Homeschool maths

Homeschooling has become a household word during pandemic lockdown, but both my grandchildren have been homeschooled since before the pandemic. Surrey historically has too few state school places, and Joshua was not offered one locally so would have had an hour's commute to a school in the next town. My daughter Jenny runs a pre-school and has a good educational brain, so took the brave - and so far very good - decision to homeschool. Since they all live with us (we are what in pandemic times is called a 'multigenerational household') I get to see homeschooling in action.

Although I spent much of my career teaching, it was in industry and at Masters university level so I am involved mostly in opining wisely about strategy rather than actually teaching school level. Since I studied physics and spent my career in physics, computing and mathematics I am handy to have around to answer questions but despite having been a school governor for ten years and chair of governors for four, the National Curriculum still defeats me in terms of relating its aims to its implementation: I pretty well understand its aims and objectives but all that sensible reasoned argument and strategic thinking is reduced to what amounts to the nightmare PowerPoint slide from Hell - hundreds of bullet points with no stated dependencies, background reasoning or flow of content - and my systems-oriented mind just can't wrap around that and trace the bullet point lists back to underlying reality. So I tend to be there in case of questions but my own contributions are more short inspirational practical design, build and test experiments and computing projects.

Being there for homeschooling is illuminating, though, and the questions that arise are interesting.

By far the hardest, for me, is maths: not because the maths is hard - I can without boasting say I am very, very good at maths, and the maths at homeschool level is mostly quite easy - but because it is harder to explain why something is being taught in the way that it is. I think this is because maths is an answer in search of a question, whereas science, computing and engineering pose questions in search of solutions. In science, in particular, we are taught to formulate a 'well posed' question: one that is clearly stated, limited in scope, well defined and in such a way that its suggested answer may be tested in a clear and unambiguous way. We pose the question in such a way that we can recognise when the answer is reasonable. Along the way we make use of maths as a tool, both to express the question and to answer it, but the maths is chosen to fit the problem and is often precisely defined in clear mathematical terms. Although science can get complicated, its questions are relatively simple to state, and when stated are quite unambiguous - at least in the physical sciences. Maths is different, because it is a tool that can be used in varied and contrasting ways - and so if we are to apply it usefully rather than just learn it for its own sake we must be able to decide which way to use the tool.

There is a saying, that if all you have is a hammer then everything looks like a nail. Which is funny but misleading. A hammer is a versatile tool: you can hammer in a nail but with a claw hammer you can also pull one out, so the hammer has an inverse function, a reciprocal; you can use the hammer to bash things together or to smash them apart, to hammer something flat and smooth or to make dents in it. So one might ruin the hammer joke by saying that if you want to hammer in a nail, use a hammer - but if you want to pull a nail out, then also use a hammer but in reverse, and if you just want to smash things up then use a hammer too. Maths is like that - each mathematical tool you learn can be used in many and varied ways, and often has an inverse that undoes what you or someone else did. If we learn maths in the abstract, without learning how and when to apply it, then we indeed risk being in the position of thinking everything is a nail. We need to learn how to use the tools, but like a good crafstman we will use them best if we learn and practice using them in their many and varied ways, not just to bang nails. Nowhere shows this more than social media.

There are many social media maths memes guaranteed to spark fury that is rivalled only by extreme politics. One goes like this:

"My child was asked to write 5x3 using the rule of repeated addition, and wrote 5+5+5 but the teacher marked it wrong and said it should be 3+3+3+3+3!"

Cue outrage.

The thing is, rules like 'repeated addition' are designed to model real-world applications - with multiplication each such rule models a different way of doing things - but if we abstract to numbers alone then that meaning, the application, is lost: 3x5 = 5x3 = 5+5+5 and so on. If we write 5x3 in words it adds back a suggestion of something the simple arithmetic lost: "five times three" - five lots of three; and if I add words to elucidate: "five times three items" then we can begin to see that in the real world five lots of three items differs from three lots of five items. If you send me to get five party bags, each with three different party favour items - one bag for each of the five children at the party - then you will probably not be happy if I come back with three bags, each containing five items: I solved a problem you did not have, and failed to solve the problem you did have, even though I got 15 items.

