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Finite Number Structures


It is common to hear that “math education goes bad” or “students are very weak” and so many “complaints” about the inefficiency of our schools. But in terms of mathematics, what would be an interesting example of the poor quality of current educational practices?

We have the impression that there are countless interesting and serious examples of inappropriate educational practices in mathematics. A striking example is that in some countries there is, at least not to our knowledge, a student support program that shows early talent in mathematics, physics and chemistry, especially mathematics, or an unusual interest in and a serious referral of these talented individuals to a life of dedication to mathematics and science, in order to create scientific intellectual frameworks without which there can be no generation of new knowledge, new technologies and autonomy of a country in the creation of wealth.

It is known that a certain percentage of children, it appears that about 3% (even if our reasoning is still less than that) is gifted, some of whom could turn out to be talented mathematicians in 10 or 15 years. We have never heard in some countries of any serious program to support the training of these citizens whose intelligence could begin to generate in 10 or 15 years precious knowledge that would lead one way or another to the production of new knowledge in universities, research institutes, material wealth in high-tech industries and enterprises. This, it seems to us, is one of the biggest strategic mistakes a country can make, for which it already pays dearly at the beginning of the information and knowledge age. Without wishing to lengthen the subject, let us remember the passage from India which, almost unnoticed, has recently become one of the most important centers of intelligence applied to software creation. Or from Bill Gates and his foundation, which about two years ago has given $ 1 billion to a Boston institution that "hunts" for young and young talent in math, physics and chemistry. Or John Kennedy, who, terrified by Yuri Gagarin's first outing around the earth, immediately set up schools for geniuses, one of which in the Bronx, New York, generated several Nobel Prizes in Science.

But here we will deal with an easier problem to analyze. Let us take an example of what we believe to be a serious mistake in the teaching of mathematics and thus in the intellectual training of students. At some point, the notion of number appears to students, and it does not matter here to determine in which grade, whether in a form presented by mathematics teachers, by influence of parents, or for whatever reason. What interests us is to analyze what happens after that. It is famous for the difficulty of most students in dealing with the rules of signals, the rules of fraction operations, the rules of manipulation with decimals, etc. What is behind all this difficulty and all this waste of time and waste? We believe the main reason is the mathematical unpreparedness of teachers, which is largely acquired in unprepared and obsolete faculties of mathematics. Here we will give an example, to illustrate this thesis, which we believe to be quite “unknown” to those who, either good or bad, worry about the mathematical ignorance of a people.

It's simple: what is the first “number” model practiced in schools? No doubt it is the "natural number" model. We need not discuss the rigorous concept of natural number to explain our thesis. We need a few intuitive remarks. The natural numbers model is a model infinite of numbers. So far no news, is it? Well, that's where the "obvious" lies one of the problems. This number set model, that is, the natural number model, is a model that can be understood intuitively as a mathematical model of an infinite cyclic abstract process. We can think of the counting process which is quite intuitive and historically primitive. The natural number model is therefore a model of a description of an infinite cyclical process that obviously never quite completes its cycle. For this reason, perhaps, it is never interpreted in this way at the fundamental and middle levels. Adding one unit to another in an endless repetitive sequence simulates an infinite cyclical process in which an idea (that of adding 1) simply repeats itself monotonously, absolutely the same way at each step. This is actually very simple, everyone "knows" what it is about, that's not our problem. The problem, in our view, is that, we do not know why, one cannot see the obvious fact that finite cycles also exist, and are abundant in nature, including the very world of ideas. Finite cyclic processes such as day and night, seasons, days, months, years, heartbeat, astronomical phenomena, etc., are fundamental to human life and imagination. Now, then, we have to ask for finite number models.

What are the number models that describe these finite cyclic processes? Why does the education system insist on a single model, and does it concentrate its efforts on a infinite cyclic process model, inexplicably ignoring the models of finite cyclic processes?

Representing day by symbol 0 and night by symbol 1, and passing from one to another as the result of adding 1, we get the interesting “algebra” 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 0. And here is a "surprise": 1 + 1 = 0. That is, a period after night is a day. Another similar surprise we get when we represent the seasons of spring, summer, fall, and winter by 0, 1, 2, 3, and the change from one to another as the result of adding with 1. Thus, the new “algebra” that appears as follows: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 2,…, 3 + 1 = 0. That is, a season after a winter is a spring.

From the standpoint of those with mathematical interest and curiosity, there are inevitable questions: What the hell kind of structures are these? Does it mean that it makes sense to operate with numerical symbols as if it were a mere game but, of course, motivated by real situations? How do you play with this game for real, that is, exploring all the necessary consequences from the above algebra? For example, we can easily tell from our experience with the natural that 3 times 1 remains 3 in the seasons system. But 2 times 2 has to be necessarily 0. Perhaps the most exciting question is: what happens differently in each of these finite systems as we increase the cycle? For example, in the system of a 5 state cycle (now it doesn't matter anymore for those who have mathematical curiosity if this cycle exists in nature) we have 1 + 1 + 1 + 1 + 1 = 0. What is your multiplication table like? How do your addition and multiplication tables differ (or resemble) the tables in other cycles of size 2, 3, and 4?

The reader then realizes that it takes no miracle to investigate interesting mathematics at an elementary level, that is, at the elementary and high school level. Or is an 11-year-old boy or girl unconditionally refusing to play to find out for himself how to fill in these little tables? Of course, if teachers have no idea what is going on with these finite systems, and have no curiosity about finite cycles of nature, a student in our middle and high schools will never know that a finite system is more natural. of numbers than the infinite (much more complex) system of natural numbers. Notice the reader that the system of debris in dividing a natural number by the natural no strictly obeys the algebra that we “accidentally” discover from nature's finite natural cycles. We will give no further details here, but the curious reader and mathematical lover can easily see this for himself. For example, if one number has remainder 3 when divided by 5, and another has remainder 1, then its sum has remainder 4 when divided by 5. Similarly, remainder 3 plus remainder 2 gives remainder 0.

So here is a serious example of deformation in the mathematics education system.

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