**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.

**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.

© Yan Kow Cheong, November 1, 2017.

]]>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.

**Pi Day vs. Abacus Day**

**A 14-Month Year for Singapore ONLY!**

*Where Are You in Pi?*

*Heavenly* Pi

**The Numerology (or Pseudoscience) of Pi**

**In Remembrace of the Late Singapore PM **

**Biblical Pi vs. Mathematical Pi**

*Happy Pi Day!*

© Yan Kow Cheong, March 14, 2016.

]]>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

**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?

**7. Does Xmas! have 25 digits?**

1! = 1, 2! = 1 × 2 = 2, 5! = 1 × 2 × 3 × 4 × 5 = 120—a 3-digit number, and 10! = 1 × 2 ×⋯× 10 = 3,628,800—a 7-digit number.

(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.

**References**

Gould T. (2013). *You’re all just jealous of my jetpack*. New York: Drawn & Quarterly.

Yan, K.C. (2011). *Christmaths: A creative problem solving math book*. Singapore: MathPlus Publishing.

Zettwoch, D., Huizenga, J., May, T. & Weaver, R. (2013). *Amazing facts… & beyond! with Leon Beyond.* Minneapolis: Uncivilized Books.

** A Xmas Bonus**: 25

**Selected Hints & Answers**

2. 12 ways. Hint: Make an organized list.

3. 100 cm.

4. 300 crossings.

5. About 30 million gallons of ink, 500 square miles of paper, and $15 trillion could be saved.

6. Hint.

7. (b) *n* = 22, 23, 24.

9. 2^{25} – 1.

11. 4.

12. *Hint*: Why as one gets older, time appears to fly faster.

© Yan Kow Cheong, December 25, 2015.

]]>

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.

© Yan Kow Cheong, August 20, 2015.

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.

**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?

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?

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 = 3 ^{3} + 4^{3}**

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 = 10101_{3}.

**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 *x*² – *y*² = 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 2* ^{k}* does not contain the digit 1 is 91.

© Yan Kow Cheong, April 26, 2015.

**Selected Answers/Hints**

**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 – *y*² = 91 has 8 integer solutions.

**9.** *Hint*: Use a computer to verify the result.

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!

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!

**References**

Yan, K. C. (2015). *The stack model method: A creative and intuitive approach to solving word problems (Grades 5–6)*. Singapore: MathPlus Publishing.

Yan, K. C. (2015). *The stack model method: A creative and intuitive approach to solving word problems (Grades 3–4)*. Singapore: MathPlus Publishing.

© Yan Kow Cheong, January 10, 2015.

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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.

**• 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.**

• 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.

**• 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.

© Yan Kow Cheong, November 10, 2014.

]]>The first official title on the Singapore model method to hit the local shelves was one co-published by the Singapore’s Ministry of Education (MOE) and Panpac Education, which the MOE christened a “monograph” to the surprise of those in academia. Thank God, they didn’t call it *Principia Singapura*!

This wallet-unfriendly—over-promise, under-deliver— title did fairly well, considering that it was the first official publication by the MOE to feature the merits of the Singapore’s model method to a lay audience. Half of the book over-praises the achievements of the MOE in reversing the declining math performance of local students in the seventies and eighties, almost indirectly attributing Singapore’s success in TIMMS and PISA to the model method, although there has never been any research whatsoever to suggest that there is a correlation between the use of the model method and students’ performances in international comparison studies.

Busy and stressed local parents and teachers are simply not interested in reading the first part of this “monograph”; they’re looking for some practical teaching strategies that could help them coach their kids, particularly in applying the model method to solving word problems. However, to their utter disappointment, they found out that assessment (or supplementary) math books featuring challenging word problems are a better choice in helping them master the problem-solving strategy, from the numerous graded worked examples and detailed (and often alternative) solutions provided—and most of them cost a fraction of the price of the “monograph.”

Not long after the MOE’s publication, the Singapore public was spoilt with another local title on the bar method. Unfortunately, the editorial team working on *Bar Modeling* then didn’t take advantage of the lack of breadth and depth of the MOE’s “monograph” to offer a better book in meeting the needs and desires of local parents and overseas math educators, especially those not versed with the bar model method.

Based on some investigation and feedback why Dr. Yeap Ban Har didn’t seize the opportunity to publish a better book than the one co-published by the MOE, it sounds like Dr. Yap had submitted his manuscript one or two years prior to the MOE’s publication, but by the time his publisher realized that the MOE had released a [better?] book similar to theirs, they had little time to react (or maybe they just over-reacted to the untimely news?); as a result, they seemed to have only made some cosmetic changes to the original manuscript. Sounds like what we call in local educational publishing as an example of “editors sitting on the manuscript” for ages or years only to decide publishing it when a competitor has already beaten them to the finishing line.

This is really a missed opportunity, not to say, a pity that the editorial team failed to leverage on the weaknesses or inadequacies of the MOE title to deliver a better book to a mathematically hungry audience, at an affordable price.

Early this year, we’re blessed with another title on the bar method, and this time round, it’s reasonably affordable, considering that the contents are familiar to most local teachers, tutors, and educated parents. This 96-page publication—no re-hashed Dr. Kho articles and authors’ detailed mathematical achievements—comprises four topics to showcase the use of the model method: *Whole Numbers, Fractions, Ratio, and Percentage*.

As in Dr. Yeap book, the questions unfortunately offer only one model drawing, which may give novices the impression that no alternative bar or model drawings are possible for a given question. The relatively easy questions would help local students gain confidence in solving routine word problems that lend themselves to the model method; however, self-motivated problem solvers would find themselves ill-equipped to solve non-routine questions that favor the visualization strategy.

