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 amathematician’s-eye-viewof 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.

I am loving the book – just started to read it after skimming. Something caught my eye early on, namely:

What you don’t know by heart, and so can’t access instantly, you can’t use.

Computers allow instant access to certain ideas without memorization. Take, for example, the problem from your list – you write:

simplify 36/54

one cannot begin unless the relevant direct facts are known by heart so that we have a chance of recognising that they are needed (e.g. 36 = 4*9, 54 = 6*9)

Suppose a kid wants to solve the problem without asking the computer to fully solve it. You can type “simplify 36/54” into Wolfram Alpha and get the answer, but say, the kid wants to do some intermediate steps by hand, or to see the structure of the problem well.

The said kid can ask the computer to factor 36 and 54, and then cross out the common factors: http://www.wolframalpha.com/input/?i=factor+36

I picked this example because I’ve been interviewing homeschoolers about their “times tables” – quite a few chose never to memorize them, using these and other techniques instead: http://naturalmath.wikispaces.com/Child-Led+Multiplication+Study

Dear Maria, should kids learn poems by heart, or is this also outdated thanks to new technology? Should learning poetry by heart be excluded from school practice? Try to type into Google a few words from any well known poem — and you will instantly get the whole poem. I tried that with “those evening bells” and with a quote from Larkin — it worked in both cases.

Also, in my opinion, parents’ responses is dangerous material and require careful analysis. I quote ar random a parent’s testimony from your website, and rest my case:

Linda K:

We have always been very relaxed in our schooling, and I had pretty much written off long ago her memorizing the tables in a traditional fashion, but I didn’t worry about it. Now, she has decided that, “it is time” for her to learn them. Much to her surprise, she has already learned much of the tables from exposure and use, in spite of herself.Alexandre, I don’t think learning by heart is outdated (in any area). What I think is outdated is the idea that learning certain lists of facts by heart is absolutely necessary. It’s the idea of universal lists that is going away, not the idea of learning by heart. In other words, now different people can learn different things by heart, while before, everybody who wanted to enter certain practices had to learn the same bodies of facts.

There are now technological tools that enable fluency without memorizing EVERYTHING in a certain area. However, memorizing SOME items is still very much needed for several key aspects of learning mathematics (or for writing poetry), for example:

– knowing enough iconic/representative examples (“keywords”) that enable the use of technology (such as a search you mentioned)

– understanding concepts through large enough, well-structured example spaces kept in front of “the mind’s eye” for the examination

– faster speed and larger example spaces in the particular areas where you do a lot of work

– identifying yourself as a member of a particular subculture (even though use of mobile devices for reference at parties is now socially acceptable)

Agreed about the need of analysis for data. I will keep you posted about this – we are not at that stage yet, just doing some pilots.

OT – can you please enable subscription to comments on the blog? It helps a lot with replies.

I don’t think you can ‘rest your case’ on this example because it supports the other side. Focus on insight and understanding, while commenting to the student on the tremendous gains in utility and efficiency that come from fluent recall – especially in examinations !!! – as I do, frequently – and many pupils – not all – will eventually realise the advantages of ‘instant recall’.

So you get the best of both worlds!

I have taken a quick look. There is much good stuff here. In particular, the need to learn essential ideas ‘by heart’ is highly apt. However, the ‘experts’ are all ‘top-downers’ and don’t understand the real difficulties involved in learning by most kids in school, in particular why teachers and students are virtually forced to learn procedurally because they don’t or can’t make sense of the mathematical ideas to connect them together in powerful ways. Here one needs to take into account how we learn based on our previous experiences and the build-up of our ideas over the years and how aspects learnt at one stage may by highly supportive at the next while other aspects become problematic and cause cognitive and emotional difficulties. The report released this week by OfStEd illustrates this fundamental myopia.

Regarding “learning by heart”: I can quote Vladimir Radzivilovsky, a brilliant mathematics teacher, who loves to say

The brain is like the stomach: it can digest only stuff which is already inside.Fine, so why not take a child who is too young to know the six times table, even by your standards, and posed to him or her a question on these lines:

“Here is a fraction with big numbers that you have never seen before: 36/54. Do you think it could be equal to a simpler fraction which you have come across before? Which one?”

