New math teaching "technology" 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 measurement and learns algebraic rules before arithmetic. Experimental results at schools in the US in 2004 showed that students studying Davydov’s curriculum could solve math problems using symbols, schematic diagrams and develop algebraic reasoning ability right from 1st grade. In 2010, The US National Science Foundation decided to subsidize 2 million USD for universities in this country to research and integrate teaching rational numbers at the elementary level with Davydov math teaching “technology”.

What advantages does the Russian-style math teaching method have over the traditional American method and the rest of the world so that Americans have to spend million to study it? Welcome to part 4.

According to mathematician Keith (the main author of this series), analyzing from all three angles above, the American and Russian approaches to math have some differences.

The 12-year American math program places knowledge and ability to compute real numbers as the end point, and the first years are the progression from positive integers, fractions, rational/negative integers, and the system. Real number systems are introduced in later classes mainly in algebra.

In contrast, Russia’s Davydov (also known as Elkonin-Davydov) program set its sights on the real-measurement system from its inception.

Psycho-educationist Davydov believes that starting with natural numbers (counts) will lead to difficulties later, when students work with rational numbers, real numbers or algebra exercises.

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

According to Keith, if learning is based on the acquisition of spontaneous concepts, starting with counting, the familiar sequence from natural numbers to rational numbers will arise automatically. However, jumping to real numbers is a difficult move both mathematically and cognitively. It was not until the end of the 19th century that mathematicians really found that going from natural to real was a difficulty. “Filling in the gaps on the rational number line” is a tricky business when the already “dense” rational axis, as mathematicians call it, doesn’t seem to have any vacancies.

When geometry and trigonometry are no longer ambiguous, the American curriculum still avoids the problem of “what is a real number” by bringing the real number system into the algebra section, where the focus on mastering the rule is more important. is to master the concept.

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

Another advantage of Davydov’s method is that it doesn’t present the more complicated problems of introducing multiplication and division that the counting-start-with-count method does, since multiplication and division are natural concepts. nature in the world of length, volume, mass… and the part-whole relationships between them.

The results of the successful application of the Elkonin-Davydov elementary math program in the US led the National Science Foundation to decide to give a large grant to scientists in this country to study the integrated approach of starting-with- Metrics to teach rational numbers into the American curriculum. According to Education Week, a grant of up to $ 2 million within 5 years is for a research team from three universities of New York (NYU), Iowa (ISU) and the Illinois Institute of Technology (IIT). The focus of the study is on multiplication, division, fractions, and ratios.

The Davydov curriculum has potential for American educators because “measurement concepts help students learn not only numbers, but also quantities, how to measure units to build a foundation for relationships. system in multiplication,” said Martin Simon, a professor of math education at New York University and a member of the sponsored research team.

The National Science Foundation (NSF) is the only federal agency in the United States whose mandate is to support all branches of engineering and basic science, with the exception of medicine.

Second difference: teaching method.

The method used by most American teachers usually consists of an instructional lecture with examples and exercises to practice specific skills illustrated in class by the instructor. The Davydov math curriculum, on the other hand, requires students to engage in solving multiple problems with incremental measures of performance and cognitive development. The problems posed are all designed in a careful sequence.

Davydov’s problem-solving teaching method is a combination of pedagogical science and psychology, based on the cognitive theories of the famous Russian developmental psychologist Vygotsky (1896-1934). Vygotsky’s theory of socio-cultural activity on the development of the human mind, born in the early 20th century, has revolutionized cognitive science and is the foundation for many researches and theories on development. different perceptions in the world since then.

According to Vygotsky, cognitive development occurs when we encounter a problem that previous methods of solving are not enough to deal with it. That is his conclusion after much research on the development of primitive people, children and traditional tribes. You can read more in Studies on the history of behavior: Ape, primitive, and child by Vygotsky and Luria in 1993.

