The Case for Chunking (or Why Recall and Reasoning are Best Buddies)

Civilisation advances by extending the number of important operations which we can perform without thinking of them.
A N Whitehead

Recently, I’ve been thinking about the driving lessons I took when I was 17. By the time I first stepped into his car, my instructor had been teaching people to drive for decades. Every sentence he said felt reassuringly rehearsed. This was especially true of a maxim that he repeated whenever I was struggling with an aspect of driving:

“First, we think. Then we practise until we don’t have to.”

Sometimes he’d justify this little maxim by describing hypothetical situations:

“When you see a football rolling into the road in front of you, I want you thinking about the kid that might be chasing after it. You can’t do that if you’re thinking about how to change gear or how to use the brakes. How can you think about the important stuff if you’re still thinking about the little stuff?”

In other words, my driving instructor recognised the importance of chunking to the development of expertise.

Chunking is the process through which individual pieces of information are consolidated into larger meaningful units. For example, in my first few driving lessons, I had to pay conscious attention to operating the clutch, moving the gear stick and assessing whether my speed required a gear change. Through conscious thought and practice, these separate actions each became automatic and were eventually chunked into a single unit: changing gear.

As I developed as a driver, changing gear, using the steering wheel and working the brakes were then chunked into a larger single unit: manoeuvring the car.

Eventually, these aspects were chunked into an even larger single unit: driving competently.

I am able to drive to a legal standard precisely because various skills and bits of knowledge were consolidated into larger and larger chunks through conscious effort and practice. This chunking of bits of knowledge and skills into larger and larger meaningful units so that we can do more and more complex things is a pretty powerful way to think about the learning process.

And this brings me onto the year 4 multiplication table check. In relation to the check, an education journalist recently posted the following tweet:

At first glance, we might agree that this is a remarkable state of affairs. After all, don’t we want pupils to grasp that 8 x 7 can be reimagined as (8 x 5) + (8 x 2) or as (8 x 10) – (8 x 3)? Isn’t this reasoning about the distributive property exactly the sort of thing we want pupils to become familiar with? It certainly is. Nevertheless, it seems that the government doesn’t want pupils to use such reasoning to work out basic multiplication facts forever. They want pupils to have chunked these reasoning steps into a single multiplication fact in each case by the end of year 4, as evidenced by the need for rapid recall to pass the multiplication table check. Why might the government see this as necessary?

If we look at the components of the national curriculum that are commonly taught in year 5, it is clear that pupils need to put their knowledge of basic multiplication facts to a lot of use. And if pupils are reasoning their way to basic multiplication facts like 8 x 7 (i.e. reaching the answer of 56 through multiple steps), then we are adding extra steps to any chain of reasoning that they undertake.

Let’s consider an example of what this means. Imagine that we want to teach pupils that we can work out 8 x 69 using the distributive property. We might consider this as one particular chain of reasoning:

8 x 7 = 56 → 8 x 70 = 560 → 8 x 69 = (8 x 70) – (8 x 1) = 552

This is tricky stuff. It takes time and exactly the sort of understanding of the distributive property to which pupils will have been introduced when they were initially learning about multiplication. Every pupil that has to go through multiple steps to find the answer to 8 x 7 is forced to add extra steps to an already complicated chain of reasoning. In contrast, those that can fluently recall 8 x 7 can focus on this more advanced application of the distributive property.

Let’s consider another example. Imagine that we want pupils to simplify ⁴²/₄₈. If they have to use multiple steps to divide each of these numbers by 6, they are less likely to focus on the underlying mathematics that allows this process to work. (One way of thinking about this process is to recognise the equivalence between 1 and ⁶/₆ and to know that dividing by 1 leaves a value unchanged → ⁴²/₄₈ ÷ ⁶/₆ = ⁷/₈)

In other words, pupils should absolutely be taught to reason their way to basic multiplication facts. But this is part of the learning process, not the end goal. At some point, pupils should be encouraged through practice to recall each basic multiplication fact without having to work them out. (Michael Pershan has a cracking blog on this subject: http://notepad.michaelpershan.com/what-people-get-wrong-about-memorizing-math-facts/)

