Singapore Math Books on the Bar Model Method

In recent years, because of the popularity of Singapore math books being promoted and used in many countries, suddenly local publishers seemed to have been hit by an aha! moment. They realized that it’s timely (or simply long overdue?) that they should come up with a general or pop book on the Singapore’s model (or bar) method for the lay public, especially among those green to the problem-solving visualization strategy.

Monograph à la Singapour

The first official title on the Singapore model method to hit the local shelves was one co-published by the Singapore’s Ministry of Education (MOE) and Panpac Education, which the MOE christened a “monograph” to the surprise of those in academia. Thank God, they didn’t call it Principia Singapura!

The Singapore Model Method
A wallet-unfriendly title that focuses on the ABC of the Singapore’s problem-solving visualization strategy

This wallet-unfriendly—over-promise, under-deliver— title did fairly well, considering that it was the first official publication by the MOE to feature the merits of the Singapore’s model method to a lay audience. Half of the book over-praises the achievements of the MOE in reversing the declining math performance of local students in the seventies and eighties, almost indirectly attributing Singapore’s success in TIMMS and PISA to the model method, although there has never been any research whatsoever to suggest that there is a correlation between the use of the model method and students’ performances in international comparison studies.

Busy and stressed local parents and teachers are simply not interested in reading the first part of this “monograph”; they’re looking for some practical teaching strategies that could help them coach their kids, particularly in applying the model method to solving word problems. However, to their utter disappointment, they found out that assessment (or supplementary) math books featuring challenging word problems are a better choice in helping them master the problem-solving strategy, from the numerous graded worked examples and detailed (and often alternative) solutions provided—and most of them cost a fraction of the price of the “monograph.”

A Missed Opportunity for a Better Strategy

Not long after the MOE’s publication, the Singapore public was spoilt with another local title on the bar method. Unfortunately, the editorial team working on Bar Modeling then didn’t take advantage of the lack of breadth and depth of the MOE’s “monograph” to offer a better book in meeting the needs and desires of local parents and overseas math educators, especially those not versed with the bar model method.

Bar Modeling
Another wallet-unfriendly title that ill-prepares local parents and teachers to mastering the model, or bar, method in solving non-routine word problems

Based on some investigation and feedback why Dr. Yeap Ban Har didn’t seize the opportunity to publish a better book than the one co-published by the MOE, it sounds like Dr. Yap had submitted his manuscript one or two years prior to the MOE’s publication, but by the time his publisher realized that the MOE had released a [better?] book similar to theirs, they had little time to react (or maybe they just over-reacted to the untimely news?); as a result, they seemed to have only made some cosmetic changes to the original manuscript. Sounds like what we call in local educational publishing as an example of “editors sitting on the manuscript” for ages or years only to decide publishing it when a competitor has already beaten them to the finishing line.

This is really a missed opportunity, not to say,  a pity that the editorial team failed to leverage on the weaknesses or inadequacies of the MOE title to deliver a better book to a mathematically hungry audience, at an affordable price.

Is Another Bar Model Method Book Needed?

Early this year, we’re blessed with another title on the bar method, and this time round, it’s reasonably affordable, considering that the contents are familiar to most local teachers, tutors, and educated parents. This 96-page publication—no re-hashed Dr. Kho articles and authors’ detailed mathematical achievements—comprises four topics to showcase the use of the model method: Whole Numbers, Fractions, Ratio, and Percentage.

As in Dr. Yeap book, the questions unfortunately offer only one model drawing, which may give novices the impression that no alternative bar or model drawings are possible for a given question. The relatively easy questions would help local students gain confidence in solving routine word problems that lend themselves to the model method; however, self-motivated problem solvers would find themselves ill-equipped to solve non-routine questions that favor the visualization strategy.

In the preface, the authors emphasized some pedagogical or conceptual points about the model method, which are arguably debatable. For example, on page three, they wrote:

“In the teaching of algebra, teachers are encouraged to build on the Bar Model Method to help students and formulate equations when solving algebraic equations.”

Are we not supposed to wean students off the model method, as they start taking algebraic food for their mathematical diet? Of course, we want a smooth transition, or seamless process, that bridges the intuitive visual model method to the abstract algebraic method.

