© Abramson Math, 2022

How does Abramson’s method help you learn math?

Contents:

“Who needs this math anyway?!” Introduction

This is a question, unfortunately, painfully familiar to many parents. Probably, when you were a school student, you asked it yourself. In this text, we will explain what is wrong with the usual school math curriculum, how to teach math to children so that it will be useful, and what to do if you decide to get back to studying mathematics as an adult.

If back in school you thought that mathematics was important and interesting — congratulations, you were lucky! “Lucky?” you would probably ask indignantly. “There was a lot of effort invested into my achievements in math!” And you would be partially correct. However, you should know that you were probably successful not because of, but in spite of, how the school curriculum had been designed.

Let’s be honest, many modern adults cannot really answer the question “why do you need math?” apart from passing the school-leaving exam. They tend to remember the algebra and geometry classes with horror.


It is especially sad when your child starts asking you this question, and you have no idea what to say. After all the knowledge that school gives you can be easily substituted with the help of modern electronic devices, and the children are really good at it.

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But if you follow this logic, you could conclude that most of the information that a child receives in school, in the best case scenario, will help them pass the school-leaving exam, but will not be useful in real life. How often did you have to solve irrational inequalities or trigonometric equations? Have you ever met Paramecium caudatum, or was it really important for you to remember the name of that guy from that book who drowned that poor dog?
For the school student struggling with a particularly difficult math problem, all the lofty words that their parents and teachers say about the development of thinking and future career possibilities mean next to nothing. They need different motivation that has nothing to do with distant realities. Even the idea of “studying math to get a good job in IT in the future” probably will not work. When these children become adults in a world where computer programs are written by other programs with little help from a human. Even today, logic is more important for an IT specialist than math, but these subjects are connected rather weakly in the school curriculum. Correctly organized teaching of mathematics (logical curriculum, attentive teacher, interesting problems, and so on) improves the quality of thinking. But where can you find that?

Adults often regret that they could not fully master the school math program. But to start learning math as an adult is also a difficult task in itself. The issue here is not motivation but the imperfection of the curriculum: it seems that it is designed for children and is too boring for an adult. I will tell you a secret: children are usually equally bored of calculating who took, gave, or ate how many apples.


But if math is explained in an interesting way that allows a person to see the familiar world in a new light, then studying math can be fun.


The next logical question is how to achieve that. How to teach math in a way that won't only be one interesting lesson, but would stay with a student for a long time?


First of all, let’s see what is wrong with the school math curriculum and why even the information presented interestingly is often poorly absorbed and remembered.

Do's and don'ts of teaching math.

“... In my childhood as I was studying arithmetics and solving the all-familiar problems about a person who is traveling from one city to another constantly changing means of transportation and therefore their speed, I kept asking my teacher about the character of this imaginary traveler, their age, aptitudes, habits, and the purpose of travel. My poor teacher had to improvise. Depending on the details that he had given me, I changed my answer. In some cases, I allowed for hold-ups at the taverns and conversations with friends; in other cases, I increased the given speed instead. It is obvious that when a person is in a real hurry, their desire to be in time is obeyed not only by their own legs and horses but even by amobiler and ships ..." Max Frei, Tubur game.

