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.
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?
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?
6. Number of Xmas Cards
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?
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.
11. A Xmas Quickie or Toughie
What is the sum of the last two digits of 1! + 2! + 3! +⋯+ 24! + 25!?
12. An Ever-Early Xmas
Show that as one celebrates more and more Christmases (or, as one gets older and wiser), Christmas seems to come earlier every year.
Gould T. (2013). You’re all just jealous of my jetpack. New York: Drawn & Quarterly.
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.
Singapore Math via Humor
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
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.
RIP: Lee Kuan Yew (1923–2015)
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:
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!
Fortune via Misfortune—From 4D to 5C
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.
Number Theory over Numerology
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
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.
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!
11. Earn as You Learn
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.
• 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.
• 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.
If you had been on Facebook for some time, it’s very likely that you would have come across a certain type of numbers puzzles streaming your feed. Clearly, they’re not quite the typical numerical puzzles that often appear in school math books; rather, they’re closer to the types of questions posed in aptitude or IQ tests. One such number puzzle is the following.
Unlike in textbooks that often present these logic puzzles in an uninteresting way, by seeing these colorful number puzzles on Facebook or Pinterest, and being hinted that only a small percent of the problem solvers apparently managed to get the correct answer, this entices readers to give it a try to see how well they’ll fare vis-à-vis their oft-mathematically challenged Facebook friends. Here’s another such number puzzle.
Arguably, these numbers-and-words puzzles have a certain charm to it, because they often require just simple logic to solving them, unlike similar brainteasers that may require some knowledge of elementary or middle-school math. What about the following Facebook numerical puzzle?
What is 1 + 1?
What do you make of this one?
If these on-line numbers puzzles indirectly help promote logic and number sense among math-anxious social-media addicts; and along the way, provide them with some fun and entertainment, let’s have more of them in cyberspace!
Let me leave you with half a dozen of these numerical puzzles, stolen from my Facebook feed. Note that these types of brain-unfriendly math or logic questions may have more than one mathematically or logically valid answer, depending on the rule or formula you use—they serve as numerical catalysts for promoting creative thinking in mathematics.
In my formative years, I don’t recall any elementary school teachers sharing some mathematical puns with us. I suppose that the arithmetic of yesteryear was often taught in the most uninteresting way by many who probably didn’t look forward to teaching it to a bunch of noisy kids.
Here are some pointers I would share with my students at the start of a secondary math olympiad programme. In Singapore, most mathletes attending these enrichment classes are usually selected by the form or math teacher, who tends to choose the best three math students from each class to form a small group of 15 to 20 participants. They would then graduate to represent the school after attending a six-, eight-, or ten-session training programme, depending on the mathematical needs and wants of the school.
• Take your time! Very few contestants can solve all given problems within the time limit. For instance, in the Singapore Mathematics Olympiad (SMO), both at the junior and senior levels, [unconfirmed] feedback based on different schools’ results hints to the fact that those who can win a medal hover around five percent.
Interestingly but disturbingly, an SMO mathlete who can get six or seven out of 35 questions correct may still win a bronze medal, revealing how unmoderated olympiad math papers had been in recent years, going by the abnormally high rate of failures among the participants. This is primarily due to the fact that few, if any, faculty members who set these competition papers, are familiar with what elementary and middle school teachers are covering in local schools.
• Try the “easier” questions first. The questions aren’t set in ascending order of difficulty. It’s not uncommon to see easier questions in the second half of the paper.
• Olympiad questions don’t “crack” immediately. Be patient. Try various approaches. Experiment with simple cases. Working backwards from the desired result in some cases is helpful.
• If you’re using a contests book, and you’re stuck, glance at the “Hints” section. Sometimes a problem requires an unusual idea or technique.
• Even if you can solve a problem, read the hints and solutions. The hints may contain some ideas or insights that didn’t occur in your solution, and they may discuss intuitive, strategic, or tactical approaches that can be used elsewhere.
Remember that modeled or elegant solutions often conceal the torturous or tedious process of investigation, false starts, inspiration and attention to detail that led to them. Be aware of the behind-the-scenes hours-long dirty mathematical work! When you read the modeled solutions, try to reconstruct the thinking that went into them. Ask yourself, “What were the key ideas?” “How can I apply these ideas further?”
• Go back to the original problem later, and see if you can solve it in a different way, or in a different context. When all else fails, remember the reliable old friend, the guess-and-check strategy (or heuristic, as it’s being arguably called in Singapore)—for instance, substituting the optional answers-numbers given in an MCQ into some given equation or expression may yield the answer sooner than later.
Meaningful or creative problem solving takes practice, with insightful or elegant solutions not being the norm. Don’t get discouraged if you don’t seem to make any headway at first. The key isn’t to give up; come back to the question after a day or a week. Stickability and perseverance are two long-time buddies for full-time problem solvers.
A rectangular enclosure is 30 meters wide and 50 meters long. Calculate its area in square meters.
The My Pals Are Here Math Series
The My Pals Are Here series has been rumored to have been edited and ghostwritten by a hundred odd editors and freelancers in the last decade.
Lack of mathematical rigor was initially targeted against Dr. Fong Ho Kheong and his two co-authors by American profs in the first or/and second editions —probably by those who were “ghost advisors or consultants” for Everyday Math.
Deconstructing the Singapore Model Method
1. It’s a problem-solving strategy—a subset of the “Draw a diagram” strategy.
2. It’s a hybrid of China’s line method and Russia’s box (or US’s bar) method.
3. It’s the “Draw a diagram” strategy, which has attained a brand status in mathematics education circles.
4. It’s a problem-solving method that isn’t recommended for visually challenged or impaired learners.
5. It allows questions traditionally set at higher grades (using algebra) to be posed at lower grades (using bars).
Painless Singapore Math
Perhaps that’s how we’d promote Singapore math to an often-mathophobic audience!
A Singapore Ex-Minister’s Math Book
Dr. Yeo had the handwriting below depicted in his recreational math book, regarding his two granddaughters, Rebecca and Kathryn.