Teaching Mathematics: What, When and Why

An in-depth examination of mathematics education, topic by topic

Discovery Based Learning

You might think that discovery based learning (DBL) is some fancy new approach to teaching mathematics. But I for one have been using DBL since the 60s. And I’m pretty sure my teachers in the 50s used this approach. But a discussion reported in the BAD MATHS blog shows me that some teachers think it is a waste of time, or worse, harmful. So let’s have a look.

The internet has vast swathes of discussion but I want some examples. Ah here’s one. The class is divided into small groups (already my alarm bells are ringing). But look, young Jimmy has discovered that if he pushes the toy car harder then it goes faster, Newtonian dynamics here we come. If you are familiar with four-year-old boys then you will realise that Jimmy already knew this. What he has actually discovered is that by stating the bleeding obvious, he will make his teacher happy.

Soon it hits me. Much of what passes for DBL in the internet is actually unguided learning. There are decades of research showing that unguided learning does not work. So there is still a need for teachers (sorry to university administrators).

So, what do I mean by discovery based learning. At the easy end of the spectrum consider this. The class has already learnt r \hbox{cis}(\theta) \times \rho \,\hbox{cis} (\phi) = r\rho \,\hbox{cis}(\theta+\phi) and as a group we are working some examples. By the time we have found [r \,\hbox{cis}(\theta)]^2 = r^2 \, \hbox{cis}(2\theta) and [r \,\hbox{cis}(\theta)]^3 = r^3 \, \hbox{cis}(3\theta) one of the more expressive students, let’s call her Betty, will declaim the general rule. Here I have a little fun; for the rest of the year, I will refer to the result as Betty’s theorem, to her delight. Of course, you need a class accustomed to back and forth and a good sprinkling of Bettys.

Let’s return to r \hbox{cis}(\theta) \times \rho \,\hbox{cis} (\phi) = r\rho \,\hbox{cis}(\theta+\phi); can this be “discovered”? Well, the student who understands what the left side means, will quickly convert it to r\rho\left\{ \cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi) +i[\sin(\theta)\cos(\phi)+\cos(\theta)\sin(\phi)] \right\}. Back in the day, all students were expected to know the compound angle formulae, so most students would recognise the patterns and discover the simple result. Ironically, the loudest voices calling for discovery based learning are the same voices that have deleted from the syllabuses those aspects that are required for discovery. The patterns of mathematics are now relegated to a formula sheet or an electronic device; for discovery they need to be lodged in the brain.

Usually a little more guidance is needed. Suppose my class is now content with first order differential equations and it’s time to consider second order linear with constant coefficients, perhaps starting with y''-y' -6=0. The class will probably need a nudge: what about first order linear with constant coefficients? A few examples show that the answer will always be a multiple of an exponential function, so someone will suggest an exponential function and after trial and error two respectable solutions will appear. When I was attending Bonehead College1 for my TTTCSSC2 there was some excitement around this example. A fellow student had used exactly this approach and his class had discovered general solutions even for higher order cases, and even with repeated roots in the auxiliary equation.

Sometimes the “discovery” part can be just a simple observation and the teacher gets the job of explaining. Here is an example from 1970. The students were asked to graph xy=1 and \quad\frac{\displaystyle x^2}{\displaystyle 2\;}-\frac{\displaystyle y^2}{\displaystyle 2\;}=1. Students noticed that the graphs are exactly congruent. Why? Well the second equation can be re-written \frac{\displaystyle x-y}{\sqrt{2}} \,\frac{\displaystyle x+y}{\sqrt{2}} = 1. So substituting x'=\frac{1}{\sqrt{2}} x-\frac{1}{\sqrt{2}} y and y'= \frac{1}{\sqrt{2}}x + \frac{1}{\sqrt{2}}y converts the second equation to x'y'=1. We have equations that convert the point (x,y) to (x',y'), rotating that point by 45^\circ about the origin. This was a sideways entry to matrices and linear algebra. In those days students learned a library of standard graphs, so their graphing task was a familiar one.

You might have noticed that my examples use a special case of the general idea being pursued. This is often how discovery happens, even at the research level. I spent an Edinburgh winter watching my office roommate researching some recondite issue in functional analysis. The topic was beyond me so I was more interested in her approach. We would discuss various special cases that she suggested. Come spring, her paper was written and it was time for sightseeing.

