At the start of August, Barbara Oakley, a Professor of Engineering, wrote an op-ed for the New York Times about maths teaching. It is a largely sensible piece about the fact that improvement in maths requires practice, not all of which is particularly pleasant.
Oakley’s article is still creating a stir in the maths blogosphere. Dan Meyer, a education software designer*, has now devoted two blog posts to attacking it. This is notable because Meyer has recently appeared to withdraw from the maths wars, having once been a constructivist/reform/inquiry/fuzzy maths partisan (see his 2010 TED talk and his blog motto, ‘less helpful’).
Meyer’s latest blog post drags up ideas about ‘conceptual understanding’ and ‘maths zombies’ that will be familiar to those of us who have been around the block a few times. I have written a lot of posts about conceptual understanding and it’s actually quite difficult to pin down. If you have the time, try reading some of the ‘productive failure’ literature. This often assesses ‘conceptual knowledge’ and yet the questions that are used to assess it will probably surprise you – they often look for all the world like recall of relatively simple declarative knowledge. This is why I am more impressed with evidence of transfer – but that’s a different blog post.
Understanding is perhaps a useful idea. We have a folk perception of what it means that I have written about before. I don’t want to be in a position of denying the obvious. However, I don’t think understanding is qualitatively different to knowledge. Otherwise, what is it? I also think that it provides a bit of a crutch for some maths teachers. The teacher down the hall may have her students acing all of the standardised tests but you can always console yourself by claiming that her students don’t understand the maths as well as yours. This cannot be easily disproved, particularly since the only objective measure we have has already been dismissed. So you can tell yourself this good night story and sleep easy.
Understanding is also necessary to the constructivist narrative. It has been comprehensively shown, from a range of different types of research, that explicit teaching is more effective than inquiry methods. Constructivists therefore need conceptual understanding as a fallback option. “OK,” they say, “a little bit of explicit instruction may be effective for learning mere procedures, but an approach based upon problem solving is more effective for developing conceptual understanding.”
However, there is very little evidence to support such a position. If it were true, there should be loads of it.
I think Oakley walks into something of a trap here; one that has also ensnared me in the past. She talks of there being too much of a focus on conceptual understanding in maths lessons. I now wouldn’t describe it like this. I think that there is too much of a focus on flawed teaching methods that are justified on the basis that they deliver conceptual understanding when they do not.
When Meyer talks of conceptual understanding versus procedural understanding, he is actually talking about the difference between experts and novices. Consider this claim of Meyer’s:
“A student who has procedural fluency but lacks conceptual understanding …
- Can accurately subtract 2018-1999 using a standard algorithm, but doesn’t recognize that counting up would be more efficient.
- Can accurately compute the area of a triangle, but doesn’t recognize how its formula was derived or how it can be extended to other shapes. (eg. trapezoids, parallelograms, etc.)
- Can accurately calculate the discriminant of y = x2 + 2 to determine that it doesn’t have any real roots, but couldn’t draw a quick sketch of the parabola to figure that out more efficiently.”
These students are the mythical ‘maths zombies’ who are great at maths but don’t understand it. Yet if you examine all of these examples, they are just examples of students who are earlier along the novice to expert continuum. If you have been practising column subtraction and you are presented with 2018-1999 then, even if you could work it out by counting up, you will probably draw on the most familiar strategy because you lack the bandwidth to analyse it at the meta level. If I really felt it was important for students to be able to determine when counting up is more efficient than column subtraction then I would teach this to students explicitly. I would use contrasting cases where I presented two examples such as 3239 – 2747 and 2018 – 1999 and I would work both problems, both ways, drawing a distinction between the two.
Framed in Meyer’s terms, the idea that there are teaching methods that are better for conceptual understanding than explicit teaching is equivalent to the idea that there are teaching methods that can short circuit the novice-expert continuum and teach expert performance from the outset. This is the progressivist dream and one that, after a couple of hundred years, is yet to be realised.
Oakley comments that many of the engineers who work with her come from education systems that stress procedural fluency and a certain amount of rote learning. Is it really feasible that these people are all maths zombies who don’t understand what they are doing and who have been harmed by the way they were taught maths?
No. It’s not feasible at all. They started out as novices just like everybody else, they worked really hard and now they’re experts. Go figure.
*Meyer makes a point that Oakley and Paul Morgan, a critic of conceptual understanding, are from ‘outside the field’ of maths teaching, so I thought I would return the favour.