I have met many people over the years who have stated that they coped fine with mathematics without being able to recall multiplication facts. This isn’t a surprise. I have no doubt that some are capable of overcoming almost any impediment in almost any situation. The issue really is that not all will overcome these impediments. Having spent a decent chunk of my career working specifically with those who have struggled academically, I am certain that it is these pupils who are most impeded by a lack of foundational knowledge on which to rely. The learning of number bonds and multiplication facts to fluency has often been the catalyst that has led to positive changes in what children deem themselves capable of in mathematics. I make no apology for advocating the fluent recall of multiplication facts as an aim for the vast majority of pupils.

Of course, there are other questions to consider in relation to the multiplication table check:

Are the expectations of the national curriculum in year 5 reasonable?
Do accountability measures such as the year 4 multiplication table check achieve what they aim to?
Will teachers prioritise the learning of multiplication facts with children for whom other aspects of maths might be more urgent (i.e. number bonds inside 20)?

These are interesting questions, and this blog is not a defence of the check itself. Instead, this blog is simply a reaction to the belief that fluent retrieval of multiplication facts (or other foundational knowledge) is somehow at odds with mathematical reasoning. It isn’t. Regardless of one’s views on the multiplication table check, it is perfectly sensible to want pupils to fluently recall basic multiplication facts if we also want them to apply these facts as elements of more advanced reasoning.

In short, chunking knowledge and skills into larger and larger single units is essential to learning, and the development of arithmetic is no exception to this. Pupils find it much harder to reason with basic multiplication facts if they are still reasoning their way to basic multiplication facts. Aiming for eventual rapid recall of basic multiplication facts is a perfectly sensible aim within any primary mathematics curriculum.

Early mathematics: where should you start?

Do you teach mathematics? If so, do you know the difference between perceptual subitising and conceptual subitising? Do you know what the cardinal principle is? Do you know how five-frames and ten-frames can play a role in developing children’s sense of number? If any of these leave you stumped, then you are not alone. From my experience, too few maths teachers at KS2, KS3 and beyond understand the mathematics journey that their students have been on and the potential foundational mis-steps that might be causing their students difficulty. Until fairly recently, I knew next to nothing about early mathematics. I’m far from an expert now, but the knowledge that I have gained on the subject has come a decade later than it should have done and has transformed how I view the teaching of mathematics. If, like me, you would like to learn more about this crucial area, then here is my guide on where to begin, which tries to take into consideration the amount of time that you have to dedicate to it:

If you effectively have no time to dedicate to this goal at present, then get the ball rolling by following these people on Twitter:

@berniewestacott

Mr Westacott combines infectious enthusiasm with expertise and insight. Here he is discussing the use of manipulatives with @mrbartonmaths: https://www.youtube.com/watch?v=HwQYVwbOgdw&list=PL7BJ-1MkmUZ9A6m4qYXgISzZy63CQT4sh

@Kieran_M_Ed

Kieran knows an astonishing amount about mathematics, and he is generous in his willingness to share his expertise. His excellent website, podcast and links to his books can be found here:

https://www.thinkingdeeply.info/

@helenjwc

Dr Williams is a thoughtful and incisive advocate of research-informed practice in EYFS. Here she is discussing early maths, again with @mrbartonmaths:

http://www.mrbartonmaths.com/blog/helen-williams-early-years-teaching-and-manipulatives/

@mattematics

This gent knows his stuff. Follow him and then bookmark any tweet he makes on the subject of mathematics.

If you have 2-3 hours only, explore the Learning Trajectories website and read Making Numbers:

This website is arguably the quickest way for a novice of early mathematics learning to become better informed. The Learning Trajectories approach is to attempt to specify a progression that learning can follow for various areas of early mathematics, from subitising to spatial visualisation. For each area, there are several levels with each demonstrated through brief videos and learning activities. If I ran a primary teacher training course, I would give every trainee and hour or two, at least, to explore this website. It is a goldmine.

