1 What’s Sophisticated about Elementary Mathematics? Plenty—That’s Why Elementary Schools Need Math Teachers By Hung-Hsi Wu tant ideas like place value and fractions is hard indeed. As a mathematician who has spent the past 16 years trying to - improve math education—including delivering intensive profes ome 13 years ago, when the idea of creating a cadre of sional development sessions to elementary-grades teachers—I mathematics teachers for the upper elementary grades am an advocate for having math instruction delivered by math (who, like their counterparts in higher grades, would teachers as early as possible, starting no later than fourth grade.* teach only mathematics) first made its way to the halls of S But I also understand that until you appreciate the importance the California legislature, the idea was, well, pooh-poohed. One and complexity of elementary mathematics, it will not be apparent legislator said something like: “All you have to do is add, subtract, why such math teachers are necessary. multiply, and divide. How hard is that?” In this article, I address two “simple” topics to give you an idea The fact is, there’s a lot more to teaching math than teaching of the advanced content knowledge that is needed to teach math how to do calculations. And getting children to understand impor - effectively. Our first topic—adding two whole numbers—is espe- cially easy. The difficulty here is mostly in motivating and engag- Hung-Hsi Wu is a professor emeritus of mathematics at the University of California, Berkeley. He served on the National Mathematics Advisory ing students so that they come to understand the standard addi- Panel and has written extensively on mathematics textbooks and teacher tion algorithm and, as a result, develop a deeper appreciation of preparation. Since 2000, he has conducted professional development place value (which is an absolutely critical topic in elementary institutes for elementary and middle school teachers. He has worked with the state of California to rewrite its mathematics standards and assess- There have been calls for math teachers in the education literature, among them the * ments, and was a member of the Mathematics Steering Committee that (see pages 397–398, available at www.nap. National Research Council’s Adding It Up contributed to revising the math framework for the National Assessment edu/catalog.php?record_id=9822#toc) and the National Mathematics Advisory Panel’s of Educational Progress. He was also a member of the National Research Foundations for Success (see Recommendation 20 on page xxii, available at www.ed. ILLUSTRATED BY ROLAND SARKANY Council’s Mathematics Learning Study Committee. gov/about/bdscomm/list/mathpanel/report/final-report.pdf). AMERICAN EDUCATOR | FALL 2009 4

2 not so easy. math). This discussion of addition may not convince you that Then you get to play the magician. Tell them that what they are math teachers are a necessity in the first through third grades, but doing is called “adding numbers.” In this case, they are adding 31 it will give you a deeper appreciation of the important mathemati- to 45, written as 45 + 31 (teach them to write addition horizontally cal foundation that is being laid in the early grades. as well as vertically from the beginning), and what it means is that Our second topic—division of fractions—is substantially it is the number they get by starting with 45 and counting 31 more harder, though it’s still part of the elementary mathematics con- steps. Show them they do not have to count so strenuously to get tent as it should be taught in fifth and sixth grades. This is a topic the answer to 45 + 31 because they can do two simple additions that, in my experience, many adults struggle with. My goal here is instead, one being 4 + 3 and the other 5 + 1, and these give the two twofold: (1) to show you that elementary math can be quite digits of the correct answer 76. sophisticated, and (2) to deepen your knowledge of division and fractions. Along the way, I think it will become apparent why mathematicians consider facility with fractions essential to, and excellent preparation for, algebra. By the end, I hope you will join All whole-number computations are me in calling for the creation of a cadre of teachers who specialize nothing but a sequence of single-digit in the teaching of mathematics in grades 4–6. For simplicity, we will refer to them as ele- , to distinguish them from math teachers computations artfully put together. This mentary teachers who are asked to teach all subjects. is the kind of thinking students will need Adding Whole Numbers to succeed in algebra and advanced - Consider the seemingly mundane skill of adding two whole num bers. Take, for example, the following. mathematics. 45 + 31 76 You can demonstrate this effectively by collecting the 45 pen- nies and putting them into bags of 10; there will be 4 such bags Nothing could be simpler. This is usually a second-grade lesson, with 5 stragglers. Do the same with the other 31 pennies. Then with practice continuing in the third grade. But if you were the place these bags and stragglers on the mat again, and ask them teacher, how would you convince your students that this is worth how many pennies there are. It won’t take long for them to figure learning? Too often, children are given the impression that they out that there are 4 + 3 bags of 10, and 5 + 1 stragglers. must learn certain mathematical skills because the teacher tells They will figure out that 7 bags of 10 together with 6 stragglers them they must. So they go through the motions with little per - total 76 again. Now ask them to compare counting the bags and sonal involvement. This easily leads to learning by rote. How, then, stragglers with the magic you performed just a minute ago. If they can we avoid this pitfall for the case at hand? One way is to teach don’t see the connection (and some won’t), patiently explain it to them what it means to add numbers, why it is worth knowing, why them. Of course, this is the time to review place value. (To better it is hard if it is not done right, and finally, why it can be fun if they understand place value, and to prepare for the occasional . learn how to add the right way advanced student, see the sidebar on page 9.) Then, you can use All this can be accomplished if you begin your lesson with a place value to explain that when they add the 4 bags of 10 to the 3 story, like this: Alan has saved 45 pennies and Beth has saved 31. bags of 10, they are actually adding 40 and 30. They want to buy a small package of stickers that costs 75 cents, and they must find out if they have enough money together. To 45 40 + 5 40 + 5 45 act this out, you can show children two bags of pennies, one bag + 31 + 30 + 1 + 31 + 30 + 1 → → → containing 45 and the other 31. Now dump them on the mat and 76 ? ? 70 + 6 76 explain that they have to count how many there are in this pile. Chances are, they will mess up as they count. Let them mess up Now, they will listen more carefully to your incantations of place before telling them there is an easier way. Go back to the bags of value because you have given them more incentive to learn about 45 and 31, and explain to them that it is enough to begin with 45 this important topic. and continue to count the pennies in the bag of 31. In other words, As mentioned above, addition of whole numbers is done to find out how many are in 45 and 31 together, start with 45 and mainly in grades 2 and 3. Often, the addition algorithm is taught . To show just go 31 more steps; the number we land on is the answer by rote, but some teachers do manage to explain it in terms of them that making these steps corresponds exactly to counting, do place value, as we have just done. Many educators believe that the a simple case with them. If there are 3 pennies in the smaller bag real difficulty of this algorithm arises when “carrying” is neces - instead of 31, then going 3 steps from 45 lands at 48 because sary, but conceptually, carrying is just a sidelight, a little wrinkle on the fabric. The key idea is contained in the case of adding with- 45 48. 47 → → 46 → out carrying. If we succeed in getting students to thoroughly So 48 is the total number of pennies in the two bags of 45 and 3. understand addition without carrying, then they will be in an Now ask them to count like this for 45 and 31; chances are, most excellent position to handle carrying too. (However, in my experi- of them will find this a bit easier but many will still mess up. You ence, the standard textbooks and teaching in most second- or can help them get to 76, but they probably will get frustrated. That third-grade classrooms focus on carrying before students are is good: here is something they want to learn, but they find it is ready, and that is a pity.) 5 AMERICAN EDUCATOR | FALL 2009