Why is it hard for school students to start learning math?
There are several answers.
  • 1
    Usually, drilling is encouraged. There is no room for understanding.
    Math problems in school often require not so much an understanding of the essence of the problem as monotonous cramming. Teachers tend to drill students in repetitive actions, sometimes even lowering the grade if somebody managed to solve the problem in some other way that was not provided in the curriculum.
  • 2
    There is usually no connection between problems in the textbook and a child's real life.
    Many math problems in textbooks have nothing to do with everyday life, but they also fail to provide enough fuel for a child’s imagination. Even if a problem has fairy tale creatures in it, it doesn’t automatically become interesting for children. In the soviet book “The Golden Key, or the Adventures of Buratino” based on “The Adventures of Pinocchio” by Carlo Collodi the main character Buratino, when asked to solve a math problem where somebody has hypothetically taken away one of the apples, answers that he would never give away an apple. Children tend to view math problems that do not account for their personal qualities and life experience as boring and sometimes even absurd.
  • 3
    The solutions to the problems are often evaluated incorrectly
    One of the most unpleasant things for school students is when their grade is lowered because of a mistake that has nothing to do with math. In primary school, one teacher usually teaches several subjects, and when they see that a correct solution is written crookedly and illegibly, they can give a lower grade. Parents tend to share these stories on social networks: “I checked my child’s homework — everything is correct, but the grade is lower than expected.”
All these issues make children simply learn math as a foreign language — they try to memorize a sequence of actions that will lead them to a result instead of understanding the underlying logic.

To make things worse, the school curriculum is designed in chronological order. It means that the topics are given in the same order as the consequent mathematical ideas were discovered. It seems logical. After all, simple things must have been discovered first. But it is not so.

Historically the driving force of math has always been practical issues: how to calculate an area and a form of a land plot taking into account its fertility, how to navigate at sea, how to build a house, and so on. These problems require particular branches of mathematics, which are not necessarily the most simple.

It can sound counterintuitive, but the logic of studying the subject should not be identical to the sequence of stages in its development.

For example, fractions appear in the school curriculum before the negative integers only because historically they have emerged earlier — as a response to the needs of traders and farmers. But negative numbers are very easy. You just have to realize that a minus times a minus equals a plus.

Besides, since children learn addition and subtraction before multiplication and division, it would make more sense to study negatives before fractions because you cannot understand the latter without division.

Where should you start?

Yakov Abramson:


Yakov Abramson
There are a lot of nonsense and discrepancies in the school curriculum. I wanted to change the very system on which it is based, using, first of all, the inner logic of the subject itself and second, accounting for the age-related psychological capabilities of primary school students. This led me to the complete overhaul of the program.
Children who study with Abramson Math are able to work with negatives already in the second grade. They learn such notions as a power of a number, polynomials, absolute value (a number that indicates the distance between the origin and point X on the graph), and many others way before their peers.

While still working only with integers, children in primary school can already add, multiply, and compose functions.
Also, before learning fractions, one can learn to solve systems of equations. A teacher just has to choose coefficients so that the solution will also be an integer. Children in primary school can learn how to add and multiply integers with vectors. And if you introduce them to one-halves (half-integers), you could teach them to calculate the squares of some polygons.

Later, with the introduction of fractions, it would be much easier for the children to start using them in the algorithms they already know well.

Another aspect of learning math from scratch concerns the moment when children get to know different numeral systems. The traditional school curriculum immediately restricts a student within the decimal system. Meanwhile, in order to visually demonstrate at least a three-digit number — the hundreds digit, a child would need to have a bundle of 100 sticks (or 10 bundles with 10 sticks in each). It is not an easy task, especially given that for a better grasp of the ideas, children should make these bundles themselves.

In the binary system, to visualize even a five-digit number, a child would not require more than 63 bricks. It means that they can easily visualize multi-digit numbers and work with them.
Abramson Math offers the following program: children start studying the binary numeral system, then switch to the ternary, and so on. First, they do it with actual material objects, then using only symbols — digits.

Children quickly learn to add and subtract multi-digit numbers and master conversion from one numeral system into another (for example, from the ternary into the binary). Later on, students can make and use times tables in different numeral systems.
Thus children learn that the decimal system is not the only possible one but one out of many variants of writing down numbers, which has been chosen only because people were using their fingers to count (it is no coincidence that we still say “to count on the fingers of one hand”).

Abramson Math is based on this logic: the new action or idea is an organic continuation of the previous ones. This sequence allows a student to see mathematics not as a number of discoveries from the past that do not allow for any creativity, but as an orderly system, which laws they understand and can consciously use for their own good. It makes it easier for children to understand the subject and even to fall in love with math.

