Professor Calculus and Tintinology

Humor, Math Publishing, Mathematical Humor, Recreational Math, , , , , , , , , , , , , , , , , , , , , , ,
Remembering Prof. Calculus’s Inventions & Innovations

Meet Professor Cuthbert Calculus, a polymath—a geeky jack of all trades—who was at home in many disciplines ranging from astronautics and astrophysics to geometry and geodetics to mathematics and thermodynamics.

The beloved clumsy and eccentric professor, or the absent-minded mad scientist par excellence, applied his mathematical and scientific knowledge to launch his futuristic inventions and innovations.

A Tribute to Calculus

To celebrate Professor Calculus, who first appeared 80 years ago in Hergé’s 12th Tintin book, Red Rackham’s Treasure (1943), the recently published Professor Calculus: Science’s Forgotten Genius (2023) by Albert Algoud—the original French edition first came out in 1994—paid a funny tribute to one of the most endearing scientific geniuses in children’s literature.

Algoud’s English edition takes a voyeuristic look at Prof. Calculus’s achievements and the real-life innovations and people that inspired one of Hergé’s most popular characters. Think of Isaac Newton, Gottfried Wilhelm Leibniz, Auguste Piccard, Thomas Edison, Alexander Graham Bell, and Konstantin Tsiolkovsky.

Over the course of 54 years, Hergé (Georges Remi) who’s born in Brussels in 1907, completed over 20 titles in The Adventures of Tintin series, which is now considered to be one of the greatest, if not the greatest, comics series of all time.

The Harry Potter Series of Yesteryear

More than 230 million copies of Tintin’s series have been sold, which proves that comic books have the same power to entertain children and adults in the 21st century as they did in the early 20th.

Tintin’s Series for Girls

Moons ago, when my daughters were in elementary or middle school, I introduced them to some Tintin’s titles. To my surprise, they’d quasi-zero interest in Tintin & Milou’s (or Snowy’s) stories, because they couldn’t relate to the male or chauvinistic, not to say, colonialist, characters.

Flipping through the pages, and taking a deeper look at Hergé’s characters, I realized that the girls weren’t wrong. The number of pages in each book depicting female characters is probably fewer than the fingers in a hand.

It’s no surprise that Hergé had been accused of sexism for the almost complete lack of female characters in his “boys books.” In fact, few people would disagree today that most (white) men from that era were racist, sexist, and misogynist.

Hergé himself denied being a misogynist, saying that “for me, women have nothing to do in a world like Tintin’s, which is the realm of male friendship.”

Tintin’s Gender

On a lighter note, Tintin himself has been labeled as being strongly feminized, especially in relation to Captain Haddock, with the hypothesis that he’s of virtually indeterminate gender.

More than half a dozen years ago, French philosopher Vincent Cespede made the headlines when he claimed that Tintin is “an androgynous redhead with blue eyes” and “presumably asexual.”

A Christmas Gift for Boys (and Girls?)

If you’ve a teenage son, grandson, or nephew, I think giving away a few Tintin’s titles would be an apt Christmas gift for them; however, for a daughter, granddaughter, or goddaughter, you’d probably want to think twice or thrice before prematurely exposing them to any of Hergé’s predominantly male white characters.

In today’s politicians’ slogan of inclusiveness or political correctness, I wonder whether The Adventures of Tintin series might even be banned in some feminist or woke circles. After all, if we recall that the Harry Potter series was not only banned from some school’s library after educators consulted with exorcists in the US and Rome, but also in some conservative Christian and radical Islamist circles.

The same could also be said for math classics like Edwin Abbott Abbott’s Flatland (1884), which directly or indirectly exposes social classes and white supremacy. Or titles like George Orwell’s 1984 that was banned in both Soviet Union and China.

The Calculus Affair

Tintin’s puzzle: How to rescue Prof. Calculus who’s in danger.

In The Calculus Affair, Tintin witnessed some very strange events like the simultaneous shattering of windows, mirrors, and chandeliers, leaving him bewildered. After a shooting and a break-in, Tintin knew Prof. Calculus was in danger, but he had only one clue—an unusual packet of cigarettes.

Could he solve the mystery before a terrible weapon fell into the wrong hands? Could he act in time to protect Professor Calculus?

Calculus’s brain. Fr: Albert Algoud’s Professor Calculus: Science’s Forgotten Genius (2023)

The Adventures of Tintin Series

If you’re a Tintin fan old and new, especially a lover of graphic novels, mysteries, and historical adventures, how many of the 20-odd titles of Hergé’s The Adventures of Tintin series have you read?

What’s your favorite Tintin’s title?

Tintin in the Land of the Soviets Aimed at communicating an anti-communist sentiment to the young reader? Banned in Ukraine?

Tintin in America Placed in a restricted access section of the Jones Library in Amherst, MA in 2014 for its “racist imagery.”

Cigars of the Pharaoh

The Blue Lotus Would Japanese nationalists call for its ban?

The Broken Ear

The Black Island

King Ottakar’s Sceptre

The Crab with the Golden Claws

The Shooting Star

The Secret of the Unicorn

Red Rackham’s Treasure First printed 80 years ago.

The Seven Crystal Balls

Prisoners of the Sun

Land of Black Gold

Destination Moon

Explorers of the Moon

The Calculus Affair A mathematician’s favorite?

The Red Sea Sharks

Tintin in Tibet Praised by the Dalai Lama, who awarded it the Light of Truth Award. Banned in China?

The Castafiore Emerald

Flight 714 to Sydney

The Adventures of Tintin and the Picaros

Tintin and Alph-Art

Tintin in the Congo The racist content: colonial attitude vis-à-vis Congolese people and for glorifying big-game hunting.

In my childhood days, I read most of Tintin’s titles in its original French, which is probably the language of choice to reading them, especially for those who’re bilingual, trilingual, or multilingual. I hope to reread the series in another language in my golden years.

Adventuresomely yours

© Yan Kow Cheong, October 8, 2023.

Formula One Singapore: Blessing or Curse?

Humor, Mathematical Humor, Recreational Math, Theomatics, , , , , , , , , , , , , , , , , , , , , ,

Since 2008, except in 2020 and 2021, when the night race was cancelled due to Covid-19, Singapore Grand Prix has had its highs and lows.

Most locals would hardly shed a crocodile tear should the three-day noise-pollution event cease to be held in the “fine” city in future, albeit a record 302,000 “fans” turned up for the 2022 F1 Singapore GP.

In 2019, when the hazy event venue was at an unhealthy level, race organizers were giving away thousands of free F1 tickets to beef up the number of attendees for the F1 night race to avoid the sight of empty seats. Who says that begging and betting are mutually exclusive?

The environmentally unfriendly event appeals mostly to diehard F1 fans, as F1 fatigue had already set in among locals who’d attended a few more canned events after 2008.

The Singapore F1 night race looks more like a curse than a blessing for a segment of the population, especially retail shop and restaurant owners (with cancelled meal orders and table reservations), and service providers (few gym or tuition classes, haircuts, etc.) in the Marina Bay area.

Some of them have their sales dwindled this week due to customers’ difficulties of navigating around road closures, or the latter’s decision to give their venue of choice a miss to avoid any inconvenience.

And religious services and recreational activities in the area had to be cancelled as a result of noise pollution from this weekend event. Even the holy souls or health freaks, who need to be in the vicinity, rain or shine, pollution or not, would have to find alternative parking space to attend to their weekly rituals.

A Hell of a Race

Deemed the “most difficult race of the year,” Singapore’s Marina Bay circuit is notorious for its plethora of 90-degree corners along the 23-turn lap. And F1 drivers’ annual complaining mantra is the sauna- or oven-like conditions of warm and sweaty Singapore.

God in the Wheels—F1 Goes Spiritual

Pray for the F1 Singapore Grand Prix

In the aftermath of F1 race organizers seeking protection from God, gods, or goddesses, back in 2016, I’d irreverently coined “F1 Blessing”:

F1 Blessing: When religious leaders from various faiths come together annually to pray for the Singapore Grand Prix and to bless the Formula One night race.

