I believe PISA is meticulous in conducting its tests and reflects a good evaluation of standards of today's maths education. And yet I think if countries like the UK simply try to climb up today's PISA assessment, they'd be doing the wrong thing.

The playing field of today's maths education is restricted to manual calculating procedures allied to the limited problem-solving that they can support. Today's mainstream real-world maths is much broader: applying the process of maths--using the best computational mechanisation--to much harder problems. The skills it requires are rather different, but if anything more conceptual, more intellectual and definitely more creative.

That's a playing field on which Brits and the like could do relatively much better than on the playing field of procedural hand-calculating. It's a playing field on which drilling kids for hours a day on their algebra isn't going to win.

Now let's be clear. I'm not saying that that's universally what's happening in Asia. In fact there's great innovation in the process of schooling and particularly the learning of maths in the region (famously Singapore). Nor am I in any way writing off Asian problem-solving ability which I think, correctly and creatively trained, could be top-notch too. What I am saying is that if Brits really put their minds to modern computer-based maths, they are just as able to compete with their Asian counterparts--whereas I don't think culturally we will do so well at drilling the needlessly pre-abstracted and often irrelevant current subject. I think that non-conformity, creativeness and looking around the rules is key to British (and many other Western) cultures and a great competitive strength if tethered appropriately, opposite to the cultural imperatives present in many of the countries performing well in today's maths PISA test, countries that may struggle to imbue such charactertistics.

And crucially, it's many of these abilities and the computer-based maths subject we desperately need in the workplace, and in life--not for the most part the subject we're largely failing to succeed at of hand-calculating procedures. (My recent talk opening the CBM summit at UNICEF details the argument).

A central question in all this is precisely what outcomes we wish for our students after their years of maths study? This is a question which we have been addressing from first principles in formulating CBM, unencumbered by constraints PISA necessarily has of not going too far ahead of today's curriculum and needing accurate quantitative assessment of it. For the brave, here is an early (hard to digest) draft which spans 10 dimensions. I won't detail all the ideas here but point out the importance of confidence, knowing how to operationally manage the application of maths, and understanding the separation between maths concepts (like significance) and use of a wide variety of specific tools (like a hypothesis test).

Intelligently ranking countries as PISA does is very helpful in pushing progress in education because succeeding at today's maths education or tomorrow's computer-based variety needs well-directed effort and focus and competition. But in the end, however well education is delivered, it must deliver the right subject.

Notice that our first CBM country Estonia is already high on PISA. They recognise that despite their achievements, they need to lead the change to maths. Actually, many of the countries near the top of today's rankings have been most active in pursuing the CBM approach.

Now the UK is doing well with Estonia in leading the coding education agenda. But why oh why does the UK government choose to separate coding in primary education from maths with which it should be so intertwined (as has the US)? They need to be closely associated as I pointed out last year. And it's particularly galling that they're not in the country where a mathematician invented the computer...

Playing badly on the wrong field is hardly smart. As the playing field shifts, let's lead the change, not be laggards at a game we can succeed well in.

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This really has at least 4-dimensions of consequence:

Firstly, it's a unique way to excite students about maths by marrying it up with coding. Coders will be able to use the power of Mathematica's maths out of the box, not only enriching what they can do but also showing off the power and importance of maths. Attaching maths to something already enjoyable to make it better and more enjoyable I think will be very encouraging in learning more maths. And you never know, politicians and policy-makers might even start to see the connection between coding, maths and fun--rather as I outlined in an earlier blogpost

Secondly, it's cheap. For $25 + some bits and pieces, you can be up and running. One reason I was excited to be able to announce this today is because we've been hosted UNICEF's building for our summit and I think we'll have a great solution for maths, coding and CBM in developing countries.