Sarah teaches the homeschool maths: she does it very well and there is a lot of laughing, some puzzling, and some asking Grandad. It is often like an episode of The Joy of Maths - a happy fun time that is so different to the maths lessons I recall from (non-home..) school. Most of the questions to Grandad are more about what is the question than what is the answer: and I recognise that from my own work where figuring out the right question - and the right way to pose it - is much more of the job than simply computing the answer. She uses White Rose Maths - which I find inspiringly good and most homeschool parents seem to find infuriatingly unlike 'the maths they learnt at school'. White Rose often pose their maths problems in cryptic ways, that are designed to provoke questioning, discussion and argument (I think this is one reason people find it so infuriating, because at school we were set quite narrowly stated mathematical questions and not supposed to ask what they meant): here is an example:

"Jack and Annie are practising their 8-times table:

• Jack says: "to multiply any number by 8, you can multiply it by 4 and then double it
• Annie says: "to multiply any number by 8, you can double the number 3 times

Who do you agree with?"

(White Rose always adds: "talk about it with a partner", which is a clue that they want you to talk about it, not to furiously argue that the child should just learn the damn 8-times table and to hell with discussing it).

Here, both Jack and Annie are right (White Rose can be tricky like that..), but why would we want children to learn these alternate ways, when they could - as some on social media suggested - indeed 'just learn the 8-times table'? The reason is that at this stage the child is indeed meant to know their 8-times table: but this is not a test of mental arithmetic, it is an exercise in breaking down problems into manageable chunks - called, expressively, 'chunking' (and known by that name to computer engineers who use it to address complex problems). Here, both Jack and Annie demonstrate ways to break down the numbers involved into smaller chunks - 'multiply by 4 then double' - and with these small numbers we can all check the answers and see that Jack and Annie are indeed right. (Oh God I have heard and read so much White Rose Maths that I am now starting to add 'Jack and Annie' into everything I write, and will soon start to invite you to 'pause the video and have a little think'). But back to the question posed - increasingly angrily - on social media - why not 'just learn the 8-times table?'. Well, let's try it with the 144-times table, shall we - surely we all 'just learnt' that? As the numbers get bigger so we have to go beyond the simple basics that we learnt by heart and start to break the problem down, into chunks that we, each in our own individual ways - or the computers we may be programming - find easier and more efficient to implement. (Notoriously, computers are great at their 2-times table and absolute rubbish at much else, so a good computer engineer will chunk anything down into 2-times table problems if they can - and can usually do the not-as-interesting-as-they-think programmer's party trick of reciting their 2-times table up to 65,536..).

But back to the claw hammer. Division is a strange operation, because it is defined only by reference to the different operation of multiplication: "division is the inverse of multiplication" - it does not exist on its own. Division poses the question: "by what number did you multiply to get this result?" - so 8 divided by 4 is asking the question: "by what number did you multiply 4 to get 8?" (don't argue with me about this, them's the rules, and I don't make the rules, maths is a hard master and if you argue with it you lose). So division is the claw of a hammer, to undo what you did by banging in the nail of multliplication. Yet division has real-world interpretations that do not depend on having multiplied: it can model real-world operations, the classic examples being 'grouping' and 'sharing' - which give the 'same' answers but to different questions. In grouping, 8 divided by 4 is asking: "how many groups of 4 can you make, from 8?": whereas in sharing, 8 divided by 4 is asking: "if you share 8 things equally between 4 groups, how many does each group contain?" - so in grouping 4 is the number of groups but in sharing 4 is the number in each group - and neither is the inverse of multiplication, because nobody asked: "by what number did you have to multiply 4 to get 8?". If it's hard to follow this when expressed in abstract numbers, make the numbers meaningful - chidren, for example: then you can imagine 8 children forming groups of 4, or forming 4 groups - and nobody asked how you ended up with 8 children.

So White Rose Maths is not just teaching maths: it is inviting children to think about the problems to which that maths might be applied, and about different and varied ways of applying the same maths tools to real-world problems. It is inviting children to think about what questions to ask, and how to ask them of maths, in mathematical form, and how to solve them by breaking them down into manageable chunks: it is asking them to think about maths - and that is harder than doing it, and more inspiring.

Now, pause the blog, and have a little think.

(Argue furiously about it with a partner..)