A friend of mine left a high-tech career in mid-life to work in mathematical education. In September 2000, just before the school year began, he called me: There is a project to promote mathematical education in elementary schools; come join. The project was in a development town called Maalot, in the far north of Israel. (Israeli development towns, built in the 1950s to settle new immigrants, are usually considered to be rather backward.)

I am a professional mathematician and, although I have been strongly interested in teaching (which is the reason that my friend had the idea of offering me the job), I had not set foot in an elementary school since I was a child. So I consulted whomever I could. The advice I got was uniform: Don't do it. Elementary math education is a profession in itself. There is no connection between it and teaching math at the university level.

In hindsight, sobriety should have dictated listening to this advice. Yet, had I listened, I would have missed one of the most fascinating adventures of my life.

The banner I was carrying at that time was that of "experience." The children should experience abstract concepts concretely, I thought, after which the abstractions should occur by themselves. I took the kids out to the playground. We measured lengths of shadows and compared them to the lengths of the objects themselves, then used this information to calculate the height of trees according to their shadows. (This idea is borrowed from Thales, who was born in the 7th century B.C.*) Then we measured the length and width of the classroom in various ways to find how many floor tiles fit into one square meter, and what the ratio was between the length of the classroom in meters and its length in tiles.

I learned the price of conceit the hard way—most of my lessons were a mess. I remember my first day of insight well: I took a fourth-grade class to the playground to draw circles on the pavement and then measure the diameters and circumferences. It soon became apparent that the kids were mainly using this opportunity to have fun outside, at which point the teacher I was working with suggested that we go back in and discuss what we had done. We drew circles on the board and, with the active participation of the children, discovered the ratio of perimeter to diameter. For me, this was a first glimpse of the power of common class discussion.

Fortunately, at around the same time, I started teaching first grade. This was a wonderful experience. First-graders are still open-minded; they go along with you wherever you lead them. Their reactions are direct, and they make it apparent to you what is working and what is not. First grade is the best place to learn about teaching. Further, by great good fortune, I was paired with an excellent teacher. She was ready to accompany me in the joint adventure, to our mutual gain. I would open the lesson, introduce an activity or an idea, and she would intervene when she felt that my didactics were less than perfect. Usually, that was when I had skipped a stage.

Since then, I have been learning intensively from each lesson and every conversation with teachers. I learn from unsuccessful lessons no less than from the better ones; mostly, I learn from those lessons that limp in the beginning, until the right thing is done and they take off.

### What Did I Learn?

I learned a lot about how to approach young kids and the way children think. I learned the importance of being systematic, a characteristic that my teaching so desperately lacked in the beginning. I learned that concepts that adults perceive as a whole are in fact constructed from many small components, built one on top of the other, and none of them can be skipped. I learned that explaining in elementary school is usually futile; the youngster should experience the concepts for him or herself. In that notion, I was right from the beginning. It's just that I had no idea what "experiencing" really means. It does not refer to complex notions. Learning through experience relates to the most basic concepts, like that of number or of "smaller than" and "larger than," which can be revealed through the counting of objects.

But what surprised me most was that I learned mathematics. Actually, a lot of it. This would not be the case had I gone to teach in a high school. The mathematical concepts there are known to a professional mathematician. In elementary school, it's the teaching of the most basic principles that counts—the nature of numbers, the meaning of the arithmetical operations, the principles of the decimal system. About these, it is rare for a mathematician to stop and think.

I also learned that the understanding of these principles is inseparable from the didactics. Good teaching means bringing the children to experience the fine fundamental points.

### Layers

Elementary mathematics is structured in the same way as high-level mathematics. Namely, it is layered. Each layer is built on top of the previous one. Just as in a complicated proof, the order in which the components are put together is important. So, in elementary mathematics, it is essential not to skip stages.

The difference is that the layers in elementary school are those at the bottom of the tower. The structures that are built are not tall. But, as if to compensate, there can be difficulties. They are often hidden, as if they were built under water, meaning that it is not always easy to realize what they are. Elementary mathematics is usually not sophisticated, but it is deep.

Here is an example. A first-grade class was given a picture of five apples, three of them green and two red. The children were supposed to tell arithmetical "stories," one on addition and one on subtraction. The importance of telling such stories cannot be overestimated. In order to understand the meaning of the operations, it is not enough to hear or read such stories. One has to be able to invent them on his or her own.

The addition story posed no difficulty: "I had three green apples and two red apples, how many apples did I have altogether?" But when they came to the subtraction story contained in the picture of three green and two red apples, confusion prevailed—and, as usually happens in elementary school, it manifested as inattention. Eventually, one of the children said: "I had five apples. I ate two. How many do I have left?" The problem was that this wasn't the "correct" story. It wasn't based on the drawing. The drawing didn't show two apples disappearing, by being eaten or any other way. That is why the children found the task difficult.

