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Singapore Math

Monday, October 16, 2023

A Singapore Grade 4 Geometry Question

On Facebook, someone recently posted the above primary 4 (or grade 4) geometry question, asking for help from fellow parents. How would you do it (without calling on ChatGPT)?

If you’re a parent, homeschooler, or tutor, how would you explain it to an eight- or nine-year-old child, who’s struggling with non-drill-and-kill questions on area and perimeter?

Give It a Try First!

Try figuring out the answer on your own first before peeping at two parents’ quick-and-dirty solutions below. Better still, could you present an idiot-proof or peasant-friendly solution that even a smart dog or cat could understand?

 Solution by Belinda Sim


6 width and 6 long = 3units  of the rectangle.

1 rectangle -> 16cm.

3 rectangle 

16x3 = 48cm.

Solution by Jesline Ang

Stretch Your Mind!

Many moons ago, I was commissioned to write Mind Stretchers 2*, a grade 5 problem-solving math book, which was popular in a number of local schools and tuition centers—disturbingly, they’re guiltlessly or blatantly photostating entire chapter questions as part of their worksheets.

And I still recall that on the topic of Perimeter, I’d posed a number of routine and nonroutine questions. What surprised or tickled me then was when the Managing Editor asked me whether or not some of these questions are solvable, because at first glance, they look like there are missing information to solving them.

These geometry questions can give students (and probably their oft-math-anxious parents and teachers) some goose pimples out of fear or panic if they can’t figure out the answers offhand. 

Understandably, without a cool mind and some patience (or perseverance), even perimeter and area questions at the elementary (or olympiad) level can prove to be a challenge to math educators if they’re not trained to tackling them using the right approach.

Frustration and fear usually set in, especially for nonroutine questions that don’t normally appear in drill-and-kill school textbooks and workbooks.

Premature Testing

The danger of setting these types of brain-unfriendly questions too early, or having them prematurely in a class test, especially when most students have yet to fully grasp the concepts of area and perimeter, could be detrimental to the mental health of most average math students.

Parents who freeze at the sight of these questions would often exhibit a knee-jerk reaction, by getting a tutor for their child, if they’d afford it.

Tuition or No Tuition?

Most stressed parents often reason that if they couldn’t even solve these grade 4 questions, things would only get worse in later years, as the higher-grade topics get more complicated and the questions become more challenging. They just don’t want to see their child struggle in math, especially when they themselves had had a negative experience of the world’s most disliked school subject.

Even for math teachers and tutors, there’s nothing to be ashamed of if at first or second reading, they’re clueless how to tackle these nonroutines. Yes, they do appear in mid- or final-year exams arguably as a social filter to separate the nerd from the herd, but once math educators and homeschoolers know how to approach these questions from the right angle, before long, these math problems would become routine to them.

The 4F’s of Mathematical Problem Solving

Few math teachers and writers would admit that (school or olympiad) math involves lots of fears, false starts, frustrations, and failures. The sooner parents and homeschoolers (and their children and grandchildren) are aware of this oft-unspoken problem-solving process or ritual, the pressure or expectation level for all parties ought to go down significantly in reducing any unnecessary mental or mathematical stress.

Pain is (always) part of the joy of creative mathematical problem solving—the no-pain-no-gain mantra is axiomatic in math at all levels.

Geometrically & perseveringly yours

© Yan Kow Cheong, October 16, 2023.

* When the wallet-friendly title was out of print, a senior editorial staff emailed me that he’d pay me a cosmetic S$150 to buy the copyrights. I never bothered to reply to his laughably ridiculous suggestion. I wouldn’t be surprised that they’d since plagiarized the content to be used and reused for other purposes.

Monday, September 25, 2023

How to Celebrate Zero Day

Zero causes so much pain and suffering to tens of millions of students and their parents every day. Indeed, zero is the real troublemaker among the numbers! It's not too late to get even with the Roman Catholic Church’s once-deemed blasphemous infidel number (again).

Below are ten idiot-proof activities you’d do to celebrate Zero Day.

10. Avoid writing the now-redundant zero before a decimal point: .23, .583, .0045, …—move away from dated deception, faux religion, or numerological superstition.

9. Use the letter “o” or “O” when you mean the number or numeral “0”—for example, when giving a bank account or ID number to someone requesting it on the phone for verification purposes.

