For the majority of us who aren’t born or blessed with a mathematical or symbol-minded brain, but nevertheless appreciate the austere beauty of mathematics, writing about mathematics and math education is the second best thing we’d do to console ourselves that we needn’t be first-rate mathematicians to enjoy the language of science and technology, or to appreciate the science of patterns.
Some mathematicians write novels under a pseudonym to avoid any suspicion from their faculty bosses; others compose limericks and haikus as a creative outlet to showcase their hidden poetic talents. And for the rest of us who are neither novelists nor poets, maybe submitting some definitions to Urban Dictionary, by coining new mathematical words, or redefining old ones, could be the first step to activating that atrophied right part of our brain, which is allegedly responsible for creativity.
On this Pi Day, let me share with fellow math educators eleven approved definitions related to the irrational and transcendental pi. Don’t ask me how many times I got rejected and needed to resubmit some of these definitions again, before the Urban Dictionary editors decided to approve them.
Pre-Pi Day
Pre-Pi Day seems to have been serially downvoted and subsequently deleted to prevent digital abuse, because the approved entry can no longer be accessed.
Rejection isn’t failure. We keep refining or redefining any rejected definitions until the editors have zero excuses to reject the resubmitted entries. I wished I’d share some recipe for these approved pi definitions, but any attempt to offer some tips to increase a math educator’s chances of getting these math words or terms approved would probably be futile, to say the least.
Over time, although I’ve managed to reduce the odds of rejection, however, some submissions inevitably end up in the little red book of the mean editors—maybe these word doctors had a bad day, or simply because I was submitting some “mathematical crap” that caused me to receive emails like the following:
Urban Dictionary – Pi-rated was not published
Thanks for your definition of Pi-rated!
A few volunteer editors read your definition and decided to not publish it. Don’t take it personally!
Pi-rated The term to describe any faux facts about the irrational number pi.
On Pi Day, our teacher tricked us with some pi-rated math: pi is a rational number (22/7); pi has a different value on the moon that on earth; pi has a value of three in the Bible.
It’s never too late to be mathematically playful, by playing your part in submitting some irreverent mathematical definitions to enliven your math lessons.
Even if math educators are politically apathetic or have near-zero interest in world politics, they can’t discount lightly the irrational thoughts, words, and actions of President Trump and Chairman Kim, because the very presence of these two world “leaders” is two too many—the world is far less secure with these two fellows around, especially when at the push of a nuclear button, millions of innocent civilians would be heading to the other side of eternity sooner than later.
Being neither a mathematician nor a politician, I’d be the most unqualified “math educator” to hypothesize how numerate or “logical” or rational these two disruptive politicians are. However, my educational hypothesis about math education in both North Korea and the US is that North Korea Math is probably less inch-deep-mile-wide than US Math.
On January 2, 2013, I tweeted in tongue-in-cheek the following:
If North Korea were to take part in TIMSS, would it be a surprise if its K–12 #math students outperform their counterparts in the US? #TIMSS
If we factor in the educational budget of each participating country in TIMSS and their local teachers’ limited resources, a politically incorrect ranking would probably look as follows:
There are no ranking errors: Singapore isn’t in the top ten (with or without private tuition).
The Rocket Man and the Mentally Deranged US Dotard
Which “political unthinking” is more dangerous: “Think like Kim” or “Think like Trump”? Who is a more unpredictable or deadly bully? Is “Fatty Kim the Third“—a derogatory term for the well-fed dictator whose own people are starving in the millions—a mere toothless bully vis-à-vis his American counterpart?
Having zero mercy for your political foes, by torturing them and their family members; and poisoning, hanging, or murdering your siblings and relatives, who you suspect are against you.
Or, tweeting and taunting illegal immigrants, radical Islamists, and the LGBT community, which tends to trigger symptoms of insomnia, irrational fear, anxiety, depression, and trauma—the so-called Trump Stress Disorder (TSD)—which unconfirmed reports suggest that they’re more likely to die sooner of heart attack, or to be victims of racial or ethnic persecution.
Kim’s Digital Murder
Below is a quick-and-dirty e-card I tweeted on August 5, 2018 on dictator-murderer Kim, who had his army of hackers or digital terrorists use Photoshop to “digitally murder” his uncle.
If Kim Jong-Un were a radical Muslim-convert, North Korea could become ISIS’s new HQ! #politics #war some.ly/dCh7Y7b
Mathematical Intercourse between Trump and Kim
Another half-baked mathematical e-card I made and tweeted during the Trump-Kim tête-à-tête in Singapore is the following:
On a nonmathematical note,
and the question is: What do you get if you cross a Trump with a Kim? and the answer is: Nothing. You can't cross a dictator with a murderer.
Or, on a mathematical note,
and the question is: What do you get if you cross a mosquito Kim with an overweight Trump? and the answer is: Nothing. You can't cross a vector with a scalar.
A Bromance of Two Dictators
Singapore as the political matchmaker.
Trump claimed that he and Kim “fell in love” after exchanging letters—it sounds like two egomaniacs trying to outwit each other with their insincere sweet words, by stroking each other’s fragile ego.
