Conclusion
It would be useless to try to write a prescriptive book about teaching mathematics.
The author William Somerset Maugham is supposed to have said: “There are three rules for writing the novel. Unfortunately, no one knows what they are.”
The same applies to teaching. Teaching is a human encounter between teacher and learners, with all the subtlety and complexity that comes with that. However, there are some helpful things we might consider, as this website has tried to show.
0.1 Remembering and understanding
Most adults seem to remember very little of their school mathematics education.
It is sometimes argued that it is still of value to have been through it, even if little is retained, but it can be difficult to say exactly what that value consists of. Once high-stakes examinations are over, perhaps it is not the little details that really matter in the long run but the BIG Ideas of what mathematics is about, and it is those that the teacher might wish would linger.
It is often a bit of a mystery why people remember some things and forget others.
The author Bill Bryson has noted1 that he
can remember the entire starting line-up of the 1964 St Louis Cardinals baseball team – something that has been of no importance to me since 1964 and wasn’t actually very useful then – and yet I cannot recollect the number of my own mobile phone, where I parked my car in any large car park, what was the third of three things my wife told me to get at the supermarket, or any of a great many other things that are unquestioningly more urgent and necessary than remembering the starting players for the 1964 Cardinals (who were, incidentally, Tim McCarver, Bill White, Julián Javier, Dick Groat, Ken Boyer, Lous Brock, Curt Flood and Mike Shannon).
Presumably Bryson cared about the baseball game, and that made him super-attentive to details that even then were not central to his enjoyment.
Not every mathematics lesson can be as much fun as a baseball game is to a committed fan. But if we can help learners at least to find satisfaction in seeing the connections between different parts of mathematics, perhaps they will also find some of those things unexpectedly memorable.
People often say that if you understand mathematics, then there is very little to remember. So much can be worked out from other things that we need not clutter learners’ heads with lots of facts that can be easily derived from others when needed.
The fictional detective Sherlock Holmes’s view of memory cluttering is often quoted:2
I consider that a man’s brain originally is like a little empty attic, and you have to stock it with such furniture as you choose. A fool takes in all the lumber of every sort that he comes across, so that the knowledge which might be useful to him gets crowded out, or at best is jumbled up with a lot of other things, so that he has difficulty laying his hands upon it. Now the skilful workman is very careful indeed as to what he takes into his brain-attic. He will have nothing but the tools which may help him in doing his work, but of these he has a large assortment, and all in the most perfect order. It is a mistake to think that that little room has elastic walls and can distend to any extent. Depend upon it there comes a time when for every addition of knowledge you forget something that you knew before. It is of the highest importance, therefore, not to have useless facts elbowing out the useful ones.
We now know from cognitive science that, although our brains must be finite, since they fit inside our skulls, our long-term memory is effectively infinite in capacity. There is no prospect of filling it up, and so we need not worry about ‘useless’ facts consuming space.
There is however something to Holmes’s concern about mental clutter. Unless we spend time organising and actively building the connections between the things we know, it will be extremely difficult for us to find what is useful when we need it. A giant heap of random books may contain the same information as a well-stocked library, but it is far less useful.
0.2 Organising knowledge
Having the right information but poorly organised can be catastrophic.
The political advisers Ayesha Hazarika and Tom Hamilton have described how they would try to trip up the Prime Minister during the very public weekly debate of ‘Prime Minister’s Questions’ in the UK Houses of Parliament by rapidly switching topics. This worked because they knew that his notes were organised alphabetically into separate files:3
[The questioner] would try to ask him something that would be in one file, and then the next in the other one … alphabetical order was a weakness … instead of saying, I’m going to ask about whatever, taxation, all the questions about tax, it was all things that have got worse or all things that have increased, I would go through six different areas of policy … Prime Ministers when they have read their briefs are reading them in silos, housing policy, health policy, environmental policy, and the Leader of the Opposition has the freedom to turn this into “we’re going to go across all of these policies”, and it doesn’t fit in with the way the briefing of the PM is structured. The first time I did that I know it really caught Tony Blair, that we were going from one subject to another.
