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

If the alphabet a, ă, â, b, c . is the first lesson when learning Vietnamese characters, then the numbers 1, 2, 3, 4 . are the beginning lesson in learning mathematics. Starting a mathematical journey with counting and natural numbers is also a tradition of teaching and learning math in many countries around the world. In the light of modern science, what are the advantages and disadvantages of this approach in terms of education and awareness? And is this the only way to build first math concepts for first grade students?

Invite readers to part 2 of the series on mathematical thinking and new methods to help students develop algebra thinking. The series shows the perspective of Keith Devlin, world famous mathematician who is a former Stanford math professor.

Part 2: Starting math, learning to count or measure first?

Another way to lay the foundations for a mathematical castle

We know that our ancestors began counting different types of antiques such as the V-mark on sticks and bones, sketching on cave walls, piles of gravel . at least 35,000 years ago, advanced. to more complex representations like the Sumerian clay 8,000 years ago, the emergence of abstract numbers and writing symbols to describe them about 6-7 thousand years ago. This development led to positive integers with addition and eventually rational numbers with addition, the development that we thought was due to trade-desires / needs to track assets and trade with people. other ethnic groups.

Through archaeological evidence, it is also evident that our ancestors developed systems of length and area to measure land, crop crops, design and construction of buildings. From a modern perspective, this is very similar to the way the real number system starts, although it is still unclear when those activities become numbers as we are aware of today.

Apparently, when guiding children to take the first steps on a long journey to mathematical thinking based on daily experience and human cognitive abilities, there are two possible ways to get started: the world is fragmented. including sizes of sets and a continuous world of lengths and volumes. The first one leads to natural numbers and counting, and the second to real numbers and measurements.

The current American math curriculum begins in the first way: following a linear order from positive integers and additions to negative numbers, rational numbers, and the real number system as the destination of the research process. This gives rise to the assumption that basic natural numbers are more natural than real numbers.

However, that is not the way things were historically revealed. If you build real numbers from natural numbers, you will face a long and complex process that took mathematicians 2000 years to complete the task at the end of the 19th century. Real numbers are a more elusive concept than cognitive natural numbers, or one that creates cognitive ones. Humans have a natural gift for generalizing discrete numbers from daily experience (the size of discrete object groups) and the natural sense of continuous quantities such as length and volume (area). is less natural) and generalizations in this area lead to positive real numbers.

In other words, contrary to the mathematical perspective, from a cognitive point of view, natural numbers are not more fundamental or natural than real numbers. They all arise directly from experience in our daily world. Moreover, they arise in parallel from different cognitive processes, which are used for different purposes without being dependent on one another.

In fact, there is little evidence from the current neurophysiological field that real-perceived numbers of sequential numbers are more fundamental than natural ones built on a sense of continuous numbers by capacity. our language. For more details, read recent books and articles by researchers such as Stanislaw Dehaene (author, French cognitive neurologist) or Brian Butterworth (famous neuropsychologist). The British studied many fields, including mathematical psychology).

Which platform is better?

If we start with measurement, positive numbers and rational numbers will arise as special points on the continuous number line. And starting with counting, real numbers will arise by "filling in the blanks" on the rational number line. And in both cases, you have to manage with the best negative numbers possible when a need arises.

From a learning perspective, no method offers significant advantages compared to the other. Choose and live with the consequences when choosing the corresponding curriculum.

It is true that mathematically it is much more difficult to construct the concept of real numbers from natural numbers than to recognize natural numbers and rational numbers on real numbers, but the problem here is not about calculating mathematics. The form in which the problem is human perception is based on daily experience.

In the United States and many other countries, teachers have traditionally chosen to start teaching math by counting, and the starting point for a mathematical journey is a natural number. But there is at least one serious attempt to design a math curriculum that completely follows that method, and that is the focus of the rest of this article. Not just because I think one way is better than the other but in essence, but whatever method we apply is more likely that we will work better and better understand what we are doing in the field of education. if we are aware of an alternative approach.

In fact, knowledge of a different approach can help us guide students through difficult areas such as the concept of multiplication, the subject of an old article in this section of mine (Devlin's Angle section on the website. American Mathematical Association - MAA). According to Piaget and others also mentioned quite a lot, helping students understand multiplication in the curriculum starting with counting is extremely difficult. The curriculum that begins with measurement is the opposite, simply because the annoying and complicated complications of multiplication when counting do not occur.

Ways of teaching the concept of multiplication for elementary students according to the current national capacity standards in American mathematics (CCSS).

Jean Piaget (1896-1980) was a famous Swiss psychologist in the 20th century, who made important contributions to the theory of cognitive development in children and constructivist learning theory.Constructivism sees learning as a positive constructive process. Learners accumulate knowledge based on personal experience and the application of new knowledge in practice instead of passively receiving knowledge like traditional learning.

Could the way forward to be more successful in early math education is to apply a hybrid approach, building concepts based on both human intuition at the same time?

I suppose, this can happen to some extent. American children starting with counting also use length, volume and other real-life measurements in their lives, and children starting with measurements can certainly count, add and subtract natural numbers. before going to school. But I am not aware of any formal curriculum that tries to combine these two approaches.

No matter which of the two approaches is applied, the main goal of the 12 years of maths in the world today is the same: to equip future citizens with real-world knowledge and fluency. procedural fluency with real numbers. In the American school system, this is done in a way that evolved from the early stages of natural numbers, integers, rational numbers in "arithmetic" and real numbers in "algebra".

It should be noted that putting real numbers within the "algebra" subject to the US approach is to ensure a purely procedural measure, avoiding many of the difficulties involved in concept building. real numbers from rational numbers. So, finally, the method that starts with counting must also rely on our daily intuition and experience with continuous measurements.

(Continue).

1. Maths not only come from everyday experience, but are also games of logical thinking (Part 1)