Conceptual Understanding

We are in a short period where North American EduTwitter seems to be (re)discovering UK EduTwitter. The result is something of a timewarp, with arguments that you no longer hear from UK educationalists being given voice by their North American counterparts.

One of these is the conceptual understanding argument. When confronted with the effectiveness of explicit forms of teaching, those who dislike explicit forms of teaching first retreat to what they think is the high ground of conceptual understanding. Once that is lost, they head for the fortress of motivation before falling back to the wall of unfalsifiability. I have been constructing siege engines outside this wall for the past couple of years and so I had forgotten about the conceptual understanding argument. So let’s journey back across the battlefield.

Firstly, we have to consider what we mean by understanding. Back in 2015, I suggested that there were two distinct forms of understanding – subject and objective. Subjective understanding is a feeling – the ‘Aha!’ moment. I suggested it comes from being able to sufficiently grapple with a concept using a combination of long-term and working memory. In other words, subjective understanding is equivalent to not being cognitively overloaded. It may be misleading because we may rely on inaccurate schemas, and it is hard to assess – we may try to use self-reports or we may try to measure cognitive load directly.

If this view of subjective understanding is correct, applying the principles of cognitive load theory should benefit, not hinder, an individual’s ability to understand. By making maths facts, for example, automatic, we free-up some cognitive load to grapple with other aspects of an issue. You could say we are drilling for understanding.

Objective understanding is a perception of the understanding of another. This is assessed by asking questions or through task performance.

What sort of tasks demonstrate understanding? This is where the research becomes pretty interesting.

Pointedly, much of the research discusses ‘conceptual knowledge’ rather than ‘understanding’ and this signals that there is really no clear demarcation like between knowledge and understanding. Back in 2018 I became aware of a paper by Crooks and Alibali surveying research into conceptual knowledge in mathematics. This then influenced my own PhD research.

To give an example of the issue, consider the principle of equivalence. Many students draw a generalisation about the ‘=’ sign in an equation to the effect that it means ‘put the answer here’. This is, in many ways, a reasonable inference to draw. If you complete many problems where you have to answer ’10 + 13 =’ and ’27 – 14 =’ then it is unsurprising that you may come to this conclusion. However, the ‘=’ sign means that the left hand side is equivalent to the right-hand side. It is this principle that enables us to answer a question such as ’10 + ? = 12 + 17′.

An understanding of the principle of equivalence is an important goal of elementary maths teaching and therefore something that researchers are keen to examine. However, Crooks and Alibali maintain that efforts to assess this understanding often amount to asking students what the ‘=’ means and accepting some kind of definition as evidence of understanding. However, I reckon most maths teachers would back themselves to teach students to parrot a definition, so does this really measure understanding at all? The real test is in the application. If students can answer questions such as ’10 + ? = 12 + 17′ then that would demonstrate a better understanding than recalling a definition. If they can successfully rearrange an equation such as ‘3a – 7 = 9’ then they would demonstrate an even more comprehensive understanding of this principle i.e. that an operation performed on one side of the equation must be match by one performed on the other side.

It is for this reason that we can perhaps discount much of the research evidence around conceptual knowledge. Instead, I would propose that the ability to transfer concepts to a range of difference contexts is a better proxy for this kind of understanding. That’s why we used transfer as a measure of conceptual understanding in my published research.

It is therefore not good enough to dismiss evidence of the effectiveness of explicit teaching that is drawn from assessment outcomes, most of which will include items that assess some level of transfer, on the basis that these assessments do not assess conceptual understanding. If you have evidence that alternatives to explicit teaching lead to superior understanding, then let’s see it.

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