Maths: Conceptual understanding first, or procedural fluency?

Should you teach conceptual understanding first, or focus on raw procedural fluency?

This question drives endless debate in maths education, but its answer is very straightforward: it depends.

I can demonstrate this quickly and easily with a single example, by teaching you how to multiply logadeons (e.g. 5-:-9,) something you’re probably not familiar with already.

Observe the following examples:

8-:-20     *   2-:-5       = 10-:-25

9-:-20     *   2-:-5       = 11-:-25

100-:-50 *   30-:-7     = 130-:-57

19-:-20    *   5-:-5      = 24-:-25

By this point, you can probably multiply logadeons together quite comfortably.  If you’d like to give it a go, try these two (answers at the end.)

30-:-17  * 4-:-3

17-:-0.5 * 9-:-2

But even if you can evaluate those correctly, you’re probably still not comfortable about all this; you probably don’t feel like you understand it, and for two reasons:

The first, is *why* does multiplying two logadeons result in us adding the digits?  It’s reasonable to assume there is a perfectly valid reason, just as is the case for adding indices when we multiply numbers in index form, but we don’t yet understand why it’s the case.

The second, is what the hell is a logadeon anyway??  I’ve spent a hundred or so words now discussing the multiplication of something that probably feels like a mental black hole in your mind; it’s very difficult focus on the process, and leave behind that question: “What on Earth is a logadeon?  What on Earth is he talking about?!”

Let’s switch to multiplying fractions instead.

If our goal is to teach pupils how to multiply together two fractions, then I would argue they first need to understand what we mean by the word ‘fraction,’ otherwise we’re saying meaningless things from their perspective (they also need to understand something of what we mean by the word ‘multiply.’)  This is to say, they need to understand fractions as a concept.  Arguably, they should be able to conceptualise fractions as parts of a whole, as ratios, as representing division of two numbers, and as positions on the number line, all of this, before we try to teach pupils how to perform arithmetic operations on this concept we call ‘fraction.’

Assuming we’re successful in that, we now wish to teach them multiplication.  Traditionally, we might simply explain that we multiply numerators together, and denominators together, and nothing more.  There is a reason that process works, and it can be explained in several ways.  It is also possible to relate the arithmetic process of multiplying fractions to other conceptualisations, such as a visual representation.  Here, we run into problems, though.  I’ve heard people argue that this visual representation offers an explanation, a why for the arithmetic process; it doesn’t.  It offers no proof, therefore no ‘why,’ simply an alternative means of conceptualisation, or to think of it differently, it’s another process for multiplying fractions, one that is significantly less efficient than the arithmetic process.

Visualisation of fraction multiplication

Now, is the proof necessary?  Are the alternative conceptualisations necessary?  We *must* conclude that no, they are not.  We must conclude that no, they are not, first because pupils can comfortably learn to multiply fractions without them, but more, because there is an almost infinity of proofs and conceptualisations within mathematics that can be related to the things we teach, we cannot possibly teach them all, and we therefore cannot possibly deem them all necessary.

But are they desirable?  Absolutely, and this is an important clarification.

Where something is necessary, such as developing the understanding of a concept before we move on to discuss what we can do with that concept, we have no choice.

Where something is desirable, such as providing proof for a process, or alternative conceptualisations, we have a lot of important decisions to make, and we must make them with careful and deliberate intentionality.

Often, standard processes are simple.  Proofs are relatively complex, as are alternative conceptualisations.  Try proving why the formula for the volume of the pyramid works, for example, which requires calculus.  Things get even messier for the cone, and sphere.  Deriving the quadratic formula is far, far more complex than using it to solve quadratics.  Pedagogically, it’s very difficult to turn these proofs into cognitive work for pupils, as well, making it tricky to fit them meaningfully into lessons (though certainly not impossible.)

In conclusion, there are times when we have to teach the concept first, and there are times when we often do not pay this close enough attention, jumping into processes before the concept is locked in place e.g. vectors, equations, irrational numbers… almost everything really.

There are other times, however, when this is being conflated with proof and alternative conceptualisations.  These should usually probably come after fluency with processes and algorithms is achieved, since those processes are often more simple than their derivations.  They must also be selected and taught carefully, since we have finite time: which proofs and which alternative conceptualisations will most help pupils develop a relational understanding of mathematics?  And when in a sequence of learning – which can take place over years – is teaching them going to be most successful?

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Answers:

30-:-17  * 4-:-3 = 34-:-20

17-:-0.5 * 9-:-2 = 26-:-2.5

And logadeons are not a thing, but sometimes it’s good to remember what it’s like to learn something without pre-existing expert knowledge.