Category Archives: Problem Solving

Three problem-solving strategies commonly used in Singapore schools are focused on: Singapore model (or bar) method; Stack method; Sakamoto method.

The Joy of Swiftematics

Humor, Mathematical Humor, Problem Solving, Recreational Math, Singapore Math, Theomatics, , , , , , , , , , , , , , , , , , , , , , , , , ,

Last July, millions across Asia competed for just 300,000 tickets to see Taylor Swift in the “fine” city of Singapore, which will host the only stop in Southeast Asia for the singer’s Eras Tour. Organizers said 22-plus million people registered for pre-sale tickets while online registrations passed the one million mark.

And last night, even pop singer Swift, who kicked off her six sold-out shows at the 55,000-seat National Stadium, couldn’t avoid creating some light-hearted political rift or jealousy among some ASEAN members.

Time magazine’s Person of the Year in 2023

Last month, after Thai Prime Minister Srettha Thavisin “complained” that Singapore had brokered a deal to “pay the pop star up to US$3 million for each of her six concerts—in exchange for keeping the shows exclusive to Singapore in Southeast Asia,” the Singapore Tourism Board admitted it “supported the event through a grant,” sans revealing its size or any conditions attached to it. Guesstimate the amount of grant that was given to stage these coveted events.

Even one unhappy politician from the Philippines said “this isn’t what good neighbors do” and called for his country to register its opposition with Singapore’s embassy. Go ahead, Mr. Joey Salceda.

Cartoon by Shannon Wheeler. #NewYorkerCartoons

Political instability, radical ideology that threatens violence to Western values, and poor infrastructure are oft-unspoken key factors for concert promoters to convince the pop superstar to give regional hubs like Bangkok, Manila, Kuala Lumpur, and Jakarta a miss as part of her “Eras Tour.”

Math in Pop Culture

With so much excitement (and concern from conservative or puritan parents) about Swift’s six-show tour in Singapore, how could math educators seize the opportunity to excite otherwise mathematically indifferent or apathetic students with some Swift-related math questions or activities?

For instance, what about coining some math or dismal science terms like Swiftematics and Swiftonomics to promote some creative problem posing?

A Singapore Math Definition of Swiftematics

Text © Anon.

Could the Boyfriend Make It on Time?

Posing real-life Swift-related math questions is only limited by our imagination. Below is a nontrivial question that was posted on Facebook, whose solution is anything but straightforward.

Posted by Judy Smith Hallett on “Maths Jokes Daily”

Swift’s Carbon Footprint

In 2022, Swift topped the list of celebrities with the highest private jet CO₂ emissions. If her jet pollution were about x times more than the average person’s total annual emissions, estimate x.

The next item is a Swift- or math-friendly question posted by news anchor Peter Busch.

A Math-Friendly Question for Swifties

The Numerology of Taylor Swift

Last month, after reading about Swift’s serial infatuation with her “lucky” number 13, I made an attempt to define Swiftie Math, which is based on the numerology (or pseudoscience) of Taylor Swift.

Since I’ve yet to receive any approval or rejection of the term—whether the editors see it fit for publication—I’d skip posting it online for now.

The Swift-Biden Conspiracy

Theomatically, MAGA evangelicals (or MAGA Xtians, where X ≠ Christ)—a subset of Christian nationalists—in red-pilled states haven’t failed to warn netizens about the “satanic” influences of Taylor Swift’s songs, but have hypocritically or selectively remained silent about the fraudulent, criminal, and sexual activities of their “political savior.”

Does Taylor Swift CAST SPELLS On Her Listeners?!

Conspiracies about the singer’s alleged support for President Biden have been rife in political and religious circles to paint Ms. Swift as an “ambassador of Satan,” who’s shown zero sign in supporting Trump and his cult.

Photo © 2024 CNN

Puritan Trumpublicans are hell-bent to warn millions of Swifties from unknowingly becoming witches lest they and their idol lose their souls, but, interestingly, hardly anything from these patriots calling for a nationwide corporate prayer for the soul of their beloved un-Christian ex-president.

Musically & mathematically yours

References

Taylor Swift named Time’s ‘Person of the Year’ https://www.cnn.com/2023/12/06/media/taylor-swift-time-person-of-the-year?cid=ios_app

‘Cruel Summer’ for Taylor Swift fans in Asia as Singapore shows sell out https://www.cnn.com/2023/07/14/business/taylor-swift-tour-singapore-asia-popularity-intl-hnk-dst

Taylor Swift’s journey from country icon to pop superstar https://www.cnn.com/2022/10/21/entertainment/gallery/taylor-swift/index.html

Does Taylor Swift CAST SPELLS On Her Listeners?! https://youtu.be/SDmzNDrj2NI?si=-gWMOb7CwsswytYV

© Yan Kow Cheong, March 3, 2024.