When it comes to school mathematics, having knowledge organised according to the BIG Ideas helps learners to find what they need and make the most of the connections. If some of the smaller ideas are lost and forgotten over time, or under the stress of a high-stakes examination, they can nevertheless be recovered from the central, BIG Ideas, provided those are deeply understood.
School mathematics curricula do not contain a lot of BIG Ideas; they contain a few BIG Ideas and a lot of corollaries. A corollary is a theorem that follows easily and naturally from a previous theorem.
I think teaching students to remember easy corollaries is generally just filling their heads with mental clutter.4 To take an obvious example, it would be ridiculous to teach these three equations as three separate formulae to remember:
\[\text{distance} = \text{speed} \times \text{time} \qquad \text{speed} = \frac{\text{distance}}{\text{time}} \qquad \text{time} = \frac{\text{distance}}{\text{speed}}.\]
With knowledge of multipliers (Chapter 1), each can be easily derived from either of the other two. Obtaining the second and the third from the first is easier than starting with either of the other equations, which gives a reason for prioritising the first one. But any of the three will do.
The physicist Frank Wilczek recalls a textbook from his youth that contained a chapter entitled “Ohm’s Three Laws”. He was familiar with Ohm’s law, which connects the voltage \(V\), current \(I\) and resistance \(R\) in the formula \(V = IR\), but he “was very curious to find out what Ohm’s other two laws were”.
He was disappointed to discover that Ohm’s second law was \(I = \displaystyle \frac{V}{R}\).
He writes, “I conjectured that Ohm’s third law might be \(R = \displaystyle \frac{V}{I}\), which turned out to be correct”.5
Teaching “Ohm’s Three Laws” may give the impression of thoroughness, and seem like the teacher is preparing the learners comprehensively for every eventuality. But it takes three times as long and is far less powerful. It is giving the learner ‘three fishes’, rather than a fishing rod that they can use to catch their own fish.
School mathematics is full of corollaries, and this is fine, but they need to be taught as corollaries, and not as though they are something new, surprising and additional, which needs to be separately remembered. Learners should spend time deducing corollaries, rather than remembering them.
Someone who understands mathematics well can easily generate supposedly helpful rules for others that they would not dream of using themselves.6 This can appear to be a kindness, but generally I think is not.
Imagine being asked to remember this (true) statement/theorem:
If you add a negative number to a positive number, the answer is positive if the positive number is greater than the absolute value of the negative number, but negative if the positive number is less than the absolute value of the negative number.
If you understand addition and subtraction of negative numbers, it is easy to create this sentence, or immediately nod along with it when you first hear it. We could easily invent others like this for other possible calculations with directed numbers. They are correct, and learners might be grateful for them, but I think they are unlikely to be very helpful to anyone.
A learner might have observed the patterns with multiplication and division shown in the left-hand side of Figure 1 and wonder about the cells marked with a question mark in the addition and subtraction grids on the right-hand side of Figure 1.
\[ \begin{array}{cc} \begin{array}{|c|c|c|} \hline \times & \mathrm{positive} & \mathrm{negative} \\ \hline \mathrm{positive} & \mathrm{positive} & \mathrm{negative} \\ \hline \mathrm{negative} & \mathrm{negative} & \mathrm{positive} \\ \hline \end{array} & \begin{array}{|c|c|c|} \hline + & \mathrm{positive} & \mathrm{negative} \\ \hline \mathrm{positive} & \mathrm{positive} & ? \\ \hline \mathrm{negative} & ? & \mathrm{negative} \\ \hline \end{array} \\ \\ \begin{array}{|c|c|c|} \hline \div & \mathrm{positive} & \mathrm{negative} \\ \hline \mathrm{positive} & \mathrm{positive} & \mathrm{negative} \\ \hline \mathrm{negative} & \mathrm{negative} & \mathrm{positive} \\ \hline \end{array} & \begin{array}{|c|c|c|} \hline - & \mathrm{positive} & \mathrm{negative} \\ \hline \mathrm{positive} & ? & \mathrm{positive} \\ \hline \mathrm{negative} & \mathrm{negative} & ? \\ \hline \end{array} \end{array} \]
However, they are unlikely to find the rule above easy to understand, remember or apply.
