Monthly Archives: October 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


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.


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

3 units + 16  = 5 units – 6


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.


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)


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.


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.

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

Mathematical Fiction Is Not Optional


“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.


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.


“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


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


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.


Murderous math that doesn’t kill!

Hungry ghosts don’t do Singapore math

In Singapore, every year around this time, folks who believe in hungry ghosts celebrate the one-month-long “Hungry Ghost Festival” (also known as the “Seventh Month”). The Seventh Month is like an Asian equivalent of Halloween, extended to one month—just spookier.

If you think that these spiritual vagabonds encircling the island are mere fictions or imaginations of some superstitious or irrational local folks who have put their blind faith in them, you’re in for a shock. These evil spirits can drive the hell out of ghosts agnostics, including those who deny the existence of such spiritual beings.


Hell money superstitious [or innumerate] folks can buy for a few bucks to pacify the “hungry ghosts.”

During the fearful Seventh Month, devotees would put on hold major life decisions, be it about getting married, purchasing a house, or signing a business deal. If you belong to the rational type, there’s no better time in Singapore to tie the knot (albeit there’s no guarantee that all your guests would show up on your D-Day); in fact, you can get the best deal of the year if your wedding day also happens to fall on a Friday 13—an “unlucky date” in an “unlucky month.”

Problem solving in the Seventh Month

I have no statistical data of the number of math teachers, who are hardcore Seventh Month disciples, who would play it safe, by going on some “mathematical fast” or diet during this fearful “inaupicious month.” As for the rest of us, let’s not allow fear, irrationality, or superstition to paralyze us from indulging into some creative mathematical problem solving.

Let’s see how the following “ghost” word problem may be solved using the Stack Method, a commonly used problem-solving strategy, slowing gaining popularity among math educators outside Singapore (which has often proved to be as good as, if not better than, the bar method in a number of problem-situations).

During the annual one-month-long Hungry Ghost Festival, a devotee used 1/4 and $45 of the amount in his PayHell account to buy an e-book entitled That Place Called Hades. He then donated 1/3 and $3 of the remaining amount to an on-line mortuary, whose members help to intercede for long-lost wicked souls. In the end, his PayHell account showed that he only had $55 left. How much money did he have at first?

Try solving this, using the Singapore model, or bar, method, before peeking at the quick-and-dirty stack-method solutions below.


From the stack drawing,
2 units = 55 + 13 + 15 + 15 = 98
4 units = 2 × 98 = 196

He had $196 in his PayHell account at first.

Alternatively, we may represent the stack drawing as follows:


From the model drawing,
2 units = 15 + 15 + 13 + 55 = 98
4 units = 2 × 98 = 196

The devotee had $196 in his account at first.

Another way of solving the “ghost question” is depicted below.


From the stack drawing,
6u = 55 + 13 + 15 + 15 = 98
12u = 2 × 98 = 196

He had $196 in his PayHell account at first.

A prayerful exercise for the lost souls

Let me end with a “wicked problem” I initially included in Aha! Math, a recreational math title I wrote for elementary math students. My challenge to you is to solve this rate question, using the Singapore bar method; better still, what about using the stack method? Happy problem solving!


How would you use the model, or bar, method to solve this “wicked problem”?
Yan, K. C. (2006). Aha! math! Singapore: SNP Panpac Education. 
© Yan Kow Cheong, August 28, 2013.


A businessman won this “lucky” urn with a $488,888 bid at a recent Hungry Ghost Festival auction.

A Before-and-After Singapore Math Problem


A Singapore math primer for grades 4–6 students, teachers, and parents

In Model Drawing for Challenging Word Problems, one of the better Singapore math primers to have been written by a non-Singaporean author for an American audience in recent years, under “Whole Numbers,” Lorraine Walker exemplified the following before-and-after problem, as we commonly call it in Singapore.

Mary had served $117, but her sister Suzanne had saved only $36. After they both earned the same amount of money washing dishes one weekend, Mary noticed she had twice as much money as Suzanne. What was the combined total they earned by doing dishes?