The Million Mistake Club

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Shot from Ed Rush’s “God Talks” (2023)

If we’ve been an employee rather than an entrepreneur or businessperson for most parts of our working life, most of us would probably not be eligible to join the Million Dollar Club.

Even for those of you who’re frequent flyers, the chances that you’d qualify for a Million Mile Club are probably not that high, too.

On the other hand, for an obscene number of us who’ve made at least a million mistakes, we’re likely to meet the criteria of becoming a member of the “Million Mistake Club.”

If you’re the lucky ones who’d never qualify for this notorious club, congratulations to you for living a life defined by manifold successes and few failures, either because you’re blessed with an Einstein brain (or born with a silver spoon in the mouth), or you’re simply averse to taking any risks (which would help reduce your failure rate).

An Age Problem

On average, guesstimate how young or old someone in most parts of the world would be by the time they make their millionth mistake. In their late 20s or early 30s? Go ahead and figure this out—it’s good for your left brain!

In TrumpLand (plagued by an unrighteous party and ruled by a self-righteous party), PutinLand (misruled by self-glory and orthodox church history), or XiLand (ruled by capitalist dictatorship and hostage diplomacy), what percentage of their politicians and prisoners (or even prosperity pastors) might have already joined the “Billion Mistake Club,” or are about to do so if they’d live beyond the three- or four-scores-and-ten lifespan?

A nation of aged oft-morally corrupt politicians and presidents

The Positives of a Million Mistake Club Member

Do you qualify as a member of the Million Mistake Club? If you’re a life member like me, what are the chances that you’d be a lot nicer to others by then?

Wouldn’t the world be a better rather than a bitter place if we celebrated members of the Million Mistake Club (instead of canceling or condemning them)?

Think of ex-convicts who’re now a free man or woman after paying for their mistakes. Or those who’ve been pardoned or released early for good behavior.

A screen shot stolen while attending a service

Why not uninhibitedly post on social media that you’ve joined the Million Mistake Club, and as a result your friends, fans, and followers could positively expect a different you—someone who’s less critical or condemning.

A new you who’s more understanding, forgiving and gracious, and less on comparing, competing, and complaining.

Of course, it’s easier preached than practiced, but nevertheless embarking on a life of contentment and consensus is the beginning of a life journey that would bear much fruit now and in future.

Remember: Your past doesn’t define you. What counts in the end: It’s not so much how you start, but how you finish.

Freely & truly yours

© Yan Kow Cheong, October 24, 2023.

Taliban Math

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A math definition that has since been hijacked by digital terrorists.

Early this week, we read in the news that the Taliban were stopping female Afghan students heading to the university from leaving the country to study in Dubai.

What kind of society or ideology would prevent girls and women from pursuing an education that would empower them to live productive or fruitful lives, and to help raise the literacy and numeracy rate of their country?

If a government don’t respect the human rights and fundamental freedom of women and girls, it speaks volumes what kind of radical thinking is behind their spiritual or ideological decision and action.

Just when the civilized world thought that forcing people to be attired in a certain way based on man-made laws is bad, banning them from attending secondary schools and universities makes a mockery of all the rites and rituals that are practiced to moralize or de-infidel-ize them.

Meme © Anon.

In most developed or semi-democratic societies, not sending your children to school, or depriving them of a formal education, for no valid reasons, could land parents and caregivers in deep trouble with the authorities.

Singapore: Lifelong Opportunity for All Minority Muslim Girls & Women

In multicultural Singapore, even educated parents can’t simply homeschool a child just because they want to without a valid reason from the Ministry of Education. For example, religious-minded parents can’t conveniently send their children to a madrasah for their formal education if they can’t convince the authorities that their children’s educational, emotional, and social needs would be well taken care of.

Art ©Edelman Rodriguez @edelstudio

A government that fail to provide formal education for its citizens, male and female (or whatever other labels some might prefer to be identified themselves with), or discriminate against girls and women, or alienate certain racial or religious groups, and minorities, makes us wonder how far these people have been radicalized, or are ideologically brainwashed or spiritually blinded.

When rogue rulers or radical religious leaders in a theocratic state control the lives of millions of men, women, and children based on radical ideology, by dictating them how they ought to live and be taught, one can imagine what kind of citizentry they’d produce for future generations.

