Manifesto on the Teaching of Mathematics

My general teaching philosophy can be summarised in three principles or axioms regarding learning. They concern the source, the process, and the goal of learning respectively.

My first axiom is this: In a perfect world students pursue learning not because it is prescribed to them but rather out of a genuine desire to figure things out. We must therefore teach as if our students were of this kind. Only by aspiring to this ideal can we bring it closer to being realised.

It follows that we must not introduce any topic for which we cannot first convince the students that they should want to pursue it. This is a standard very rarely met in mathematics. Everyone likes to tell themselves that they are giving motivations for what they teach, but very little of what passes for motivation stands up to critical scrutiny as a motivation in the sense of the learning ideal outlined above. In all such cases, therefore, the student has no reason to pursue the topic in question other than obedience to the dictatorial authority of the teacher. In my view we cannot fault a student who hates mathematics in such circumstances; if anything, I would sooner fault a student who did not.

My second axiom concerns the process of learning. It says: We learn when we are challenged, when we push ourselves. If you’re not stuck you’re not learning. If it’s not a struggle you’re not doing it right.

It follows that we must always look for new points of view and pursue open-ended questions. The role of the teacher is not to make life easy for the student by giving crystal clear lectures and predictable tests. Instead the role of the teacher is to guide and encourage the student’s own process of learning by setting suitable challenges and by stimulating thought and reflection.

The final axiom of my teaching philosophy is that the goal of teaching is independent thought. We want students to be able to think and reason and apply what they know in new situations. We do not want to create robots or parrots or one-trick ponies.

It follows that when we learn something we must always inquire why it is so, and that we must answer this question according to our own judgement, not by mimicking some external standards of rigour and proof. It also follows that we must always seek out the broader meaning of what we are studying through its applications and interconnections with other ideas.

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I have coined the phrase “Intellectual Mathematics” for the teaching philosophy I have in mind, because its fundamental principle is to treat students with the greatest possible intellectual respect.

This is the opposite of traditional mathematics teaching, which treats students like circus animals who need to be taught to jump through hoops by means of mindless drill training.

In the traditional approach the essence of the teacher’s role is authority. The teacher holds the carrot and the stick and that’s why you have to do as he says.

In the Intellectual Mathematics approach the essence of the teacher’s role is inspiration, and the goal of teaching is to stimulate thought and reflection. The teacher disavows the notion that he has the right to boss people around. Instead he considers it his responsibility to nourish in the students a desire to pursue their studies out of their own intrinsic motivation and interest.

It follows that in Intellectual Mathematics a topic is introduced only when the student can be convinced of the value of doing so. This is the opposite of the traditional approach where topics are routinely introduced at a stage where they serve no credible purpose whatsoever, simply because some curriculum designer decided that the students “need to have seen it” a year or two down the line.

Traditional curriculum designers butcher mathematics the way colonialists used to divide conquered continents: with crude and clinical cuts that are profoundly insensitive to any and all organic connections between the constituent parts. Such an approach makes sense for those who take their own authority for granted.

Intellectual Mathematics does not use such totalitarian techniques. Borders are not drawn where they do not belong and organic connections are respected. Mathematics is not severed from physics, nor differential equations from calculus, and so on, regardless of the administrative efficiencies of such compartmentalisation. You do not read every other line of a Shakespeare play in one class, and then the remaining lines in another class the following year. But in mathematics we routinely do precisely this. Such an approach is incompatible with intellectual respect for the students.

In traditional mathematics, things are taught because they are replicable and testable. The teacher is so dependent on the drillmaster paradigm that only topics that fit it can be taught. If a topic doesn’t allow for fifty-eight near-identical drill problems at the end of the section, then that topic is unteachable in traditional mathematics. It will not be taught no matter how important or crucial for understanding the true purpose of the entire subject. Conversely, topics that do lend themselves to endless drill problems will often be taught for this reason alone, despite being utterly pointless and contrived.

In Intellectual Mathematics, presenting topics in an inherently interesting and meaningful way is the first and foremost consideration. The purpose of the problems at the end of the section is not to force students through a repetitive obstacle course, but to convince the students of the value and importance of what they are studying.