The solution offered is as follows:


© 2010 Crystal Springs Books

The author shared that she did two things to make the model look much clearer:

• To add color in the “After” model;
• To slide the unit bars to the right.

This is fine if students have easy access to colored pens, and know which parts to shift, but in practice this may not always be too convenient or easy, especially if the question gets somewhat more complicated.Let me share a quick-and-dirty solution how most [elementary math] teachers and tutors in Singapore would most likely approach this before-and-after problem if they were in charge of a group of average or above-average grades 4–5 students.


From the model drawing,

1 unit = $117 – $36 = $81
1 unit – $36 = $81 – $36 = $45

2 × $45 = $90

They earned a total of $90 by doing dishes.

Analysis of the model method

Notice that the placement of the bars matters—whether a bar representing an unknown quantity is placed before or after another bar representing a known quantity.

In our model, had we placed the [shaded] bar representing the unknown unit on the right, it would have been harder to deduce the relationship straightaway; besides, no sliding or shifting is necessary. So, placing the bar correctly helps us to figure out the relationship between the unknown unit and the known quantities easier and faster.

In general, shading and dotting the bars are preferable to coloring and sliding them, especially when the problem gets harder, with more than two conditions being involved.

The Stack Method

This word problem also lends itself very well to the Stack Method. In fact, one can argue that it may even be a better method of solution than the bar model, especially among visually inclined below-average students.

Take a look at a quick-and-dirty stack solution below, which may look similar to the bar method, but conceptually they involve different thinking processes. To a novice, it may appear that the stack method is just the bar method being depicted vertically, but it’s not. Perhaps in this question, the contrast isn’t too obvious, but for harder problems, the stack method can be seen to be more advantageous, offering a more elegant solution than the traditional bar method.


From the stack model diagram, note that the difference $81(= $117 – $36) must stand for the extra unit belonging to Mary.

1 unit = $81
$36 + ▅ = $81
▅ = $81 – $36 = $45
2 ▅ = 2 × $45 = $90

So, they had a total of $90.

The Sakamoto Method

This before-and-after problem also lends itself pretty well to the Sakamoto method, if the students have already learned the topic on Ratio. Try it out!

Let me leave you with three practice questions I lifted up from a set of before-and-after grades 4–6 problems I plan to publish in a new title I’m currently working on, all of which encourage readers to apply both the bar and the stack methods (and the Sakamoto method, if they’re familiar with it) to solving them.


Use the model and the stack methods to solve these questions.

1. At first, Joseph had $900 and Ruth had $500. After buying the same watch, Joseph has now three times as much money as Ruth. How much did the watch cost?

2. Moses and Aaron went shopping with a total of $170. After Moses spent 3/7 of his money and Aaron spent $38, they had the same amount of money left. How much money had Aaron at first?

3. Paul and Ryan went on a holiday trip with a total of $280. After Paul had spent 4/7 of his money and Ryan had spent $52, the amount Paul had left was 1/4 of what Ryan had left. How much money did Ryan have at first?

1. $300 2. $86 3. $196

Walker, L. (2010). Model drawing for challenging word problems: Finding solutions the Singapore way. Peterborough, NH: Crystal Springs Books.

© Yan Kow Cheong, August 4, 2013.

Numbering Our Days


Thou shalt remember thy PIN! © 2010 Summersdale Publishers Ltd

Teach us to number our days aright, that we may gain a heart of wisdom.
— Psalm 90:12

Every second counts. Every minute counts. These sound more like clichés to many of us that few would pay attention to. Most of us live our lives as if there’ll be many more tomorrows. For the rest of us who are nearing, or have lived past, the half life, mortality is no longer a topic we can conveniently dismiss. Some try to ignore it, or pretend that age is just a number, or that they’re young at heart—they “psycho themselves” to think or speak like folks from the Positive Thinking or New Age movement.

