Last update - 02:50 27/08/2004
Pi in the sky
"Heshbon lehorim" ("Arithmetic for Parents: A Book for Adults on Mathematics for Children") by Ron Aharoni, Schocken, 165 pages, NIS 75.
By Ariel Rubinstein
Once a year, Israelis wake up in the morning, open their newspapers, read reports about surveys comparing the level of knowledge in mathematics of our genius children with that of children from countries, which we more or less respect, and discover to our horror that we are slipping badly. Israeli kids, we sadly conclude, are just not all they are cracked up to be.
Since the level of knowledge in mathematics is customarily linked with intellectual prowess in general, the myth of Jewish genius suffers a massive blow. Since mathematics is customarily considered the "queen" of the sciences, serious doubts are raised as to whether the Jewish People of Zion can truly build a Temple of Academia and Science to which the nations of the world will flock. And since the quality of our human resources and technological progress are vital for the maintenance of our edge over those nations that are bent on destroying us, our very survival is threatened.
This year's headlines in scorching red on the subject set off alarm bells in the minds of the members of a wonderful group of mathematicians and academics who have devoted an immense amount of time and energy to shaking up the school system and getting it out of its doldrums. They were moved to take action in light of their conclusion that Israel is stuck with an unsuccessful curriculum, and that only an across-the-board reform can substantially change the present dismal situation.
The basic assumption behind this welcome flurry of activity is that the Jewish genius is still alive and kicking in the State of Israel and that Israelis are essentially the victims of "systemic failure," a low level of professionalism among elementary school teachers, and "technocratic ineptitude." I am not sure that I support this assumption. Over the past few decades, the State of Israel has undergone dramatic changes. I simply do not understand why it makes any sense to expect a society that has so profoundly altered its demographic character to retain those same traits it had in the 1950s and 1960s.
Nor should the fact that Israeli Jews no longer place education at the top of their agenda come as any great surprise. It is unclear how a society whose fine young men and women must invest all their energy and skill in fighting stone-throwers on the Trans-Samaria Highway, and whose best creative and idealistic minds focus their talents on ways of catching terrorists near the Tomb of the Patriarchs and Matriarchs in Hebron, can maintain its qualitative edge as well in mathematics instruction.
Man with a mission
But let us leave those issues aside and return to "Arithmetic for Parents." Prof. Ron Aharoni is one of the more prominent members of the above group, whose mission is to save the teaching of mathematics in the country. He teaches mathematics at the Technion - Israel Institute of Technology in Haifa, and went to the trouble of participating in an educational project in the northern town of Ma'alot, preferring to teach schoolchildren addition and subtraction rather than giving lectures on combinatorics to brilliant students at the Technion. He also developed ideas for alternative curricula, and has gone to great lengths to produce a book (in Hebrew) for parents that is subtitled "A Book for Adults on Mathematics for Children."
"Arithmetic for Parents" is designed for worried, energetic and concerned parents who feel that they are able and need to provide their children with a basic understanding of mathematics that goes beyond what the schools are offering. The book can also serve teachers in elementary schools and in seminaries who seek to help improve teaching methods in the field.
At the core of the book are a number of ideas for the improvement of mathematical understanding and skills among children in the first stages of absorbing this subject. The author believes that what is considered straightforward and simple is actually very profound. In order to understand arithmetic operations, there is a need for studying many different examples from daily life. You cannot base any understanding of the equation "2 + 3 = 5" on the knowledge that this is what the pocket calculator says. This arithmetic statement is a generalization of the fact that whenever you add 2 units of something to 3 units of the same something, you will always receive a set of 5 units of that same something. You cannot make children understand that statement through abstract methods. Instead, you can help them to generalize, for example, by the fact that when you place two oranges in a basket and then add another three oranges, you will have five oranges; or the fact that if there are three houses in a neighborhood and another two are built, there will be five houses there; or, if you have two Cuisenaire rods (bdidim) and you add another three rods, the total length of the rods will be that of five rods.
Aharoni stresses the need for illustrating the various meanings of arithmetic operations. For instance, subtraction has several meanings: distancing (you have 5 apples, somebody eats 2; how many will you have left?); separation (there are 5 children in a group, 2 of them are girls; how many boys are there?); and comparison (there are 5 girls and 2 boys in a group; how many more girls are there than boys?). It would be interesting to obtain Aharoni's response to the following question that a sharp-witted child might ask him: "There are 5 `illegal' Jewish settlements, and 2 are dismantled. How many are left?" (The correct answer is 6.)
