On the beauty of creative mathematics

Some years ago I wrote about the dangers of mathematical modelling. There I explained how using formal models might lead us to ignore aspects of realty that are not represented in the model. This is a particular risk when we're using models without understanding them deeply. With this I don't want to take sides in the mathematicians vs engineers wars - I'm quite sure there's space for both. This time I'd rather want to focus on the idea that understanding a model well allows us to creatively exploit it. My goal is to ultimately make a case that both programmable and creative maths should have their space in teaching. This might be a false dichotomy, but let me expand on what I mean.

Think of these (vaguely indicative) mathematical pairs:

Origami	Japanese Traditional Geometry
Graph Theory	Mathematical Optimisation (and in particular Integer Programming)
Stereometry	Volume integration
Knot theory	Topology
Geometry	Linear algebra

What do they have in common? Here's my answer. Anyway, in my subjective experience, the first column contains a set of mentally stimulating problems that have provoked the development of intuitive heuristics. The second are extended formal theories that generalise and aim to solve the problems in the space of the former. My table is only illustrative and alternative formal theories exist. I'm not saying these theories are not creative. Rather my idea is that they are way too complex to provoke untrained intuition and interest.

This is why I am a strong believer that in order to provoke interest for mathematics in its complexity, the problems in the first column need to be presented to children and they need to be allowed time to play with their own heuristics. Based on my personal experience, I hypothesise that then children should probably be taught advanced heuristics, which is a form of history of mathematics, but also incremental provocation to improve the self-taught intuition they've originally developed. I'm thinking of these as a form of build-up as preparation for getting introduced to complex mathematical theories. This build-up should allow students to develop a comprehensive understanding about the complex model that formulas only partially represent. To illustrate this, my thinking is that certain intuition - regardless whether or turns out to be misleading - is an important instrument for the development of knowledge that is actively usable (contrast to passive knowledge, e.g. in languages). From the perspective of threshold concepts, this might be a more informed way to enter the oscillation that precedes the Eureka effect of mastering the relevant threshold concept.