Why are we taught mathematics?
And by the way, what do math?
As a math teacher, but above all as a person who loves math and that math brought much (in my training and in every day life), I am concerned about the current lack of vocations for this discipline (and science in general). I especially asked about the role and utility of mathematics for me who had studied them (I had done by simple taste) and especially to those to whom I was going to teach them, often forced to study math, they rarely passionate. Three questions seem to me fundamental:
- Why is mathematics education compulsory (beyond knowledge count)? Why the math she were in France as much importance to the point of serving the same selection criteria for occupations that use the most?
- What pleases in math for those who love? What makes it a nightmare for others?
- What do math?
Explanation of these questions:
- The first question seems all the more crucial that we tend to briefly answer that math are used everywhere, so that students know very well that a large part of them are touch more past contests and examinations. To be honest, mathematical theories learned in school will not be used as such in most of those studying. If in many trades MAY be used math if desired, there is also much that we can do without. Hence the importance of the first question.
- The second question provides a different approach since most of you are forced to study math (and the teachers are forced to teach to students who do not want it), there he was using do not hate them or love them a little?
- In fact, after so many years studying math, you know what it is? And do you know what to expect after your studies if you keep the math?
These questions seem crucial, and I feel that the time when I was in school no one had ever asked me to think about it. You have the chance, try to answer by yourself. It is a way of becoming an actor of your own learning, and not to suffer it. You can then read, if you like, responses (personal and subjective) I would give right now.
I) Doing mathematics
In high school, when we do not understand, we tend to summarize a chapter by chapter the last formula, the most difficult, and that we have not understood in general. Thus making you forget that the most important of mathematical activity was elsewhere, in all the preliminary ...
Here (with examples) the presentation of various aspects of the mathematical approach.
1) The definition
The deeper issues are sometimes the simplest. precisely define the objects we handle always (it's called "axiomatizing") to determine a starting point to demonstrate more complex properties. Do you know for example what is a number? (Or simply "an integer, a rational number, a real number). A question that may arise is: who will invent a Martian math without knowing anything about mathematics land, would it lead to invent the same numbers? (Of course, he would not give them the same name). Or: what is a map? a vector ? a function? the form of an object? Can we move a cube to a ball by pulling it? And a ball at a donut? What is probability? so try to calculate the probability that a randomly taken on a rope circle is shorter than the side of an equilateral triangle inscribed in the circle?This exercise is called Bertrand paradox. When you have the solution (there are several, with different results.
What is a surface or volume? Intuitively, this is something that measures the size of a figure (in the plane or space) and that checks a simple property: the meeting of the volume number (2, 10, or a finite number) is disjointed figures the sum of the volumes of these various figures. However, Banach and Tarski showed that one could (theoretically) cut a ball in a finite number of pieces, move these pieces and rearrange them (like a puzzle) into a ball of radius twice as large. So that you could theoretically cut into pieces and a pea into a ball the size of the sun !!! This paradox shows that the notion of volume is delicate and needs to be properly defined (and "simple" forms only). Curiously, this paradox does not occur in the plane with the surfaces ...
2) Ratings
The second important aspect of mathematics, overlooked by the novice, is that of notations: math is a language that creates new words and new rules that describe more precisely notions. "Details" as the introduction of Hindu-Arabic numerals (before, only the big officials knew multiplications ask!), Equal signs, +, x, etc. and to give a name (s) of unknown did make the giant leaps mathematical putting within reach of the college once very difficult operations. The notion of derivative, such as today learns any first occupied at the origin of several books high school !!! So every notations used today is actually a digest that summarizes centuries of evolution, enrichment, research. Hence the importance of rigor and precision in their use. This work continues on the ratings there ...
3) Demonstration
Once clarified the concepts, remains to agree on the rules of reasoning and therefore demonstration. This concept has evolved (eg the Greeks used "demonstrations by figures ..."). The question we might ask is: can we prove it? In other words, do any mathematical property that is not false is provable? This is Gödel and his famous incompleteness theorem that answered in the negative in any theory, regardless of the number of initial axioms that we chose, there are properties that can not be demonstrated, but as leur contraire ne peut pas être démontré non plus... Ceci veut dire qu'il existe des propriétés vraies qu'on ne peut pas démontrer, et des propriétés dont on ne peut pas dire si elles sont vraies ou fausses, mais qu'en plus on a le choix de les considérer vraies ou fausse !
4) The actual calculations
This is the best-known aspect of mathematics: solving problems and equations, the practical calculation of these solutions. These equations can come of physics (you will see a number next year: Newton's laws of mechanics, equations of an electrical circuit), computer (if we synthetic images, how to calculate the moving a character?). Do you know for example what navigation and GPS use geometry you do since college? Cryptography (encoding of data so that it can not be read by matter which) uses math as one approaches specialty TS math? The search engine Google uses and solves 500 million equations systems with as many unknowns?Moreover, it is often these applications (which have enormous economic and strategic issues) who ordered the development of new mathematical branches, and which mathematicians are and will always highly sought after.
5) existence of problems
That said, when the actual calculations fail, when that does not have options, we simply question the existence of solutions to a problem.You know solving an equation of degree 2. There are formulas for solving equations of degree 3 and 4. It was long sought formulas for fifth degree before Galois shows that "there is no formula to solve the equations of degree 5, and in addition that the solution exists theoretically, one can only calculate approximate values "for example, we have that we can not express the roots of the polynomial x ^ 5 - 4x + 2, while a simple study of first function allows to give guidance. In fact, in the field of college geometry, are there other transformations preserving distances (isometrics) than those seen in college (symmetries, rotation, translation)? The specialty courses in response TS math ...
II) To love or not math
What one can love in math:
- Things are simple in the sense that they are either true or false. When one is in the right, we will not have a hard time convincing others. As we answered a question in general we know if the response is good, we do not need to come back, so that you can go further.
- There is elegance in mathematical theories, by their simplicity and brevity. Knowledge does not accumulate as in other areas. In math we can do a lot by learning little by heart (understanding rather than learn). For example, the complete TS program takes a single sheet (one-sided), the spe math program (12 hours per week) with all demonstrations and curricular concepts held by a double sheet!
- This is a very powerful instrument. Can quickly achieve results and applications that seemed out of reach (and unfortunately high schools programs do not give time to develop ongoing). Besides, master the math can address much more easily other sciences (especially physics).
- It is a formal, logical game which stimulates thinking (a bit like the logical game, chess, Sudoku, but of course more complex) and, once you have mastered a few basics, it can be very rewarding.
Making it difficult for some math:
- Math is a language: to do interesting things he must first master the grammar and spelling. So there are rules that must be learned to be applied automatically, without question them, not to mention, as in French. Moreover, school math is sometimes closer in the minds of grammar courses as other science courses. It may be noted that these same reasons, spelling and grammar are usually also hated that math ...
- It takes discipline, which is a form of discipline here than elsewhere must apply the rules, not to be content with approximations. It is learned, but it is difficult. So that most are discouraged by the well before this obstacle math, and achieve rewarding moments those in which we understand.
- Math is challenging: if we want to understand the language, it takes practice and it takes (to all, even the best) efforts, patience and work outside the classroom. We can not hope to escape being who all retained. There is certainly a big bonus when you come to the end of a problem, but it's true that there are now more immediate distractions ...