The teacher might be tempted to try to make rules like this into catchy songs, but I think this is unnecessary. Instead, by thinking structurally (see Chapter 4), learners will find that they do not need such rules or songs to make sense of directed numbers.
0.3 Prioritising general mathematical knowledge
When learners understand addition of fractions well, they hardly need to be told how to subtract fractions.
They will need to practise the process, of course, as well as switching between addition and subtraction, and figuring out which operation is needed when. But to teach subtraction of fractions after teaching addition of fractions, as though it were a new process to be learned, is a missed opportunity to capitalise on learners’ sense-making powers.
When we teach the BIG Ideas well, we have to spend a long time on them, and we should expect learners to have difficulties. We will want to address misconceptions and spend lots of time thinking about the meanings.
When we move on to the small ideas - which still need to be taught - we should find the students thinking, “Of course!” The teacher will not want to say, “This should be easy”, but would be delighted if the learners were saying “I could have figured that out for myself. It is trivial!”
The biologist Richard Dawkins7 has described learning to drive and being taught:
not how to develop skills in general, but how to do particular things … I was taught how to reverse round a particular corner in Banbury, which happened to be the favourite corner the examiner headed for when testing that particular skill: ‘Wait till that lamppost is level with the back window, then swing hard around’.
This method was of no use on any other corner, or even in any other vehicle, so was just an examination-passing trick, rather than a more broadly applicable skill. This is the antithesis of a BIG Idea.
0.4 The power of preparation
I have alluded repeatedly throughout this website to Abraham Lincoln’s metaphor of succeeding with felling a tree by spending four hours sharpening the axe, leading to completing the cutting down in just two more hours.
We can imagine an observer arriving during those final two hours. Cutting down trees seems so effortless - your technique with the axe seems so elegant! But when they go back to their forest and try it for themselves, they seem to have very little impact on their tree, and they find it hard to understand why.
The problem is that they missed the uninteresting ‘warming up’ phase – they do not know, or they forget, all the time you spent sharpening the axe before you ‘really’ began. The felling was the bit they focused on, and they missed the essential preparation that went before it.
Lots of things are like this. Doing a great job of painting a wall is not so much dependent on the expert arm movement that draws the brush smoothly up and down the surface, and more to do with all the time spent in preparation of the surfaces – the cleaning, filling holes, sanding down. When all of that is done, the actual painting is the easy bit.
Perhaps you disagree with my choice of BIG Ideas. That is OK. I am sure there is more than one way to frame the BIG Ideas in school mathematics. For me, it matters more to have some well-thought-through BIG Ideas that you are focused on, than exactly what BIG Ideas you have chosen. Of course, if you think ‘stem and leaf diagrams’ is one of the main BIG Ideas in school mathematics, then I must disagree! But I’m sure there is room for healthy debate over the details.
But the absence of any clear set of BIG Ideas, where any idea is on a more or less equal footing with any other, seems to me a suboptimal and unmathematical approach. A learner might go home and be asked by their carers, “What did you learn this week in maths?” and they might say, “Well, we did collecting like terms, and Pythagoras’ Theorem and box plots”. The impression could be that these are random ingredients, all muddled together, where nothing particularly sticks out as being of singular importance. Any of these things might be remembered or forgotten, and the eventual result in an examination will be something like a trivia pub quiz: the more little pieces you happen to remember, the better you will do.
Instead, I want to conceive of the school mathematics curriculum as an interconnected, structured whole, where a small number of key, BIG Ideas dominate over the rest. Over time, most of us will forget many of the individual pieces – hopefully not too many of them before a high-stakes examination – but beyond school, unless the person has a mathematically-focused life, most of the little pieces will end up being lost.