The Three-Scores-and-Ten Lifespan

The Holy Scriptures tell us that the majority of us are approximately given a three-scores-and-ten lifespan; for a blessed minority, it’d be extended to four scores and ten—90 years. The names of two ex-political ethical leaders cross my mind: Nelson Mandela and Lee Kuan Yew.

Even with medical breakthroughs in recent decades, a look at the obituary pages in the papers every day shows that the average lifespan of a man or woman has remained fairly constant for centuries—even with women living an average of 3 to 5 years longer than men, depending on which continent of the world they live in.


Is age a mere psychological figure? “I am seventy years young!”

The Billionth Heartbeat

For most part of human history, the billionth heartbeat has defined the length of a man’s days. Even today, that nine-zero figure remains fairly constant in a number of African or developing countries. But, thanks to medical advances and better standards of living, many in developed nations can live up to about two to three billion heartbeats.

A Satanic Alert

If you’re more of an apocalyptic type, then you’re more likely to define your mortality in terms of some multiple of the beast number. This means you’ve about 888 months during your earthly stay to live in a manner that could reduce your odds of joining folks like Idi Amin Dada, Saddam Hussein, and Adolf Hitler.


Not all dozing folks on the bus are dead! © 2003 Summersdale Publishers Ltd

Mortality and Eternity

Show me, O Lord, my life’s end and the number of my days; let me know how fleeting is my life. —Psalm 39:4

It’s easy to count our age and the number of years we’ve lived, but it’s difficult to count the number of our remaining days—we simply can’t count from the future.

Perhaps, we need to pray the psalmist’s prayer: Teach us to number our days aright, that we may gain a heart of wisdom.

In Job 14:5, we read: Man’s days are determined; you have decreed the number of his months and have set limits he cannot exceed.

Elsewhere, Jesus told Peter: “I alone control the length of a man’s days.”

No doubt, our days are numbered, and how can we make wise use of them? How do we frame our finite days in the light of eternity? How do we break away from living unremarkably average lives? How can we plan not just for a big, meaningful day or event, but also for a big, meaningful life?

It’s high time we stop kidding ourselves: We don’t have 500 years to live. If we realize that our average lifespan of “three scores and ten” years on earth—about two to three billion heartbeats, depending on our location, position, or station in life—are insignificant in the light of eternity, we’ll value what are important: Love God and His people.


What would you define as your “prime years”?

Math Educators, What’s Your Legacy?

With The Lord a day is like a thousand years, and a thousand years are like a day.
— 2 Peter 3:8

For us, math educators, how can we move more to the right side of the bell curve when it comes to impacting the lives of others in some areas of mathematics education? How can we say NO to living mediocre mathematical lives, although we may not presently have all the necessary tools in our mathematical toolkit to reach out to those whom we long to positively influence?


Is fifty the new thirty-five?

A Spiritual Formula for Longevity

Let me leave you with some verses that may hold the key to seeing you live beyond the  billionth heartbeat—three verses that may be worth keeping in your heart.

Ephesians 6:2–3 

“Honor your father and mother”—which is the first commandment with a promise—”that it may go well with you and that you may enjoy long life in the earth.”

Exodus 20:12 (The 6th Commandment):

“Honor your father and your mother, so that you may live long in the land the Lord your God is giving you.”

Number your days. More than a million people die every single week. Think about it. You made it another week! You’re better off than a million folks. You can thank God that you’re alive. Make “numbering your days” a priority in your mathematical journey!


1. How many times does the heart beat in a person’s lifetime? How do the figures vary for those living in developed and developing countries?

2. Show that most folks have an average of 888 months to live on this side of eternity.

3. Based on a two- or three-billion-heartbeat lifespan, or depending on the continent you are living in, what fraction of your lifetime have you lived? How do you plan to spend the remaining of it meaningfully?


Solomon, R. M. (2012). Reflections on time & eternity. Singapore: Genesis Books.

Summersdale Publishers Ltd (2011). Old is the new young. UK: Summersdale Publishers Ltd.

Fraser, B. (2010). You know you’re having a senior moment when…. West Sussex, UK: Summersdale Publishers Ltd.

