In this paper I have discussed how the ‘key idea’ Understand the mathematical structures that underpin the multiplication of fractions features in the Guidance document and related NCETM documents. Similar observations can be made about the treatment of other such ideas, for example the ‘core concept’ expressions and equations, which I might put in another blog.
So, on the one hand, I have acknowledged the attempts to identify various components of a mathematical concept and the different ways in which a concept can be interpreted. Such endeavours are extremely worthwhile, since they can inform us about the range of ideas that students need to engage with. It is worth pointing out that they can also help us recognise where students have constructed such ideas themselves, so that we can build on them more effectively.
However, I have questioned the claim that such ideas can be arranged into a ‘rigorous, coherent and connected’ curriculum, even to the point where we can devise a ‘detailed termly breakdown of the knowledge, skills and understanding required for Key Stage 3 mathematics’ (page A10). The claim does not stack up. First, using the example of scaling, I have argued that if ideas have not been carefully defined, it makes little sense to talk of them being deeply understood. Second, I have argued that the notion that mathematical concepts, however carefully defined, can be treated as more or less discrete, fixed and ordered, is an oversimplification which may not be helpful. The hollowness of the claim that ‘key ideas need to be understood deeply before moving on’ (p A11) is shown by the example of scaling, where various NCETM documents indicate, quite sensibly, that students’ encounters with scaling should be spread over several years.
I worry that this rhetoric is used to legitimise the ‘small steps’ approach to teaching which NCETM promotes, particularly at primary level. In effect, this approach rejects the notion that concepts have many facets and that students bring with them a diversity of ideas and interpretations. So, instead of students being enriched by expressing and evaluating such ideas, elicited by the teacher, they are marshalled along a carefully smoothed path where there is little scope for gaining an overview or insight or feeling connected to the maths.
How has the Guidance document, with its puzzling advice come about? In the days of the National Strategy, the DfE would provide guidance on the National Curriculum but it scrupulously avoided giving advice on how to teach specific content. NCETM has been funded by the DfE since its inception but the relationship has changed. Part of NCETM’s function now is to do the DfE’s bidding, including giving very particular advice on teaching. Given the baleful political influence on the DfE since 2010, this is a pity since it tarnishes NCETM’s reputation and stifles discussion and innovation.
Thankfully, this gloomy condition does not apply to all NCETM initiatives. I have recently been involved in an NCETM project developing algebra materials. There I have had the pleasure of working with highly experienced teachers affiliated to NCETM and we have been free to develop the materials based on our knowledge and experience, not through the lens of a particular ideology. The wonderful thing is, the structures of NCETM allow us to disseminate the materials to a large group of experienced teachers, in the form of NCETM PD Leads, who in turn can ask teachers to trial the materials through the auspices of their local NCETM hubs. The result, I hope, will be a set of well thought out and well trialled materials that will provide teachers with a fresh look at how they might, but not how they must or should, develop students’ understanding of algebra. There are NCETM documents that do the same kind of thing, but there are many that don’t.
References
I refer to extracts from six NCETM documents in this paper. Some have quite complex titles, so they are listed here with their pdf file names. The documents should all be available on the NCETM website.
A: Mathematics_guidance_key_stage_3.pdf
C: ncetm_spine3_segment06_y4.pdf
D: ncetm_spine2_segment17_y4.pdf
E: ncetm_spine2_segment27_y6.pdf
F: ncetm_spine3_segment09_y6.pdf