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

Stephen Huggett: Multiple choice exams in undergraduate mathematics

S. Huggett, Multiple choice exams in undergraduate mathematics, The De Morgan Journal, 2 no. 1 (2012), 127-132.

From the Introduction:

In addition to a rigorous practical test called the general flying test, candidates for a private pilot’s licence have to pass written exams in subjects including meteorology, navigation, aircraft, and communications. These written exams are multiple choice, which seems appropriate. The trainee pilots are acquiring skills supported by background knowledge in breadth not depth, and this can be tested by asking them to choose the right option from a limited list under a time constraint. It is not necessary, of course, for pilots to understand the underlying theoretical concepts.

In contrast, students of mathematics are certainly expected to understand underlying theoretical concepts. To a certain extent, this understanding can also be tested using multiple choice exams. Clearly, mathematicians need skills too, of which one of the most important is the ability to perform calculations accurately. This can also be tested using multiple choice exams.

Given that no one method of assessment is good for all of the understanding and skills expected of a student, one should use a variety of different assessment methods in a degree programme, including things such as vivas, projects, and conventional written exams. There is no claim here that multiple choice exams can do everything!

Read the rest of the paper. 

Draft Mathematics Curriculum

On 23 May 2012 Department for Education published Draft Programme of Study for Primary Mathematics. From the official announcement:

The Secretary of State has written to Tim Oates, the Chair of the Expert Panel, with his response to the panel’s recommendations for the primary curriculum. The Secretary of State has also confirmed that he will write again to the panel about the secondary curriculum in due course. You can view a copy of the letter from Michael Gove to Tim Oates regarding the National Curriculum update. Draft Programme of Study for […] mathematics has also been published. These drafts are a starting point for discussion with key stakeholders at this stage, but there will be a full public consultation on revised drafts which will start towards the end of this year.

This blog could be a natural place to start an in-depth discussion of the new curriculum. The following (independently developed) draft curriculum could be useful for such a discussion:

A. D. Gardiner, A draft school mathematics curriculum for all written from a humane mathematical perspective: Key Stages 1–4, The De Morgan Journal, 2 no. 3 (2012),  pp. 1–138.

Abstract: This draft was hammered out by a small group, which included experienced school teachers, textbook authors, curriculum administrators, and mathematicians. In particular, many helpful suggestions from Tony Barnard, Richard Browne, Rosemary Emanuel, and David Rayner have contributed to the current version. It offers a mathematician’s-eye-view of school mathematics to age 16, which we hope will serve as a useful focus for wider discussion and debate.

Comments are most welcome and should be sent to

Anthony.D.Gardiner >>>at<<< gmail.com

Alternatively, leave a comment at this post.

David Pierce: Induction and Recursion

D. Pierce, Induction and Recursion, The De Morgan Journal, 2 no. 1 (2012),  99-125.

From the Introduction:

In mathematics we use repeated activity in several ways:

  1. to define sets;
  2. to prove that all elements of those sets have certain properties;
  3. to define functions on those sets.

These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; it can also highlight important concepts and results such as Fermat’s (Little) Theorem, freeness in a category, and Goedel’s Incompleteness Theorem.
The main purpose of the present article is to show this.

In the `Preface for the Teacher’ of his Foundations of Analysis of 1929, Landau discusses to the confusion just mentioned, but without full attention to the logic of the situation. The present article may be considered as a sketch of how Landau’s book might be updated.