### Does My Class Look Big In This?

It's all a matter of style... or is it?

I'm almost certainly going to ruffle a few feathers with this outing, so brace yourself, dear reader.

It will come as no surprise to regular readers of my various musings that I'm a massive advocate of education. It's my deepest passion, and the reason that I embarked upon this project. Above all, I advocate a scientific, evidence-based approach to all things- education included - and this is the motivation for this current outing. This one's taken a fair while to write, not because of the writing of it, but because of the research involved and because I wanted to be sure I didn't miss anything. I'd already pretty much written it and then decided that I hated the approach I'd taken and decided to scrap it and start again only a couple of weeks ago.

There's been a battle raging in educational circles for some time. Those outside education will probably not even be aware of it, but it sometimes gets quite intense, with lines drawn, colours raised and blood spilled - figuratively, of course - and the passion borders on religiosity, which seems to make this a fitting topic for this blog. It's also going to be something of a treatment of some manifestations of our old friend cognitive bias.

There's a common idea that began to surface some time in the early 1900s - shortly after the introduction of the first intelligence tests by French psychologist Alfred Binet - that essentially states that we all learn in different ways. It's an extremely attractive idea, and seems on the face of it to be fairly obvious.

Those who've been paying attention to my output will currently be hearing the awuga waltz playing in their heads, I suspect (and hope). When something's obvious, that's often the first sign that we need to be taking a closer look at it. The name given to the idea that something being obvious means that it must be true is 'appeal to intuition'. I've come across some discussion of whether this should be considered a fallacy, but all such discussion is specious, as a brief exposition should demonstrate readily.

It's intuitively true that time runs the same for every observer except that, if this were actually true, satellite navigation would be a pipe dream. It's intuitively true that something can't be in two places at once except, if that were true, the technology I'm employing to share my prattlings with you today wouldn't even rise to the level of fantasy. It's intuitively true that I can't walk through a wall - and indeed I've never managed it yet- except that, if this were actually true, we couldn't exist, because fusion in stars wouldn't occur, meaning that elements heavier than beryllium couldn't be synthesised.

So, returning to this idea, popularly known as 'learning styles', the initial inklings followed directly from Binet's work, first by Maria Montessori, well-known for the Montessori Method of education. Montessori began to develop specialised educational materials based on her work with children with special educational needs. After initially gaining some traction, not least via an association with an Italian Baron and Baroness, this idea fell into disfavour, festering in the recesses of public consciousness.

Then, in the 1950s, it began to resurface. By the 1970s, it had begun to find favour globally until, by the early noughties, it was fairly widely accepted.

I don't intend to delve into the history of this idea in this post (and probably not in any other), not least because it would make this offering massively unwieldy, so I'll leave it there, noting that the wiki on this topic is pretty comprehensive.

So what is 'learning styles' really?

Broadly, it's the notion that we all have particular preferences for the way that educational material is presented, and that these preferences translate to differentials in the way that effective learning is achieved. The idea is that some of us learn best when material is presented visually, others learn best from text and still others learn best from tactile presentations or by doing.

It's certainly a meretricious notion, but does it hold water?

While it's certainly true that students will tend to latch onto those modes of learning that they feel most comfortable with, it's less clear that this is actually a good way of structuring learning, as a little careful thought should elucidate, so let's pull it apart and give it some real consideration.

The best place to start is, I think, a simple principle that can be presented in many ways. Such things aren't that easy to find, it turns out, but we're quite lucky because we have one to hand that we've encountered a few times hereabouts, and it's one that is familiar to schoolchildren everywhere: Pythagoras' Theorem.

We'll begin with presenting this as simple text in the way that it was first presented to me.
The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.
This seems to be fairly straightforward, albeit with a couple of undefined terms. Of course, once you know that a right angle is an angle of 90°, or a square angle, and that the hypotenuse is the side of the triangle opposite the right angle, it becomes fairly clear. However, it might not be visceral, so let's try a more mathematical approach.