Trump-Putin, Trump-Xi, Trump-Sisi, Trump-Erdogan, and now Trump-Kim. It appears that dictators do attract each other! A political hypothesis math educators pursuing a PhD in math education might wish to test is: Dictatorship is quadratic!
From Dictator Putin to Emperor Xi to soon-to-be Pharaoh Sisi, Tweeter-in-Chief or Pinocchio-in-Chief Trump, all these power-hungry men have no limits to controlling more yes-men and yes-women, while expecting blind obedience—those who don’t toe the line are likely to be fired prematurely.
In Trump & Kim We Trust
The Symbolic Trump-Kim Meeting in Singapore
Last June, the Singapore government forked out a wallet-unfriendly $20 million to hold the symbolic meeting between President Trump and Chairman Kim.
A Political Math exercise I tweeted then was: Guesstimate how much on average each taxpayer in Singapore “contributed” to footing the $20 million bill for the Trump-Kim meeting. bit.ly/2sVT6jG
Fire and Fury on Kim and His Gang of Killers
@juche_school1
Unlike his dictatorial and murderous grandfather and father who had longed to meeting a US President while they’re still alive, Kim Jong-un is the luckiest of the unholy trinity in finding a “good friend” in Donald Trump.
Political pundits think that North Korea needs more than trade sanctions for its nuclear and missile programs and the threat they pose to the world. A regime change to deliver North Koreans from the tyranny of the Kim dynasty ought to be in the political pipeline.
Unlike Vietnam which tries to threaten Trump and Kim impersonators to stop their “mocking acts,” it’s rather surprising that Singapore didn’t ban these pseudo-tyrants from walking around in town to have some political fun with both locals and tourists.
Abel & Cain 4.0
Thou shalt not kill thy brother.
On the right is an e-card I wrote and tweeted around the time when Kim Jong Un wanted so badly to exterminate his half-brother, Kim Jong Nam.
And below is an approved entry I contributed on “Kim Jong-un and Kim Jong-nam,” which could no longer be publicly accessed online:
Kim Jong-un and Kim Jong-namThe modern-day version of the biblical Abel and Cain, with the chances of the two Kims not meeting their late father and grandfather in hell near to zero.Brothers-rivals Kim Jong-un and Kim Jong-nam serve as ideal plot characters for a Korean spy movie.
Maybe Kim’s hackers felt that the days of its publication should be numbered.
Obama vs. Kim
Unlike a dozen-odd mean tweets on Trump and Kim, any entries on Obama and Kim were in short supply. One I tweeted about them in 2014 in the aftermath of a racist comment on President Obama is the following:
An Unrighteous Deed, IndeedIf Obama is like a "monkey in a tropical forest," then Kim Jong-un must be a "fat pig in Siberia." #North-Korea (@MathPlus on 27/12/14)
Politico-Mathematica à la Singapour
Below are some politically incorrect “political math” questions that teachers could creatively tweak to pitch to their oft-politically challenged students, by conveying the message that math and politics do mix.
1. Parallelism between Two Irrational Personalities
List a dozen parallels between President Trump and Chairman Kim.
For example, Trump and Kim each have been conferred with high-sounding titles for their “contributions” to mankind.
Please call me “Dr. President Trump.”
2. Modeling with Trump and Kim
(a) Trump’s Tweets—Firing by Twitter
Model President Trump’s tweets, which provide a rich source of comedy, into a little juicy formula, which would predict his tweet-before-you-think posts in coming years (assuming that he would still be allowed to tweet behind bars should he be convicted for some political or business collusion with foreign powers).
(b) Kim’s Murders—Murder by Numbers
Formulate a “wicked algorithm” that would guesstimate the number of political critics or foes the Kim dynasty had ordered to be imprisoned, tortured, or killed every year since the Korean War.
3. [Fake] Nobel Prize Winners
What are the odds that the Tweeter-in-Chief and the Murderer-in-Chief might share the coveted the Nobel Peace Prize, if their peaceful actions are perceived to help avert World War III, which could irrationally or maniacally be triggered by pressing their nuclear button?
If someone like Yassir Arafat, who supported terrorism against Israel, could win a Nobel Prize, it’s not far-fetched that both Trump and Kim might be “honored” for fakely bringing world peace to an already-violent world, made insecure by radical Islamists.
4. Political Math
Below are some politico-mathematica questions I posted in recent months.
