A Digression on Breadth, Depth, Pace and Mastery

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Tricoloring (1)

For a more recent post on these issues, go here

This post explores the emerging picture of mastery-based differentiation for high attainers and compares it with a model we used in the National G&T Programme, back in the day.

It is a rare venture into pedagogical territory by a non-practitioner, so may not bear close scrutiny from the practitioner’s perspective. But it seeks to pose intelligent questions from a theoretical position and so promote further debate.

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Breadth, depth and pace

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Quality standards

In the original National Quality Standards in Gifted and Talented Education (2005) one aspect of exemplary ‘Effective Provision in the Classroom’ was:

‘Teaching and learning are suitably challenging and varied, incorporating the breadth, depth and pace required to progress high achievement. Pupils routinely work independently and self-reliantly.’

In the 2010 version it was still in place:

‘Lessons consistently challenge and inspire pupils, incorporating the breadth, depth and pace required to support exceptional rates of progress. Pupils routinely work creatively, independently and self-reliantly.’

These broad standards were further developed in the associated Classroom Quality Standards (2007) which offered a more sophisticated model of effective practice.

The original quality standards were developed by small expert working groups, reporting to wider advisory groups and were carefully trialled in primary and secondary classrooms.

They were designed not to be prescriptive but, rather, to provide a flexible framework within which schools could develop and refine their own preferred practice.

Defining the terms

What did we mean by breadth, depth and pace?

  • Breadth (sometimes called enrichment) gives learners access to additional material beyond the standard programme of study. They might explore additional dimensions of the same topic, or an entirely new topic. They might need to make cross-curricular connections, and/or to apply their knowledge and skills in an unfamiliar context.
  • Depth (sometimes called extension) involves delving further into the same topic, or considering it from a different perspective. It might foreground problem solving. Learners might need to acquire new knowledge and skills and may anticipate material that typically occurs later in the programme of study.
  • Pace (sometimes called acceleration) takes two different forms. It may be acceleration of the learner, for example advancing an individual to a higher year group in a subject where they are particularly strong. More often, it is acceleration of the learning, enabling learners to move through the programme of study at a relatively faster pace than some or all of their peers. Acceleration of learning can take place at a ‘micro’ level in differentiated lesson planning, or in a ‘macro’ sense, typically through setting. Both versions of acceleration will cause the learner to complete the programme of study sooner and they may be entered early for an associated test or examination.

It should be readily apparent that these concepts are not distinct but overlapping.  There might be an element of faster pace in extension, or increased depth in acceleration for example. A single learning opportunity may include two, or possibly all three. It is not always straightforward to disentangle them completely.

Applying these terms

From the learner’s perspective, one of these three elements can be dominant, with the preferred strategy determined by that learner’s attainment, progress and wider needs.

  • Enrichment might be dominant if the learner is an all-rounder, relatively strong in this subject but with equal or even greater strength elsewhere.
  • Extension might be dominant if the learner shows particular aptitude or interest in specific aspects of the programme of study.
  • Acceleration might be dominant if the learner is exceptionally strong in this subject, or has independently acquired and introduced knowledge or skills that are not normally encountered until later in this or a subsequent key stage.

Equally though, the richest learning experience is likely to involve a blend of all three elements in different combinations: restricting advanced learners to one or two of them might not always be in their best interests. Moreover, some high attainers will thrive with a comparatively ‘balanced scorecard’

The intensity or degree of enrichment, extension or acceleration will also vary according to the learners’ needs. Even in a top set decisions about how broadly to explore, how deeply to probe or how far and how fast to press forward must reflect their starting point and the progress achieved to date.

Acceleration of the learner may be appropriate if he or she is exceptionally advanced.  Social and emotional maturity will need to be taken into account, but all learners are different – this should not be used as a blanket excuse for failing to apply the approach.

There must be evidence that the learner is in full command of the programme of study to date and that restricting his pace is having a detrimental effect. A pedagogical preference for moving along the class at the same pace should never over-ride the learner’s needs.

Both variants of acceleration demand careful long-term planning, so the learner can continue on a fast track where appropriate, or step off without loss of esteem. It will be frustrating for a high attainer expected to ‘mark time’ when continuity is lost. This may be particularly problematic on transfer and transition between settings.

