Tag Archives: Singapore math textbooks

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.

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.