Wednesday, 25 March 2015

Learn the formula or learn the idea? What does deep connected mathematical understanding look like?

Consider the area of this sector:



Students may learn a formula for this area, something like:


Memorising formulas strikes me as a fundamentally non-mathematical activity. Memory is notoriously unreliable and mathematics is the complete opposite of this - being certain about things through deductive reasoning (and other means.) Obviously there are occasions in mathematics when we do want to memorise (or otherwise refer to) a formula. Knowing from memory that the area of a circle is given by πr², for example is more efficient than deriving it from more basic ideas every time you want to use it - and efficiency is certainly a desirable trait in mathematics.

However, care must be taken that students' use of formulas does not obscure a deeper sense of what is going on mathematically and constrain them to inflexible ways of thinking. I believe that students should spend valuable time making sense of the underlying ideas before they learn a particular rule or formula. If we leap to formulas (or other rules) too soon in students' encountering of a new idea, tasks for students can quickly become about 'performing' substitutions and calculations, rather than grappling more deeply with underlying ideas. In some sense, a formula can be a 'spoiler' for mathematical thinking - it should be the end result of exploring an idea, rather than a rule presented in the same way as arbitrary knowledge.

In contrast to the formula for the area of a circle, I do not encourage students to learn a formula for the area of a sector, because it is so easily deducible at any moment in time from simpler ideas:


All a student really needs to remember is 'a sector is a fraction of a circle' (along with some other ideas they should certainly know very well) and from this they can quickly deduce what the formula must be. The 'must be' here is important - given simpler known facts (some of which are arbitrary conventional facts and others of which are themselves necessary must-be maths facts) this fact is deducible, and easily so in this case. I believe we should encourage all students to think in this more powerful way.

There is a qualitative difference in learning to think about an idea such as sector area in this connected way, rather than relying on a formula. Rather than memorising this particular result, developing a strong awareness of how it fits into this network of ideas is more reliable, more mathematical and more flexible.

For me, this scratches the surface of what deep, inter-connected mathematical understanding is all about. Indeed, we could view the above diagram as part of a larger diagram, such as the following:


I have highlighted in red facts that are arbitrary, conventional facts, and in green those I consider to be necessary, must-be mathematical facts. (I do not claim this to be the complete set of all underpinning ideas - perhaps you can think of some I have missed?)

A student who develops a robust well-understood network of mathematical ideas relies only minimally on memory, has a clearer idea of appropriate and inappropriate uses of particular rules, and can more easily adapt these ideas when necessary. If students merely learn a formula without developing a sense of its connection to other mathematical ideas, their flexibility of thinking is obstructed. Formulas (and more generally summarised rules - think 'two minuses make a plus') can serve to obscure mathematical thinking and understanding.

It may be tempting to think that formulas are reliable. After all, they are algorithmic, they save the human mind from working from scratch every time. However, whilst they are 100% reliable when used by a computer - for example in a spreadsheet - for a human mind, they are only as reliable as memory and arithmetic allow.

Thinking mathematically does not mean 'thinking like a computer'! A computer can remember, follow rules and perform calculations perfectly accurately, but it does not have mathematical intelligence beyond this (at the time of writing - though check out waitbutwhy's articles about AI.) Unfortunately our current systems can often seem to prize these things from students, as if being mathematical is somehow akin to 'being like a machine'. We should dignify our students' minds with greater respect than this.

More on formulas vs. ideas to follow...

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