Comprehensive coverage

Thoughts on mathematical education

A quick calculation shows that teaching mathematics has more than one fractional line, and that there is no substitute for hard work

A classroom in Australia - 1932 - when they still knew what hard work was
A classroom in Australia - 1932 - when they still knew what hard work was

Ron Aharoni Galileo

The implausible efficiency of mathematics
In 1960, Nobel laureate physicist Eugene Wigner published an article whose name precedes him, in a literal sense - many know his name, even if they have not read the article itself: "The Unreasonable Efficiency of Mathematics". Indeed, a kind of miracle is how useful mathematics is in formulating scientific theories and predicting phenomena in the world.

Apparently, mathematics is just a tool in the hands of other sciences. In fact, it often leads the way. A small formula occupies an entire world, and indicates where the physical Torah should go. This is not the only reason why it is important to study mathematics.

To think mathematically means to think abstractly and precisely, which is useful in any field. Mathematics is the key to all scientific and technological professions, both because of its direct use and because of its indirect uses - the development of abstract thinking.

If so, it is no wonder that the State of Israel is alarmed every few years in the face of the less than promising achievements of its students in mathematics, and every few years educators and researchers are required to find a solution to this. Like everyone else, I also have no solution to offer.

Studying is work, and there is no substitute for hard work. The main secret is in the learning environment and as a mathematician, and even one who has spent the last ten years as a teacher, I don't have much to say on the subject.

How to relieve math anxiety
A well-known obstacle in teaching mathematics is anxiety, which paralyzes and prevents effective learning. Educators, who recognize this, try to prevent math anxiety through fun activities, self-discovery activities and sophisticated and beautiful exercises.

This is a trend that has dominated mathematical education since the late eighties of the last century, a time when the American Teachers Association published a book called "Standards". The intention is that the children do not need to know certain material, but only ways of thinking. The "standards" mean certain thinking abilities that the child is supposed to reach: setting and testing hypotheses, estimating, and discovering legalities.

This approach, whose common name is "exploration", or "self-discovery", took over mathematical education in America in the early nineties and immediately began to permeate mathematical education in all other countries of the world as well.

Mathematicians who hear this idea are shocked. Mathematics is a layered body of knowledge, much more so than any other field of knowledge. Every mathematician is clear that thinking develops while studying mathematics, not from studying principles of thinking.

You can't learn on a blank! And every mathematician is clear that mathematics is not an unordered collection of activities. And also, if every child is allowed to discover the facts of mathematics without systematic guidance, he will not go far. But the mathematicians were not involved (almost) in this educational reform.

In fact, few outside the world of mathematics education knew about the revolution. This is one of the problems in education: it is very easy to make changes in it, and most bystanders, including parents, do not even know about these changes.

However, after a few years it became clear that the "exploration" revolution led to a tremendous endarlemosis. The students of the United States simply did not know math. In 1994, several hundred mathematicians and scientists (most of them parents of children affected by the method) were alarmed and published a large ad in the "Washington Post" against the new method.

It was the opening shot for the "math wars" that have been rocking math education in the United States and around the world ever since. They are not over yet. The "research" approach came to Israel quite late, after the extent of the failure abroad became clear, so it did not cause too much damage.

Three principles
Because indeed, the secret is not in fun activities, but in understanding. A child enjoys learning when he understands. And he understands when he studies the material systematically, according to the correct order, from the light to the heavy, and especially from the tangible to the abstract. In my opinion, you can actually summarize everything that is important in mathematical education in three principles:
1. Methodology
2. Walking from the concrete to the abstract
3. Exact verbal formulations.

But the simplicity of the wording of these three principles is misleading. Behind each of them stands a whole world, and especially behind the first principle, systematicity. It's not as simple as it sounds. For, the principles of elementary mathematics are often hidden from view. We are so used to them that we don't notice their existence. It is easy to miss them, and the result is then building new floors on top of missing floors.

The point is that systematic teaching requires delving into minute and delicate facts, otherwise the study remains empty. For most high school students, the fact that the dash is a sign of division is empty words. They don't even really understand what a fracture is.

In algebra, the break line is preferred over the division sign, because the break line is also used as parentheses, and the saving in parentheses makes complicated expressions easier to read. And those who do not understand the connection to division get into trouble. Many high school students, even good ones, ask their teachers "just not fractions".

What's the solution?
As mentioned, there are no miracle solutions. But there is one solution that is very effective: good textbooks. Many of the authors of the textbooks believe that their role is to provide the child with interesting activities. Interesting activities are important, but they are not the main thing.

The main thing is to teach in order and in a tangible way. A textbook for elementary school that emphasizes the meaning of arithmetic operations, on concreteness, tells about the fine facts, can make a difference. Especially because the textbook teaches first of all - the teacher! The student spends time with the textbook once: the teachers return to it many times.

A good textbook is the best training teachers can get. If so, in my eyes this is the secret, and that is where the effort should be concentrated. Textbooks written according to the three principles I listed above can change the face of mathematics teaching, and have a positive effect on students' attitude to mathematics and their achievements.

