I use this example in my lectures when I explain the difference between arithmetic and harmonic means:

A car travelled from A to B with speed 40 miles per hour, and back from B to A with speed 60 miles per hour. What was the average speed of the car on the round trip?

Anatoly Vorobey and Vladimir Kramchatkin made a useful comment on Facebook on this quite standard and well-known problem:

“The answer is obviously 48 [miles per hour]. 95% can not solve this problem the first time. But if they are told in advance that the distance between A and B is 120 [miles], 95% of schoolchildren will easily solve this problem.”

A concrete number, 120 km, serves as a strong hint that students are expected to do something with this number. But, for majority of students, if a magnitude or a quantity is not assigned a concrete numerical value, it does not exist. This is one of the flaws of mathematics education at schools: no-one tells students that they have to be able to see hidden parameters in arithmetic problems. But this is not the only flaw: students are also not told how to check solutions. Checking answers frequently benefits from seeing a problem in a wider context and varying the data. The standard answer that students give to the problem with the car is 50 miles per hour, the arithmetic mean of the two speeds. But this solution immediately collapses if we slightly change the problem: what would happen if the speed of the car on its way back from B to A was 0 miles per hour?

There are two very different types of perfectionism.

Type A: interiorised perfectionism driven by personal, internal criteria. As an apocryphal story goes, one of the presidents of Harvard University was once asked what was so special in teaching at Harvard to justify their extortionate fees. His answer was of just three words: “We teach criteria”. Cambridge appears to be the only university in Britain which teaches criteria. I know a criterion when I see one — I myself was lucky to get my own education at a boarding school and an university which taught criteria. Among my mathematician colleagues (and co-authors) I know a number of Type A perfectionists. Some of my friends teach or spread criteria — by means of art classes, or mathematics circles, or lectures on history of mathematics, or poetry evenings …

Type B: external perfectionism, a Pavlovian dog reflex to meet crude criteria, at the level of metrics, rankings, “likes” on social  media — all of them imposed from outside. In modern world, most perfectionists belong to Type B. IMHO, the best inoculation against the soul-destroying Type B perfectionism is development of a system of deeply interiorised personal criteria. In principle, this is what education should give to every child. It fails. Moreover, the vast majority of schools and universities spread the disease.

There are two very different types of perfectionism.

Type A: Interiorised perfectionism driven by personal, internal criteria. As an apocryphal story goes, one of the presidents of Harvard University was once asked what was so special in teaching at Harvard to justify their extortionate fees. His answer was of just three words: “We teach criteria”. Cambridge appears to be the only university in Britain which teaches criteria. I know — I myself was lucky to get my own education at a boarding school and an university which taught criteria. Among my mathematician colleagues (and co-authors) I know a number of Type A perfectionists. Some of my friends on Facebook teach or spread criteria — by means of art classes, or mathematics circles, or lectures on history of mathematics, or poetry evenings …

Type B: External perfectionism, a Pavlovian dog reflex to meet crude criteria, at the level of metrics, rankings, “likes” on social media — all of them imposed from outside. In modern world, most perfectionists belong to Type B. IMHO, the best inoculation against the soul-destroying Type B perfectionism is development of a system of deeply interiorised personal criteria. In principle, this is what education should give to every child. It fails. Moreover, the vast majority of schools and universities spread the disease.