Who Invented the Model method?

Because one of the authors had previously worked with Dr. Kho Tek Hong, they mentioned that he was a “pioneer of the model method.” True, he was heading the team that made up of household names like Hector Chee and Sin Kwai Meng, among others, who helped promote the model method to teachers in the mid-eighties, but to claim that Dr. Kho was the originator or inventor of the bar method sounds like stretching the truth. Understandably, it’s not well-known that the so-called model method was already used by Russian or American math educators, decades before it was first unveiled among local math teachers.

I’ll elaborate more on this “acknowledgement” or “credit” matter in a future post—why the bar model method is “math baked in Singapore,” mixing recipes from China, US, Japan, Russia, and probably from a few others like Israel and UK.

Mathematical Problem Solving—The Bar Model Method
A wallet-friendlier book on the Singapore model method, but it fails to take advantage of the weaknesses of similar local and foreign titles on the bar method

Mr. Aden Gan‘s No-Frills Two-Book Series

Let me end with two local titles which I believe offer a more comprehensive treatment of the Singapore model method to laypersons, who just want to grasp the main concepts, and to start applying the visual strategy to solving word problems. I personally don’t know the author, nor do I have any vested interest in promoting these two books, but I think they’re so far the best value-for-money titles in the local market, which could empower both parents and teachers new to the model method to appreciate how powerful the problem-solving visualization strategy is in solving non-routine word problems.

A number of locals may feel uneasy in purchasing these two math books published by EPH, the publishing arm of Popular outlets, because EPH’s assessment math books are notoriously known to be editorially half-baked, and EPH every now and then churns out reprinted or rehashed titles whose contents are out of syllabus. However, my choice is still on these two wallet-friendly local books if you seriously want to learn some basics or mechanics on the Singapore model (or bar) method—and if editorial and artistic concerns are secondary to your elementary math education.

Singapore Model Method
A no-frills two-assessment-book series that gives you enough basic tools to solve a number of grades 5–6 non-routine questions

References

Curriculum Planning & Development Division Ministry of Education, Singapore (2009). The Singapore model method. Singapore: EPB Pan Pacific.

Gan, A. (2014). More model methods and advanced strategies for P5 and P6. Singapore: Educational Publishing House Pte. Ltd.

Gan, A. (2011). Upper primary maths model, methods, techniques and strategies. Singapore: Educational Publishing House Pte Ltd.

Lieu, Y. M. & Soo, V. L. (2014). Mathematical problem solving — The bar model method. Singapore: Scholastic Education International (Singapore) Private Limited.

© Yan Kow Cheong, August 5, 2014.

7000+ Drill-and-Kill Singapore Math Questions

Based on the average number of questions contained in an assessment (or supplementary) math book these days, it can be estimated that most students in Singapore would have solved some seven thousand math questions by the time they had graduated from elementary school. Indeed, an awful lot, to say the least.

I try to understand why local math writers feel that they’ve to provide such an unhealthy number of these drill-and-kill questions or routines for students to practice. Surely, their publishers are behind this “numerical obscenity,” arguing that many kiasu parents would feel that they’re buying a value-for-money book. Imagine the emotional or psychological harm it could do to a child who barely has any interest in the subject.

Even for those with a fluency with numbers, wouldn’t a thousand (or even half of that number) questions bore them to death? Many would end up getting a distorted picture of what elementary math is all about—imitate-practice-imitate more-practice more.

Primary One Mathematics Tutor—Book 1A
Imagine that’s just Book 1A for grade one! How many more drills are there for Book 1B?

The Junk Food of Singapore’s Math Education

Drill-and-kill math questions are like the junk food of math education. Every now and then, we all visit these fast food outlets, but to grow up on a diet that consists mainly of burgers, French flies, and Coke for most days of the week, that can only be a good recipe for unavoidable obesity and all sorts of preventable diseases that would pop up sooner than later. Likewise, over-exposing students to these math drills would only produce an army of drill-and-kill specialists, who wouldn’t be able to think critically and creatively in mathematics—and later in life.

