Category Archives: Singapore Math

News and Views about the Good, the Bad, and the Ugly about Singapore Math

Make a Ten

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On June 4, 2023, @PicturesFoIder tweeted the following grade one question:

If you were a dad or mum who’s not familiar with teaching and learning math that focuses on relational understanding, not just on instructional understanding, most teachers and homeschoolers would sympathize with you. You’re not alone!

Make a ten is a simple but not simplistic strategy, commonly used by Singapore math and Common Core math elementary school teachers to teach the operations of addition and subtraction meaningfully rather than procedurally through rote learning.

Angry parents would say, “Why make math more complicated? Wouldn’t that (noble?) way of teaching frighten kids rather than motivate them to do math?” They do have a point, don’t they?

Summary page from CPDD’s “Primary Mathematics Textbook 1A” (2021)

Of course, it’s easier said than done, because given a choice, most of us, teachers, tutors, and parents, would find it more convenient or faster to getting children learn math by rote, by rationalizing that they’d naturally know why the procedure works in later years. For now, just teach them the hows—don’t bother about the whys.

Summary page from CPDD’s “Primary Mathematics Textbook 1A” (2021)

In most parts of the world, teaching math the algorithmic way is the default mode of teaching. A common reply or complaint is “Who’s the time and patience to ensure that 30-odd students in a class have really understood why the procedures for adding and subtracting whole numbers make sense to them?”

The Hows and the Whys

What percentage of grades 1–2 teachers worldwide teach both the hows and whys of addition and subtraction? Do they consciously tell children why they need to learn the algorithms rather than using their fingers to count? Or why is it to their benefit that they learn multiplication and division as a shortcut of addition and division, respectively, not to say, the algorithms to perform these operations?

Just because the majority of us didn’t learn school math relationally or meaningfully in our formative years doesn’t mean we’d also subject schoolchildren to the same boring or uncreative pedagogical ritual due to limited time, or to mimimize conceptual overload or potential confusion.

Teach Not the Way You Learned!

School teachers and parents of yesteryear most likely didn’t know or learn about concepts like “make a ten” and “draw a model,” but our present generation of math educators do know. So there’s no excuse not to introduce them to schoolchildren.

We often underestimate young children thinking that they’d not understand or appreciate the whys, because most are already trying or struggling to make sense of the hows. Valid as this argument may be, patiently (and painfully) providing a good mathematical foundation in the early years would bear much fruit in later years, because understanding trumps rote learning any time.

Relationally and meaningfully yours

© Yan Kow Cheong, June 11, 2023.

The Pseudo-Proportional Problem

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On May 23, 2023, @OrwellNGoode tweeted the rate question below, with the “teacher” marking the student’s answer of 20 minutes wrong, and showing why it’d be 15 minutes instead.

Who the heck takes 10 minutes to saw a board?

I’ve no idea what grade level this math question was being assigned to. Although questions on ratios and rates are formally introduced in grades 5–6 in most parts of the world, however, it’s not uncommon to spot these types of mathematical quickies in grades 1–4 Singapore math olympiad papers to trap the unwary.

Assuming that the word problem didn’t come from a bot or from ChatGPT, the teacher’s intuitive reasoning may be said to be ratio-nally sound but rationally incorrect in solving this pseudo-proportional math question.

Logically or mathematically speaking, few math teachers would disagree that the student was right and the teacher was wrong.

This arguably “badly worded” or “ill-posed” math question provides a fertile ground for a number of possible (valid or creative?) answers, probably much to the annoyance of most math teachers and editors, who often feel uncomfortable or jittery about questions with more than one possible answer.

Indeed, there is no shortage of supporters to defend a “15 minutes” answer. For example, since there is zero mention that the length of each board is of equal length (and we can’t assume it to be so), or as each sawing might take place in a different direction, the “logical” answer of 20 minutes can’t be taken as mathematical gospel truth.

That three pieces need two cuts or sawings is unanimous among problem solvers. The bone of contention is the assumed length of the second cut. Say, if the second cut was half as long as the first one, then it’d take half the time of the first cut, in which case the answer of 15 minutes would be practically plausible.

