Tony Gardiner on CPA and (Anti)Differentiation

In 2022, in one of his first comments on this blog, Tony Gardiner introduced us to his three Mathsteasers books. These excellent books are extension companions to the (also very good) Maths — No Problem! series of primary school textbooks. The Mathsteasers also contain an appendix, an extended essay by Tony on the philosophy underlying the MNP series and his Mathsteasers companions. Although written in regard to MNP and, more broadly, the English curriculum, Tony’s writing, as always, contains general truths, beautifully expressed. 

After reading Tony’s essay, I asked Tony if he would permit me to post his appendix-essay on this blog. Tony was fine for me to do so, possibly with suitable modifications, and he contacted the publisher in order to gain their permission. Tony reported back that the publisher CEO “responded encouragingly” but was busy at the time, and the idea fell by the wayside. But now, with Tony gone, and in the spirit of What the Hell, I figure to just post the thing. I can’t imagine the publisher really cares and, if they do, I hope they’ll treat it as an advertisement for their fine publications. 

For regular readers, some of Tony’s themes will be familiar, and it was tempting to extract some of his punchy lines. But, I’ll let Tony’s essay speak for itself; it is long but it is worth it. I’ve added a few links but otherwise the text is as Tony wrote it. Also, a note of clarification on the English Key Stages, to which Tony refers: Key Stage 1 is the first two years of schooling (ages 5-7), KS2 is Years 3-6 (ages 7-11) and KS3 is Years 7-9 (ages 11-14).

FOR THE TEACHER OR PARENT

The last 40 years have seen wild fluctuations in official policies relating to “differentiation”. This Appendix seeks to explain — for the benefit of all potential users — the approach which underpins these Mathsteaser extension booklets. The booklets constitute a supplement to the Maths — No Problem! Primary Textbook Series for Years 4–6, so some of the explanatory comments refer to specific textbooks. However, the series follows the English National Curriculum, so these Mathsteasers also form a valuable resource for schools using other schemes of work, and for parents and pupils in search of more substantial challenges on standard curriculum themes no matter what programme they may be following.

 

General background

Primary school lays the foundations for all subsequent work in mathematics. Pupils develop confidence and fluency in working with whole numbers, with counting, and with place value (working with numerals, words, measures, the four operations, and with standard algorithms); they learn to solve word problems; they learn to recognise, to describe and draw, and to compare and sort different shapes; and they learn to use the relevant vocabulary. Above all, pupils need to learn to think flexibly.

The Maths — No Problem! texts address the basic national curriculum, developing each topic as a progression:

• from Concrete objects and operations,

• through Pictorial representations of these objects and operations,

• to the underlying Abstract mathematics.

This progression principle (from Concrete examples, via carefully chosen Pictorial stepping stones, to the Abstract mathematics) is often referred to by the simple acronym C-P-A.

Each year’s textbook consists of around 14 chapters, and each chapter has a single theme, which is addressed through a sequence of between 5 and 30 lessons.

Each lesson has the same basic structure. lt starts by introducing the new topic, or method through a picture, and an Anchor Task: these embody the relevant ideas in concrete form (often incorporating the theme of the chapter in several different ways). The concrete realisation of the underlying ideas captured in the picture and in the associated Anchor Task is explored throuqh a structured class discussion, before moving on to a more formal treatment of the material (in the Let’s Learn and Guided Practice sections of each lesson).

The later phases of each C-P-A cycle emerge over a sequence of lessons. Each sequence begins with a concrete image. The ideas extracted from this concrete image are then embodied in one or more carefully chosen pictorial representations. We give an example (together with references, in brackets, to the Maths — No Problem! textbooks for those who may be familiar with them):

– When extending the idea of place value, we might begin with a picture of 1000 seedlings, arranged in ten “10 by 10″ square trays (e.g. page 1 of Lesson 1 in Textbook 4A)

– This slightly messy concrete image can then be modified by stacking these ten “10 by 10” square trays on top of each other, so that the same 1000 seedlings form a more compact “10 by 10 by 10″ cube (e.g. on page 5, in Lesson 2).