8. Use “nothing” to half-truthfully convey the idea of zero, even though the number zero is “anything but nothing.”

7. Fight for the Year Zero to be included in the timeline, to avoid confusing billions of people that the year 2023 is conventionally in the 21st century rather than in the 20th century.

6. Canvass for the number zero instead of the number one to be the first number in the set of whole numbers, especially in some puritan or radical circles, where the devil number is excluded.

5. Mobilize semi- or quasi-innumerate folks to use a different numeral when zero is used as a place number and as a number itself—strip zero of its double personality. For instance, use Ø in lieu of 0: 3Ø14Ø to emphasize place-value.

4. Be a zerobreaker, by defying the eleventh commandment, Thou shalt not divide by zero! without life-threatening consequences—that one has the (civil or political or religious) rights to divide by zero, although the answer may prove insensible or unreasonable.

3. Puncture the egos of haughty geeks who think they know it all, by exposing mathematical loopholes in their reasoning, resulting from illegally (or unknowingly) dividing by zero—humble them with fallacies and paradoxes, such as 1 = 0 or π = 3.

2. Have a law that fines companies from offering 0% interest rate as a gimmick to lure gullible customers.

1. Warn people to flee from the free—beware of free gifts, free seminars, and free air tickets, which seldom have zero strings attached.

Now is the time to teach Zero who’s in charge here—zero is your servant, not your master.

Nøtøriøuly yours

© Yan Kow Cheong, September 25, 2023.

Monday, August 21, 2023

A Numbers Puzzle

On X (formerly Twitter), @wiseconnector posed the following question:

How many numbers can you find?

It’s never too late for an eye check-up and also an opportunity to hone your logical and visualization skills.

This logic or math question can serve as an icebreaker at a birthday party even for those who proudly or unashamedly boast of being “hopeless at math or numbers.” 

Try it! The answer is anything but obvious.

Ask a toddler or kindergartener, and also your seniors at home, especially if you’re concerned that they might succumb to dementia or Alzheimer’s in their later years, based on your family health history—a [generational?] curse you can break in the name of Jesus if you don’t want your children and their children to go through this mental illness in their golden years.

Summarize the answers to this puzzle from different age groups in a statistical diagram, say, a pie or bar chart. The information is likely to surprise you, giving you some insights what is going on in people’s minds when they try solving this kind of trick or tricky math question.

This little mathematical quickie has the potential to unite rather than divide people, by bringing them and their loved ones and friends much joy and laughter—never mind if they can’t always agree on the correct answer.

Logically & visually yours

© Yan Kow Cheong, August 21, 2023

Monday, August 7, 2023

Racism in Singapore Math Publishing


Does the comedic rant on “Racism in Maths Problem Sums” serve as a “proof” that inequality or racism is rampant in math and math education?

Often times, math educators pose and solve these artificial or contrived word problems without giving much thought to them, especially when they’re oblivious that they might trigger mixed or negative feelings among highly sensitive schoolchildren and parents that belong to certain racial or minority or economically disadvantaged groups.

Math in Multicultural Singapore

It’s no harm nor too late for the Ministry of Education (MOE) to occasionally review their approved textbooks presently used in local schools, which might perceivably promote some form of racial inequality in multiethnic or multicultural Singapore.

Not too long ago, to promote multiculturalism in math education, there was this unofficial or unwritten directive from the MOE for local math teachers and textbook writers to mindfully or multiculturally use people’s or children’s names from different racial groups in their word problems or problem sums. An educational move that would likely be banned and dubbed “woke math” in some political circles or polarized parts of the globe today.

Because Singapore has four official languages (English, Mandarin, Malay, Tamil), math educators are encouraged to permute or vary their choice of names in order not to appear racially prejudiced, especially when the majority of textbook authors have traditionally been Chinese.

In the name of racial or religious harmony, commendable or encouraging as the curricular or political move was then, its implementation wasn’t a walk in the park, because in practice, it’s not a mere change of substituting a Chinese name with a minority or non-Chinese one.

The present trend appears to revert to generic names in math textbooks and assessment titles for reasons unknown to me. Is it part of a grand marketing scheme to internationalizing (or colonizing?) the Singapore math brand to as many countries as possible?

XX, XY, or Others?