In the preface, the authors emphasized some pedagogical or conceptual points about the model method, which are arguably debatable. For example, on page three, they wrote:

“In the teaching of algebra, teachers are encouraged to build on the Bar Model Method to help students and formulate equations when solving algebraic equations.”

Are we not supposed to wean students off the model method, as they start taking algebraic food for their mathematical diet? Of course, we want a smooth transition, or seamless process, that bridges the intuitive visual model method to the abstract algebraic method.

Because one of the authors had previously worked with Dr. Kho Tek Hong, they mentioned that he was a “pioneer of the model method.” True, he was heading the team that made up of household names like Hector Chee and Sin Kwai Meng, among others, who helped promote the model method to teachers in the mid-eighties, but to claim that Dr. Kho was the originator or inventor of the bar method sounds like stretching the truth. Understandably, it’s not well-known that the so-called model method was already used by Russian or American math educators, decades before it was first unveiled among local math teachers.

I’ll elaborate more on this “acknowledgement” or “credit” matter in a future post—why the bar model method is “math baked in Singapore,” mixing recipes from China, US, Japan, Russia, and probably from a few others like Israel and UK.

Let me end with two local titles which I believe offer a more comprehensive treatment of the Singapore model method to laypersons, who just want to grasp the main concepts, and to start applying the visual strategy to solving word problems. I personally don’t know the author, nor do I have any vested interest in promoting these two books, but I think they’re so far the best value-for-money titles in the local market, which could empower both parents and teachers new to the model method to appreciate how powerful the problem-solving visualization strategy is in solving non-routine word problems.

A number of locals may feel uneasy in purchasing these two math books published by EPH, the publishing arm of Popular outlets, because EPH’s assessment math books are notoriously known to be editorially half-baked, and EPH every now and then churns out reprinted or rehashed titles whose contents are out of syllabus. However, my choice is still on these two wallet-friendly local books if you seriously want to learn some basics or mechanics on the Singapore model (or bar) method—and if editorial and artistic concerns are secondary to your elementary math education.

**References**

Curriculum Planning & Development Division Ministry of Education, Singapore (2009). *The Singapore model method*. Singapore: EPB Pan Pacific.

Gan, A. (2014). *More model methods and advanced strategies for P5 and P6*. Singapore: Educational Publishing House Pte. Ltd.

Gan, A. (2011). *Upper primary maths model, methods, techniques and strategies*. Singapore: Educational Publishing House Pte Ltd.

Lieu, Y. M. & Soo, V. L. (2014). *Mathematical problem solving — The bar model method*. Singapore: Scholastic Education International (Singapore) Private Limited.

© Yan Kow Cheong, August 5, 2014.

]]>I try to understand why local math writers feel that they’ve to provide such an unhealthy number of these drill-and-kill questions or routines for students to practice. Surely, their publishers are behind this “numerical obscenity,” arguing that many *kiasu* parents would feel that they’re buying a value-for-money book. Imagine the emotional or psychological harm it could do to a child who barely has any interest in the subject.

Even for those with a fluency with numbers, wouldn’t a thousand (or even half of that number) questions bore them to death? Many would end up getting a distorted picture of what elementary math is all about—*imitate-practice-imitate more-practice more.*

Drill-and-kill math questions are like the junk food of math education. Every now and then, we all visit these fast food outlets, but to grow up on a diet that consists mainly of burgers, French flies, and Coke for most days of the week, that can only be a good recipe for unavoidable obesity and all sorts of preventable diseases that would pop up sooner than later. Likewise, over-exposing students to these math drills would only produce an army of drill-and-kill specialists, who wouldn’t be able to think critically and creatively in mathematics—and later in life.

For Singapore’s Golden Jubilee, when the nation celebrates its 50th year of independence in 2015, other than giving items like a special medallion, a multi-functional shawl, and a set of baby clothes, one of the suggested gifts by the mathematical brethren to be given to Singaporean babies born next year should be a check amounting to not less than a thousand bucks. This would help defray the average cost incurred by parents who would need to buy those elementary math supplementary titles during the child’s formative years—it’s a rite of passage for every child studying in a Singapore government school to expose himself or herself to these drill-and-kill questions. Even if parents are opposed to these math drills, school teachers or tutors might still recommend the child to buy one!

Of course, it’s better to produce tens of thousands of drill-and-kill specialists than an army of innumerates. In the short term, this raises the self-esteem of the children, as they feel that they can solve most of these routine questions, by parroting the solutions of the worked examples. We all learn by imitation, but at least the questions need to slowly move from routine to non-routine. We can’t afford to have hundreds of questions that test the same skills or sub-skills almost without end. We need a balanced percent of drills, non-drills, and challenging questions—not a disproportionate unhealthy number of drill-and-kill questions, which would only affect our long-term mathematical health.

One *good* thing about these wallet-friendly drill-and-kill math titles is that they can easily replace a lousy or lazy math teacher, or even a mathophobic parent. After all, we can’t expect too much from a mathematical diet that could help sustain (or even strengthen) us for a while. At best, they’re like Kumon Math, which subscribes to the philosophy that more practice and more drill could eventually help the child to become self-motivated, and to raise his or her self-esteem in the short term.

If mathematical proficiency is credited to be achieved through practice and more practice, would you buy a few of these math practice titles for your child? Or, as a teacher or tutor, would you recommend one for your student or tutee? Don’t you think that this is a case of “more is less”? Could the child end up being mathematically undernourished when it comes to creative problem solving and higher-order thinking? Or maybe the child could even be numerically constipated by an overdose of these drill-and-kill questions?

© Yan Kow Cheong, July 4, 2014.

]]>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.

*Happy Creative Problem Solving!*

© Yan Kow Cheong, June 12, 2014.

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