Fraser, B. (2003). You know you’re getting old when…. West Sussex, UK: Summersdale Publishers Ltd.

© Yan Kow Cheong, July 27, 2013.


A funny book with a touch of seriousness.

Problem Solving Made Difficult


The US edition of a grade 5 Singapore math supplementary title.

Recently, while revising a grade 5 supplementary book I wrote for Marshall Cavendish, I saw that other than the answer, there was no solution or hint provided to the following question.

If Ann gave $2 to Beth, Beth would have twice as much as Ann.
If Beth gave $2 to Ann, they would have the same amount of money.
How much did each person have?

Most grade 7 Singapore math textbooks and assessment books would normally carry a few of these typical word problems, whereby students are expected to use an algebraic method to solve them. For instance, using algebra, students would form two linear equations in x and y, before solving them by the elimination, or substitution, method. A pretty standard application of solving a pair of simultaneous linear equations, by an analytic method.

However, it’s not uncommon to see these types of word problems appearing in lower-grade supplementary titles, whereby students could solve them, using the Singapore model, or bar, method; and the Sakamoto method. In other words, these grade 7 and 8 questions could be solved by grade 5 and 6 students, using a non-algebraic method.

Algebra versus Model Drawing

Conceptually speaking, I think a grade 6 or 7 student who can solve the above word problem, using a model drawing, appears to exhibit a higher level of mathematical maturity than one who simply uses two variables to represent the unknowns, before forming two simultaneous linear equations to solve them. Of course, because the numbers in this question are relatively small, it’s not surprising to catch a number of average students relying on the trial-and-error method to find the answer.

Try to solve the question, using both algebra and a model; then compare the two methods of solution. Which one do you think demands a deeper or higher level of reasoning or thinking skills?

Depicted below is a model drawing of the above grade 5 word problem.


From the model drawing,

1 unit = 2 + 2 + 2 + 2 = 8
1 unit + 2 = 10
1 unit + 6 = 14

Ann had $10.
Beth had $14.

Generalizing the Problem

A minor change in the question, by altering the “number of times” Beth would have as much money as Ann, reveals an interesting pattern: the model drawing remains unchanged, except for the varying number of units that represent the same quantity.Here are two modified versions of the original grade 5 question.

If Ann gave $2 to Beth, Beth would have three times as much as Ann.

If Beth gave $2 to Ann, they would have the same amount of money.
How much did each person have?

Answer: Ann–$6; Beth–$10.

If Ann gave $2 to Beth, Beth would have five times as much as Ann.
If Beth gave $2 to Ann, they would have the same amount of money.
How much did each person have?

Answer: Ann–$4; Beth–$8.

From Problem Solving to Problem Posing

The two modified questions could serve as good practice for students to become skilled in model drawing, and to help them deduce numerical relationships confidently from them. Besides, they provide a good opportunity to challenge students to pose similar questions, by altering the “number of times” Beth would have as much money as Ann. Which numerical values would work, and what ones wouldn’t, in order for the model drawing to make sense, or for the question to remain solvable?


Let me end, by tickling you with another grade 5 question, similar to the previous three word problems.

If Ann gave $2 to Beth, Beth would have three times as much as Ann.
If Beth gave $2 to Ann, they would have twice as much money as Beth.
How much did each person have?

Answer: Ann–$4.40; Beth–$5.20.

How do you still use the model method to solve this slightly modified ratio question? Test it on your better students or colleagues! It’s slightly harder, because any obvious result isn’t easily deduced from the model drawing, as compared to the ones posed earlier on. Besides, unlike the three previous word problems whose answers are integers, this last problem has a decimal answer—it just doesn’t lend itself well to the guess-and-check strategy.

Share with us how your students or colleagues fare on this last question. Remember: No algebra allowed!

© Yan Kow Cheong, July 12, 2013.

Singapore math authors-millionaires


Mr. Chow’s new revised grade 7 textbook—a US edition is also available, which competes with an equivalent title in the “Math in Focus” series.

