### 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.

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