**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 first lessons learned in math.** Starting the math journey with counting and natural numbers is also a tradition of teaching and learning mathematics 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 grade math concepts?

Welcome to part 2 of a series on mathematical thinking and new ways to help students develop algebraic thinking. The series presents the perspective of Keith Devlin, a world-renowned mathematician who is a former Stanford University math professor.

We know that our ancestors started counting different types of artifacts such as V marks on sticks and bones, sketches on cave walls, piles of pebbles… at least 35,000 years ago, progressing forward. to more complex representations such as Sumerian clay 8,000 years ago, the appearance of abstract numbers and written symbols to describe them about 6-7 thousand years ago. This evolution leads to positive integers with addition and eventually to rational numbers with addition, a development we think is driven by commerce-the desire/need to track assets and trade with people. other ethnic groups.

From archaeological evidence, it is also clear that our ancestors developed systems of measuring length and area to measure land, crop crops, design and construct buildings. From a modern perspective, this is very similar to how the real number system began, although it is unclear when those operations became numbers to the extent we perceive them today.

It seems that, when guiding children to take the first steps on the long road to mathematical thinking based on everyday experience and human cognition, there are two possible ways to start: the discrete world includes the size of the sets and the continuous world of lengths and volumes. The former leads to natural numbers and counting, and the second leads to real numbers and measurements.

The current American math curriculum begins in the first way: going in a linear order from positive integers and addition to negative numbers, rational numbers and the real number system as the goal of the study. This gives rise to the assumption that the natural numbers are more fundamental or more natural than the real numbers.

That’s not how things are historically revealed, though. If you construct real numbers from natural numbers, you will face a long and complicated process that took mathematicians 2,000 years to complete in the late 19th century. The real number is a more elusive concept than the cognitively natural number, or one makes the other cognitively. Humans have a natural gift for generalizing discrete counts from everyday experience (the size of discrete groups of objects) and a natural sense of continuous quantities such as length and volume (area). analysis is less natural) and generalizations in this area lead to positive real numbers.

In other words, contrary to the mathematical point of view, from the cognitive point of view, natural numbers are not more fundamental or natural than real numbers. They all arise directly from our everyday world experience. Moreover, they arise in parallel from different cognitive processes, used for different purposes, independent of each other.

In fact, there is little evidence from current neurophysiology that real numbers—perception of continuous numbers—are more fundamental than natural numbers built on continuous number perception by capacity. our language. For more details, check out recent books and articles by researchers such as Stanislaw Dehaene (French author, cognitive neuroscientist) or Brian Butterworth (renowned neuropsychologist) British people studied many fields, including mathematical psychology).

If we start with measurement, positive rationals and counts will arise as special points on the continuous number line. And starting with counting, real numbers will be generated by “filling in the blanks” on the rational number line. And in both cases, you have to deal with negative numbers as best you can when the need arises.

From a learning perspective, neither method offers significant advantages over the other. Make choices and live with the consequences of choosing the respective 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 and rational numbers on the real number line, but the point here is not to construct mathematically the form in which the problem is human perception based on everyday experience.

In the US and many other countries, teachers have always chosen to start teaching math with counting, and the starting point of the math journey is natural numbers. But there is at least one serious attempt to design a math curriculum that follows the other entirely, and that is the focus for the rest of this article. Not just because I think one way is intrinsically better than the other but because, no matter what approach is applied, it is more likely that we will do a better job and better understand what we are doing in the teaching profession. members, if we are conscious of an alternative approach.

Indeed, knowledge of an alternative 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 (the Devlin’s Angle section of the website). Mathematical Association of America (MAA). As Piaget and others have also mentioned quite a lot, it is extremely difficult to help students understand multiplication in a textbook that begins with counting. A textbook that begins with measurement is just the opposite, simply because the thorny intricacies of counting multiplication don’t appear.

Ways to teach the concept of multiplication to elementary school students according to the current US National Competency Standards in Mathematics (CCSS).

Jean Piaget (1896-1980) was a famous Swiss psychologist of the 20th century, who made many important contributions to theories of cognitive development in children and constructivist learning theory. **Constructivism** views learning as an active 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 adopt a hybrid approach, building concepts based on both human perception at the same time?

I suppose, no matter what, this is possible to some extent. American children who start with counting also use length, volume, and other real number measurements in their lives, and children who start with measurement can certainly count, add, and subtract natural numbers. before going to school. But I am not aware of a formal curriculum that attempts to combine the two approaches.

Regardless of which of the two approaches is applied, the primary goal of 12 years of mathematics education in the world today is the same: to equip future citizens with an understanding of real numbers and a proficient use of them. **procedural fluency** with real numbers. In the American school system, this is done in a way that evolves from the early stages of natural numbers, integers, rational numbers in “arithmetic” and real numbers in “algebra”.

It should be noted that leaving real numbers in the field of “algebra” according to the American approach is to ensure a purely procedural measure, avoiding many of the difficulties involved in conceptual construction. real numbers from rational numbers. So, in the end, the method that begins with counting must also rely on our intuition and our daily experience with continuous measurements.

*(To be continued)…*