The marginalization and objectification of girls and women is condoned in many Mohammedan milieux or Islamist circles, and these practices hardly ever make the headlines, compared to the misinformation or disinformation on the repression of Uyghurs and genocide in Xinjiang.

As someone aptly commented, it sounds like Americans “don’t like Chinese and also don’t like Muslims, but they seem (or pretend) to like Chinese Muslims.”

An e-card that proves ISIS values math education.

For “infidels,” selling or marketing Singapore math titles to the Taliban or Boko Haram (which outwardly or publicly abhor and denounce anything Western or Christian) is like running a half-priced campaign for ice cubes in Alaska.

Politics 1 Math Education 0

Since the U.S. pulled out of Afghanistan, after trying to free the people from terrorism and radical Islamism for two-odd decades, arguably, both Trump and Biden are directly or indirectly responsible for the current deprivation of education among girls and women in the country.

It’s a back-to-square-one situation before 9/11, when radical Islamists and terrorists religiously relegated girls and women to domestic slaves.

The Western media put a premium on “radical democracy” but pay lip service to the lack of educational and job opportunities for millions of girls and women in Afghanistan and many so-called “moderate Muslim” countries.

Radical Math Questions

Below are some previously x-ed (or tweeted) politico- or religio-mathematical questions non-NATO [no-action-talk-only] math educators would like to reflect on if they wished to play an active part in the education emancipation of girls and women in many oppressive or rogue regimes around the world.

1. Guesstimate how many millions of girls and women in Afghanistan would be denied of their human and educational rights, as the Taliban start enforcing their man-made Sharia law to oppress or enslave them in most spheres of life.

2. How many Afghan girls and women would be robbed of an education under the Taliban in 2021 and beyond, as radical Islamists and terrorists force them to be subservient to men?

3. If Section 377A humiliates and hurts gay people, doesn’t the men-designed Sharia also “discriminate or disadvantage girls and women”? Shouldn’t those discriminatory laws against them be repealed?

4. [Fake] Math News: Trump & the Taliban. Afghan judges are deliberating what sharia punishments should be meted out to Trump for his financial, political & sexual crimes. What are the odds that he’d be spared of barbaric amputation, caning or stoning?

5. How many Malala Yousafzai’s Afghanistan and other Muslim-majority nations like Iran and Saudi Arabia—where women and girls are often discriminated and enslaved—would need to fight for women’s and children’s rights against the Taliban and the ayatollahs?

Photo © Anon. Before and After the Taliban

© Yan Kow Cheong, September 3, 2023

Algebra for Babies & Toddlers

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A math definition that appeared to have been hijacked by Al-Qaeda or the Taliban

An unspoken commandment among parents and homeschoolers is: Thou shalt not introduce algebra to young kids without close adult supervision.

Looking at the unhealthy number of pre-school math titles in local bookstores, some Singapore math authors have set questions that directly or indirectly help promote algebraic thinking among toddlers and kindergarteners, particularly via the bar model method and number patterns, whether they’re pedagogically conscious of it or not.

Kiasu parents or tiger mums would buy assessment (or supplementary) math titles (often disguised as “parents’ guides”) to give their kids an “unfair advantage” over their peers.

On closer look, disappointingly, these preschool “enrichment math” books are often mere rehashed primary one (or grade one) assessment math titles.

Fr: Cartoon from Judy Smith Hallett

I decided not to showcase any covers of these oft-drill-and-kill kindergarten math titles here to avoid any perception that I’m endorsing some local authors or their publishers.

Notion, Not Notation

Debatably, it’s no harm getting preschoolers to start thinking algebraically long before they’re formally taught generalized arithmetic. Yes to pre-algebraic thinking but no to algebraic notation or equation for kindergarteners.

Personally, I’ve yet to see any decent locally published K–2 Singapore math titles in bookstores (other than through some questions in children’s puzzles books), which creatively or systematically promote algebraic thinking skills.

In the last two decades, there had been a number of journal articles and a few NCTM (and even some AMS) titles that feature activities or nonroutine questions that champion pre-algebraic thinking at the kindergarten level.

It’s a pity that Pre-K and kindergarten teachers (and mathepreneurs) haven’t leveraged on these rich resources to come up with supplementary math titles to evangelize the algebraic gospel to K–2 students.