Thirdly, this is the first pass of the Wolfram language. For years it's been lurking under the umbrella of Mathematica, a key aspect not only of our technology stack but the framework, even our symbolic way of thinking about structuring ideas. And because Wolfram Language is multi-paradigm it's a great early language to learn because it avoids students getting into thinking of everything as best expressed in one structure or other. This all complements Raspberry Pi and its goals very well and so it's nice that our first manifestation of Wolfram Language is there. Others will follow.

Fourthly, it's simply amazing that Mathematica and Wolfram language can run on something as small and cheap as Raspberry Pi. Yes, by modern desktop PC standards it can be a little clunky, but functionally it's all there--all the thousands of functions (even including my show-off special function HypergeometricPFQRegularized[ ]!). One further consequence: because Raspberry Pi is small and cheap enough to act as an embedded computer, we for the first time we have a quick-to-deploy yet full-power embedded solution.

Really looking forward to seeing what the world's students (and their tinkering parents!) come up with with this new super-combo and how it can help to drive CBM forward.

P.S. This rather completes our fruity announcements for the moment--from Apple to Blackberry to Raspberry Pi (though not as my daughter keeps calling it the Apple Pi).

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So I am very pleased that we're able to collaborate with UNICEF on our 3rd CBM summit, holding it at their headquarters in New York City on November 21-22.

That collaboration means a few things. Firstly, it demonstrates UNICEF's recognition of maths as crucial to improving the lives of all children, and particularly in the sort of developing countries in which UNICEF's role is central. Great credit to UNICEF and Chris Fabian (their Innovation unit chief) for being so proactive in getting this. Secondly it will broaden horizons on CBM, by bringing new groups into the action-plan, shaping the outcomes we're trying to achieve and the reality of deployment in many different environments.

I am really looking forward to this summit and also how it will push us to get some "in gestation" projects ready. Look out for a new **visualisation of the maths process**, what's currently a **10-dimensional outcome tree** and demos of draft **Estonian CBM modules** amongst many outside contributions.

This promises to be a unique gathering for fixing the world's maths education--not to mention your country's, state's or industry's. Policy-makers and key maths education voices: please come! Or suggest who should :-).

]]>As I understood his central point it was that practising hand calculations is akin to practising music pieces--it's simply the way to learn to play. Also there was some attempt to draw the analogy between listening to music and CBM, whereas playing was like traditional hand-calculating maths.

I think music education can teach us quite a bit but believe his analysis and conclusions were wrong.

We need to start from outcomes. What do we hope to achieve from people learning maths and music?

For most people, music is enriching. And for some, generating that music adds enrichment. For a few, it may even be financially enriching too, if they become professional. But I don't think that latter case is why most people study music.

The objective of learning an instrument is to play music. And practising playing music is a direct requirement to achieve that. It usually starts very early--as soon as you can string notes together, you're off trying to practice simple pieces. You are also supposed to practice scales and arpeggios. In my case I wasn't very punctilious at scales, primarily because I didn't see the point. If it had been explained that getting really good at the Eb major scale would aid my playing of an Eb major Haydn piano sonata, I would have been much more interested. In fact no association was made between the scale being practiced and the key of the piece I was trying to play.

Back to maths. My argument for CBM is that practising hand-calculating doesn't relate to the real-world outcomes in any direct way. It's not akin to practising a piece of music because the real-world outcome is disconnected. In fact my adversary in the debate agreed completely with my analysis of real-world maths: that it's computer-based. He just believed practising hand-calculating was the way to get there. I don't. In fact for all the reasons I've gone into before, I think it's detrimental for a start because it de-prioritises much more important, much more real-world outcome-connected material.

Far from just learning that practice is important, we should learn from music education that repeated practice or experience of the actual outcomes (in maths--real-world problem-solving) is vital. CBM aims to do just that.

We shouldn't forget that one big difference between music and maths is compulsion. For the most part you only learn an instrument if you (and/or your parents) want to. Everyone is made to learn maths. In music if you want to play pieces, you need to practice them; that motivates you. In maths, if you have no idea why you'd learn it, can't see an outcome you're interested in, why would you practice? And in fact the practice prescribed is largely dissociated from outcomes you'll face; so you'd have a point!

