Conrad Wolfram’s silver bullet

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Someone popped up on my timeline to ask me about Conrad Wolfram’s computer-based maths. I had never heard of it before but it is apparently becoming popular in Australia. Despite my ignorance, I had a funny feeling as to what this might look like and a suspicion as to why it may be popular.

The first item I found was a TED talk; never a good sign. It is full of beliefs and assertions such as, “I believe that correctly using computers is the silver bullet for making math education work.” It is mainly argument from direct experience and utility. As ever in these polemics, mathematics is not treated as a serious academic subject but rather as a mundane tool to be judged only by its usefulness for solving problems. Any appeal to evidence from cognitive science or educational research is entirely absent. Instead, we have a lot of truthiness:

“And one of the reasons it’s so important — so it’s very important to get computers in exams. And then we can ask questions, real questions, questions like, what’s the best life insurance policy to get? — real questions that people have in their everyday lives.”

Why would we ask this? Because it kinda feels like the right thing to do. It’s what Wolfram reckons. That’s about it. It’s like saying, “Forget Shakespeare, kids, your English exam is going to involve writing an email complaining to your broadband provider. Forget Beethoven, your music exam is going to involve advertising jingles.” I’m sure people say things like this all the time. After all, everyone and anyone feels qualified to pronounce on education. But that’s not enough if you are calling for schools to change their methods. You need to back up your claims. And the evidence suggests that you cannot gain a good understanding of mathematics through this approach.

One reason is explained by cognitive scientist Dan Willingham in an article on flexible knowledge. When we first learn something new, it tends to be locked to the specific contexts in which we learnt it. In order to be able to abstract that knowledge, we need to see it in a number of different contexts. Mathematics is the ultimate abstraction and that’s what makes it so powerful. If we only ever perform our mathematics in mundane, everyday contexts then we risk two things; we will overload working memory with contextual detail and we will lock that maths to the context in which we learnt it. This is why problem-based learning and its variations are so ineffective.

Wolfram has a black box approach to calculation. He has invented a model of ‘maths’ that consists of four steps: define questions, translate to maths, compute answers and interpret results. Apparently, the ‘compute answers’ bit can be done by computers these days and so there is very little need for students to be able to do it. Effectively, this step becomes a black box – you feed it input and collect output. Instead, students should spend time on the other components like thinking of really good questions etc.

But how is this supposed to work? How are students intended to translate something into maths when they don’t know the maths needed to compute the answers? How are they meant to interpret the results? You cannot just leave a great big sink hole in the middle of mathematics understanding and assume everything will be fine. Maths is all connected. The best evidence we have is that conceptual and procedural knowledge develop in tandem; the one supports the other. This has been a substantive field of research and yet this argument is dismissed in the computer-based maths ‘brochure’ in another folksy analogy about car mechanics and photography. What Wolfram seems to miss is that mathematics is not simply a tool like a camera, it is a form of knowledge, organised as such in the mind. You think with it.

Precisely the same logic is needed to translate problems into maths and to interpret the results as is needed to compute the answers. Computers can be of enormous help in performing large numbers of calculations but only when programmed by someone who understands the maths. As part of my university course, I had to programme a mainframe computer to calculate the wavefunction of an electron in a potential well. I literally had to tell it what to do and so I had to know the procedure that I wanted it to perform. The programme took a while to debug and, again, I needed to know the maths in order to do this.

As is often the case, I find myself wondering whether Wolfram has read the research and disagrees with it or whether he is not even aware of the evidence that refutes much of what he claims. The trope about problem-based learning appears so often that I wonder whether it occurs to people independently as some kind of flash of insight; individuals who then think that they are the first person to ever have the idea. You can sense the missionary zeal. ‘Why haven’t educators though of this themselves?’ they must wonder, before concluding that we must be a bit limited.

In this case, I am not sure that Wolfram is that familiar with the research. For instance, it is now extremely well known that there is no evidence for the utility of various learning styles theories, and yet, in answer to the question, “Not all students learn in the same way. How does CBM cater for diversity?” the computer-based maths brochure reassures us that, “In CBM, we include a range of teaching and learning styles in the lessons, from data gathering to presentations, from individual work to group tasks, from typed answers to diagrams, posters and videos.” So there’s that.

Wolfram’s computer-based maths is just the latest in a long line of revolutionary approaches that is at odds with the scientific evidence. The difficult part is in explaining why such reheated ideas gain traction in the first place. Why do they end up on TED talks? Why do Australian administrators reach for them?

In my view, it’s the sales-pitch. Maths is hard. Many people struggle with it and often much of the blame lies with poor teaching. The solution, of course, is a more rigorous, research-informed explicit approach rather than this fashionable nonsense. But if you did fail in maths at school, these narratives are attractive, even if it takes quite a stretch of the imagination to believe that solving mundane problems about insurance would have been the ‘silver bullet’.

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5 thoughts on “Conrad Wolfram’s silver bullet

  1. Bart says:

    I remember the TED talk many years ago but put it down to a businessman wanting to promote his product, mathematica.
    I agree with you. Lot’s of garbage about authentic tasks. Also quite a bit of misrepresenting what goes on in modern classrooms eg. who hasn’t made the link or given a question on passwords when covering permutations.
    What I find the biggest disconnect is the insinuation that understanding doesn’t matter just solving problems (and I’m not convinced it is even the best approach for that).
    I have taught a lot of kids to program. Both you and Wolfram reference abstraction but where I agree he gets it wrong is thinking that the least complex parts can be bypassed (the black box). From my experience that fundamental understanding is key to good programmers and software engineers. Some kids can ‘mod’ a game and create something that looks great but all they have done is change a few variables. If they are required to change something at a lower level of abstraction, they are completely lost. In fact some of them have very little idea of what is possible and how you would even go about solving these ‘sub’ problems if they haven’t spent time learning programming fundamentals.
    To continue with the abstraction argument. If I need to learn some complicated piece of maths that I don’t fully understand for a problem I have I can look up the parts that help me understand it. I can keep going back until I am at concepts that I fully understand. If this level and the level I need a too far apart I won’t be able to understand the original piece of maths. Maybe I can still use it but I don’t understand it (and isn’t that what education is about??). I think this is the issue with a lot of project based learning with novices. Their understanding of concepts doesn’t seem to actually seem to increase – I could ask an even simpler question in the same vein and they still have no idea.

    • ‘Problem solving’ is often used as the sum total of the object of education. Few shills, such as Wolfram talk about the benefit of deep education (the only term I can think of for depth of understanding in, for instance, maths), but the real benefit of understanding is finding problems. Solving them is relatively easy in most contexts. But enabling children to learn how to ask questions and analyse situations to find what the problem is is the essential first step in everything from business to research and down to party planning.

  2. David F says:

    Administrators reach for this stuff because they don’t know any better themselves. Their conferences are filled with TED-talk-like presentations, where they get excited about innovation X and are always happy to listen to something claiming to be a “silver bullet”. Then, they get frustrated when they come back to their schools and are met with eye-rolling teachers.

  3. P Wing says:

    He isn’t actually a mathematician, though, is he? He just has an undergraduate degree in natural sciences and mathematics, from what I can see.

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