Thinking about conceptual understanding

I read an interesting blogpost earlier today by US teacher Barry Garelick, a post which takes to task the suggestion that

Math Learning is a 3-legged stool: 1) Procedural knowledge- Processes & shortcuts; 2) Conceptual knowledge-the WHY behind them; 3) Application knowledge- HOW they are used. We’ve focused on the first one for too long.

Garelick proposes that a focus on why a procedure works can be an activity with high opportunity cost, suggesting that this kind of understanding is much easier the more expert you become and, importantly, doesn’t always contribute towards reaching that expertise.

I very much agree. I think there are procedures whose ‘whys’ are accessible (the example of multiplying and dividing by powers of 10 that Barry uses is a good one) but there are also procedures whose whys are harder to get at. In my experience, for instance, differentiation from first principles is not as accessible as the procedure for differentiating . There is absolutely no harm done by teaching children how to differentiate before deriving the formula from first principles. You might disagree with my specific example – fine – but there are more. It’s very easy to come up with a list of procedures whose explanations involve more sophisticated knowledge than their use.

To this end, anyone who proposes ‘why before how’ in every case, or ‘why alongside how’ in every case is prioritising an ideology over subject matter and pragmatism. But this isn’t what I wanted to write about after reading Barry’s post. What I wanted to pick up on was the explanation of conceptual understanding as why a procedure works, an explanation that I see very often.

You see, understanding why a procedure works is not the same as conceptual understanding, and as long as teachers think this, their thinking about subject will be limited. To grasp conceptual understanding, we need to be clear on what the concepts are in mathematics. To be clear on the concepts in maths, we need to first understand what a concept is.

A concept is an abstract idea, something that is, by definition, not concrete. In English literature, for instance a concept might be ‘theme’. Hard to define, best understood through multiple examples or instances (all’s fair in love and war, the circle of life, power corrupts, …). In history, for instance, there is the concept of ’empire’. Again, hard to define, increasingly understood through examples (Roman, Persian, British, Russian, …) and non-examples (the European Union, the African continent). I’m not sure one can ever fully understand a concept, because there’s always more to it. Rather, understanding of a concept is just something that gets deeper over time, through increasing exposure to and analysis of concrete examples.

So what are the concepts in maths? What are our abstract ideas? ‘Addition’ is a great example. What it means to add is a really hard thing to explain. Try it now – what is addition? (Not ‘how can we add?’ but ‘what does it mean to add?’) You might have come up with something that conveys the idea of ‘grouping together’ somehow, or counting somehow, and then we’re starting to get to it. But it’s not easy. What we can do, however, is exemplify the concept with many examples across many areas of mathematics. In this sequence of examples we see the idea of counting items by grouping them together:

Natural numbers:

Decimals:

Fractions:

Unknowns:

Expressions:

Surds:

Studying these examples helps you to ‘see’ the basics of the concept of addition. A pupil would meet them over a number of years and so their conceptual understanding will deepen over a number of years

The examples in the sequence above were chosen to communicate more about the concept of addition. Each example follows the structure: one ‘thing’ plus two ‘things’ is equal to three ‘things’. What varied was the ‘thing’, or, more rigorously, the ‘counting unit’. By considering these examples together, our understanding of the concept of addition is deepened, to see that we can easily add things in the same counting unit. A pupil who has this understanding is arguably less likely to make a mistake like this: as their conceptual understanding of addition provides an organising framework for the procedure they learn for adding surds.

We could contrast the previous sequence with examples like this:

which tell us something more about addition – that we must exchange ‘things’ for ‘other things’ that are in the same counting unit before we add. This is the same principle that underpins the column method for adding, where we add ones, then tens, then hundreds, and so on.

As we are exposed to more and more concrete examples of addition, we gain a deeper understanding of the concept of addition and its various principles, or underlying structure. This is conceptual understanding and it’s a spectrum rather than a have/have-not measurable. It comes over time and we can do things that help or hinder it (such as teaching examples of a concept in a related way, rather than totally different ways).

To finish, I want to come back to the original point of contention, that understanding why a procedure works is conceptual understanding. The why of a procedure is a part of both procedural and conceptual understanding and is not, in itself, the latter. I see why people want to think about maths as more than procedures, because it is more than procedures, but by defining conceptual understanding in these terms we are at risk of still looking at maths solely through the a procedural lens.

To truly see more in mathematics, we need to step underneath procedures and look at the subject’s organising concepts. Doing so is a great exercise for a teacher as it helps us to organise our teaching in a coherent and deliberate way – teach every instance of addition in a similar or related way and we will help pupils to understand the concept of addition. What we must remember, though, is that these concepts are abstract, inherently difficult to grasp, and need plenty of the concrete to get at them. We can’t ‘teach’ conceptual understanding in the way that we ‘teach’ procedural. In the classroom, the main way we get at concepts is through procedures and their careful, coherent teaching. We can make conceptual principles gradually more explicit as we go, but conceptual understanding is a bit like an infinite sum – we can get closer and closer but will never really get there.