Procepts in Mathematics education

Todays blog post is a bit different to some of my views on politics, or current affairs. Its back closer to my student life as a Mathematics Masters student.
Inevitably when one enjoy mathematics and physics one gets involved in giving tutorial classes or helping friends and family. Mathematics is something that we all agree is very important, yet no one knows why.
Anyway today I want to talk about a ‘procept’, which I think reminds me of some of David Hestenes work on ‘modelling’ theory.

I came across the following paper in Mathematics Education which introduces the concept of a procept.
David Tall’s seminal paper on procepts
PROCEPT: An elementary procept is the amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or object.
A procept consists of a collection of elementary procepts which have the same object.

“A proceptual known fact should be distinguished from a rote learned fact by virtue of its rich
inner structure which may be decomposed and recomposed to produce derived facts. For
instance, faced with 4+5, a child might see 5 as “one more than 4” and might know the double
4+4=8 to derive the fact that 4+5 is “one more”, namely 9. For the proceptual thinker this gives
a powerful feedback loop which uses proceptual known facts to derive new known facts”

Take when you learn Noetherian Rings in Commutative Algebra, well its a struggle at first. Yet eventually you learn what this is, you accept the definition and you may think for instance that a Noetherian ring is a finitely generated ideal, or one with an ascending chain of ideals which eventually become stationary. There is a lot locked up in that short statement, knowing what an ideal is etc takes a long time. Yet its amazing that one eventually gets to the point when one can discuss very complicated topics, such as ‘categories’ and ‘functors’ and one is happy to have a procept understanding of these. A functor is both a morphism between two categories and an object in itself. There are also various types of functors – but this is for another discussion.
Some of that may be too technical for some readers, yet I see it in most areas of expertise. A problem in exams and for most disciplines is ‘passing the test’ versus true understanding.

On tricks for exams versus rich cognitive units.
Another article by David Tall Plenary talk
“Consider, for example, the notion of ‘linear relationship’ between
two variables. This might be
expressed in a variety of ways
• an equation in the form y=mx+c,
• a linear relation Ax+By+C=0,
• a line through two given points,
• a line with given slope through a given point,
• a straight-line graph,
• a table of values,
and so on. Crowley (2000) (reported in Crowley & Tall, 1999) reveals
how successful students
develop the idea of ‘linear relationship’ as a rich cognitive unit
encompassing most of these links as
a single entity, whilst the less successful simply carry around a
‘cognitive kit-bag’ of isolated tricks
to carry out specific algorithms. The cognitive kit-bag may get the
student through the examination,
but it is too diffuse to build on in later courses and students may
soon reach a point where the ideas
they are handling place too great a cognitive burden, leading
inexorably to failure.”

Now sometimes I cut corners and fail to develop the necessary
intuitions and range of techniques. I’m quite embarrassed sometimes at
the basic things that trick me.
Yet no one really carries around an infinite supply of answers, the
important thing is to think how to solve these sorts of problems.
The discussion in the article of the usage of computers got me
thinking about how they should be used to complement thinking rather
than replace them. I do however result to using Mathematica when I
come across a nasty integral. I’m not sure though if thats laziness or
a substitute for understanding.
When I now thinking of a derivative I for instance have a few
sets of ideas about them:
a) tangent to a curve or a surface
b) the standard analytic version of the limits etc
c) as part of an algebra of derivatives
d) rate of change
Obviously, there are other ways to define a derivative (I’ve been
told there are 167 ways – but don’t quote me on that).
The most important thing I took from that is the need for formal
theories. For instance thinking of Tensors as objects that change
under certain co-ordinate changes versus the following
Tensor. I’d take
the following, or perhaps more abstractly as an object satisfying the
universal mapping property.
Yet when writing notes on tensors I do also think in terms of ‘this
term eats the first r terms, and that term eats the rest’. So there is
a blend between intuition and formalism there.
The concept of a ‘procept’ is very useful, because it makes you understand how deeply you need to go into something to truly get an intuition for it. Definitions and vocab are needed, but so is an understanding (which is often developed through oral and visual explanation) of the concepts.
I’ll write some more about this, maybe thinking in terms of Physics too, but its a very power concept to have when learning mathematics. In some sense Marvin Minsky was correct when saying ‘to truly understand things, you need to know it in 2 or 3 different ways’