The yin and yang of maths education - Part 7. The difference in the nature of the red route and the blue route.

There are two arrows from zone C to zone B.  How are they similar?  How are they different?

When the purple arrows are in operation, the brain is constructing a need and place for the knowledge which it acquires before it receives it.

But what of the red arrow?  Is the opposite true?  Is knowlege being presented before the brain is ready to receive it?

I would say sometimes and sometimes not.  I would also say that where this is happening it is not necessarily disastrous.

Sometimes and sometimes not: A wise and experienced teacher will not tell students what they are to learn, they will guide them through experiences and help them develop their own insights which will naturally lead their brains to demand the knowledge which is to be taught.  This is appropriate in many areas of mathematics where knowledge is constructed in an axiomatic way.  In these cases there will be little difference between the red and the purple routes.  However not all mathematical skills lend themselves to axiomatic construction.  This is particularly true of human skills such as resililance and application to contexts, but is also true of other ontological aspects of mathatics such as the definitions of root concepts.

Where this (knowledge being presented) is happening it is not necessarily disastrous. This is particularly true if the student has developed a mind which is used to acting in a constructivist way.  Such a student will look at the knowledge being presented and will work actively to see if they can construct a place and a context for it given what is already authentic to them.  It is useful that students develop the ability to do this.