(a) Political Math: Guesstimate how many false or misleading claims Donald Trump will make by the time he leaves the White House in 2020 (or earlier if he is impeached and imprisoned)—7546 white lies in 700 days. #Singapore #math #politics #estimation #humor (@MathPlus on 30/12/18)
(b) Political Math: Which event has the higher odds of ever happening: Donald Trump winning the Nobel Peace Prize or being canonized as “Saint Trump”? If Arafat can win it, so can Trump! bit.ly/2IwDuP1 #Vatican #peace #sainthood #North-Korea #Singapore #math #miracle #humor (@MathPlus on 18/2/19)
(c) Political Math: What are the odds that if we had had President Hillary Clinton instead of President Donald Trump, she too would have fired the ex-FBI director James Comey, and that her opponents would now be calling for her impeachment and prosecution? #politics #math #hypocrisy (@MathPlus on 3/1/19)
How Trump Does Calculus
(d) Political Math: What are the odds that President Trump would not seek re-election in November 2020 in the aftermath of his impeachment by the House for colluding with Putin and gang? #statistics #SingaporeMath #math #odds #collusion #election #politics #Trump #Putin #humor (@MathPlus on 30/11/18)
5. Murderous Math
Here are some deadly math toughies that may not be apt for politically or religiously immature souls. Caution: Arm yourself, if need be, if you feel that you may be a victim of some form of “mathematical rebellion” from your hostile audience, who may be on a different political or spiritual wavelength as you.
(a) Murderous Singapore Math: What are the odds that there would be a coup in North Korea when Dictator Kim traveled to Hanoi for another symbolic meeting with President Trump? Is China, Iran, Syria, Russia, or Venezuela keen to take him?
Murderous Zeros: If filthy rich Saudis can buy a morally bankrupt fellow like Donald Trump to keep his silence, guesstimate how many “loyal zeros” he is worth to his murderers. #Khashoggi #MBS #murder #guesstimation #math (@Zero_Math on 21/11/18)
Murderous Math: A World Without North Korea—What are the chances that in the aftermath of a North Korean nuclear attack on the US, the Kim dynasty would cease to exist (as the US and allies retaliate to wipe out North Korea from the map)? #war #North-Korea #apocalypse #math (@Zero_Math on 6/12/18)
6. Food & Dog Diplomacy
(a) Singapore Math: What are the chances that North Korea might have a McDonald franchise before having a US embassy? #McKim #humor (@MathPlus on 15/8/17)
McKim was approved on July 13, 2017, and had since been probably “hijacked” by the hackers of the guardian deity of planet Earth.
McKimThe local burger McDonald plans to offer to middle-class North Koreans once Dictator Kim Jong Un and gang give them the green light to operate their first outlet in Pyongyang.The Trump camp thinks that food diplomacy may be a first step to getting the Kim dictatorship to give up its nuclearization program—they've secretly approached McDonald to come up with a North Korean recipe for McKim.
(b) Dog Diplomacy: Chinese Communists give pandas; North Korean Communists give dogs. bbc.co.uk/news/world-asi… #politics #North-Korea #Communism (@MathPlus on 26/11/18)
(c) Faith in god Kim
Kim’s likely promise to Trump: “I want to denuclearize”—as the sanctions hurt and my dynasty must prevail. But like China, Iran, and Russia, where lying and cheating are in their DNA, can the world trust North Korea & the Kim dynasty? #Singapore #politics (@Zero_Math on 10/6/18)
God Has the Final Say in Trump’s Destiny—Not Men or CNN
Let me end on a positive note on how I re-christened or re-defined “President Trump” on 9/11, the date when he’s miraculously elected in 2016.
How God used an ungodly man to effect political change.
Bibliography & References
Monti, R. A. (2018). Donald Trump in 100 facts. UK: Amberley Publishing.
Pater, R. (2016). The politics of design. Amsterdam: BIS Publishers.
A Creative & Disruptive Math Title Coming Your Way*
*Agents keen to represent publishers confident enough to sell an obscene number of If Trump Were Your Math Teachercould contact K C Yan at his e-mail coordinates. A trumpillion thanks!
In Singapore, the durian is officially the only tropical fruit that is banned inside a public train or bus—to critics, it smells worse than urine combined with a pair of used socks.
Presently, transport officials are likely to confiscate the notorious fruit should someone be found conspicuously with it, until recurring public complaints force politicians to implement a fine for those caught carrying one in forbidden places.
If anyone in Singapore can be fined for failing to flush a public toilet, it’s not far-fetched to expect a penalty in a-not-too-distant future for those who inconsiderately propagate the pungent aroma of durians among Singaporeans.
A proof that Singapore is a “fine” city.
Dubbed the “King of fruits” by locals, enjoying the durian is arguably an acquired taste; however, it may cause premature death when eaten together with some types of food or drinks—check this out with your doctor to avoid going to the other side of eternity sooner than later.
For math educators who can’t stand the pungent smell of durian, much less taste it, how can they creatively make use of this much-loved or much-disliked fruit in their mathematics teaching?
In the aftermath of a church in Sarawak, Malaysia erecting a Christmas durian tree, the following estimation questions crossed my mind:
1. Guesstimate the number of durians that were used to make the Christmas tree depicted below.
A durian Christmas tree at a church in Sarawak, Malaysia. Source: @nobisha by norizan sharif on 25/12/18
2. Estimate how much the durian business in Malaysia meant for the China market is worth every year.
3. Estimate how many durians a ten-hectare durian plantation could produce every year.
4. What percentage of the Asian population love to eat the pungent-smelly durian?
Singapore Math and Durian
Below are two irreverent tweets I posted to poke fun at the notoriety of the durian among fruit lovers, who are often tickled by durianians who wouldn’t think twice about forking out more than fifty bucks for one über-smelly durian.