A creative solution from @A_MGregory

It looks like we’re only limited by our imagination or creativity to rationalize why the answer can’t in practice be 15 minutes or any other duration, by using a different (creative) reasoning from the flawed one provided by the “teacher.”

Like most artificial or impractical word problems in school math, this rate question debatably falls short of design thinking and is thus open to different interpretations or assumptions, which might also weaponize some “anti-woke” math educators to ban or censor these types of “confusing or tricky” math questions.

Ironically, this is why injecting a dose of realism or creativity to these oft-ill-posed or contrived math questions would help open up the minds of uncritical or unquestioning math educators.

Don’t just answer the questions, question the questions.

© Yan Kow Cheong, May 29, 2023.

Right Answer, Wrong Method

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A grade 3 question with right answer marked wrong.

A grade 3 Singapore math question made some waves among netizens in yesterday’s local paper. https://bit.ly/3VOzpHP

Debatably, the root of the discussion centers around the “right” way to do multiplication.

Process v. Product

In Singapore, it’s not uncommon for elementary school teachers and tutors to witness a student’s incorrect method of solution to a word problem that produces the correct answer. This frustrating situation arises far too often than what many of them would want to admit.

Due to a shortage of time, most school teachers pay lip service to the method of solution, pretending not to see that the students’ workings fall short of what is expected of them. These time-starved civil servants would simply look at the children’s answers and conveniently mark them “correct.”

In some cases or circles, it’d not be surprising too that some teachers or tutors themselves are oblivious of the correct method or procedure to solve a (routine or nonroutine) math question.

Personally, I’d want to give these school teachers the benefit of the doubt that they’re merely “lazy” rather than because they’re conceptually blind to the mathematical stain on their students’ worksheets.

Getting students to master a math concept with understanding requires time and effort (and also patience and pain), and most stressed teachers can’t afford either one in the name of having to “rush to complete the syllabus.”

For example, students’ or parents’ parroting that “multiplication is repeated addition” gives teachers and tutors quasi-zero clues whether they’ve understood the multiplication concept or not.

With regard to this grade three routine question that made the headlines, writing the correct procedure, without the teacher explaining to the student why he or she went wrong, only solves half the problem.

Understandably, some parents would argue that teachers shouldn’t be too rigid or radical about their children’s half-baked solutions to avoid dampening their self-esteem. For mathematically puritan math educators, the process is more important than the product.

Procedural proficiency with little understanding of math concepts would only produce elementary school drill-and-kill math graduates—the whys are as important as the hows.

An early penalty is better than a later one when it comes to a child’s learning of basic mathematical concepts in their formative years, which must be accompanied by a teacher’s or tutor’s explanation of the incorrect solution.

And in this grade three routine multiplication question, order matters.

© Yan Kow Cheong, May 11, 2023.

TikTok Math

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Long before the TikTok saga, when the quid-pro-quo ex-president and his gang were entertaining the idea how they’d extract any financial or political gain from the mainland Chinese company on the pretext of security concerns, I was thinking how local math teachers (who’re dissuaded from using social media to share their personal or professional views on math or math education) could leverage on this fast-growing platform to popularize the gospel of mathematics to billions of people.

During the Covid-19 pandemic, I’d christened “TikTok Math” as follows:

Is “TokTok Math” a four-letter word for MAGA patriots?

The TikTok Factor

Last month, following TikTok’s CEO’s four-odd hours meeting with US lawmakers, I tweeted the following:

TikTok’s “theoretical threat”: Blessing or curse? Unlike Singapore that welcomes (or sometimes reluctantly allows) foreign competition as its people mostly benefit from their presence, the US—and the yes-nations—would rather use politics or paranoia than creative power to be a leader.

TikTok “Theoretical Threat”

Does it pay to be a technology leader if for decades your competitor has dominated the industry?

The TikTok saga is an eye-opener to the outside world, as it shows that even the United States, when technologically challenged or militarily threatened, would abandon its own playbook of business ethics, or concoct some theoretical or imaginary security threats, to neutralize the competition or enemy.