There's something else music can teach us--about how assessment works. (Lord) Jim Knight pointed this out to me. At least in the UK, you take a "Grade" exam when *you're* ready, not along with everyone else whatever your level. The exams are closely tied to the outcomes, mainly playing pieces live to examiners. There's some sight-reading (you'll need that if you want to learn new things), some scales and some questions on listening to music. Most people do well in the exams because they're ready, yet they are still highly-regarded, not dumbed down.

Why not adopt this sort of model in maths?

]]>I'm very excited to announce that computerbasedmath.org has found the first country ready for our completely new kind of maths education: it's Estonia. (...and here's the press release).

I thought Estonia could be first. They are very active on using technology (first to publish cabinet decisions immediately online, first to include programming in their mainstream curriculum), have ambition to improve their (already well respected) STEM aptitude and lack the dogma and resistance to change of many larger countries. There aren't so many countries with all those characteristics.

In our first Estonia project we will work with them to rewrite key years of school probability and statistics from scratch. This is an area that's just crazy to do without a computer, even harmful. It's an area that's only come to the fore since computers because it only makes sense with lots of data. No-one in real life does these hand analyses or works with only 5 data points, so why do we make our students? Why get students emulating what computers do so much better (computing) rather than concentrate on imaginative thinking, analysis and problem-solving that students ought to be able to do so much better even than today's computers?

Worse, in a subject like probability and statistics, current maths education often forces you to learn the wrong tools for the job.

Take the Normal Distribution--one of very few options taught to students for data analysis. Approximating your data with it rarely gets you the most accurate solution; it can be wildly wrong. Instead why not learn to select the best of 100+ other distributions and test their predictions against each other? Or why use a distribution at all, when you can work out results directly from each and every data point?

The reason is historical. Normal Distributions (and Poissons) are easiest to calculate, appear appropriate in over-simplified problems and you can't practically compare lots of distributions or work directly with data by hand. But with a computer you can and you should.

Out in the real world, there are real consequences to drilling students in de- or mis- contextualised techniques--and with the idea that each school problem has one right technique, and each technique has particular patterns of problem. Take the miscalculating of large swathes of financial risk analysis: people applied Normal distributions because they knew of them, had been trained to expect them but that didn't make for effective representations for the data.

This reminds me of an old adage. "If all you have is a hammer, everything looks like a nail". In maths context--the fact that every school problem can be solved with a small subset of maths tools leads to a false expectation that in the real-world this same subset will suffice. CBM broadens the toolkit dramatically by not insisting students should make all the tools that they use, freeing time for using more in a wider variety of situations.

Estonia is the first place where we're starting to change all of this though many other countries have voiced interest in pursuing CBM and being within the first group.

But it's slow work on several fronts for a little while. Even though several of us have been thinking along CBM lines for ages, we're constantly questioning whether a particular way of thinking or doing is in fact now the best way or simply a legacy of the pre-computer era. Indeed what outcomes are we trying to achieve and how does learning tools of maths fit with learning how to solve problems?

And education changes slowly, though now is the most vibrant and exciting time of change in my lifetime. Even with this, I expect it to be a couple of decades until the world's mainstream maths subject is universally computer-based maths rather than today's "history of hand-calculating". But today is an important step.

Countries that start the change early will reap many benefits from being first--a bit like the changes that universal education brought to countries who were first, but in microcosm for maths.

In fact it's more of a macrocosm. It affects lucrative problem-solving STEM jobs where pushing the boundaries of modeling is crucial to success. But it can make happier citizens too--able to assess risk, understand complex finances better, have an in-built mathematical 6th sense by which to understand life.

]]>But where does programming fit with ICT, computer science and maths? How central a subject is it?