Here's the same theorem presented as a mathematical equation: $a^2+b^2=c^2$

Interestingly, to the non-mathematician, this conveys less information than the purely textual example given above. To somebody comfortable with mathematical presentations, however, this conveys every bit as much information and, moreover, requires no additional information, and contains no ambiguous or ill-defined terms.

As an aside that some might find interesting (who am I kidding, anybody who really finds this interesting probably already knows it), this is also the root not only of how relativity theory works, but is also the foundation for Fermat's Last Theorem, one of the great unsolved problems of mathematics, until a solution was found by Andrew Wiles*.

OK, so it could be reasonably argued that both of those presentations are textual, albeit in different languages. How about a visual presentation? Here's one that we've used a couple of times before:

Well suddenly, it all becomes clear, so it must be the case that we ALL learn best from visual representations, yes?

Let's circle back to that, because doing so will be instructive for our purposes.

Let's finally do something that's pretty much impossible when your medium is a two-dimensional surface, but which we must attempt to get past, namely the kinaesthetic presentation. Let's see what we can learn by doing:

Of course, this is still a visual presentation but, with a little imagination, we should be able to get the gist of it. We could easily imagine building this from wood and glass and filled with liquid so that it could be physically turned. Indeed, this has been done:

This covers a fair bit of the content of learning styles, and it can seem that there's something here for everybody, no matter what one's preference for information acquisition might be. It would be hard, on the basis of the foregoing, to see what objections might be raised against it.

That fact alone should serve as a warning. As we've discussed in quite a few different contexts, the things that should be most vigorously challenged are those that we think are obvious, and we should think hard about whether our underlying assumptions are actually correct.

Let's jumble them up a bit, and see where that gets us. Let's start with the purely visual iteration.
What can we learn from this alone? The short answer is absolutely nothing. Without the context given by the textual example that preceded it, this only works as a way of learning Pythagoras' Theorem if you're already aware of the underlying geometry. It isn't clear, for example, that the squares on the adjacent sides bear any relationship to the square on the hypotenuse, let alone that their areas are equal. If you don't know what's going on, you can learn nothing about the length of each side based purely on this image. It requires additional information.

How about the mathematical example? Let's see it again. $a^2+b^2=c^2$

What can we learn here? Again, in isolation, and without context, we can learn nothing. It requires additional information.

OK, so what about the text?
The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.
Have we learned anything useful from this on its own? Of course not because, as we've already highlighted, there are ill-defined or undefined terms in there and, absent those definitions, which must themselves be learned, we can't hope to make head nor tail of this.

Only one of those representations, on its own and without reference to any outside information source, actually gives us the meat of Pythagoras, namely the kinaesthetic, or 'touch-based' approach. From that, it's really clear what's going on. It's obvious to pretty much anybody, yes? So must it be, then, that we all have the same learning style?

Let's think about it a bit more carefully.

How about if we combine these modes. We can take the prosaic statement of Pythagoras' Theorem  and combine it with, say, the algebraic formulation, and it becomes reasonably clear. Or we can take the algebraic formulation alongside the visual geometric representation and the information conveyed seems complete. The same is true if we combine the text and the geometric representation.

So what about if we combine all of the above into a single unified lesson on Pythagoras' Theorem? The answer should be reasonably clear. If we were to structure all of this information into one lesson, all the cues would be there, and pretty much everybody should be able to walk away understanding how it works, and able to see a route to calculating the length of an side of a right triangle given only the lengths of the other two sides. One could even see that, given only the length of one side and the one none-right angle adjacent to it, one could construct a complete right-triangle.

I'm once again aware that this is turning into a somewhat lengthy offering, so I'll finish with an anecdote.

I was watching the latest episode of one of my favourite shows the other night, Suits. This isn't my usual kind of thing, and I tend to stay away from this sort of drama, but the central character has an eidetic memory, and this is something that's always fascinated me (were I to choose one trait allowable by evolution, this would be it), so I have a tendency to watch anything containing such a character, because it's always interesting to see how it gets treated and how the memory is used in the plot.