The Baby [Math] Gift for Singapore’s Jubilee

For Singapore’s Golden Jubilee, when the nation celebrates its 50th year of independence in 2015, other than giving items like a special medallion, a multi-functional shawl, and a set of baby clothes, one of the suggested gifts by the mathematical brethren to be given to Singaporean babies born next year should be a check amounting to not less than a thousand bucks. This would help defray the average cost incurred by parents who would need to buy those elementary math supplementary titles during the child’s formative years—it’s a rite of passage for every child studying in a Singapore government school to expose himself or herself to these drill-and-kill questions. Even if parents are opposed to these math drills, school teachers or tutors might still recommend the child to buy one!

Primary Four Mathematics Tutor—Book 4A
It looks like an average grade 4 child has to solve not less than a thousand odd questions before he or she could graduate to grade 5.

Drill-and-Kill Specialists or Innumerates

Of course, it’s better to produce tens of thousands of drill-and-kill specialists than an army of innumerates. In the short term, this raises the self-esteem of the children, as they feel that they can solve most of these routine questions, by parroting the solutions of the worked examples.  We all learn by imitation, but at least the questions need to slowly move from routine to non-routine. We can’t afford to have hundreds of questions that test the same skills or sub-skills almost without end. We need a balanced percent of drills, non-drills, and challenging questions—not a disproportionate unhealthy number of drill-and-kill questions, which would only affect our long-term mathematical health.

Primary Five Mathematics Tutor—Book 5A
Interestingly, many average students find books like the above far more useful than their oft-boring school textbooks and workbooks.

The Good Part of Drill-and-Kill

One good thing about these wallet-friendly drill-and-kill math titles is that they can easily replace a lousy or lazy math teacher, or even a mathophobic parent. After all, we can’t expect too much from a mathematical diet that could help sustain (or even strengthen) us for a while. At best, they’re like Kumon Math, which subscribes to the philosophy that more practice and more drill could eventually help the child to become self-motivated, and to raise his or her self-esteem in the short term.

Maths Practice 1000+—Grade 5
Some of these reprinted or rehashed “math drills” titles cost on average less than one cent per question—no doubt, many [gullible?] parents think or feel that they’re getting a pretty good deal from buying these wallet-friendly books—cheap and “good”!

Are You a Disciple of Drill-and-Kill?

If mathematical proficiency is credited to be achieved through practice and more practice, would you buy a few of these math practice titles for your child? Or, as a teacher or tutor, would you recommend one for your student or tutee? Don’t you think that this is a case of “more is less”? Could the child end up being mathematically undernourished when it comes to creative problem solving and higher-order thinking? Or maybe the child could even be numerically constipated by an overdose of these drill-and-kill questions?

Maths Practice 1000+—Grade 6
Caution: The above title may be hazardous to your long-term mathematical health—it could atrophy your higher-order, creative, and critical thinking skills in mathematics!

© Yan Kow Cheong, July 4, 2014.

Number Patterns on Social Media

If you had been on Facebook for some time, it’s very likely that you would have come across a certain type of numbers puzzles streaming your feed. Clearly, they’re not quite the typical numerical puzzles that often appear in school math books; rather, they’re closer to the types of questions posed in aptitude or IQ tests. One such number puzzle is the following.

20140611-133825-49105422.jpg

Unlike in textbooks that often present these logic puzzles in an uninteresting way, by seeing these colorful number puzzles on Facebook or Pinterest, and being hinted that only a small percent of the problem solvers apparently managed to get the correct answer, this entices readers to give it a try to see how well they’ll fare vis-à-vis their oft-mathematically challenged Facebook friends. Here’s another such number puzzle.

20140611-150315-54195123.jpg

Arguably, these numbers-and-words puzzles have a certain charm to it, because they often require just simple logic to solving them, unlike similar brainteasers that may require some knowledge of elementary or middle-school math. What about the following Facebook numerical puzzle?

20140611-150011-54011255.jpg

 

What is 1 + 1?

What do you make of this one?

20140611-154155-56515670.jpg

If these on-line numbers puzzles indirectly help promote logic and number sense among math-anxious social-media addicts; and along the way, provide them with some fun and entertainment, let’s have more of them in cyberspace!