3 31251, we will focus only on 45 and 31 to enhance clarity. We have, Understanding the addition algorithm in terms of place value— then, the following. for example, that 45 + 31 is 40 + 30 and 5 + 1—is appropriate for ... = (4 × 45723 + 31251 1000) + 10000) + (5 × beginners, but it cannot stop there. The essence of the addition ... 10000) + (1 × + (3 × 1000) + - algorithm, like all standard algorithms, lies in the abstract under ... = (4 × 10000) + 10000) + (3 × the arithmetic computations with whole numbers, standing that ... + (5 × 1000) + (1 × 1000) + no matter how large, can all be reduced to computations with ... 10000 + (5 + 1) × (4 + 3) 1000 + = × single-digit numbers . (For more on this, see the sidebar on page Again, the last equality makes use of the distributive law. If we 10.) In other words, students’ ultimate understanding of these compare the two expressions (4 + 3) × 10 + (5 + 1) and (4 + 3) × algorithms must transcend place value to arrive at the recognition 10000 + (5 + 1) × 1000, we see clearly that the same single-digit that all whole-number computations are nothing but a sequence additions (4 + 3) and (5 + 1) are in both of them, and that the dif- of single-digit computations artfully put together. This is the kind ference between these expressions lies merely in whether these of thinking students will need to succeed in algebra and advanced single-digit sums are multiplied by 10 or 1,000 or 10,000, the place mathematics. More precisely, students should get to the point of values of the respective digits. This clearly illustrates the primacy recognizing that 45 + 31 is no more than the combination of two of single-digit computations in the addition algorithm. single-digit computations, 4 + 3 and 5 + 1. Whether the 4 stands Returning to our original second-grade lesson of 45 + 31, let’s for 40 or 40,000 and the 3 stands for 30 or 30,000 is completely review what you have accomplished. You have irrelevant. shown students what addition means; this is To drive home this point, consider the following two addition important because we want to promote problems. 45 723 45 the good practice among students 251 31 + 31 + - that through precise defini 76 974 76 get to know tions, they what they will do before The problem on the left is the one we have been working with, and doing it . T h e n y o u parts of the problem on the right are tantalizingly similar, except ma d e t h e m wa nt t o that the 4 and 5 in the first row are no longer 40 and 5 but 40,000 l e a r n i t , a n d m a d e and 5,000, respectively. Similarly, the 3 and 1 in the second row them realize that the are not 30 and 1 but 30,000 and 1,000, respectively. Yet, do the most obvious method changes in the place values of these four single-digit numbers (4, (counting) is not the 5, 3, and 1) change the addition? Not at all, because the result is easiest. Best of all, you still the same two digits, 7 and 6, and that is the point. opened their eyes to the We are now able to directly address the main concern of this magic of learning: acquiring article, which is the need for math teachers at least starting in the power of making something complicated much simpler. grade 4. In grade 4, the multiplication algorithm has to be Instead of tedious, error-prone counting, you used the concept of explained. A teacher knowledgeable in mathematics would know place value to introduce the idea of breaking up a task digit by digit that this is the time to cast a backward glance at the addition algo- and adding only two single-digit numbers in succession. A couple rithm to make sure students finally grasp a real understanding of of years later, the fourth-grade math teacher will have the oppor- what this algorithm is all about : just a sequence of single-digit tunity to explain and make explicit the idea that to add any whole computations. Why is this knowledge so critical at this point? numbers, no matter how large, all the children need to do is add Because it leads seamlessly to the explanation of why students single-digit numbers. must memorize the multiplication table (of single-digit numbers) The main goal of the elementary mathematics curriculum is to to automaticity before they do multidigit multiplication: in the provide children with a good foundation for mathematics. In this same way that knowing how to add single-digit numbers enables context, the addition algorithm, when taught as described above them to add any two numbers, no matter how large, knowing how in grades 2–4, serves as a splendid introduction. It teaches chil- to multiply single-digit numbers enables them to multiply any if possible, always break dren an important skill in mathematics: two numbers, no matter how large. We want students to be up a complicated task into a sequence of simple ones . This is why exposed, as early as possible, to the idea that beyond the nuts and we do not look at 45 and 31, but only 4 and 3, and 5 and 1. bolts of mathematics, there are unifying undercurrents that con- Of course, they will encounter somewhere down the road nect disparate pieces. something like 45 + 37, but they will be in a position to under - Let us go a step further to make explicit the role of single-digit stand that the carrying step is actually adding a 1 to the 10s col- computations in the additions of 45 + 31 and 45723 + 31251. If umn. Despite how it is presented in most U.S. textbooks, carrying students have been given the proper foundation in second grade, the main idea of the addition algorithm. The main idea is not is then in fourth grade, a math teacher will be able to give the fol- to break up any addition into the additions of single-digit num- lowing explanation. bers and then, drawing on our understanding of place value, put = (4 × 10) + 5 + (3 × 10) + 1 45 + 31 these simple computations together to get the final answer. If = (4 × 10) + (3 × 10) + 5 + 1 you can make your students understand that, you are doing fan- × = (4 + 3) 10 + (5 + 1) tastically well as a teacher, because you have taught them impor- In the last equality, we used the distributive law—i.e., (b + c)a = know and tant mathematics. They now have an important skill ba + ca—to rewrite (4 × 10) + (3 × 10) as (4 + 3) × 10. For 45723 + AMERICAN EDUCATOR | FALL 2009 6