Prof. Ron Aharoni is a faculty member in the Faculty of Mathematics at the Technion. Author of the books "Calculation for parents - a book for adults about children's mathematics" (Shoken, 2004) and "Mathematics, Poetry and Beauty" (Ha Kibbutz Ha'Ehud, 2008). Another book of his, "The Cat We Are Not There" (on the definition of the concept of "philosophy") is about to be published by Magnes.

18 תגובות

  1. .

    There is no point in teaching everyone math

    Time and time again, comprehensive tests and studies on high school graduates have produced the self-evident and predictable results - namely, that most of our young scholars know very little mathematics. Maybe they know how to add and subtract, somehow, maybe a little multiplication and a drop of division; But not much more than that.

    The immediate conditioned reflex response to studies such as these is one of horror. "Yah, how we have failed in our schools", sigh some community leaders; "We need changes, reforms, to ensure that our young people learn the advanced mathematics they need in order to get by in the modern world. Look at the Japanese schools, etc., etc., etc."

    First of all, I am completely sick of hearing about the Finnish, Japanese, Taiwanese, Russian, German, Swiss, French schools, and everything related to them. With all these wonderful schools of theirs, they fail to produce societies that are somehow more attractive than ours. The sense of personal freedom, self-fulfillment, creativity, vitality, optimism, liveliness, tolerance, compromise, openness, and all the other wonderful qualities that are components of our worldview - all of these are conspicuously absent or, at best, lit in a small flame in all these countries presented to us Supposedly as a role model. If their schools do such a great job, the results don't seem to last long enough to affect the lives of their citizens.

    But the real question is: Who cares about the results of studies like these? Isn't it a bit hypocritical to accept the ignorance of high school graduates in mathematics, when in fact the entire adult population in the country (and in fact in the entire world) knows about the subject to the same extent? Does anyone really believe that if the teaching and administrative staff of our local schools were tested, the results would be different than the students? With the exception of math teachers, and perhaps physics and chemistry, does anyone else have any reason to know or use math in their everyday or professional life? Do the managers of the big companies, or the partners of the leading law firms, or the members of the state or national legislatures, know the subject better? Some of them were able to deal with the problem below, which is a typical textbook problem: "The alcohol concentration had to be 52 percent. How many milliliters of a 60 percent solution must be added to 200 milliliters of a 20 percent solution to get the appropriate concentration?" I mean, how many people could give you an idea of ​​what a milliliter is, let alone how to calculate concentrations of different solutions and who cares.

    Talk about the new king's clothes! A certain group of educators decides which tiny part of the mathematical knowledge space should be taught to everyone in school, and everyone says "Amen" and puts it in the curriculum. Much more remains outside than is brought in, but no one knows how to do it better, and no one stands up and says: "This is nonsense. The areas you chose to teach are as irrelevant as the ones you left out. The king is completely naked!"

    I guess I'm already saying it, as loud as I can. I say it over and over and over again, as long as I have breath: there is no point in teaching 99 percent of the math curriculum that is taught in schools today, certainly not to someone who is not very interested in the subject. I have such a good training and such extensive knowledge in mathematics as can be acquired in a PhD program at a good university, so I am not talking about a field in which I am ignorant. The fact is that the effort to teach this material is a futile, wasteful effort, and may lead, as is the case today, to a general hatred of mathematics, and to resentment at the lack of meaning practiced in our schools.


  2. Why don't we adopt the way of many countries to separate male and female students to avoid distractions. The philosophers of three thousand years ago already understood that there is a contradiction between the pursuit of the evening and the desire for wisdom. I would also suggest strict censorship of advertisements to remove the madness of buying products and satisfying lesser desires and thus make room for high desires and ambitions that exist in every person, but in force and not always in practice. I would appreciate a response from thoughtful people.

  3. The "Hidan" website was destroyed, because a. Annoyed by comment #9, the destroyer of all exact sciences has arrived.

  4. Hello. First of all, I think it is very nice that Prof. Ron Aharoni decided to engage in parallel academic research in the improvement of mathematical education in the State of Israel! The question of why mathematics works in the world is a question for which there is no agreed solution. This is actually one of the most connected puzzles of mathematics. I think that learning mathematics is a proper kind that can also be a systematic study but will also keep the animals of the children's curiosity alive. This is of course not an easy task. Ron Aharoni decided to adopt the education system of Singapore. I am convinced that these are excellent books. I would be happy to share my conclusions with the discussion participants as they appear on Gan Adam's website. (link by clicking on my name) I think you can also find answers there to the question that Virgin Wigner asked about the improbable efficiency of mathematics.

    Moshe Klein

  5. teacher:
    I think there is a lot in your words.
    When I gave private lessons, I always tried to make it clear to the students that solving a problem is often a series of balanced formulations of the problem's data, where the last formulation is actually the answer.
    This difficulty in reading comprehension is also expressed in some of the responses to my articles "to be a mathematician"And the centrality of language in our thinking is expressed both in this article and in the article "The first word".