What's termed ICT seems to be "how to operate your computer...or generic applications on it...or even past computing forms like calculators". Frankly children are often good at operating the latest tech--usually better than their teachers. Primary schools need to help, verifying that they can do basic operations and offer remedial, individual help if not, but this "operating your computer" should not be a subject per se and is far from programming in subject-matter and required skillset.

What about computer science? It's the specialist subject of how you optimise programs, programming, build large-scale software or even design new programming languages. Important though this is, attaching programming only to CS is too narrow a viewpoint.

Instead, programming is much more fundamental to STEM: it's the way you communicate technical ideas and processes in the modern world. It's as central as that.

You can view it as a superset evolution of mathematical notation, far more general and with the immediate consequence of machine computable results. Programs are the way you write down maths.

And so I believe programming is an integral, core part of maths education. It's the hand-writing of technical ideas and just like hand-writing is in the early years attached to learning English (if you're in England!), so core, basic programming should be attached to maths.

To be clear, I'm not talking calligraphy, but basic hand-writing. Calligraphy is the CS end--the subject in which you study programming in its own right, its nuances, detailed optimisation. Hand-writing is the basic tool, to let everyone communicate. Just like hand-writing is more generally applied than in English, programming is more generally applicable than is today's perception of maths' applicability in schools (though not than maths' actual utility). Whether in geography, economics or science, technical problem solving needs maths and the way you write down and do anything but trivial arithmetic is with programming.

I'm not knocking the new efforts with programming. Far from it. I'm all for getting programming into education under whatever guise is easiest. If making ICT "rigourous" is the politically expedient way, starting there is fine so long as we recognise it just as the start.

It would be folly indeed if in the very country where a mathematician invented the computer and effectively the concept of programming, we should fail to see the crucial integration of programming with maths education.

(Perhaps if Alan Turing had lived longer, computer science would have been generally considered a part of maths, not a separate discipline--just like mechanics or statistics usually are today).

]]>It's amazing how little cross-pollination there is between computational areas. Each area has largely had systems with their own lingo and customs and only the types of computation with which they have become familiar.

We can do a simple demo of graph layout of stock correlation to a group of financial engineers and they are impressed. Well, we do have a very nice implementation, but the algorithms are well established and standard fitment in areas like social network analysis.

Finance is clearly an area where the analytics needs rebuilding, particularly for risk. In truth, it's a mixture between questionable analysis and antiquated reporting. So it's not just straight computation we're talking here either. It's high-level language, instant interactive reporting and linguistic interfaces to name a few. But what it really needs is the coherence of having an all-in-one system with intelligent automation that builds trust.

This is just the start of taking *Mathematica* technology and doing much deeper deployment in finance and other, different verticals.

When governments talk maths, they seem intent on convolving hand-calculating with rigour, rigour with understanding, calculating with numeracy, maths with calculating, rote-procedure learning with the vital conceptual and intellectual requirements of today's real-world maths.

I read the response letter first. It fitted this mould rather too well.

Then I scanned the curriculum itself. It seemed much better. I agree with many problem-solving aspirations and indeed many of the skills cited. I like its not-too-prescriptive approach, as I understand it, giving leeway for lots of different ways to achieve the teaching outcomes including (though this is not specifically cited) basing it on technology. Yes, I'd like this to be much more radical and programming to be included as a core skill, but I understand the difficulty of hard-coding it at this stage. I also understand why there's little reference to technology on the basis that its use isn't an outcome but a highly appropriate (I'd argue essential) tool to reach the outcome--outcomes which I think could have been bolder if computers were the default assumption for calculating.

Where my support starts to diverge is with procedures for multiplying fractions (when did you last use this formally eg. 3/16 x 7/8?) and there's a gaping chasm by the time we get to long-division (ever need to use that?).

Not only are these examples mechanics-led outcomes, not problem-centric (in the end it's problems that maths is there to solve not its own mechanics), but the mechanics in question is in practice obsolete ie. it's not in use in the real world nor do I believe it empowers understanding that is.

