A. Borovik: Sublime Symmetry: Mathematics and Art

A new paper in The De Morgan Gazette:

Form the Introduction:

This paper is a text of a talk at the opening of the Exhibition Sublime Symmetry: The Mathematics behind De Morgan’s Ceramic Designs  in the delighful Towneley Hall  Burnley, on 5 March 2016. The Exhibition is the first one in Sublime Symmetry Tour  organised by The De Morgan Foundation.

I use this opportunity to bring Sublime Symmetry Tour to the attention of the British mathematics community, and list Tour venues:

06 March to 05 June 2016 at Towneley Hall, Burnley
11 June to 04 September 2016 at Cannon Hall, Barnsley
10 September to 04 December 2016 at Torre Abbey, Torbay
10 December 2016 to 04 March 2017 at the New Walk Gallery, Leicester
12 March to 03 September 2017 at the William Morris Gallery, Walthamstow

William De Morgan, Peacock Dish. The De Morgan Foundation

William De Morgan, Peacock Dish. The De Morgan Foundation.

Yagmur Denizhan: Performance-based control of learning agents and self-fulfilling reductionism.

Yagmur Denizhan: Performance-based control of learning agents and self-fulfilling reductionism. Systema 2 no. 2 (2014) 61-70. ISSN 2305-6991. The article licensed under the Attribution-NonCommercial-NoDerivatives 4.0 International License. A PDF file is here.

Abstract: This paper presents a systemic analysis made in an attempt to explain why half a century after the prime years of cybernetics students started behaving as the reductionist cybernetic model of the mind would predict. It reveals that self-adaptation of human agents can constitute a longer-term feedback effect that vitiates the efficiency and operability of the performance-based control approach.

From the Introduction:

What led me to the line of thought underlying this article  was a strange situation I encountered sometime in 2007 or 2008. It was a new attitude in my sophomore class that I never observed before during my (by then) 18 years’ career. During the lectures whenever I asked some conceptual question in order to check the state of comprehension of the class, many students were returning rather incomprehensible bulks of concepts, not even in the form of a proper sentence; a behaviour one could expect from an inattentive school child who is all of a sudden asked to summarise what the teacher was talking about, but with the important difference that –as I could clearly see– my students were listening to me and I was not even forcing them to answer. After observing several examples of such responses I deciphered the underlying algorithm. Instead of trying to understand the meaning of my question, searching for a proper answer within their newly acquired body of knowledge and then expressing the outcome in a grammatically correct sentence, they were identifying some concepts in my question as keywords, scanning my sentences within the last few minutes for other concepts with high statistical correlation with these keywords, and then throwing the outcome back at me in a rather unordered form: a rather poorly packaged piece of Artificial Intelligence.
It was a strange experience to witness my students as the embodied proof of the hypothesis of cognitive reductionism that “thinking is a form of computation”. Stranger, though, was the question why all of a sudden half a century after the prime years of cybernetic reductionism we were seemingly having its central thesis1 actualised.

Rebecca Hanson: National Assessment Reform – Where are we now?

R. Hanson, National Assessment Reform – Where are we now? The De Morgan Gazette 5 no. 5 (2014), 33-39.

This short report summarises the pending changes to national assessment at 4/5, 6/7, 10/11, 15/16 and 17/18.  It attempts to list the key concerns about the reforms and to describe the likely imminent calls for modifications.

It can also be downloaded as a word document here:
National Assessment Reform Where are we now 1 Sept 2014

If you have any questions you can contact the author.

Misha Gavrilovich: Point-set topology as diagram chasing computations

M. Gavrilovich, Point-set topology as diagram chasing computations, The De Morgan Gazette 5 no. 4 (2014), 23-32.


We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms \(T_0\) and \(T_1\) in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain).

We also offer a couple of brief speculations on cognitive and AI aspects of this observation, particularly that in point-set topology some arguments read as diagram chasing computations with finite preorders.

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: