The passing of a Great Giant - Jerome Bruner

Jerome Bruner, a great psychologist, has died recently.  We thank him for his theories that guided the development of Singapore Mathematics education.

One of his greatest theories that is useful for children's learning is the Concrete-Pictorial-Abstract (Enactive-Iconic-Symbolic) approach.

It is impossible to use only words to explain simple mathematical concepts to a young child, since those concepts cannot be further explained using words.  You will run out of words to explain!  What definitely does not work is to start from use just words and they don't "get it", then scold them for being stupid.

They have to learn buy touching and playing with things, then from looking at pictures and then using abstract words or reasoning.

[MathEd] Mathematics is not Memorisation

Article
Memorizers are the lowest achievers and other Common Core math surprises


My Comments
Not very surprising to me, actually.

Quotes
"Mathematics is a broad and multidimensional subject. Real mathematics is about inquiry, communication, connections, and visual ideas. We don’t need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms. All of these ways of working are encouraged by the Common Core."

[OlymUSec_20150412BDP] Guessing Cheryl’s Birthday

Question

Introduction

     This “Primary 5 mathematics” (actually an upper secondary Olympiad) logic puzzle has gone viral.  It has been making its rounds in various forums in Singapore and overseas, stumping adults and children alike.  It is actually a parody of an old puzzle.  Can it even be solved?  It seems that there is no information given by each parties that we can exploit.  Actually there is!  In a subtle way ...

Solution

     In the beginning, everybody knows that Albert knows only the month and Bernard knows only the numerical day of the month.

     When Albert tells us “I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.” he is leaking out information (from his knowledge of the month) that the day of the month appears more than once and cannot be (June 18 or May 19).  Actually, the original phrasing is more like “If I don’t know when Cheryl’s birthday is, then Bernard does not know too.”.  The person who set this question merely changed the names of the people and the dates, without appreciating the subtle but crucial difference between a statement of fact and an implication (an “if ... then ... ” statement). 

     Ruling out June 18 and May 19, we also know that Albert knows that the birthday month is neither June nor May.  Otherwise, how would he have been so confident in saying that he knows Bernard would not know Cheryl’s exact birthday?  So we can eliminate those months.

     Bernard acknowledges the above state of affairs and the embedded hint.  With the choice narrowed down and with his knowledge of the numerical date, he now knows Cheryl’s birthday.  Since we know that Bernard knows Cheryl’s birthday, we know that it cannot be a numerical date that appears more than once (otherwise he would not have been able to know).  So we can cross out July 14 and August 14.

     Now Albert would telepathically thank Bernard for this helpful hint.  Because now he is able to deduce Cheryl’s birthday with his knowledge of the month.  That would mean that this cannot be a month with two candidate dates.  We blot out the August dates and see for ourselves the only remaining possibility.

Conclusion: Cheryl’s birthday is  July 16.

Remarks

     This puzzle was solved using the process of elimination and analysing our knowledge of what each party knows and can know.  Thus we successively narrow down the possibilities until the answer becomes obvious.  Here we learn that

     knowledge of other people’s knowledge can itself give us knowledge. 

This principle is actually employed in cryptology (the use of secret codes) which finds applications in fields like banking, the military (cf. interesting story of how the German Enigma code was broken in WorldWar II) and communications.  As an example, radio communication can tell the enemy of troop positions and warn of an impending attack, and that is why radio silence is imployed as a precaution.  Sensitive information in certain organisations is restricted on a “need to know” basis.

Suitable Levels
* Upper Secondary Olympiad
* other syllabuses that involve knowledge or epistemology
* application of mathematical principles in real life
* for all people interested in logic puzzles