– This concrete image of a “cube of seedlings” can then be further simplified, so that each “10 by 10 by 10″ cube is replaced by a simple pictorial substitute: a single disc labelled “1000” (e.g. on page 6, where the relevant disc is coloured blue). This single labelled disc can then become our standard pictorial representation of “1000″, and can be manipulated as if it were a mathematical coin of that value.

– The simplified pictorial representation of “1000” may then be set in a larger context by introducing discs for “100”, for “10”, and for “1” (as on page 9, Lesson 3, where we meet the discs labelled “100”, “10” and “1”, each with its assigned colour which then remains fixed).

ln each case, the pictorial representations serve as stepping stones between the original concrete image and the (potentially alien) simplicity of the ultimate abstract formulation. For example:

– Lesson 3 in Textbook 4A introduces a collection of 2 blue discs, each labelled “1000”; 1 purple disc labelled “100”; 2 yellow discs, each labelled “10”; and 6 pink discs, each labelled “1”. This collection, and the associated class discussion, is then used as a stepping stone between the pictorial version — of eleven differently coloured and labelled discs — and the abstract representation of this collection of discs as the 4-digit number “2126”

– Later, the process of “adding” two numbers can be interpreted via the concrete–pictorial construction of combining the two collections of relevant discs, and “re-grouping” (e.g on page 49 of Textbook 4A the re-grouping requires us to replace “eleven 1s” by “one 10 and one 1″).

– This concrete–pictorial construction is then reinterpreted in terms of the abstract standard “column addition” algorithm.

The abstract formulation may at first be seen as an efficient summary of the pictorial procedure. Eventually it becomes a direct way of manipulating the given information abstractly — learning to work mentally (in the head, or on paper), without the support of concrete materials or pictorial images.

The support that is needed for everyone to achieve the important final step (to the abstract) varies from topic to topic, and from pupil to pupil. It also becomes more challenging as pupils progress. Whether a pupil has moved beyond the concrete phase, and can handle an idea (with all its associated “variations”) in their head, often becomes clear only later — when the idea is needed to solve unfamiliar problems, or is used as the basis for more advanced methods. Until then, we need to repeatedly check that all pupils can relate the formal language and procedures of elementary mathematics to their standard concrete–pictorial realisations.

Those who recognise the “abstract” character of elementary mathematics may be tempted to rush the preliminary phases (or even to skip them altogether) encouraging some to move swiftly on to formal processes, while deciding that for others mastery of formal processes is less important. The Maths — No Problem! materials take the opposite view on both counts. They insist (i) that everyone should be supported in mastering the abstract formulation of the basic ideas and processes of elementary mathematics; and (ii) that all learners benefit from anchoring these abstract ideas and procedures in a rich repertoire of concrete–pictorial representations.

The initial concrete phase shows mathematics rooted in the world around us. But the power of its ideas and methods — even at primary level — only becomes apparent when pupils re-interpret this external version in a new form, by constructing an internal, mental universe. All subsequent progression depends on embracing this transition, and on learning to operate within this mental universe. ln short, despite its concrete roots, elementary mathematics is inescapably abstract. The pictorial stepping stones, which are used systematically in the Maths — No Problem! textbook series, have been devised to help all pupils make the transition from C, through P, to A.

Once an abstract idea, or process (such as place value), has been established as the endpoint of a “C-P-A” sequence, and becomes sufficiently familiar, it can be used as the “concrete” starting point for other C-P-A cycles” For example:

– Once the idea of multiplication as “repeated addition” has been grasped: one can use this idea as the basis for counting the unit squares in an m by n rectangle row-by-row, and so extract the familiar formula for the area of a rectangle.

– One can then count unit cubes in a cuboid layer-by-layer, and so arrive at the familiar formula for the volume of a cuboid.