Careful monitoring is also required, to ensure that the learner continues to benefit, is comfortable and remains on target to achieve the highest grades. No good purpose is served by ‘hothousing’.

Mastery and depth

The Expert Panel

The recent evolution of a mastery approach can be tracked back to the Report of the Expert Panel for the National Curriculum Review (December 2011).

‘Amongst the international systems which we have examined, there are several that appear to focus on fewer things in greater depth in primary education, and pay particular attention to all pupils having an adequate understanding of these key elements prior to moving to the next body of content – they are ‘ready to progress’…

… it is important to understand that this model applies principally to primary education. Many of the systems in which this model is used progressively change in secondary education to more selective and differentiated routes. Spread of attainment then appears to increase in many of these systems, but still with higher overall standards than we currently achieve in England…

There are issues regarding ‘stretch and challenge’ for those pupils who, for a particular body of content, grasp material more swiftly than others. There are different responses to this in different national settings, but frequently there is a focus on additional activities that allow greater application and practice, additional topic study within the same area of content, and engagement in demonstration and discussion with others

These views cohere with our notion of a revised model that focuses on inclusion, mastery and progress. However, more work needs to be done around these issues, both with respect to children with learning difficulties and those regarded as high attainers.’

For reasons best known to itself, the Panel never undertook that further work in relation to high attainers, or at least it was never published. This has created a gap in the essential groundwork necessary for the adoption of a mastery-driven approach.

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National curriculum

Aspects of this thinking became embodied in the national curriculum, but there are some important checks and balances.

The inclusion statement requires differentiation for high attainers:

‘Teachers should set high expectations for every pupil. They should plan stretching work for pupils whose attainment is significantly above the expected standard.’

The primary programmes of study for all the core subjects remind everyone that:

Within each key stage, schools therefore have the flexibility to introduce content earlier or later than set out in the programme of study. In addition, schools can introduce key stage content during an earlier key stage, if appropriate.’

But, in mathematics, both the primary and secondary PoS say:

‘The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.’

These three statements are carefully worded and, in circumstances where all apply, they need to be properly reconciled.

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NCETM champions the maths mastery movement

The National Centre for Excellence in the Teaching of Mathematics (NCETM), a Government-funded entity responsible for raising levels of achievement in maths, has emerged as a cheerleader for and champion of a maths mastery approach.

It has published a paper ‘Mastery approaches to mathematics and the new national curriculum’ (October 2014).

Its Director, Charlie Stripp, has also written two blog posts on the topic:

The October 2014 paper argues (my emphasis):

‘Though there are many differences between the education systems of England and those of east and south-east Asia, we can learn from the ‘mastery’ approach to teaching commonly followed in these countries. Certain principles and features characterise this approach…

… The large majority of pupils progress through the curriculum content at the same pace. Differentiation is achieved by emphasising deep knowledge and through individual support and intervention.’

It continues:

‘Taking a mastery approach, differentiation occurs in the support and intervention provided to different pupils, not in the topics taught, particularly at earlier stages. There is no differentiation in content taught, but the questioning and scaffolding individual pupils receive in class as they work through problems will differ, with higher attainers challenged through more demanding problems which deepen their knowledge of the same content.’

In his October 2014 post, Stripp opines:

‘Put crudely, standard approaches to differentiation commonly used in our primary school maths lessons involve some children being identified as ‘mathematically weak’ and being taught a reduced curriculum with ‘easier’ work to do, whilst others are identified as ‘mathematically able’ and given extension tasks….

…For the children identified as ‘mathematically able’:

  1. Extension work, unless very skilfully managed, can encourage the idea that success in maths is like a race, with a constant need to rush ahead, or it can involve unfocused investigative work that contributes little to pupils’ understanding. This means extension work can often result in superficial learning. Secure progress in learning maths is based on developing procedural fluency and a deep understanding of concepts in parallel, enabling connections to be made between mathematical ideas. Without deep learning that develops both of these aspects, progress cannot be sustained.
  2. Being identified as ‘able’ can limit pupils’ future progress by making them unwilling to tackle maths they find demanding because they don’t want to challenge their perception of themselves as being ‘clever’ and therefore finding maths easy….