Today, the world’s policeman, which has caused more pain than inciting peace, has lost its moral compass on the global stage. A divided polarized nation that doesn’t walk its talk, with promises often turning into pains, or where perks and profits trump people and principles.

The democracy deliverer has inflicted more human pain and suffering than any other country since the last world war. Think of Vietnam, Iraq, and Afghanistan, to name a few nations they “delivered” against communism, totalitarianism, or terrorism.

You’d easily parrot your list of nations whose peoples’ lives have gotten worse in the name of democracy, political freedom, rule of law, or religion. All these universal human ideals have been traded for poverty and pain.

Even with quality or selective immigration, the US is struggling to maintain its superpower status, much less commanding respect from the world. Its gospel of justice, equality, equity, opportunity, morality, or democracy falls short when we see that the lives of half of its [mostly nonwhite] population have gotten worse in the last three or four decades, when their incomes haven’t kept up with the standard of living.

For those in the Chinese diaspora, communist China or the CCP is arguably autocratic, anticompetitive, and antidemocratic, but these marxists-capitalists aren’t that stupid to ask TikTok or ByteDance to hand over the personal of millions of Americans just because they can.

Hearing the shallow or oft-laughable arguments of some of the technologically challenged US lawmakers or politicians questioning TikTok’s CEO, it’s crystal clear that they’ve quasi-zero idea how the mainland Chinese or Asian or African psyche operates in business settings.

On March 20, 2023, @MathPlus asked the following question:

Yes-states & Yes-nations

A few days earlier, I’d hypothesized that

It looks like President Biden & Co. are a second-rate team that can’t think creatively to contain China, by banning their app and the sale of chips, instead of outsmarting them with better products. https://www.bbc.com/news/technology-64973156 #US #politics #TikTok #ban #China #competition #5G #fear

TikTokers v. Trumpists

Way back in 2020, before the last US presidential election, when Microsoft hinted that they’d be keen to buying TikTok, to poke fun at Trump, I’d posted the following:

Survival Odds: Which group would see their wish come true: TikTokers or Trumpists? Which hashtag would triumph: #SaveTikTok or #SaveTrump? Who would have the last laugh on 11/3: Xi or Trump? #ban #TokTok #Trumpism #US #politics #technology #China #Microsoft #math #odds #Covid-19

On 2/8/20, @Zero_Math tweeted the following:

Political Math: Should President Trump ban TikTok for “security reasons,” guesstimate how many millions of young and old voters he’d lose on 11/3, as he diverts attention on his failure to contain the pandemic crisis. #math #Covid-19 #US #politics #TikTok

Waco: A Tale of Two Faux Messiahs

Comparing David Koresh, 33, a “prophet” of the Branch Davidians cult in 1993 to Donald Trump, 76, Trumpublicans’ “anointed one” and “political messiah” of rural America, before the ex-president’s lies-plagued rally in Waco, under “Political Math,” I’d tweeted:

Guesstimate how many TikTokers would be reserving seats for Trump’s campaign rally in Waco, Texas, without showing up—a prank proposed by his niece, who’s suing him for lying and cheating. cnn.com/2023/03/25/politics/texas-trump-2024-rally/index.html

A cartoon by John McNamee

TikTok & Math

TikTok’s global popularity or notoriety provides math educators a fertile ground for creative mathematical problem posing and problem solving in the midst of polarized politicians’ and puritan parents’ oft-paranoiac security or mental health concerns.

Let not politics, paranoia, or phobia hijack math and math education, because the double-edged TikTok could be a mathematical blessing rather than a curse, thanks to the creativity of tens of thousands of math educators worldwide.

√–1 TikTok Math, ∴ √–1 am.

© Yan Kow Cheong, April 16, 2023.

What’s Your Life PhD?

