Does this approach work?

The JUMP math program, founded by John Mighton, is a primary-level approach to mathematics teaching. The JUMP motto is that “every child can learn mathematics and love it”, and a maximin-type policy is enforced, with classes progresses only once all students have mastered the material. JUMP works by carefully building up basic skills, The Myth of Ability,

namely guided discovery and the core ethic tha. The data unambiguously show that, at least for primary students, this approach works wonders. I am proposing a change in how we teach secondary mathematics, with various other tweaks in emphasis and technique. There may be wrinkles that arise from this difference in emphasis, but I would be surprised if the results were dramatically different.

Does maximin fail engaged students?

Maximin does not forbid writing questions for engaged students. It only forbids questions which only engaged students can enjoy. But bonus questions can help engage the class as a whole. In the JUMP program, bonus questions are used as a carrot, dangled just beyond the standard material, enticing students to finish that material and move onto something “fun”. In his book The Myth of Ability, John Mighton even recalls breaking up a fight by threatening to withhold one of these bonus questions! The point is that bonus questions can be for everyone, from the least engaged (who will be incentivised) to the most enganged (who will be challenged). There is no need for streaming!

Do your examples pass the maximin test?

The revised questions seem challenging, but I intend them to be taken in the spirit of JUMP’s bonus questions: a carrot for a globally engaged cohort who trust the teacher or text. But bonus questions are not a blank cheque. They only work once students have a basic level of self-efficacy and engagement. Even then, they still need to be designed deliberately, using a careful model of what students know and where their zone of proximal development is, chunking a difficult passage of reasoning into small, achievable steps. This is consistent with guided discovery and the constructivist philosophy of teaching.

I’ve tried to chunk my questions this way, and aim for relatively low prerequisites. Without embedding them in a broader classroom context, perhaps the only thing they demonstrate clearly is that high-school calculus has exciting and nontrivial applications. But this raises the question: if we abandon the scrubland, how will students develop basic skills necessary to do bonus problems?

How do students develop basic skills?

As with JUMP, we can treat the aesthetic and real-world highlights as bonus material, motivating students to slog through repetitious, facility-building exercises. But do these have to be shrubs? The JUMP program shows that, with a careful constructivist mindset, building basic skills need not be an interest-killing affair. Let’s try a similar thing with our earlier patch of scrubland. Consider these two questions.

  1. Differentiate the following and state the maximal domain:

    (a) $\sin^{-1}(x/3)\qquad$ (b) $\cos^{-1}(x^2)\qquad$ (c) $\tan^{-1}(\sqrt{x})$.

  2. (a) Using the chain rule, find the derivative of $f(x) = \cos(\sin^{-1}(x))$.

    (b) Deduce from trigonometric identities that $f(x) = \sqrt{1-x^2}$. Do the domains match? Differentiate and check that your answer agrees with (a).

    (c) Implicitly differentiate $\cos^{-1} y = \sin^{-1} x$ and isolate the expression for $dy/dx$. Is this consistent with your previous answers?

Which is better? I would argue that the second passes the shrub test while the first does not, though they develop the same skills: differentiating inverse trigonometric functions, using the chain rule, and identifying domains.

What should we assess?

The first question has its own dubious virtues. The format (that is, lack of content!) leads to easy scaleability, which as noted earlier, explains its appeal to textbook writers. But it has two apparent pedagogical virtues: the psychological hack of associative learning by repetition, and its functional manifestation, the speedy execution of algorithms. This is the logic of drill.

But what is the point of memorising inverse trigonometric derivatives? Or being able to perform the differentiation quickly? It’s not clear. I don’t know these derivatives by heart, and I don’t need to; I use them rarely enough in my research that I just derive them if required. Being able to derive them is the more useful skill! In general, we place far too much emphasis on memorising useless facts and reproducing them (or performing the associated algorithms) speedily under exam conditions.

How should assessment look? Consider this inspiring assignment question (from a graph theory class with David Wood):

Define a numeric measure of the complexity of a graph. Compute it for several examples, and prove some properties about your measure. You do not need to write more than 1 or 2 pages to get full marks.

This is exactly the sort of exploratory, open-ended thinking we need to encourage. A scrubland of drill trains students for the scrubland of assessment. It does nothing to prepare them for the variety, complexity, and open-endedness of real life.

Does this work for pre-calculus topics?

Of course! I literally chose calculus at random from the textbook. Any mathematical topic has gorgeous theorems, freakish specimens, and a rich history of real-world motivation and use. Teaching trigonometry? Perhaps Euclid is a little dry, but what about non-Euclidean geometry? This is a wonderful place for guided discovery and exploration. And what could be more empowering than determining the size of the earth from the shadow of a stick, as Eratosthenes did in 100 BC. What about algebra? Try the unsolvability of the quintic, compass and straightedge constructions, and applications to dimensional analysis. Logarithms? Here we have the density of prime numbers, slide rules, algorithmic complexity and Fermi estimates. The list goes on.

Hopefully the idea is clear. Pick a topic, any topic. Figure out its history, where it comes from, what it’s really used for and what mathematicians or scientists find beautiful about it. Now incorporate that material into the syllabus. Mathematical skills live in a big constructivist framework, and according to the maximin test, attention needs to be paid to developing that framework so that all students can benefit. But these goals are not at odds!