Because Abramson Math accounts for the aspects of the psychology of learning, children naturally go from using existing algorithms to creating their own. It helps them develop critical thinking, and an ability to search for different, maybe even unordinary ways of solving problems, which can help not only in math but in other spheres of life as well.
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How did Yakov Abramson design his method from scratch?

This unique method came to life largely due to a lucky combination of events. The first such event was when a sixth-grader Yakov Abramson almost on a dare gets enrolled in a specialized math school. Second, Yakov’s grandfather was Piotr Galperin — a psychology professor, who was a passionate professional. Thus, growing up, the future author of a revolutionary method was equally influenced by both mathematics and psychology.

Yakov has always been interested in psychology. Although he cast his lot with the exact sciences and not with behavioral sciences or humanities, in some way he has followed in his grandfather’s footsteps, as he has used Galperin’s theory in practice building it into his teaching method.

Yakov Abramson:


Yakov Abramson
Since early childhood, I have heard conversations about psychology at home. My grandfather has already invented a theory of the systematic formation of mental actions and concepts. When I was still a school student, I had already realized that there was something wrong with the school math curriculum. There was something illogical in the way the material was structured. I wanted to change that.

It was clear, that you can’t immediately change the large textbook industry and retrain teachers who have been studying with these textbooks and are willing only to reproduce the familiar material. The breakthrough was possible only on a limited scale within single schools. So I tried to do it, checking my hypotheses and teaching in a new way but drawing upon my grandfather’s theory.

How is the math learning process changing with age?

In the middle of the 20th century, professor Piotr Galperin created theory of the systematic formation of mental actions and concepts. According to it, there are three types of learning:

1. Trial and error method, also known as the guess and check method. It is not very effective timewise but, for example, in the Montessori schools it is the only acceptable method.

2. Ready-made algorithms. This is a specially developed method that teaches with the help of ready-made schemes. It is the most effective method timewise. Unfortunately, it doesn’t allow for the development of a student's creative potential — why think of something new, when you have a template to follow?

3. Creation of algorithms. Here a student builds an algorithm themselves. It is a complex method that combines the elements of the first two types.

Abramson Math combines the second and the third types in its step-by-step approach. First two years children use only ready-made algorithms. As children in primary school have a more developed capacity for repeating actions, they get satisfaction from the very fact that they have successfully repeated an algorithm. Besides, there are a lot of problems involving construction, division, ferrying, and crossing out a given set of points with a continuous stroke of the pen in the form of a broken line from a given number of links.
Such organization of a lesson is seen by a child as a game, and because of that, they can handle abstract concepts. To support students’ creativity and their emotional involvement in the learning process, they are regularly presented with logical puzzles and so-called problems testing creative thinking.

As the students grow older, gradually the importance is shifted from the question “How to solve?” to the question “Why do we solve this way? What other options are there?” Synthesis gives way to analysis. The ready-made algorithms make room for the original ones: students with the help of a teacher begin to search for effective ways to solve the problems.

Yakov Abramson:


Yakov Abramson
People are curious by nature. Children do not have a negative educational experience yet, when every mistake is seen as a tragedy, so their curiosity is not thwarted. If everything is done right, children enjoy math classes: they love finding themselves in unknown situations, solving problems, and finding ways. When they work in a group, they inspire each other. Enthusiasm can be very inspiring.

Can adults learn math from scratch?

Of course, they can! Moreover, adults have bigger and faster-working brains, so Abramson Math would be even easier for them than for kids.

The only hurdle that adults could face is that they usually already have established and often false ideas about mathematics and it is quite difficult to influence them. It could be helpful if you are open to new knowledge and are prepared to make efforts to change your own familiar attitudes.