…I do think much of what I’m saying here also applies at secondary level.

Countries at the top of the table for attainment in mathematics education employ a mastery approach to teaching mathematics. Teachers in these countries do not differentiate their maths teaching by restricting the mathematics that ‘weaker’ children experience, whilst encouraging ‘able’ children to ‘get ahead’ through extension tasks… Instead, countries employing a mastery approach expose almost all of the children to the same curriculum content at the same pace…’

The April 2015 post continues in a similar vein, commenting directly on the references in the PoS quoted above (my emphases):

‘The sentence: ‘Pupils who grasp concepts rapidly should be challenged through rich and sophisticated problems before any acceleration through new content’, directly discourages acceleration through content, instead requiring challenge through ‘rich and sophisticated (which I interpret as mathematically deeper) problems’. Engaging with ‘rich and sophisticated problems’ involves reasoning mathematically and applying maths to solve problems, addressing all three curriculum aims. All pupils should encounter such problems; different pupils engage with problems at different depths, but all pupils benefit

…Meeting the needs of all pupils without differentiation of lesson content requires ensuring that both (i) when a pupil is slow to grasp an aspect of the curriculum, he or she is supported to master it and (ii) all pupils should be challenged to understand more deeply…

The success of teaching for mastery in the Far East (and in the schools employing such teaching here in England) suggests that all pupils benefit more from deeper understanding than from acceleration to new material. Deeper understanding can be achieved for all pupils by questioning that asks them to articulate HOW and WHY different mathematical techniques work, and to make deep mathematical connections. These questions can be accessed by pupils at different depths and we have seen the Shanghai teachers, and many English primary teachers who are adopting a teaching for mastery approach, use them very skilfully to really challenge even the highest attaining pupils.’

The NCETM is producing guidance on assessment without levels, showing how to establish when a learner

‘…has ‘mastered’ the curriculum content (meaning he or she is meeting national expectations and so ready to progress) and when a pupil is ‘working deeper’ (meaning he or she is exceeding national expectations in terms of depth of understanding).’

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Commentary

NCETM wants to establish a distinction between depth via problem-solving (good) and depth via extension tasks (bad)

There is some unhelpful terminological confusion in the assumption that extension tasks necessarily require learners to anticipate material not yet covered by the majority of the class.

Leaving that aside, notice how the relatively balanced wording in the programme of study is gradually adjusted until the balance has disappeared.

The PoS says ‘the majority of pupils will move through…at broadly the same pace’ and that they ‘should be challenged through being offered rich and sophisticated problems before any acceleration through new content).

This is first translated into

‘…the large majority of pupils progress through the curriculum content at the same pace’ (NCETM paper) then it becomes

‘…expose almost all of the children to the same curriculum content at the same pace’ (Stripp’s initial post) and finally emerges as

‘Meeting the needs of all pupils without differentiation of lesson content’ and

‘…all pupils benefit more from deeper understanding than from acceleration to new material.’ (Stripp’s second post).

Any non-mathematician will tell you that the difference between the majority (over 50%) and all (100%) may be close to 50%.

Such a minority could very comfortably include all children achieving L3 equivalent at KS1 or L5 equivalent at KS2, or all those deemed high attainers in the Primary and Secondary Performance Tables.

The NCETM pretends that this minority does not exist.

It does not consider the scope for acceleration towards new content subsequent to the delivery of ‘rich and sophisticated problems’.

Instead it argues that the statement in the PoS ‘directly discourages acceleration through content’ when it does no such thing.

This is propaganda, but why is NCETM advancing it?

One possibility, not fully developed in these commentaries, is the notion that teachers find it easier to work in this way. In order to be successful ‘extension work’ demands exceptionally skilful management.

On the other hand, Stripp celebrates the fact that Shanghai teachers:

…were very skilled at questioning and challenging children to engage more deeply with maths within the context of whole class teaching.’

It is a moot point whether such questioning, combined with the capacity to develop ‘rich and sophisticated problems’, is any more straightforward for teachers to master than the capacity to devise suitable extension tasks, especially when one approach is relatively more familiar than the other.