The food is now in the child’s stomach, which contains, however, no knowledge of the 6 times table. How will the child digest this food?

This is you a very good problem for a [suitable] child because it is Richly Complex rather than Falsely Simple.

I particularly enjoyed your Background and “21 themes” sections (especially since I had previously ploughed through Ofsted’s “Made to measure”). Your work certainly reflects my own perspectives on mathematics education, and I will be recommending it to others. I failed to make it through the whole curriculum document (is very brief, brief and fuller required?); although this could easily reflect my pressure for time more than anything else. When looking at a new curriculum I like to see a section entitled something like “Main differences” – a summary of what’s out and what’s in. Would that fit in?

This proposal for an inclusive, humane mathematics curriculum is a timely contribution to the debate about the form of the national curriculum for mathematics.

I think the draft does a nice job of dealing with the various conflicting demands on what a national curriculum should do. In particular, I think the report balances nicely the needs of all students with those who will go on to advanced study in the subject providing an interesting perspective on the traditional argument between acceleration and enrichment.

I have two main comments. The first is to do with the treatment of geometry. I absolutely endorse the inclusion of straightedge and compass construction in the school curriculum, as I think this provides an important platform for the development of rigour in proof. However, I was surprised that the proposal did not mention dynamic geometry software. Many students have great difficulty with the physical precision and coordination required to execute these constructions, and I do not think there is much value in spending a lot of time on this aspect, as opposed to the mathematics itself. Indeed, the danger with physical construction is that it leads students to think in terms of what is drawn, rather than the Platonic ideals that the drawings represent (which is drawn out so clearly in the proposed curriculum).

The second point is to do with what we can learn from other mathematics curricula. In recent years, I have become convinced about the power of the Russian mathematics curriculum, particularly in the way it builds in the “big ideas” of mathematics from the earliest years. This is done well in the current draft with regard to, for example, inverse problems, but I think this could be drawn out more clearly. For example, if we accept inverse and invariance as two of the really “big” ideas of mathematics, these could be introduced in the earliest years by introducing subtraction at the same time as addition, so that students are taught that 3 + 4 = 7 is the same number fact as 7 – 4=3 and also learn 7+ 0 = 7 and 7 – 0 = 7.

I am a mathematical engineer, originally educated in Russia, with 33 years of research experience in applied mathematics. For 16 years I also taught mathematics to engineering students at a London University of widening participation (South Bank). As such I worked with “victims” of the current approach to school mathematics. I will never forget the best student in my first cohort asking “Why do you keep talking about logic? Mathematics has nothing to do with logic.”

I have developed a system for turning these educational pariahs (often considered unteachable) into normal engineering students. However, I have encountered a great reluctance on behalf of colleagues to learn to explain things to students. You talk so eloquently about making links, Most colleagues are not familiar with the concept.

For this reason I suggest that your flawless plan could benefit from more emphasis on necessity to write comprehensive textbooks, to show the school teachers at least one complete set of Hows. The most inventive of them could then embellish the system but everyone would have a good starting point.

While my expertise is in teaching adults, if you think there is anything I could do to help promote good maths teaching at schools please let me know.

Since I serve on the LMS Education Committee, I had a chance to observe the work of the team led by Tony Gardiner. Two points impressed me:

(a) Structural integrity, cohesion, consistency of the draft. Using an architectural simile, it is a building which has proper load bearing walls: every next step in learning is meant to be supported by the previously achieved understanding.

(b) The draft sets a benchmark of quality, and, hopefully, it will make it useful in the forthcoming discussion of mathematics curriculum reform.

I write this from a position of a lecturer in mathematics for Foundation Studies. Many my students have no A levels in mathematics; I am at the receiving end of Key Stage 4 mathematics education — in particular, I have to explain to my students long division, among other things. I share Larissa Fradkin’s views and can add that, from the universities’ perspective, an exposure to disciplined thinking is more important than any particular technical skill.