Optimistic book
Aharoni believes in continual drilling. Just as Ze'ev Vilnai taught us that you can only learn to love your country if you continually explore it by foot, Aharoni believes that, to understand and love mathematics, children must themselves perform multiplication operations such as 769 x 139 instead of pressing the buttons of a pocket calculator.
To an extent, this book is a reaction to the method based on using the rods, which has already been met by scathing criticism. It must be admitted that the scorners, including myself, must first express scathing criticism of their own ineffectiveness and indifference. I remember how I could not understand some of the questions in my children's math textbooks. When I saw their bags of colored rods, I could not fathom how the identification of a number with a certain color could make any children understand anything beyond blurring the concept of the number. After all, the very essence of a number is that it is the abstraction of a unit, from which its size, smell and color have been removed. However, just as the members of Israel's defense community comfort themselves with the thought that they know how to calculate the dangers of the nuclear reactor in Dimona, and just as we comfort ourselves with the thought that the prime minister knows what he is doing, similarly we can comfort ourselves with the knowledge that the experts in education know better than we do how mathematics should be taught, even if all their theories fly in the face of human intuition and plain old horse sense.
Although his scorn of the research studies of experts in education might go overboard a little, Aharoni recognizes the sociology of academia, which can sometimes push scholars to very mistaken conclusions, and he thus believes in his own intuition and in his own understanding more than he believes in the findings of some article in an academic journal on teaching mathematics. More power to him!
This is a very optimistic book, which carries the message that the author has faith in his fellow human beings, as well as faith in parents and children. The author genuinely believes that parents can separate their children from TV soap operas and their Game-Boys and can get them to practice their arithmetic. He genuinely believes that the question "Is the argument 1 + 1 = 2 an axiom or a statement?" will challenge the thinking of the parents who will use this book. He genuinely thinks that it is really possible to implement his view that, "even while going for a drive, you can recite it [the multiplication table]. You can take turns. You, the parent, will say: `1 x 6 = 6.' Then your child will say: `2 x 6 = 12,' and so forth." And he genuinely thinks that you can teach the multiplication table by putting memo notes on the refrigerator door.
Beauty of mathematics
"Arithmetic for Parents" is full of admiration for the beauty of mathematics. For instance, the book describes itself as teaching "the intricacies that are at the very heart of mathematics and which give mathematics its beauty and the teaching of mathematics its meaning." Aharoni sometimes gets a little carried away with himself. At times his fine-sounding ideas float somewhere between "The Adventures of a Mathematics Professor in an Elementary School" and "A Brief Introduction to Mathematics without Mathematics." Naivete is also expressed by the "cute" illustrations that are strongly reminiscent of the now-defunct Davar magazine for children. However, what I see as the book's central problem is the author's excessive belief in the ability of parents to help their children understand mathematics.
I have grave doubts as to whether intelligent parents who do not have a solid grounding in math can ever really absorb the "intricacies" that Aharoni wants children to learn. Aharoni would do himself a favor if he produced a workbook (perhaps an electronic one) that could be used by teachers with initiative and very diligent parents (I would not be surprised if he has already prepared one).
Aharoni also plays on our feelings of nostalgia for our childhood. He opens the book with a reference to the trauma of learning mathematics that still haunt most adults. My elementary school trauma was created when the principal, Mr. Shamgar, entered the classroom, placed one of my classmates, a new immigrant by the name of Yaakov, on the desk, and beat him on his bottom with a ruler. And my best memory is that same Mr. Shamgar entering our class one morning. I was in second grade and Hanukkah was fast approaching. Mr. Shamgar did something that day that Aharoni would have praised more than I did at the time: The principal asked us how many candles we light on the first night of Hanukkah, how many we light on the second night, and so forth. He drew a Hanukkah menorah on the blackboard and wrote down the numerals from 2 to 9 above its branches. He then asked us to add up all the numbers (the menorah itself gave us a hint as to discovering the formula for the total). I remember how I astonished him when I shouted out my answer: 44. "And how did you figure that out?" asked Shamgar, who had become very excited. "That's what it says on the box of Hanukkah candles," I replied.
I read this book a day after Naomi Shemer's death. The newspapers were full of eulogies about an Israel that no longer exists and which perhaps may never have existed. "Arithmetic for Parents" belongs to an Israel that had faith in its spiritual ability, which was naive, that was always ready to volunteer, that replaced the culture of the steam engine with the spirit of working industriously for the good of all society. If Israeli schoolchildren enhance their achievements in arithmetic, the credit should go to naive dreamers like Aharoni.
Prof. Ariel Rubinstein teaches economics at Tel Aviv University.