4 dard algorithm for long division, as shown in the following the reasoning behind it—and they will have used both to deepen example. their appreciation of place value. 4 ) Dividing fractions 6 24 –24 I’ve had plenty of encounters with well-educated adults who can’t 0 divide fractions without a calculator, or who can, but have no idea why the old rule “invert and multiply” works. With that in mind, What we tell children is that to divide 24 by 6, we look for the I’ll break this topic into three parts: we’ll review division, then number which, when multiplied by 6, gives 24. (Of course, chil - fractions, and finally the division of fractions. Along the way, the dren who have memorized the multiplication table of single-digit answer to our larger question—what’s sophisticated about ele- numbers will do this easily; those who haven’t will struggle.) mentary mathematics?—will become apparent, as will the ways In a similar fashion, the meaning of 36/12 = 3 is that in which mastering fractions prepares students for algebra. 3 36 = × 12, and the meaning of 252/9 = 28 is that 252 = × 9, etc. 28 Let’s begin with the division of whole numbers, which would There is a subtle point here that is usually slurred over in the upper elementary grades but should be pointed out: We want students to be exposed, as early in our examples, the dividend (be it 24, 36, or 252) is a multiple of the divisor, since otherwise the quotient can- as possible, to the idea that beyond the not be a whole number. That said, now we can use abstract † symbols to express this new understanding of the division nuts and bolts of mathematics, there are of whole numbers as follows: for whole numbers m and unifying undercurrents that connect n, where m is a multiple of n and n is nonzero, the mean- ing of the division m/n = is that m = q × n. q disparate pieces. Beginning in fifth grade, we should teach students to reconceptualize division from this point of view. Their math teachers should help them revisit division from the perspec - normally be taught in third grade. What does 24/6 = 4 tive of this new knowledge and reshape their thinking accordingly. mean? In the primary grades, we teach two meanings of Such is the normal progression of learning. division of whole numbers: partitive division* and mea- Note that this reconceptualization is not a rejection of students’ surement division. For brevity, let us concentrate only on mea- understanding of the division of whole numbers in their earlier surement division, in which the meaning of 24/6 = 4 is that by sepa- grades. On the contrary, it evolves from that understanding and rating 24 into equal groups of 6, we find that there are 4 groups in makes it more precise. This reconceptualization is important all. So the 4 tells how many groups of 6s there are in 24. quotient because the meaning of division, when reformulated this way, By fifth grade, students should be ready to apply their under- turns out to be universal in mathematics, in the following sense: standing of measurement division to a more symbolic format. This two numbers (i.e., not just whole numbers) and if m and n are any will prepare them for the division of fractions, for which the idea of n is nonzero, then the definition of “m divided by n equals q” is “dividing into equal groups” often is not very helpful in calculating means m = q q that m = q × n. In other words, m/n = × n. answers. (For example, the division of ¹⁄₇ by ¹⁄₂ does not lend itself to any easy interpretation of dividing ¹⁄₇ into equal groups of ¹⁄₂. Being able to draw or visualize where ¹⁄₇ and ¹⁄₂ fall on the number - e now turn to fractions, a main source of math pho line is helpful in estimating the answer, but not in arriving at the bia. In the early grades, grades 2–4 more or less, precise answer, ²⁄₇.) Any understanding of fraction division, there- - students mainly acquire the vocabulary of frac fore, has to start from a more abstract level. With this in mind, we tions and use it for descriptive purposes (e.g., ¹⁄₄ of W express the separation of 24 objects into 4 groups of 6s symbolically a pie). It is only in grades 5 and up that serious learning of the as 24 = 6 + 6 + 6 + 6, which is, of course, equal to 4 × 6, by the very of fractions takes place—and that’s when students’ mathematics - definition of whole-number multiplication. Thus, the division state fear of fractions sets in. statement 24 = 4 × 6. multiplication ment 24/6 = 4 implies the From a curricular perspective, this fear can be traced to at least At this point, we must investigate whether the multiplication two sources. The first is the loss of a natural reference point when of the information in the division all statement 24 = 4 × 6 captures students work with fractions. In learning to deal with the math- statement 24/6 = 4. It does, because if we know 24 = 4 × 6, then we ematics of whole numbers in grades 1–4, children always have a know 24 = 6 + 6 + 6 + 6, and therefore 24 can be separated into 4 natural reference point: their fingers. But for fractions, the cur - groups of 6s. By the measurement meaning of division, this says ricular decision in the United States has been to use a pizza or a 24/6 = 4. Consequently, the multiplication statement 24 = 4 × 6 pie as the reference point. Unfortunately, while pies may be useful carries exactly the same information as the division statement in the lower grades, they are an awkward model for fractions big- meaning 4 24/6 = 4. Put another way, the × 6. is 24 = 4 of 24/6 = ger than 1 or for any arithmetic operations with fractions. For This is the symbolic reformulation of the concept of division of example, how do you multiply two pieces of pie or use a pie to whole numbers that we seek. solve speed or ratio problems? This meaning of division is actually very clear from the stan- A second source of the fear of fractions is the inherently abstract † This definitely would be appropriate for fifth-graders once the idea of using symbols An example of partitive division is to put 24 items in 6 bags (each with an equal * for abbreviations is introduced and many examples are given for illustration. number of items), and find that each bag has 4 items. 7 AMERICAN EDUCATOR | FALL 2009

5 the fractions with denominators equal to 3: the first division point nature of the concept of a fraction. Whereas students’ intuition of to the right of 0 is what is called ¹⁄₃, and the succeeding points of whole numbers can be grounded in counting their fingers, learning the sequence are then ²⁄₃, ³⁄₃, ⁴⁄₃, etc. The same is true for ¹⁄ fractions requires a mental substitute for their fingers. By its very , , ²⁄ , ³⁄ ⁿ ⁿ ⁿ nature, this mental substitute has to be abstract because most frac - etc., for any nonzero whole number n. Thus, whole numbers tions (e.g., ¹⁹⁄₁₃ or ²⁵¹⁄₆₀₄) tend not to show up in the real world. clearly fall within the collection of numbers called fractions. If we Because fractions are students’ first serious excursion into reflect the fractions to the left of 0 on the number line, the mirror abstraction,* understanding fractions is the most critical step in image of the fraction is by definition the negative fraction m / ⁿ † understanding rational numbers – In . Therefore, positive and negative fractions are now just and in preparing for algebra. / m ⁿ points on the number line. Most students would find marking off . order to learn fractions, students need to know what a fraction is a point ¹⁄₂ of a unit to the left of 0 to be much less confusing than Typically, our present math education lets them down at this criti- contemplating a negative ¹⁄₂ piece of pie. cal juncture. All too often, instead of providing guidance for stu- dents’ first steps in the realm of abstraction, we try in every con- The number line is especially helpful in teaching students ceivable way to ignore this need and pretend that there is no about the theorem on equivalent fractions, the single most impor - tant fact in the subject. To state it formally, for all whole numbers k k, m, and n (where k ≠ 0 and n ≠ 0), = / / . In other words, m m k ⁿ ⁿ k m m and represent the same point on the number line. Let / / k ⁿ ⁿ Because fractions are students’ first serious us consider an example to get a better idea: suppose m = 4, n = 3, and k = 5. Then the theorem asserts that excursion into abstraction, understanding 4 5 × 4 20 5 × 4 = = . and, of course, 5 × 3 5 × 3 15 3 fractions is the most critical step in preparing The number line makes the equality clear. To see how ⁴⁄₃ equals for algebra. ²⁰⁄₁₅, draw a number line and divide the space between 0 and 1, as well