It’s an open secret that two of the well-paid math authors in Singapore are Dr. Fong Ho Kheong and Mr. Chow Wai Keung—two non-Singaporeans who have made it to the Millionaire Dollar Club. Also on the Forbes’ Singapore Math List are local folks like Dr. Y. H. Leong, Andrew Er, Fabian Ng, and Lee-Ann Goh, albeit their names are most likely alien to those outside Singapore.

Obscure writing, obscene royalties

A talking point in the local mathematical community is that both millionaires-authors “can’t write”—their titles are notoriously heavily edited or ghostwritten by editors. For instance, there is a decade-long local joke that over a hundred editors have their “editorial footprints” on Dr. Fong’s dozen odd titles.

Form or substance

As for Mr. Victor Chow, his critics remarked that his series of no-frills Discovering Maths titles—apparently a canned version of his ill-written books, which have been poorly received in Hong Kong—is ironically (or miraculously?) doing pretty well in Singapore, in spite of the fact that the competitors’ authors have been household names in math education for decades—many of whom are still teaching teachers.

Many attributed the decent or successful adoption of the Discovering Mathematics series in local schools, primarily because of better sales and marketing strategies by the publisher, as compared to those used by its competitors—form has allegedly triumphed over substance, thanks to lateral (and often shady) marketing.

Interestingly, that many in academia and in local publishing circles subscribe to the above views or rumors, whether because they’re jealous and envious of their “obscene” royalties, is understandable. Apparently, they rationalized that Dr. Fong’s and Mr. Chow’s “below-average writing skills” didn’t match their deserved earnings.


Dr. Fong co-authored latest grade 1 textbook, based on the new Singapore syllabus.

A mix of jealousy and envy and …

Having had the opportunity to speak with some of Dr. Fong’s ex-colleagues, and those who know him personally, it sounds to me that jealousy and envy feature high in discrediting him for “earning so much,” as they feel that they “can lecture better” and “have written more quality research papers” than him.

The argument is that writing textbooks (even successful ones) are for second-rate math educators and mathematicians—unspokenly, first-rate math folks write papers and speak at conferences; second- and third-rate folks write textbooks, or become consultants of these textbooks.

What is seldom talked about is that a number of these so-called seasoned lecturers feel marginalized or “blacklisted” by local publishers for not approaching them—many are still waiting for publishers to line up outside their offices to beg them to write for them. As a result, it’s not surprising that a number of them condescendingly blame local publishers and editors for choosing second-rate writers to author the school textbooks.

Dr. Fong—Singapore’s math popularizer

What we seldom hear, though, is that albeit Dr. Fong might arguably be a “boring presenter or lecturer,” as remarked by his critics, he nevertheless had the guts to promote his books in public, unlike his fellow ex-colleagues who think that it’s a “degrading job” to become a salesperson in promoting their titles at math conferences. Today, who’s having the last laugh to the bank?

In fact, it’s probably not an exaggeration to say that other than Fabian Ng and one or two ghostwriters, it’s Dr. Fong who helped popularize the Singapore model method and the problem-solving strategies locally, through his supplementary math books and public talks in the nineties, written for both students and parents. Yes, long before the Andrew Er’s and Yeap Ban Har’s books were spotted in the local market.


An assessment title of yesteryear—a forerunner of Dr. Fong & Company’s titles.

When risky wasn’t the new safe yet

At the other end of the wealth distribution curve, we’ve dozens of local math writers who wouldn’t dare to being a full-time author, simply because they’re more likely to end up begging than earning enough royalty to pay their bills. Unless you’re a shrewd textbook author-entrepreneur like Dr. Fong, the rest of us write more for our egos than expect any financial rewards, albeit few would admit it.

Negative royalties

I’ve also heard of local math authors who had earned “negative royalties,” which means they owed the publisher instead—they had sold zero copies, and dozens of free copies were given, as part of some book promotion or launch.

Math can make you rich!

Dr. Fong and Mr. Chow both show that you needn’t be the best writer in town, not even a decent one, but if you work hard and smart, and ignore your critics; and if you’ve faith that your publisher has a good sales and marketing strategy, it’s possible to make a decent living in math education.