This transition (from the external “concrete” world, to an internal, abstract “mental universe”) is never done-and-dusted. Even when learners seem confident with abstract methods, we still need to check that the underlying mental abstractions are understood, and to make sure that pupils retain the crucial links between these abstractions and their underlying meaning. ln particular, although this collection of Mathsteasers is aimed at those who appear to have made a successful transition (from C, through P, to A), one can be fairly sure that the challenge of facing unfamiliar problems will reveal instances where these apparently “successful” pupils will need to pause, and think again about basic techniques before moving on.

Basic techniques are the hardware — the visible building blocks — of elementary mathematics. They tend to be what gets listed in curricula, and what is tested in examinations. So it is important for pupils to learn to handle them with confidence. But there are two reasons why they need more than this.

• First, what gets tested (and hence what teachers may be tempted to focus on) tend to be the easier, more predictable versions of these techniques; in contrast, what is generally needed for subsequent progression are certain harder, more elusive variations on these techniques.

• Second, if we restrict attention to mastering separate techniques (perhaps because these are what is specified in a syllabus), we risk neglecting the underlying logic that allows learners to organise these techniques into a coherent whole. Elementary mathematics then comes across as a collection of unrelated rules and procedures. In contrast, emphasising connections allows the subject to become more coherent as one progresses with the connections acting like “mortar”, that binds the separate building blocks together in the mind, transforming a mere collection of bricks into a robust, coherent structure.

It is all-too-easy to neglect harder variations, and to gloss over important connections and thereby to encourage a fragmented view of mathematics. Number words are part of everyday language; but the connections between language, digits, numerals, and place value have to be explicitly taught. “Combining”, or “putting together”, or “taking away”, belong to everyday experience; but sustained teaching is required to link these ideas to addition and subtraction within the mental universe of elementary mathematics. When “successful” pupils face the challenge of more searching problems (as with these Mathsteasers), we should not be surprised to find that some of these underlying connections need to be revisited, or strengthened.

 

Consequences for “differentiation”

The background above may help to explain the approach we take here to differentiation (i.e. provision for pupils with different levels of apparent attainment) Since the 1970s, there has been a tendency to think of elementary mathematics as a uniform “ladder”, up which all pupils have to climb, if at different speeds. Those who appeared to be making good progress were therefore routinely encouraged to rush ahead at their own pace.

However, mathematics is more like a web than a ladder, with key ideas and methods linked together in complex ways. The power of this “web” depends both on mastery of each key idea, and on the number, and the robustness, of the connections between these key ideas and methods. Medium term progression in mathematics is not a matter of mastering each idea superficially, and then galloping ahead; rather it requires:

– a firm grasp of each basic theme (including the important variations on that theme);

together with

– a steadily growing fluency in constructing and exploiting the links between apparently different ideas.

Hence the goal for all pupils is to achieve a clear understanding of each topic, together with a grasp of the links and connections between them. And the greater a pupil’s potential, the more important it is to establish both the technical foundations and the interconnections as they progress.

Rather than appeal to less familiar examples from these Mathsteasers, we choose our examples from familiar core material noting only that the underlying principle remains the same. Textbook 4A is an extended illustration of precisely this theme: namely that pupils need to master basic techniques, and develop fluency in connecting different ideas, if they are to successfully handle the mathematical ideas and methods that lie ahead. For example, the dramatic advantages of place value are only available for use

– if pupils learn to move flexibly between a number as given and the same number with parts renamed (e.g. Maths — No Problem! Textbook 4A, page 53, 56, 59, etc.); and

– if simple digital calculations such as 2 x 3 = 6 can also be understood as telling us how to calculate 2 x 30, 20 x 3, etc. (e.g. Textbook 4A, page 161).

Grasping such connections constitutes a significant challenge for all pupils! And those who rise to the challenge will generally be those who slowed down sufficiently to master the relevant variations in earlier material, rather than those who achieved superficial early success before rushing on ahead.

