In my fourth and final instalment of my reaction to the Ofsted Maths review, I’ll be considering the missing puzzle piece – mathematical thinking. Find my earlier reactions here on problem solving, representations and talk.
At Ark Curriculum Plus, we're keen to avoid any unnecessary polarisation, as we know how unhelpful this can be to teachers. We've enjoyed listening to the maths education community about where they agree and disagree with the findings of the Ofsted research review and reflecting on how it resonates with the research literature and the experience of teachers in schools across our partnership.
As I come to the end of this blog series, and focus on mathematical thinking, I've tried hard to include the highlights as well as possible areas for improvement, but the quest for a balanced commentary has been particularly challenging here.
So, let’s delve in with one of my favourite quotes from the review, which resonates closely with our thinking at Mathematics Mastery: “an intense focus on underlying knowledge structures and connections rather than the surface coherence of activities and teaching. This means that teachers are planning for what pupils will be thinking about or with, not what they will be ‘doing’.” This quote is jam-packed with thought-provoking elements. It gets us contemplating that thinking mathematically often involves reshaping previously held ideas (re-evaluating our mathematical “structures” and “connections”). It also sits under the heading ‘Planning for what pupils will be thinking about’. We feel an important skill of teachers is being able to recognise and plan for learners’ mathematical thinking. So, I read on with great excitement. Tell me more! Disappointingly though, I feel the report missed a trick by moving on and focusing on broader comments on knowledge alone, rather than delving into mathematical thinking skills too.
A key aim of the National Curriculum is to ‘reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language’.
It’s great to see a section dedicated to ‘intelligent variation’ in the review which promotes ‘pattern-seeking habits’ ‘connections between varying problems’ and ‘important patterns and rules’. But we feel there’s so much more to mathematical thinking than brief references to patterns and connections alone. We were hoping to see references to Cuoco et al’s ‘habits of mind’ which encourage pupils to be ‘pattern sniffers’ ‘’experimenters’ ‘describers’ ‘tinkerers’ ‘inventors’ ‘visualizers’ ‘conjecturers’ and more!
The EEF found that:
“To solve problems, learners need to develop generic mathematical strategies, sometimes known as ‘processes’ or ‘generic mathematical skills’ (HMI, 1985), or as ‘strategic competence’, which Kilpatrick et al. (2001) define as the “ability to formulate, represent, and solve mathematical problems” (p. 5). These include actions such as specialising and generalising, and conjecturing and proving (Mason & Johnston-Wilder, 2006, pp. 74-77). The development of these strategies appears to be supported by teachers highlighting when they or their learners spontaneously use them; for example, by naming them and asking for other examples of their use (Mason, 2008).”
(Hodgen et al. 2018, page 16-17).
The EEF highlights further mathematical thinking skills that we can foster in our pupils, citing the influential and highly regarded ‘maths giant’ John Mason. As the principal author of the aptly named ‘Thinking Mathematically’, I am surprised he was not cited in Ofsted’s review or colleagues. It’s also worth noting that Mason’s co-author, Kaye Stacey, receives a single reference in the review but wrote ‘What is mathematical thinking and why is it important?’ - another maths giant’s core messaging feels disappointingly absent in the review.
The Ofsted review states, ‘Many pupils start school with some mathematical knowledge’. It reads as a risky statement – potentially alluding to the ‘other’ pupils as blank slates or empty vessels. All pupils start school with mathematical experiences. Gattegno, Freudenthal, Mason & Johnston-Wilder (to name a few) have all spoken about pupils’ natural ‘powers of the mind’. From a very early age, before starting school or early years settings, children demonstrate that they can imagine, sort and classify objects and experiences and compare them, make conjectures about what might happen, use specific examples and generalise from these.
This is mathematical thinking! These skills are central to problem solving. A quick search of the review document finds many results of pattern seeking but no mention of ‘imagining’, ‘sorting’ ‘generalisation’ or ‘conjecturing’. Have the complexities and extent of mathematical thinking research truly been reviewed? The impact of teacher questioning feels limited by the single reference in the review: ‘Questioning, as long as teachers take care with language and timing, can also aid instruction’ Being the first and last reference to teacher questioning, it’s impact feels limited. Instead, I’d suggest checking out Watson (a nod to another overlooked ‘maths giant’) and Mason’s practical suggestions in ‘Questions and prompts for mathematical thinking’.
In the words of Ofsted, I’d like to do a ‘Deep Dive’ on their interpretation of ‘efficiency’ in relation to calculation methods. Readers are repeatedly urged against the merits of informal/mental methods:
“However, teachers need to be cautious when considering curriculum approaches that are heavily weighted towards encouraging informal and self-generated methods. These approaches may purport to develop pupils’ understanding, but the evidence shows that when pupils use a variety of informal procedures, it can inhibit understanding later on.”
"The ideal pen and paper methods in the 4 operations and for working with fractions are efficient, accurate and clear. The resulting neatness and logical approach helps to minimise the risk of pupils making accidental errors”
We’ve interpreted this as more of a focus on presentation than number sense (the latter is another phrase lacking in this review), to mitigate against the ‘risk’ of mental methods, it’s encouraged “to plan to use informal methods for only a short amount of time”
What is risky is referring to the formal methods as “ideal” – this is an overgeneralisation because efficiency is entirely dependent on the numbers being worked with. A way that pupils can minimise their risk of “making accidental errors” is by applying number sense instead of the ‘ideal’ formal method of subtraction to solve 5000 – 1999, for example. This stance is at odds with the Teacher Assessment Framework statement in KS1. Instead of one single ‘ideal’ method, pupils should “add and subtract any 2 two-digit numbers using an efficient strategy, explaining their method verbally, in pictures or using apparatus”.
We’re disappointed that unlike the Ofsted science research review which foregrounds disciplinary knowledge, we feel that the Ofsted maths review appears to have missed out on key evidence supporting development of mathematical thinking for every child.
Sadly, I won’t have been able do justice to every maths giant in the space of four blogs, but my aim was to celebrate, through gentle reminders and nods, some of those who have dedicated their careers and decades to improving mathematics education – the essence of ‘ambition for all’.
There’s hope! I call for OFSTED’s subject report, following this Autumn, to share more comprehensive guiding principles to truly support schools facing difficult curriculum choices amidst COVID disruption. Let’s put our pupils first.
Cuoco, A., Paul Goldenberg, E. and Mark, J., 1996. Habits of mind: An organizing principle for mathematics curricula. The Journal of Mathematical Behavior, 15(4), pp.375-402.
Gattegno, C. 1971. What we owe children: the subordination of teaching to learning. London, Routledge Kegan Paul
Freudenthal, H. (1973) Mathematics as an Educational Task. Springer
Hodgen, J., Foster, C., Marks, R., & Brown, M. (2018). Evidence for Review of Mathematics Teaching: Improving Mathematics in Key Stages Two and Three: Evidence Review. London: Education Endowment Foundation, accessed at: https://educationendowmentfoun...
Mason, J., Burton, L. and Stacey, K., 2011. Thinking mathematically: Pearson Education
Mason, J. & Johnston-Wilder, S (2004) Fundamental Constructs in Mathematics Education
Stacey, K. (2007). What is mathematical thinking and why is it important? Retrieved from http://www.criced.tsukuba.ac.j...
Watson, A. & Mason, J. (1998) Questions and Prompts for Mathematical Thinking. Derby: ATM