Simply sign up for free, set up a class (it doesn’t need to be populated with students, just given a name) and away you go:

https://www.learningtrajectories.org/class/55937/developmental-progression

Making Numbers by Rose Griffiths, Jenni Back and Sue Gifford

I can’t imagine a more welcoming, accessible introduction to early maths pedagogy than Making Numbers. Full of simply expressed ideas that belie the depth and utility of the underlying concepts, this book is a relatively quick read, but one that you will return to repeatedly, especially if you teach maths in reception or Key Stage 1. The only drawback to this book is its cost. I’d recommend persuading your maths coordinator to fork out for it as this book would be a useful addition to any CPD library.

If you have 6 hours, also read this book:

Understanding Mathematics for Young Children by Derek Haylock and Anne Cockburn

This book is the perfect place to start. It is accessible and informative and provides one framework for getting to grips with the idea of ‘understanding’ in mathematics. It discusses early number, operations, the principles of arithmetic, shape & space and problem-solving in highly practical ways, but also manages to grapple with what it means to think mathematically.

Highlight: within the Understanding Shape and Space chapter, there is an analysis of this topic that progressively moves through subtle changes to shapes, illuminating the crucial mathematical thinking in finding differences and similarities. It is an implicit lesson in the use of variation and an insightful look at what is meant by ‘equivalence’ in mathematics.

If you have 10 hours, also read these two books:

Teaching and Learning Early Number by various authors and edited by Ian Thompson

A collection of short essay-like chapters by various researchers into early maths, this book unpicks many of the complexities of children’s early ideas. Each chapter can be read as a standalone exploration of a given topic, which makes it an easy read, but it can feel a little disjointed. Regardless, it’s well worth your time.

Highlight: Thompson’s chapter on mental calculations, in particular the extra complexity hidden in some supposedly simpler methods of calculation, discusses why getting stuck on certain strategies of addition and subtraction can lead to unnecessary struggles over the longer term.

Teaching Mathematics 3-5 by Sue Gifford

While the books mentioned above provide a useful route to understanding early mathematics for all primary and secondary educators, Teaching Mathematics 3-5 is probably the most useful for an educator working with children across the EYFS age range. Gifford presents a research-informed view of early mathematics, addressing the need for a holistic view of children’s learning that respects the social, emotional and physical aspects of the journey to understanding. She emphasises the need for adult-initiated mathematics learning (contrasting this with adult-led activities) within stimulating, thoughtfully constructed learning environments. In addition, Teaching Mathematics 3-5 is full of practical suggestions and snippets from real interactions with children. Often these snippets are of children’s misconceptions, making the book a welcoming read for those new to the profession and one that will resonate with more experienced teachers.

Highlight: The brief section on playfulness and humour in mathematics learning stands out as a subject that is too rarely discussed on Edutwitter or in other texts, and yet is a valuable component of expert teaching.

If you have 20 hours, also read these two books:

Growing Mathematical Minds by Jennifer S McCray, Jie-Qi Chen and Janet Eisenbard Sorkin

This is an attempt to join the findings of early mathematics research to the practicalities of classroom teaching by giving teachers the chance to enter conversation with researchers. While I don’t agree with every interpretation made within the book, it is a worthwhile exploration of how research can impact real settings. I particularly appreciated Siegler’s ‘overlapping waves’ model for the way that children use different calculation strategies under different circumstances helps tie the complexity of real learning to the occasionally simplified categories in academic research.

Hands On, Minds On by Claire E Cameron.

This book details the research on executive function, motor skills and spatial skills and how they relate to early learning, the last of these in particular being implicated as having a relationship with later mathematics learning. It’s a fascinating look into the foundations of all school learning.

Visible Maths by Pete Mattock.

