Wednesday, June 8, 2016

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.

Saturday, February 27, 2016

[S1_20160227FZCK] Factorisation by Chunking

Problem / Question
 

Solution

Commentary
     Here I illustrate the usefulness of chunking to factorise (AmE: factor) an algebraic expression.  Observe that  3a – 2b  is a repeated part of the expression.  I call it a “chunk”.  To make it clear, I rewrite  (3a – 2b)²  as   (3a – 2b)(3a – 2b)  so that you can see it as two copies of the same chunk.  I highlight in yellow one copy of  (3a – 2b)  from each of 
(3a – 2b) (3a – 2b)   and  -3(3a – 2b).  The remaining stuff are highlighted in blue and green.  Take out the yellow chunk as common factor by writing it out on the left in the third line, shown in yellow.  You can pull out the common factor by writing it out to the right if you want, but here I chose to put it on the left.  The result would be equivalent anyway.  Once you have written out the common factor,  you write out the other stuff (shown highlighted in blue and green) into another other bracket.
     Once you understand how it works, you can actually do the second line mentally and write down the answer straightaway.  Chunking is a very useful technique in mathematics.  Here are some more examples of the technique of chunking: (1), (2), (3).

H04. Look for pattern(s)
H10. Simplify the problem
H11. Solve part of the problem

Suitable Levels
Lower Secondary Mathematics (Sec 1 ~ grade 7)
GCE ‘O’ Level “Elementary” Mathematics
* other syllabuses that involve algebra and factorisation (factoring)
* any learner who is interested





Tuesday, February 23, 2016

[Pri20160223FPDM] Pernicious Portion Problem? Shift Happens!

Problem / Question

Strategy
     This seems to be a confounding question on decimals.  What shall we do with the triangles?  Is there a short cut?

     Yes!  What you can do is to imagine putting the two triangles together to form a rectangle.  And then the solution becomes easy!  This is because the area is unchanged and hence the proportion of the shaded area is unchanged, is the same as before.  We can make use of fractions and convert it to a decimal.


Solution


H02. Use a diagram / model
H04. Look for pattern(s)
H09. Restate the problem in another way
H10. Simplify the problem

Suitable Levels
Primary School Mathematics
* other syllabuses that involve fractions and decimals
* any learner who is interested


Monday, February 22, 2016

[Maths Education] Mathematical Journalling

     Nowadays, I see some schools / textbooks asking students to search the internet and to write on a certain problem on their “mathematics journal”.  It's so very guided.  It's so artificial.  The questions should come from the learners themselves, out of their own curiosity.  The learn then seeks to answer their own questions.  The journal can serve as to document and summarise their process of learning.

     The best maths journals are self-initiated.  Great mathematician Karl Friedrich Gauss and renowned scientist Richard Feynman kept math journals on their own accord, not because some teacher told them to do it. 

     When I was a student, I borrowed books from the National Library on things out of the normal curriculum.  I kept notes of things I learned.  I also did my own investigations.  I accidently discovered quadratic equations when I was in Primary 4.  I read guidebooks, asked my friend's elder brothers and sisters, my Chinese teacher (!) and other people to find out more.  I did not like factorisation by trial-and-error.  Neither did I like completing the square nor using the quadratic formula.  So I did my own research to find a sure-fire way to factorise without trial-and-error.  I finally managed to find a way, but my method had an uncanny similarity to the quadratic formula.  It was a Pyrrhic victory, but it was fun!  I thoroughly enjoyed it.

     If students need to be told or goaded to write mathematics journals, then we as educators need to ask ourselves:  Why?  What is their conception of mathematics and education?  What experiences have they gone through that lead them to these beliefs?

     Some food for thought, eh?







Thursday, February 18, 2016

[P6_20160217RTTU] Books on Bookshelves

Problem


Introduction
     Here we have a numerically challenging problem that involves ratios, and it ultimately reduces to an algebraic problem with two unknowns.  Nevertheless, we are spoilt for choice as regards to methods of solution:-
     (1)   Bar Diagram Modelling
     (2)   explicit letter-symbolic Algebra
     (3)   “p” and “u”  (parts and units)
     (4)   Distinguished Ratio Units
     Despite the fact that Bar Diagram Modelling made “Singapore mathematics” famous, let us remember that it is only one of the ways of solving problem by diagramming, which is just one of the eleven Primary School heuristics recommended by the Singapore Ministry of Education.
     The methods have a lot in common, and they differ mainly in the form of presentation.  However, standard Bar modelling is impractical under high-stakes high-stress examination conditions for this problem, not least because one would have to cut the bars into many pieces.  One should not cut off one’s feet just so as to fit the shoes (削足适履), as one Chinese saying goes.  We need to be flexible and open-minded.  I present a solution using my own Distinguished Ratio Units.

