Mathematics: what is its knowledge?

…he is raging against methods of instruction that really aren’t very advanced, and really don’t lead to any great mathematical development.  We certainly can do better, and we will do better; not by abandoning aspects of traditional teaching that have been effective and giving in to a fantasy, but by bringing traditional methods under the scrutiny of modern science, refining and improving them until they are fit for the 21st century.

The red x is where I’d placed maths on the axes.

Axes (2) - Maths

Substantive Knowledge

The stuff you think maths is.  It’s knowing how to count, how to perform arithmetic, knowing the different kinds of number (natural, negative, integer, rational, irrational, imaginary, complex…), knowing the theorems, trigonometry, what a function is and so on, so forth.

Notice that there is procedural knowledge in here (‘know how’) as well as declarative (‘know that’).  Pytagoras’ Theorem states that

The sum of the squares of the legs of a right-angle triangle is equal to the square of the hypotenuse.

If a person memorised that, and assuming they knew what a right-angle triangle, legs, hypotenuse and squares were, then it would not be rote knowledge; they would indeed know Pythagoras’ Theorem.  Some people might even argue that a person who didn’t know any of those concepts, but who still memorised the words above, would know the theorem, and their knowledge would not be rote.

Either way, this would be distinct from knowing how to make use the theorem in constructing right-angle triangles, or in calculating the lengths of unknown sides.  One is declarative knowledge, one is procedural, both are substantive.

Disciplinary Knowledge

I placed mathematics where I did because, by and large, in maths at school we only ever teach the substantive knowledge.  We teach the mathematical facts, theorems, symbols, language and procedures that have been discovered and invented over the millennia, and that’s about it.  I didn’t place it on the x-axis because some teachers do make some effort to run some kind of investigation from time to time… but really, we don’t institutionally teach the disciplinary knowledge of mathematics; we don’t teach or explain just how the many women and men of its history discovered all those facts, invented that language, and proved those theorems.

This is why people struggle to understand the world of the mathematician.  Mathematicians obviously do not sit around all day ‘doing sums’ and ‘solving for x’.  So what do they do?

Mathematicians might be driven by a range of motives, circumstances and interests, but in general, they come up with questions they think are interesting.  One such example would be:

“Is there a formula that can generate prime numbers?”

If the answers no, fair enough.  If it’s yes, we would probably go on to wonder what they formula was, and try to figure that out.

Mathematics can provide an answer to the question in a variety of stages.

Stage 1: The question is answered Yes or No, but it is possible for us to be able to prove that something is possible (a yes answer) without actually finding the thing we now know exists!

Stage 2: If yes, a formula is found that works in some special cases.  The breadth of special cases might expand over time.  Special cases might be ‘primes with fewer than 10 digits,’ or ‘whose digits sum to a multiple of 9’ as random examples.

Stage 3: If yes, a formula is found that works for all cases.  Any prime number can be generated by this formula.  We say that it is fully generalised.

I chose this example because it is a question that some people have found interesting, and yet we don’t know either way whether such a formula or pattern exists (insofar as I’m aware…)

How would a mathematician go about finding the answer to their question?  She might try to work with a few numbers that are small and manageable to begin with, so see if she can find a pattern.  If she finds a pattern that continues to generate primes, she might then formulate a conjecture, being “This is the formula that will generate primes.”  She might then work to prove the conjecture true, using a variety of methods of proof.  If successful, she would have contributed to the human body of knowledge, and will publish her work.

This isn’t the only thing a mathematician will do with their time, this is just one example (any professors of mathematics out there, please feel free to chip in around this point…!)

Alternatives might include thinking about how to apply their specialist knowledge of mathematics to real world problems, such as those in science, finance, engineering and so on.  It might not be that the mathematician thinks first ‘I want to apply this to finance,’ but upon learning a bit more about the world of finance realises ‘Oh!  There’s something I’ve been doing that might be useful here!’

It might also include deciding how to handle new kinds of number.  Descartes found roots of negatives so absurd that he chose to describe them as imaginary.  The term stuck, even though they are quite real.  We didn’t have to include imaginary numbers in what we consider number – indeed for thousands of years we didn’t – it was the work of mathematicians expanding the sphere of their discipline’s knowledge who argued eventually why they must be included, showed how they can behave like other types of number, and how they can be submitted to arithmetic.  In the process they literally opened up a new dimension of numbers, giving way to a new infinity of complex numbers, and with it, all of our modern electronics.


As best I can suggest, then, the disciplinary knowledge of mathematics includes, but is not limited to investigation, conjecture, proof, problem modelling and problem solving – all the things we don’t teach in school.

Mathematics Reform

Most attempts at mathematics reform are geared around trying to move maths’ cross from the bottom right, to the top left.

This is seen in the popular Lockhart’s Lament.  Lockhart saw the mathematics of school to be devoid of all of the disciplinary knowledge that made it come alive.  He concludes, quite literally, by saying he would rather school children learnt little or no substantive knowledge, but learnt something about the discipline of mathematics.

Conrad Wolfram makes the same point here.  He defines mathematics as:

  1. Posing the right questions
  2. Building a mathematical model (from the real world, into mathematical language)
  3. Computation
  4. Interpretation (back from the mathematical model, into the real world meaning)

He argues that we place all of our efforts on school maths education on point 3, computation, and ignore the other three.  He is right.