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For the majority of people around the world without a PhD, the academic title is often creatively or cynically assigned a different meaning. Talking of poking fun at those who make a living in an ivory tower—the image of an “ivory tower” is used in the Bible in the Song of Songs (7:4) to describe a woman’s purity—the lay public’s general impression or perception of most PhDs is often anything but positive. Could this be due to some subconscious “intellectual envy”?

Maybe because when they think of academics becoming politicians or of them serving as consultants or advisors for an oft-inept or corrupt government (or of an educational consultant for a publishing house hoping to boost their school adoption rate), many have mixed feelings about these exam-smart folks, who are mostly “un-street-smart,” when it comes to solving everyday life or real-world problems for their fellow citizens—their oft-halfwitted decisions often serve as a living proof of their (practical) unintelligence rather than their intelligence.

PhDs to Save the Planet from Covid-19

Below are three entries I submitted during the lockdown two-odd years ago.

Trump’s Covid-19 Con-sultant
Was Cummings, ex-PM Johnson’s “corona jerk”?
One rule for the Cummings and
one for the common people.

Be it the canned “Permanent Head Damage” or “Post Holiday Depression,” new meanings associated with the acronym are only limited by our imagination.

Boosted Jabs at PhDs

A few years ago, I started relooking at new meanings of a PhD. Two such revised definitions were:

Cheat PhD on the Cheap
God’s Doctorate Disciples

What’s your life’s PhD, especially when you respectfully compare yourself with those with big titles, most of whom often have infinitesimally positive or quasi-zero impact on those around them?

Meanwhile, why not pray, help, and do rather than just preach, hope, and delay?

© Yan Kow Cheong, March 25, 2023.

A North Korea without the Kims

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North Korea and the Kim Dynasty

Way back in 2017, when ISIS or radical Islamist ideologies were making inroads in a number of developing or war-torn countries, and North Korea then looked like the safest place on the planet from green terrorism, I coined North Korea as follows:

North Korea: Where jihadists daren’t go in trying to Islamize infidels unless they don’t mind going to hell sooner than later to meet up with ex-dictators Kim Jong Il and Kim Il Sung.

ISIS or the Taliban is no match for Dictator Kim Jong Un’s trained killers, if these Mohammedan jihadists dream of going to North Korea to set up their Caliphate.

by MathPlus May 31, 2017

The Kim Dynasty

The Kim dynasty is “like grandfather, like father, like son.” Would the world witness a gender change at the top, if a sister (and subsequently a daughter, legitimate or not) were to be reluctantly appointed, thus breaking the decades-long patriarchal political order? A modern-day Jezebel in waiting!

North Korea & Math Education

Imagine if North Korean students were to take part in TIMSS and PISA, what are the chances that they would outrank or outwit their Singaporean or mainland Chinese counterparts in both math and science?

Or, if you needed a part-time or freelance value-for-money coder (or licenced hacker), would you choose one from Singapore, India, or North Korea? The choice is pretty clear, isn’t it?

Dark Political Math

From Reel Trump to Real Kim

After Comrade Kim’s last heartbeat, what are the odds that the two Koreas might be reunited as one? Or would this reunification happen earlier, say, if the hermit nation was forced to surrender following a series of bombings on Pyongyang or on its nuclear sites by the Allies?

Broken Bromance Blossomed

Imagine that Donald J. Trump is miraculously re-elected in 2024, and the Trump-Kim lukewarm bromance is rekindled with a flurry of more “love letters.”

Reel Leaders Nominated for Nobel Peace Prize

What are the odds that they’d eventually have a tête-à-tête at the White House? A post-pandemic reunion between two vainglorious rogues that could help raise their chances of being nominated for the Nobel Peace Prize for the nth time, as their actions (or inactions) arguably prevented a WW3!

The Peacefakers Sans Clothes

What if fearing that he’d end up on the wrong side of eternity, and also be remembered as a modern-day Hitler or Stalin, comrade Putin decided to return the annexed lands, including Crimea, to Ukraine? Would the world witness a peace prize being shared by the unholy Kim-Putin-Trump trio? Or would all three be re-nominated for the Ignoble Peace Prize instead?