Meanwhile, every effort is made to associate maths mastery with other predilections and prejudices entertained by educational professionals:

  • It will have a positive impact on teacher workload, but no evidence – real or imagined – is cited to support this belief.
  • The belief that all children can be successful at maths (though with no acknowledgement that some will always be comparatively more successful than others) and an associated commitment to ‘mindset’, encouraging learners to associate success with effort and hard work rather than underlying aptitude.
  • The longstanding opposition of many in the maths education community to any form of acceleration, fuelled by alarming histories of failed prodigies at one extreme and poorly targeted early entry policies at the other. (I well remember discussing this with them as far back as the nineties.)
  • The still contested benefits of life without levels.

On this latter point, the guidance NCETM is developing appears to assume that ‘exceeding national expectations’ in maths must necessarily involve ‘working deeper’.

I have repeatedly argued that, for high attainers, such measures should acknowledge the potential contributions of breadth, depth and pace.

Indeed, following a meeting and email exchanges last December, NAHT said it wanted to employ me to help develop such guidance, as part of its bigger assessment package.

(Then nothing more – no explanation, no apology, zilch. Shame on you, Mr Hobby. That’s no way to run an organisation.)

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Conclusion

Compared with the richness of the tripartite G&T model, the emphasis placed exclusively on depth in the NCETM mastery narrative seems relatively one-dimensional and impoverished.

There is no great evidence in this NCETM material of a willingness to develop an alternative understanding of ‘stretch and challenge’ for high attainers.  Vague terms like  ‘intelligent practice’, ‘deep thinking’ and ‘deep learning’ are bandied about like magical incantations, but what do they really mean?

NCETM needs to revisit the relevant statement in the programme of study and strip away (pun intended) the ‘Chinese whispers’ (pun once more intended) in which they have cocooned it.

Teachers following the maths mastery bandwagon need meaningful free-to-access guidance that helps them construct suitably demanding and sophisticated problems and to deploy advanced questioning techniques that get the best out of their high attainers.

I do not dismiss the possibility that high attainers can thrive under a mastery model that foregrounds depth over breadth and pace, but it is a mistake to neglect breadth and pace entirely.

Shanghai might be an exception, but most of the other East Asian cradles of mastery also run parallel gifted education programmes in which accelerated maths is typically predominant. I’ve reviewed several on this Blog.

For a more recent treatment of these issues see my September 2015 post here.

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GP

April 2015

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12 thoughts on “A Digression on Breadth, Depth, Pace and Mastery

  1. Very well stated, GP. A related question is how to measure breadth and depth to understand the relative success of different schools in catering to higher attaining pupils (pace is easier). And if depth is to be emphasised, the primary assessment commission needs to consider urgently how this depth should be reflected in KS2 assessments.

  2. Thanks George. I’m rather confused now about the locus of the Commission in respect of statutory teacher assessment. I suppose the new principles and guidance on assessment without levels might explain – where there is differentiation by task – how different combinations and intensities of breadth, depth and pace ought to be reflected. But given this ‘statement of intended outputs‘ I’m not holding my breath.

    GP

  3. Thanks for that link, I hadn’t seen the “statement of intended outputs” before. It is rather bland. In general, “how best to do X without Y” is probably not the best starting point for a principled discussion about “how best to do X”! I hope the commission will call for evidence from the appropriate people, including those who responded to the consultation.

  4. The mere presence of the “It’s for their own good” bright child mindset meme is a compelling reason to expect damage.

    My indirect experience via whizzy Y7 Sprogette is that maths teaching does suffer from topic fragmentation so a more sustained focus on one topic before moving to the next gets my vote, but it wouldn’t have worked for her at the primary mixed-ability class pace. I’m not convinced there’s that much scope for depth in the primary curriculum but depths were plumbed where possible and weren’t enough, so yes removing breadth and pace from the toolbox is regressive.

  5. You might be interested in how to assess whether or not a pupil is truly advanced. Often pupils show signs of being advanced, but sometimes only possess a superficial understanding. Acceleration in this instance often leads to poor performance later. Acceleration should only really happen when all enrichment opportunities have been exhausted, this is a rare occurrence. Here are some tips form Dr. Yeap which might be helpful when assessing young children, in particular the last video. (embarrassingly just noticed a spelling mistake on the video) on this page. http://www.mathsnoproblem.co.uk/blog/effectivelesson/

  6. Thanks Andy. I understand and appreciate the guidance on ensuring that a child has ‘a relational understanding’. I’m less comfortable with the idea that acceleration should only happen when all enrichment opportunities have been exhausted. As I indicated in my post, it is sometimes difficult to disentangle enrichment (as I have defined it) from extension and/or acceleration – and it should be feasible to adjust the approach to reflect learners’ different needs.