Visible Maths is not directly related to early mathematics; in fact, much of its content is most useful for secondary teachers. Nonetheless, its explanation of how the use of manipulatives and pictures can enhance learning is very useful for teachers of early mathematics.

If you have more than 20 hours, also consider these books. (I am currently reading these, but they seem well worth the effort):

Learning and Teaching Early Maths – The learning Trajectories Approach byJulie Sarama and Douglas H Clements.

Early Childhood Mathematics Education Research: Learning Trajectories for Young Children byJulie Sarama and Douglas H Clements.

These are rather expensive, but come highly recommended from people who know considerably more than I on this subject. Both are related to the Learning Trajectories website that is discussed above. The first of the two books is more practical, while the second is a description of the research upon which the Learning Trajectories approach is based.

If you’re interested in mastery approaches to mathematics, I’d highly recommend Mastery in Primary Mathematics by Tom Garry.

And that’s about it. I hope you get as much from learning about early mathematics as I have.

Maths fluency part 2 – a useful list?

Given the positive response to my recent blog-post on the subject of mathematical fluency, I thought a few people might find this useful…

I wrote the last blog-post because I’d been considering the areas of mathematics that were (a) most frequently relied upon in later learning[i], (b) often better taught in a few minutes each day over an extended period, and (c) absent from too many children’s repertoire of fluency. With this in mind, I began working on a list of fundamentals to be taught to the point of fluency in each year group.[ii] The point of the list was to indicate which areas of the curriculum might be best tackled using a ‘little and often’ approach after initial teaching. These are indicated in italics. (Alongside these, I included some conceptual fundamentals that I thought needed to be revisited throughout the year. These are indicated in bold.) The list is very much a work in progress, and I am open to feedback. The position of each component in a given year group matches fairly well to the national curriculum, but the idea is that this list only indicates where the ‘little and often’ teaching of each component should begin; where each component should end, however, is entirely dependent on how long it takes a class to reach fluency; this means that a component might continue into the following academic year if necessary. I post the list here merely in the hope that it might be useful to others as a jumping off point for further discussion of this topic:

primarily ‘conceptual’
primarily ‘mental fluency’

Year 1

(a) Understand and use various concrete and pictorial representations of numbers up to 100 (including all of cubes, dienes and number lines)

(b) Understand and use bar models and part-part-whole diagrams as representations of number bonds inside 20; use these to know whether an unknown is the part or whole in equations such as ? – 8 = 4 and 5 + ? = 11.

(c) Recall number bonds up to 10 (i.e. without a counting strategy)

(d) Find half of even numbers up to 20 and double of integers up to 10

(e) Count up to 20 in 2s, up to 50 in 5s and up to 100 in 10s

Year 2

(f) Understand different interpretations of addition (collecting similar objects, counting on and extending) and subtraction (removing objects, counting back, shortening and finding a difference)

(g) Understand rules of commutativity for addition, subtraction, multiplication and division (i.e. addition and multiplication – order doesn’t matter; subtraction and division – order matters)

(h) Recall addition and subtraction facts inside 20 (i.e. not using a counting strategy; e.g. 7 + 6 –> double 6 then add 1 = 13 or 7 + 6 –> 7 + 3 + 3 –> 10 + 3 = 13; ideally, pure recall from practice)

(i) Use recall of addition and subtraction facts inside 20 to calculate mentally TO + O and TO – O (i.e. without a counting strategy)

(j) Using note-taking (without counting) calculate O + O + O

(k) Use concrete objects, pictures or mental strategies to calculate T + T (e.g. 70 + 20 = 90), T – T (e.g. 90 – 30 = 60), TO + T (e.g. 56 + 40 = 96) and TO – T (e.g. 89 – 40 = 49)

(l) Recall (i.e. not count up) multiplication and division facts for 2x, 5x and 10x including missing number questions (e.g. 5 x __ = 35)

Year 3

(m) Understand different interpretations of multiplication (repeated addition, increase in dimension, change in the counting unit and the scaling of a value) and division (sharing and grouping)