He then argues that we should invert this, spending our time on points 1, 2 and 4, and leveraging computers to do the computation.  He is wrong.

Where next?

The reform agenda has attempted to move maths education away from substantive knowledge and into disciplinary knowledge for decades now, and persistently we have survived those attempts… for the most part.  This has been the trend across the range of subjects; it’s the reason maths is pretty much alone in the bottom right there.

I sympathise with Lockhart’s lament, I really do, but Lockhart doesn’t have the answer.  He says himself, he would be happy to have kids leave school knowing no maths, provided they were given a glimmer of insight into the world that will now be forever closed to them.

Wolfram’s four points above are speaking to the disciplinary knowledge of mathematics – something which he knows all about.  What is hidden in amongst it all, though, is the necessary substantive knowledge.  You can’t even imagine the sorts of questions you might pose mathematically unless you know a lot of maths, and therefore appreciate both its power, and its limits.  You cannot build a model unless you intimately understand how the mathematical ideas work and fit together, and likewise for interpretation.

How do we come to understand how the mathematical ideas all fit together?  Unfortunately, over 2000 years since Euclid first said it, there is still no royal road to geometry.  If you want to learn mathematics, be able to understand mathematics and do mathematics as mathematicians do… then you must work study hard, and practice.  That means you need to learn what was discovered before, and you need to work with it in a range of contexts so that your memory and understanding of its application becomes thoroughly embedded.

Does that mean we cannot inject any of the investigation, discovery or discipline of mathematics into school?  I want to say that perhaps we can; I would certainly love to, but I’m very wary of it.  It’s too easy to surrender to the romanticism of such ideas and be blinded to how little it would actually serve children well in the long run.  What do we want them to take away from such education?  Will they remember any of it 10 years after they leave school?  What will we cut in its place?  These are serious questions that must be answered.  Still, I can see value in a society that understands what mathematics is, and what mathematicians do.  I can see harm in the complete ignorance of the mathematicians’ world.  With incredible care, and thought, I can see how a curriculum might be devised that provides this insight, without closing the road to geometry.

Lockhart’s lament is understandable because he is raging against methods of instruction that really aren’t very advanced, and really don’t lead to any great mathematical development.  We certainly can do better, and we will do better; not by abandoning aspects of traditional teaching that have been effective and giving in to a fantasy, but by bringing traditional methods under the scrutiny of modern science, refining and improving them until they are fit for the 21st century.




About Kris Boulton

Teach First 2011 maths teacher, focussed on curriculum design.
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4 Responses to Mathematics: what is its knowledge?

  1. Just to complicate matters further…

    Would it be fairer to say that mathematics in school is more a *language* than it is a *discipline*?

    I get that mathematics can be a discipline, involved in the production of knowledge, but is it the case that in order to start learning about maths as a discipline, you must first learn the language of mathematics, and that we rather confusingly call both of these things ‘mathematics’?

    A lot of the disciplines children study in secondary school – history, literature, geography – are mostly studied in English, but in the discipline of mathematics (and also in the natural sciences) the discipline is studied though the language of mathematics.

    All of this might explain why maths and languages are often seen as quite similar in their pedagogy (instruct, practice, review, etc.) because what is going on in all cases is the learning of a language: its vocabulary, grammatical rules, and so on. These are not up for debate or discussion: they are what they are (unlike, say, the causes of the First World War where there is legitimate debate within the discipline).

    Any thoughts on this? Does the language / discipline distinction make sense? And might it help us understand some of the differences that exist between maths and languages (on the one hand) and sciences, history, literature etc. (on the other?)

    • Kris Boulton says:

      I think yes and no. Take Pythagoras’ Theorem, as a simple example that everyone understands. That is not a language. It is not a part of the grammar of a language. It is a mathematical theorem: a relationship between certain well-defined mathematical objects.

      2a is equal to a + a and is also equal to 2 x a, on the other hand, is symbolic representation of an idea, or, language.

      The same is true for a being equal to 1a.

      Likewise, that = means ‘is equal to’ is linguistic

      So there *is* a language to learn, perhaps, but there is also content that I think is distinct from the linguistic.

      I think there’s some useful mileage in the comparison, but ultimately it would be deficient to draw a direct parallel between a spoken language and mathematics…

  2. ben says:

    I thinks there’s more time in the standard sequence to delve deeper than you’re giving credit for. We commonly have kids in our district finishing Calculus in 11th grade or earlier. That means in theory there are 2 years worth of instructional time available. What I think a lot of inquiry implementations get wrong is that they front load the elementary grades when the capacity for abstraction is not very developed and mostly ignore what happens by the time you hit algebra. Its not terribly interesting to spend lots of time discovering what happens when you multiply fractions for example and there’s just not that much depth there anyways. If you look at curriculum like the one offered by Art Of Problem Solving for example, its evident that once you move beyond basic computation there is a lot of space to do interesting mathematics. For K-12 mathematics to develop future mathematicians it has to gradually but more systematically at the end expose kids to proofs, problem solving and more complex problems. Its necessary but not sufficient to have a solid computational/arithmetic background.

    Lockhart’s essay is an interesting read and I admire his honesty in admitting he doesn’t know what the answer is. Wolfram’s TED Talk on the other hand is really frustrating. There are never enough details to figure what such an approach would actually look like in 2nd grade for example or how it would be developmentally feasible.

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