    If we’re discussing a Singapore approach, it would be interesting to know more about how – if at all – these methods are adapted for children within the GEP? Can you refer me to any in-depth material on that topic?

    Thanks

    GP

  7. You have made some great points in your post, which is why I chose to respond, I usually don’t. The interesting variable in all of this comes down to the definition of words. Words like mastery, pace, reasoning etc.. can be interpreted in so many ways.

    Unfortunately, I don’t think the there is any in-depth material on the GEP in Singapore, but the essence of it is that children in the gifted education programme will study the same ‘content’ as their counterparts in the other classes, but using more challenging examples. This could mean more ‘reasoning’, and ‘problem solving’ than the other children. What wouldn’t (or shouldn’t happen) in the GEP classes is that when all the children in the other classes are doing numbers up to 10,000 the kids in the GEP are doing larger numbers let’s say numbers up to 1,000,000.

    The main reasons for this, as I understand it, is two fold. In my simple example, clearly there is no advantage whatsoever to go to larger numbers as there is no new learning in using larger numbers, it’s simply more tedious. The second reason is that the children in the GEP will write the same exam at the end of primary, (called the PSLE) The only way to do VERY well in the PSLE (which is the hope for the children in the GEP) is by being able to apply concepts in unfamiliar situations, ones that the children are likely to never have encountered before. Knowing more content is not going to help, and the PSLE is a very high stakes examination.

    I suppose the underlying philosophy is that they believe that teaching children who are very advanced more content is unlikely to produce the results that one would hope for but that allowing the time to accommodate for new generalisations that they make will build a stronger foundation for future learning.

    Being Singaporean, they have based most of their ideas on the work of the big 5 learning theories and have spent decades refining it. That is not to say that it is the only way to do it, but certainly

    Keep up the great blog!

  8. Thanks for the clarification – and for the nice things you say about my posts. My guess is that the secondary GEP is rather more accelerative in nature, but I won’t press you on the point.

    GP

  9. Secondary is certainly accelerated for advanced pupils. Depending on how the children do on their PSLE exam, they will go into different streams, that’s why the primary school leaving exam is such a high stakes exam. For example, the express stream covers the same content in 4 years that the normal academic stream covers in 5 years.

  10. Thank you, really interesting post. I’ve seen Debbie Morgan speak a number of times on Mastery, and the NCETM are really keen on it. I was also worried by the conflation from ‘the majority of children’, to ‘almost all of the children’.

    I’ve seen reference to the removal of levels being a way of getting back to the idea of greater depth rather than greater coverage. I’d like to see more able children extended by solving more challenging problems and making deeper links between mathematical ideas, as well as meeting new material. I think effective teachers having been doing this for some time.

    Where I worry about teaching methods, is when the pressure on teachers is to ‘move them on to level X’, in order to ‘demonstrate progress’ (usually pressure from leadership in schools). This usually means acceleration through the coverage of the curriculum in order to show evidence of progress and pass tests, rather than to develop their thinking skills (which is the real benefit of mathematics). If the move away from levels reverses this trend, I am all for it. But if it means the more able children (however you interpret that), get fed the same diet as everyone in the class in order to make teachers lives easier, than that is neither inclusion, nor adequately meeting their needs.

  11. Thanks for taking the trouble to comment. I agree with you that the ‘levels culture’ probably led to a fair amount of inappropriate acceleration/pace – and that their removal may foster renewed emphasis on depth/extension. My fear is that the new approach may turn out to be equally as narrow as the old one, that this national drive to import a simplified mastery-driven model may not be in the best interests of a fair proportion of high attainers, who might well benefit from a much richer and more sophisticated model of ‘stretch and challenge’. I don’t think it should require any greater pedagogical expertise or workload to implement such an approach than it requires to ‘master’ mastery. But I should repeat again that these observations are not based on substantial classroom experience.

    GP

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