as between 1 and 2, into three equal parts. Count up to the 4th point on the sequence of thirds—that’s ⁴⁄₃. Then take each of what is a fraction? abstraction. When asked, , we say it is just some- the thirds and divide them into 5 equal parts (an easy way to make thing concrete, like a slice of pizza. And when this doesn’t work, 15ths). Count up until you get to the 20th point on the sequence we continue to skirt the question by offering more metaphors and of 15ths—that’s ²⁰⁄₁₅, and it’s in the same spot as ⁴⁄₃. 0 1 more analogies: What about a fraction as “part of a whole”? As | | | | | | | | | | | | | | | | | | | | | another way to write division problems? As an “expression” of the ₃ ₄ ₃ ₃ ⅓ ⅔ ⁄ ⁄ form m/n for whole numbers m and n (n > 0)? As another way to 15 10 ₅ 20 ₅ ₅ ₅ ₅ ⁄1 ⁄1 ⁄1 ⁄1 write ratios? These analogies and metaphors simply don’t cut it. Fractions have to be numbers because we will add, subtract, mul- The use of the number line has another advantage. Having tiply, and divide them. whole numbers displayed as part of fractions allows us to see more What does work well for showing students what fractions really clearly that the arithmetic of fractions is entirely analogous to the are? The number line. In the same way that fingers serve as a natu- arithmetic of whole numbers. For example, in terms of the num- ral reference point for whole numbers, the number line serves as ber line, 4 + 6 is just the total length of the concatenation (i.e., ‡ a natural reference point for fractions. linking) of a segment of length 4 and a segment of length 6. The use of the number line | | | has the immediate advantage of conferring coherence on the 6 4 study of numbers in school mathematics: a number is now defined § unambiguously to be a point on the number line. In particular, Then in the same way, we define ¹⁄₆ + ¹⁄₄ to be the total length of the concatenation of a segment of length ¹⁄₆ and a segment of regardless of whether a number is a whole number, a fraction, a rational number, or an irrational number, it takes up its natural length ¹⁄₄ (not shown in proportion with respect to the preceding place on this line. (For the definition of fractions, including how number line). | | | to find them on the number line, see the sidebar on page 12.) 1∕6 1∕4 Now, let’s describe the collection of numbers called fractions. Divide a line segment from 0 to 1 into, let’s say, 3 segments of equal We arrive at ¹⁄₆ + ¹⁄₄ = ¹⁰⁄₂₄ as we would if we were adding whole length; do the same to all the segments between any two consecu- numbers, as follows. Using the theorem on equivalent fractions, tive whole numbers. These division points together with the whole we can express ¹⁄₆ and ¹⁄₄ as fractions with the same denominator: numbers then form a sequence of equal-spaced points. These are ¹⁄₆ = ⁴⁄₂₄ and ¹⁄₄ = ⁶⁄₂₄. The segment of length ¹⁄₆ is therefore the concatenation of 4 segments each of length ¹⁄₂₄, and the segment of length ¹⁄₄ is the concatenation of 6 segments each of length ¹⁄₂₄. * Very large numbers are already an abstraction to children, but children tend not to The preceding concatenated segment is therefore the concatena- be systematically exposed to such numbers the way they are to fractions. † tion of (4 + 6) segments each of length ¹⁄₂₄, i.e., ¹⁰⁄₂₄.** In this way, Rational numbers consist of fractions and negative fractions, which of course include whole numbers. (Continued on page 10) ‡ See, for example, page 4-40 of the National Mathematics Advisory Panel s “Report ’ of the Task Group on Learning Processes,” www.ed.gov/about/bdscomm/list/math ** Naturally, the theorem on equivalent fractions implies that 10/24 = 5/12, as panel/report/learning-processes.pdf. 10/24 = (2 × 5)/(2 × 12), but contrary to common belief, the simplification is of no § We exclude complex numbers from this discussion, as they are not appropriate for great importance . Notice in particular that there was never any mention of the the elementary grades. “least common denominator.” AMERICAN EDUCATOR | FALL 2009 8

6 Understanding Place Value In this scheme, counting nine times lands us at the 9 of the first Many teachers, rightly in my opinion, believe place value is the row, and counting one more time would land us at the 0 of the foundation of elementary mathematics. It is often taught well, second row. If we want to continue counting, then the next step using manipulatives such as base-10 blocks to help children grasp lands us at the 1 of the second row, and then the 2 of the second that, for example, the 4 in 45 is actually 40 and the 3 in 345 is row, and so on. actually 300. However, this way of counting obviously suffers from the But despite the importance of place value, the rationale defect of ambiguity: there is no way to differentiate the first row behind it usually is not taught in colleges of education or in math from the second row so that, for example, going both professional development. That’s probably because two steps and twelve steps from the first 0 will the deeper explanation is not appropriate for land us at the symbol 2. The central break- most students in the first and second grades, through of the Hindu-Arabic numeral system is which is when place value is emphasized. But it is to distinguish these rows from each other by appropriate for upper-elementary students who placing the first symbol (0) to the left of all the are exploring number systems that are not base 10 symbols in the first row, the second symbol (1) (which often is done, without enough explana- to the left of all the symbols in the second row, tion, through games)—and it is certainly some- the third symbol (2) to the left of all the symbols in thing that math teachers should know. So here it the third row, etc. is: the sophisticated side of the simple idea of place value. 00 01 02 03 04 05 06 07 08 09 Let’s begin with a look at the basis of our 10 11 12 13 14 15 16 17 18 19 The most so-called Hindu-Arabic numeral system. * 20 21 22 23 24 25 26 27 28 29 basic function of a numeral system is the ability to 30 31 32 33 34 35 36 37 38 39 . . . . . count to any number, no matter how large. One . . . . . way to achieve this goal is simply to make up . . . . . symbols to stand for larger and larger numbers as 90 91 92 93 94 95 96 97 98 99 we go along. Unfortunately, such a system Now, the tenth step of counting lands us at 10, the requires memorizing too many symbols, and eleventh step at 11, etc. Likewise, the twentieth step makes devising a simple method of computation lands us at 20, the twenty-sixth step at 26, the impossible. The overriding feature of the Hindu- thirty-first step at 31, etc. By tradition, we omit the 0s Arabic numeral system, which will be our exclusive to the left of each symbol in the first row. That done, concern from now on, is the fact that it limits itself we have re-created the usual ninety-nine counting to using exactly ten symbols—0, 1, 2, 3, 4, 5, 6, 7, numbers from 1 to 99. † 8, 9—to do all the counting. Let us see, for We now see why the 2 to the left of the symbols example, how “counting nine times” is repre- on the third row stands for 20 and not 2, because the sented by 9. Starting with 0, we go nine steps and 2 on the left signifies that these are numbers on the land at 9, as shown below. third row, and we get to them only after we 9 → → 3 → 4 → 5 → 6 → 7 → 8 2 1 → 0 → have counted 20 steps from 0. Similarly, we know 31 is on the fourth row because the But, if we want to count one more time beyond the 3 on the left carries this ninth (i.e., ten times), we would need another symbol. information; after Since we are restricted to the use of only these ten symbols, counting thirty steps someone long ago got the idea of placing these same ten from 0 we land at 30, symbols next to each other to create more symbols. and one more step The most obvious way to continue the counting is, of course, lands us at 