Can a Maths Ed Theory Give Back More Than It Takes?

A year or so ago, a decades-long friend and colleague reappeared. My friend also has a strong interest in maths ed, although she takes the “Let’s all be friends” approach. Readers of this blog know well that I’ve given up on that, but still my friend and I can argue amicably about this and that. In particular, she took some issue with my “all modern maths ed sucks” post. While conceding that most educational research is bad, she was unwilling to write off the discipline entirely and she suggested a few things for me to read. I gave them a semi-decent try, and my response was “meh”. While the stuff she suggested was mostly reasonable, or even good, I felt it was, at best, addressing third order issues. One article, however, has had me pondering a little, and I thought it may be worth sharing.

Andrea diSessa is a big shot in education research, particularly in science education and maths education. The particular article that my friend suggested is from the proceedings of a 1991 conference on the psychology of mathematics education, and is titled,

If We Want to Get Ahead, We Should Get Some Theories

DiSessa’s title is clear enough, and his abstract makes it clearer:

In education, or in the learning sciences generally, theory is in a poor state. We have not reached deep theoretical understanding of knowledge or of the learning process, and it is important that we recognize this. Even more importantly, our community does not seem particularly intent or armed to change the situation. This paper is aimed at raising the issue of intent, arguing for new dedication toward theory. It is also aimed at a modest contribution to our toolkit for a more theoretically attentive practice of education research.

Theory is not my thing, and I didn’t read DiSessa’s paper closely. But the paper is well written and it seems interesting enough for the genre. DiSessa, who has a PhD in physics, spends a fair amount of time discussing how theories in physics work, and whether and how the nature and the success of such theories might shed some light on the way theories of learning might work.

For me, one passage of DiSessa’s article stuck out as worth pondering (to the extent any of this is worth pondering). DiSessa frames his quest for theories of learning with explicit reference to his background in physics:

I take three things from my experiences with physics. Each of these provides a “place to look” and a “judgment to make” with respect to the state of theory in an empirical science.

It is the third of these things that struck me.

Respectable theory, when we get it, cleanly transcends common sense.

My last point of extrapolation from physics to our expectations for theory in education really follows from discussion of the above two points. Unless we can unambiguously point to how we have transcended – in generality, precision, clarity, and justifiability – the intuitive sense of mechanism we all build in daily life observing and thinking about psychological matters, we just won’t have adequately prepared theoretical ground. I’ll pick one focus for this exposition, but I think the point is much broader. Commonsense vocabulary just won’t do the job of providing the technical terms of a theory of learning. When we stop with “beliefs,” “knowledge,” “concepts,” and so on, even with a few phrases of elaboration, we are on extremely shaky ground.

To put an edge on this, physics theorizing has always involved ontological innovation. The “force” in Newton’s theory is a new entity that simply does not exist in common sense. Even mass took on a much refined interpretation to make sense in that theory. More evidently, quantum wave functions did not exist before quantum mechanics. My presumption is that we will not have adequate theoretical purchase on learning until concepts, facts, beliefs, skills, and all the rest of our common sense about knowledge and learning become reinterpreted within a fabric of more precise and less intuitively loaded terms. Please, do not mistake: I’m not appealing for obscure language, or for proliferation of new words. I’m appealing for the clarity that can come with ontological innovation.

This seems a very clear and useful phrasing of the challenge. Does any theory of learning, does maths ed theorising in any part or as a whole, transcend common sense? Does anyone even try? If the claim of current practitioners is that maths ed theorising gives back something larger than the ideas it employs, then what is it that it gives?

DiSessa wrote his paper thirty years ago and perhaps the current education guys, even perhaps DiSessa, believe that respectable theories of learning now exist. Perhaps the Brain Boys consider that they have it all under control. If anyone can suggest a theory of learning that answers DiSessa’s challenge, can suggest a theory of learning that gives back more than it takes, I’m willing to listen and to read.

But of course I am as sceptical as ever. From what I’ve seen learning theory is still, at best, truisms. More often, it is falsisms.

33 Replies to “Can a Maths Ed Theory Give Back More Than It Takes?”

    1. Thanks, Greg. I had already noticed your paper, as a missile in one of your recent amusing-absurd Twitter battles. Since the paper is taking a whack at the inquiry guys, I figured I didn’t have much to learn from it, but I’ll give it a look. I imagine that you are correct, that I’ll end up putting it in the Truism Box.

    2. Hi, Greg. I finally got the chance to read your article. Yes, it all seemed like common sense to me. Still, itemising the common sense is preferable to the inquiry guys winging it. But it was interesting to read how you guys do battle. The article was simultaneously polite and pretty snarky. Not a bad achievement.

