning, teaching, and assessing of rational number concepts, in Rational numbers: An integration of research, T. P. Carpenter, E. Fennema, and T. A. Romberg, Editor, Erlbaum, Hillsdale, NJ, 1993, p. 327–362. 53. T. R. Post, G. Harel, et al., Intermediate teachers’ knowledge of rational number concepts, in Integrating research on teaching and learning mathematics, E. Fennema, T. P. Carpenter, and S. J. Lamon, Editor, SUNY Press, Ithaca, NY, 1991, p. 177–198. 54. C. M. Reese, K. E. Miller, et al., NAEP 1996 mathematics report card for the nations and states. Nactional Center for Education Statistics, Washington, DC, 1997. 55. L. Saldanha and P. W. Thompson. Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. Proceedings of the Annual Meeting of the Psychology of Mathematics Education - North America. North Carolina State University, Raleigh, NC, 1998. 56. J. Schwartz, Intensive quantity and referent transforming arithmetic operations, in Number concepts and operations in the middle grades, J. Hiebert and M. Behr, Editor, National Council of Teachers of Mathematics, Reston, VA, 1988, p. 41–52. 57. E. A. Silver and P. A. Kenney, An examination of relationships between the 1990 NAEP mathematics items for grade 8 and selected themes from the NCTM standards, Journal for Research in Mathematics Education 24(2) (1993) 159–167. 58. M. A. Simon, Prospective elementary teachers’ knowledge of division, Journal for Research in Mathematics Education 24(3) (1993) 233–254. DRAFT - 43

Thompson & Saldanha

To understand numbers

59. L. P. Steffe, Children’s construction of number sequences and multiplying schemes, in Number concepts and operations in the middle grades, J. Hiebert and M. Behr, Editor, National Council of Teachers of Mathematics, Reston, VA, 1988, p. 119–140. 60. L. P. Steffe, Composite units and their schemes. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL, April, 1991. 61. L. P. Steffe, Operations that generate quantity, Journal of Learning and Individual Differences 3(1) (1991) 61–82. 62. L. P. Steffe, Learning an iterative fraction scheme. Paper presented at the Conference on Learning and Teaching Fractions, Athens, GA, February, 1993. 63. L. P. Steffe, Children’s multiplying and dividing schemes: An overview, in The development of multiplicative reasoning in the learning of mathematics, G. Harel and J. Confrey, Editor, SUNY Press, Albany, NY, 1994, p. 3–39. 64. J. W. Stigler, P. Gonzales, et al., The TIMSS Videotape Classroom Study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States. National Center for Education Statistics Report NCES 99-0974, Washington, D.C., U. S. Government Printing Office, 1999. 65. D. Tall and S. Vinner, Concept images and concept definitions in mathematics with particular reference to limits and continuity, Educational Studies in Mathematics 12 (1981) 151–169. 66. A. G. Thompson, R. A. Philipp, et al., Calculational and conceptual orientations in teaching mathematics, in 1994 Yearbook of the NCTM, A. Coxford, Editor, NCTM, Reston, VA, 1994, p. 79–92. 67. I. Thompson, Young children’s idiosyncratic written algorithms for addition, Educational Studies in Mathematics 26(4) (1994) 323–346. 68. P. W. Thompson, A theoretical framework for understanding young children's concepts of whole-number numeration. Unpublished Doctoral dissertation. Department of Mathematics Education, University of Georgia, 1982. 69. P. W. Thompson. Quantitative concepts as a foundation for algebra. Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Dekalb, IL, 1988. 70. P. W. Thompson, Notations, conventions, and constraints: Contributions to effective uses of concrete materials in elementary mathematics, Journal for Research in Mathematics Education 23(2) (1992) 123–147. 71. P. W. Thompson, Quantitative reasoning, complexity, and additive structures, Educational Studies in Mathematics 25(3) (1993) 165–208. 72. P. W. Thompson, The development of the concept of speed and its relationship to concepts of rate, in The development of multiplicative reasoning in the learning of mathematics, G. Harel and J. Confrey, Editor, SUNY Press, Albany, NY, 1994, p. 179–234.

DRAFT - 44

Thompson & Saldanha

To understand numbers

73. P. W. Thompson, Notation, convention, and quantity in elementary mathematics, in Providing a foundation for teaching middle school mathematics, J. Sowder and B. Schapelle, Editor, SUNY Press, Albany, NY, 1995, p. 199–221. 74. P. W. Thompson, Imagery and the development of mathematical reasoning, in Theories of mathematical learning, L. P. Steffe, et al., Editor, Erlbaum, Hillsdale, NJ, 1996, p. 267-283. 75. P. W. Thompson and T. Dreyfus, Integers as transformations, Journal for Research in Mathematics Education 19 (1988) 115–133. 76. F. Tourniaire and S. Pulos, Proportional reasoning: A review of the literature, Educational Studies in Mathematics 16 (1985) 181–204. 77. R. Tzur, An integrated study of children's construction of improper fractions and the teacher's role in promoting that learning, Journal for Research in Mathematics Education 30(3) (1999) 390-416. 78. G. Vergnaud, A classification of cognitive tasks and operations of thought involved in addition and subtraction problems, in Addition and subtraction: A cognitive perspective, T. P. Carpenter, J. M. Moser, and T. A. Carpenter, Editor, Erlbaum, Hillsdale, NJ, 1982, p. 39–59. 79. G. Vergnaud, Multiplicative structures, in Acquisition of mathematics concepts and processes, R. Lesh and M. Landau, Editor, Academic Press, New York, 1983, p. 127–174. 80. G. Vergnaud, Multiplicative structures, in Number concepts and operations in the middle grades, J. Hiebert and M. Behr, Editor, National Council of Teachers of Mathematics, Reston, VA, 1988, p. 141–161. 81. G. Vergnaud, Multiplicative conceptual field: What and why?, in The development of multiplicative reasoning in the learning of mathematics, G. Harel and J. Confrey, Editor, SUNY Press, Albany, NY, 1994, p. 41–59. 82. D. Wearne and J. Hiebert, Place value and addition and subtraction, Arithmetic Teacher 41(5) (1994) 272–274. 83. R. Wilder, Evolution of mathematical concepts: An elementary study. Wiley, New York, 1968. 84. T. Wildi, Units and conversions: A handbook for engineers and scientists. IEEE Press, New York, 1991.

DRAFT - 45