traditional vs progressive: mathematics, logic and philosophy meet the real world

For thousands of years, human beings have been trying to figure out why the world they live in works in the way it does. But it’s only been in the last five hundred or so that a coherent picture of those explanations has begun to emerge. It’s as if people have long had many of the pieces of the jigsaw, but there was no picture on the box. Because a few crucial pieces were missing, it was impossible to put the puzzle together so that the whole thing made sense.

Some of the puzzle pieces that began to make sense to the ancient Greeks involved mathematics – notably geometry. They assumed that if the consistent principles of geometry could be reliably applied to the real world, then it was likely other mathematical principles and the principles underlying mathematics (logic) could too. So philosophers started to use logic to study the fundamental nature of things.

Unfortunately for the mathematicians, logicians and philosophers the real world didn’t always behave in ways that mathematics, logic and philosophy predicted. And that’s why we developed science as we know it today. Scientific theories are tested against observations. If the observations fit the theory we can take the theory to be true for the time being. As soon as observations don’t fit the theory, it’s back to the drawing board. As far as science is concerned we can never be 100% sure of anything, but obviously we can be pretty sure of some things, otherwise we wouldn’t be able to cure diseases, build aircraft that fly, or land probes on Mars.

unknown unknowns

Mathematics, logic and philosophy provide useful tools for helping us make sense of the real world, but those tools have limitations. One of the limitations is that the real world contains unknowns. Not only that, but as Donald Rumsfeld famously pointed out, some unknowns are unknown – we don’t always know what we don’t know. You can work out the unknowns in a set of mathematical equations – but not if you don’t know how many unknowns there are.

Education theory is a case in point. It has, from what I’ve seen, always been a bit of a mess. That’s not surprising, given that education is a heavily derived field; it encompasses a wide range of disciplines from sociology and politics to linguistics and child development. Bringing together core concepts from all relevant disciplines to apply them to education is challenging. There’s a big risk of oversimplifying theory, particularly if you take mathematics, logic or philosophy as your starting point.

That’s because it’s tempting, if you are familiar with mathematics, logic or philosophy but don’t have much experience of messier sciences like genetics, geography or medicine, to assume that the real world will fit into the mathematical, logical or philosophical grand scheme of things. It won’t. It’s also tempting to take mathematics, logic or philosophy as your starting point for developing educational theory on the assumption that rational argument will cut a clear path through the real-world jungle. It won’t.

The underlying principles of mathematics, logic and philosophy are well-established, but once real-world unknowns get involved, those underlying principles, although still valid, can’t readily be applied if you don’t know what you’re applying them to. If you haven’t identified all the causes of low school attendance, say, or if you assume you’ve identified all the causes of low school attendance when you haven’t.

traditional vs progressive

Take, for example, the ongoing debate about the relative merits of traditional vs progressive education. Critics often point out that framing educational methods as either traditional or progressive is futile for several reasons. People have different views about which methods are traditional and which are progressive, teachers don’t usually stick to methods they think of as being one type or the other, and some methods could qualify as both traditional and progressive. In short, critics claim that the traditional/progressive dichotomy is a false one.

This criticism has been hotly contested, notably by self-styled proponents of traditional methods. In a recent post, Greg Ashman contended that Steve Watson, as an author of a study comparing ‘traditional or teacher-centred’ to ‘student-centred’ approaches to teaching mathematics, was inconsistent here in claiming that the traditional/progressive dichotomy was a false one.

Watson et al got dragged into the traditional/progressive debate because of the terminology they used in their study. First off, they used the terms ‘teacher-centred’ and ‘student-centred’. In their study, ‘teacher-centred’ and ‘student-centred’ approaches are defined quite clearly. In other words ‘teacher-centred’ and ‘student-centred’ are descriptive labels that, for the purposes of the study, are applied to two specific approaches to mathematics teaching. The researchers could have labelled the two types of approach anything they liked – ‘a & b’, ‘Laurel & Hardy’ or ‘bacon & eggs’- but giving them descriptive labels has obvious advantages for researcher and reader alike. It doesn’t follow that the researchers believe that all educational methods can legitimately be divided into two mutually exclusive categories either ‘teacher-centred’ or ‘student-centred’.

Their second slip-up was using the word ‘traditional’. It’s used three times in their paper, again descriptively, to refer to usual or common practice. And again, the use of ‘traditional’ as a descriptor doesn’t mean the authors subscribe to the idea of a traditional/progressive divide. It’s worth noting that they don’t use the word ‘progressive’ at all.

words are used in different ways

Essentially, the researchers use the terms ‘teacher-centred’, ‘student-centred’ and ‘traditional’ as convenient labels for particular educational approaches in a specific context. The approaches are so highly specified that other researchers would stand a good chance of accurately replicating the study if they chose to do so.

Proponents of the traditional/progressive dichotomy are using the terms in a different way – as labels for ideas. In this case, the ideas are broad, mutually exclusive categories to which all educational approaches, they assume, can be allocated; the approaches involved are loosely specified, if indeed they are specified at all.