  1. An interesting read, Marty. No method or theory gives back more than it takes, at least in the short run. However, eventually: “All knowledge that arises from imperfect circumstances tends to perfect itself. It arises from sense impressions but gradually becomes an object of our contemplation and finally enters the realm of intellect.” – ancient Greek scholar Eudemus of Rhodes.

    1. I don’t see how “No method or theory …” helps, or can be accurate. Theories of physics are successful in ways that theories of learning, if they exist, are not.

      1. I never said it doesn’t help. I said: “No method or theory gives back more than it takes, at least in the short run.” We live in a world of uncertainties/frictions. No system can ever give back more than is entered into the system.

  2. Why should ontological novelty be an essential feature of a productive theory? I guess it usually is, but not really clear why it has to be (think this may go down the rabbit hole of Universals and natural kinds)

    For what it’s worth, I think we probably do need an ontological revolution: all the empirically dubious entities that both progressives and traditionalists are drawn to (concepts, beliefs, ideas, thinking skills, items of ‘knowledge’, etc) need ditching.

    Academic learning is about language, and nothing else. That is to say, the induction of students into vocabulary, syntax and text conventions. These are the only things you can actually ever truly observe anyone using correctly, or not.

    Of course, teachers may well have other roles – sustaining a liberal culture of curiosity and tolerance, mentoring students through character development etc, but language is the only thing that actually exists when it comes to the academic curriculum.

      1. When you say ‘nuts’ I shall assume you mean ‘transcends common sense’, and that I have therefore pleased you.

        Not really sure where to start, but let’s assume that mathematical instruction (to take just one domain of academic learning) is basically some combination of three elements: teacher modelling, student practice and teacher assessment of student performance. Each of these three things essentially involves some kind of activity, and the only possible kind of activity that we could be talking about is linguistic. When someone models a mathematical operation, they are modelling the use of particular words, and the rules around how those words be used together – their syntax. That’s what the students see and hear – words and clauses. They have no privileged access to the teacher’s mind (even assuming we know what that is), just a bunch of words being spoken and written (‘gradient’ ‘x axis’ ‘function’ ‘three’ etc), along with another, perhaps more familiar set of words and phrases that explain them (things like ‘this means …’ or ‘this is the opposite of…’). This is also true when the student practices the operation, or when a teacher assesses their competence. It’s all about words – what they mean, and how they relate to one another.

        Perhaps you are mystified by the idea that maths is a linguistic activity – I can imagine that some maths teachers might feel they’re doing something different, but I assume you’re at least familiar with this broadly accepted notion in philosophy of maths. Mathematical activity is a series of declarative clauses, mostly making identity claims, with the odd ‘if’ and ‘therefore’ scattered around. It really makes no theoretical difference if mathematicians have invented a load of funny squiggles to write it down. It’s still a genre of language.

        You asked for a theory that gives more than it takes. My hypothesis deprives us of cloudy, uncountable, unaccountable, unobservable notions like concept, belief, meaning etc. And what it gives us is that it deprives us of cloudy, uncountable, unaccountable, notions like concept, belief meaning etc.

        I should add that what is being offered here is a theory of learning – it can remain totally agnostic about whether there are such things as beliefs, concepts etc. I for one am quite sceptical about the more reductionist elements of the cognitive science crowd. But the point is, mental concepts are irrelevant for teaching. All teachers – and school systems – need worry about is whether they have successfully helped students to use certain words and text types appropriately (remembering my comments above about other, really important, cultural and mentoring roles that teachers also have, but which are not specifically academic).

        1. “But the point is, mental concepts are irrelevant for teaching.”

          Conflicting with common sense is not the same as transcending it.

          1. Struggling to believe that you Marty can make a reference to ‘common sense’ without at least some level of irony – perhaps you missed it in mine. Or maybe you think that the modern world is such a disgraceful den of idiots that these kind of folk are starting to appeal – https://en.wikipedia.org/wiki/Common_Sense_Group

            As I suspect you know, for many serious psychologists, philosophers, cognitive scientists etc today, mental concepts are an illusion, only waiting for empirical dissolution – maybe somewhat like rainbows.

            A theory of teaching and learning that I would like is one which deals with entities that I as a classroom teacher have a realistic chance of observing – hence my attraction to words, clauses and texts as the objects of such a theory, and to their correct use as the definition of learning and teaching success.

            I have no more been able to observe a student’s ‘mental concept’ of three than I have been able to measure their ‘cognitive load’, witness their ‘knowledge schema’ or meter their capacity for ‘critical thinking’.