What is the teacher’s role?

In the approach I’m suggesting, the role of the teacher seems to be supplanted by a curriculum full of sparkling ideas and fancy constructivist praxis. But learning is a relational endeavour, and the teacher remains an essential part of the process. It is through the teacher that trust is established, students are engaged, and the subject comes alive. For all of this to happen, teachers need to be on top of the material! If the syllabus now encompasses continuous but nowhere differentiable functions, numerical approximation, diffusion, and self-organised criticality, it seems like an impossible ask for all but the most specialised of high school maths teachers.

But here’s the thing: if students can learn it, so can teachers! And if the syllabus is designed in an inviting, constructivist fashion to reduce maths anxiety in students, it should also reduce anxiety in teachers. Perhaps, once they’ve left the scrubland, more teachers will be genuinely enthusiastic about what they are teaching. Their enthusiasm will be infectious.

What’s next?

A single blog post is not going to overthrow secondary maths education. Instead of replacing the curriculum (an ambitious and long-term goal), paths into the landscape can be offered as a complement to traditional textbooks. I’m currently busy with a PhD, but in the near future, I hope to start assembling a database of extensions, theorems and applications, embedded in a constructivist framwork and cross-referenced against real curricula. Watch this space!

5. Conclusion

Teachers and curriculum writers are experts on modelling what students know and how they come to know it. Scientists and mathematicians are domain experts, with specialised knowledge, taste, and judgment. We need to stop marching students through the scrubland, and introduce them to the vast topography of the real mathematical world, with its majestic summits, calm valleys, and bestiary of strange creatures. From this fertile soil, students can grow new plants, and in its caves mine for patterns, fir for the as-yet-unknown mathematical needs of the 21st century. And just as important, they can enjoy the simple pleasures of mathematics which have for so long, so unjustly and unnecessarily, been denied them.

Annotated bibliography

Here are some resources which informed my ideas, were used for illustrative examples, or both. I’ve included some brief annotations.

  • Self-organised criticality: an explanation of 1/f noise (1987), Per Bak, Chao Tang and Kurt Weisenfeld. The explosive paper that started a whole new field of study. Unfortunately, real sand piles don’t seem to behave this way, but the model remains provocative and useful.
  • A model for the dynamics of sandpile surfaces (1994), J.-P. Bouchard, M. E. Cates, J. Ravi Prakesh and S. F. Edwards. A simple model of sand piles, developing the insights of Nobel-prize winner Pierre-Gilles de Gennes.
  • Darwin’s Tree of Nature and other images of wide scope” (1981), Howard Gruber. In Aesthetics in Science, ed. Judith Weschler. A beautiful essay about scientific imagery, intuition, and the aesthetics of complexity.
  • A Mathematician’s Apology (1940), G. H. Hardy. The classic apologia for pure mathematics.
  • Problems in General Physics (1981), Igor Irodov. A concise, Soviet introduction to physics.
  • Out of the Labyrinth (2007), Ellen and Robert Kaplan. A playful book about the Harvard maths circle, with a very similar philosophy of maths teaching, learning, and doing.
  • The Myth of Ability (2003), John Mighton. An exposition of the JUMP method and the philosophy behind it. Mighton has a similar attitude to the universality of mathematical ability (and the data to back it up!), has a maximin “leave no child behind” policy, and uses bonus questions as an incentive.
  • Reinventing explanation (2014), Michael Nielsen. An inspiring and comprehensive look at explaining science using representations fit for purpose, or “cognitive media”. See also Bret Victor.
  • Sublimation of moth balls (1978), M. G. C. Peiris and K. Tennakone. A cute AJP article on the thermodynamics of moth ball sublimation.
  • A Theory of Justice (1971), John Rawls. Perhaps the most influential book on political philosophy in the 20th century.
  • The Mathematical Unconscious” (1981), Seymour Papert. In Aesthetics in Science, ed. Judith Weschler. A perceptive essay on aesthetics and mathematical intuition from one of the giants of old-school AI.
  • Up and down the ladder of abstraction (2011), Bret Victor. A brilliant essay on the use of interactive media (as opposed to symbols) to facilitate abstraction. I fundamentally disagree with Victor’s claim that a select few have the “freakish knack” for manipulating symbols, but agree with Nielsen that interactive media can and should provide a rich complement to traditional symbolic methods.
  • Abstraction, intuition, and the monad tutorial fallacy (2009), Brent Yorgey. Analogies are great, but not a silver bullet for understanding. Yorgey argues that intuition-building should precede analogy-making.

Do your examples pass the maximin test?

This is a classroom-dependent question. The pure and applied mathematics sections are designed as motivating higlights for a classroom of self-confident who are comfortable, conceptually and computationally, with derivatives. But even given these prerequisites, a question could still fail. But the examples are chunked into small, achievable steps, consistent with a guided discovery approach.

In some cases this is even made explicit. In the Australian maths curriculum, for instance, “Critical and Creative Thinking” is one of the general capabilities, and A corollary, I think, is that we should value the simple pleasure that doing mathematics can bring, though