## Tony Gardiner: National curriculum – Comments and suggested necessary changes

### Featured

Published today:

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.

• 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’.

# MathJax v2.3 now available

From The MathJax Team:
After a successful beta run, we’re happy to officially release MathJax v2.3.

Version 2.3 is available on the CDN at

http://cdn.mathjax.org/mathjax/2.3-latest/MathJax.js

and starting today the files at the

http://cdn.mathjax.org/mathjax/latest/MathJax.js

address will be switched over the v2.3; it will take 24h-48h for the changes to propagate out to the distributed cloud servers.

# GCSE: Higher level content

Reformed GCSE subject content includes three types of content: standard, underlined and bold. In the words of he document,

The expectation is that:

• All students will develop confidence and competence with the content identified by standard type
• All students will be assessed on the content identified by the standard and the underlined [here, for technical reasons, emphasised -- AB] type; more highly attaining students will develop confidence and competence with all of this content
• Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content.

The distinction between standard, underlined and bold type applies to the content statements only, not to the assessment objectives or to the mathematical formulae in the appendix.

What follows is the list of items in the  Mathematics GCSE subject content and assessment objectives which contain bold type, higher content.I think this short lists clearly marks the boundaries of GCSE — AB

# What exactly do student evaluations measure?

From a post by  Philip Stark on The Berkeley Blog:

●  student teaching evaluation scores are highly correlated with students’ grade expectations[10]

●  effectiveness scores and enjoyment scores are related[11]

●  students’ ratings of instructors can be predicted from the students’ reaction to 30 seconds of silent video of the instructor: first impressions may dictate end-of-course evaluation scores, and physical attractiveness matters[12]

●  the genders and ethnicities of the instructor and student matter, as does the age of the instructor[13]

Red the whole text.

# GCSE reform is finalised

Ofqual today (1st November 2013) [...]) announced a revised timetable for the reforms, meaning new GCSEs in English language, English literature and maths will take priority and will be introduced for first teaching from 2015.

The Department for Education will today be confirming the subject content for these subjects, following a separate consultation.

Key features of the new GCSEs in England will include:

• A new grading scale that uses the numbers 1 – 9 to identify levels of performance, with 9 being the top level. Students will get a U where performance is below the minimum required to pass the GCSE
• Tiering to be used only for subjects where untiered papers will not allow students at the lower end of the ability range to demonstrate their knowledge and skills, or will not stretch the most able. Where it is used [this apparently applie to maths -- AB], the tiering model used will be decided on a subject-by-subject basis
• A fully linear structure, with all assessment at the end of the course and content not divided into modules. This is to avoid the disruption to teaching and learning through repeated assessment, to allow students to demonstrate the full breadth of their abilities in the subject, and to allow standards to be set fairly and consistently
• Exams as the default method of assessment, except where they cannot provide valid assessment of the skills required. We will announce decisions on non-exam assessment on a subject-by-subject basis
• Exams only in the summer, apart from English language and maths, where there will also be exams in November for students who were at least 16 on the preceding 31st August. Ofqual is considering whether November exams should be available in other subjects for students of this age.

Read the full statement from Ofqual.

# Tony Gardiner: David Willetts and “Robbins Revisited”

David Willetts (Minister of State for Universities and Science) has just published a possibly significant pamphlet with the Social Market Foundation called Robbins Revisited:

Robbins was an LSE economist whose committee examined the future of the ish university system – reporting in 1963. The report was unusually perceptive and its recommendations influenced policy for the next 30 years. This report is interesting in that it indicates the relevant Minister’s desire to place current HE policy in a historical context. Half of the time the contextualising makes sense; but half of the time it seems to be influenced by the need for post hoc self-justification.

As so often with high level documents, the data are wilfully distorted (whether deliberately or through wishful thinking one cannot know) in order to fit a required political perspective. Willetts repeatedly interprets data as demonstrating “improvement” even where we know it does no such thing. And where the uncomfortable explanation is to hand, he prefers to express puzzlement – as on p.69 where he observes

“an apparent mismatch between the supply and demand for high-level computer skills. Employers currently say they cannot find the skills they need yet computer science graduates find it relatively hard to find graduate-level work”

but then fails to infer that perhaps many computer science undergraduates are accepted onto courses, and graduate, without the relevant “high-level skills”.

He makes no mention of the botched attempts to broaden studies at age 16-18 (e.g. Tomlinson), or of the fact that the A level ‘gold standard’ his colleagues defend makes sense only if it supports specialisation. He then misinterprets the English fudge of continuing with A levels while abandoning specialisation (Table 5.1) as if it were a move in the direction of the kind of breadth Robbins advocated.

However, he has a relevant qualification (pp.50-1):

“there is an important distinction to be made between the need for breadth in general, and the need for maths skills in particular. In an interview with The Listener in 1967 Robbins was asked why the numbers opting for applied and pure sciences had fallen below expectations. He blamed what he called “the terror of mathematics”, caused by poor teaching and a preoccupation in university maths departments with producing “aces”.

This issue has not gone away. Last year the Lords Science and Technology Committee expressed its shock that many Science, Technology, Engineering and Mathematics (STEM) undergraduates lacked the mathematical skills required to cope with their course at university. The National Audit Office has warned that this is an issue for student retention. Maths is a core part of science and engineering subjects – but it comes into many others [...] it is the universal analytical tool which matters more and more in today’s higher education.”.