(n) Understand and use various concrete and pictorial representations of numbers up to 1000 (including all of dienes, place value counters and number line)

(o) Calculate mentally (without a counting strategy) HTO + O and HTO – O

(p) Count in 10s up to 200 and count in 100s up to 2000

(q) Recall (without counting) multiplication and division facts for 3x, 4x and 8x, including missing numbers (e.g. 8 x __ = 56)

Year 4

(r) Use number bonds inside 20 knowledge to calculate addition and subtraction facts for tenths inside 2.0 (e.g. 0.8 + 0.7 = 1.5)

(s) Use place value knowledge (not adding/subtracting zeros) to mentally multiply and divide by 10 and 100

(t) Recall all multiplication facts up to 12 x 12

(u) Round numbers to the nearest ten and hundred using number line representation.

(v) Recognise the decimal equivalents of tenths up to 9/10 and hundredths up to 99/100, and vice versa.

Year 5

Understand and use various concrete and pictorial representations of numbers including up to 1 000 000 and numbers including tenths, hundredths and thousandths (including all of dienes, place value counters and number line)

(x) Mentally multiply multiples of 10, 100 and 1000 by other multiples of 10 and 100 (e.g. 400 x 80 = 32 000)

(y) Mentally divide multiples of 10, 100 and 1000 by single digit numbers (e.g. 3200 ÷ 8 = 400)

(z) Calculate equivalent fractions using all multiplication facts

That’s all folks. I hope it is useful.[iii] Let me know if there you consider there to be any glaring omissions or things on the list that need not be there.

[i] Will Emeny’s blogpost showing the GCSE maths curriculum in a visual model is a fascinating way of showing the areas of the curriculum upon which other learning relies most often. In the model, each area is a node and the number of connections between each node is shown by the size of the node, making it clear which areas of the curriculum are relied upon again and again. My description doesn’t do it justice, so I’d highly recommend you follow this link:

You’ve never seen the GCSE Maths curriculum like this before…

[ii]Advocates of a genuine mastery approach will likely take me to task over this implication that there are particular components of the curriculum that require fluency, rather than the whole thing; I absolutely understand their point and agree. However, many – if not most – schools teach the curriculum in a way that Mark McCourt would describe as a ‘conveyor-belt’ approach. This being the case, identifying some areas of the curriculum where fluency is at its most important may be of some use, though I recognise the pragmatism in it.

[iii] Those of you who are familiar with Edutwitter might notice some similarities between this list and a much-reduced curriculum presented in one of the thought-provoking blogs of Solomon Kingsnorth (@solomon_teach). The difference between the two that makes this post worthwhile (I hope) is that this list by no means constitutes the breadth of the curriculum as I would like to see it. It is merely a list of the stuff that is essential and yet can slip between the cracks when teaching is seen in terms of lessons, topics and year groups.

Mental mathematics fluency and the importance of perspective

Imagine this: you are one of two people entered into a race with a large amount of money as a prize. You both arrive at the starting line fancying your chances in a foot race over any distance. (In this imagined scenario, you are quite the athlete.) You are each presented with a bicycle that you can choose to ride if you so wish. Sadly, neither you nor your competitor has ever learned to ride a bicycle. Before the two of you have time to realise that your entire existence is merely a part of a blogger’s stretched metaphor, the starting pistol goes and you’re faced with a decision – whether to learn to ride the bike or whether to leave it behind and just run. What should you do? The correct decision depends entirely on one crucial fact: the distance over which the race is to be competed. If the race is 200 metres long, then obviously the sensible decision is to ditch the bike and run. If not, your opponent will cross the line and walk off with the cash prize before you even learn how the brakes work. But what if the race is to be competed over 500 miles? Then, clearly you will stand a far better chance if you spend a few hours – or perhaps even a day or two – learning how to ride the bike. It will slow you down at first, but the eventual gains will more than make up for it. The length of the race is the crucial factor. Lose sight of that, and you’re prone to inefficient decisions.