The traditional approach fosters robotic, unthinking students. It selects for obedience and punishes independent and critical thought. Intellectual Mathematics does the opposite.

I say with Rousseau: “Let the child do nothing because he is told; nothing is good for him but what he recognises as good. When you are always urging him beyond his present understanding, you think you are exercising a foresight which you really lack. To provide him with useless tools which he may never require, you deprive him of man’s most useful tool — common-sense. You would have him docile as a child; he will be a credulous dupe when he grows up. You are always saying, ‘What I ask is for your good, though you cannot understand it. ...’ All these fine speeches with which you hope to make him good, are preparing the way, so that … every kind of fool may catch him in his snare or draw him into his folly.”

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Intellectual Mathematics should not be confused with what passes for “reform” teaching. Everything we have said about the traditional approach applies equally well to “reform” approaches, because what is called “reform” pertains almost exclusively to surface form, not substance.

The basic fault of the modern “reform” movement is that it does not have the courage and confidence and ability to challenge the mathematical establishment on matters of substance. It assumes that the traditional approach is mathematically infallible, though pedagogically flawed. It therefore busies itself with concocting pedagogical schemes to make the same old medicine go down more easily. Group work! Use of technology! Inquiry learning! Flipping the classroom!

Modern “reform” efforts start at the wrong end. They put the cart before the horse, lipstick on the pig. The enterprise is doomed because it is predicated on the false assumption that the underlying curriculum is beyond rebuke. It doesn’t matter what pedagogical tricks you use if the substance you are trying to teach is poorly conceived in the first place. It is impossible to teach bad material well. That is why any reform worthy of its name needs to actually reform mathematical substance.

The Intellectual Mathematics approach starts with content and substance. It is not primarily about how to teach, but what to teach. It does not start with the question: “How can we make students understand concept X?” Rather it starts with the question: “Should we even teach concept X in the first place? If so, why?” This should be the guiding question of true reform.

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Intellectual Mathematics is written for the intellectual fulfilment of the reader. This means that it seeks the most satisfying explanations, the most vivid illustrations, and the most compelling motivations. It also means that it engages our intuition whenever possible.

Traditional mathematics is written for robots and nitpickers. It is obsessed with being technically correct at the expense of all else. Again and again the ugliest proofs and the most contrived order of presentation are favoured in the traditional approach on the sole grounds that they are the easiest to write down in a manner that cannot be faulted with respect to logical correctness.

In Intellectual Mathematics, when facing a new concept, our primary goal is to understand how and why it works. The standard by which this is judged is our own sense of satisfaction and understanding. Emotion, passion, and the joy of insight are therefore essential components of Intellectual Mathematics.

In traditional mathematics, when facing a new concept, the goal is to reach the requisite results without making any technical errors. Crossing the t’s and dotting the i’s are the alpha and omega of traditional mathematics. Traditional mathematics is anti-human. It fetishises robotic manipulation of symbols and involves no emotions except a crippling fear of genuine and free human thought.

In traditional mathematics, the character and bulk of any given proof usually has next to nothing to do with why that particular theorem is true and everything to do with incidental technicalities. Students soon get the hint that mathematics is not about actually thinking and trying to figure stuff out; rather it is clearly a formal game completely divorced from common sense.

Indeed, formal proofs are sometimes accompanied by an informal heuristic argument, only to be followed immediately by the admonition that “of course this is not a proof!” Much more than giving the student some shred of intuitive insight, such passages convey more than anything that honest thinking is futile and impermissible in mathematics because a proof only counts if it is packed with technical jargon that has nothing whatsoever to do with the specific matter at hand.

Kant said: “Enlightenment is man’s emergence from his self-imposed immaturity. Immaturity is the inability to use one understanding without guidance from another. … Sapere Aude! Dare to know! Have courage to use your own understanding! Thus is the motto of Enlightenment.”

This is also the motto of Intellectual Mathematics. Stop thinking that a mathematical argument is a bureaucratic form submitted for approval to the Gods of Pedantry. A mathematical argument is written for you. Stop worrying about what “they” want you to do. Have the courage to think for yourself.

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For further elaboration of these ideas see my blog posts tagged Pedagogical purposes and Intellectual Mathematics (synonyms of).