31. So the 3 of to simply recycle the same ten symbols over and over again, 31 signifies 30, and the 1 signifies one more step beyond 30. placing them in successive rows, as follows. With a trifle more effort, we can 0 1 2 3 4 5 6 7 8 9 carry on the same discussion to 0 1 2 3 4 5 6 7 8 9 three-digit numbers (or more). The 0 1 2 3 4 5 6 7 8 9 moral of the story is that place . . . . . . . . . . value is the natural consequence . . . . . of the way counting is done in the decimal numeral system . For a fuller discussion, including * This term is historically correct in the sense that the Hindu-Arabic numeral system numbers in arbitrary base, see was transmitted to the West from the Islamic Empire around the 12th century, and pages 7–9 of The Mathematics the Arabs themselves got it from the Hindus around the 8th century. However, recent research suggests a strong possibility that the Hindus, in turn, got it from the Chinese, on K–12 Teachers Need to Know who have had a decimal place-value system since time immemorial. See Lay Yong http://math. my Web site at Lam and Tian Se Ang, Fleeting Footsteps: Tracing the Conception of Arithmetic and berkeley.edu/~wu/School Algebra in Ancient China (Hackensack, NJ: World Scientific, 1992). . mathematics1.pdf † Historically, 0 was not among the symbols used. The emergence of 0 (around the 9th –H.W. century and beyond) is too complicated to recount here. 9 AMERICAN EDUCATOR | FALL 2009

7 (Continued from page 8) can make use of the same scaffolding as learning to divide with whole numbers; students proceed from the simple to the com- students get to see that fractions are the natural extension of whole plex. For example, a simple problem like ¹⁄₂ ÷ ¹⁄₄ = 2 could be taught numbers and not some confusing new thing. This realization using the measurement definition of division and showing stu- smoothes the transition from computing with whole numbers to dents on the number line that ¹⁄₄ appears twice in ¹⁄₂. That’s fine computing with fractions. as an introduction, but ultimately, in order to prepare for more advanced mathematics, students must grasp a more abstract— opefully this discussion has smoothed the transition and precise—definition of division with fractions. They must be for you too, because it’s time for us to skip ahead to able to answer the following question: sixth grade and tackle division with fractions. Having 4 5 9 5 learned to add, subtract, and multiply with fractions, H equal × ? Why does 6 4 6 9 students should be comfortable with fractions as numbers (just like whole numbers). So, their learning to divide with fractions In other words, why invert and multiply? To give an explanation, Teaching the Standard Algorithms This is an excellent In the context of school mathematics, an algorithm is a finite example of the kind of sequence of explicitly defined, step-by-step computational abstract thinking that is procedures that end in a clearly defined outcome. The so-called critical to success in standard algorithms for the four arithmetic operations with mathematics learning. whole numbers are perhaps the best known algorithms. Building on the At the outset, we should make clear that there is no such discussion of the addition the unique thing as standard algorithm for any of the four algorithm given in the operations +, −, ×, or ÷, because minor variations have been main article, we can incorporated into the algorithms by various countries and ethnic further illustrate this leitmotif groups. Such variations notwithstanding, the algorithms provide with the multiplication algo- shortcuts to what would otherwise be labor-intensive computa- rithm. In this case, let us assume tions, while the underlying mathematical ideas always remain that students already know the meaning of multiplication as the same. Therefore, from a mathematical perspective, the label repeated addition. The next step toward understanding “standard algorithms” is justified. multiplication requires that they know the multiplication table While it is easy to see why these algorithms were of interest by heart—i.e., that they know the multiplication of single-digit before calculators became widespread, a natural question now numbers to automaticity. We now show, precisely, how this is why we should bother to teach them. There are at least two knowledge allows them to compute the product 257 × 48. First, reasons. First, without a firm grasp of place value and of the observe that 257 = (2 × 100) + (5 × 10) + 7, so that by the logical underpinnings of the algorithms, it would be impossible distributive law [i.e., a(b + c) = ab + ac]: to detect mistakes caused by pushing the wrong buttons on a calculator. A more important reason is that, in mathematics, 257 × 4 = (2 × 4) × 100 + (5 × 4) × 10 + (7 × 4) and learning is not complete until we know both the facts and their 257 × 8 = (2 × 8) × 100 + (5 × 8) × 10 + (7 × 8). underlying reasons . For the case at hand, learning the explana- tions for these algorithms is a very compelling way to acquire Since they already know the single-digit products (2 × 4), many of the basic skills as well as the abstract reasoning that (5 × 4), (7 × 4), (2 × 8), (5 × 8), and (7 × 8), and they know how are integral to mathematics. Both these skills and the capacity to add, they can compute 257 × 4 and 257 × 8. Such being the for abstract reasoning are absolutely essential for understand- case, we further note that 48 = (4 × 10) + 8, so that again by ing fractions, decimals, and, therefore, algebra in middle the distributive law: if students do not feel comfort- school. One can flatly state that 257 × 48 = (257 × 4) × 10 + (257 × 8). able with the mathematical reasoning used to justify the standard algorithms for whole numbers, then their chances of The right side being something they already know how to success in algebra are exceedingly small . compute, they have therefore succeeded in computing 257 × 48 These algorithms also highlight one of the basic tools used starting with a knowledge of the multiplication table. (For lack by research mathematicians and scientists: namely, that of space, we omit the actual writing out of the multiplication whenever possible, one should break down a complicated task algorithm.) into simple subtasks. To be specific, the leitmotif of the Although the case of the long-division algorithm is more standard algorithms is as follows: to perform a computation sophisticated, the basic principle is the same: it is just a with multidigit numbers, break it down into several steps so sequence of single-digit computations. that each step (when suitably interpreted) is a computation For further details on the standard algorithms, see pages involving only single-digit numbers . Therefore, a virtue of the 38–90 of the first chapter of a professional development text standard algorithms is that, when properly executed, they for teachers that I am currently writing, available at http:// allow students to ignore the actual numbers being computed, math.berkeley.edu/~wu/EMI1c.pdf . no matter how large, and concentrate instead on single digits. –H.W. AMERICAN EDUCATOR | FALL 2009 10