Solution
Ans:  735 books

Commentary
     First off, we need to equalise the numerators of  2/5  and  11/4 = 5/4  and put them ratio form.   This is because the  “2”  in the  2/5  represents the same quantity as the  “5”  in  5/4.
We do this adjustment by multiplying the former through by  5  and the latter through by  2.  Thus we deduce that the original number of books in A and in B are  25  and  8  “heart” units respectively. 
     Next, we add on the  2  and  3  “triangle” units.  By doing a comparison, we can figure out that  1  “triangle” unit must be  45  more than  17  “heart” units.  So  2  “triangle” units must be equal to  34  “heart” units plus  90.  Replacing the  2  “triangle” units (shown in yellow) with their equivalent, we now know that  59  “heart” units plus 90 gives  444.  This allows us to figure out that  1  “heart” is actually  6.  Thus, we can work out what  1  “triangle” unit, and then what  5 “triangle” units are worth.

Final Remarks
     Due to the difficulty of the numbers, the solution presented above is about as streamlined as I can make it to be.  
     There is another variation that can be used – equalising the “triangle” units (akin to the technique of elimination in standard algebra).  What we do is we multiply the group with total  444  by  3  and to multiply the group with total  489  by  2.  This would give  6  triangle units on each side.  Then we can compare the “heart” units and continue from there.  This way of proceeding is not for those who fear 4-digit numbers.
     If there are nicer or more elegant ways to tackle this question, I would definitely love to hear from you.

H01. Act it out
H04. Look for pattern(s)
H05. Work backwards
H06. Use before-after concept
H09. Restate the problem in another way
H11. Solve part of the problem

Suitable Levels
Primary School Mathematics
* other syllabuses that involve whole numbers and ratios
* any problem solver who loves a challenge






Wednesday, February 17, 2016

[S1_20160117AFNS] Much Ado About Nothing?

Problem
 

Introduction
     Assuming no typing errors, this is a tricky Secondary 2 question involving an equation with algebraic fractions.  How to solve it?  How to present the solution?

Solution

Remarks
[1]   Once the LHS expression has no meaning, it would not even make sense to continue.
[2]   This is a proof by contradiction type of argument.
It turns out that not all algebraic equations are soluble (or solvable).  This problem is one case in point.  The “unknown”  m  cannot be 5/2 because that would make the expression undefined.  But if you substitute any other value, you always end up with nonsense like “15 = 0”.  So no matter what, there is no solution.  In other words, there is no value of  m  that you can substitute into the equation that makes it a true statement.

H05. Work backwards
H08. Make suppositions

Suitable Levels
Lower Secondary Mathematics (Sec 2 ~ grade 8)
GCE ‘O’ Level “Elementary” Mathematics
* other syllabuses that involve algebra
* any learner who is interested in algebra





Monday, February 8, 2016

Happy Chinese New Year 2016 (#LunarNewYear)


2016
= 12 × 168


     Wishing all my readers a Happy Chinese New Year ... or more accurately a Lunar New Year – many other Asians (e.g. the Japanese and the Koreans) also celebrate this festival.  This year  2016  is mathematically special, because it is the product of  12  and  168.  12  is a lucky number for Western people (e.g. 12 signs of the zodiac and 12 days of Christmas)  and “8” is especially auspicious in Chinese because it sounds like /  [fa in Mandarin,] which means to prosper or to grow.  168 sounds like 一路发 [yaad lou faat in Cantonese] which is to prosper all the way through life.

     I think this year will be challenging, because the Fire Monkey could be monkeying around even more with the economy and world peace.  Nevertheless if all human beings can unite together and learn to think critically, creatively and logically (and mathematics is about all these), the planet Earth can be a better place.  So best wishes to one and all!