Let me leave you with half a dozen of these numerical puzzles, stolen from my Facebook feed. Note that these types of brain-unfriendly math or logic questions may have more than one mathematically or logically valid answer, depending on the rule or formula you use—they serve as numerical catalysts for promoting creative thinking in mathematics.

20140612-114955-42595222.jpg

20140612-115024-42624274.jpg

20140612-115055-42655633.jpg

20140612-115202-42722803.jpg

20140612-115228-42748693.jpg

20140612-115317-42797136.jpg

Happy Creative Problem Solving!

© Yan Kow Cheong, June 12, 2014.

Some Fun with Stewart Francis’s Puns

20140319-171119.jpg
www.cafepress.com

In my formative years, I don’t recall any elementary school teachers sharing some mathematical puns with us. I suppose that the arithmetic of yesteryear was often taught in the most uninteresting way by many who probably didn’t look forward to teaching it to a bunch of noisy kids.

Recently, while reading Stewart Francis’s Pun direction: Over 500 of his greatest gags…and four crap ones!,  I came across a number of numerical puns that might even be appreciated by some nerdy seven-year-olds. Stewart Francis is considered to be the best Canadian comedian from Southern Ontario. Here are two dozen odd math-related puns I’ve stolen from his punny book. Hope you enjoy them!

The number of twins being born has doubled.

They also stole my calculator,
which doesn’t add up.

Four out of ten people are used in surveys. Six are not.

Crime in lifts is on the rise.

20140319-171211.jpg
“QED Gravestone Small Poster” www.cafepress.com

I recently overcame my fear of calculators
It was a twelve-step progra
m.

Women are attracted to foreign men. I’ve heard that at least uno, dos, tres times.

All seventeen of my doctors say I have an addictive personality.

There’s a slim chance my sister’s anorexic.

Truthfully, we met at a chess match, where she made the first move.

Is my wife dissatisfied with my body? A tiny part of me says yes.

I read that ten out of two people are dyslexic.

20140319-171135.jpg
From: wherethepunis.com

They now have a website for stutterers, it’s
wwwdotwwwdotwwwdotwwwdotdotdot.

I have mixed race parents, my father prefers the 100 metres.

I’m the youngest of three, my parents are both older.

Clichés are a dime a dozen.

I’m an underachiever 24-6.

I used to recycle calendars.
Those were the days.

20140319-171153.jpg
www.cafepress.com

I’ve learned two things in life.
The second, is to never cut corners.

‘Any man who lives his life in accordance to a book is a fool.’
Luke 317

I can say ‘No one likes a show off’ in forty-three languages.

I was once late because of
high-fiving a centipede.

Of the twenty-seven
students in my maths class,
I was the only one who failed.
What are the odds of that, one
in a million?

I’ve met some cynical people
in my twenty-eight years.

I was good at history.
Wait a minute, no, no I wasn’t.

I was terrible at school. I failed
maths so many times, I can’t even
count.

Reference
Francis, S. (2913). Pun direction: Over 500 of his greatest gags…and four crap ones! London: Headline Publishing Group.

© Yan Kow Cheong, March 19, 2014.

20140319-142556.jpg

A Grade 5 Bicycles-and-Tricycles Problem

20140220-124202.jpg

In an earlier post, I shared about the following chickens-and-rabbits problem.

There are 100 chickens and rabbits altogether. The chickens have 80 more legs than the rabbits. How many chickens and how many rabbits are there?

Other than using a guess-and-guess strategy and an algebraic method, both of which offering little pedagogical or creative insight, let me repeat below one of the two intuitive methods I discussed then.

Since the chickens have 80 more legs than the rabbits, this represents 80 ÷ 2 = 40 chickens.

Among the remaining (100 – 40) = 60 chickens and rabbits, the number of chicken legs must be equal to the number of rabbit legs.

Since a rabbit has twice as many legs as a chicken, the number of chickens must be twice the number of rabbits in order for the total number of legs to be equal.