I was experienced enough to know that such confusion almost always originates from having skipped a stage. In this case the missing stage was the understanding that *subtraction has more than one meaning*. There is the meaning of "diminution," where objects are removed: I had 5 balloons, 2 of them burst, how many do I have left? This is the meaning the child used in his story—his apples disappeared. But there is also the meaning of "comparison of quantities," where nothing disappears: There are 5 children in a group, 2 of them are boys. How many are girls? Or perhaps: How many more green apples than red apples are there? In these cases, too, the exercise is one of subtraction, 5–2 or 3–2, but the meaning is different. This is the meaning depicted in the drawing.

The various meanings of subtraction are an example of a fine point that has to be taught explicitly. Skipping this stage will result in later difficulties with word problems.

### The Importance of Explicit Naming

When I started teaching in elementary school, I was convinced that precise formulations and the explicit naming of principles was a matter for grownups. Children should learn things on an intuitive level, I thought. One of the greatest surprises that awaited me was to realize how wrong I was about that. Children need precise formulations. Such formulations consolidate their knowledge of the present layer and make it a safer basis on which higher layers can be built. Moreover, children love "adult" formulations and notations, and are proud of being able to use them. First-grade children who learn the notation "1/2" are happy to discover the notation for "1/3" by themselves.

The different meanings of subtraction—diminution and comparison—gave me the opportunity to realize the importance of the explicit naming of principles. I was lucky to accompany three different classes on this very same page in the first-grade textbook. The first lesson I taught was described above; we moved directly from the addition story to the subtraction story. In the second class, before getting to the subtraction story, I stopped the lesson and started an explicit discussion of the various meanings of subtraction. This went very smoothly, and the children had no difficulty identifying the type of subtraction in the picture. In the third class, I conducted an experiment. Instead of explicit discussion, I preceded the work on the page with an example: the problem of the five children of whom three were girls. This did not work. The example did not provide the children with a solid enough ground upon which to build. This was a good lesson to me on how important it is to formulate principles explicitly.

### What Arithmetic Should Be Covered in Elementary School?

The embarrassingly simple answer is: the four basic operations—addition, subtraction, multiplication, and division.

Yet, this seemingly simple answer is deceptive in two ways. One is that there are actually *five* operations. In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, *forming a unit*, taking a part of the world and declaring it to be the "whole." This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have "two," and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a "10," then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find "three-quarters" of the class, having missed the stage of the corresponding process of repeating a unit in multiplication.

But there is another reason why learning "the four operations" in elementary school is not such a simple answer. That is because the operations have two distinct components. One is their meaning, and the other their calculation. I stress this seemingly simple fact because this distinction is not always clear to education policymakers, especially the writers of textbooks. Some textbooks start with calculation. Some do not stress the difference between "2 times 3" and "3 times 2." Most of them do not make the distinction between the two types of division (6 / 2 = 3, because 3 + 3 = 6, and 6 / 2 = 3, because two goes into six three times, namely 2 + 2 + 2 = 6).

The meaning of an operation is the link between it and reality, the real world operation corresponding to it. The calculation is finding the result. But again, there is something that is not often realized: It is not really finding the result. It is finding the *decimal representation* of the result. Ancient man, when adding eight and four, drew eight lines beside four lines, and represented the result by twelve lines—there is no "calculation" here, and ancient man did not have to send his children to school to learn this. In "8 + 4 = 12," on the other hand, there is calculation, and an invisible operation is performed: that of collecting ten units into one "10." And of course there is also the "place value" writing of the number, another non-trivial principle. Thus, understanding the algorithms for calculation is tantamount to deep understanding of the decimal system. If policymakers realized this, they might be less apt to introduce the use of calculators into elementary school.

To summarize, in elementary school, children (and mathematicians) learn the *meaning of five operations* and the *decimal system*.

* * *

Another important fact should be known about elementary school mathematics: Division has a special status in the study of arithmetic. This is true in most syllabi all over the world, and with good reason. It is awarded a greater portion of teaching time than any other operation. The turnabout occurs around the middle of the fourth grade. From this point on, until the end of the sixth grade, the children are taught the meanings of division, ratio problems (which are expressed by division), and the efficient, systematic tool used when discussing division and ratios: the fraction—both the simple fraction and the decimal fraction.

Why is division so special? Because addition and subtraction are operations too simple to describe the world. When things get complicated, multiplication and division are required. A large part of our world operates according to the principles of linear relationship. In elections, for example, the number of mandates each party receives is more or less linearly related to the number of votes it received. And linear relationships are expressed by division.

Another reason for spending more time on division is that it is more difficult than the other operations. Of the four operations, it has the most meanings, it is the hardest to calculate, and the problems it can represent are the most complicated.

### The Curious Story of Mathematical Education in Israel

As an academic subject, mathematics education is very young, and all of us have had the misfortune of being its guinea pigs. Arguably, Israel has paid a higher price than anywhere else in the world for this experimentation.