Now imagine a year 3 teacher. Today’s lesson is on written addition. She has explained the column method, and she’s content that – having explored the dienes and place value counters – the children understand the method. She sets the class a question – 29 + 38 – and begins to purposefully wander around the room. Almost immediately, she is struck by what she sees from the majority of the children: they are adding the digits in the ones column – 9 + 8 – using a ‘counting on’ method, starting on the 9 and counting forward 8, keeping track of this by using their fingers. The majority seem to be getting the correct answer, though the ‘addition by counting on’ they are doing is a mentally laborious way to go about it. The teacher knows that this is far from ideal. She knows that these children will find all sorts of mathematics easier in the future if they develop a strategy for adding one-digit numbers based on known facts (e.g. seeing 9 + 8 as a near double; using compensation to see 9 + 8 as equivalent to 10 + 7; or perhaps even just knowing that 9 and 8 sum to 17). So, what should she do? Should she carry on as before? Or should she begin to spend five minutes every day – spread over several weeks – developing her class’s fluency with number bonds inside 20, eventually saving her children the mental effort of an inefficient ‘counting on’ strategy? She’s painfully aware that taking this time will slow her down; doing this might be beneficial in the long term, but there’s only so much time in the year. (The year before, for example, she taught bar charts and pictograms in a single lesson towards the end of the year just to cover everything she was supposed to.) She decides, this once, to leave the majority with their ‘counting on’ strategy. After all, it might be inefficient, but the class seem to be getting the correct answers. Thanks in part to the pressures and artificial delineations of individual lessons, topics and year groups, she has lost sight of that one crucial factor: the length of the race. Her kids have set off on their 500 mile trek, and their bicycle is at the starting line.

With the day-to-day pressures of teaching, it is all too easy to forget that our kids are in this for the long haul. We must remember that we are building something. Mathematics, in particular, is a highly hierarchical subject; if a foundation is poorly built, the effect it has on the construction above is exponential. In my experience, the most confounding gaps in children’s mathematical abilities are those that are difficult to teach in the space of a few lessons. There is much in primary school mathematics that is understood with relative ease, but requires a great deal of practice before it becomes fluent; number bonds inside 20 and the related subtraction facts are a perfect example of this.[i] I challenge anyone to ‘teach’ this fluency to children over just a few lessons. Like fluency with multiplication facts (and the related division facts), these need to be seen regularly for a considerable period – though perhaps just for a few minutes each day – for them to become fluent. This is not to advocate any particular method for teaching this. Children can practise these against the clock, keeping their answers private; they can quiz each other using cards; they can play games that expose them to these facts over and over. How they see them is, in my view, less important than how often they see them. Regardless, one thing is certain: if we don’t take every opportunity to prioritise long-term efficiency over short-term convenience, then we are doing the children in our classes a major disservice.

A few things I find useful in teaching underlying mental skills:

1. ‘Little and often’ is perhaps the most time-efficient way to develop fluency with mental mathematics. Five hours spread over ten weeks (roughly five minutes per day) seems to be far more effective than the same amount of time spread over one week.

2. Take the pressure off. It’s perfectly possible to quiz children – and have children mark their own work – without there being any stress about performance or speed. In my experience, if you give children a separate strip of paper with the questions and ask them to mark their own work when you share the answers, they soon realise that the only pressure is their own desire to improve. Children almost always take pleasure in developing their fluency over time. It often becomes the part of the lesson they like most.

3. Use routines and don’t be afraid to use the same activities and resources repeatedly. Routines allow children to engage with the actual learning rather than dealing with the superficial elements of an activity.

4. Be persistent. Beyond the rarest of exceptions, all children can learn fluency in the underlying mental mathematics skills. Have faith that this will lead to considerably more effective learning in the long term.

[i] This is not to suggest that the different interpretations of basic addition and subtraction do not require a great deal of conceptual understanding; they absolutely do, ideally through a variety of physical and visual structures. Peter Mattock’s book Visible Maths is superb on this subject.