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All entries for January 2018

## January 08, 2018

### Can’t do maths? Try Fermi problems instead! – Jeremy Burrows

I talk to Jean - an intelligent, articulate Glaswegian who left school with no qualifications - over the internet. I told her about my essay on re-engaging students who are “switched off” to maths.

“I can’t do maths.”

“Perhaps you could if you were taught using Fermi problems.” Being keen to try this approach, advocated by some of the literature, I told her how Enrico Fermi solved problems such as “how many piano tuners are there in Chicago?” without any data, using his knowledge to supply defensible estimates.

“I don’t care about piano tuners in Chicago.”

“Fermi problems can be about anything,” I said. “The teacher should choose a problem which is meaningful to their students.”

“How?”

“You’re interested in cycling, so … how long would the averagely fit person take to cycle from Land’s End to John o’ Groats?”

“I don’t know. Two weeks? Three weeks?” Jean was engaging with the problem, but she was guessing. She needed some scaffolding.

“Don’t guess: use your knowledge. How many hours a day will they spend in the saddle?”

“Eight.” This sounded too high but I didn’t want to discourage her, so I said nothing.

“What will their average speed be?”

“40 mph.”

No way! That’s the speed of a world champion, not the averagely fit person. I challenged it, and after discussion Jean settled for 15mph.

“How many miles will they cover each day, riding at 15 mph for 8 hours?”

“120” was Jean’s instant response: she was “doing maths” without even realising it!

“What’s the distance from Land’s End to John o’ Groats?”

“I don’t know. About 1,000 miles?”

The actual answer is 874. I would have accepted 1,000 as an estimate, but she had guessed. You mustn’t guess when solving Fermi problems.

“Don’t guess. How can you estimate it?”

“I don’t know.” Jean was slipping into “maths lethargy”, but I was not going to let her give up.

“Suppose I tell you there’s a sign outside King’s Cross saying Edinburgh 401 miles? So London to Edinburgh is about 400 miles.”

“1,000 miles is two and a half times that.” Jean was re-engaging, and we were back in the hunt!

“Think about the map of Britain. Is Land’s End to John o’ Groats two and a half times London to Edinburgh?”

A pause, then: “Yes, it’s about that.”

“Good. So you can defend your figure. How many days will it take to cycle 1,000 miles at 120 miles a day?”

“Eight.”

“What’s 8 times 120?”

“960. Oh! So nine.”

“Is that your answer? Nine days?”

“Yes,” said Jean. Then, “No! Some days they might not do 120 because of punctures, or hills. So, ten.”

“Is that your final answer?”

“Yes.”

“Are all your numbers defensible?”

“Yes.”

“OK. Let’s check it.”

I consulted Wikipedia. Land’s End to John o’ Groats cyclists normally take 10 - 14 days. Jean’s sense of accomplishment when I shared the link was palpable: she had just solved a Fermi problem and absolutely nailed it! In doing so she had multiplied 8 by 15 and 120 by 8; divided 1,000 by 400; estimated comparative distances; and identified and applied an appropriate rule of rounding. She had “done maths”; and I intend to incorporate Fermi problems into my teaching practice.

### Mind the Gap! (between what is taught and what is learned) – Jeremy Burrows

I had three Latin teachers. Between them they taught me everything I know about Latin … and a whole lot more besides. It is this “whole lot more” which concerns me right now. We have taught something to a class, but that doesn’t necessarily mean that all of them (or even any of them) have actually learned it. In my previous career with HMRC I was concerned with closing the “tax gap”: the difference between the amount of tax which was owed and the amount of tax which was actually collected. In my new career as a teacher I am concerned with closing the “learning gap”: the difference between the things we teach and the things our pupils actually learn.

In mathematics, this problem is particularly acute. Children do not like learning mathematics. Boaler (1996, 1997, 1998) and Nardi & Steward (2003) have conducted studies which offer some very keen insights as to why this is: it’s all down to the way in which we are teaching them. When I learned woodwork at school, they didn’t just show me how to use a drill, tell me to drill 16 holes in a piece of wood, and then move on to the next skill as soon as they judged that I had mastered drilling holes (or move on even if they didn’t judge that I had mastered drilling holes). They taught me woodworking skills in context, and I actually used them for a purpose. I made a box, with dovetailed corners and a hinged lid. I veneered a chess board on the top and a backgammon table on the inside. I never finished it, but that doesn’t matter: I learned the skills and what they were for. I used the skills in order to make something. I have not forgotten those skills.

In mathematics, we just seem to teach the skills, without any context, and then move on. We cannot take time to explore how those skills are used in practice, to set them in context, because then we wouldn’t have time to get through the whole syllabus. And we must teach the whole syllabus, mustn’t we?

Must we?

In Boaler’s comparative study, the pupils who were taught the traditional way covered the whole syllabus. The pupils whose teaching consisted of open-ended mathematical investigations did not. There were gaps in their teaching. But … they performed better in their examinations, because the “learning gap” was smaller. They were taught maths with a purpose. They “made boxes”, rather than just “drilling holes”. At the end of the day there was a closer identity between the maths they had been taught and the maths they had learned, and they understood how to use what they had learned.

Maybe we need to spend more time in the mathematics classroom “making boxes”, and not be afraid if this results in a “teaching gap”. It is, after all, the “learning gap” that matters.

References

Boaler, J. (1996) Learning to lose in the mathematics classroom: a critique of traditional schooling practices in the UK. International Journal of Qualitative Studies in Education 9 (1), 17 – 33

Boaler, J (1997) Experiencing School Mathematics: Teaching styles, sex and setting. Maidenhead (Open University Press)

Boaler, J. (1998) Open and Closed Mathematics: Student Experiences and Understandings. Journal for Research in Mathematics Education 29 (1), 41 – 62

Nardi, E. and Steward, S. (2003) Is Mathematics T.I.R.E.D? A Profile of Quiet Disaffection in the Secondary Mathematics Classroom. British Educational Research Journal 29 (3) 345 - 367