Tony Gardiner: Teaching mathematics at secondary level


A. D. Gardiner, Teaching mathematics at secondary level. The De Morgan Gazette 6 no. 1 (2014), 1-215.

From the Introduction:

This extended essay started out as a modest attempt to offer some supporting structure for teachers struggling to implement a rather unhelpful National Curriculum.  It then grew into a Mathematical manifesto that offers a broad view of secondary mathematics, which should interest both seasoned practitioners and those at the start of their teaching careers.  This is not a DIY manual on How to teach.  Instead we use the official requirements of the new National Curriculum in England as an opportunity:

  • to clarify certain crucial features of elementary mathematics and how it is learned — features which all teachers need to consider before deciding `How to teach’.

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Tony Gardiner: National curriculum – Comments and suggested necessary changes

Published today:

A. D. Gardiner, National curriculum (England), September 2013; Attainment targets and programmes of study (key stages 1–3). Comments and suggested necessary changes. The De Morgan Gazette 4 , no. 3 (2013), 13-57

From the Introduction:

The Education Order 2013 was “made” on 5 September 2013. The relevant details were “laid before parliament” on 11 September 2013, and will come into effect on 1 September 2014. Some of the details for GCSE were published on 1 November 2013. Further elaboration of GCSE assessment structure, and curriculum guidance for Key Stage 4 (Years 10–11, ages 14–16) are awaited.

It is generally agreed that the curriculum review process adopted over the last 3–4 years has been seriously flawed. Those involved worked hard, often under very difficult conditions. But the overall approach (of relying on civil servants and drafters whose responsibilities and constraints remained inscrutable) has merely demonstrated that drafting and maintaining curricula is a specialist task, requiring dedicated professionals with specialist experience.

Whatever flaws there may have been in the process, we will all have to live with the new curriculum for some years. So it is important to have an open discussion of the likely difficulties. This article is an attempt to indicate aspects of the National curriculum in England: mathematics programmes of study that will need to be handled with considerable care, and revised in the light of experience.

After three years of widespread unease about the process of the curriculum review and its apparent direction, it is remarkable that there has been almost no media coverage, and no clear professional response to the final mathematics programmes of study for ages 5–14. There is therefore a real danger that insights that emerged along the way will simply be forgotten, and that the same mistakes may then be made next time. [...]

The details laid before parliament are `statutory’; but they incorporate basic flaws, and significant contradictions between the statutory list of content (which could all-too-easily be imposed uncritically) and the declared over-arching “aims” (which could get forgotten, or ignored). Given these flaws, the fate of the new programmes of study will depend on how sensitively their implementation is handled—whether slavishly, or intelligently. Teachers—and Ofsted, senior management, etc.—need to be alert to those aspects of the stated programmes of study that incorporate predictable pitfalls.

We summarise here what seem to be the two most important flaws.

Some material in Key Stage 1 and 2 is very poorly specified (especially from Year 4 onwards).

Some items are listed unnecessarily and unrealistically early, and so may be introduced at a stage:

  • where they are not yet needed,
  • where they will not be understood,
  • where they will be badly taught, and
  • where – if the relevant requirements were relaxed – the premature material could easily be delayed without causing any subsequent problems.

The listing of content for Key Stage 3 is in some ways reasonable, but too many things are left implicit. The programme of study is less structured than, and contains less detail than, that for Key Stages 1 and 2. Hence the details of the Key Stage 3 programme need interpretation. At present:

  • the words of each bullet point are rarely elaborated;
  • the connections between themes are mostly suppressed; and
  • there is no mention of essential preliminaries.

In addition

  • the Key Stage 3 programme has no accompanying `Notes and guidance’.

In summary, if the declared goals for Key Stage 4 are to be realised,

  • we need some way of clarifying the specified content and relaxing the unnecessary and potentially damaging pressures built in to the Key Stage 1–2 curriculum as it stands; and
  • the centrally prescribed curriculum for Key Stage 3 needs to be much more clearly structured to help schools understand what it is that is currently missing at this level—initially by providing suitable non-statutory `Notes and guidance’.

The De Morgan Journal: change of the name

By a decision of the LMS Education Committee, The De Morgan Journal changes its name to The De Morgan Gazette (ISSN 2053-1451).

The last paper of the old Journal and the first paper of the new Gazette are two parts of Tony Gardiner’s analysis of changes in Mathematics GCSE:

David Wells: Can mathematicians help?

D. G. Wells,  Can mathematicians help? The De Morgan Journal 2 no. 4 (2012) 1–4.


Professional mathematicians have not made the contributions to the teaching of mathematics in schools that might have been expected, in part, at least, because of their failure to appreciate the processes of conceptualisation and reconceptualisation that lie behind good maths teaching and lead young children from naïve concepts, objectionable perhaps to the professional, in time to more sophisticated and professionally acceptable interpretations. Illustrated

by the idea that ‘Multiplication is repeated addition.’

Acceleration or Enrichment

Acceleration or enrichment: Report of a seminar held at the Royal Society
on 22 May 2000, The De Morgan Journal, 2 no. 2 (2012), 97-125.

Full title of the paper:

Acceleration or Enrichment?
Serving the needs of the top 10% in school mathematics.
Exploring the relative strengths and weaknesses of “acceleration” and “enrichment”.
Report of a seminar held at the Royal Society on 22 May 2000.

The report includes contributions from Tim Gowers, Gerry Leversha, Ian Porteous, John Smith, and Hugh Taylor.


This report was originally published in 2000 by the UK Mathematics Foundation (ISBN 0 7044 21828). It was widely red, and was surprising influential. However, it appeared only in printed form. Various moves made by the present administration have drawn attention once more to this early synthesis— which remains surprisingly fresh and relevant. Many of the issues raised tentatively at that time can now be seen to be more central. Hence it seems timely to make the report available electronically so that its lessons are accessible to those who come to the debate afresh.

