New technology 'technology' teaches math in the world: Discover algebraic rules before counting 1, 2, 3, 4 ... (Part 4)

In contrast to the American way of teaching math, the Russian Davydov math program begins with measuring and learning the rules of algebra before arithmetic. Experimental results at US schools in 2004 showed that Davydov students could solve math problems with symbols, general diagrams and develop algebraic reasoning ability right from grade 1. In 2010, The US National Science Foundation decided to grant US $ 2 million to universities in the United States to study integrated teaching of rational numbers at the elementary school level using the "technology" of Davydov math teaching.

What is the advantage of teaching Russian-style maths over traditional American methods and the rest of the world for Americans to spend 2 million dollars studying it? Invite readers to part 4.

According to mathematician Keith (the main author of this series), the analysis from all three angles above, the math approach of the US and Russia has some differences.

The first difference: math

The US 12-year math program puts the knowledge and competence of real numbers to an end, and the first years are progressions from positive integers, fractions, rational numbers / negative integers, and systems. Real statistics are included in the following grades mainly in algebra.

In contrast, Russia's Davydov program (also known as Elkonin-Davydov) places its sights on the real-number system from the beginning.

At what point does the second approach outperform the first?

Education-psychologist Davydov said that starting with natural numbers (numbers) would lead to difficulties later, when students work with rational numbers, real numbers, or algebra exercises.

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According to Keith, if learning is based on taking over spontaneous concepts, starting with counting, the familiar sequence from natural numbers to rational numbers will automatically arise. However, jumping to real numbers is a difficult step both mathematically and cognitively. It was not until the late 19th century that mathematicians actually realized that from natural numbers to real numbers was difficult. "Filling in the gaps on the rational number line" is a tricky thing, when the rational axis is "dense" as the mathematicians call it, there doesn't seem to be any gaps.

When geometry and trigonometry are no longer obscure, the American curriculum still avoids the problem of "what is real" by putting the real number system into algebra, where the focus on mastering the rules is more important. is mastering the concept.

And clearly, the Davydov method does not have such difficulties. When the real number system is the basic axis, integers and rational numbers are just special points on the real number line.

Another advantage of the Davydov method is that there are no more complex problems with the introduction of multiplication and division that the common start-to-count method is, since multiplication is a self-concept. in the world of length, volume, mass . and the whole-parts relationship between them.

The successful application of the Elkonin-Davydov elementary math program in the US has led the National Science Foundation to decide to grant a large sum of money to scientists in this country to study the integration of the start-to-equal approach. measurement to teach rational numbers in American curriculum. According to Education Week, a grant of up to $ 2 million over a five-year period goes to research teams from three New York Universities (NYU), Iowa (ISU) and the Illinois Institute of Technology (IIT). Research focuses on multiplication, division, fractions and ratios.

The Davydov curriculum is potential for American educators because "measurement concepts help students not only learn numbers but also learn quantities and measures in units to build the basis for relationships." ", says Martin Simon, a professor of mathematics at New York University, a member of the research group.

The National Science Foundation (NSF) is the only US federal agency responsible for supporting all basic engineering and science disciplines, except medicine.

The second difference: teaching methods.

The method most American teachers use usually includes a tutorial lecture with examples, exercises to practice specific skills illustrated by the instructor in the classroom. As for Davydov's math curriculum, it requires students to participate in solving many problems with measures of implementation and developmental awareness in an increasing direction. The problems raised are designed in careful order.

Davydov's problem-solving teaching method is a combination of pedagogy and psychology, based on the cognitive theories of famous Russian development psychologist Vygotsky (1896-1934). The theory of socio-cultural activities on the development of Vygotsky's human mind, which was born in the early 20th century, created a revolution in cognitive science and a foundation for many researches and development doctrines. Other perceptions in the world since then.

According to Vygotsky, cognitive development occurs when we encounter a problem that previous methods of solving were not enough to cope with. That was his conclusion after many studies on the development of primitive people, children and traditional tribes. You can find more in Studies on the history of behavior: Ape, primitive, and child by Vygotsky and Luria in 1993.

Picture 2 of New technology 'technology' teaches math in the world: Discover algebraic rules before counting 1, 2, 3, 4 ... (Part 4) Students study the Davydov program in the USA.(Photo: CRDG).

The third difference: scientific concepts

A distinguishing feature of the Davydov approach is the distinction of two types of Vygotsky concepts : scientific concepts and spontaneous concepts, also called living concepts, empirical concepts. ).

The notion of activity arises when a child generalizes the characteristics of daily experience or concrete examples, the scientific concept develops from formal experience with its characteristics. Regular experience is the experience in formal education in the classroom with the guidance of trained teachers.