8 should probably not bring up in a sixth-grade classroom, but we have to ask what it means to divide fractions in the first place. which is, nevertheless, something a math teacher should be if we do not specify the meaning of dividing fractions, The fact that aware of. The question is whether, for arbitrary fractions M and then we cannot possibly get a formula for it should be totally obvi- N (N > 0), we can always divide M by N—i.e., whether there is ous, yet this fact is not common knowledge in mathematics edu- Q so that M = Q × N. The answer, of course, is fraction always a cation. For such a definition, let us go back to the concept of divi- a a c d yes: if M = sion for whole numbers. Recall that in the case of whole numbers, and N = / ) would do. So the ) × ( , then Q = ( / / / c d b b having a clearly understood meaning for multiplication (as upshot of all this is that we can always divide a fraction M by a repeated addition) and division (as measurement division) nonzero fraction N, and the quotient, to be denoted by M/N, is allowed us to conclude that the meaning of the division statement the fraction obtained by the invert-and-multiply rule. m/n = q for whole numbers m, n, and q (n > 0) is inherent in the Once we know the meaning of division, we see there is nothing multiplication statement m = q × n. But now we are dealing with to the procedure of invert and multiply. What is sobering is that fractions, and the situation is different. To keep this article from the rhyme, “Ours is not to reason why; just invert and multiply,” becoming too long, let’s assume that we already know how to gets it all wrong. With a precise, well-reasoned definition, there is multiply fractions,* but we are still searching for the meaning of fraction division. Knowing that fractions and whole numbers are The rhyme, “Ours is not to reason why; , it would be a reasonable working on the same footing as numbers hypothesis that if m/n = q means m = q × n for whole numbers m, just invert and multiply,” gets it all wrong. n, and q, then the direct counterpart of this assertion in fractions should continue to hold. Now, if M, N, and Q are fractions (N > 0), With a precise, well-reasoned definition, we do not as yet know what M/N = Q means, although we know there is no need to wonder why—the - the meaning of M = Q × N because we know how to multiply frac tions. Therefore, the only way to make this “direct counterpart” in answer is clear. of fraction division. fractions come true is to use it as a definition In other words, we adopt the following definition: for fractions M of M by N, written M/N, is the fraction and N (N > 0), the division no need to wonder why—the answer is clear. Thus, we return to Q, so that M = Q × N. our earlier theme: before we do anything in mathematics, we must We’ll get acquainted with this definition by looking at a special make clear what it is we are doing. In other words, we must have case. Suppose - a precise definition of division before we can talk about its proper 5 9 ties. (And we must have a precise definition of fractions before we = Q for a fraction Q. 6 4 can expect students to do anything with them.) But one question remains: if division is just multiplication in What could Q be? By definition, this Q must satisfy ⁵⁄₆ = Q × ⁹⁄₄. k a different format, why do we need division at all? The correct Now, recalling that / = m m / (the theorem on equivalent frac- k ⁿ ⁿ answer is that certain situations in life require it. An example of tions), we use this fact to find Q by multiplying both sides of ⁵⁄₆ = such a problem is the following: Q × ⁹⁄₄ by ⁴⁄₉. 5 4 4 9 Q = × × × yard long ³⁄₄ A 5-yard ribbon is cut into pieces that are each ( ( ) ) 6 9 4 9 to make bows. How many bows can be made? 4) × (9 Q × = (4 9) × Students usually recognize by rote that this problem calls for a division of 5 by ³⁄₄, but not the reason why division should be used. = Q × 1 = Q To better understand the reason for dividing, suppose the prob - lem reads, instead, “A 30-yard ribbon is cut into pieces that are This is the same as ⁵⁄₆ × ⁴⁄₉ = Q. We can easily check that, indeed, each 5 yards long. How many pieces can be made?” It would fol- this Q satisfies ⁵⁄₆ = Q × ⁹⁄₄. So, we see that 5 9 4 5 low from the measurement interpretation of the division of whole = × 9 6 4 6 numbers that the answer is 30/5 = 6 pieces—i.e., there are six 5s in 30. The use of division for this purpose is well understood. and we have verified the invert-and-multiply rule in this special However, we are now dealing with pieces whose common case. But the reasoning is perfectly general, and it verifies in c c a length is a fraction ³⁄₄, and the reason for solving the problem by exactly the same way that for a nonzero fraction )/( / / / ) , if ( d b d a d dividing 5 by ³⁄₄ is more problematic for many students. But if we is equal to a fraction Q, then Q is equal to ( ) × ( / / ). Therefore, c b use the preceding definition of division, the reason emerges with the invert-and-multiply rule is always correct. clarity. Suppose Q bows can be made from the ribbon. Here Q We have been staring at the concept of the division of fractions could be a fraction, and the meaning of “Q bows” can be explained for quite a while, and we seem to be getting there because we by using an explicit example. If Q = 6 ²⁄₃, for example, then “6 ²⁄₃ have explained the invert-and-multiply rule. Therefore, it may be bows” means 6 pieces that are each ³⁄₄ yard long, plus a piece that a little deflating to say that although we are getting we close, very is the length of 2 parts when the ³⁄₄ yard is divided into 3 parts of . There is a subtle point about the definition are not quite there yet , so that the mul- If multiplication is taught correctly equal length. of fraction division that is still unsettled. This is something one tiplication of two fractions is defined clearly, one can then explain The treatment of fraction multiplication in textbooks and in the education literature * why Q bows, no matter what fraction Q is, have a total length of Q is mostly defective, but one can consult pages 62–74 of http://math.berkeley.edu/~wu × ³⁄₄ yards. Therefore, if Q bows can be made from 5 yards of rib- /EMI2a.pdf for an introduction. 11 AMERICAN EDUCATOR | FALL 2009

9 all that is to come, it must, in a grade-appropriate manner, respect bon, then 5 = Q × ³⁄₄. , this is By the definition of fraction division the basic characteristics of mathematics. What does this mean? exactly the statement that To answer this question, we have to remember that the school 5 = Q . mathematics curriculum, beginning with approximately grade 5, ³⁄₄ becomes increasingly engaged in abstraction and generality. It This is the reason why division should still be used to solve this will no longer be about how to deal with a finite collection of num- problem. Incidentally, the invert-and-multiply rule immediately bers (such as, ¹⁄₂ × (27−11) + 56 = ?), but rather about what to do leads to Q = ²⁰⁄₃, which equals 6 ²⁄₃ pieces. In greater detail, that’s collection of numbers all at once (such as, is it true infinite with an 6 pieces and a leftover piece that is the length of 2 parts when ³⁄₄ − − numbers x and 2 xy)(x² + y² − √ 2 xy) for that x⁴ + y⁴ = (x² + y² + √ all yard is divided into 3 equal parts. y?). The progression of the topics, from fractions to negative frac- The Bigger Picture tions, and on to algebra, Euclidean geometry, trigonometry, and precalculus, gives a good indication that to learn mathematics, a At this point, I hope you can see that there’s more to teaching student gradually must learn to cope with abstract concepts and elementary mathematics than is initially apparent. The fact is, precise reasoning, and must acquire a coherent overview of topics there’s much more to it than could possibly be covered in an that are, cognitively, increasingly complex and diverse. For this article. But allow me to give you a glimpse of the bigger picture—of reason, students in the upper elementary grades must be pre- what elementary mathematics is really all about. I’ll conclude with - coher pared for the tasks ahead by being slowly acclimatized to some of the latest thinking on the subject, thinking that points to reasoning ence , although always in a way that is , and precision , mathematics teachers in the upper elementary grades being our grade appropriate. Allow me to amplify each of these character - best hope for providing all students the sound mathematics foun- istics below. dation they need. Mathematics in elementary school is the foundation of all of Coherence : If you dig beneath the surface, you will find that the K–12 mathematics and beyond. Therefore, to