Department of Mathematics, Technion

A friend of mine left a high-tech career in mid-life to work in mathematical education. Two and a half years ago he called me: there is a project of promoting mathematical education in elementary schools - come join. The project was in a development town called Maalot, in the far north of Israel. (Development towns were built in Israel in the 1950-s, to settle new immigrants, and 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 never set foot in elementary school since I studied there as a child. So, I consulted whoever I could. The advice I got was uniform: don't. Elementary mathematical education is a profession in itself. There is no connection between it and teaching in university. It is very different from the teaching of gifted children (which I had also done).

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

The banner I was carrying was that of "experience". The children should experience concretely the abstract concepts, after which the abstractions should occur by themselves. I took the kids out to the yard, we measured lengths of shadows and compared them to the lengths of the objects, and used this to measure the height of trees; we measured the length of the class in terms of meters and tiles, and calculated the ratio between the two.

I learnt the price of conceit the hard way.

I remember well my first day of insight. I took a fourth grade to the yard to draw circles on the pavement and calculate the ratios of perimeters to radii. It soon became apparent that the kids were mainly using this opportunity to have fun outside, at which point the teacher suggested that we go in and discuss what we have done. We drew circles on the board, and with the active participation of the children discovered that the ratio of the perimeter to the radius is between 4 and 8. This was for me a first glimpse to the power of common discussion in class.

Fortunately, at around the same time I started teaching in first grade. This was a wonderful experience. One reason is that first graders are still naive, 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. The second reason was that I met there an excellent teacher, who was ready to accompany me in a joint adventure, from which we both gained. I would open the lesson, introduce an activity or an idea, and she would intervene when she felt that my didactics was less than perfect. Usually, when I skipped stages.

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, and mostly I learn from those lessons that limp in the beginning, and then the right thing is done and they take off.

I learnt a lot about how to approach young kids, and the way children think. I learnt the importance of systematicality, that I was so desparately lacking at the beginning. I understood that concepts that look to the adult as being made of one piece are in fact constructed from many small components, built one on top of the other. I learnt that explaining in elementary school is usually futile, the kid should experince the concepts himself. In that I was right at the beginning of my way, it is only 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 the number, or of "smaller than", or "larger than".

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

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

Elementary mathematics is structured the same way that high mathematics is. Namely, it is layered. One layer is built on top of another. 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 mathematics are those at the bottom of the tower. The structures built are not tall. But as if to compensate, there is a difficulty: 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. In a first grade class the children were given a picture of 5 apples, 3 of them green and 2 red. They were supposed to tell "arithmetical strories", 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 such stories on his own. We start from telling stories alluded to by pictures, then go on to more and more free stories.

The addition story posed no difficulty: "I had three green apples and
two red apples, how many apples did I have altogether?" But when we
got to the subtraction story, there was general confusion in class. 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 usual meaning of removal - I had 5 apples,
and ate 3. How many did I have left? (Actually, this was the story
made up eventually in class. The trouble is that it had no connection
to the picture, a point which had to be made clear to the children in
order to clarify to them that there was reason for their confusion.)
But there are at least two more meanings, that is situations
corresponding to subtraction. One is comparison (Tom has 5 apples,
Mary has 3, how many apples does Tom have more than Mary?) and the
other is separation, or classification: among 5 children 3 are girls,
how many of them are boys? This last meaning is the one appearing in
the picture. "I had 5 apples. 3 of them were green. How many were
red?"

The various meanings of subtraction is an example of a fine point which has to be taught explicitly. Skipping this stage will later evince itself in difficulties in verbal questions.

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

The different meanings of subtraction, mentioned in the last section, gave me an opportunity to realize how important is explicit naming of principles. I was lucky to accompany three different classes on this very same page in the book. The first lesson was described above. 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 a third class I made an experiment: instead of explicit discussion, I preceded the work on the page with an example - the problem on the 5 children of whom 3 were girls. This did not work. The example did not provide good enough ground to build upon. This was a good lesson to me on how important it is to formulate principles explicity.

The embarassingly simple answer is: the four operations.