But it’s not always possible to find a revealing special case. Teaching the product rule for differentiation, I have never found a simple example that is any easier to “see” than just deriving the general case. But examples may still be revealing. The argument \frac{d}{dx} x^2 = \frac{d}{dx} (x \times x) = \frac{d}{dx} x \times \frac{d}{dx} x = 1 shows that the obvious multiplicative rule does not work. Special cases are useful for hammering what is not true.

Given constraints of contact time we typically cannot include more than the occasional instance of DBL in our teaching. Is it worth the effort? On a granular level, students find the idea that they or Betty found is more memorable than just another from the teacher. More generally, they may begin to see mathematics in a new light, as a living communal activity. From my viewpoint there is a lift in the esprit de classe, that intangible feeling that makes class teaching so pleasurable.

If you are an experienced teacher you may well have examples to add. Where did DBL work and where did it fail?

Postscript

Re-reading the above led me to wonder whether we can introduce DBL into textbooks? Serendipitously, my current bedtime reading is “Introduction to the Constructive Theory of Functions” by John Todd. This book is ill-suited for casual reading as the proofs often assume one has worked problems in earlier chapters. In reviewing the book, P J Davis writes “This is not a textbook in the usual sense. … It is rather a set of blueprints for a problem course, a course in which the quantity of subject matter is secondary and the students’ active participation in the process of discovery is crucial.“

  1. Alias of the Technical Teachers’ College, Melbourne, an institution that gave practical advice to teachers, shut down by the University of Melbourne. ↩︎
  2. Trained Tertiary Teachers’ Certificate Special Short Course. And you thought I was unqualified. ↩︎

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tom

13 responses to “Discovery Based Learning”

  1. Tom Discovers Learning – BAD MATHEMATICS Avatar
    Tom Discovers Learning – BAD MATHEMATICS

    […] dealing with AMT nonsense is not enough, now Tom has a new post on his Teaching Mathematics blog, promoting the wonders and joys of Discovery Based Learning. […]

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  2. Glen Wheeler Avatar
    Glen Wheeler

    Hi Tom! I think it is dangerous to use existing terminology for something different (and especially for something personal, like in this case). I would shy away from it, and suggest you follow your own advice. Why not call it “Tom’s kind of learning” instead of “Discovery based learning”?

    Definitions are important!

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    1. tom Avatar
      tom

      Thanks Glen
      My unreliable memory is that in the 60s what I did was called “discovery based learning”. So what does the term mean currently?

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      1. Glen Wheeler Avatar
        Glen Wheeler

        Discovery based learning is part of the constructivist approach to education, and focused on students engaging in processes (and not engaging in content). Students discover their own facts based on exploration and play. They complete investigations and projects. The teacher takes the role of facilitator and does not give content, focusing instead on the processes and helping with meta-content.

        Hope that helps!

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      2. tom Avatar
        tom

        I love it. An English translation is “teachers just watch the children play”. Great scope for teachers to retire early but continue to receive full pay. I’m thinking they could use chatGPT to automatically generate lesson plans. However, when I tried the results were traditional teacher dominated lessons, but with the class broken into small groups for practice 😦

        I accept your advice to not use the terminology “discovery based learning”. The Goths at the gates have stolen the term. But neither is my approach well described as “Tom’s kind of learning”; I am just trying to preserve an ancient and effective practice. To be effective it requires some planning on the teachers part, to identify what is to be discovered! and an active sprinkling of breadcrumbs. Could we call it “guided discovery” or is the word “discovery” still a problem?

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  3. Terry Mills Avatar
    Terry Mills

    Natanson’s “Constructive theory of functions”, volumes 1-3 is easier to read; no problems to distract you from making progress, but beautifully written.

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    1. Terry Mills Avatar
      Terry Mills

      My previous comment was in connection to your reference to the book by John Todd.

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      1. tom Avatar
        tom

        Thanks Terry, yes that was clear.
        So far I have learnt that “constructive theory of functions” somehow means approximation theory. This reminds me of a visit to Bendigo decades ago. You had a student working for a Masters and her work consisted of reading chapters of a book on approximation theory and then presenting them to the faculty. Great training for the student, and the staff! Was the book Natanson?