Some Chinese first names are used by both males and females; in other instances, it can be tricky to guess the gender by just looking at some Asian or Chinese names—the task doesn’t get any easier if you’re a non-Chinese. In fact, just recently, I needed to Google a Chinese name (in Hanyu Pinyin) to make sure that I don’t mistake one gender for another. Indeed, it’s pretty embarrassing or unacceptable to address a he or she (or he-she) by a Miss or Mr. When in doubt, don’t assume—ask!

For a long time, I thought the name of one of the original math co-writers of the then MOE’s Curriculum Development Institute of Singapore (CDIS) writing team belongs to a certain gender, when I later found out of my mistaken assumption, because in the olden days, certain names were used by both sexes.

Foreign Editions: Pluses & Pitfalls

A similar situation arises when Singapore math textbooks are adopted or adapted for overseas markets. Changing names of living and nonliving things that have both British and American equivalents, or replacing local names or terms like local food or fruit with foreign substitutes that readers elsewhere are familiar with, is a common practice.

Of course, local publishers would also caution math writers to avoid using any Chinese or Asian names if their titles are being tailor-made for a Western audience, or if they don’t want to alienate potential “anti-woke” homeschoolers-customers.

Lesser known or discussed is that math publishers in Singapore have mixed feelings about local authors using their full Asian names, because in their marketing eyes, readers in the US and in the EU would prefer a Western name of the form X Y (with the first name X before the surname Y) rather than an Asian (or traditional Chinese) name of the form Z Y X (where the surname Z precedes the first name Y X) on the cover of a math textbook or supplementary math title.

Western Names Preferred

Many moons ago, I vividly recall a respected experienced publishing manager advising me to use a Western or European name, because my targeted (American) audience might not be receptive to seeing an Asian or Chinese name on the covers of my recreational and problem-solving math titles. 

Just as every (additional) equation in a general or pop math book targeted at the lay public might potentially halve its sales, the odds of an Asian math author, who insists on using their real rather than a [Western] pen name, achieving a decent sale in front of their non-white audience are significantly (or unprovenly) reduced. There is this apparently unspoken bias towards nonwhite authors appearing on covers of math titles.

Personally, I’m not sure of the extent of this alleged or perceived discrimination vis-à-vis Asian math authors, but I’ve faith to believe that those who’ve taken a risk to order my wallet-friendly, brain-unfriendly books online, or to buy them in a bricks-and-mortar bookstore, have got their money worth, except for a few one-star haters or sour grapes who revel in posting vitriolic book comments on Amazon.

At the other extreme, it’s also not uncommon for math publishers in Singapore or Asia to conveniently or unethically omit the names of the authors, consultants, or foreign advisors (if they can get away with it), especially if foreign editions exist, or sale of copyrights occur at book fairs, with zero knowledge from the authors (until or unless morally convicted or vindictive editors leaked out the news).

Recently, upon requesting my author’s overdue complimentary copies to one of my foreign editions titles, I discovered that both the author’s name and his American Curriculum Advisor of Challenging Word Problems (Grade 6) were missing from the title page and imprint page, respectively, which raises serious IP and ethical concerns on the part of the publisher and potential customers and readers.

Without a name on the cover or title page, when previous editions had carried them, it just makes one wonder about the rationale of these unconsulted omissions, when the parties involved in the writing or reviewing of the content were kept in the dark. This only opens up the floodgate to unauthorized use, or helps promote plagiarism or piracy, where the party that often benefits is the lawyer.

The Chinese Advantage (in Singapore)

Years ago, a non-Chinese co-author on Singapore’s bar modeling approached me, and wanted me to be his co-author or consultant for one of his upcoming math projects. He’d pay me for being his faux partner, because understandably having a Chinese name associated to a new book or website would add credibility to the project.

Businesswise, I couldn’t disagree with him, and I was praying that he’d find someone else more qualified than me, who wouldn’t mind being his lifelong business buddy.

Indeed, the Singapore math publishing industry is an ethical minefield that few have traversed without compromising their character or reputation. This is one of the oft-unspoken reasons why unlike other developing (and developed) countries that pay lip service to copyrights, Singapore is a first-class economy with a third-class educational publishing industry.

Ethically and multiculturally yours

© Yan Kow Cheong, August 7, 2023.