The form and duration of Abramson Math lessons are designed for primary schoolers. But for the adult who decided to get to know math better or start learning it from scratch, this design would only be beneficial. The easy-to-understand inner logic of the course, where every topic stems from the previous one, makes it simple for both adults and high school students, who want to improve their math fundamentals, to remember, rethink, and finally understand something that they had been studying, but failed to understand, and forgotten.

How does Abramson Math work? Details and examples.

For primary school, the author of the method gave up traditional pens and notebooks. Instead, he uses the dry-erase boards squared from one side, markers, and board dusters.

The thing is that children’s hands are not yet used to holding pens or pencils, so they spend too much precious time writing something down in their notebooks. Besides, for any brain to control an unusual activity is a source of additional stress that distracts from the main problem in hand. When children do not have an opportunity to write comfortably, they get tired faster and consequently lose the ability to concentrate on the topic of the lesson.
With the dry-erase boards, children can easily write big numbers, draw paintings, and simply erase any mistake they made. They raise their boards showing the answers to the teacher and receive their grades in the form of stars. It allows the teacher to keep the fast pace of the lesson and work with all of the children simultaneously, and not only with the one presenting in front of the class. Besides, not getting a star is psychologically more comfortable than getting an F. The things you write down in your notebook will stay there, but the dry-erase board will not even keep a trace of your mistakes.
As it was mentioned earlier, students learn numeral systems with actual material objects. We use bricks, lined sheets of paper, or dry-erase boards. Numbers represent the objects understandable to children, for example, the number of mice or cats.

In the binary system, two mice equal one cat, two cats equal one dog, two dogs equal one fox, two foxes — one wolf, and so on. They count them by hand, using bricks and putting them together.

We begin to teach addition also with the binary system because it is simple and obvious there. The numbers are already represented digit-by-digit and are added separately — mice with mice, cats with cats, so that the sum will be in the same numeral system. Children tend to master this procedure of adding “in a column” faster with longer numbers, so it doesn’t make sense to begin with adding two-digit or three-digit numbers. It is better to start with multi-digit (ten and more digits) numbers.
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Example. Learning the binary numeral system with Abramson Math.

Let’s assume that you have to present a binary number 10011 in the form of animals. Ones and zeros represent the presence or absence of an animal: 1 — the animal is present, 0 — the animal is absent.

The largest animal is on the left, and the smallest — on the right (in the decimal system, we write numbers in the same way: in the number 51, the leftmost digit is the biggest one — tens, the rightmost is the smallest — ones).

So, we have the number 10011. We know that the rightmost digit is the smallest animal — a mouse. We see that the rightmost number is 1, which means that the mouse is present. To the left is another 1 — that is one cat. The next digit is 0. It means that the next biggest animal, a dog, is absent. Another 0 means that the fox is absent as well. The leftmost 1 represents one wolf.
For the children, we present this process in reverse. They see 19 bricks in front of them — 19 mice. 18 mice broke up into couples and turned into 9 cats. One mouse is left. 8 cats also broke up into couples, leaving one cat, and turned into 4 dogs. 2 couples of dogs turned into two foxes, and two foxes made one wolf.
As a result, we have the following list:
  • 1 wolf
  • 0 foxes
  • 0 dogs
  • 1 cat
  • 1 mouse
Which in binary notation looks like 10011.
“...In fact, trolls traditionally count like this: one, two, three, many, and people assume this means they can have no grasp of higher numbers. They don’t realise that many can BE a number. As in: one, two, three, many, many-one, many-two, many-three, many many, many-many-one, many-many-two, many-many-three, many many many, many-many-many-one, many-many-many-two, many-many-three, LOTS.” Terry Pratchett, Men at Arms.
Abramson Math — free lesson
For children 6+ years old and their parents. In the first lesson, your child with your help will learn about numeral systems, how binary, ternary, and other numeral systems are organized, and how to convert numbers from the binary system to the decimal one and back.
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How do Yakov Abramson’s students become winners of math olympiads?