But if the BIG Ideas are a strong enough, consistent focus, they will be retained and will allow the person to live with an awareness of mathematics as being a connected set of ideas that hang together and make sense, and which they feel positive about.
0.5 Further Resources
0.5.1 Books
Burkhardt, H., Pead, D., & Stacey, K. (2024). Learning and teaching for mathematical literacy: Making mathematics useful for everyone. Routledge.
Morgan, J. (2019). A compendium of mathematical methods: A handbook for school teachers. John Catt Educational Ltd.
Watson, A., & Mason, J. (1998). Questions and prompts for mathematical thinking. Association of Teachers of Mathematics.
Watson, A., & Mason, J. (2006). Mathematics as a constructive activity: Learners generating examples. Routledge.
Williams, H. J. (2022). Playful mathematics: For children 3 to 7. Corwin UK.
0.5.2 Websites
The LUMEN Curriculum, a completely free set of mathematics teaching resources for ages 11-14: lboro.ac.uk/lumen
A fantastic website for locating high-quality mathematical tasks: resourceaholic.com
One hundred outstanding formative assessment lessons: map.mathshell.org
A fantastic resource full of rich mathematical problems: nrich.maths.org
0.5.3 Other books by Colin Foster
Foster, C. (2027). Statistics Without Calculations: A guide for qualitative researchers in the social sciences. Cambridge University Press.
Foster, C. (2017). Questions Pupils Ask. Mathematical Association.
Foster, C. (2013). The Essential Guide to Secondary Mathematics: Successful and enjoyable teaching and learning. Routledge.
Foster, C. (2012). Ideas for Sixth-Form Mathematics: Further pure mathematics & mechanics. Association of Teachers of Mathematics.
Foster, C. (2012). Ideas for Sixth-Form Mathematics: Pure mathematics & statistics. Association of Teachers of Mathematics.
Foster, C. (2012). Flowchart Investigations: Explorations in mathematics. Mathematical Association
Foster, C. (2011). Resources for Teaching Mathematics 11–14. Continuum.
Foster, C. (2010). Resources for Teaching Mathematics 14–16. Continuum.
Foster, C. (2009). Mathematics for Every Occasion. Association of Teachers of Mathematics.
Foster, C. (2008). Variety in Mathematics Lessons. Association of Teachers of Mathematics.
Foster, C. (2008). 50 Mathematics Lessons: Rich and engaging ideas for secondary mathematics. Continuum.
Foster, C. (2003). Instant Maths Ideas for Key Stage 3 Teachers: Number and algebra. Nelson Thornes.
Foster, C. (2003). Instant Maths Ideas for Key Stage 3 Teachers: Shape and space. Nelson Thornes.
Foster, C. (2003). Instant Maths Ideas for Key Stage 3 Teachers: Data handling, numeracy and ICT. Nelson Thornes.
More details of these books are available at foster77.co.uk/books
Details of and links to Colin Foster’s articles are available at foster77.co.uk/articles
Colin Foster’s blog is available at foster77.co.uk/blog
Further details about Colin Foster’s work are available at foster77.co.uk
Notes
Bryson, B. (2019). The Body: A Guide for Occupants. Random House, p. 64.↩︎
Doyle, A. C. (2001). A Study in Scarlet. Penguin Classics.↩︎
Hazarika, A., & Hamilton, T. (2018). Punch and Judy Politics: An Insiders’ Guide to Prime Minister’s Questions. Biteback Publishing.↩︎
Foster, C. (2022). Methods that are just mental clutter. Mathematics in School, 51(2), 20–22. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Methods%20that%20are%20just%20mental%20clutter.pdf↩︎
Wilczek, F. (2010). The lightness of being: Mass, ether, and the unification of forces. Basic Books, p. 24.↩︎
Foster, C. (2019). Doing it with understanding. Mathematics Teaching, 267, 8–10. https://www.foster77.co.uk/MT26703.pdf↩︎
Dawkins, R. (2014). An appetite for wonder: The making of a scientist. Random House.↩︎