# Conveying the full complexity of mathematics as early as possible in an individually adapted way

This is an idea that I got from a fellow student about a year ago. It is not my idea and I would probably hardly ever had this idea because I might lack the necessary mathematical insight. But still, I was gratefully able to listen to the crucial ideas of this fellow student and I have not forgotten this particular idea here and it is probably quite important, at least if the child has the potential and is intelligent enough to develop its knowledge to the level of a university degree level mathematician or beyond.

The idea is that the child, if it is cognitively able to master this additional information, should as early as possible realize how complex mathematics really is and especially the extent of mathematics' complexity when being derived from its foundations, axioms and theorems.

In terms of conveying the full complexity of mathematics, there are two learning paradigms and a third, "compromise paradigm":

## Speed-oriented learning paradigm

- Begin as early as possible with conventional basic counting and conventional basic arithmetic tuition and go through arithmetic and algebra as quickly as possible and reach university level mathematics as quickly and as early as possible

## Strictly axiomatic learning paradigm

- Present only the most fundamental axioms and let the child develop mathematics only by deriving theorems and mathematical concepts out of these axioms.

Probably the most extreme and unusual learning paradigm would be to teach mathematics based on axioms only and only assist in the child at deriving theorems from these axioms and derive more specific mathematical theories from the theorems until after several years of study, even ultra-specific (and "easy") mathematical concepts like numbers are reached. According to this learning paradigm, one would begin with one of the fundamental axiomatic systems (see http://en.wikipedia.org/wiki/Axiomatic_system), for example the Zermelo-Fraenkel set of axioms (see http://en.wikipedia.org/wiki/Zermelo-Fraenkel_set_theory), and just base every mathematical tuition just on such an axiomatic system. There would be no counting, no arithmetic, no conventional algebra (and not even theorems as the foundation), but everything would be taught and developed only out of the most fundamental axioms beginning as early as possible in childhood. This would be exactly the opposite how mathematics is taught today. I am not sure if that is the most adapted form of tuition even for a child that is extremely intelligent, for example having the potential to reach an IQ of 190 or even more in adulthood (see also intelligence measured by IQ and mathematical intelligence), because it is probably too detached from the very young child's brain that is still significantly dependent on its senses and would probably not be able to understand such a high level of abstraction from the beginning. It would probably mean that important opportunities, that a more moderate, compromising approach would offer, would have been missed (for the compromising approach, see the next section).

The advantage of such a teaching method would probably be that the chances that a completely new axiomatic system is developed by a mathematician who was able to have such a tuition increases because such a mathematician would have been used to derive all theorems and the whole mathematics out of the foundation of that axiomatic system since early childhood. See also new type(s) of mathematics thanks to a better didactics of conventional mathematics? and how can the mathematical imagination of a child be preserved and enhanced?

## Learning paradigm as a compromise between the two

- Teach axioms as early and as much as possible as long as the child understands it
- Introduce every mathematical concept as generalized and as early as possible
- Foundations / axioms of mathematics from early childhood
- Theory and examples as broadly as possible from early childhood
- As abstract, generalized and formal as possible from early childhood as long as the child is able to follow
- Examples:
- Introduce all types of numbers (not only the natural numbers, but all real numbers and complex numbers too and also the even more abstract numbers) as early as possible
- Instead of introducing just simply arithmetic, give full information and overview about as many algebraic structures as possible (see http://en.wikipedia.org/wiki/Algebraic_structure) as early as possible in childhood, for example an overview of the laws of commutativity, associativity and distributivity in the context of algebraic structures like groups, fields and rings and don't simplify mathematically as long as the child is able to follow.

Some questions in terms of priorities remain: Is it better to say teach as quickly as possible in a conventional way, for example go through the natural numbers, integers, real numbers, complex numbers and beyond as quickly as possible (say 10 years at the age of maybe 0-10 years) or introduce all number types from the beginning, at the age of 0 years? I think the example shows that this might be a little bit too ambitious because even the most intelligent mathematician on the surface of Earth would not have been able to understand an overview of algebraic structures let alone the even more abstract and formal foundations/axioms of mathematics at age 0. But maybe at age 3? Therefore, a compromise between the two learning paradigms might be necessary even if a baby and toddler has the very rare potential to become the most intelligent mathematician on Earth's surface.

The limit of simplification would have to be assessed constantly and individually for every child in order to constantly minimize the degree of simplification during the whole learning process. Because of such a constant assessment and effort to prevent unnecessary oversimplification for every child, the learning process would take place exactly at the edge of mathematical simplification for every child individually.

Robotics and the advanced mathematical concepts used in robotics might also help to motivate the child to learn advanced mathematical topics from early childhood and therefore also mathematical concepts that are not simplified or oversimplified and a mutually beneficial and motivating process between pure mathematics and robotics could develop and the child would be more willing to learn mathematics in the least simplified way from the beginning.

Copyright by Lukas Girtanner, 2005-2010 (the conventional copyright valid for all pages), and also (at least partially at least in terms of the idea itself) by a fellow university student, 2009 (-2010?), although I have not contacted/asked him about the copyright issue. See also Should the content of lukasgirtanner.com be conventionally copyrighted?.

If you have other unique, new or fundamental ideas about the didactics of pure mathematics that are similar to the idea on this page here, please contact me at moc.liamg -ta- rennatrigsakul, I would really appreciate it and publish it here.