Conrad Wolfram




PISA results: Let's win on the right playing field, not lose on the wrong one

Today's maths PISA results are predictable in the successes that many Asian countries show and the mediocrity of many of the traditional Western countries--like the UK. 

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.


Excited: Raspberry Pi gets free Mathematica + Wolfram language

I was very excited at our CBM summit this morning with Eben Upton to announce that Mathematica will be bundled on the Raspberry Pi computer for free, and so will the new Wolfram Language--also announced today.


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).



Collaborating with UNICEF: next month's CBM summit on fixing world's maths

Fixing maths education is becoming ever more central to individual life-chances and our societal needs.

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 :-).


Maths v. Music Education

I was debating Computer-Based Maths education (CBM) with a sceptic before the summer and he brought up the analogy of music education to support various claims he was making of maths.

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?


Announcement: our first CBM country

I'm very excited to announce that 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.