Not all students that study with Yakov Abramson have outstanding math abilities. However, they successfully participate and win math olympiads, sometimes beating not only their gifted peers but even the older children. What is the secret? The thing is, while others strictly follow the school curriculum, Abramson’s students manage to cover many different topics in the same amount of time and as a result, know more than their peers.

Besides, what is more important — children learn how to navigate new and unknown situations. Students constantly face such situations in class, which require them to intensify their mental effort, and that is exactly what they have to face at the olympiads.

In one year of Abramson Math children learn addition, subtraction, multiplication, division (with remainder), introduction to geometry, squares, logical puzzles, and many other topics.

In the second grade, they learn how to work with negatives, how to build functions and how to work with them, graph transformations, ​​addition and multiplication of polynomials, formulas for abridged multiplication, permutations, and geometry.

As a result, these children can consciously use mathematical knowledge, so olympiads in math and even in other exact sciences come easily to them, and ordinary kids become prize-winners.

Yakov Abramson:


Yakov Abramson
Our goal was not to give prime schoolers the program of the seventh or even higher grade. We wanted to build a curriculum based on the logical sequence of topics and children's psychological and age-related abilities of children.

In particular, we introduce our students early to logic, sets, and geometry. They can start developing the very idea of mathematical proof. Generally, we brought to primary school many of the concepts that are traditionally part of the high school and sometimes even university curriculum.

We see that children easily master these concepts, just as middle schoolers do three or four years later. Besides, younger children learn complex ideas with greater pleasure and interest.

This approach motivates children and gives them background for the more rampant and fast development. I am talking first of all about intellectual development — children develop what we call a “mathematical maturity” — they don’t just know particular topics, but they can navigate the new material.

How to help your child study math?

Our website offers online math courses, so younger children would require help from an adult. You need to teach a child how to enter the platform, how to watch the videos and help them with the text.

If your child is already seven years old, try solving additional problems with them. You can print them out, solve them together, and then watch the video on this topic with the strategies for a solution and correct answers. You can commend them for the solved problems and analyze those that caused difficulties.

Take note: a child should not get tired in the process. The lesson should not become annoying. It is better to spend only 20 minutes which the child would enjoy, than a long tedious class. If you see that your child is already tired, it is better to skip lessons on that day. However, it is best to keep a high frequency of lessons — several times a week, without significant breaks.
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How to overcome the psychological barrier while studying mathematics?

As a rule, psychological barriers appear when you experience failure in the process of studying the subject.

The main advice here both for a child and for an adult is to try and understand that failures are fine! You cannot achieve any results without learning from your own mistakes.

Children in this situation require help and support from their parents or other involved adults, who would not criticize them for the mistake, but instead would encourage their interest in the subject and insistence in achieving their goals, who helps grapple with the subject and delve deeper into it. Children need to understand that lessons are the place for training and that they can make and sometimes should make mistakes here to better understand what kind of actions lead to wrong results.

Concerning adults, it is important to be aware of and understand their motivation. You need to answer yourself: what problems would I solve, and what goals would I achieve if I improve my knowledge in mathematics?

Some goals can be very practical: to get a higher-paid and more interesting job requiring mathematics; to enroll in accounting or an advanced programming course (Big Data or genetic programming). For others, goals could be less practical but more important on a personal level — to widen one’s horizons, train one’s brain, or develop new neural connections.

Even if you simply want “to prove to that math teacher from your primary school that you were not stupid, and she really could not teach” — that would work too. This math teacher may have disappeared from your everyday life twenty years ago, but she might still live in your subconscious, criticizing you and gnawing at your self-confidence. To prove her wrong would be incredibly useful for your self-esteem!
Would you like a free lesson?
See for yourself how novel, interesting, and easy to understand math can be with the first lesson of Abramson Math.

The lesson is devoted to numeral systems (binary and decimal).

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