# MathJax v2.3 beta release

From Peter Krautzberger of The MathJax Team:

We are entering beta testing for MathJax v2.3 today. This release focused on new webfonts options and improvements. It also includes improved localization features and improvements to our Native MathML output (to work around shortcomings in Firefox and Safari).

A copy of the beta release is available via our CDN at http://beta.mathjax.org/mathjax/latest/MathJax.js and also via a stable https address. Please note that the beta now has its own subdomain. You can also download the beta as a zip file at https://github.com/mathjax/MathJax/archive/v2.3-beta.zip.

If you are able to test the release, we would very much appreciate your feedback. I’ve copied the upcoming announcement with more information below.

# Pierre Deligne: Early Years

Some quotes from  Interview with Abel Laureate Pierre Deligne, by Martin Raussen (Aalborg, Denmark) and Christian Skau (Trondheim, Norway), European Mathematical Society  Newsletter, September 2013, pp. 15-23.

You were born in 1944, at the end of the Second World War in Brussels. We are curious to hear about your first mathematical experiences: In what respect were they fostered by your own family or by school? Can you remember some of your first mathematical experiences?

I was lucky that my brother was seven years older than me. When I looked at the thermometer and realized that there were positive and negative numbers, he would try to explain to me that minus one times minus one is plus one. That was a big surprise. Later when he was in high school he told me about the second degree equation. When he was at the university he gave me some notes about the third degree equation, and there was a strange formula for solving it. I found it very interesting.

When I was a Boy Scout, I had a stroke of extraordinary good luck. I had a friend there whose father. Monsieur Nijs, was a high school teacher. He helped me in a number of ways; in particular, he gave me my first real mathematical book, namely Set Theory by Bourbaki, which is not an obvious choice to give to a young boy. I was 14 years old at the time. I spent at least a year digesting that book. I guess I had some other lectures on the side, too.

Having the chance to learn mathematics at one’s own rhythm has the benefit that one revives surprises of past centuries. I had already read elsewhere how rational numbers, then real numbers, could be defined starting from the integers. But I remember wondering how integers could be defined from set theory, looking a little ahead in Bourbaki, and admiring how one could first define what it means for two sets to have the “same number of elements”, and derive from this the notion of integers.

I was also given a book on complex variables by a friend of the family. To see that the story of complex variables was so different from the story of real variables was a big surprise: once differentiable, it is analytic (has a power series expansion), and so on. All those things that you might have found boring at school were giving me a tremendous joy.

Then this teacher, Monsieur Nijs, put me in contact with professor Jacques Tits at the University of Brussels. I could follow some of his courses and seminars, though I still was in high school.

It is quite amazing to hear that you studied Bourbaki, which is usually considered quite difficult, already at that age.

Can you tell us a bit about your formal school education? Was that interesting for you, or were you rather bored?

I had an excellent elementary school teacher. I think I learned a lot more in elementary school than I did in high school: how to read, how to write, arithmetic and much more. I remember how this teacher made an experiment in mathematics which made me think about proofs, surfaces and lengths. The problem was to compare the surface of a half sphere with that of the disc with the same radius. To do so, he covered both surfaces with a spiralling rope. The half sphere required twice as much rope. This made me think a lot: how could one measure a surface with a length? How to be sure that the surface of the half sphere was indeed twice that of the disc?

When I was in high school, I liked problems in geometry. Proofs in geometry make sense at that age because surprising statements have not too difficult proofs. Once we were past the axioms, I enjoyed very much doing such exercises. I think that geometry is the only part of mathematics where proofs make sense at the high school level. Moreover, writing a proof is another excellent exercise. This does not only concern mathematics, you also have to write in correct French – in my case – in order to argue why things are true. There is a stronger connection between language and mathematics in geometry than for instance in algebra, where you have a set of equations. The logic and the power of language are not so apparent.

You went to the lectures of Jacques Tits when you were only 16 years old. There is a story that one week you could not attend because you participated in a school tip…?

Yes. I was told this story much later. When Tits came to give his lecture he asked: Where is Deligne? When it was explained to him that I was on a school trip, the lecture was postponed to the next week.

He must already have recognised you as a brilliant student. Jacques Tits is also a recipient of the Abel Prize. He received it together with John Griggs Thompson five years ago for his great discoveries in group theory. He was surely an influential teacher for you?

Yes; especially in the early years. In teaching, the most important can be what you don’t do. For instance, Tits had to explain that the centre of a group is an invariant subgroup. He started a proof, then stopped and said in essence: ‘An invariant subgroup is a subgroup stable by all inner automorphisms. I have been able to define the centre. It is hence stable by all symmetries of the data. So it is obvious that it is invariant.”

For me, this was a revelation: the power of the idea of symmetry. That Tits did not need to go through a step-by-step proof, but instead could just say that symmetry makes the result obvious. has influenced me a lot. I have a very big respect for symmetry, and in almost every of my papers there is a symmetry-based argument.

Can you remember how Tits discovered your mathematical talent?

That I cannot tell, but I think it was Monsieur Nijs who told him to take good care of me. At that time, there were three really active mathematicians in Brussels: apart from Tits himself, Professors Franz Bingen and Lucien Waelbroeck. They organised a seminar with a different subject each year. I attended these seminars and I learned about different topics such as Banach algebras, which were Waelbroeck’s speciality, and algebraic geometry.

Then, I guess, the three of them decided it was time for me to go to Paris.Tits introduced me to Grothendieck and told me to attend his lectures as well as Serre’s. That was an excellent advice.