The Leadershit Series for Rogue Leaders

Fake Missile for Anti-NATO*

Would the US and allies (yes-nations) only stop playing the more-sanctions game when Kim Jong-un’s patience ran out—when he decided to launch a “fake” missile targeting one of its neighbours, which would force them to take the rogue nation seriously or to treat the Kim dynasty with respect or reverence?

Peacefully & prayerfully yours

* NATO = No Action, Talk Only

© Yan Kow Cheong, March 3, 2023.

Numbers vs. Letters

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A while ago, I tweeted the following math or language or brain question, hoping for a layman answer from math educators or linguists or brain specialists, who might offer a quick-and-dirty explanation to that puzzle.

Tweet from @MathPlus

Another nontrivial question is: “For a number of us who’d no choice but to learn or master a few languages or dialects to survive, why do we feel at home decades later still vocalizing or reciting numbers in the (foreign) language we used while we’re growing up rather than in our mother tongue or lingua franca?”

Personally, I find it easier to recite or utter a sequence of consecutive numbers, or to work with mathematical symbols, in French rather than in English or Chinese—or in my Hakka dialect. I find it puzzling because French has now been relegated to my third or fourth language, and I hardly ever use it in my daily communication, or in any tête-à-tête meetings, other than occasionally dropping some French jargon in my writing to appear like a faux Francophone.

Although today English is my second language and lingua franca, French remains my language of choice when it comes to self-talking (or maybe even daydreaming or dreaming) in numbers or numerals.

It looks like if we learn numbers and symbols in a certain language or dialect in our formative years, we’re brainwired to recall or recite numbers in that particular language later in our adult life. This occurs especially when we’re on our own, even though we may be equally versed or quasi-fluent in other languages or dialects.

Like cycling, driving, or swimming, it appears that reciting numbers in the language of our childhood days in later years is something that stays with us for life.

When self-talking about numbers, do the majority of you who’re forced to be bilingual, trilingual, or multilingual to survive (or thrive) in school and in the workplace also share my experience? Sounds like it’s a neuro rather than a numero question we’re trying to address here!

Linguistically yours

© Yan Kow Cheong, February 11, 2023.

Lunar New Year

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The Spring Festival

Today is the last day of the annual 15-day Chinese New Year (or Lunar New Year) festival in China and Chinese communities around the world.

The Lunar New Year is so-called because the dates of celebration follow the phases of the moon—the new moon could fall on dates between January 21 and February 20, which is similar to Easter that could take place between March 22 and April 25.

Due to its “movable” date, the Lunar New Year (which is unspokenly steeped in superstition and divination, but unquestionably or expectantly celebrated by a billion-odd mainland Chinese and the forty-plus million faithfuls in the Chinese diaspora as part of Chinese tradition) serves as a rich recreational math or calendrical activity for teachers or educators worldwide.

In the Year of the Ox (or “Covidox”), which ushered in a palindromic date (12/02/2021), I pondered: “Any sexy formula that tells us when the Chinese New Year falls in a given year? Not calendrical recipes meant for symbol-minded geeks, but one for the majority of us, the simple-minded folks who’d key in the year and out come the CNY date & day of the week.”

With superstitious couples unfairly or irrationally treating baby tigers and bulls as “inauspicious,” but don’t mind baby bunnies, could supposedly conservative or puritan “fine” Singapore with a frightening low fertility rate of 1.2—below its replacement rate of 2.1, which could see its population heading the way of the dodo sans selective immigration and baby bonus cash incentives—expect a mini-baby boom in the Year of the Rabbit?

With few Covid restrictions still in place, would Singaporeans and permanent residents (and tax fugitives fearing political persecution or prison) be more excited this year to play their part in producing an above-average number of newborns-bunnies? And with Valentine’s Day around the corner, could the nation expect an overbooking of hospital beds or single wards in November?

I completely forgot that I wrote A Dozen Numerical Deeds for the Chinese New Year eight [sounds like a numerologically Sino-auspicious number?] years ago. If you want to keep the spirit of giving alive in the new bunny year, help yourself with some of the suggested gifts to bless others, Chinese and non-Chinese.