It's a mildly entertaining legal drama set in a high-power legal practice in New York. There was a scene in it in which an associate gets bawled out for handing off a piece of work to a younger associate. She raises the excuse that the younger associate is better at the kind of work this involved. It was pointed out that the reason for giving her the work to do in the first place was that she needed to improve these skills.

And this brings me to the central point. All of these 'learning styles' are skills. Indeed, learning is itself a skill - something that must be learned. Different ways of absorbing information must themselves be learned. If anything, the notion of learning styles should be seen as a guide to how NOT to do it. If you have a student whose learning style is visual, it's actually counter-productive to rely solely on visual learning cues, because this encourages cognitive pathways that exclude other routes to learning, and are ultimately destructive to the learning process. Far better to focus on the kinaesthetic and text-based approaches, using visual cues only as a secondary or tertiary approach as an additional learning aid. The same is true of the other styles, in that the preferred approach is almost certainly best used as the last resort.

Most of my readers know me as somebody who can distil very difficult concepts into simple, accessible terms. This can be quite a challenge, especially when dealing with cutting-edge physics and other lofty topics. In reality, most of those topics are best presented in a very specific way for clarity, in the language which best describes the processes involved; mathematics.

Mathematics is difficult for many, and their eyes glaze over at the first sign of an exponent. Indeed, I used to find this myself, but forced myself to keep plugging away, even if it took me a week to get through the meat of an equation so that I could solve it. It's still a slow process for me now, but the depth of understanding I have of the processes I've had a fairly visceral grasp of for years is increased by orders of magnitude just from understanding the structure of the equations and how the quantities in the universe are related numerically. I know from discussions I've had with people who've questioned me on some of what I've presented in this blog that, as simple and straightforward as my presentations are, there are still who firmly grasp the wrong end of the stick. There's only one way to remove ambiguity, and that's to move away from natural language entirely, and cast the entire discussion in terms of the mathematics. In other words, there are some kinds of information for which there is only one rigorous way to present them.

Now, I have no doubt that, in some settings, especially in settings in which special educational needs are manifest, that the framework of learning styles is useful, and that there are students who benefit from it greatly, in that they can achieve a level of education that might have been denied them via more general teaching methods but, as a general teaching method, it sucks.

More importantly, from the perspective of somebody interested in ensuring that evidence is the final arbiter of anything taken as fact, there is no evidence beyond the entirely unreliable anecdotal that it's anything other than an attractive cop-out.

Of course a student who likes visual representations is going to say that they found that learning via visual cues was easier. This is really not the point. I'm perfectly happy to accept as true the idea that different people find learning more readily when the material is presented in different ways. However, the simple fact is that, as a general approach to learning, it's counter-productive. Offering a sop in this manner closes the student off to the acquisition of skills in other areas and, for those kinds of information that lend themselves to only one way of being unambiguously presented, it closes them off from those subject areas entirely.

All of these are skills, and can be acquired.

If you're really interested in what the evidence says about effective learning techniques, I recommend keeping abreast of the research. Here's a nice presentation from the Learning Scientists.

Nits and crits welcome as always.

*Fermat's Last Theorem is a famous problem in mathematics. It states that there are no solutions to the equation $a^x+b^x=c^x$ where $x$ is any integer greater than two. It was one of the famous 'Hilbert' problems - named after David Hilbert, the German mathematician who listed them - a list of 23 of the great unsolved problems of mathematics compiled in 1900. This problem was ostensibly solved in 1994, although my good friend Phil, author of Definitions and Axioms, had this to say back in 2015:
Wiles' proof can only be checked by a tiny handful of mathematicians on this planet. That puts it very much in a "doubtful" place by the standards of pure maths. I doubt the average professional number theorist could check Wiles' proof.
Wiles won the Abel prize for his proof in 2016.

It's fairly certain that Pythagoras had little to do with this theorem. It was certainly understood as a general principle before Pythagoras, but the formal description of the principle comes from the Pythagorean school.