The raison-d’être of premature algebra teaching

In Singapore, a mecca for brain-unfriendly, budget-friendly assessment (or supplementary) math titles, it looks rather surprising that local Singapore writers have so far not come up with an “Algebra for Babies or Toddlers” when local libraries already carry catchy foreign titles like Bayesian Probability for Babies and Pythagorean Theorem for Babies.

Ripe Harvest but Few Workers

The time is ripe for creative math educators, local or foreign-born, to publish a creative algebra series for toddlers and kindergarteners of kiasu parents, but it looks like the writers who’d help pluck up the fruits are few. An untapped market for publishers that want to move away from canned or drill-and-kill preschool math titles.

Opportunistically & creatively yours

© Yan Kow Cheong, August 27, 2023.

The Clock Problem

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On July 12, @PicturesFoIder x-ed (or tweeted) the following picture:

Picture © Anon.

Is this another ill-posed math question? Or just another arguably creative solution that put the teacher or tutor in a catch-22 response?

Let’s look at a sample of comments for and against the given answer.

They don’t want a digital clock!
This is the correct answer for anyone that is somehow confused! 🤔

Teacher needed to say clock with hour and minute hands.

The question says “small clock”, not “analog clock”, therefore the answer is correct.

This is everything what’s wrong with current educational system.
It sure does prepare you.
To think in the frameworks they want you to think. For example “there is only one right solution to a problem and that ain’t it”

thats what happens when you let kids use ipads at a young age

This student should be transferred to art school immediately

On one hand I’m scared that the new generation can’t read physical clocks, on the other hand, I’m surprised by the out of the box thinking

If my child received a X for that answer, I would challenge it. There is nothing at all wrong. It is a small clock showing ten past eleven. 100% accurate. IF they wanted a conventional clock face that should have been stated. I’d have given 2 ticks for innovative thinking!

The question doesn’t specify that it meant “analog clock” plus it says “10 minutes past 11:00” which implies digital time as opposed to “10 minutes past 11 o’clock” which would imply analog time.

I would have drawn an analog and digital clock with a note saying the request was ambiguous and next time be more specific. Also how small? Another ambiguous request

How many of these responses would you agree or disagree with? Valid or invalid, or preposterous in some instances, most of these comments can’t be discounted offhand.

Followers or Oddballs?

At a time when politicians, pastors, or even prisoners are often hypocritically or insincerely pushing for an overhaul of their rigid educational system (from which they themselves benefited much)—which promotes rote learning or regurgitation, or prepares students to the test—are math teachers ready for students’ unconventional or disruptive solutions, which often border on the ridiculous or irreverent?

If a child (or a trained chimpanzee) presented the solution below to the above problem, what would your response or reaction be?

Picture © Anon.

Would you mark it wrong or partially correct, because he or she had failed to take account that time on a clock is determined by the hour hand alone, with the minute hand acting as a mere convenience? Or in layman terms, the hour hand had also moved when the minute hand took a sixty-degree turn.

Or would you take this opportunity to introduce nonroutine (or more subtle faux) questions like the ones below?

1. What is the angle measure between the hands of a clock at 10 minutes past 11:00?

2. A clock reads ten minutes past eleven. What time would the clock read if the hands of the clock were interchanged?

3. Are there other times of the day when the hands of a clock would also show the same angle measure as when they were at 11:10?

The Positives of Ill-Posed Questions

An ill-posed question, or the unexpected answers to such a flawed question, is a gold mine for creative mathematical problem posing. It not only provides an off-the-wall sense of humor, but also gives math educators an opportunity to address students’ mathematical loopholes or their half-baked understanding of concepts.

Positively & creatively yours

© Yan Kow Cheong, August 13, 2023.

The Mask of Math

Humor, Math Publishing, Problem Solving, Recreational Math, Singapore Math, Theomatics, , , , , , , , , , , , , , , , , , , , , ,
Mask Art as Therapy. Original un-memed photo from Hunny & Lummy’s “Masks of Singapore” (2021).

What mathematical or nonmathematical crisis are you presently facing or undergoing? Mid-life crisis? Existential crisis? Financial crisis? Relational crisis? Post-pandemic crisis?

Have you forgotten what it means to enjoy math? If you’re a school teacher or university lecturer, are you planning to leave the [Singapore’s or US’s or XYZ’s ] rigid educational system to pursue your mathematical dream?

If you’re an editor, are you longing for the day when you don’t have to handle those quasi-uneditable manuscripts once you’ve paid up your mortgage or send your children to college?

And if you’re a writer, do you long (or pray?) for those pseudo-math editors to get promoted to their next level of incompetency, where they’re less likely to adulterate your manuscript?

