Maths, memorisation and understanding

Much of the debate around the Leaving Certificate in Ireland centres on the issue of ‘rote learning’. Many commentators will express a desire that education should emphasise ‘understanding’, ‘critical thinking’ and ‘creativity’, the implication being that ‘skills’ like these can be developed in a way that doesn’t involve memorisation of some kind.

But let’s look at mathematics and think about how a student becomes ‘good at maths’.  Maths is an interesting discipline because to be good at it requires learning rules ‘off by heart’ in the same way that one learns the rules of grammar when learning a language; but ultimately it also requires one to develop a deeper understanding of what all of those symbols and rules actually mean.

For example, a simple rule that a maths student must know is this:

A*(B+C)=A*B+A*C

This is a rule you just need to know.  Of course you can agonise all you want about why this rule is true but to get on the road to being ‘good at maths’, angst is not necessary at this point. As Fields’s medallist, Timothy Gowers has said, “It is quite possible to use mathematical concepts correctly without being able to say exactly what they mean. This might sound like a bad idea, but the use is often easier to teach and a deeper understanding of the meaning often follows of its own accord.

As you progress into the world of mathematics, you will meet all sorts of rules that, once again, you can fret about or you can just accept them as you would a rule of grammar. For example, an important rule that my engineering students often need to able to recall automatically is

If   y=x^n then x = y^(1/n)

Again you can worry about what raising a number to the power of 1/n actually means (especially if n is an irrational number!), but you can also just accept that, once you know the rule, all sorts of algebraic manipulations are possible, as is working out the numerical value of something like 51/6  on a calculator or spreadsheet software. Either way, if you are in the midst of doing an actual calculation, algebraic or numerical, it does simplify the whole process for you if you have remembered the basic rules of powers. If you haven’t remembered the rules (as I find is often the case with my students) you get bogged down, you take wrong turns, and any chance of creative thinking is lost.

As you delve deeper into mathematics and do more advanced topics like calculus, you can still reach a reasonable degree of competence by simply knowing rules and practicing the application of those rules. But eventually you will come up against problems, especially problems where you want to actually apply mathematics, where you will need to go further and go a bit deeper.

It turns out, for example, that calculus, especially differential calculus, is, in many situations, the natural ‘language’ of a host of problems that arise in my subject, chemical engineering. In order to be able to express problems in mathematical form, you do have to have an understanding of what d/dx means. Being adept at the mechanics of differentiation is not enough.  ‘Understanding’ here means remembering that d/dx of something represents the rate at which that something is changing with respect to x. But it’s a deep kind of remembering, one that is formed not by repeating some sort of refrain but by doing lots of practice in which the idea of d/dx representing a rate of change becomes ordinary, not an abstract thing to be learned off.

Ultimately the road to  being ‘good at maths’ is  a long and twisty one and it  really does help to know the rules of the road. Some times those rules are quite deep and  for a while at least you may just have to accept them, or, better still, have them explained to you by a good teacher. But maths is fundamentally  abstract and some rules will only become part of your consciousness by lots of practice.

The fundamental point though is that to become good at maths, you do need to remember quite a bit of stuff!

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About Greg Foley

A lecturer in Biotechnology in Dublin City University for more than 25 years. Trained as a Chemical Engineer in UCD (BE and PhD) and Cornell (MS). Does research on analysis and design of membrane filtration systems.
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4 Responses to Maths, memorisation and understanding

  1. Pingback: Ninth Level Ireland » Blog Archive » Maths, memorisation and understanding

  2. Anonymous says:

    I think there is an issue here around what it means to “do maths”, Too often students think they don’t have a mathematical brain, because they couldn’t memorize and perform procedural tasks at speed.
    Your reaction rates example is a good one. Asking a student to give a real world example of how some piece of mathematics might be applied in their field can be a caustic test of both the course and the students in any service teaching situation.

  3. batchloaf says:

    Hi Greg,

    Once in a while, a few lines of poetry or song lyrics I learned by heart in secondary school will pop into my mind and I’ll realise that the intervening years have given me whatever it was I needed to finally understand them. It feels good when it happens, maybe because it feels like I’m unlocking additional value in knowledge I already have stowed away in there. It’s just as well I did learn them off then, even if I wasn’t ready to understand them, because I certainly don’t think I could learn very much off by heart now. In fact, I find it really hard to remember anything I don’t understand now.

    That definitely goes for some mathematical rules too – especially (in my experience) formulas in trigonometry, calculus, complex numbers. I find it so much easier to understand all of those now, but thanks be to goodness that I learned them off while I still could!

    Ted

    • Greg Foley says:

      Ted,

      I think the vast majority of people would agree with you. Unfortunately there seems to be an educational ideology about that devalues not just memory but knowdlege. You can see it in the new Junior Cycle curriculum which seems to me to be very light on content and heavy on ‘skills’.

      Greg

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