And what’s even more amazing is that both are foreign-born writers, who have seized the opportunity to make it big in Singapore, when the majority, some of whom are no doubt smarter and better than them, have let their intellectual or mathematical pride and arrogance prevent them from contributing more to raising the standard of mathematics education in Singapore.

© Yan Kow Cheong, June 30, 2013.

Postscript: The author (@Zero_Math and @MathPlus) is a self-professed zeronaire, who is “infinitely jealous and envious” of these authors-millionaires, who have shown us that with hard work (and some luck by the side) “one can get rich with math,” infinitesimal as the chances may be.

The legitimacy of the bar method


“In Step Maths” grades 1-6 used to be a popular series among local schools—a far more user-friendly series than the “My Pals Are Here” and ‘Math in Focus” series.

During this haziest and most polluted week in Singapore, while looking out for some teaching tips in some dated teaching guides, I came across the following grade 3 Singapore math question, which looks more like a grade 5 question to me:

A number represented by the letter B, divided by 6 and then added to 6, gives the same answer as when the same number B is divided by 9 and then added to 9. What is the number B?

How would you solve it, using the Singapore model, or bar, method? Give it a try before peeping at the solution below, which is the one given in the guide. Would you accept the teacher’s guide’s solution as one that effectively uses the power of the bar model in arriving at the answer?


A bar-modeled solution to the above grade 3 word problem.

Is there an abuse or misuse of the bar method?

Personally, I’m not too comfortable with the given solution, as I feel it lacks some legitimacy in the effective use of the bar method in arriving at the answer. What do you think? Do you sense a misuse or abuse of the visualization strategy? How would you use the bar method, or any non-algebraic method, in solving this question? Share your thoughts with us on whether the bar method has legitimately been applied to solve this grade 3 word problem.

Gunasingham, V. (2004). In Step Maths Teacher’s Guide 3A. Singapore: SNP Panpac Pte Ltd.

© Yan Kow Cheong, June 21, 2013.

Is Singapore math frickin’ hard?

There is a millennium myth that Singapore math is hard or tough—that only geeks from some remote parts of Asia (or from some red little dot on the world map) should do it. The mass media (and math educators, too) have implicitly mythologized that those who are presently struggling with school math, should avoid Singapore math, in whatever form it’s being presented, totally. A grave mistake, indeed!

Is Singapore math a mere fad?

After reading dozens of tweets and blog posts on the pluses and minuses of Singapore math, it sounds as if Singapore math has a certain mystique around it—some kind of foreign math bestowed by some creatures from outer space to terrorize those who wished math were an optional subject in elementary school.

Not to say, folks who totally reject Singapore math on the basis that it’s just another fad in math education, or another marketing gimmick to promote an allegedly “better foreign curriculum” to math educators. They believe that “back-to-basics math,” whatever that phrase means to them, with all its memorizing and drill-and-kill exercises, is a necessary mathematical evil to get kids to learn arithmetic.

Singapore math OR/AND Everyday Math

Singapore math? Sure, no problem! It’s no big deal!

Well, it’s a big deal for traditional publishers, which may lose tons of money if more states and schools continue to embrace this “foreign brand” of math education.

Poor writing and teaching from a number of us could have indirectly contributed to the white lie that if you can’t cope with Everyday Math or Saxon Math, or whatever math textbook your school or state is currently using, Singapore math is worse! You might as well forget about it!


One of the first few Singapore supplementary titles to promote the model method in the mid-nineties.

The bar method as a powerful problem-solving strategy

Objectively speaking, Singapore math doesn’t come close to most pedagogical insights or creative ideas featured in journals and periodicals published by the MAA and the NTCM. Personally, I must admit that I become a better teacher, writer, and editor, thanks to these first-class publications. The Singapore model method, although a key component of the Singapore math curriculum, is just one of the problem-solving strategies we use every day, as part of our problem-solving toolkit.