These observations suggest a very different model of provision for those who find mathematics at Key Stages 1 and 2 (and early Key Stage 3) relatively straightforward. During this period (from age 5 to age 13 or so) the crucial initial goal is to ensure that key ideas and methods are mastered sufficiently flexibly to support the plethora of techniques and interconnections that multiply later in Key Stage 2 and in Key Stage 3. This is an area where we in the UK have tended to cut corners relative to mathematics teaching in Eastern Europe, or in the Far East. ln particular, we have tended to focus our teaching on the simplest cases required by an “attainment target” or “level”, while ignoring the range of variations, which are generally what subsequent progress depends on (but which are less likely to be tested directly at age 11).

The Maths — No Problem! series embraces this spirit of “more haste, less speed”, seeking to keep each class of pupils together for as long as possible, rather than exacerbating divergence by encouraging some to rush ahead, only to discover later the fragility of their understanding of key ideas. The texts support all pupils by treating related topics in a uniform way; with each topic being pursued in depth as part of an extended sequence of lessons. As a result, those who find a topic difficult meet the underlying basic ideas over and over again in slightly different contexts — giving them and their teachers plenty of opportunity to ensure that they progress well beyond the concrete phase, and that they are then relatively well placed to use the ideas later. At the same time, those who appear to handle the ideas with little difficulty can grapple with those variations which will determine whether these ideas can be effectively used in subsequent years. Some “extension” work of this kind can be done within the standard class lesson. However, where something different (and more demanding) is needed, this collection of Mathsteasers offers a sequence of harder problems, aligned to the core texts.

When devising extension work for use in class, teachers need to exploit and strengthen the connections which pervade the curriculum, and which proliferate as we move through Key Stage 2. We give a few examples from the Maths — No Problem! textbooks. ln Textbook 4A such connections may seem fairly evident; but they are mostly unstated, and need to be made explicit. For example:

• The differently coloured discs representing “1”, “10”, “100”, and “1000” can be manipulated as if they were a form of “mathematical coinage”; but they also correspond to the different values of the abstract “numeral places”.

• The number patterns that arise from counting in 6s, in 7s, and in 9s (e.g. on page 30 of Textbook 4A) prepare for the x6, x7, and x9 multiplication tables (on pages 96, 101, and 104–109 of Textbook 4A) and to the inverse process of division: by 6, and by 9 (on pages 121, 123, and 125).

• Mental strategies for addition (e.g. on pages 62–63 of Textbook 4A) need to be linked to earlier work on place value, which emphasised the advantages of looking out for “opportunities to make 10″ (e.g. as 1 + 9), to “make 100” (e g. as 98 + 2, or as 73 + 27), or to “make 1000”.

• The inverse process to “renaming” (e.g. decomposing a simple “10” into “1 + 9″, or “2 + 8“) becomes important when we come to subtraction–with–renaming (e.9. on pages 72–73 of Textbook 4A, where “80” has to be re-interpreted as “70 + (ten 1s)”).

ln other words, ideas and methods are never isolated: they prepare the way for subsequent developments; and they depend on previous material having been mastered flexibly rather than having been completed quickly and superficially before rushing on.

The English national curriculum states explicitly

• that pupils [in a particular cohort] should “move through the curriculum at broadly the same pace”,

• that “decisions about when to progress [to subsequent material] should always be based upon the security of pupils’ understanding and their readiness to progress to the next stage”, and

• that schools are free to adjust the sequencing of the material within each Key Stage (hence allowing separately listed items to be re-grouped to form extended teaching sequences).

Schools need to remember these injunctions when trying to balance conflicting requirements. The Maths — No Problem! texts show in detail how these curriculum requirements can be satisfied.

We end these comments on “differentiation” by explaining briefly not just how, but why the approach adopted here differs from what has been advocated in England over the last 40 years or more (for example, in the Cockcroft Report, or in the Gifted and Talented strand of the Excellence in Cities programme from 1998–2006).

The Cockcroft Report (paragraphs 342, 436) treated the “seven year gap” at age 11 as God-given, rather than as an artefact of instruction, which might be partly within our control. And the official strategy from 1997 (until around 2012) assumed that the only way to nurture able pupils was to oblige schools to “accelerate” the early progress of some pupils, while accepting lower expectations for others. These policies were based on two assumptions:

• that ability is “God-given”: hence those possessing (or lacking) it should be “identified”, and should then be treated differently from other pupils; and

• that the curriculum is a single ladder (of simple rungs, or levels) up which everyone has to climb.