But this seemingly simple answer is deceptive, in two ways. One is
that in fact there are *five* operations. Beside the four
classical operations there is a fifth one, more basic and important:
that of
*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 "ten",
and then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learnt and stressed explicitly. I met children who in fifth grade knew how to find a quarter of a class of size 20, but had difficulties understanding what is "3 quarters" of the class, having missed the stage of the corresponding process of repeating a unit in multiplication.

But there is another reason why "the four operations"' is not such a simple answer to the question of what is learnt in elementary school. It is that 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 educationalists, and mainly to 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 2 goes three times into 6, 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 not realized by many
educationalists: it is not really finding the result. It is finding
the *decimal representation* of the result. Ancient man, when
adding 8 and 4, drew 8 lines beside 4 lines, and represented the
result by 12 lines - there is no "calculation" here, and the ancient
man did not have to send his children to school to learn that. In
"8+4=12", on the other hand, there is calculation, and an operation
was performed: that of collecting 10 units into one "ten". And of
course there is the "place value" writing of the number, another
non-triviall principle.

Were this fact clear to educationalists, they would not have suggested the use of calculators in class. Learning how to calculate means acquiring deep understanding of the decimal system.

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

Another important fact has to be known about the subject matter of arithmetic in elementary school. At some point, around the middle of fourth grade, a switch occurs. From that point on, the main operation studied is division: its calculation, fractions, which are the basic tool for its study, decimal fractions and ratios.

This is true in most syllabi, all over the world, and has a good reason. Division is the most difficult operation, from points of view of both meaning and calculation. Another reason for the importance of division is that much of the reality around us obeys rules of proportionality.

Mathematical education, as an academic subject, is very young, and our generation has had the misfortune of being its guinea pig. But possibly nowhere else was the outcome so bizarre as in Israel of the last quarter of a century.

Two British researchers, Clement and Ellerton, wondered how is it that educational reforms are exported from the US to other countries after having failed miserably in their country of origin. Their explanation is that people who take their Ph.D degrees in the US during the time of the reform return to their countries, and bring with them the gospel, disregarding its failure.

The two main American reforms, the "new math" of the 60-s and the constructivism oriented "standards" of the 80-s and 90-s, were exported to all the rest of the Western world. In Israel the second reform was stopped at the last moment: opposition of a few mathematicians to a constructivistic curriculum for elementary schools forced the ministry of education to practically withdraw it. But the first reform, the "new math", was brought to Israel in a most extreme and strange form, and spread in most of its schools, for a full quarter of a century. It was implemented in a system of books, used in 85% of the schools, following an approach named by its authors "structuralistic".

What this means is that no concept or operation is taught directly, or through its meaning. For every concept there is a "representation", namely a substitute, whose study is supposed to lead to the understanding of the original concept. The four operations are taught using the Quisenaire rods. A face-like picture in which three numbers are put, two in the places of the eyes and one at the mouth, is supposed to teach the child when is he supposed to add and when to subtract: if the two numbers are at the eyes, and the missing number is at the place of the mouth, it is addition. If one number is at an eye and another at the place of the mouth, it is subtraction (the children recite: "eye and eye is plus, mouth and eye is minus"). Division is taught as the reverse operation of multiplication, using so called "multiplication rectangles". Most extreme is the teaching of the decimal system. Instead of the principle of the collection of tens, strange creatures are invented called "bodytails", which have bodies representing the tens and tails representing the units. They are used not only to represent the numbers, but also to perform the operations, in ways that even professional mathematicians find hard to fathom. And this is just the beginning of a long list of such devices.

Israel dropped from first place in the world in the international tests in 1964 to 28-th place in 1999, among countries considered to belong to the third world.

It is only recently that different systems of books have been introduced to elementary schools in Israel, and presently the "structuralistic" books are under attack by mathematicians and teachers, though by almost no academic educationalists.

Singapore has been leading consistently in mathematical education for over a decade, as given evidence by the results of the international tests. This has led people in many countries to examine their books. For me the books were a discovery. They are direct, devoid of any sophistication for the sake of sophistication, and full of mathematical wisdom. At present there is a movement in Israel for introducing these books, which is gathering more and more momentum.