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Singapore PSLE Math baffled Anxious Parents

One afternoon 5 friends rented 3 bikes from 5 p.m. to 6:30 p.m. and took turns to ride on them. At any time, 3 of them cycled while the other 2 friends rested.

If each of them had the same amount of cycling time, how many minutes did each person ride on a bike?

Note: PSLE (Primary Schools Leaving Exams) is the Singapore National Exams for all 12 year-old pupils at Primary 6 year end. The result of which will determine which secondary school the pupil is qualified to enter the following year. Math subject, besides Science, English and mother tongue (Chinese or Malay or Tamil) are tested in PSLE.

[Answer] Try before you scroll down below ….

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Answer = each student rides 18 mins per bike (= 90 mins /5 ).

The “3” bikes are tricky “smokes” not relevant, it could be any “n” (<6) bikes , as long as total 90 mins, and each student rides same duration.

Why do the French excel in maths ?

It is not only owed to the Ecole Normale Superieure where the 11 Fields medalists were educated, but the prominent “Math Culture” in French society.

This is similar to the International Math Olympiad (IMO) “craze” in China since 1980s till today, where the parents send their primary school kids to drill in IMO boot- camps, because that is a “direct-entry” gateway to enter top university, bypassing the highly competitive “Gao-kao” 高考 (University Entrance Exams for 500,000 places among 7 million students each year, only 7% successful chance !! vs Singapore 40%).

https://www.theguardian.com/world/2012/may/31/europa-french-maths-ecoles-normales-superieures

Quora: read how this middle-age (48) French recounts his “French Math” education since 6 to 15 years old:

The curriculum was designed in the late 60s in part by a group of real mathematicians, the Bourbaki. It was very abstract. I learned about basic set theory when I was 6. I did learn about the basic operators but not before I was able to perform them in arbitrary bases from 2 to 10. I was taught the properties of ordering relationship, equivalence classes and partitions at age 11, vector calculus at age 13. Basic differential calculus was taught at age 15, complex numbers and integral calculus at 17. This was carried over to physics where we were happily solving ODEs for mechanics and electrical circuits at age 16. This was for almost everyone except the most dyed-in-the-wool literary person. I was taught elementary proof techniques at age 13. By age 15 I knew about formal logic, contradictions and recursive proof.

https://www.quora.com/How-does-the-publics-perception-of-mathematics-differ-between-France-and-America?from=timeline

How Mathematicians Think

Hadamard estimated that :

About 90% of mathematicians think visually, 10% think formally.

Usually, they think in steps:

  1. Get the right idea, often think vaguely about structural issues, leading to some kind of strategic vision;
  2. Tactics to implement it;
  3. Rewrite everything in formal terms to present a clean, logical story. (Gauss’s removal of ‘scaffolding’ – middle working steps)

Source: [NLB #510.922]