Another dichotomy characterises the traditional/progressive divide; teacher-centred vs student-centred methods. In his post on the subject, Greg appears to make three assumptions about Watson et al’s use of the terms ‘teacher-centred’ and ‘student-centred’ to denote two specific types of educational method;

• because they use the same terms as the traditional/progressive dichotomy proponents, they must be using those terms in the same way as the traditional/progressive dichotomy proponents, therefore
• whatever they claim to the contrary, they evidently do subscribe to the traditional/progressive dichotomy, and
• if the researchers apply the terms to two distinct types of educational approach, all educational methods must fit into one of the two mutually exclusive categories.

Commenting on his post, Greg says “to prove that it is a false dichotomy then you would have to show that one can use child-centred or teacher-centred approaches at the same time or that there is a third alternative that is commonly used”.  I pointed out that whether child-centred and teacher-centred are mutually exclusive depends on what you mean by ‘at the same time’ (same moment? same lesson?) and suggested collaborative approaches as a third alternative. Greg obviously didn’t accept that but omitted to explain why.

Collaborative approaches to teaching and learning were used extensively at the primary school I attended in the 1960s, and I’ve found them very effective for educating my own children. Collaboration between teacher and student could be described as neither teacher-centred nor student-centred, or as both. By definition it isn’t either one or the other.

tired of talking about traditional/progressive?

Many teachers say they are tired of never-ending debates about traditional/progressive methods and of arguments about whether or not the traditional/progressive dichotomy is a false one. I can understand why; the debates often generate more heat than light whilst going round in the same well-worn circles. So why am I bothering to write about it?

The reason is that simple dichotomies have intuitive appeal and can be very persuasive to people who don’t have the time or energy to think about them in detail. It’s all too easy to frame our thinking in terms of left/right, black/white or traditional/progressive and to overlook the fact that the world doesn’t fit neatly into those simple categories and that the categories might not be mutually exclusive. Proponents of particular policies, worldviews or educational approaches can marshal a good deal of support by simplistic framing even if that completely overlooks the complex messiness of the real world and has significant negative outcomes for real people.

The effectiveness of education, in the English speaking world at least, has been undermined by the overuse for decades of the traditional/progressive dichotomy. When I was training as a teacher, if it wasn’t progressive (whatever that meant) it was bad; for some teachers now, if it isn’t traditional (whatever that means) it’s bad. What we all need is a range of educational methods that are effective in enabling students to learn. Whether those methods can be described as traditional or progressive is not only neither here nor there, trying to fit methods into those categories serves, as far as I can see, no useful purpose whatsoever for most of us.

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14 thoughts on “traditional vs progressive: mathematics, logic and philosophy meet the real world

  1. I think this is a good post. Clear, lucid, persuasive. Except for the last paragraph which suggests that the debate isn’t necessary? I suspect that the trad/prog divide is effectively one of underlying philosophy with respect to the relationship between the student, the teacher and the workplace, although I also suspect that most teachers don’t see it like that. Hence, I think that the debate/discussion is an important one. On the other hand, labelling a specific practice as trad/prog is probably not very helpful. The key question is really about effectiveness of the practice. Individual teachers, though, should examine their biases and prejudices, imo.

      • …nor of lumping teachers together in one camp or another. It’s unhelpful to find a label and define oneself by that so that it would be a guilty sin carrying out a practice that would be deemed to belong to the other camp, even if it seemed to be the right approach at that time.

  2. This is not great writing. It is pretty weird. There seems to be a terrible conflation here of ancient Greek geometers and modern day mathematicians and the only way to tie the first point to the conclusion would be to read it as today’s mathematicians, logicians and philosophers are a lazy bunch who haven’t kept up with scientific progress since before Galileo.

    You don’t have to look hard to find philosophers dealing with real world issues and most mathematicians work on real world problems. In an article decrying labeling as at least potentially problematic the broad brush only makes sense as a rhetorical technique to appeal to the unsophisticated reader.

    • I’m not conflating Greek geometers with modern day mathematicians. Nor am I saying that mathematicians, logicians and philosophers aren’t, or can’t, work on real world problems.

      What I am saying that applying only mathematical, logical and philosophical principles to real world problems doesn’t work because maths, logic & philosophy all decide, from the outset, what entities there are in the problems they are tackling.

      That approach to real world problems doesn’t work because the real world contains unknown unknowns. The scientific method has evolved as a way of tackling the unknown unknowns.

      But teachers who claim that educational approaches fall into two mutually exclusive categories, either traditional or progressive, often appeal to logic or philosophy to support that claim. This is despite the fact that a short exercise involving sorting some educational approaches into one or other of the categories will show that they don’t fit neatly, i.e. the categories might be mutually exclusive in the minds of the people using them, but are not when it comes to what actually happens in classrooms.

      Collaborative approaches is a case in point. Are collaborative educational methods traditional or progressive?