This saps student's time, energy and motivation. But I'm concerned about a far more serious problem: the lowly government portrayal of maths.

Should placing long-division or learning your times tables really be portrayed as the pinnacle of achievement in maths at primary school? Worse still, why imply that those tedious procedures are what maths is primarily about?

This is about the worst maths marketing you can do to prospective students--and in the long term to parents. Perhaps it's a good short term vote-winner for some, like brands that consistently do special offers improving sales short-term, but it's not a good long-term strategy for building a quality image of maths in our society or one that's aspirational. It's using long-division as a badge of honour of what the government call rigour when in fact it's a prime example of mindless manual processing.

And more than ever, it presents a broadening chasm between government's view of maths and the real-world subject.

The nub of real maths isn't rote-learning procedures nor does it depend upon them. It's not calculating, but the highly challenging mathematising of ever more complex situations for a computer to calculate, de-mathematising the results and validating their worth. It's creative, applied, powers some of the most successful ideas and developments of recent centuries and can even be fun and engaging!

A useful analogy is with survival skills. In the past your life would depend on rubbing sticks together to make a fire. Now those aren't likely to be life or death. Instead basic survival is how to cross the road or handle money. What are today's maths survival skills? What's at the pinnacle of today's maths?

Instead of rote learning long-division procedures, let's get students applying the power of calculus, picking holes in government statistics, designing a traffic system or cracking secret codes (so topical this month with Alan Turing's anniversary and his computer-based code breaking). All are possible, all train both creativity, conceptual understanding and have practical results. But they need computers to do most of the calculating--just like we do in the real world.

I hope these sorts of examples will all be encouraged under this new curriculum, and crucially that the assessments will highly value the skills they require, utilising computers so problems can be harder, more realistic and far more engaging.

One country will take this computer-based approach first and leapfrog others' technical education. This change will happen. The question is when not if. I worry that the UK won't be in the leading edge of this, but in so many ways it still could be.

Meanwhile, I won't be much help with my daughter's long-division homework. I've actually never learnt how to do long-division; I don't think it's disadvantaged me one iota.

P.S. Paradoxically, I would include times tables in a curriculum: they're still useful. Surrounded as I am by computers, I still find mentally hand-estimating helps me make quick judgements on information and I often do approximate multiplications to achieve that. Of course I could get my computer to do this---but for the moment, it's that bit slower. I do not think times tables give one valuable inherent understanding; but they are useful. Long-division doesn't and it isn't.

]]>It was great to welcome David Cameron, British Prime Minister, officially to open our new Wolfram Centre in Oxfordshire, UK today.

Rather than a traditional plaque unveiling, we went virtual: an iPad button wirelessly firing off a sequence on a nearby TV, the ending "plaque" presenting live data captured at the moment of unveiling--the current weather, FTSE level, star chart and even the PM's age of 16562 days.

More seriously, we talked two topics I believe are key Britain's hi-tech role: making government data truly accessible (to citizens and government(!) alike) and resetting maths education to be computer-based--both more conceptual and more practical.

It's interesting how much the first chimed with the PM's 2010 TED talk about people empowerment in a "post bureaucratic age". It was fun showing how Wolfram|Alpha queries and interactive CDF could serve this agenda (including through Siri), and how the problem-centred approach of computerbasedmath.org might give the UK an opportunity to leapfrog other countries in STEM.

It's clear that the PM is keen to see Britain as a bold new tech and information hub, able to punch above its weight in reshaping the value-chain of knowledge, or what I've described before as the "computational knowledge economy".

In our unusual kind of way, I believe we can contribute unique facets to driving this agenda.

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I really like the badge our team came up with for computerbasedmath.org. If you stand on the power and automation of computers, you really can reach to infinity! Maths has been truely aspirational to world development and so it can be to each individual too.

Our next challenge: make a 3D printout.

P.S. If you like our plans why not show your support by adding this to your website? Available here.

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