  6. jubilee:
    The article is very beautiful and explains well what mathematics is.
    However - I don't know what "math teaching" he is talking about and complaining about. He mentions K-12 there - do you know what that is? If your description of the program is correct - it does not seem to me that this is what is customary in Israel or that it has ever been customary there.
    Those who study mathematics do not study formulas.
    Usually they try to teach him to understand.
    Maybe they don't always succeed (and part of the blame lies with the teachers) but I have yet to come across a student who simply taught him the formula for the area of ​​the triangle by heart and didn't try to teach him the proof of it or maybe even try to make him find it himself.
    Beyond that - when you say that the article convinced you otherwise - it is not clear what the opposite is.

    Also his conclusion that the fact that mathematics studies are compulsory studies is the one that causes problems seems to me to be unfounded and wrong.
    In fact he might as well argue that it is compulsory education law that makes people want to be uneducated. Just nonsense.

  7. The stimuli that distract the mind such as television and the Internet and computer games, are the ones that interfere with studies.

  8. You have to invest much more in understanding than in learning the technique to reach a solution.

    Today the great pressure is to know how to get to the right answer and where the tests lead.

    It is possible to select the students with theoretical questions about mathematics such as after calculating a ratio of a triangle to compare it to something from the real world and for the student to know whether it is a larger or smaller ratio.

    May the mathematics be logical and practical in thinking to understand the trends of everything, the process that the function does. So really the Tamid will acquire mathematical thinking skills. After all, it is not really necessary to know how to solve exercises in industries, but to understand the thinking and act accordingly. There is no point in knowing how to open derivatives if you don't understand what the derived operation does and what the result we got actually means.

    I bet that more than half of the students in Israel who have completed full matriculation do not know what the result in the derivative means. Not in relation to the function itself and not in relation to anything else.

  9. If we ignore the "frightening" word mathematics, and use the calming word "measurement" we will reach surprising results
    Studying measurement is an understandable practical study, even though there are numbers in it.
    After learning to measure distances such as the length of a pencil = 21 cm and the length of a broom stick = 147 cm
    Immediately understand the concept of "scary" as a relative number.
    Right triangles are then drawn, and through measurements the ratio numbers between their sides are obtained
    which have scary names like sine tangents, etc
    In the next step, you can learn the Pythagorean theorem, and the sky is the limit.
    In short, measurement theory is the key to success.

    Best regards
    A. Asbar

  10. It is not for nothing that the retirement conditions are very difficult. There just aren't enough good math teachers out there.

  11. Where will the success come from if the teacher who studied 30 years ago still continues to teach and they don't allow him to retire and replace him with a fresher teacher - the retirement conditions are very difficult and they don't allow the teacher to retire until after 36 years or maybe after his death
    In my opinion, the Ministry of Education must allow early retirement and compensate those who retire and introduce young and fresh players into the lineup, and then success will come

  12. To learn mathematics you need to have mastery in reading and writing and reading comprehension and it is impossible without them
    In my opinion 1) you need to know how to read well and then 2) understand what you read - 3) build a plan or strategy through which we will reach solutions 4) you need to know how to check the final product (the answer)
    Where are our students stuck? In elementary school, they deal with many marginal subjects and miss the main thing, after all, reading and understanding - the student finishes elementary school and barely knows how to read and write, and in middle school, he is required to study mathematics at a high level that he was not used to, so in elementary school, the failure that cannot be corrected in regular studies is revealed

    From my many experiences in mathematics, in my opinion, we need to invest heavily in the elementary schools, and after the success in the elementary schools, greater success is built in the middle and high schools - (((There is no success in the high schools if there is no success in the elementary schools)))

  13. In the end, those who don't want to learn will never understand mathematics.
    Those who want to learn must first learn to break their heads.

  14. Why not transfer the mathematical education to computer training?
    I think that part of the problem in the classrooms is a student's inability to ask questions that go beyond not understanding a specific exercise, such as - "Teacher, what is a sinus?" "What do you do with an integral derivative" etc. It's a combination of group dynamics in high school mixed with the pressure of a tired teacher to get enough material and some problems of teaching level.
    Why not record solution steps for each exercise in the material being studied, allow the students to click in each exercise on options such as "explanation of basic terms" "practical implications of the material being studied (why it is actually good)" "another way to solve the exercise" explanation by another teacher, etc.
    The computer also makes it possible to statistically examine the level of each class student in each field, including the rate of progress - to identify individual problems and alert them and invest in a private lesson.
    In my opinion, this can also reduce the amount of formal tests and thus reduce math anxiety.

    Have studies and experiments been done in the field?

  15. What you called standards is accepted to be simply called 'stantsim'. Children today are taught heaps of tricks, techniques for solving questions and they have no idea that there is a possibility to really understand the material, they only experiment with a technical solution without understanding and thus the material is always scary.

  16. Interesting article, I am far from familiar with the world of studies, but as a high school student I suffered from mathematics and it was only in my late twenties that I discovered an affinity for the field. There are many good heads in Israel, it's a shame that so many are missed.

Leave a Reply

Email will not be published. Required fields are marked *

This site uses Akismat to prevent spam messages. Click here to learn how your response data is processed.