Math & Mask

Beyond the mask that we wear to function in our daily lives as math educators (lecturers, teachers, tutors, editors, writers, consultants, managing editors, publishing managers, …), who are we?

Do you see yourself enjoying the mathematical journey while you’re building your career or struggling to pay the bill? When you take off your daily masks, when you don’t feel the pressure to pretend, when you’d simply be yourself, what does it feel like? What does it smell like? What does it taste like? What does it sound like?

A Commandment to Deal with the Mask of Pride

Mathematical Synesthesia

Can you visualize the color of infinity? Taste the number zero? Smell the fragrance of pi? Or you think these synesthetic experiences are only reserved for autistics or idiot savants?

We all came into this world with zero, and we’ll also leave it with zero but the [mathematical] spirit of life we’ve lived in our lifetime. Are you always waiting for permission to write that math book? Or hoping that when you retire, you’d have the time (and space) to explore and pursue that math pet project?

Are you petrified that others might witness that you’ve been a victim of the imposter syndrome, as you get promoted and being tagged with bigger flowery job titles? Still struggling to fake it until you make it?

Unmask Your Math

To make a mark in math or math education in the local, regional, or international community, you need to strip your mask away. People want to see and work with vulnerable or fallible folks, who’re prepared to make a fool of themselves, to be a laughable stock or mathematical clown, and not to take themselves seriously.

What are you waiting for? Not some other time when you’ve accumulated enough zeros in your bank account, or next semester (or pandemic?), but today. Because when you’re financially free, you’re unlikely to have the energy to do that math thing you so desire.

Don’t die with a book inside you! Or miss tithing one or two years of your life to volunteer as a math teacher in some low-GDP countries to help raise the numeracy level of the locals. Or fail to resurrect that off-atrophied “math & art” project for a solo exhibition. It’s better to fail or experience the journey than regret on your deathbed.

Remember: Let not pride, insecurity, or failure prevent you from fulfilling your God-given purpose on this side of eternity, as you embark on your mathematical journey.

You needn’t do it alone: Seek Him and His wisdom for your mathematical needs and wants. Be fearless and free.

Fearlessly & faithfully yours

© Yan Kow Cheong, July 23, 2023.

A Question on Inequality

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Fr: Ralph McConahy on Facebook

Many years ago, I read about the co-authors of a handbook for mathematics teachers in primary schools warning readers not to use the sign “<” or “>” (because the symbols were removed from the primary school syllabus); instead, they suggested using phrases like “more than” and “less than.”

For example, teachers were to avoid setting questions in these formats:

34 is 6 > ☐
8 > 43 is ☐

Instead, they’d rephrase them as “34 is 6 more than ☐.” and “8 more than 43 is ☐.”

Similarly, they’d refrain from posing inequalities questions such as the following:

7 < ☐ is 15.
9 < 25 is ☐.

And also avoid problem sums like the one below:

What is the largest (or greatest) whole number that can be placed in the box to make the statement true? 8 + ☐ < 40

Why the Ban (with or without a Fine)?

Based on teachers’ feedback that young (or even older) schoolchildren are often confused about the similarity of the two symbols < and >, that’s likely a key why that prompted local curriculum math specialists in the “fine” city to ban these “unequal symbols” in primary school mathematics moons ago.

Inequality Metaphors from the Sunshine State

Over the years, to reduce the confusion between < and >, some elementary math authors have come up with some witty ways to help schoolchildren remember which is which.

For instance, students are often taught to see the symbols as hungry alligators or crocodiles with gaping mouths—these reptiles always want to eat the larger numbers, so the open mouth will always face this.

Observe that the < looks somewhat like a lopsided L, which reminds us that it denotes less than. Or, in any true statement, the large open mouth of the symbol is on the side of the greater quantity, and the small point is on the side of the lesser quantity.

No More Ban

Like last year’s repeal of Section 377A in pseudo-puritan Singapore, based on the CPDD’s Primary Mathematics Textbook 2A (2022), the inequality signs too are now free to roam the pages of any MOE-approved primary 2–6 textbooks and workbooks.

In the aftermath of zero ban on inequality signs, questions that involve comparing and ordering numbers would no longer be symbolically penalized or criminalized for using the “>” and “<” signs (until further notice).