On the other hand, for vested interests on the part of some publishers, little has been done to promote other problem-solving strategies, such as the Stack Method and the Sakamoto method, which are as powerful, if not more elegant, than the bar method—in fact, more and more local students and teachers are using them as they see their advantages over the model method in a number of problem situations.


Fabien Ng in his heyday was a household name in mathematics education in Singapore—it looks like he’s since almost disappeared from the local publishing scene.

Don’t throw out the mathematical baby yet!

You may not wish to adopt the Singapore math curriculum, or even part of it; but at least consider the model and stack methods, not to say, the Sakamoto method, as part of your arsenal of problem-solving strategies (or heuristics, as we call them here). Don’t let traditional publishers (or “math editors” with a limited repertoire of problem-solving strategies) prevent you from acquiring new mathematical tools to improve your mathematical problem-solving skills.

In Singapore, the model, or bar, method is formally taught until grade six to solve a number of word problems, because from grade seven onwards, we want the students to switch over to algebra. However, this doesn’t mean that we’d totally ban the use of the model method in higher grades, because in a number of cases, the model or stack method often offers a more elegant or intuitive method of solution than its algebraic counterpart.

© Yan Kow Cheong, May 27, 2013.


A booklet comprising of PSLE (grade 6) past exam papers, which cost me only 88 cents in the eighties.

A Singapore Grade Two Tricky Question

A classic elementary math problem that folks from a number of professions, from psychologists to professors to priests like to ask is the following:

A bat and a ball cost $1.10 in total. 
The bat costs a dollar more than the ball. 
How much does the ball cost?

For novice problem solvers, the immediate, intuitive answer is 10 cents. Yet the correct response is 5 cents. Why is that so?

If the ball is 10 cents, then the bar has to cost $1.10, which totals $1.20. Why do most of us jump to the wrong conclusion—that the ball costs 10 cents?

We should expect few students to bother checking whether the intuitive answer of 10 cents could possibly be wrong. Research by Professor Shane Frederick (2005) finds that this is the most popular answer even among bright college students, be they from MIT or Harvard.

A few years ago, I included a similar question for a grade 2 supplementary title, as it was in vogue in some local text papers. See the question below.

Recently, while revising the book, I saw that the model drawing had been somewhat modified by the editor. Although a model drawing would likely help a grade 2 child to better visualize what is happening, however, a better shading, or the use of a dotted line, would have made the model easier to understand. Can you improve the model drawing?


Try to solve the above question in a different way, using the same model drawing.

Interestingly, I find out that even after warning students of the danger of simply accepting the obvious answer, or reminding them of the importance of checking their answer, variations of the above question do not seem to help them improve their scores. I recently tickled my Fan page readers with the following mathematical trickie.

Two cousins together are 11. 
One is 10 years older than the other. 
Find out how old both of them are.

Let me end with this Cognitive Reflection Test (CTR), which is made up of tricky questions whose answers tend to trap the unwary, and which may be suitably given to problem solvers in lower grades.

1. If it takes 5 machines 5 minutes to make 5 bearings, how long would it take 100 machines to make 100 bearings?

2. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

3. A frog is climbing up a wall which is 12 m high. Every day, it climbs up 3 m but slips down 2 m. How many days will it take the frog to first reach the top of the wall?

4. A cyclist traveled from P to Q at 20 km/h, and went back at 10 km/h. What is his average speed for the entire journey?

5. It costs $5 to cut a log into 6 pieces. How much will it cost to cut the log into 12 pieces?

Expected incorrect answers

1. 100 minutes. 2. 24 days. 3. 12 days. 4. 15 km/h 5. $10

Correct answers


Donnelly, R. (2013). The art of thinking clearly. UK: Sceptre.

Yan, K. C. (2012). Mathematical quickies & trickies. Singapore: MathPlus Publishing.

Postscript: What’s your CTR score? Here is something to ponder about: Those with a high CTR score are often atheists; those with low CTR results tend to believe in God or some deity.

© Yan Kow Cheong, May 7, 2013.


For practice on the Singapore model method, this title may help—visit