Let us anticipate one of the possible criticisms that may be made of the proposal of our previous article. From our point of view this is the only relevant criticism that the proposal admits. But in fact such criticism is possible only because our intention was to present an idea in a very simple and straightforward way. Today we will give the missing details and we believe that after that it will be very difficult for anyone (who likes math and doesn't want children to be mediocre) to refuse the path of finite number systems.

The way we motivate finite number systems allows the following criticism: "spring + 1 = summer" is not a valid equation because it is a sum of inhomogeneous quantities…! Of course, we are speaking intuitively and we are not concerned with rigor with the word "quantity". But the fact is that it is strange to think of the "sum of a season with a pure number". So we do the following: We fix the spring season in our thinking and point it out to P. Then we will only think about the seasons that will follow after a few periods have passed. For example, P + 1 = V, where we indicate only the following spring season after a period of three months. This means that our children will play to find out what happens when we repeat this reasoning. They will find a very interesting table: P + 1 = V, P + 1 + 1 = O (autumn), P + 1 + 1 + 1 = I, P + 1 + 1 + 1 + 1 = P. With a little help from the teacher (that's what the teacher is for) children will "discover" that imagination "1 + 1 + 1 + 1" produces the same as imagination "doing nothing," or if you need the teacher's help, " 1 + 1 + 1 + 1 = 0 ". They will find that they can play with the symbols "=", "+", "0" and "1". There is no longer the problem of "relating inhomogeneous quantities". This is a great discovery, not just for kids, say 11, but for anyone curious about math. The teacher can emphasize the beautiful fact that only four symbols were used in this number system: "0, 1, +, =". And there is a great opportunity for the teacher to lead the children through a very useful experience with the idea of simplification so fundamental in mathematics. The simple "1 + 1" experiment, "1 + 1 + 1", immediately leads to a new problem: the need for simplification and economy in the use of symbols and operations.

One of the major problems for beginners in mathematics is the problem of getting used to the use of symbols for ideas. Each number needs a different symbol. One of the reasons why the infinite system of natural numbers is extremely complex for children is that it needs infinite symbols to represent numbers. The child has to solve, among others, the problem of how he could produce these infinite symbols necessary for the representation of the natural ones. This is not easy and has not been fully resolved by mankind before the year 1000 AD: it is the famous Indo-Arabic positional system whose discovery can be compared to the creation of the digital computer. The Indo-Arabic positional system and the digital computer allowed deep revolutions in human knowledge simply because they made the results of complicated calculations available. The reader can get a very simple idea of this by trying to calculate 13 x 29 using roman numerals.

In the case of the positional system, the simplest idea that can be associated with it is the idea of simplifying the representation of numbers. That is, how can we simplify, for example, the representation "1 + 1". No one will have the patience to write "1 + 1 + 1 + 1 + 1 + 1 +… + 1" all the time and still have to calculate additions and multiplications with these expressions. This is, we venture to say, the primary problem of the elementary school math teacher: getting children to incorporate symbols and use them in the representation of numbers. But there is an extra difficulty: all this has to be done economically and simply, otherwise it will make calculations and progress in the study of mathematics unfeasible. By simplifying the expression 1 + 1 = 2, 1 + 1 + 1 = 3, we will be helping the child to solve the fundamental problem of number representation which, in the case of natural numbers, will require the ingenious Indo-Arabic solution of positional system. This allows the child to experience all the fundamentals of a number system in a simple way and to play freely with possible generalizations.

Note the reader that the system "1 + 1 = 2, 1 + 1 + 1 = 3, 1 + 1 + 1 + 1 = 0" has only six symbols (0, 1, 2, 3, +, =), can be completely motivated by the seasons system, and provides two mathematically very interesting tables with which children can play until they get tired, and when they get tired of them, they can search the following cyclic systems of 5, 6, 7, etc., States. It is important for the teacher to introduce the multiplication table because multiplication is part of the solution of the simplification problem: 1 + 1 = 2.1, 1 + 1 + 1 = 3.1. But this immediately brings a new problem and an important discovery: Could not the new symbols 2 and 3 also be "added" and "multiplied"? It is very important that this "play with a finite system" has a well understood end: the cycle of the seasons were numerically represented, the problem of numerical representation of a cyclic system with few possible states was solved, the problem was solved. the problem of simplification and the economy of symbols and operations, and a very natural path of continuation for thought has been opened, that is, the path of the representation of all possible finite cycles. The teacher should ensure that the two tables below are fully incorporated into each child's experience, making sure that they are able to interpret and explain how each entry in the table was achieved. We see only one way for the teacher to make sure that the children have incorporated this experience: by checking what they will be able to play with the five state cycle.

The net result of all this is one: children will realize that symbols can be related to "great freedom". This is the great lesson of mathematics, that is, we don't have to "pick things up with our hands to reason about their properties." We can reason through symbolic representations, but there are several natural problems that must be solved before these representations can be made successfully. Children already have to face these problems when they begin their experiment with numbers. And nothing more natural than starting with the simplest number systems, the finite systems.

The well-informed math teacher knows that there is another extraordinary gain to this mathematics teaching strategy: children can naturally begin with measuring symmetries, a fundamental notion of contemporary science. The simplest forms of symmetry are finite cycles. Finite number systems are nothing but measurements of the simplest symmetries. We are not against the experimentation of children with natural numbers, but we must bear in mind that this system represents an infinite cycle and, therefore, the difficulties to be overcome are non-trivial, highlighting the problem of generating infinite symbols. Why not allow children the previous experience of finite cycle representation?

+ | 0 | 1 | . | 0 | 1 |

0 | 0 | 1 | 0 | 0 | 0 |

1 | 1 | 0 | 1 | 0 | 1 |

+ | 0 | 1 | 2 | 3 | . | 0 | 1 | 2 | 3 |

0 | 0 | 1 | 2 | 3 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 2 | 3 | 0 | 1 | 0 | 1 | 2 | 3 |

2 | 2 | 3 | 0 | 1 | 2 | 0 | 2 | 0 | 2 |

3 | 3 | 0 | 1 | 2 | 3 | 0 | 3 | 2 | 1 |

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