While the thrust of the report’s argument remains relevant today, its peculiar context needs to be understood in order to make sense of its apparent preoccupations. These were determined by the gifted and talented policy’ adopted by the incoming administration in 1997, and certain details need to be interpreted in this context. There are indications throughout that many of those involved would probably have preferred the underlying principles to be applied more generally than simply to “the top 10%”, and to address the wider question of how best to nurture those aged 5–16 so as to generate larger numbers of able young mathematicians at age 16–18 and beyond. The focus in the report’s title and subtitle on “acceleration” and on “the top 10%” stemmed from the fact that those schools and Local Authorities who opted at that time to take part in the Gifted and Talented strand of the Excellence in Cities programme were obliged to make lists of their top 10% of pupils; and the only provision made for these pupils day-to-day was to encourage schools to “accelerate” them on to standard work designed for ordinary older pupils. The wider mathematics community was remarkably united in insisting that this was a bad move. This point was repeatedly and strongly put to Ministers and civil servants. But the advice was stubbornly resisted; (indeed, some of those responsible at that time are still busy pushing the same linez.

The present administration seems determined once more to make special efforts to nurture larger numbers of able young mathematicians, and faces the same problem of understanding the underlying issues. Since this report played a significant role in crystallising the views of many of our best mathematics teachers and educationists, it may be helpful to make it freely available—both as a historical document and as a contribution to current debate.

Read the whole paper. 

Tony Gardiner: Nurturing able young mathematicians

A. D. Gardiner, Nurturing able young mathematicians, The De Morgan Journal  2 no. 7 (2012), 87-96.


We summarise the developments of the last 20 years—highlighting the key underlying assumptions, and indicating certain unfortunate consequences. We show how official policy has been based on

  • persistent failure: (i) to develop and to implement a suitably challenging curriculum, and (ii) to provide ordinary teachers with good texts, suitable subject-specific professional development, and appropriate assessment targets;
  • a misconception of the curriculum as a one-dimensional ‘ladder’ (with each topic nominally the same for everyone, with uniform expectations for all pupils at a given ‘level’), up which pupils progress at their personal rate, and
  • associated accountability measures that have unintended consequences.

We then outline the alternative conception of a two-dimensional “*-curriculum”, in which each theme in the standard curriculum sequence is explored (and where necessary, assessed) to different depths, and where those who manage to dig deeper and to lay stronger foundations emerge naturally as the ones who are well-placed to subsequently progress further. In such a model, able pupils in Years 5 and 6 would not be pushed ahead to achieve a premature and superficial mastery of ‘Level 6’ material, but would spend time exploring harder problems at ‘Level 4’ and ‘Level 5’ (so-called 4* and 5* material). Similarly, able students in Years 10 and 11 would not be entered early for an accessible but superficial GCSE, but would instead be expected to master core GCSE material more deeply, so as to make the subsequent transition to A level in Year 12 straightforward.

Read the rest of the paper. 

Tony Gardiner: Observations on the LMS Response to Draft Programme of Study in Mathematics, Key Stages 1–2

A. D.Gardiner, Observations on the LMS Response to Draft Programme of Study in Mathematics, Key Stages 1–2, The De Morgan Journal  2 no. 3 (2012), 139–148.


The general response to the draft primary curriculum has been highly critical in some respects. But all responses appear to accept the fundamental idea that there is considerable scope for ‘raised aspirations’. This is remarkably positive.

Many responses also appear to welcome the idea of a clearer focus on core ideas and methods. For example, a survey completed by 5500 primary teachers revealed surprising support (~55%) for delaying calculator use until late primary. And—apart from one or two interest groups—there has been surprisingly little special pleading for the idea of preserving ‘data handling’ as a separate Attainment Target: it would seem that many respondents accept the need for a reduced profile in Key Stages 1-2.

In short, the underlying balance of opinion is now clearer in some respects than one might have anticipated. So the criticisms alluded to in the first paragraph should not be classified as ‘obstructionist’, but as reflecting a desire to give the new curriculum a reasonable chance of succeeding.

The summary of these criticisms provided by the LMS has been widely appreciated and focuses on six main points:

  1. There is an official insistence that a curriculum should concentrate on ‘what’ should be taught rather than `how’ it should be taught. This makes sense but can be taken too far: in mathematics the way a topic is developed over time may be designed to remain as part of students’ mental superstructure. But the official line should make it even clearer to specify something even more basic than `what’—namely `how many hours’ are to be devoted to mathematics in each School Year (the time devoted to mathematics in English schools is low).
  2. A main-school curriculum represents an 11 year journey. One cannot assess an outline of the early years without a clear idea of the mathematical destination it is leading towards. Since the primary curriculum (and the associated `leaks’ about developments at secondary level) raise very awkward questions, one cannot assess a draft for KS1-2 in isolation.
  3. The current draft is insensitive to `the way human beings learn’—in that it fails to convey the way in which the `mental universe of mathematics’ emerges from practical engagement with measures, shapes and quantities.
  4. The current draft is too ambitious—with unreal expectations in Years 1-2, and forcing material into Years 5–6 that belongs more properly in Years 7-8.
  5. The current draft still `nibbles’ at the same material year-after-year, instead of preparing the ground well whilst delaying the formal introduction of hard ideas, and then making significant progress when they are eventually introduced.
  6. Like so much in education, the success of any change depends on maintaining the support of teachers. For it is teachers who must interpret and present the changes to parents, and who implement them in classrooms. This support will be difficult to generate and to sustain without delaying to allow a more realistic schedule, and without a clearer sense of the associated assessment, accountability, and training structures.

Read the rest of the paper