Head Teacher Update - Summer Term 2023
MATHEMATICS: Teaching mathematical concepts and uency
2023-05-11 11:28:17
If we are to develop mathematical fluency that sticks, then we must begin by teaching pupils key subject concepts before practising how to do them. Peter Mattock explains conceptual mathematics teaching
Over the last few years there has been an increasing focus on primary pupils learning certain facts and processes.
Spurred by the introduction of things like marks only being available for “approved” methods in key stage 2 SATs and the multiplication “check” in year 4, there is the potential that teachers may be driven towards using mathematics learning time for “drilling” these facts and processes.
While this may prove effective in ensuring pupils pass these assessments, there remain question marks over the longer term impact of such approaches.
The Trends in international mathematics and science study for 2019 (Richardson et al, 2020) notes: “Compared to 2015, England’s performance significantly improved in mathematics at year 5 ... and remained stable in mathematics at year 9.”
This indicates that changes in approach to key stage 2 mathematics between 2015 and 2019 did not translate into lasting benefits in mathematical understanding.
Although the pupils in year 9 in 2019 would perhaps not have gained the full benefit of any changes to approach from 2015, this is still suggestive that a different approach may be needed that can continue to support pupils in year 5 to achieve the desired fluency that will “stick” as they move through their school life.
Don’t worry, this is not a dive into the arguments about “progressive/ inquiry” versus “traditional/explicit” approaches to teaching. I am satisfied within myself (and believe many teachers agree with me) that both are required at different times.
No, this is about whether pupils arrive at these processes and facts through pure rote memorisation or from a position of understanding the concepts from which these other things arise.
Take, for example, the process of short division – one of the approved “formal” methods for key stage 2 SATs. When it comes to division, most primary teachers of mathematics are familiar with modelling division as either “grouping” or “sharing”.
However, there appears to be less clarity over how the short division algorithm arises from (one of) these models.
In fact, it seems that there is little effort in many places to develop the algorithm from the model, with it being much more likely that the two will be treated in a relatively isolated way (i.e. pupils will learn about models for division to help with identifying division problems, but then learn the short division algorithm as an abstract process to calculate the results of divisions).
For example...
To illustrate this let’s examine the division 432÷3.
Typically, the language used to describe calculating the result of this division using the short division algorithm starts something like: “How many 3s are there in 4?”
This is broadly in line with a grouping model for division, although whether this is recognised explicitly is unclear. Whether the grouping model is explicitly referenced or not in this process, the bigger issue is that it is perhaps not the best model to show what is happening with the short division algorithm!
The question itself suggests how many 3s in 4, but of course, the 4 in this scenario is not actually 4, but rather 400.
However, the question “how many 3s in 400?” is a much more complicated one and the answer is significantly different to the “one, remainder one” that would be required for the short division algorithm.
The much more applicable model to the short division algorithm is actually the sharing model. The way the short division algorithm arises from this model for division can be seen clearly if the division is modelled using Dienes blocks or, as in the illustration below, with place value counters. The process here is:
■ We start by sharing our four “hundreds” across the three shares, resulting in one “hundred” in each.
■ The remaining hundred is then exchanged into 10s, leaving us with 13 value counters worth 10.
■ Of these 12 are divided so we have four in each of the three shares.
■ The remaining 10 is then exchanged into 1s, leaving us with 12 value counters worth 1.
■ These 1s are, likewise, divided across the three shares so that we have four in each.
■ This gives one “100”, four “10s” and four “1s” in each of the three shares, meaning the result is 144.
Not only does this approach to the short division algorithm avoid the sort of issues with place value highlighted above, it is also much more clearly linked to models of division that pupils should already be familiar with.
Indeed, provided pupils are already familiar with division as sharing (and the necessary understanding of place value) then they should not be particularly surprised by this extension of the model to dealing with larger numbers, it develops quite naturally from their earlier experiences of division.
Walking and running
This example illustrates what it means to approach maths conceptually. By first ensuring that pupils understand the concept of division through the way we model it, we can then support pupils in developing the required procedural understanding.
This is possible for every concept in school level maths. Each of them begins with an idea from which the associated facts and procedures arise, and so by beginning with the idea, teachers of maths can create a connected and coherent journey of study, including the facts and procedures associated at natural points in the journey.
Many vocal maths educators have espoused the view that the best way for pupils to achieve this conceptual understanding is by first focusing on a more procedural understanding, or “instrumental understanding” (see Skemp, 1976).
Their argument goes that when a pupil is so comfortable with a procedure that they no longer really have to attend to its “surface” features, the pupil is then in a position to examine the deeper features of the concept.
Taken at face value, I can see why teachers are sympathetic towards this view – the old adage “you can’t run before you can walk” springs to mind. However, as I write in my book Conceptual Maths, there are two reasons I do not share this view.
First, much of my experience of maths education has been of a very instrumental approach in which pupils are often practising procedures for much of the time that they don’t spend listening to their teacher.
While I can see ways to improve this practice so that learners take more from the experience, I don’t see it giving the gains in pupil understanding or motivation to continue studying mathematics.
And second, as I have intimated, the procedures attached to different concepts arise from the structure of the concept itself. To rely on knowledge of the procedure to provide understanding of the concepts seems to be backwards in approach. In addition, because there are many different procedures associated with each concept, it would seem to be time intensive to have to study many of them to the point of automaticity before being able to use this experience to gain a window into the underpinning structure. Instead, it would seem much more logical to first secure and exploit the structure to gain insight into the associated procedures.
For me, the “walking” is that careful build up and securing of the idea, pupils slowly and cautiously building an understanding of a concept that can be utilised to work flexibly with processes and show insight into problems.
And this is what I hope Conceptual Maths will support teachers in doing. By examining virtually every concept that is taught at school level and offering teachers of mathematics a detailed insight into the structure of each concept, as well as questions and tasks to reinforce this relational understanding, I hope to empower them to change and develop their practice in order to support pupils on a coherent journey through mathematics that makes sense to them by teaching “about” (rather than just “how to do”) maths.
• Peter Mattock has been teaching maths for more than 15 years. He is a specialist leader of education and an accredited maths professional development lead. His book, Conceptual Maths, published by Crown House, is available via www.crownhouse.co.uk/conceptual-maths
Further information & resources
■ Richardson et al: Trends in international mathematics and science study (TIMSS) 2019: National report for England, DfE, 2020: https://bit.ly/3Aynfc8
■ Skemp: Relational understanding and instrumental understanding, Mathematics Teaching (77), 1976: https://bit.ly/3oPJAze
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