20140220-180550.jpg

From the model drawing,

3 units = 100 − 40 = 60
1 unit = 60 ÷ 3 = 20

Number of rabbits = 1 unit = 20
Number of chickens = 2 units + 40 = 2 × 20 + 40 = 80

The Bicycles-and-Tricycles Problem

Again, if we decided to ban any trial-and-error or algebraic method, how would you apply the intuitive method discussed above to solve a similar word problem on bicycles and tricycles?

There are 60 bicycles and tricycles altogether. The bicycles have 35 more wheels than the tricycles. How many bicycles and tricycles are there?

Go ahead and give it a try. What do you discover? Do you make any headway? In solving the bicycles-and-tricycles question, I find that there are no fewer of half a dozen methods or strategies, which could be introduced to elementary school students, three of which lend themselves easily to the model, or bar, method, excluding the Sakamoto method.

© Yan Kow Cheong, March 4, 2014.

Some Advice for Singapore Mathletes

20140205-183957.jpg

Here are some pointers I would share with my students at the start of a secondary math olympiad programme. In Singapore, most mathletes attending these enrichment classes are usually selected by the form or math teacher, who tends to choose the best three math students from each class to form a small group of 15 to 20 participants. They would then graduate to represent the school after attending a six-, eight-, or ten-session training programme, depending on the mathematical needs and wants of the school.

• Take your time! Very few contestants can solve all given problems within the time limit. For instance, in the Singapore Mathematics Olympiad (SMO), both at the junior and senior levels, [unconfirmed] feedback based on different schools’ results hints to the fact that those who can win a medal hover around five percent.

Interestingly but disturbingly, an SMO mathlete who can get six or seven out of 35 questions correct may still win a bronze medal, revealing how unmoderated olympiad math papers had been in recent years, going by the abnormally high rate of failures among the participants. This is primarily due to the fact that few, if any, faculty members who set these competition papers, are familiar with what elementary and middle school teachers are covering in local schools.

• Try the “easier” questions first. The questions aren’t set in ascending order of difficulty. It’s not uncommon to see easier questions in the second half of the paper.

• Olympiad questions don’t “crack” immediately. Be patient. Try various approaches. Experiment with simple cases. Working backwards from the desired result in some cases is helpful.

• If you’re using a contests book, and you’re stuck, glance at the “Hints” section. Sometimes a problem requires an unusual idea or technique.

• Even if you can solve a problem, read the hints and solutions. The hints may contain some ideas or insights that didn’t occur in your solution, and they may discuss intuitive, strategic, or tactical approaches that can be used elsewhere.

Remember that modeled or elegant solutions often conceal the torturous or tedious process of investigation, false starts, inspiration and attention to detail that led to them. Be aware of the behind-the-scenes hours-long dirty mathematical work! When you read the modeled solutions, try to reconstruct the thinking that went into them. Ask yourself, “What were the key ideas?” “How can I apply these ideas further?”

• Go back to the original problem later, and see if you can solve it in a different way, or in a different context. When all else fails, remember the reliable old friend, the guess-and-check strategy (or heuristic, as it’s being arguably called in Singapore)—for instance, substituting the optional answers-numbers given in an MCQ into some given equation or expression may yield the answer sooner than later.

Meaningful or creative problem solving takes practice, with insightful or elegant solutions not being the norm. Don’t get discouraged if you don’t seem to make any headway at first. The key isn’t to give up; come back to the question after a day or a week. Stickability and perseverance are two long-time buddies for full-time problem solvers.

Happy problem solving!

20140205-184012.jpg

© Yan Kow Cheong, Feb. 7, 2014.

The Lighter Side of Singapore Math (Part 5)

Elementary Math from an Advanced Standpoint

A Grade 4 Question

Ken has 69 planks that are of standard size. He would need 5 such planks to make a bookshelf. What is the most number of bookshelves Ken can make?

Method 1

Let x be the number of bookshelves Ken can make.

5x ≤ 69
5x/5 ≤ 69/5
x ≤ 13 4/5

So the maximum whole number that satisfies the inequality x ≤ 13 4/5 is 13.

Hence the most number of bookshelves is 13.

 

Method 2

Using the floor function, the most number of bookshelves is ⎣69/5⎦ = 13.

 

A Grade Two Question

Verify that 2 × 2 = 4.