Make a short trip to Malaysia or Thailand during the peak durian season. Try to get hold of a dozen-odd types of durian from the local market or some durian plantations owners. Compare their prices, weights, textures, pH levels, smells, or tastes; and make some conjectures based on nasal, oral, and tactile factors. Does the number of spikes of some durian type exhibit Fibonacci-like behaviors?
2. Death by Durian
Model how many “durian bombs” pseudo-jihadists planning a terrorist hoax in some public places like a college campus or shopping mall would need to simulate some panic or irrational fear among the undergraduates or shoppers.
What are the odds that one of Singapore’s neighboring frenemies could one day use the durian as a low-tech weaponry to neutralize her, just as man-made haze pollution from unfriendly neighbors could potentially be weaponized to suffocate an entire nation?
3. A “Fine” Durian
Imagine that you have been assigned to draft a set of rules that would penalize those caught with durians in forbidden public areas in Singapore. Model a “fines guideline” that wouldn’t unfairly punish those who selfishly insist on polluting their milieux with the nose-unfriendly smell of durians.
New Year, New Entries
On a more positive or non-apocalyptic note, for this new year, some of you might wish to redefine Durian Math or add a new twist to it, as you discover new ways to infuse the term in your math lessons.
A blessed New Year 2019 to all math educators around the world.
Esplanade Theatre—Singapore’s “The Durian.” Photo source: visitsingapore.com
Recently, I was peeping at some postings on the FacebookPSLE Parents group, and I came across the following question:
Philip had 6 times as many stickers as Rick. After Philip had given 75 stickers to Rick, he had thrice as many stickers as Rick. How many stickers did they have altogether?
Here are two solutions that caught my attention to the above primary or grade 6 word problem.
Solution by Izam Marwasi Solution by Jenny Tan
Pseudo-Bar Model Method?
Arguably, the solution by the first problem solver offered to parents looks algebraic, to say the least. Some of you may point out that the first part uses the “unitary method,” but it’s the second part that uses algebra. Fair, I can accept this argument.
Since formal algebra, in particular the solving of algebraic equations, isn’t taught in primary or grade six, did the contributor “mistake” his solution for some form of bar model solution, although no diagram was provided? It’s not uncommon to see a number of pseudo-bar model solutions on social media or on the Websites of tuition centers, when in fact, they are algebraic, with or without any model drawings.
Many parents, secondary school teachers, or tutors, who aren’t versed with the bar model method, subconsciously use the algebraic method, with a bar model, which on closer look, reveals that the mental processes are indeed algebraic. No doubt this would create confusion in the young minds, who haven’t been exposed to formal algebra.
Does the Second Solution Pay Lip Service to Design Thinking?
What do you make of the second solution? Did you get it on first reading? Do you think an average grade five or six student would understand the logic behind the model drawing? From a pedagogical standpoint, the second solution is anything but algebraic. Although it makes use of the bar model method, I wonder what proportion of parents and their children could grasp the workings, without some frustration or struggle.
One common valid complaint by both parents and teachers is that in most assessment (or supplementary) math books that promote bar modeling, even with worked-out solutions to these oft-brain-unfriendly word problems, they’re often clueless how the problem solver knew in the first place that the bar model ought to be presented in a certain way—it’s almost as if the author knew the answer, then worked backwards to construct the model.
Indeed, as math educators, in particular, math writers, we haven’t done a good job in this area in trying to make explicit the mental processes involved in constructing the model drawings. Failure to make sense of the bar models has created more anxiety and fear in the minds of many otherwise above-average math students and their oft-kiasu parents.
Poor Presentation Isn’t an Option
Like in advanced mathematics, the poor excuse is that we shouldn’t be doing math like we’re writing essays! No one is asking the problem solver or math writer to write essays or long-winded explanations. We’re only asking them to make their logic clear: a good presentation forces them to make their thinking clearer to others, and that would help them to avoid ambiguity. Pedantry and ambiguity, no; clarity and simplicity, yes!
Clear Writing Is Clear Thinking
It’s hard work to write well, or to present one’s solution unambiguously. But that’s no excuse that we can afford to be a poor writer, and not a good thinker. As math educators or contributors, we’ve an obligation to our readers to make our presentation as clear as possible. It’s not enough to present a half-baked solution, on the basis that the emphasis in solving a math problem is to get the correct answer, and not waste the time to write grammatically correct sentences or explanations.
I Am Not a Textbook Math Author, Why Bother to Be Precise?
As teachers, we dread about grading students’ ill-written solutions, because most of us don’t want to give them a zero for an incorrect answer. However, if we’re convinced based on their argument that they do know what they’re doing, or show mathematical understanding or maturity of the concepts being tested, then we’d only minus a few marks for careless computation.
Poorly constructed or ill-presented arguments, mathematical or otherwise, don’t make us look professional. Articulating the thinking processes of our logical arguments helps us to develop our intellectual maturity; and last but not least, it makes us become a better thinker—and a better writer, too.
