Guidance or wishful thinking?
A critique of the NCETM document Mathematics Guidance: Key Stage 3
Dietmar Küchemann
The DfE/NCETM documentA, Mathematics Guidance: Key Stage 3, was published (online) in September 2021 (see Figure 1). This Guidance document provides some useful insights into the content of school mathematics. However, it is, in my view, beset by wishful thinking. It is not clear which individuals wrote the document in its final form but I presume they came from within the DfE. These anonymous authors do though provide a list of people ‘involved in the production of this publication’. This includes people that I know and respect and in no way do I wish to suggest that they are responsible for the shortcomings that I am about to describe.
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| Figure 1: A tweet introducing the Guidance document |
The weakness of the document lies primarily in its dogmatic nature. This leads to advice that is ill-defined or not thought through and so impossible to apply.
We are presented with a single view of teaching and learning which is never questioned. The dogmatism is nicely illustrated here:
A curriculum compatible with teaching for mastery rejects superficial short-term coverage in favour of developing deep, connected understanding of key ideas. (page A11)
The passage may sound like ‘common sense’, as is true for much of the document, but leaving the content, and its false binary choice, aside for a moment, notice the form of the language. There is no human agent here, offering a viewpoint. The subject of the sentence is the curriculum itself, which somehow can’t help itself from actively rejecting one thing and developing another. It is as if a ‘mastery curriculum’ is a force of nature, an immutable fact of life.
What first caught my eye was another piece of common sense:
A fundamental principle of teaching effectively in mathematics is that key ideas need to be understood deeply before moving on. A curriculum which encourages teachers to move on to the next topic too quickly, before key ideas are deeply understood, results in superficial learning. While such an approach to ‘covering’ the curriculum at a rapid pace may seem to work in the short term, in the long term it is an inefficient use of precious curriculum time, because it leads to the same key ideas being retaught year after year. (page A11)
In part this rang true. I well remember, from my school teaching days, the frustration when students didn’t understand what I was trying to teach because, it seemed, they hadn’t understood what they had been taught previously. “Didn’t we do that last year?!” I was clinging to the belief, If they’ve been taught it, they must have learnt it, even though this clearly didn’t hold.
A response to this, as expressed in this and numerous other NCETM documents, is to spend more time on a topic so that ‘key ideas are deeply understood’. This makes sense if ideas are seen as fairly discrete, fixed and ordered, with one idea providing a solid foundation for the next. It might work for learning to ski or to hit a golf ball, but is this how mathematical ideas are best characterised? It seems to me, a more fitting view is to see mathematical ideas as forming a network, where idea A informs idea B, but where B can also inform A. So the more we learn, the more a given idea evolves, or at times is restructured, by being connected, and connected more securely, to more and more ideas. At what point is idea A ‘deeply understood’? Probably not before we’ve also grappled for some time with idea B and a host of other ideas!
So is the common sense notion that ‘key ideas need to be understood deeply before moving on’ really a fundamental principle, or an over-simplification that leads to teaching that is not as effective as it could be?
Early on, the authors give this example (Figure 2) of the ‘deep, embedded and sustainable’ understanding that is needed ‘to make sense of the structures underlying the multiplication and division of fractions’ (page A12).
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| Figure 2: Prerequisite knowledge.... (Guidance, p A12) |
The claims in this example would make sense if it were saying that students would develop a better understanding of the structures underlying multiplication and division of fractions, the more they developed an understanding of these criteria. However, I struggle with the idea that there is some kind of absolute standard of making sense of these structures, and that this will click into place once certain criteria, which are assumed to be clearly defined, have been achieved.
The authors move on from this example without further comment, but return to it in the next section of the Guidance document. This section (of 250 pages, no less!) is titled Sample Key Stage 3 curriculum framework and looks at some ‘key mathematical ideas’ with suggestions on how they might be taught in Key Stage 3. Reading about these key ideas provides useful food for thought about the nature of school mathematics. But when it comes to teaching, they can be viewed in different ways. Do we line them up, ready to be mastered one by one, or do we see them as coexisting, waiting to be explored and connected, and with more complex situations throwing light on and giving meaning to the simpler ones?
We discuss this Sample section in the next post.