Generously yours

© Yan Kow Cheong, February 5, 2023.

Toilet Math

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I’ve been discreetly and randomly working on the tentative The Little Toilet Math Book in the bathroom, where my best thinking often takes place. I suppose a number of you too get your aha’s or eureka’s there, albeit understandably you wouldn’t admit it.

Yesterday, as I was passing by a community center, I happily saw a toilet rolls collection box. I immediately took a photo shot of its front and side views. And my mind started rolling in some real-world questions that could match the pictures.

Below are a sample of these toilet math (🧻🔢) questions:

  1. If residents in the “fine” city of Singapore were to recycle their toilet rolls, guesstimate how many trees could be saved every year.
  2. How much more space could be used if donors were considerate enough to flatten their toilet rolls before dumping them into the collection or recycle box?
  3. If citizens and residents decided using water rather than toilet rolls to clean themselves up, how many millions of liters of water they would need per day after visiting a public toilet (excluding those hundreds of thousands of men who hardly ever wash their hands after relieving themselves or doing their dirty business)?
  4. For a household, would using a bidet or water at home be dearer or cheaper than relying on toilet rolls in the long run (including medical bills for piles and other related health issues that are directly or indirectly due to paper uncleanliness or poor hygiene)?
  5. Imagine that Singapore could no longer import toilet paper from its neighbors. Wouldn’t switching to NEWater—Singapore’s homemade drinkable water, which comes from polluted sea water and “toilet water” mixed with bacteria-killing agents, and which disputably smacks of or smells like sewage—cost for for the population than using toilet paper for doing their business?

Often times, for a community project to succeed, not only do we need that little extra effort but also an iota of thought to ensure that all stakeholders mindfully (or inconveniently) play their part.

Let’s all do the right thing and the thing right, because success often lies not so much in the doing but rather in the caring.

Remember: There is no Planet B (for Generations 𝛼, 𝛽, and 𝛾).

Environmentally and mindfully yours

© Yan Kow Cheong, January 23, 2023

The Toilet Rolls Collection Box
A Side View of the Box

Pi versus WHO

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Pi recently took a page from Singapore’s only ruling party’s decades-long successful template how to stay in power perpetually, with quasi-zero public protests, strikes, and riots.

Will Pi win her viral legal case against the World Health Organization (WHO)? Millions of math teachers worldwide are unspokenly hoping that the WHO won’t alienate or betray Pi, just as it did for the marginalized and unrecognized variant Xi.

Would Pi be vindicated for unfairly being discriminated by those who’re playing politics to prevent her from being admitted as a bona-fide corona variant worthy of WHO’s recognition?

Imagine how excited math educators globally would be if Pi won her legal case against the WHO.

Come Pi Day or Pi Approximation Day, if the 16th letter of the Greek alphabet were officially declared a WHO-approved variant by then, schools and universities would be electrified in conducting an eclectic mix of pi talks and lectures to promote Pi’s ambivalent “murderous” new status.

I, for one, have started to work on Murderous Pi—the title of a math e-book to celebrate Pi’s evolution from a humble mathematical constant to a proud corona variant, which would be plagued with devilish or wicked problems on mathematics’s most beloved irrational number.

Are you looking forward for Pi’s long-overdue recognition by WHO’s health professionals in the Lunar New Year of the Rabbit?

Until she’s virally credited for her fatally mixed impact on earthlings before she’d rest in peace to make way for Rho and Sigma, who’re lining up to be WHO-recognized, Pi’s rejection would be perceived as an irrational [mathematical & viral] injustice or betrayal, whose aftermath paranoiac behaviors among math educators and virologists worldwide could potentially be unpeaceful, to say the least.

Like the “infidel” Zero, who’s ostracized by the Church Fathers of yesteryear, may the “agnostic” or “devilish” Pi be vindicated as she longs to gain formal approval to be in the company of fellow corona variants.

© Yan Kow Cheong, January 17, 2023

Will Pi Be WHO-Recognized?