Thousands of students around the world celebrate Pi Day today, but local math students in Singapore can only dream of being part of this annual mathematical event. Singapore math students, teachers, and parents don’t (and can’t) celebrate Pi Day, as long as they officially follow the British style of writing their dates (DD/MM/YY).
What makes matters worse is that this year, Pi Day falls on the first day of the one-week school break, which makes it almost impossible for hardcore math teachers, who want to buck the calendrical trend, to get their students excited about the properties and beauties of the number Pi.
Until Singapore switches to the American style of writing dates (MM/DD/YY), which may not happen, at least during my lifetime, however, this shouldn’t prevent us from evangelizing the gospel of Pi among the local student population.
Here are seven e-gifts of the holy Pi, which I started musing about 314 minutes ago on this Pi Day.
Vintage Christmas—Just like Baby Jesus two millennia ago!
Christmas is a golden and joyful opportunity for number enthusiasts and math geeks to sharpen their creative mathematical problem-solving skills.
Here are 12 CHRISTmaths cookies that may help you shake your brain a little bit in the midst of Christmas festivities.
Warning: Refrain from forwarding this post to relatives or friends living in countries, which are intolerant of Christmas and Christianity, such as Brunei, Saudi Arabia, and Somalia, as it’s haram for “infidels” to take part in any kind of Christmas celebrations. And I assume that includes reading any on-line materials deemed un-Islamic or un-Mohammedan, which might lead believers astray from the faith.
1. Unlucky Turkeys
Estimate the number of turkeys that make their way to the supermarkets every year.
2. A Xmas Candy
Mary wanted to buy a candy that costs 25 cents. A dated vending machine would take one-cent, five-cent, and ten-cent coins in any combination. How many different ways can she use the coins to pay for the candy?
Remember to scan your Christmas item!
3. The Dimensions of a Cross
A square of side 25 cm has four of its corners cut off to form a cross. What is the perimeter of the cross?
4. The Number of Crossings
Two lines can cross one time, three lines three times, four lines six times, and five lines ten times. If there are 25 lines, what would be the maximum number of crossings be?
5. An Eco-Xmas
If all instances of the word “CHRISTMAS” were replaced with “XMAS,” how much ink and paper (or Xmas trees) could you save every year? How much money could be channelled back to feeding the poor and the hungry during the festive season?
In an age of Xmas e-cards and video cards, how many Christmas greetings cards are still being sent worldwide? How many trees are being saved every festive season?
(a) Without a calculator, how would you verify whether the number 25! has precisely 25 digits or not.
(b) Which positive integers n (other than the trivial case n = 1) for which n! has exactly n digits?
GST (or VAT) with no thanks to Father Xmas
8. Xmas Trees
Guesstimate how big a forest would 25 million Christmas trees occupy.
9. Folding papers
Fold a single piece of paper perfectly in half, from left to right. How many creases will there be after the 25th fold, when you continue folding so that all the rectangles are folded into two halves each time?
10. Pre-Xmas Tax
Imagine Singapore were to implement a pre-Christmas tax on all kinds of Christmas marketing before the first week of December. Estimate how many extra million dollars would the Income Tax department collect every festive season.
Math educators, especially stressed [often self-inflicted] local teachers in Singapore, are always on the look-out for something funny or humorous to spice up their oft-boring math lessons. At least, this is the general feeling I get when I meet up with fellow teachers, who seem to be short of fertile resources; however, most are dead serious to do whatever it takes to make their teaching lessons fun and memorable.
It’s often said that local Singapore math teachers are the world’s most hardworking (and arguably the world’s “most qualified” as well)—apparently, they teach the most number of hours, as compared with their peers in other countries—but for the majority of them, their drill-and-kill lessons are boring like a piece of wood. It’s as if the part of their brain responsible for creativity and fun had long been atrophied. A large number of them look like their enthusiasm for the subject have extinguished decades ago, and teaching math until their last paycheck seems like a decent job to paying the mortgages and to pampering themselves with one or two dear overseas trips every other year with their loved ones.
Indeed, Singapore math has never been known to be interesting, fun, or creative, at least this is the canned perception of thousands of local math teachers and tutors—they just want to over-prepare their students to be exam-smart and to score well. The task of educating their students to love or appreciate the beauty and power of the subject is often relegated to outsiders (enrichment and olympiad math trainers), who supposedly have more time to enrich their students with their extra-mathematical activities.
A prisoner of war in World War II, Sidney Harris is one of the few artists who seems to have got a good grasp of math and science. While school math may not be funny, math needn’t be serious for the rest of us, who may not tell the difference between mathematical writing and mathematics writing, or between ratio and proportion. Let Sidney Harris show you why a lot of things about serious math are dead funny. Mathematicians tend to take themselves very seriously, which is itself a funny thing, but S. Harris shows us through his cartoons how these symbol-minded men and women are a funny awful lot.
Angel: “I’m beginning to understand eternity, but infinity is still beyond me.”
Mathematical humor is a serious (and dangerous) business, which few want to invest their time in, because it often requires an indecent number of man- or woman-hours to put their grey matter to work in order to produce something even half-decently original or creative. The choice is yours: mediocrity or creativity?
