New math teaching "technology" 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 other countries often start learning math by counting numbers to master the 4 basic operations, and then move from arithmetic to algebra. In addition to this popular approach, there is a completely different approach to math in the world: starting math from measuring length and volume and learning to discover algebraic rules before coming to children. number. That is the way of teaching math of the country that has produced many scientific and technical inventions that have changed human history in the past 300 years.

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

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

This long-term series is compiled from an article by author Keith Devlin on the American Mathematical Association (MAA) website and related studies and works. Keith Devlin is a famous American mathematician, former Stanford University math professor.

The algebra-to-arithmetic approach is the method in contemporary textbooks that begins with measurement instead of counting that author Keith Devlin mentioned in part 2.

The curriculum will be the main focus in the rest of the series developed in the second half of the 20th century led by educators and psychologists Vasily Davydov (1930-1988), DBElkonin (Daniil Elkonin, 1904) -1984).

The Davydov curriculum today is often called after Davydov with contributions from many others, the most famous being 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, a country with a leading mathematics background in the world. Mathematics and scientific thinking are the foundation for this country to pioneer in creating many scientific and technical inventions for the world in the past nearly 300 years: computer technology, space travel, systems. street lights, helicopters, television…

Two of the few experimental studies on the Davydov elementary math program in the US observed by Keith have shown very encouraging results.

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

According to the report, the children involved were able to consistently problem-solve in remarkable challenges. The ability to maintain strong concentration, a necessary condition for success, is also gradually developed. The time required for students to meet the challenges is approximately one year.

Surprisingly , “when they complete the program, they can solve math problems normally only available to American high school students” .

Some argue that, in the age of low-cost electronic computers, children don’t need to learn how to calculate, and that the time spent in calculations will actually hinder the conceptual learning of math. In response to these complaints, Professor Schmittau said : “It is unacceptable that concept formation and the ability to solve difficult problems must be compromised with learning computation. Children learn the Davydov program. Not only do they achieve a high degree of rule-applying and math knowledge, but they are also able to analyze and solve problems that would normally be classified as difficult for American high school students. computers, and they solve each computational error in terms of concept formation without clinging to rules” .

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

Meaningful learning, also known as deep learning, is understanding concepts, associating new knowledge with old knowledge, and applying new knowledge in practice, as opposed to memorization, which is simply 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 the Curriculum Research & Development Group (CRDG) project. University of Hawai’i School of Education. The two study authors are Barbara J. Dougherty, PhD in mathematics pedagogy, CRDG director, and Dr. Hannah Slovin, a CRDC member.

The results of the second study are also very positive. “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 be applied in many contexts”.

New math teaching "technology" in the world: Discover algebraic rules before counting 1, 2, 3, 4... (Part 3)
Photo of students studying Measure Up math program. (Photo: CRDG).

New math teaching "technology" in the world: Discover algebraic rules before counting 1, 2, 3, 4... (Part 3)
Measure Up math students’ notes in algebraic notation. (Photo: CRDG).

The Davydov curriculum adopts the second approach: concrete numbers are abstractions of measurement results. However, if we only measure the length of a certain object or count the number of a certain group of objects, we only get spontaneous concepts of numbers. The Davydov method is also known as the scientific concept method. According to this method, in order to have a scientific concept of numbers, students need to go through a pre-numerical period before learning specific numbers.

The following is a description of the Davydov program according to the work Logical and psychological problems of elementary mathematics as an academic subject, 1975a.

The first part in the class is non-numerical exercises in terms of dimensions such as length, volume, and mass designed in the direction of increasing complexity.

The first step is a “pre-math” step that prepares students for these exercises.

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

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

In this step, the unknown quantities are not specific numbers. Using algebraic notation (in this case, alphabetic characters) before using numerical symbols will help students focus on abstractions right from the start. The physical situation is created to introduce “abstract” algebraic elements in a meaningful way, young children do not see them as abstract but very real.

New math teaching "technology" in the world: Discover algebraic rules before counting 1, 2, 3, 4... (Part 3)
The image illustrates the volume comparison situation.

The purpose of the above situation is to help the child explore the relationship of equality, relation of comparison.

Next is an exercise on the part-whole relationship. Students learn how to make unequal quantities equal or make equal quantities unequal.

From the volume A > B situation, children can achieve balance by adding to volume B or subtracting from volume A. Then they observe that, whether they choose to add or subtract, the amount adds or subtracting is the same thing, and that amount is called disparity—one of the first math concepts that the program’s students learn.

A = B + X
X = A – B
A = B + (A – B)

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

For example, after working with masses and seeing that mass Y is the whole and masses A and Q are the parts that make up the whole, they are encouraged to show it with a simple diagram. Simplify an upside down V like this:

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

Then continue to rewrite the expressions in more formal ways:

Y = A + Q, Q + A = Y, Y – Q = A, Y – A = Q

The above expressions begin the stage of setting definite numerical values for “variables” to solve equations derived from practical problems.

Numbers (here 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 for solving algebraic equations, but also increase their ability to reason directly about part-whole relationships.

Coming to multiplication and division, the Davydov curriculum asks students to connect new multiplication and division actions with prior knowledge of measurement and place value, add and subtract, and apply operations. multiply and divide into problems involving measurement systems, other base systems (base systems other than decimal are taught in 1st grade), area and perimeter, how to solve more complex equations .

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

In other words, they learn new math operations both from a real-world basis and with connections to previously learned math knowledge. Students must explore two new math operations and the systematic connections between them and previously learned concepts. They are constantly presented with problems where they have to build connections with old knowledge.

Each new problem differs from the previous and subsequent problems in some practical ways. This is in contrast to the American program, where problems are presented in sets with each set focusing on a single process.

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

According to Keith, Davydov’s curriculum is grounded in practice, but the starting point is the world of continuous measurement rather than the world of discrete counting.

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

(To be continued)…