Absolute vs Relative

A recent episode of Radio 4’s “More or Less”, addressed the issue of Progress 8, which is obviously interesting to me as a teacher. However, it was the discussion about poverty in the UK which most caught my attention.

The Trades Union Congress (TUC) recently hit the headlines by pointing to statistics which showed that the number of children from working households who are in poverty has significantly increased in the last ten years. They claim that the main drivers of this have been cuts to in-work benefits and restrictions on public-sector pay. The government’s response: It doesn’t recognise the TUC analysis; there are one million fewer people living in absolute poverty.

There a few extra details in the programme, but the gist is that both claims are correct. Relative poverty is increasing, but absolute poverty is decreasing. So the question really about which we value as a society? In the UK, my feeling is that the focus should be on relative poverty (although what I have written next has made me question this slightly!). Indeed, I’m surprised that there are many people at all living in absolute poverty: I know I live in a social bubble, but I suspect that the government figures are not based on the international definition as set by the World Bank.

Returning to education, I feel that a similar debate that has been ‘raging’ on twitter for a few months now (perhaps even longer), boils down to the same issue.

Is Ofsted biased against schools in more deprived areas? Clearly, many people on twitter are convinced by Stephen Tierney’s recent blog post  on the topic and regular references to this graph:

It shows that schools with a high proportion of White British children receiving Free School Meals are judged, on average, much worse than schools who have fewer children in this group. The immediate conclusion is that Ofted is biased against these schools. Surely the proportions should be the same for all types of schools? No.

Why not? Because Ofsted’s standards are absolute, not relative. As Jason Bradbury and Sean Harford explain, the evidence shows that when looking at schools with the same Progress 8 measures, inspectors actually give more generous judgements to these ‘most deprived’ schools.

This thorough treatment of the issue points out that there are many reasons why it’s difficult for schools in ‘deprived’ areas to attain the same absolute standards as schools in more affluent areas. However, this doesn’t mean that we should instead use relative judgements: that would be to accept that it’s ok for children growing up in disadvantaged areas to go to schools with lower standards.

What the analysis does show, however, is that it’s much harder to run a good or outstanding school in underprivileged areas. As a result, perhaps management and staff in these schools should be rewarded / treated with leniency to a greater extent than those in prosperous areas? Similarly, should these schools be funded more generously?

Overall, this has got me thinking about whether we need to get better at teaching the key idea of ‘Absolute vs Relative’ in maths classrooms. Up until now, I haven’t taught it explicitly… another one to add to my scheme of work, perhaps.

(Disclaimer: although I am rather convinced by Ofsted’s blog, I don’t think it proves beyond all doubt that there is no bias: judgements clearly account for progress 8 weaknesses, but to what extent?)

Things that are NOT TRUE

A colleague at a previous school produced this poster (it goes on to 29 points: click the picture to download the full version)

If you’ll allow me a brief aside, credit for this poster goes to Robin Bhattacharyya. Robin was the most knowledgeable person I have ever met and at the time, one of my heroes. Originally a University Challenge champion in 1995, he went back for more in 2011 Christmas special, in which he led his college to victory over a more recent hero of mine, Daisy Christodoulou. (strictly, it was Trinity vs Warwick but from memory, it may as well have been Bhattacharyya vs Christodoulou!)

Anyway, back to the maths… A recent tweet from Stuart Price reminded me of this poster.

Stuart was focussing in on one specific type of misunderstanding which features heavily in Robin’s poster: the over- or under-generalising of the distributive property.  Trying to head off these problems early, I’ve been making a specific effort to teach year 8 about distributivity this week. It has gone better than in the past, so I thought I’d share what (I think) I’ve learned.

In the past, the phrase “multiplication is distributive over subtraction” sounds clunky and just saying it, let alone understanding it, has been trouble enough for my pupils. I’ve replaced it with “multiplication distributes over subtraction“. This small change has had a surprisingly large effect. I think this is partly because it relates more closely to the usual English usage of the verb ‘to distribute’; indeed, this also allows for visualisation: the multiplication is literally being distributed over the subtraction.

Secondly, Having checked whether several operations distribute over each other, I asked my pupils to generalise which operations distribute over others. In this section, Colombe introduced a metaphor which helped her peers to remember the general rule: Multiplication and Division are the government: they distribute (resources) to the citizens, Addition and Subtraction but not to themselves. This also ties in very nicely with Order of Operations.

I also used a three-act-math approach in these lessons, introducing the topic with a classic moment from countdown (if you’re a maths teacher, you must have seen this!)

This was my hook: how on earth did he do this? Did you notice that he didn’t actually know the intermediate numbers? This gained interest, and later on in the lessons I tried to set some questions in which pupils applied the distributive property to simplify some expressions, leading up to (318 x 75 – 50) / 25, but it didn’t go well. If I had used the strategy that John Mason suggested on Craig Barton’s podcast, of imagining what I was going to say and how the lesson would proceed, I feel like I may have anticipated this. It wasn’t clear that the equivalent of 318 x 3 – 2 was the ‘final answer’ in the simplification problems I had created. In my updated lesson plan, I have removed these questions, leaving just the explanation of the countdown genius as a teacher-led section of the lesson.