Example: The public has no idea whether the F1 blessing requires the holy men to go through a list of prayer items; if not, what exactly are they praying about?—safety of drivers? good sale of tickets? no crazy spectators crossing the racing track when the race is on? God knows!

by MathPlus September 09, 2016

F1 Prayers

Let’s pray these three F1 prayers for 2023:

1. Pray that few diehard (or better still, zero) fans at the Singapore Grand Prix would be infected with Pirola, the newly recognized variant of Omicron (Covid-19 virus strain BA.2.86), and that no foreign spectators would bring any WHO-undetected variants into the local community.

2. In past events, we’d had unexpected guests like lizards and snakes at the Formula One Singapore event. Pray that no reptiles, giant hornets, or extraterrestrials would show up on Sunday.

3. Pray that all corrupt men and women, be they billionaires, ministers-millionaires, or organizers, who’re behind the “success” of the F1 Singapore Grand Prix, would be exposed, fined, and imprisoned for their illicit financial gains.

F1 Math

A math quickie on Singapore’s “Highest Noise Pollution Day”: Local drivers have so far failed to make the grade at the Singapore GP. Which is more likely: A Singaporean F1 driver making it to the top ten, or Singapore getting into the World Cup final?

Prayerfully and environmentally yours

© Yan Kow Cheong, September 17, 2023.

Taliban Math

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

Bar Modeling, Math Publishing, Problem Solving, Recreational Math, Singapore Math, Stack Modeling, , , , , , , , , , ,
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

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

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.

Calculus for Mature Students

Humor, Mathematical Humor, Recreational Math, Singapore Math, Theomatics, , , , , , , , , , , , , , , , , , , , , ,
Math Meme from Math Lady Hazel

Like math, calculus needn’t sulk (to any degree)! In the hands of an excited middle-school or high-school math teacher, or with access to some creatively written (or online free) resources, the ABCs of calculus can even be taught to elementary school kids.

Think of Mr. Jaime Escalante who had successfully taught calculus to cohorts of Mexican-American students. There is zero excuse why we can’t emulate him to teaching it to financially disadvantaged or minority groups.

What’s Calculus to You?

Do you parrot the textbook definition of calculus to your students? The mathematics of (instantaneous) change. Or do you share it as the branch of mathematics that measures “how far an object has been going fast,” and “how fast an object has gone far”?

Moons ago, I cheekily approached or indirectly defined calculus via the division of zero as follows:

0 ÷ 0: The Raison d’être of Calculus

With some dose of irreverence, the bête noire of high-school or college math could turn out to be a much beloved topic even among the so-called innumerates or mathematically challenged.

Calculus for the Numerati

It’s debatably said that without an exposure of some delta-epsilon calculus, no man or woman can honestly claim to be “mathematically educated” or “mathematically civilized.” Sounds like mathematical pride or arrogance, isn’t it?

Or just an example of “mathematical elitism” à la Trump for those fakes who declare themselves as being a “very stable genius.” Even Einstein had remarked that calculus was “the greatest advance in thought that a single individual was ever privileged to make.”

Fr: Ryan Truong on Facebook’s “Mathematical Mathematics Memes”

Recently, while working on the Urban Calculus manuscript, I forced myself to reread some of the out-of-print pop calculus titles like David Berlinski’s A Tour of the Calculus, Steven Strogatz’s The Calculus of Friendship, and Mary Stopes-Roe’s Mathematics with Love to get an intuitive feel of the subject again.

For a long time, the thought of taking up the challenge to read Newton’s The Principia (even its annotated version) frightens me, because the complexity of the content is beyond me. I’ve a good excuse not to borrow the thick copy from the university library unless I want to look like a “mathematical snob” carrying it around, or use it as a temporary doorstop.

When Comics and Calculus Converge

Calculus for All

Let’s play our part in sharing the mathematical gospel of Newton and Leibniz that calculus needn’t be a four-letter word—how these two mathematical greats had exorcized the demon out of the division of zero.

Why not strive to be the “James Escalante” of your school, state, or country? You’d be the changemaker or mathematical savior in motivating some undecided or mature students to read a calculus course or module in college?

Differentially and integrally yours

© Yan Kow Cheong, July 30, 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

Humor, Mathematical Humor, Problem Solving, Singapore Math, , , , , , , , , , , , ,
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

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

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

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

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.