Basic hint: Use the FOIL method.
Intermediate hint: Use area.
Advanced hint: Use axioms (à la Whitehead and Russell)

Solution

Using the FOIL method

2 × 2
= (1 + 1) × (1 + 1)
= 1 × 1 + 1 × 1 + 1 × 1 + 1 × 1
= 1 + 1 + 1 + 1
= 4   QED

 

A Grade Four Question

A rectangular enclosure is 30 meters wide and 50 meters long. Calculate its area in square meters.

Solution

20131228-144507.jpg

 

The My Pals Are Here Math Series

The My Pals Are Here series has been rumored to have been edited and ghostwritten by a hundred odd editors and freelancers in the last decade.

Lack of mathematical rigor was initially targeted against Dr. Fong Ho Kheong and his two co-authors by American profs in the first or/and second editions —probably by those who were “ghost advisors or consultants” for Everyday Math.

 

Deconstructing the Singapore Model Method

1. It’s a problem-solving strategy—a subset of the “Draw a diagram” strategy.

2. It’s a hybrid of China’s line method and Russia’s box (or US’s bar) method.

3. It’s the “Draw a diagram” strategy, which has attained a brand status in mathematics education circles.

4. It’s a problem-solving method that isn’t recommended for visually challenged or impaired learners.

5. It allows questions traditionally set at higher grades (using algebra) to be posed at lower grades (using bars).

 

Painless Singapore Math

Perhaps that’s how we’d promote Singapore math to an often-mathophobic audience!

20131222-143047.jpg

 

A Singapore Ex-Minister’s Math Book

20131229-155500.jpg

Dr. Yeo had the handwriting below depicted in his recreational math book, regarding his two granddaughters, Rebecca and Kathryn.

20131229-155535.jpg

 

References

Yan, K. C. (2013). The lighter side of Singapore math (Part 4). December 2013. http://www.singaporemathplus.com/2013/12/the-lighter-side-of-singapore-math-part.html

Yeo, A. (2006). The pleasures of pi, e and other interesting numbers. Singapore: World Scientific.

© Yan Kow Cheong, December 30, 2013.

Life’s Simple Mathematical Pleasures

Stealing the idea from Nancy Vu’s “Just Little Things,” I tried to reflect on some mathematical things that give us delight and pleasure as we go about our hectic 24/7/365 lives.

Here are some of my personal mathematical pleasures.

20131110-181000.jpg

20131110-181013.jpg

20131110-181159.jpg

20131110-181026.jpg

 

 

 

 

 

 

20131110-181137.jpg

20131110-181050.jpg

 

20131110-181149.jpg

 

20131110-181229.jpg

 

20131110-181038.jpg

20131110-181219.jpg

 

20131110-181209.jpg

 

 

 

 

 

 

 

It’s now your turn to reflect on some aha! mathematical moments  to share with the rest of us.

References

Wu, N. (2013). Just little things: A celebration of life’s simple pleasures. New York: A Perigee Book.

JustLittleThings.net

© Yan Kow Cheong, November 10, 2013.

The Singapore Excess-and-Shortage Problem

In Singapore, in grades four and five, there is one type of word problems that seldom fail to appear in most local problem-solving math books and school test papers, but almost inexistent in local textbooks and workbooks. This is another proof that most Singapore math textbooks ill-prepare local students to tackle non-routine questions, which are often used to filter the nerd from the herd, or at least stream the “better students” into the A-band classes.

Here are two examples of these “excess-and-shortage word problems.”

Some oranges are to be shared among a group of children. If each child gets 3 oranges, there will be 2 oranges left. If each child gets 4 oranges, there will be a shortage of 2 oranges. How many children are there in the group?

A math book costs $9 and a science book costs $7. If Steve spends all his money in the science books, he still has $6 left. However, if he buys the same number of math books, he needs another $8 more.
(a) How many books is Steve buying?
(b) How much money does he have?

A Numerical Recipe

Depicted below is a page from a grade 3/4 olympiad math book. It seems that the author preferred to give a quick-and-easy numerical recipe to solving these types of excess-and-shortage problems—it’s probably more convenient and less time-consuming to do so than to give a didactic exposition how one could logically or intuitively solve these questions with insight.

