Not even close

Last year, I showed how the level of math-anxiety is so great in certain sections of The Institute that students will put down crazy answers in tests, quizzes and exams just to get it over with. In that case the majority of my students stoutly asserted that 44 concrete blocks occupied half a billion cubic metres.  Now it's another year, another batch of students sitting a midterm in Quantitative Methods = remedial maths. I wasn't about to ask the concrete block question again but I wasn't going to shy away from casting the "math" questions in contexts approximating real life.

Q13. The volume of a sphere is given by the formula 4/3πr³. If a soccer ball is 22cm in diameter, what is its volume in litres?  Everyone in the room has handled a 1 lt carton of milk or orange juice and a soccer ball or seen one on TV.
A. It is just bonkers to assert that the volume is 44,000lt. That's about the size of the largest road tanker we have on the roads in Ireland. Nevertheless 2 students offered this as an answer. Actually they offered 44,602.2 lt.  Which showed that they didn't know the difference between a diameter and a radius and that one had probably copied the other's answer.

Q16. Given that the period (tick . . tock) T of a pendulum is given by T = 2π*√(L/g), that g the acceleration due to gravity is 9.8ms^-2 and the length L of the pendulum is 50cm, how long does it take for the pendulum to swing one complete cycle?
A. I can't claim that everyone will have seen a pendulum up close: only a minority of a homes have a grandfather clock. But we have all been in a children's play ground with swings. The almost universal answer to this question was 14 seconds; because they treated the length as 50 rather than 0.5m. You can run 100m in 14 seconds; how could a pendulum half a metre long take that long to swing forward and then back?? In the post-mortem I whipped out my car-keys and swung them from their lanyard - which is about 50cm long - to show that it takes a tad under 1.5 seconds.

Somehow, I/we have to get kids to realise that the units count! Merely plugging numbers into a formula is the easy part, evaluating the answer for sense is another level of competence altogether. We should have a class in guesstimation, like in Mathsemantics Home Ed class.  It might include numbers like counting the spaces for receptor proteins in a typical mammalian cell. More:

• The full moon was super big on Monday, how far from your eye do you have to hold a dime to eclipse it?
• Is it possible for all 7.5 billion people to stand comfortably in Co Carlow = 900 sq.km?
• in the Republic of Ireland 70,000 sq.km?
• many of them all small
• If a cell is mostly water and so are you, how many cells are there in a bloke weighing 70kg is the average diameter is 30μm?
• How thick is a sheet of 80gsm A4 xerox paper if a 500 sheet ream weighs 2.5kg?
• How many blades of grass did Louis Agassiz inspect during his Summer beetle collecting?

These are all 'first order' problems not requiring anyone to integrate data from several different domains. More complex puzzles are often called Fermi Problems and are much beloved by interviewers at Google, Apple and Megacorp. How many piano tuners in Chicago? etc. Here's a clearing-house site for lots of links on the subject. And here's a list from U.Maryland.  Finally a question turned into a judgement by Dorothy Parker "If all the girls who attended the Yale prom were laid end to end, I wouldn't be a bit surprised." [More]