Humorously and irreverently yours
References
Adams, D. S. (2014). Lab math. New York: Cold Spring Harbor Laboratory Press.
Harris, S. (1970). What’s so funny about science? Los Altos, Ca.: Wm. Kaufmann, Inc.
Check out an inexpensive (but risky) way to make a Singapore math lesson less boring: The Use of Humor in Mathematics. The author would be glad to visit local schools and tuition centers to conduct in-service three-hour math courses for fellow primary and secondary math teachers, who long to bring some humor to their everyday mathematical classrooms—as part of their annual 100 hours professional upgrading. Please use his e-mail coordinates on the Contact page.
In the aftermath of the death of Singapore’s founding father, Mr. Lee Kuan Yew (1923–2015), a number of numerological tidbits (or numerical curiosities, to put it mildly) floated on social media, which got a number of apparently self-professed innumerates pretty excited. Here are three such postings that I saw in my Facebook feed and on WhatsApp.
The WhatsApp message gives the impression that it was the works of some “pseudo-mathematician,” but it could very well have been the digital footprints of a “mathematical crank” or an amateur-numerologist, who wanted to tickle mathophobics with such numerical coincidences.
Did Singapore’s numerologists (or pseudo-mathematicians) fail to point out some of the following numerological absurdities?
The digital root of Mr. Lee’s birth year is 1 + 9 + 2 + 3 = 15, which stands for the last two digits of the year he experienced his last heartbeat.
The pollution index for that week was in an unhealthy range, and the average PSI for the six-day mourning period was about 91.
Or, were there exactly 91 priests on vigil at an undisclosed Roman Catholic Church, who were interceding for Mr. Lee to ensure that his heavenly destination is 100% secured, although his manifold deeds to the nation inarguably exceeds the number of his political faux pas, especially vis-à-vis his political enemies or opponents?
Or, did 91 senior monks and nuns (or did I mistake them for disciples of Shintoism?) resort to “synchronized chanting” to ensure that the highest level of enlightenment be bestowed on the late Mr. Lee, who might be reincarnated as a future Buddha for his numerous selfish deeds towards his oft-ungrateful and unappreciative fellow citizens?
And did any police personnel verify whether there were 91,000 odd mourners in black attire on that Black Sunday, not to say, 91 VIPs or Heads of States who attended the eulogy, depending on one’s definition of a VIP?
The Numerology of the Old Guard
One Facebook numerological factoid that circulated in the first post-LKY week was the following:
Singapore’s political fathers who outlived the biblical three-scores-and-ten lifespan
At face value, these nonagenarians had their blessed lives prolonged up to “four scores and ten and one” years. Sounds like their good earthly or political deeds were good karma for their longetivity? Are they the recipients of the following success equation?
Sacrifice + Service + Incorruptibility + Risk = Political Success + Longevity
Observe that simply taking the difference between the birth year and the death year of Mr. S Rajaratnam suggests that he died at the age of 91; however, if we look closely at the month dates (Feb. 25, 1915 – Feb. 22, 2006), he was still 90 years old, when he passed away. The same argument goes for Dr. Toh Chin Chye (Dec. 10, 1921 – Feb. 3, 2012), who wasn’t yet 91, when he died. So, always take the pseudoscience of numerology with a grain of salt. As with fengshui charlatans, a degree of skepticism towards numerologists of all sizes and shapes isn’t an option—wear your critical-thinking cap when meeting, or reading about, these paranormal folks!
To rational non-punters or non-gamblers, betting on someone’s death date, whether he or she was poor or rich on this side of eternity, seems like an extreme case of bad taste, or simply showing zero respect for the deceased and their family members. However, in superstitious circles, that practice isn’t uncommon among mathematically challenged or superstitious punters, who think that bad luck paranormally translates into good omen, if they bet on the digits derived from the death date or age of a recently deceased person.
In fact, during the nation’s six-day mourning period for its founder, besides the long queues of those who wanted to pay their last respects to Mr. Lee at the Parliament House, another common sight islandwide were meters-long lines of 4D or TOTO punters, who wanted to cash in on the “lucky digits” to retire prematurely, hoping to lay hold of the traditional 5Cs (cash, car, condo, credit card, country club), coveted by hundreds of thousands of materialistic Singaporeans.
Instead of promoting a numerological or pseudoscientific gospel based on Mr. Kuan Yew’s death date or age, which only helps to propagate superstition and pseudoscience, perhaps a “mathematically healthy” exercise would be to leverage on the D-day to teach our students and their parents some basic numerical properties—for example, conducting a recreational math session on “Number Theory 101” for secondary 1–4 (or grades 7–10) students might prove more meaningful or fruitful than dabbling in some numerological prestidigitation, or unhealthy divination.
A Search for Patterns
91 is the product of two primes: 91 = 7 × 13
91 = 1² + 2² + 3² + 4² + 5² + 6²
91 is also the sum of three squares: 1² + 3² + 9²
Are there other ways of writing the number 91 as a sum of squares?