New math teaching "technology" in the world: Discover algebraic rules before counting 1, 2, 3, 4... (Part 4) Students study Davydov program in the US. (Photo: CRDG).

A prominent feature in Davydov’s approach is the distinction between two types of concepts according to Vygotsky: scientific concepts and spontaneous concepts, also known as living concepts and experiential concepts. ).

The concept of living arises when children generalize features in everyday experience or concrete examples, the scientific concept develops from formal experience with its features. Formal experience is experience in formal education in the classroom with the guidance of trained teachers.

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

For example, children learn positive integers by counting groups of objects, thereby acquiring 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 that their number is three. After counting numbers, children have a spontaneous concept of “three”. Before entering grade 1, children have the ability to recognize numbers through the instruction of parents and people around them, that is, they have a spontaneous concept of numbers.

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

And learning to play chess will lead to a ” scientific” understanding of the game. As stated in part 1, in Keith’s experience as a student and senior math teacher, the scientific method is the most effective and possibly the only way to learn a highly abstract subject. as integral.

One question Keith asked in part 1 was, where and what kind of math does abstract-it-from-the-world end? Where does learn-it-by-the-rules, scientific concepts, start?

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

From an educational perspective, the above question should be rewritten into a more useful sentence, which parts of mathematics should be taught in terms of spontaneous concepts and which parts should be taught in terms of scientific concepts?

In the familiar American way of thinking, the spontaneous method is the way to go all the way through math, at least through 8th grade, and possibly all the way up to 12th grade.

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

Davydov believes that learning mathematics according to the “scientific” method from general-to-specific (also known as abstract to specific, general-to-specific) will lead to understanding and mathematical competence. better spontaneously in the long run. If very young children begin math with abstractions, they will be better prepared to use formal abstractions in later school years, and develop thinking in a way that helps solve problems. more complex problems.

Formality, a formula in mathematics is the use of letters of the alphabet that follows certain rules.

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

“There is nothing in the intellectual capacities of elementary school children to prevent the algebraicization of elementary school math. In fact, it will enhance and enhance the abilities children already have to learn math.” , Davydov writes in Logical and psychological problems of elementary mathematics as an academic subject, 1975a.

One thing Keith emphasizes is that Davydov’s concept-scientific approach is not the same as teaching mathematics in an abstract, axiomatic fashion (new knowledge is inferred from given axioms, how mathematics is taught and learned). according to the rules above).

And so, Keith’s comparison of math to chess from the beginning of the series to the present is no longer applicable, like all comparisons that sooner or later become lame, however useful they may be at first. anyhow.

In Davydov’s scientific conceptual approach, the theoretical basis is based on actual experience, and there are many such real experiences. Before coming to the explicit math, students in the Davydov curriculum will spend more time on practical activities at the start than students in the American curriculum. Then, when mathematical concepts are actually introduced, they are presented in a scientific way. Students are able to associate a scientific concept with real-world experience not because the concept arose randomly from actual experience, but because they were guided through a variety of rich real-world experiences that brought preparedness, so they can immediately see how the concepts apply to the real world.

From the perspective of cognitive metaphor mentioned in part 1, the cognitive mapping in the scientific method is built from new to old perception, in contrast to Lakoff and Nunez’s learning system going from old to new.

An example of learning a scientific concept from Davydov’s point of view is the concept of real numbers that comes from the learning situations outlined in section 3. After comparing the volumes and masses of measured objects, students Study the syllabus Davydov records his comments as wildcards and transforms the clauses according to the teacher’s instructions.

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

After students master the big, small, equal, part-whole relationships, they are led to the problem of quantifying the variables in the equation. When weighing and measuring specific masses and volumes in units, students will understand the conceptual essence of real numbers as representing the relationship between a unit and a certain quantity, which is an abstract measurement of dimension. length, volume, mass, etc.

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

That’s how Davydov teaches students to grasp the scientific concept of real numbers!

(To be continued)…