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      2. Terry Mills Avatar
        Terry Mills

        Yes. Constructive function theory deals with classifying a set of functions by how well the functions can be approximated usually by polynomials of a certain degree.

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  4. George LILLEY Avatar
    George LILLEY

    Thanks Tom, your paragraph including –
    “From my viewpoint there is a lift in the esprit de classe, that intangible feeling that makes class teaching so pleasurable.” pretty much describes my 20 year experience in VCE maths

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  5. Glen Wheeler Avatar
    Glen Wheeler

    Hi Tom! I don’t seem to be able to reply (the button is missing) but this comment is continuing the thread above.

    I’m sorry, my suggestion “Tom’s kind of learning” was only in jest. You’re better situated than me to suggest a more serious name, but from looking at the post, it seems to me that you are using a classical approach: You identify learning outcomes, and then structure your lesson to give students a good chance at attaining them. That’s called “constructive alignment”. Your approach is mostly “direct instruction” and “explicit teaching”. I think the notion of group work cuts across most of these descriptors.

    In my view, a lot of your discussion about how identities or formulae might be discovered is part of a standard approach to content and would be needed to engage students in the delivery by a teacher or lecturer. In the example you gave about differential equations (which I just happen to be teaching right now) I would lead students through that line of thought and give them chances to make leaps and bounds themselves. It is the approach of many people I’ve seen, and it works well. When students make those leaps, they get fired up. When they don’t, it’s fine, because the teacher is there to keep the lesson moving forward.

    That’s one of the major philosophical differences with what you’re talking about (and me, in that paragraph above) and DBL. For DBL teachers, there is no notion of “lesson moving forward”. Content is created by students. If they spend two hours on trialing different kinds of polynomials (and having a blast), that’s fine. If one group works out a theorem and another is doing examples, that’s fine. If one student is still working out how to graph functions, another is coding numerical methods, and yet another is proving a-priori estimates… that’s fine.

    I *can* see a use for DBL. For example if you have a group of students interested in a topic and you want to give them a chance to explore. Say a meeting after school of the math club. They can do some DBL. But in my view it should *not* be in regular school hours and it should absolutely under no circumstance be used as a substitute for standard lessons.

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    1. tom Avatar
      tom

      Hi Glen
      Not sure how to run this wordpress ordering system …

      Thanks for educating me; your post is a lovely summary of the area. I’ve been assuming that what I do “is part of a standard approach to content” so glad to hear your confirmation. Within this standard approach the “give them chances to make leaps and bounds themselves” part still needs a name? I do fear that many teachers are missing out on this technique. So perhaps labelling this thread as “Discovery Based Learning” was a good choice as such teachers may happen upon that aspect.

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      1. Glen Wheeler Avatar
        Glen Wheeler

        Hi Tom,

        Like groupwork (and its various forms), I think this cuts across those “philosophical” descriptors. I haven’t tried to give it a name myself. If teachers are indeed missing this basic strategy to delivering content, that speaks very poorly of their own education, reflecting badly on the university courses that they are likely graduates of.

        In my classes, there is a solid number of teachers-to-be (I believe the entire cohort of math teachers in training at my university), and from my experience they are aware of it (and bright-eyed and bushy-tailed). Perhaps as with many things if they lose their passion then they forget, or perhaps there is something else going on.

        A little more seriously, in a ’98 article Beck attempted to bring order to the chaos around teaching strategies. You can read it if you’re interested (I can send it to you e.g.), but from what I can see we are talking about a combination of particular strategies. Beck comments specifically on this, saying that it heightens the effectiveness of the instruction to use a variety of methods. I agree with this personally, as if you tried to teach an entire class using only interactive elements I don’t think it would be particularly good. Novel, but not good.

        I’d guess associative, deliberative, and expositional. But I think I’ve had enough of this article! I’m wondering what the point of making a name is. I’m not going to try and sell it to anyone, and I don’t think I’ll use it myself. An exercise in procrastination to prevent me from finishing my marking!

        But one final plea: Don’t use DBL. The best case is that someone may think they are doing DBL and actually deliver an effective class. So, learning by accident. I think we can do better than that! Call a spade a spade, and be honest in our teaching.

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