91 = 33 + 43
Non-Numerological Questions to Promote Problem-Solving Skills
Let’s look at an “inauspicious number” of elementary- and middle-school (primary 5–secondary 4) math questions, which could help promote numeracy rather than numerology among students and teachers.
1. Sum of Integers
Show that the number 91 may be represented as the sum of consecutive whole numbers. In how many ways can this be done?
2. The Recurring Decimal
What fraction represents the recurring decimal 0.919191…?
3. Palindromic in Base n
For what base(s) will the decimal number 91 be a palindromic number (a number that reads the same when its digits are reversed)? For example, 91 = 101013.
4. The Billion Heartbeat
Does a 91-year-lifespan last less or more than a billion heartbeats?
5. Day of the Week
Mr. Lee Kuan Yew (September 16, 1923–March 23, 2015) died on a Monday. Using the 28-year cycle of the Gregorian calendar, which day of the week was he born?
6. One Equation, Two Variables
If x and y are integers, how many solutions does the equation x² – y² = 91 have?
7. Singapore’s New Orchid
A new orchid—Singapore’s national flower—had been named after Mr. Lee: Aranda Lee Kuan Yew. Using the code A = x, B = x + 1, C = x + 2, …, , does there exist an integer x such that ARANDA sums up to 91? In other words, does there exist a numerological system such that A + R + A + N + D + A = 91?
8. Singapore’s Coin Goes Octal
The alleged involvement of Mr. Lee in Singapore’s “lucky” octagonal one-dollar coin
There is an apocryphal story that had circulated for many years linking Mr. Lee Kuan Yew with Singapore’s octagonal one-dollar coin. A high-ranking monk had apparently told Mr. Lee that Singapore’s fortune would continue to rise only if Singaporeans were to carry a bagua—the eight-sided fengshui symbol. That prediction allegedly prompted the Monetary Authority of Singapore to issue the octagonal shape of the nation’s one-dollar coin.
That rumor was later put to rest by no other than self-declared agnostic Mr. Lee himself in one of his books, Hard Truths. He remarked that he had zero faith in horoscopes, much less the pseudoscience of fengshui.
What is the sum of the interior angles of the Singapore’s eight-sided coin?
9. Show that the largest number k for which the decimal expansion of 2k does not contain the digit 1 is 91.
1. One example is 91 = 1 + 2 + 3 +⋯+ 13. 2. 91/99. 5. Mr. Lee was born on a Sunday. 6. Hint: Show that x² – y² = 91 has 8 integer solutions. 9.Hint: Use a computer to verify the result.
Gain that competitive edge, by being a creative Singapore math educator and problem solver! Title available on App Store and Google play.
One Singapore’s problem-solving strategy that is gaining currency among more and more local teachers in Singapore is the Stack Model Method, which has proved to be conceptually more advantageous—a more intuitive and creative strategy—than the bar model method. On a lighter note, let’s look at a dozen benefits one could derive should one fearlessly embrace this visualization problem-solving strategy to solve word problems.
1. As a Form of Therapy
Like bar modeling, getting involved in stack modeling may act as a form of visual therapy, especially among visual learners, and for those who need a diagram or model to make sense of a problem-situation. Indeed, a model drawing is often worth more than a dozen lines of algebraic symbols.
2. A Possible Cure to Dementia
Like Sudoku and crossword puzzles, practicing the science and art of stack modeling may help arrest one’s schizophrenia or dementia, particularly those who fear that their grey matter might play tricks on them in their golden years.
3. Prevention of Visual or Spatial Atrophy
For folks wishing to enhance their visualization skills, stack modeling could potentially turn their worry of short-term visual apathy and long-term visual atrophy into aha! moments of advanced visual literacy.
4. A Disruptive Methodology and Pedagogy
When most Singapore coaches and teachers are no longer excited or thrilled about the Singapore’s model method, what they need is a more powerful and intuitive problem-solving strategy like the stack model method to give them that competitive edge over their peers, all of whom are involved in the business of Singapore math—from training and coaching to consulting and ghostwriting.
5. A Platform for Creative Thinking in Mathematics
Getting acquainted to the stack model method would not only help one to hone one’s visualization skills, but it’ll also refine one’s problem-solving and creative thinking skills. Being mindful that competing stack models could be designed to figure out the answer, the challenge is to come up with the most elegant stack model that could vow even the mathophobics!
6. Look-See Proofs for Kids
Stalk modeling could help remove any “mathematical cataract” from one’s mind’s eye to better “see” how the parts relate to the whole. The way stack models are drawn (up-and-down and sideways) often allows one to see numerical relationships that would otherwise be difficult to visualize if bar models were used instead.
7. The Beauty and Power of Model Diagrams
Even those who are agnostic to the Singapore math curriculum, a “Stack Modeling” lesson could help enliven the beauty and power of model diagrams in creative problem solving. The stack model method could act as a catalyst to “seeing” the connection between parts and whole—normally, the same result would be tediously or boringly derived by analytic or algebraic means, understood only by students in higher grades.