prepare students for Defining Fractions order to avoid The precise definition of a fraction as a point on the number misunderstanding. line is a refinement of, not a radical deviation from, the usual What we should concept of a fraction as a “part of a whole.” As I will explain, specify, instead, is this refinement produces increased simplicity, flexibility, and that the whole is precision. the length of the Let us begin with a line, which is usually taken to be a ], unit segment [ 0 , 1 horizontal one, and fix two points on it. The one on the left will rather than the 1 , and the one on the right by be denoted by 0 . (Because we will segment itself. When we say not take up negative numbers, our discussion will focus entirely [ , 1 ] is divided into “equal 0 on the half-line to the right of 0 .) Now as we move from 0 to the parts,” what we should say is right, we mark off successive points, each of which is as far apart ] is divided into that [ 0 , 1 is from 1 0 (like a ruler). Label these points from its neighbors as segments of equal length. The , 3 , etc. by the whole numbers 0 , 1 , 2 fraction ¹⁄₃ therefore would be the length of 0 1 2 3 etc. any segment so that three segments of the same length, when | | | | pieced together, form a segment of length 1 . Since all segments between consecutive whole numbers have length 1 , when we We begin with an informal discussion. If we adopt the usual likewise divide each of the segments between consecutive approach to fractions, the “whole” would be taken to be the 3 whole numbers into segments of equal length, the length of 0 segment from , called the unit segment, to be denoted by to 1 . In particular, each of each of these shorter segments is also ¹⁄₃ [ 0 , 1 ]. The number 1 is called the unit. Then a fraction such as ¹⁄₃ and is therefore ¹⁄₃ the following thickened segments has length , would be, by common consent, ] is 1 1 0 part when the whole [ . a legitimate representation of ¹⁄₃ divided into 3 equal parts. So far so good. But if we try to press forward with mathematics, we immediately run into trouble 0 1 2 3 etc. because a fraction is a number—not a shape or a geometric | | | | | | | | | | | | 1 0 ] therefore cannot be the whole. , figure. The unit segment [ The language of “equal parts” is also problematic because in Now concentrate on the thickened segment on the far left. anything other than line segments, it usually is not clear what . Since the ¹⁄₃ The distance of its right endpoint from 0 is naturally “equal parts” means. For example, if the whole is a ham, does value of each whole number on the number line reveals its “equal parts” mean parts with equal weights, equal lengths, distance from 0 (e.g., the distance of the point labeled 3 is equal amounts of meat, equal amounts of bones, etc.? So, we ), logic demands that we label the right end- exactly 3 from 0 are forced to introduce more precision into our discussion in point of this segment by the fraction ¹⁄₃ , and we call this AMERICAN EDUCATOR | FALL 2009 12

10 ubiquity of the general principle of reducing a complicated task main topics of the elementary curriculum are not a collection of to a collection of simple subtasks. This principle runs right through unrelated facts; rather, they form a whole tapestry where each all the standard algorithms, and also all the algorithms for deci- item exists as part of a larger design. Unfortunately, elementary mals. In middle and high school mathematics, it also is the guid- school students do not always get to see such coherence. For ing principle in the discussion of congruence and similarity, example, although whole numbers and fractions are intimately provided these concepts are presented correctly. It also should be related so that their arithmetic operations are essentially the same, the guiding principle in the discussion of quadratic functions and too often whole numbers and fractions are taught as if they were their graphs, thereby making the basic technique of completing unrelated topics. The comment I frequently hear that “fractions the square both enlightening and inevitable. Similarly, we saw are such different numbers” is a good indication that elementary how one embracing definition of division clarifies the meaning of mathematics education, as it stands, cannot go forward without the division of whole numbers and fractions and, as students significant reform, such as the introduction of math teachers. should be taught in later grades, all rational numbers, real num- Another example of the current incoherence is the fact that finite bers, and complex numbers. decimals are a special class of fractions, yet even in the upper elementary grades, decimals often are taught as a topic separate : Children should learn about this mathematics tapestry Precision from fractions. As a result, students end up quite confused having in a language that does not leave room for misunderstanding or - to learn three different kinds of numbers (whole numbers, frac guesswork. It should be a language sufficiently precise so that they tions, and decimals), whereas learning about fractions should can reconstruct the tapestry step by step, if necessary. Too often, automatically make them see that the other two are just more of such precision of language is not achieved. For example, if you the same. These are only two of many possible examples of our tell a sixth-grader that two objects are similar if they are the same splintered curriculum and the great harm it does to students’ shape but not necessarily the same size, it raises the question of learning. what “same shape” means. A precise definition of similarity using Another manifestation of the coherence of mathematics is the ⁷⁄₃ (again, in ¹⁄₃ of multiple th 7 is the we can also say that . We also denote ¹⁄₃ segment the “standard representation” of self-explanatory language). 0 this thickened segment by [ , ¹⁄₃ ], because the notation clearly What we have done to fractions with denominators equal to exhibits the left endpoint as 0 and the right endpoint as ¹⁄₃ . To can be done to any fraction. In this way, we transform the 3 summarize, we have described how the naive notion of as ¹⁄₃ naive concept of a fraction as a part of a whole into the clearly equal parts” can be “ 1 part when the whole is divided into 3 defined concept of a fraction as a point on the number line. refined in successive stages and made into a point on the There are many advantages of this indispensable transforma- number line, as shown below. 0 1 2 3 etc. tion, but there are three that should be brought out right away. | | On the number line, all points are on equal footing, so that | | | | in the preceding picture, for example, there is no conceptual ⅓ ¹¹⁄₃ because both numbers are equally and difference between ²⁄₃ In a formal mathematical setting, we now use this particular easy to access. The essence of this message is that, when a point as the official representative of ¹⁄₃ . In other words, fraction is clearly defined as a point on the number line, the whatever mathematical statement we wish to make about the conceptual difference between so-called proper and improper , it should be done in terms of this point. This ¹⁄₃ fraction fractions completely disappears. So the first major advantage of agreement enforces uniformity of language and lends clarity to understanding fractions as points on the number line is that all ¹⁄₃ . At the same time, the any mathematical discussion about . Now we can discuss all fractions all fractions are created equal preceding discussion also gives us confidence that we can relate at once with ease, whether proper or improper. In this small this point on the number line to our everyday experience with way, the concept of a fraction begins to simplify, and learning ¹⁄₃, should that need arise. about fractions gets easier. What we have done to the representation of can be done ¹⁄₃ The second major advantage is that such a concept of 3 to any fraction with a denominator equal to ; for example, the 1 Once we specify what the unit fractions is inherently flexible. ²⁄₃ would be the marked point to the standard representation of . stands for, all fractions can be interpreted in terms of the unit right of 1 on the line above, and that of ³⁄₃ itself. In would be ¹⁄₃ 1 Now we are ready for that ham. If we let stand for the weight general, ⁄₃ m we identify any for any whole number m with its ¹⁄₃ would represent a piece of ham that is a of the ham, then , and we agree to let be written as ⁰⁄₃ . standard representation 0 third of the whole ham in weight. If, on the other hand, we let 1 Here, then, are the first several fractions with denominators stand for the volume of the ham, then the same fraction will equal to . 