The approach allowed some pupils to progress more quickly in the short term. But it proved to be ineffective — and counterproductive — for almost all pupils in the longer term Strugglers were no longer expected to progress to those formal methods on which subsequent progress (in upper primary and lower secondary school) depends. And more able pupils, who moved on too quickly, often failed to master basic ideas and procedures sufficiently robustly, or in sufficient depth: many made strong early “progress” only to discover later that the shallowness of their understanding hindered subsequent progress. In other words, accelerated short-term progress regularly proved fragile, and ultimately illusory. For example:

• An official Ofsted report noted that more than 37 000 pupils whose measured performance in mathematics in Year 6 was “excellent” (Level 5+) subsequently performed rather poorly at age 16.

• In the large-scale international comparison TIMSS, the apparent performance of English pupils at Key Stage I and 2 improved impressively after 2000 (perhaps as a result of the efforts linked to the Numeracy Strategy). There was a significant jump in the average Year 5 TIMSS scores for pupils in England in 2003, with further modest improvements in 2007, and in 2011. But this “sustained improvement” in Year 5 led to no significant improvement in the Year 9 scores in the TIMSS assessments four years later in 2007 and 2011 — even though these Year 9 cohorts were the same as the Year 5 cohorts in 2003 and 2007. So, whatever had “improved” in Year 5 in 2003 and 2007 had no discernible consequences four years later.

• Despite the extra attention given to “gifted and talented” pupils for the last twenty years, the percentage of English Year 9 and Year 10 students who managed to solve “moderately hard” problems (as measured in TIMSS and is PISA) has declined over the same period to an uncomfortably low level.

• ln other words, the policy of acceleration at Key Stages 1 and 2 somehow failed to prepare our most able pupils for the mathematical challenges which should be a natural part of Key Stage 3 for such students.

As long as each pupil’s progress continues to be monitored via an assessment of “levels”, or via performance on the next official assessment, schools may still be tempted to limit the expected degree of mastery to what is needed in the short-term — even though we know that this is often too superficial to support pupils’ subsequent (medium-term) progress. ln contrast to the view of mathematics as a succession of simple “levels”, or “rungs” on a ladder, the recurring “C-P-A” leitmotif in the Maths — No Problem! textbooks develops each topic as multi-layered (like an onion), and as multi-linked (as in a web) Each basic idea is developed from previously learned material, and brings with it an accompanying web of related methods — all of which need to be integrated into a single mental schema in order to prepare pupils for subsequent progression.

ln short, effective education systems

• recognise that later progression depends on all pupils mastering current material in greater depth,

• and use whole-class teaching both to achieve this objective for all pupils, and to postpone for as long as possible the point at which classes begin to diverge.

So standard classroom provision should

• develop ways of ensuring that pupils who grasp basic ideas quickly can routinely master the associated techniques in sufficient depth to support subsequent progression, and

• build up a repertoire of harder variations, which explore the multi-faceted aspects of each new idea or method.

These Mathsteasers may be seen as a structured extension of such provision for those who need material beyond what is part of the standard, shared classroom.

 

ln conclusion

We have seen that a pupil’s ability to use, or to extend, a given idea, depends on them having mastered the obvious prerequisite methods, on them having moved beyond the concrete-pictorial to the abstract mental formalism, and on them having grasped the necessary variations and connections. At one extreme, simple counting starts with objects, or fingers, but has to progress to a mental operation, combining the digits 0-9 and place value to enable pupils to count “in the head” — forwards and backwards, in ones. or in twos, or in tens. At the other extreme, the work on multiplying and dividing decimals in Chapter 4 of the Year 6 Maths — No Problem! texts depends on a combination of robust mastery of the corresponding operations and techniques for whole numbers, and a willingness to think flexibly about place value; this in turn depends on an effective understanding of early place value, starting with a clear initial grasp of all aspects of the single digit numbers 0–9, followed by extensive engagement first with numbers 0–20, then to 100, then to 1000 and beyond. For a robust understanding of decimals, the principles of place value need to be fully mastered first in the context of two-digit and three-digit integers, and exercised in tackling all sorts of variations: counting forwards and backwards, in ones, in twos, in tens, in hundreds, in twenties, etc.; crossing boundaries (19 to 20; 99 to 100; 119 to 120; 199 to 200; etc, both forwards and backwards). All this work must later be repeated with simple decimals — extending the ideas and the notation to the right of the decimal point (from tens and hundreds, to tenths and hundredths).