I feel a real problem is to give learners at any stage

confidencein their ability to perform and continue to learn at their own level, and this also means to be able investigate freely patterns and structures, and so improve their performance, which may not reach mastery, but will get by at their own level. I got a good degree but was not then so confident in my ability tomathematisewhich may mean not so much to solve quickly the given problem, but to search happily, till one says “of course”. Some of my most successful research students were those who seemed to want to knowwhysomething was true.In football, we do not expect everyone to be a potential premier division players, but vast numbers play happily, and I hope confidently, at their level. So I would like to see the word

confidencehaving it’s due place in the report.You are entirely correct: this is important and much underrated. But the question is whether this and numerous other affective concerns belong in in a draft written curriculum. Once one starts mentioning such things, the list grows and grows. Our goal was to try to capture the sequencing and flavour of mathematical content that might make this kind of key objective more accessible for more students.

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This is an extremely interesting attempt at an ambitious goal though I do wonder whether it will receive the attention it deserves in view of the plethora of reports released recently. See Tony Gardiner himself on “A mathematician’s view of the current education scene in the UK” and my 24/11 post in response.

Next, two related points of criticism. Under the heading, “Learning by heart, fluency, automaticity”, he writes that, “Pupils certainly need to be on top of that limited collection of basic facts and techniques in terms which most elementary mathematics can be understood.” [Page 19]

My immediate reaction is to wonder why he prefers this assertion to the ‘dual’ assertion: “Pupils certainly need to be on top of that limited collection of basic insights and understandings in terms which most basic facts and techniques can be understood.”

I have tutored many pupils from private schools, preparing for the 13+ or for the examinations for school such as City of London or Westminster, who are perfectly capable of dividing one fraction by another (for example) by the usual rote rule but have no understanding whatsoever of why the rule works. Have they by learning the rule by heart and subsequently displaying fluency and automaticity, made any worthwhile step towards being the young mathematicians that I’m sure Tony Gardiner would like them to be? I think not.

Shortly afterwards he remarks, discussing the tackling of unfamiliar problems, how, “On a mundane level, when faced with routine inverse problems … such as simplify 36/54 […] one cannot begin unless THE relevant direct facts are known by heart so that we have a chance of recognising that they are needed (e.g. 36 = 4 x 9; 54 = 6 x 9 …).” [My emphasis]

I will query the definite article and refer back to his earlier admirable statement about ‘levels’: “This is a natural consequence of the recognition that each statement or idea can be mastered to different depths, and our consequent rejection of the interpretation of the statements as if they were rungs in some shared ‘ladder’ up which each pupil climbs at his or her preferred rate.”

Indeed. Strong pupils, of whatever age, may well simplify 36/54 instantly by recognising that both numbers appear in their 9 times table – or indeed by recognising that 36 and 54 and multiples of 18. Being ‘strong’, we hope that behind their automaticity lies a genuine understanding of fractions and ratios.

Weaker pupils however, for example primary pupils, might consult quite a different set of facts. For example, they might notice that 36 and 54 are both even, so they can cancel the fraction, once. They might also notice, by digit sum, that both numbers are divisible by 3; or having already halved both they might notice that 18 and 27 are both multiples of 3 and so they work their way towards the answer 2/3.

It is then a matter of insight – not easily achieved by most pupils – that because both numbers are multiples of both 2 and 3, they must be multiples of 6 also. By no means all of my tutees whom I mentioned earlier have achieved this fundamental insight – in which case, what is the mathematical value of their fluency in cancelling by rote by a factor of 6?

POSTSCRIPT: Page 4 claims that, “The act of devising a curriculum is inevitably a top-down process, in which the drafters select and interpret certain the higher objectives.”

David Tall (27 May 2012) implicitly queried this claim, and I agree with him. It would be better to say, “The act of devising a curriculum is inevitably a top-down process AND a bottom-up process, as drafters select and interpret certain higher objectives and experts in children’s mathematical development relate them to children’s capacities at different ages.”