Thursday, July 27, 2023

Musk vs. Math

In the aftermath of Twitter being rechristened to X, the letter or symbol X can now boast of a new nonmathematical meaning.

The Lawsuit of the Century

Would the world soon be witnessing the legal case of the year: X vs. Math Educators? Or Musk vs. Math?

Meme © Anon.

What on earth is behind the choice of the letter or symbol or logo X to rebrand a once-beloved app, whose shelf life now looks shaky, to say the least?

The X of Singapore Math Publishing

Earlier, I’d x-ed (or “tweeted”) tongue-in-cheek that

“Changing Twitter’s iconic bird logo to a white “X” hoping to launch a "super app" is like Singapore math editors—who’re responsible for their publishers’ MOE-rejected titles—rationalizing that by joining a competitor, their odds of approval would be higher.”

Most mediocre math editors are often deluded into thinking that their new higher position or pay in another publishing house would magically make them a better “word doctor,” thus increasing the chances of their manuscripts getting approved by the Ministry of Education.

Twitterdamus: The Fall of Twitter

Late last year, when the founder of SpaceX started downsizing Twitter’s workforce, the temptation to coin Twitterdamus was just too strong.

I tweeted the above definition with this hypothesized question: “Twitter Math: Guesstimate the number of Twitterdamuses who secretly wished to witness the bankruptcy of Twitter in a-not-too-distant future.”

Threat from Threads

Below are some tweets I recently posted related to the privacy or security of our personal data. 

Your metadata has now a choice. Who’s your less trusted or beloved billionaire? [Cartoon from FRusty.creates]

Threads’s Threat: Is the Threads app an Alibaba version of Twitter? Sounds like the social media equivalent of Trump’s frivolous lawsuits against those who expose him! Why isn’t Twitter also suing Weibo—China’s copycat?

When Free is often costly: If the product or service is free, you are the product. Think of Google, Facebook, Instagram, YouTube, and Twitter, which have sold or resold your personal data multiple times for billions.

A prayer for millions of anti-𝕏 followers who sorely miss the little blue bird: Let 𝕏 Be Twitter Again!

© Yan Kow Cheong, July 27, 2023.

Sunday, July 16, 2023

Mortal Math

Photo © Anon.

Few would disagree that the above math or science question has now gained deadly relevance or importance in this year’s cruel summer, especially during this week’s abnormally hot weather.

EcoMath 4 Everyone

Is it apt for math teachers to use this week’s extreme weather condition plaguing some parts of the planet to help some middle-school students brush up their Celsius-Fahrenheit conversion skills, and to challenge them with some real-world problems in the aftermath of climate change? 

A typical grade nine (or secondary three) ecomath question is the following:

On a certain day, the Celsius thermometer and the Fahrenheit thermometer registered the same numerical value for the temperature. Given that 86°F = 30°C and 104°F = 40°C, what was the temperature that day?

Post-Pandemic Holeydays 

Guesstimate how many tourists regret flying to some parts of the US and the EU at this time of the year—when popular tourist attractions are closed during the hottest hours to protect visitors from dehydration or to reduce the odds of their visiting an emergency room.

How much would tourism-related businesses in heat-stricken regions suffer financially this summer?

What percents of unlucky tourists would need to extend their “holidays” due to prolonged heat-related hospital stay?

The Polluters Preaching to the Poor  

Estimate the number of deaths in the animal kingdom resulting from our hotter planet, which is primarily driven by materialistic consumers in the developed world, who now hypocritically want to lecture the developing world about human-induced climate change and to some extent the naturally-occurring weather pattern known as El Niño.

Why should China, India, Indonesia, and other emerging economies listen to the West, which are historically responsible for the dire climatic situation we’re in today? Why must much-maligned totalitarian China and undemocratic others donate to international climate funds set up by “guilty” high-GDP big polluters to help poorer nations struggling with climate change?

With the world paying lip service on cutting down on oil and gas for alternative sources of renewable or clean energy to grow their economies, low yield of crops and the increasing risk of wildfires and floods, not to say, frequent electricity blackouts in poorer or corrupt developing nations, are likely to be the new (new) normal.

Heat Kills

According to the Centers for Disease Control (CDC), more than 700 people in the US lose their lives to extreme heat each year, with close to 68,000 people being sent to emergency rooms.