ln general, once the ground has been prepared, each basic theme in the Maths — No Problem! texts receives more extended engagement, and greater variety than we in the UK have traditionally provided. As long as this extended engagement is handled sensitively and consistently, all pupils benefit: those who find the basic ideas elusive benefit from meeting the same ideas over and over again in different contexts; and those who find things easier benefit from grasping key ideas more robustly and in greater depth. There is no sense in which more able pupils are being “held back”. ln mathematics as in building construction:

if you ultimately want to build higher,
you’d better first dig deeper.

Hence teachers should focus their efforts for as long as possible (and certainly for the first three years of primary school) on enriching standard material by devising natural variations on the theme of each Chapter — both to support those who find the material hard, and to challenge those who think they find it easy. These booklets of Mathsteasers are offered for the benefit of those who in later Key Stage 2 need a supplement of additional material to extend and deepen their grasp of elementary mathematics with a view to optimising progression at secondary level.

We end by encouraging teachers at all levels to reflect upon ten general themes, to discuss (and refine) them with their colleagues, and to incorporate the resulting ideas into their teaching.

1. Mathematics can be daunting; but everyone can make progress with perseverance. It is important to pace the introduction of new material so as to establish this basic message.

2. The mental universe of mathematics emerges from practical engagement with measures, with shapes, and with quantities. Hence one should look for and exploit opportunities for pupils to talk, to explain, to calculate, to draw, to measure, and to make things for themselves.

3. It is important to establish — and to continually check through oral and mental work — pupils’ grasp of the inner structure of elementary mathematics.

4. One should actively develop pupils’ powers of remembering, and gradually extend the range and scope of important facts and methods that pupils understand and know by heart.

5. What is learned builds on what is already known. ln particular, early work lays the foundations for mathematics in Key Stage 2 and beyond. So colleagues teaching upper primary years should be encouraged to suggest what might have benefitted from more, or perhaps slightly different, attention in earlier years.

6. Always look for alternatives to acceleration (for example, by regularly using resources such as this collection). Aim for all pupils to achieve a robust fluency and sufficient depth to prepare for later progression. The easier a pupil finds a topic, or a group of topics, the more important it is for them to master the topic in serious depth.

7. Ensure that initial class discussion addresses each topic in sufficient depth. Follow this up with carefully designed sets of graded exercises, that range from the very simple to the general, and that routinely explore the more demanding “indirect” variations which are needed for subsequent applications.

8. a) Recognise the link between

– each direct operation or process (such as counting forwards, or addition, or multiplication, or multiplying numerator and denominator to obtain an equivalent fraction), and

– the corresponding inverse operation or process (such as counting backwards, or subtraction, or division, or cancelling common factors to simplify such fractions as or ).

   b) Regularly expect pupils to combine simple steps to solve multi-step problems. Fluency in implementing simple procedures and direct calculations is essential; but its sole purpose is to provide a foundation for solving multi-step problems and harder inverse problems. ln particular, resist the temptation to “help” by breaking down multi-step problems, and harder inverse problems, into more manageable (direct) steps: this art has to be learned the hard way.

9. Routinely establish the link between language and mathematics — including simple word problems alongside technical exercises — so that pupils learn to identify, and to extract relevant information from short two or three sentence problems given in words.

10. Regularly include short, non-routine problems (including two-step, and multi-step problems), that cultivate pupils’ willingness to face the unexpected, and to think how to link known techniques into effective solution chains.