A page from Terry Chew’s “Maths Olympiad” (2007).

Strictly speaking, it’s incorrect to categorize these questions under the main heading of “Excess-and-Shortage Problems,” because it’s not uncommon to have situations, when the conditions may involve two cases of shortage, or two instances of excess.

In other words, these incorrectly called “excess-and-shortage” questions are made up of three types:
・Both conditions lead to an excess.
・Both conditions lead to a lack or shortage.
・One condition leads to an excess, the other to a shortage.

One Problem, Three [Non-Algebraic] Methods of Solution

Let’s consider one of these excess-and-shortage word problems, looking at how it would normally be solved by elementary math students, who have no training in formal algebra.

Jerry bought some candies for his students. If he gave each student 3 candies, he would have 16 candies left. If he gave each student 5 candies, he would be short of 6 candies.
(a) How many students are there?
(b) How many candies did Jerry buy?

If the above question were posed as a grade 7 math problem in Singapore, most students would solve it by algebra. However, in lower grades, a model (or intuitive) method is often presented. A survey of Singapore math assessment titles and test papers reveals that there are no fewer than half a dozen problem-solving strategies currently being used by teachers, tutors, and parents. Let’s look at three common methods of solution.

Method 1

20131026-231338.jpg

Difference in the number of candies = 5 – 3 = 2

The 16 extra candies are distributed among 16 ÷ 2 = 8 students, and the needed 6 candies among another 6 ÷ 2 = 3 students.

Total number of students = 8 + 3 = 11

(a) There are 11 students.

(b) Number of candies = 3 × 11 + 16 = 49 or  5 × 11 –  6 = 49

Jerry bought 49 candies.

Method 2

Let 1 unit represent the number of students.

20131027-213637.jpg

Since the number of candies remains the same in both cases, we have

3 units + 16  = 5 units – 6

20131026-231524.jpg

From the model,
2 units = 16 + 6 = 22
1 unit = 22 ÷ 2 = 11
3 units + 16 = 3 × 11 + 16 = 49

(a) There are 11 students.
(b) Jerry bought 49 candies.

Method 2 is similar to the Sakamoto method. Do you see why?

Method 3

The difference in the number of candies is 5 – 3 = 2.

20131026-231534.jpg

The extra 16 candies and the needed 6 candies give a total of 16 + 6 = 22 candies, which are then distributed, so that all students each received 2 extra candies.

The number of students is 22 ÷ 2 = 11.

The number of candies is 11 × 3 + 16 = 49, or 11 × 5 – 6 = 49.

Similar, Yet Different

Feedback from teachers, tutors, and parents suggests that even above-average students are often confused and challenged by the variety of these so-called shortage-and-excess problems, not including word problems that are set at a contest level. This is one main reason why a formulaic recipe may often do more harm than good in instilling confidence in students’ mathematical problem-solving skills.

Here are two grade 4 examples with a twist:

When a carton of apples were packed into bags of 4, there would be 3 apples left over. When the same number of apples were packed into bags of 6, there would still be 3 apples left over. What could be the least number of apples in the carton? (15)

Rose had some money to buy some plastic files. If she bought 12 files, she would need $8 more. If she bought 9 files, she would be left with $5. How much money did Rose have? ($44)

Conclusion

Exposing students of mixed abilities to various types of these excess-and-shortage word problems, and to different methods of solution, will help them gain confidence in, and sharpen, their problem-solving skills. Moreover, promoting non-algebraic (or intuitive) methods also allows these non-routine questions to be set in lower grades, whereby a diagram, or a model drawing, often lends itself easily to the solution.

References

Chew, T. (2008). Maths olympiad: Unleash the maths olympian in you — Intermediate (Pr 4 & 5, 10 – 12 years old). Singapore: Singapore Asian Publications.

Chew, T. (2007). Maths olympiad: Unleash the maths olympian in you — Beginner (Pr 3 & 4, 9 – 10 years old). Singapore: Singapore Asian Publications.

Yan, K. C. (2011). Primary mathematics challenging word problems. Singapore: Marshall Cavendish Education.