This difference is more or less (but not entirely) similar to the two concepts discussed by Keith in Part 1 (Mathematics not only comes from everyday experience but also games of logical thinking): the other The difference between math is learned by abstracting from the world and math is learned by following the same rules as playing chess.

For example, children learn positive integers by counting groups of objects, thereby occupying a spontaneous concept that comes from generalizing the number of groups of objects of the same number. Three people, three apples all have in common is that their number is all three. After counting the numbers, children have a spontaneous concept of "three". Children before 1st grade were able to recognize numbers through the teaching of their parents and surrounding people, meaning they had a spontaneous notion of numbers.

Picture 3 of New technology 'technology' teaches math in the world: Discover algebraic rules before counting 1, 2, 3, 4 ... (Part 4)

Learning to play chess will lead to a " scientific" understanding of this game. As stated in part 1, based on Keith's experience as a senior math teacher and teacher, the scientific method is the most effective and possibly the only way to study a highly abstract subject. as integral.

One question Keith once posed in Part 1 is, where does this kind of abstract-it-from-the-world, spontaneous concept come to and the type of math? Where do you learn the rules of science (learn-it-by-the-rules, scientific concepts)?

As stated, it's an obscure naive question because in reality, the world is a constantly changing spectrum rather than a breakthrough.

From an educational perspective, the above question should be rewritten into a more useful sentence than, which parts of mathematics should be taught in a spontaneous notion and which parts should be taught in accordance with scientific notions?

In the traditional way of thinking in America, spontaneous method is a way to go all the way to mathematics, at least up to grade 8, and possibly all the way up to grade 12.

In the Davydov curriculum, the scientific concept method is applied from day one.

Davydov said that learning math from the "scientific" method from general to specific (also called abstract to specific, general-to-specific) will lead to mathematical knowledge and competence. preferably in the long run spontaneously. If very young children start learning math with abstractions, they will be better prepared to use formal abstractions in later years of school, and develop thinking in a way that helps them solve it. More complex math problems.

Formality, formulas in math are the use of letters in the alphabet following certain rules.

Picture 4 of New technology 'technology' teaches math in the world: Discover algebraic rules before counting 1, 2, 3, 4 ... (Part 4)
Photographs of experiments and notes of students studying the curriculum Davydov.They use algebraic notation before learning specific numbers.(Image: Maria Mellone)

"Nothing in the intellectual capacities of elementary school children prevents the digitization of elementary school mathematics. In fact, this will bring and increase the possibilities children have to learn math." , Davydov wrote in Logical and said problems of elementary mathematics as an academic subject in 1975a.

One thing Keith emphasized is that Davydov's scientific-conceptual approach is not the same as teaching abstract, axiomatic mathematics (new knowledge derived from given axioms, ways of teaching and learning math). follow-rules above).

And so, Keith's comparison of learning to play chess from the beginning of the series to this day is no longer usable, like all comparisons sooner or later become lame even though at first they are useful. anyway.

In Davydov's scientific concept method, the theoretical basis is based on real experience, and there are many such real experiences. Before coming to explicit mathematics knowledge, students studying the Davydov curriculum will spend more time for practical activities at the beginning than those studying the American curriculum. Later, when mathematical concepts were actually introduced, they were presented in a scientific way. Students are able to associate a scientific concept with a practical experience not because that concept arises randomly from real experience but because students are guided through a variety of hands-on, hands-on experiences. preparedness, from which they can immediately see how concepts apply to the real world.

From the perspective of the cognitive metaphor mentioned in Part 1, the cognitive mapping in scientific method is built from new to old awareness, as opposed to Lakoff and Nunez's learning system from old to new.

An example of learning the concept of science from Davydov's perspective is the concept of real numbers derived from the learning situations outlined in section 3. After comparing the volume, the volume of objects being measured, the student Davydov's syllabus records his remarks in the form of wildcards and transforms the clauses according to the teacher's instructions.

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After students master large, small, equal, part-whole relationships, they are guided to the problem of quantifying variables in the equation. When weighing a specific volume or volume of units, students will understand the essence of the concept of real numbers that represents the relationship between a unit and a given quantity, an abstract measurement of dimension. length, volume, mass, etc.

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That's how Davydov teaches students to grasp the science of real numbers!


  • 'Technology' teaches new maths in the world: Discover algebraic rules before counting 1, 2, 3, 4 . (Part 3)
  • New technology 'technology' teaches math in the world: Discover algebraic rules before counting 1, 2, 3, 4 . (Part 2)
  • Maths not only come from everyday experience, but are also games of logical thinking (Part 1)

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