Below are a sample of three “uninhibited” Singapore math primary two inequality questions:

Which sign will you use, > or <?
(a) 45 ☐ 42
(b) 81 ☐ 71
(c) 317 ☐ 407
(d) 734 ☐ 724

Fill in the boxes with ‘<’ or ‘>’.
(a) 35 ☐ 53
(b) 65 ☐ 62
(c) 79 ☐ 68

It’s not uncommon to see once-banned open-ended questions now gracing the pages of primary math textbooks, such as the following:

In 38 > 33 + ☐, what could the missing number be?

It looks like we’ve come some way in restoring the inequality signs in the (lower) primary school syllabus. Now that the mathematical resurrection of these symbols has taken place, does their confusion among schoolchildren still remain a concern for both teachers and parents?

An Inequality Quiz

Let’s end with a math quiz that tests our basic knowledge of inequalities.

1. How many types of inequalities in elementary school math are there?

2. Which metaphor(s) would you use to help children who are prone to mistake one inequality sign for another?

3. Name half a dozen math inequalities in real life that schoolchildren could relate to.

4. “An inequality is an equation that forbids the use of an equal sign.” True or False.

Symbolically yours

© Yan Kow Cheong, July 9, 2023.

Singapore’s PSLE Math Paper

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There is an educational (or psychological or emotional) price a country has to pay if it wants its students to continually rank among the top three in international comparative studies like TIMSS and PISA, or in regional or international math contests and competitions.

An irreverent definition of Singapore’s most dreaded school exam paper

Understandably, parents in Singapore are unhappy about the difficult PSLE (grade 6) math questions that are used to assess their children, before they’d graduate from primary (grades 1–6) school to secondary (middle) school.

And the oft-politically correct or modeled answers from the city-state’s Ministry of Education (MOE) hardly ever pacify or satisfy teachers, parents, and caregivers; in most cases, the canned suggestions or quasi-laughable solutions only make them angrier or more cynical.

PC Slogans for Kiasu Parents

Be it the mantra that “every (local) school is a good school,” or that parents need to help or educate their children believe that “their self-worth or value isn’t dependent on their exam grades” is easier preached than practiced.

When politicians or MOE officials preach to parents that they needn’t be paranoid about their children’s PSLE exam or math score, because it’s not the end of the world, it’s like ex-loansharks-turned-philanthropists or ethically challenged ex-CEOs- or ex-bankers-turned-preachers now telling the financially struggling public that money isn’t everything, or that they’d not make money their god. For the haves to tell the have-nots, it’s utter hypocrisy, to say the least.

Answers and oft-ill-edited half-baked solutions are usually from tutors or teachers-moonlighters.

Tuition: A Necessary Evil for the Nation

Without compulsory tuition or heavy parental involvement, the majority of elementary math students in local schools would likely struggle to score a decent grade in their PSLE math paper.

Singapore’s PSLE math paper with its quota of brain-unfriendly questions looks like a necessary evil that would help define or maintain the “fine” city’s high standard of mathematics regionally and internationally.

From Mid-Year to Mock Exams

This year, Primary 6 students didn’t have to sit for mid-year exams at school, because last year, Education Minister Chan Chun Sing had said that the move would allow them to “focus more on their learning and less on marks.”

In the aftermath of the MOE’s move to do away with all mid-year exams for primary and secondary schools, tuition centers saw a golden opportunity to lure kiasu parents with their mock mid-year exams, whose questions are generally harder (not better) than those set in the PSLE math paper.

Other than parental or peer pressure to excel, most students’ undue stress could be traced to the difficulty of math questions set by neighborhood schools (driven by school rankings) and tuition centers (powered by profits), which are generally harder than those that appear in the PSLE math paper. Yes, they’re the two big culprits that set an unhealthy number of nonroutine questions that often demoralize the kids, by making them feel like they still “aren’t that good in math.”

A Promised Land for Geeks—and Tax Fugitives

Singapore is a “promised land” for those who’re born or blessed with the “mathematical gene” or for those who’d afford a private tutor. However, for the majority of average or math-anxious school children, we can only pray that PSLE math wouldn’t become their bête noire, and that God would keep them motivated and focused as they go through this oft-stressful rite of passage of their schooling years.

I might sound like a mathocrite (short for “mathematical hypocrite”) in giving mathematical or parental advice; nevertheless, let me end with this educational slogan that is worth reiterating, because I believe that the sooner we put it into practice, the less stressful (or peaceful) our life will be: Our math scores or grades don’t define us—in or out of school, and certainly not in life.

Grade-consciouslessly yours

© Yan Kow Cheong, June 18, 2023.

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

Problem Solving, Recreational Math, Singapore Math, , , , , , , , , , , ,

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.