3 now be a piece of ham that is a third of the whole ham in 0 1 2 3 etc. volume—e.g., in cubic inches. | | | | | | | | | | | | This brings us to the third major advantage: the increase in ₄ ₅ ₃ 7 ₃ ₃ ₃ 6∕3 ∕3 8∕3 9∕3 10∕3 11∕3 ⁄ ⁄ 0∕3 ⅓ ⅔ ⁄ flexibility mandates an increase in precision . Gone is the loose reference to “equal parts” in such a setting, because one must Notice that it is easy to describe each of these fractions. For ask, equal parts in terms of what unit? th division point when the number line is is the 7 example, ⁷⁄₃ –H.W. divided into thirds (in self-explanatory language). Equivalently, 13 AMERICAN EDUCATOR | FALL 2009

11 Learning cannot take place in the classroom if students are kept in the concept of dilation from a point A would not allow for such the dark about why they must do what they are told to do. confusion (as students will see that an object changes size, but not shape, when each point of the object is pushed away from or by the same scaling factor). A pulled into he characteristics of coherence, precision, and reason- Another example of the need for precision manifests itself in ing are not just niceties; they are a prerequisite to mak- the way we present concepts. It is worth repeating that before we ing school mathematics learnable. Too often, all three do anything in mathematics, we must make clear what it is that are absent from elementary curricula (at least as they T we are doing by providing precise definitions. There is no better are sketched out in both state standards and nationally marketed example of the need for precision than the way fractions are gen- textbooks).* As a result, too often they also are absent from the erally taught in schools. Too often, fractions are taught without elementary classroom. The fact that many elementary teachers definitions, so that students are always in the dark about what lack the knowledge to teach mathematics with coherence, preci- fractions are. Thus, students multiply fractions without knowing sion, and reasoning is a systemic problem with grave conse - what multiplication means and, of course, they invert and multi- quences. Let us note that this is not the fault of our elementary ply, but dare not ask why. It is safe to hypothesize that such con- teachers. Indeed, it is altogether unrealistic to expect our general- ceptual opaqueness is largely responsible for the notorious ist elementary teachers to possess this kind of mathematical knowledge—especially considering all the advanced knowledge of how to teach reading that such teachers must acquire. Com- It is unrealistic to expect our generalist pounding this problem, the pre-service professional development † in mathematics is far from adequate. There appears to be no hope elementary teachers to possess this kind of solving the problem of giving all children the mathematics edu- of mathematical knowledge—especially cation they need without breaking away from our traditional practice of having generalist elementary school teachers. considering the advanced knowledge The need for elementary teachers to be mathematically profi- - cient is emphasized in the recent report of the National Mathemat they must acquire to teach reading. ‡ ics Advisory Panel. Given that there are over 2 million elementary teachers, the problem of raising the mathematical proficiency of nonlearning of fractions—and, as a result, for great difficulty as all elementary teachers is so enormous as to be beyond compre- students begin algebra. hension. A viable alternative is to produce a much smaller corps with strong content knowledge who of mathematics teachers Reasoning : Above all, it is important that elementary school math- would be solely in charge of teaching mathematics at least begin- ematics, like all mathematics, be built on reasoning. Reasoning is ning with grade 4. The National Mathematics Advisory Panel has the power that enables us to move from one step to the next. When taken up this issue. While the absence of research evidence about students are given this power, they gain confidence that mathe- the effectiveness of such mathematics teachers precluded any matics is something they can do, because it is done according to recommendation from that body, the use of mathematics teachers some clearly stated, learnable, objective criteria. When students in elementary school was suggested for exactly the same practical § are emboldened to make moves on their own in mathematics, reasons. Indeed, this is an idea that each state should seriously they become sequential thinkers, and sequential thinking drives consider because, for the time being, there seems to be no other problem solving. If one realizes that almost the whole of mathe- way of providing our children with a proper foundation for math- matics is problem solving, the centrality of reasoning in mathe- ematics learning. matics becomes all too apparent. We have neglected far too long the teaching of mathematics in When reasoning is absent, mathematics becomes a black box, elementary school. The notion that “all you have to do is add, sub- and fear and loathing set in. An example of this absence is some tract, multiply, and divide” is hopelessly outdated. We owe it to children’s failure to shift successive rows one digit to the left when our children to adequately prepare them for the technological multiplying whole numbers, such as on the left below. society they live in, and we have to start doing that in elementary 826 826 school. We must teach them mathematics the right way, and the × 473 × 473 only way to achieve this goal is to create a corps of teachers who 2478 2478 have the requisite knowledge to get it done. ☐ 5782 5782 3304 + + 3304 s “Report of the Task ’ See, for example, the National Mathematics Advisory Panel * 390698 11564 Group on Conceptual Knowledge and Skills,” especially Appendix B, www.ed.gov/ If no reason is ever given for the shift, it is natural that children about/bdscomm/list/mathpanel/report/conceptual-knowledge.pdf, and “Report of the Subcommittee on Instructional Materials,” www.ed.gov/about/bdscomm/list/ would take matters into their own hands by making up new rules. mathpanel/report/instructional-materials.pdf. Worse, such children miss an excellent opportunity to deepen their † See, for example, the National Council on Teacher Quality ’ s No Common Denomina- understanding of place value and see that, in this example, the tor , www.nctq.org/p/publications/reports.jsp. multiplication 4 × 8 is actually 400 × 800, and that this is the basic ‡ See Recommendation 7 on page xviii and Recommendations 17 and 19 on page xxi reason underlying the shift. Another notorious example is the addi- in Foundations for Success: The Final Report of the National Mathematics Advisory Panel , www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf. tion of fractions by just adding the numerators and the denomina - § , Recommendation 20, page xxii, see note above for URL. Foundations for Success tors, something that happens not infrequently even in college. AMERICAN EDUCATOR | FALL 2009 14