'Technology' teaches new maths in the world: Discover algebraic rules before counting 1, 2, 3, 4 ... (Part 3)

In Part 2, we learned that children in the United States and many countries start to learn math by counting numbers to get to know the four basic calculations, and then move from arithmetic to algebra. In addition to this popular approach, the world has a completely different approach to math: starting math from measuring length, volume, and learning about algebraic rules before coming to your children. number. It is the country's way of teaching mathematics that has produced many scientific and technological inventions that change the history of mankind in the past 300 years.

What is the new method of teaching math specifically and what are the results of testing this new method in the US? Invite readers to part 3:

The "technology" of designing mathematics develops abstract thinking from algebra to arithmetic of the Russians

This lengthy series draws from an article by Keith Devlin on the American Mathematical Association (MAA) website and related research and works. Keith Devlin is a famous American mathematician, former Stanford University professor of mathematics.

Experiment with new math teaching methods in America

The algebra-to-arithmetic approach is a method in the contemporary curriculum that starts with measurement instead of counting the number that Keith Devlin mentioned in part 2.

The syllabus will be the main focus of the rest of the series developed in the second half of the 20th century with two heads being educators and psychologists Vasily Davydov (1930-1988), DBElkonin (Daniil Elkonin, 1904 –1984).

The Davydov curriculum today is often called the name Davydov with the participation of many others, the most famous is DBElkonin. The foundation of the Elkonin-Davydov program is the cognitive theories of Lev Semenovich Vygotsky (1896-1934), a major developmental psychologist.

Vaisly Davydov, DB Elkonin and Lev Vygotsky are all famous scientists from Russia, the country with the world's leading mathematics. Mathematics and scientific thinking are the foundation for this country to pioneer the creation of many scientific and technical inventions for the world in nearly 300 years: computer technology, space travel, systems. street lights, helicopters, television .

Two of the few experimental studies of the Davydov elementary math program in the United States observed by Keith were very encouraging.

The first study, led by Jean Schmittau, a professor of mathematics pedagogy at New York National University of Education in Binghamton, led the three-year Davydov elementary math curriculum at a school in New York.

According to the report, the participating children can continually solve problems in notable challenges. The ability to maintain strong concentration, the conditions necessary for their success, are also growing. The time it takes for students to meet the challenges is about a year.

Surprisingly, "when they complete the program, they can solve common problems for US high school students only . "

Some argue that, in the age of cheap electronic computers, children do not need to learn how to calculate, and the time spent on calculations will actually hinder conceptual math. In response to these complaints, Professor Schmittau said: "It is impossible to accept the concept of conceptualization and the ability to solve difficult problems that have to be compromised with computer learning. Davydov children Not only do they gain the ability to apply mathematical rules and knowledge at a high level, but they can also analyze and solve problems that are often classified as difficult for American high school students. computers, and they solve each conceptual computing error without sticking to the rules . "

In addition, when they master computing skills, they also develop mathematical thinking and the ability to make new connections, the foundation of meaningful learning.

Meaningful learning is also known as deep learning, which is understanding concepts, associating new knowledge with old knowledge and applying new knowledge into practice, as opposed to memorizing as merely memorizing. .

The second study at two schools in Hawai'i used the Measure Up curriculum, a version of the Davydov curriculum for American children in Curriculum Research & Development Group (CRDG) Hawai'i University Pedagogical School. The two research authors are Barbara J. Dougherty, PhD in mathematics pedagogy, CRDG director and Dr. Hannah Slovin is a CRDC member.

The second research result is also very encouraging. "The problem-solving methods (problems) emphasize that young children are able to use algebraic symbols and generalized diagrams to solve problems. Diagrams and related symbols can represent the structure of a mathematical situation and are applied in many contexts ".

Snapshot of student learning math program Measure Up.(Photo: CRDG).

Take notes using algebraic notation for students of Mathematics Measure Up.(Photo: CRDG).

Math design develops "abstract" algebra thinking from grade 1

The Davydov curriculum applies a second approach: specific numbers are an abstraction of measurement results. However, if we only measure the length of an object or count the number of a certain group of objects, we only have the concept of spontaneous concepts. The Davydov method is also called the scientific concepts method. According to this method, to get a scientific concept of numbers, students need to go through a pre-digital stage before learning specific numbers.

Below is a description of the Davydov program according to Logical and said problems of elementary mathematics as an academic subject in 1975a.

The first part of the classroom is non-numerical (non-numerical) exercises about dimensions such as length, volume, and volume that are designed to increase complexity.

The first step is the "pre-math" step to prepare students to do these exercises.

In grade 1, the teacher will ask students to describe and identify the physical characteristics (length, volume, mass) of some comparable objects.

During this phase, when describing the results, students write statements such as A> B, B = C, A> C. A and B are unknown quantities being compared.

In this step, unknown quantities are not specific numbers. Using algebraic notation (this is the letter) before using numerical symbols helps students focus on abstractions in the first place. The physical situation was created to introduce meaningful "abstract" algebraic elements, children not seeing them abstract but very real.

Artwork of situation comparing volume.

The purpose of the above scenario is to help children discover equal and comparative relationships.

Next is the whole-body relation exercise. Students learn how to make unequal quantities become equal or make equal quantities unequal.

From a volume situation A> B, children can gain balance by adding to volume B or subtracting from volume A. Then they observe that, whether they choose to add or subtract, add or minus is the same, and that amount is called the difference - one of the first mathematical concepts that students in the program learn.

A> B
A = B + X
X = A - B
A = B + (A - B)

Only after students have mastered the pre-numerical knowledge of dimensions and whole-part relationships will they continue with tasks that require quantification.

For example, after working on mass and seeing that the mass Y is the whole and the masses A and Q are the components that make up the whole, they are encouraged to show it with a single diagram. Simple inverted V-shape like this:

Then continue rewriting the expressions in more formal ways:

Y = A + Q, Q + A = Y, Y - Q = A, Y - A = Q

The above expressions are a prelude to the stage of defining numerical values ​​for "variables" to solve equations derived from practical problems.

Numbers (here the real numbers) are introduced in the second half of first grade as abstract measurements of length, volume, mass and the like.

As a result, students not only do not need to memorize the rules of solving algebraic equations, but also increase their ability to reason directly about partial-total relationships.

By multiplication and division, the Davydov curriculum requires students to connect multiplication and division actions with previous knowledge of the measurement and value of place values, addition and subtraction, and application of operations. multiplying and dividing into problems involving measurement systems, other base systems (other base systems than the decimal system taught in grade 1), area and circumference, and solving more complex equations .

An exercise in the Davydov elementary math program in the US.(Photo: CRDG).

In other words, they learn new maths from a practical base and a connection to previously learned math knowledge. Students must explore two new mathematical operations and systematic connections between them and previously learned concepts. They are constantly presented with problems that they have to build in connection with the old knowledge.

Each new problem is different from the previous and then problems in a number of practical ways. This is in contrast to the US program, where problems are presented in sets with each focus focusing on a single process.

As a result, students have to constantly think about what they are doing to make that a reality for them. Many such design problems help children build connections between new actions (multiplication and division) with previous knowledge of addition and subtraction, positional systems, and equations. All of these problems will help children integrate their knowledge into a single conceptual system.

According to Keith, the Davydov curriculum is grounded in practice, but the starting point is that the measurement world is more continuous than the discrete counting world.

Both measurements and counting provide good specific starting points for a mathematical journey. People are born with the ability to make comments and reasoning about length, area, volume . as well as the ability to compare the size of groups of objects. Each capacity directly leads to a number concept but two different concepts corresponding to real numbers and numbers.

(Continue).

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