The American "new math" reform of the 1960s was brought to Israel in a most strange and extreme form, called "structuralism" by its authors. It was imposed on practically all Israeli schools for a full quarter of a century. Its story may reflect on the politics of education, not only in Israel.

Two Australian researchers, Ken Clements and Nerida Ellerton, wondered why so many American and British educational "reforms" have been exported to other countries, despite having failed miserably in their country of origin. Their explanation was that people, studying for their PhD degrees in the U.S. or the U.K. at the time of the reform, returned home to their own countries bringing the untested gospel of reform with them.^{†}

In this case, new math "structuralism" meant that no concept or operation was taught directly or through its meaning. For every concept there was a "representation," or substitute, whose study was supposed to lead to an understanding of the original concept. The four operations were taught using Cuisenaire rods. A face-like picture into which three numbers are put, two in the places of the eyes and one at the mouth, was supposed to teach children when to add and when to subtract. If the two numbers were at the eyes, and the missing number was at the mouth, it was an addition problem. If one number was at an eye and another at the mouth, it was a subtraction problem. (The children were made to recite: "Eye and eye is plus; mouth and eye is minus.") Division was taught as the reverse operation of multiplication, using so-called "multiplication rectangles." Most extreme was the teaching of the decimal system. Instead of the principle of the collection of tens, strange creatures were invented called "bodytails," which had bodies representing the tens and tails representing the units. And those are just a few of many such devices.

In international mathematics assessments, Israel dropped from first in the world in 1964 to 29th place in 1999—behind everyone except the developing nations.

* * *

Mathematics education in Israel has recently undergone a profound change. Together with other mathematicians and classroom teachers, we established a nonprofit organization, the Israeli Foundation for Math Achievement for All, to work for the improvement of mathematical education in Israel. We successfully lobbied the Ministry of Education to remove the "structuralist" textbooks from schools, replacing them with a more traditional curriculum that puts the focus on the content to be learned, rather than on how it should be taught.

Currently, the foundation is working directly in about 10 percent of the nation's Hebrew-speaking schools (a parallel movement has begun in Arabic-speaking schools). These schools are using translations of mathematics textbooks from Singapore, one of the world's highest performing nations in international assessments of mathematics achievement. These texts are direct, devoid of the use of sophistication for the sake of sophistication, and are based on mathematical wisdom. Mathematical concepts and procedures are introduced carefully, step by step, so as to minimize the possibility of missed stages and future confusion. The texts focus on the meaning of mathematical operations before they teach how to calculate using those operations.

Our approach is based on two principles: Start with the concrete *and* a lot of class discussion. There is much less individual work in workbooks than there used to be. In a typical lesson, the children experience some principle together, in a concrete way, and verbalize what they have experienced. For example, if the children calculate 23–5, they will have two groups of 10 sticks, bound by rubber bands, and 3 loose sticks. They are then asked to explain how 5 sticks can be subtracted, including why it is necessary to unbind one of the groups of 10 in order to accomplish this. They are then asked to write what they did, in vertical form, and relate this to the concrete process of subtraction. Although students do a lot of the talking, they sit facing a teacher at the front of the classroom, where she can guide the discussion and lead the children to the right concepts.

Our organization offers a lot of support to teachers as they implement the new methods and materials (every class is visited at least once a month), with professional development provided on a constant basis. We find that the teachers learn a lot of mathematics along with the children—just as I did.

### "How" vs. "What"

I started by making the point that elementary mathematics has a lot of depth, with many subtle and sometimes hidden principles. To summarize, let me return to this point.

For the last 50 years, education policymakers have been researching *how* to teach mathematics. Our experience with this approach makes it clear that the *what* should come before the *how*. Sound teaching is based, first of all, on the understanding of the fine points of elementary mathematics and on the systematic unfolding of its concepts. "Fine points" do not mean sophistication. Quite the contrary; they mean that even ideas that may look obvious should be experienced and verbalized.

The current trend in education is to make children happy in their studies, so as to prevent "math anxiety." My experience is that children are happiest when they truly understand the principles of mathematics, not when we make believe that they do.

*Ron Aharoni is a professor of mathematics at the Technion-Israel Institute of Technology in Haifa, Israel, and author of *Arithmetic for Parents*, a book for grownups on children's mathematics, published by Schocken Press in 2004. This article is adapted from a 2003 address to the British Mathematical Colloquium in Birmingham, England.*

*Thales was the first mathematician in history to be mentioned by name. He used this method to calculate the height of the pyramids. (back to article)

^{ †}Clements, M.A. and Ellerton, N.F. (1996) *Mathematics education research: Past, present and future*. Bangkok: UNESCO. Also see Clements, M.A. (2003). An outsider's view of North American school mathematics curriculum trends. In G.M.A. Stanic and J. Kilpatrick (Eds.), *A history of school mathematics* (Vol. 2). Reston, VA.: National Council of Teachers of Mathematics. (back to article)

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