Guesstimate how many locals and foreigners, especially the homeless, the migrant workers, and the refugees, in all the global regions now experiencing a heatwave would find it hard to avoid a heat-related hospital visit this year. How many of them might fail to cheat death simply because of their helplessness or others’ indifference?

The Dodo Effect

The world has already warmed by about 1.1°C since the industrial era began and scientists forecast temperatures will keep rising unless governments and businesses around the world have the (political or moral) will to cut greenhouse gas emissions.

How many fish in rivers and canals, or seabirds and whales, would disappear or starve thanks to “global warring”? How many animal and plant species are going the way of the dodo in coming years and decades?

Singapore’s First Winter

Source Photo: © Anonymous

Way back in 2016, I’d tweeted tongue-in-cheek about Singapore’s experiencing her first winter:

Singapore's 1st Winter: Will fine dining up and more folks aspiring to be middle-class cause Singapore to witness its #Singageddon? #climate (1/25/16)

Fake News: Singapore experienced its first winter this week—tens of thousands of Singaporeans took no-pay leave when the temperature dropped to zero degrees! #climate #humor #zero #winter Source Photo: © Anonymous (1/14/18)

At the current rate in the global rise (or drop) in temperature, the chances of those of us (who’ll be blessed to live beyond the three-scores-and-ten lifespan) seeing snow in Singapore are anything but zero.

Environmentally & morally yours

© Yan Kow Cheong, July 16, 2023.

Thursday, June 29, 2023

A Grade 3 Singapore Math Question

Not too long ago, a mother posted the above grade three question on Facebook, and sought help from fellow parents: “Hi all, can anyone advise this P3 qn? Thanks!”

Before you read some of the replies or suggestions below, if you’re a parent or tutor, how would you solve this fraction question (with the mathematical knowledge of an average grade three student)?

A parent’s reply was: “This is a question that wants u to compare the fraction with half. Whichever fraction is smaller than half is the answer.

“To compare against half, take the denominator and divide by 2. If the numerator is less than the divided value, it is smaller than half.”

The mother replied, “蔡锦森 oh ic! Nv thought of that! Gosh, how to explain to a P3 ? 😝”

Another parent or tutor joined in: “So for such questions, there are 2 types. One type requires the student to compare against half. The other is to compare the difference with 1 whole. E.g. Which fraction is closest to 1? 7/8, 3/4, 6/7 etc”

Later, another member commented: “Normalize all the denominators and compare.”

Would you use any of the suggested strategies to solve the problem?

An Apt Question for Grade Three Students? 

In the Singapore math curriculum, the guidelines for elementary school students (or primary school pupils) learning fractions at different grades are as follows:

Primary 2 (Grade 2): Add and subtract like fractions whose sum is less than 1.

Primary 3 (Grade 3): Learn equivalent fractions; add and subtract related fractions whose sum is less than 1.

Primary 4 (Grade 4): Along with improper and mixed numbers; add and subtract related fractions whose sum exceeds 1.

As I’ve zero idea about the source of this test question, I leave it to the reader to decide for themselves whether or not this grade three question is in line with the local syllabus. If not, why? Or do you think this question is somewhat ill-posed?

What If?

Imagine if the same question were to be assigned to a group of grade five students. How would they approach it? 

Would most of them resort to a knee-jerk solution by finding the LCM of the four fractions, before comparing their numerators, rather than thinking whether a less-tedious or shorter method might do the job?

Or would they conveniently convert the fractions into decimals (with or without a calculator)?

The Art of Problem Posing

I wonder whether rephrasing the above question might make it mathematically (or fractionally?) more meaningful for students to apply their knowledge of comparing fractions. Say, what if we asked them to compare a set of fractions vis-à-vis a base fraction like 1/2?

Which of the following fractions is greater than 1/2?

A. 2/5     B. 3/7    C. 6/10     D. 4/9

Again, here too, the aim is to discourage students from religiously applying the fraction-decimal conversion, or from tediously converting these fractions to a common denominator.

At higher grades, we’d use larger fractions that would make finding equivalent fractions painfully tedious or time-consuming.

Meanwhile, may I encourage you to share how you’d tackle this fraction question without adding more pressure on students (who might already be suffering from some form of “fraction trauma” as a result of being subject to premature abstraction and procedural methods)?

Methodically & meaningfully yours.

© Yan Kow Cheong, June 29, 2023.