© Singapore Math, October 27, 2013.

PMCWP4-2
See Worked Example 2 on page 8; try questions 7-8 on page 12.

Mathematical Fiction Is Not Optional

Picture

“The Parrot’s Theorem” (translated from “Le Théorème du Perroquet”) was an instant bestseller in France when it was published in 1998.

Sylvia Nasar’s A Beautiful Mind and G. H. Hardy’s A Mathematician Apology are two nonfiction mathematical classics for both mathematicians and mathematics educators. Lesser known are the mathematical novels which often feature characters whose speciality is number theory, also known as higher arithmetic, and elevated as math’s purest abstract branch.

Mathematics à la Tom Clancy

Two novels that revolve around famous unsolved problems in mathematics are Philibert Schogt’s The Wild Numbers and Apostolis Doxiadis’s Uncle Petros & Goldbach’s Conjecture.

If you’re looking for math, women, sex, and back-stabbing, The Wild Numbers is a math melodrama unlikely to disappoint.

Picture

Winner of the New South Wales Premier’s Prize

 

Fictional math

Who are these mathematical fiction books targeted? Math and science teachers? Educated laypersons? Pure mathematicians may like to read them, yet at the same time they may complain that the mathematics discussed in these books is anything but rigorous.

These books seldom fail to convey the following subtle messages:

• The thin line between mathematical genius and madness.

• The search for mathematical truth at all costs, and the heavy price of finding it.

• The arrogance and pride of pure mathematicians who look down on their peers, most of whom work as applied mathematicians and research scientists.

• The relatively high divorce rate among first-rate mathematicians as compared to their peers in other disciplines.

• Mathematics is apparently a young’s man game; one has past one’s prime if one hasn’t written one’s best paper by the age of 40.

• Mathematicians are from Mars; math educators are from Venus.

• Pure mathematicians (or number theorists) are first-rate mathematicians; applied mathematicians are second- or third-rate mathematicians. To the left of the “mathematical intelligence” bell curve are math educators from schools of education, and high-school math teachers.

Picture

“Reality Conditions” is collection of 16 short stories, which is ideal for leisure reading—it’s suitable for promoting quantitative literacy, or it’d serve as the basis for a creative course on “Mathematics in Fiction.”

The joy of reading mathematics

Let’s rekindle the joy of appreciating mathematics for mathematics’s sake. Let’s welcome poetry, design thinking, and creativity, whatever ingredient that may help to draw the community into recognizing and appreciating the language of science and of technology. These “pure-math-for-poets” titles have a place in our mathematics curriculum, as they could help promote the humanistic element of mathematics.

Here are ten titles you may wish to introduce to your students, as part of a mathematics appreciation or enrichment course.

The New York Times Book of Mathematics

The Best Writing on Mathematics 2010 

Clifton Fadiman’s Fantasia Mathematica

Clifton Fadiman’s The Mathematical Magpie

Don DeLillo’s Ratner’s Star

Edwin A. Abbott’s Flatland

Hiroshi Yuki’s Math Girls

John Green’s An Abundance of Katherines

Philip J. Davis’s The Thread: A Mathematical Yarn

Thomas Pynchon’s Gravity’s Rainbow

Picture

Juvenile fiction—A child prodigy and his friend tried to create a mathematical formula to explain his love relationships.

References

Green, J. (2006). An abundance of Katherines. New York: Dutton Books.

Guedj, D. (2000). The parrot’s theorem. London: Orion Books Ltd.

Kolata, G. & Hoffman, P. (eds.) (2013). The New York Times book of mathematics: More than 100 years of writing by the numbers. New York: Sterling.

Hiroshi, Y. (2011). Math girls. Austin, Texas: Bento Books.

Pitici, M. (ed.) (2011). The best writing on mathematics 2010. Princeton, New Jersey: Princeton University Press.

Wallace, D. F. (2012). Both flesh and not: Essays. New York: Little, Brown and Company.

Woolfe, S. (1996). Leaning towards infinity: A novel. NSW, Australia: Random House Australia Pty Ltd.

© Yan Kow Cheong, September 12, 2013.

Picture

Murderous math that doesn’t kill!