Another issue that came up as part of these lessons was a good question from my pupils: If it doesn’t distribute, what does happen? My answer at the time was fluffy… something like “it depends on the situation, usually you just apply the operation to one of the numbers”. I feel like I need to work on this, perhaps I need to add another sequence of lessons to my combining operations thread.

Overall though, I feel that this is the first time I’ve been even vaguely successful when teaching the distributive property, mostly because I managed to distill it, with the help of a pupil’s metaphor, to this:

One final reason why I think it went better than in the past: I was more committed. Previously, I would always question the value of teaching such lessons, because I suspected that the pupil’s future teachers wouldn’t make reference to the distributive property. Now I know that I will be teaching these pupils for several years and as HoD, I can better integrate this topic into our scheme of work, it is worth the investment.

How a Mathematician Solves a Problem

One of my colleagues at a previous school emailed round this problem and it caught my attention. Possibly just the bright colours, but it has a simple beauty to it, reminding me of the sort of problems that Ed Southall often shares.

I got stuck into it straight away, first putting together the triangles in a rather naive way that wasn’t any use at all. Realising that I wasn’t getting anywhere, I changed tack and went for the ‘brute force’ approach of applying trigonometry to get the answer.

Key point: if one approach isn’t working, don’t get stuck in a rut. Ask yourself what other mathematical concepts you know that may be related to the problem, and try those instead.

Getting an answer of 180 degrees suggested to me that there would be a neater approach., probably involving putting all three of these triangles into one large triangle. I started trying to do so.

Key point: Get as many ideas on paper as possible, don’t give up if your first couple of attempts don’t work out.

This is a bit of a mess, but in retrospect it’s my favourite page. Trying out a load of options quickly can give you a feel for what might work. I eventually got the idea that two of the angles needed to be on the ‘base’ of the triangle, and it was then a matter of varying the two angles and the distance between them until the third angle appeared at the top.

Key Point: Sometimes drawing a geometry problem more neatly will help you more than you might expect.

Then I decided to start trying to draw this out a bit more neatly; this helped me notice that I could create a key isosceles triangle.

Finally, of course you should write your work up neatly in order to present it to your readers (in my case, a triumphal email to my colleague!)

Key point: Rough working is great, but make sure you write up your solutions neatly at the end. Of course, your teacher will be even happier if they can see your rough thought process too.

I looked at my work at the end of the process.
a) I felt that sense of satisfaction that reminded me why I like maths.
b) I thought it would make a good lesson for my pupils.

In summary, here are generalised versions of my strategies:

  • If one approach isn’t working, don’t get stuck in a rut. Ask yourself what other mathematical concepts you know that may be related to the problem, and try those instead.
  • When you’re confident you’re going in the right direction, get as many ideas on paper as possible, don’t give up quickly.
  • Sometimes drawing a geometry problem more neatly will help you more than you might expect.

I know that the idea of general problem-solving skills (even within the domain of maths) is controversial. Of course, I wouldn’t have been able to solve this problem if I didn’t have a secure understanding of properties of triangles (obviously), proportional reasoning (to scale up and down the triangles) and trigonometry (to give me faith that it was worth continuing!)

However, I do feel that some of the general strategies suggested in  Thinking Mathematically have improved my ability to solve this kind of problem. It was interesting to listen to Dylan William’s thoughts on this topic: Craig Barton suggested that he was moving away from the idea that general problem-solving skills are useful and Dylan was less convinced that he learned general skills from George Polya’s classic book. He did admit, however, that ‘trying a simpler case’ (in the terms of Thinking Mathematically: ‘specialising’) is a pretty useful general principle. This, alongside writing down what you know and want and drawing a bigger, clearer diagram, are the main general strategies which I’ve seen maths students use effectively.

Personally, I’m fairly confident that I wouldn’t have been so effective at solving this problem 10 years ago, when I’d just graduated with a first class maths degree. If nothing else, teaching makes you a better mathematician.

Want to teach your pupils general problem-solving skills? Take a look at my resources here: http://lukepearce.eu/problem-solving/

#mathsresourcechat

I love collaboration, which is why I am very happy to have discovered the educational twitter community. I enjoy the general ideas and big-picture teaching discussions that take place, but I particularly love the nitty gritty: discussions of the practical side of teaching.

I find that I get the most ideas that I can apply straight into my lessons when people share resources and others makes suggestions for improvement.

I’ve done it a few times myself recently, and had really useful feedback from other twitter users. A few other people are doing it, but I’d love it if it was even more common. I find some such posts on #mathschat and #mathscpdchat, but there is also a lot of more general patter.

So to focus in on the detail, I’m going to start using #mathsresourcechat and encouraging others to do the same.

As the examples show, posting a picture of the resources makes it particularly easy for others to see and comment.

Go forth and start resource discussions!