On the ordinary and gifted children
“You can call any child gifted if they are curious. Not all 100% of children have this trait, but those who have it can be taught.”


On the issues with the common school curriculum
“Actually, from the logical, psychological, pedagogical and many other points of view, one has to teach negative numbers before decimals. Negative numbers are very easy. You just have to realize that a minus times a minus equals a plus. There is no need to memorize anything, because it is all very natural. You can explain it in many ways, and even first-graders would understand.

But decimals are a really complex concept. You have to cultivate a mathematical culture, before you can turn to them. That is why I try to thoroughly develop this kind of culture in my students. They need to develop abstract thinking skills, learn how to work with letters, with polynomials, functions, and so on before beginning to work with decimals.”


Abramson Math in the primary school for ordinary children
In the first module of the second grade we are building integers, consequently, we can explore negative numbers, functions, and graphs. But our graphs consist only of points because we don’t have yet decimals or real numbers. So we can’t have any lines — only points. But these are still graphs, graphs based on the array of integers. Graphs of straight lines, absolute values, quadratic function. Then we turn to the graphs transformations, shifting graphs up and down, to the left and to the right, and reversing them. Addition and multiplication of the functions, polynomials, addition and multiplication of the polynomials, formulas of abridged multiplication, and their application for the quick count. Quick count technique.
In the second grade, we learn how to count quickly. We introduce dyadic rational numbers — one-halves, quarters, one-eights, and so on. We continue with planimetrics — we are going to have inscribed and circumscribed circles. Congruence of triangles on different criteria: three medians, three bisectors, and so on. Then we turn to the language of mathematical logic, and the theory of sets.
In the third grade — divisibility, criteria of divisibility, Diophantine equations, building rational numbers, a system of equations with 2 unknowns, and text problems for building equations. Consequently, once we have decimals — homothety, similarity.
In the fourth grade — elements of combinatorics, method of mathematical induction, vectors on the plane, 2x2 matrices, linear plane transformations, determinants, and their geometric meaning. This is what we will study in four years, in other words, the primary school curriculum.


How to hold students’ attention?
“How can you keep this emotional suspense? How do you make them not want to stop at all? How do you make them constantly strive for knowledge like for the carrot? What should you do to keep them in the race? That is the actual art. Keep them not too far away or you might lose them. But not too close, or the tempo will die down. You have to find that boundary, the boundary of the zone of proximal development, the boundary of their capabilities.

1.
Quick change of tasks, low repeatability. Children become weary when they keep doing monotonous, similar actions. A person gets tired of the monotony, of similar tasks. When they are faced with problems, where they unexpectedly reach a dead end, where they have to look out for some solution, where there is a constant search, always something new… Then time flies, and they don’t feel tired.

2.
Practising by overlapping. Usually, teachers tend to keep working on one particular topic until all the students master it 100%. I do things differently. Say, we are learning addition, for example, column addition. Students haven’t mastered it yet, and they still make mistakes, but we already start multiplication. Because they will still have to practice addition when they are multiplying. They are still working on addition while learning something else. So there is an overlap. We start with something new and keep practicing the previous topic. We don’t wait while all 100% of the students start adding numbers without mistakes. If 30-40% mastered it, we move to multiplication. And when they do multiplication they keep working on addition, as they have to add numbers over and over again. In the same fashion, I don’t wait until all 100% of students master subtraction. 60-70% have mastered it? Great, we move to division. As they divide numbers they still have to do subtraction over and over again. But there is no monotony in it, and they don’t get tired, as they are always doing something new. In the same way, you can teach how to solve a system of linear equations.

3.
A human is designed in such a way that they love being successful. And the children, who are developing faster than their peers, are bound to be successful in the various competitions. They like it, they like getting diplomas and presents. They come home, and their parents commend them. It is natural. But it is an emotional reinforcement. It helps their teacher as well. They want to keep learning this subject since everybody like getting carrots.

4.
Continuity. I give homework for the weekend, for the school breaks, and the summer breaks. As they say, constant dropping wears away a stone. Constant intellectual activity leads to the need for such an activity, boosts mental stamina, and in the end changes the personality of a student. You can’t see the difference during the first 2-3 months, but after a year it is clearly visible. This constant mental workload pays off.

5.
Continuity helps develop skills of independent work with the text. Because they have to work with the notes that I give them. But that’s later, this is a bridge to the secondary school, beginning with 6th grade.

6.
It is useful to… not explain anything! I never explain how to solve the problems, never in front of the class. I think that the less the teacher speaks, the better! The less he explains in the class the better. You shouldn’t explain anything! You can give some hints, but they can also be different.”


Working with little children
“We just speak with them. They do not write anything down. They don’t have notebooks. They have dry-erase boards, they draw with markers on them, and show me their answers. It is all fast-paced because they still have weak hands. It would take them a long time to actually write something down, so we don’t work in notebooks. However, some of them can write, so they write things down in notebooks whenever they want. But I do not demand it. We have only verbal communication, they only write something with markers on the dry-erase boards. They write something, show it to me, and erase it. But for their parents, I do write things down.”

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