8. A Simple but Not Simplistic Strategy
Like Trial and Error, or Guess and Check, the stack model method shows that Draw a Diagram is a simple, but not simplistic, problem-solving strategy. The stack model reinforces the idea that often “less is more.” The simplicity of a stack model can reveal much hidden information that is often lost in an algebraic argument.
9. A Branded Problem-Solving Strategy
For math educators who might think that Singapore math, or the bar model method, in particular, is a mere fad in mathematics education, the stack model method further disproves that myth. Like bar modeling, stack modeling shows that a simple problem-solving strategy like the “draw a diagram” has what it takes to attaining brand status, especially when we consider the types of challenging word problems that lend themselves to both bar and stack models, and which could also be assigned to a younger audience.
10. Stack Modeling as a Creative Art
To the novice problem solver, stack modeling is a science; to the seasoned problem solver, stack modeling is an art— the challenge is to come up with more than one stack model to arrive at the answer. Remember: Not all stack models are created equal!
If you are a mathepreneur, you can easily steal the ideas in The Stack Model Method: An Intuitive and Creative Approach to Solving Word Problems to write a more expensive Singapore math book on the subject. There are dozens of ethically challenged ghost writers and cash-strapped undergrads from China, India, and the Philippines, who can help you professionally plagiarize any types of editable contents! You earn as you learn! Of course, you need to mail them your copy, or buy a new copy for them to do the “creative work” at a fractional cost! Make sure you don’t get caught, though!
12. Green Math à la Singapour
Ecologically speaking, stack modeling, which generally uses less space than bar modeling, could help math educators save millions of ink and square miles of paper [aka trees]. In economic terms, millions of dollars could be saved by the right choice of model drawing. In other words, stack modeling could act as a catalyst to help one play one’s part in reducing one’s carbon footprints!
From Bar to Stack Modeling
With a bit of imagination, I bet you could come up with another dozen benefits of stack modeling. The stack model method is no longer an option, nor should it be treated as a mere visualization strategy to be discussed only during an enrichment math lesson.
The stack method is going to be a problem-solving strategy of choice, as more math educators worldwide invest the time to learn and apply it to solve non-routine questions in elementary math. Be among the first creative problem solvers to embrace the stack model method, as you gain that methodological or pedagogical edge over your fellow math educators!
Most of us may not admit it, but we’ve all fallen victim to the lure of innumeracy—the mathematical equivalent of illiteracy—consciously or unconsciously. Here are twenty of my favorite innumerate events I often witness among my numerate and semi-numerate friends, colleagues, and relatives.
• Taking a 45-minute train journey to save a few dollars at Carrefour or Walmart.
• Lining up for hours (or even days, if you’re in China?) to buy an iPhone or iPad.
• Paying a numerologist or geomancy crank to divine your “lucky” and “unlucky” days.
What is the smallest and the largest four-digit number?
• Visiting a feng-shui master to offer advice how best to arrange your furniture at home, or in your office, to ward off negative or “unwanted energies.”
• Buying similar items in bulk at discounted prices, which you don’t need but because they’re cheap.
• Offering foods to idols [aka gods and goddesses] in the hope that they’ll bring you good luck and prosperity in return.
• Offering gifts to hungry [angry?] ghosts to appease them lest they come back to harm you and your loved ones.
• Buying insurance policies against alien abduction, meteorites, biological warfare, or the enslavement of the apocalyptic Beast.
• Filling up lucky draw vouchers, by providing your personal particulars for future pests-marketeers and time-sharing consultants.
The Hello Kitty Syndrome in Singapore—Purchase of no more than four sets per customer will start past midnight!
• Betting on horses, football, stocks, and the like—any get-rich activities that may cut short a 30-year working life, slaving for your mean or half-ethical bosses 9-to-6 every day.
• Buying lottery tickets to short-circuiting hard work, or to retiring prematurely.
• Going on annual pilgrimages to seeking blessing from some deities, prophets, saints, or animal spirits.
• Outsourcing your thinking to self-help gurus or motivational coaches.
• Going for prices that end in 99 cents, or acquiring auctioned items that are priced at $88 or $888—the number 8 is deemed auspicious among superstitious Chinese.
An NIE motto to innumerate undergrads: “Always give more than 100%!”
• Replying to spam mails from conmen and “widows” from Nigeria, Russia, or China, who are exceedingly generous to transfer half of their inherited money to your bank account.
• Taking a half-day leave from work, or faking sickness to visit the doctor, to line up for hours to buy McDonald Hello Kitties.
• Lining up overnight to buy the latest model of a game console, or to secure an apartment unit of a newly built condominium.
• Enrolling for courses that cost over a thousand bucks to learn “Effective Study Habits of Highly Successful Students.”
• Postponing all important meetings, or avoiding air traveling, on a Friday the thirteenth.
• Canceling all major business dealings, weddings, or product launches during the Ghost (or Seventh) Month.
Now is your turn to share with the mathematical brethren at least half a dozen of your pet innumerate activities—those numerical idiocies or idiosyncrasies— that you (or your loved ones) were indulged in at some not-too-distant point in the past.
© Sidney Harris There is nothing new under the mathematical sun! 