            But maybe Maths PhDs have superpowers us ordinary mortals lack? Some kind of special platonic x-ray vision? Or, maybe, commutative sense?

  3. I’m not sure I have understood the intended thrust of this thread, so forgive me if this is way off script.

    When exactly is “theory” appropriate, or useful?

    My take on this is that in most fields of human endeavour, “theory” is very much a Johnny-come-lately, arriving late in the day, after centuries of inscrutable, often misguided, sometimes semi-magical *craftsmanship* – rooted in hard graft, dead ends, and accident.

    Think about smelting iron or steel or brass, or whatever, in ancient times. Some (maybe all) practitioners down the ages had their own *personal theory* about what was going on “underneath the hood”. But they were craftsmen, whose primary goal was – and whose social status depended on their ability – to produce something that worked, or that improved on what had been done before.

    Maths teaching is currently in a similarly primitive state. We remain craftsmen. We may each know slightly different ways of proceeding – and we may trust our own approach because (a) we are familiar with it; and (b) it “sort-of-works”. Yet even the most effective among us would hesitate to claim that we really understood.

    Since around 1975 we have we begun to accumulate a new cadre of “maths education researchers” (familiar with the research literature, but often with limited classroom experience). For 20 or so years, this group remained fairly close to the classroom and to the craft of teaching; but since 1990-2000(?), the two worlds have increasingly drifted apart.

    I have the impression that their professional status now obliges researchers to work within a declared “theoretical framework”. Like the dissenting theologies of the 17th century, these theoretical frameworks come in different flavours. Yet unlike those theologies, writing up a study within a declared theoretical framework does not imply serious “commitment” to that framework! So researchers are free to adjust to changes in the prevailing direction of the wind. (I attended nine successive ICMEs 1976-2008, and was often astonished to find that the dominant theoretical framework had changed, without any obvious discussion as to why, or what the consequences might be.)

    From here is seems that modern (western) education researchers are no longer like the old practitioners – looking for rules that work, even if they have as yet no clear theoretical underpinning. Neither are they like modern physicists – embracing a theory only when it genuinely succeeds in successfully explaining why-things-are-as-they-are. Instead, educational theories appear more like fashion accessories – styles that are briefly shared with an identifiable group of others, adopted to blend in with whatever trend is in the ascendancy. The old *craft* tradition of using one’s insights to produce “something that works”, whose efficacy can then be compared with rival approaches (as with textbooks) also seems to have died the death.

    All of this is done at massive public expense – perhaps because those who allocate funds have been told that maths (and hence maths teaching) is important, but do not have sufficient confidence to cut through the “mathematicalcrap”.

    1. Wow. Accurate and fair, but not something I suspect too many in either “world” would have suggested as a summary of the way the current system (in Australia at least) finds itself struggling.

      Some of the very best teachers know very little (and care much less) about educational research and probably rarely utter the word “pedagogy”.

      If teaching were (Marty, did I get the grammar right there?) treated as an Art rather than a Science… does that kind of veto much of the Maths Ed Theory?

    2. Thanks very much, Tony. You’ve painted as well as one can how and why the world of edutheory is now so entirely nuts.

    3. I’m late to answer (hadn’t checked the blog in about a week), but thank you very much for this post Tony! I’m thinking of going back to study mathematics education (in part because there might be a job at the end for me if I do so; I already have a doctorate in mathematics and I teach mathematics at the post-secondary level but there is no permanent position that is open for me), and what you describe rings very true to me when I look at what mathematics education academics do. Namely, that there used to be a strong connection between the practice of teaching mathematics and mathematics education, but that much of this connection has now evaporated. (If I study mathematics education, my goal would be to produce something that can be of use to actual teachers.) Also, that mathematics education research, being social science research, is now intended to take place within a theoretical or conceptual framework, but that the connection between the framework and the research is sometimes unclear. This is what Greg Ashman describes as education academics smoking Gitanes while quoting French philosophers; I think Greg displays a bit of francophobia there, but he also touches on something real. And finally, that much of education research, including mathematics education research, seems to be about fashion; I believe that right now STEM education as well as the use of information technology are in vogue in mathematics education. (Just this Monday I attended a seminar about the use of 3D printing for teaching geometry to primary school students. I’m not sure how useful that will prove to be, but this is what interests many mathematics education researchers these days.)

      This said, I do believe teaching is not merely an art, and that there is a science to it, that can be discovered through research. I also think that “mathematics education”, and the research about it, includes a lot of things, some that are much closer to the actual act of teaching, and therefore cannot be said to all “suck”. I consider the analysis of textbooks and other teaching material that is done here on Marty’s blog to be part of mathematics education, and therefore I view the blog as a mathematics education blog and (though he may very well yell at me for saying so) consider Marty to be a mathematics education specialist.