      • If you meant teachers not mathematicians perhaps that is what you should have said. As written you are talking about what mathematicians do in the present and wrong.

        Most happily engage with real world problems. I am sure you are aware a lot of well written scientific conclusions use the language of mathematics such as p values and confidence intervals to make sense of newly discovered but uncertain knowns. They also have probability and information theory to make sense of unknowns.

        Your argument by example also seems to be exactly what a mathematician, philosopher or logician might call an existence proof.

        Yes you have made the point that using broad or inappropriate labels to describe things is not a great idea.

      • The post is quite clearly about a specific debate amongst teachers about whether or not educational methods can be divided into the categories ‘traditional’ and ‘progressive’. I’ve included links to the relevant posts. I’d suggest taking a look.

        I’m not saying mathematicians do anything wrong; I’m saying that applying only mathematical, logical and philosophical principles to real world problems is pretty well doomed to fail.

      • The core knowledge about education theory is disputed because the principles of the disciplines it draws on are not taught. Educational ideas are extracted from various other domains and applied to educational settings, often ineptly.

        Teaching a rigorous, academically grounded body of knowledge doesn’t mean everybody has to learn the same thing; every other academic field has different emphases and approaches.

        Your faith in medicine is touching; there are major, fundamental disagreements about approaches to medicine that are the subject of heated debate.

        I’m not claiming that all doctors and lawyers are good ones – far from it. But most professions are fairly tightly accredited and regulated in a way that has never applied to teaching.

        The reason why the theory doesn’t help with the practice is most to be because the theory isn’t up to the task, or that it’s taught badly; good training should be of practical use.

  3. Yes I understand you are making a point about what teachers think. That is that some exclude other alternatives than just traditional and progressive or attempt to place all items into these two categories. But that is why I don’t follow your logic. You do talk about Mathematicians (3rd paragraph) as if they were doing something relevant to your argument. While you refer to two others having this discussion it is not clear why you think they are doing what you decry and arguing for it from a mathematical basis. It is just not clear at all how your references to mathematics or mathematicians relate to the individuals you refer to.

    This is why this is such a terrible piece of writing. The references to ancient Greeks and mathematicians clearly appeal to some – does it just sound good. But it is not clear at all that it is relevant to the case you refer to.

    In general it is not clear in general why people can’t focus on two categories, leaving the unknowns in one third category and debate just the merits of those two. As in your example of school attendance, there would be nothing wrong with dealing with two well understood causes of low attendance, if they were significant, even if others exist.

    • I refer to the mathematicians, the Greek ones. The ones I talk about in paragraphs 1&2. It was those mathematicians and logicians & philosophers who found the real world didn’t fit their theories.

      I’m not referring to the Greeks and their mathematicians because they ‘appeal to some’ or because it sounds good. Their way of thinking formed the foundation for Western thought. You can trace that lineage directly back to Pythagoras, Plato and Aristotle. People still use Plato’s and Aristotle’s approach to solving problems in the real world, even though they might not be aware they’re doing so, and that causes a lot of problems.

      Karl Popper nails it in the first part of The Open Society and its Enemies, but there’s a summary here http://www.iep.utm.edu/popp-pol/

      With regard to people focussing on two categories, it is clear why they can’t. People tend to reduce their cognitive load wherever possible, and two categories is/not are about the smallest number you can get whilst still maintaining a plurality. Two categories often work fine if they are broad labels used in general conversation, but for the purposes of research or policy-making you need to address all salient categories and there are often more than two.

      • I think people can understand what a false dichotomy is and why it is problematic without having to worry about the history of science and Popper, Aristotle and Plato. You seem to be doing exactly what you decry – saying Popper was right, Plato wrong and what someone is doing was unknowingly using Plato’s ideas.

        Going back to the original discussion this was meant to address you have one guy, Watson, saying don’t debate a false dichotomy and another, Ashman, saying there is always a dichotomy – here what Watson is proposing verses the traditional approach Watson himself refers to. I think the merits of Greg’s complaint can’t be dismissed with but Plato or but Popper.

      • I agree that people generally understand what a false dichotomy is. But in this case the disagreement was over whether the traditional/progressive dichotomy was false or not. Most people disagreeing about whether or not it was false didn’t see the disagreement merely as a matter of opinion either; it’s generated some heated arguments. And false dichotomies and other categorisations imposed on the world from the standpoint of ‘pure reason’ have caused immense problems in science, and a string of heated arguments going back to Plato.

        You imply that you don’t think what Plato or Popper said is relevant to Greg’s argument. But as I pointed out, I’ve also addressed Greg’s argument directly. Greg said “to prove that it is a false dichotomy then you would have to show that one can use child-centred or teacher-centred approaches at the same time or that there is a third alternative that is commonly used.” I did both and he hasn’t responded.

        If you think he’s right, perhaps you could explain why collaboration between teacher and child does not constitute ‘a third alternative that is commonly used’?

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