Raising the bar

The evolution of school math, from the Mesozoic era to the present day

To write this post, I looked through all the current math textbooks used in schools in Kazakhstan, a couple of Soviet textbooks and a couple of American ones.

The content has become more advanced

In the Mesozoic era, when the current dinosaurs of Kazakhstani mathematics went to school, limits, derivatives and integrals were an elective. This advancement of school mathematics towards modern mathematics was achieved thanks to the efforts of A.N. Kolmogorov.

Soon these sections, which together make up the basics of analysis, were included in the main program of the 9th and 10th grades. Any innovations were met with strong resistance under the pretext that the existing program was overloaded. Therefore, subsequent additions were made only after 30 years. This is the theory of probability and vectors, and with them accompanying tools such as combinatorics and the scalar product.

Conclusion on the content: it has become more complex, closer to modern mathematics and adequately reflects the basics. The exception is symmetric polynomials: I have been doing mathematics for 50 years, and I have never needed them either in theory or in practice.

Many topics have been shifted down 1-2 grades

The 1st grade textbook assumes that students can read. Can everyone? How justified is it to devote 50 pages to studying the first 10 numbers?

In grade 2, letter symbols and an equation with one unknown appear. Isn’t it too early?

Verbal multi-step problems are introduced in the 3rd grade. This is normal for their age, but they require special teaching methods. Many students have difficulties at this level that do not disappear until they graduate from school. A simple method I used with my children allowed them to overcome these challenges with ease. It also ensured a smooth transition to equations with unknowns. Read on Facebook.

In grade 4, the use of formulas and equations turns from occasional to regular. Is everyone ready for this? Many of the American students I taught had an organic aversion to formulas.

In the 6th grade, technical difficulties increase. The pictures below explain the situation. Imagine that you only have a shovel and you need to climb a high brick wall (picture on the left). Anyone will understand the idea: to make a mound of earth (picture on the right), but not everyone will implement this idea.

Idea vs techniques

To master elementary school mathematics, understanding ideas and working occasionally is enough. Starting from the 6th grade and especially in high school, systematic work is necessary to consolidate the technique. This explains the following paradox: less capable but diligent students often learn better than more capable fidgeters. I often tell my students: “Your knowledge is not the books on your shelves or what you can find on the Internet. It's what you carry in your head."

The textbooks I looked at helped me understand another paradox, which I will explain with an example.

Last academic year I had a student who, by my standards, I considered capable. He quickly grasped my explanations, could reproduce quite complex calculations, but for some reason sometimes he floundered in simple questions. One of them is absolute value. It turns out that it is given in the 6th grade. The explanation for the paradoxical contradiction between person’s abilities and his misunderstanding of simple things, in my opinion, is this: a person who did not understand something at one time tends to ignore this something later. The reasons may be different: associated negative emotions, or the impression that this is well known, or reduced demands on oneself.

The trouble with most high school students is that there are many holes of this kind left in their knowledge from previous classes. In this regard, the illustrations below: on the left there is a slender building of science, without holes or flaws, on the right there is a building with holes that easily collapses under increased load.

Knowledge building

The vaunted pedagogical principle of relying on student interest fails disgracefully in the upper grades. The psychology of people is such that they enjoy learning what they understand. And if understanding is zero, and the technical obstacles leading to understanding are great, then there is zero interest. It is a vicious circle: without interest there is no study, without study there is no interest.

Conclusions on material distribution. Even with the changes described, the program is feasible for good students. On the other hand, for poorly prepared students from about 6-7 grades it becomes hopelessly difficult. At the 6-7 grade level, the vertical structure of mathematics becomes pronounced, which leads to the need to master all previous material, in order to understand new material. Intuitive understanding, which is sufficient in elementary school, is not enough in high school. There is an urgent need to improve the memory and logic of high school students.

Teaching methodology

Primary school students do not have developed analytical skills. The entire methodology is built on the “Do as I do” principle; students are given many standard exercises to reinforce the material.

In high school, the material is much more complex and following the same methodology leads to a contradiction between the theory and exercises. Since the theory is advanced, theoretical level exercises are required to consolidate it. The theory is given, but theoretical thinking is not instilled. Instead, textbook authors provide many exercises of a computational nature. Some exercises have no practical applications and are completely meaningless from a theoretical point of view. Below is a screenshot of one such exercise from a book for grade 11. Giving such exercises is the best way to discourage interest in mathematics.

Example

Republican National Test questions are compiled in accordance with the program and are quite advanced.

In the United States, there is no single program for the entire country. Each state has its own program, and schools have some freedom within that program. There are many competing textbooks, while we only have 1-2 textbooks for each class.

Our textbooks lag far behind American ones in terms of intuitive explanations, illustrations, and exercises, both applied and conceptual. The design of our textbooks is much worse. On the other hand, American textbooks are too verbose (some reach 1500 pages). Such verbosity in itself creates a problem, especially when textbooks in all subjects are like this. I once emphasized that mathematics is a speculative science and excessive verbalization interferes with the study of mathematics.

Many parents can help their children learn languages ​​and literature. When it comes to math, most parents' options are limited to the first few grades. Children are like plasticine; in skillful hands they can reach the pinnacle of science. In high school, this plasticine is completely in the hands of teachers and parents are powerless to change anything.

I am convinced that the transition from a 10-year school to an 11-year school was caused by the inability of teachers to meet 10 years with the existing methodology. I am also convinced that children of average ability, with the right methodology, can do it within 10 years, as my children did.