  4. Many theories rely on the principle of induction; that is the future will resemble the past. An experiment is conducted and, if we assume this principle, then the results of the experiment become a theory. However, it seems that there is no *logical* reason for assuming this principle.

    1. It depends which lens you choose to use to look at the history of science. Bacon was a big believer in induction basically being “what researchers do” until Karl Popper wrote “Conjectures and Refutations” (a paper that I believe is still relevant today) and then Thomas Kuhn and Imre Lakatos tried (and had mixed success) to determine what was actually happening in research circles rather than what should ideally be happening.

      Feyerabend basically concluded that all scientists do their own thing and the so called “scientific method” does not actually exist.

      Still… as interesting as all that may be, I’m much less clear on how helpful it is to the crafter/teacher who just wants tp do their job and do it well.

          1. The point is that the principle of induction if often invoked, albeit implicitly, even though there seems to be no logical basis for it.

            1. My point is that I don’t treat you as if you were a terrapin, even though there seems to be no logical basis for it.

  5. During the time I played in the Maths Ed space, I could not get past the truisms that 20 years of teaching had made clear to me. Thus I rarely understood the point in the papers I read, nor the requirement to create/work within a theoretical framework when writing papers. I could not see how it was going to help me, or anyone, do a better job. It felt totally un-natural.

    Highly successful teachers have a strong idea how they came to understand whatever piece of mathematics being taught, and they use that know-how to craft questions that lead students to have similar thoughts that lead them to understanding.

    If a teacher does not understand the mathematics, and have a decent sense of how they came to understand it, then …

    My sense is there are numerous ways to come to understand any given bit of mathematics. Which complicates the process of overly systematizing the learning of mathematics, and makes it fun!!

    I personally cannot understand how theory can transcend this sort of common sense.

    Skemp, in the 70s, was good to read.

    1. Thanks, ST. I also find it difficult to imagine even how a theory of learning might transcend common sense. But, as Tony notes, we’re in the Stone Age (or Dark Ages?). Perhaps in a few centuries, or millennia, it’ll all be obvious.

  6. Not sure how much I believe it myself, but I think there’s a case for the research being done on mixed and spaced practice. It is common-sensical that reviewing material a while after not engaging with it will help with retention; it is common-sensical that mixing up the kinds of tasks one is learning during a practice session will help with retention and application. But perhaps where this research transcends common-sense is in helping to more precisely and systematically detail (i) how the timing of a spaced practice schedule affects the amount or level of retention, (ii) how much variation, and in what dimensions, of mixed practice affects retention and application. So I see it as more about taking a common-sensical idea, but fleshing out the details (and this “fleshing out” is not something that can be done effectively just by armchair reflection on common-sensical platitudes about learning) so we can get a more precise idea of how and when it works.

  7. Tony has summarised this beautifully – the Emperor’s New Clothes indeed!

    My take is likely too simple, but I see the fact that we KNOW we have no theory of learning as helpful. It reflects a Dylan Wiliam comment that I love that we really don’t know how learning takes place. It cuts away from all those who say ‘teaching MUST be done this way’. And lends support to teachers having autonomy to respond to situations. Of course it’s not completely open slather …..

    My personal very simple and not fleshed out theory of teaching is that if something has been done for a long time, there is likely wisdom to it (as per Nassim Nicholas Taleb). So we fundamentally do what has been done over millennia, ie what is now called direct instruction, but for hundreds of years was just called teaching (and use rote memorisation etc).

    At the same time, teaching is an incredibly complex activity so I respond to individual classes by trying out things around the edges and seeing if they appear to work. To date this has led me to the conclusion that computers are a bit helpful for demonstration but very much a net negative for student use, among other things.

    In the absence of a theory, common sense based upon experience seems as good a guide as we will get. We are indeed craftspeople, but that can’t be admitted – it would undercut so much of the universities’ power base.

    By the way 1, this podcast with Dylan Wiliam discusses his attitude to PISA (note I find the interviewer’s questions less helpful than Dylan Wiliam’s answers)   https://www.aei.org/podcast/dylan-wiliam-on-pisa-assessment-and-de-implementation/

    By the way 2, my Dip Ed had the wonderful Dick Gunstone teaching us just how hard in Physics it was to teach something that transcended common sense. This included techniques on how to actually help students change their underlying mindsets. Actually understanding how Newton’s laws transcend common sense is something I suspect many Physics teachers may not really grasp.

    1. Thanks, JJ. Teacher autonomy is not an automatic good. Plenty of the decline in Australian education came from the glorification of teacher autonomy.

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