Category Archives: Problem Solving

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

The Mathematics of Trump and Kim

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

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.

Below are two irreverent entries that were approved by the Urban Dictionary editors about Trump and Kim.

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
Edelman Rodriguez’s “Play Date” @edelstudio (12/6/18)

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.

Thou Shalt Not Impersonate Thy Leader
Two dictators who dream of the Nobel Peace Prize.
  Photo ©BBC & Ed Jones/Getty

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-nam

The 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, Indeed

If 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.

McKim

The 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.

Defector: Kim Jong Un ordered execution by flamethrower https://www.cnn.com/videos/world/2019/02/22/north-korea-defectors-todd-pkg-tsr-vpx.cnn

Vietnam deports Kim Jong-un impersonator ahead of summit http://www.bbc.co.uk/news/world-asia-47354761

Vietnam detains impersonators of Kim Jong Un and Trump http://www.bbc.co.uk/news/world-asia-47337945

Vietnamese barber marks summit with free Trump-Kim haircuts http://bit.ly/2tu4Rij

Trump-Kim T-shirts hottest item ahead of Hanoi summit as shops cash in http://bit.ly/2EveqDV

Jim Carrey targets Trump with blunt political cartoons https://www.cnn.com/style/article/jim-carrey-trump-political-cartoons/index.html

© Yan Kow Cheong, February 27, 2019.

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 Teacher could contact K C Yan at his e-mail coordinates. A trumpillion thanks!

The Fake Bar Model Method

Bar Modeling, Problem Solving, Singapore Math, Singapore Model Method, , , , , , , , , , ,

Recently, I was peeping at some postings on the Facebook PSLE 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 contributed by Izam Marwasi Solution by Izam Marwasi
Solution by Jenny Tan 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.

© Yan Kow Cheong, November 1, 2017.

The 12 Problems of CHRISTmaths

Mathematical Humor, Problem Solving, Recreational Math, Singapore Math, Theomatics, , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
Vintage Christmas—Just like Baby Jesus two millennia ago! 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?

Christmastize your code!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?

XMaths Tech© T. Gauld’s You’re all just jealous of my jetpack (2013)

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?

GST with no thanks to Father XmasGST (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.

Folding a Santa Claus© Anonymous Folding a Santa Claus

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.

Xmas Möbius Strips Christmas Möbius Strips

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 CHRISTmaths Toughies from Singapore ?? http://tinyurl.com/q9w3ne9

 

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. 225 – 1.

11. 4.

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

2012-12-21 23.13.28

© Yan Kow Cheong, December 25, 2015.

 

Singapore Math Books on the Bar Model Method

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

In recent years, because of the popularity of Singapore math books being promoted and used in many countries, suddenly local publishers seemed to have been hit by an aha! moment. They realized that it’s timely (or simply long overdue?) that they should come up with a general or pop book on the Singapore’s model (or bar) method for the lay public, especially among those green to the problem-solving visualization strategy.

A Monograph à la Singapour

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!

The Singapore Model MethodA wallet-unfriendly title that focuses on the ABC of the Singapore’s problem-solving visualization strategy

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.”

A Missed Opportunity for a Better Strategy

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.

Bar ModelingAnother wallet-unfriendly title that ill-prepares local parents and teachers to mastering the model, or bar, method in solving non-routine word problems

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.

Is Another Bar Model Method Book Needed?

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.

Who Invented the Model 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.

Mathematical Problem Solving—The Bar Model MethodA wallet-friendlier book on the Singapore model method, but it fails to take advantage of the weaknesses of similar local and foreign titles on the bar method

Mr. Aden Gan‘s No-Frills Two-Book Series

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.

Singapore Model MethodA no-frills two-assessment-book series that gives you enough basic tools to solve a number of grades 5–6 non-routine questions

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.

7000+ Drill-and-Kill Singapore Math Questions

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

Based on the average number of questions contained in an assessment (or supplementary) math book these days, it can be estimated that most students in Singapore would have solved some seven thousand math questions by the time they had graduated from elementary school. Indeed, an awful lot, to say the least.

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.

Primary One Mathematics Tutor—Book 1A Imagine that’s just Book 1A for grade one! How many more drills are there for Book 1B?

The Junk Food of Singapore’s Math Education

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.

The Baby [Math] Gift for Singapore’s Jubilee

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!

Primary Four Mathematics Tutor—Book 4A It looks like an average grade 4 child has to solve not less than a thousand odd questions before he or she could graduate to grade 5.

Drill-and-Kill Specialists or Innumerates

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.

Primary Five Mathematics Tutor—Book 5A Interestingly, many average students find books like the above far more useful than their oft-boring school textbooks and workbooks.

The Good Part of Drill-and-Kill

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.

Maths Practice 1000+—Grade 5 Some of these reprinted or rehashed “math drills” titles cost on average less than one cent per question—no doubt, many [gullible?] parents think or feel that they’re getting a pretty good deal from buying these wallet-friendly books—cheap and “good”!

Are You a Disciple of Drill-and-Kill?

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?

Maths Practice 1000+—Grade 6 Caution: The above title may be hazardous to your long-term mathematical health—it could atrophy your higher-order, creative, and critical thinking skills in mathematics!

© Yan Kow Cheong, July 4, 2014.

Number Patterns on Social Media

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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.

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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.

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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?

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What is 1 + 1?

What do you make of this one?

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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.

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Happy Creative Problem Solving!

© Yan Kow Cheong, June 12, 2014.

A Grade 5 Bicycles-and-Tricycles Problem

Problem Solving, Singapore Math, Singapore Model Method, , , , , , , ,

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In an earlier post, I shared about the following chickens-and-rabbits problem.

There are 100 chickens and rabbits altogether. The chickens have 80 more legs than the rabbits. How many chickens and how many rabbits are there?

Other than using a guess-and-guess strategy and an algebraic method, both of which offering little pedagogical or creative insight, let me repeat below one of the two intuitive methods I discussed then.

Since the chickens have 80 more legs than the rabbits, this represents 80 ÷ 2 = 40 chickens.

Among the remaining (100 – 40) = 60 chickens and rabbits, the number of chicken legs must be equal to the number of rabbit legs.

Since a rabbit has twice as many legs as a chicken, the number of chickens must be twice the number of rabbits in order for the total number of legs to be equal.

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From the model drawing,

3 units = 100 − 40 = 60
1 unit = 60 ÷ 3 = 20

Number of rabbits = 1 unit = 20
Number of chickens = 2 units + 40 = 2 × 20 + 40 = 80

The Bicycles-and-Tricycles Problem

Again, if we decided to ban any trial-and-error or algebraic method, how would you apply the intuitive method discussed above to solve a similar word problem on bicycles and tricycles?

There are 60 bicycles and tricycles altogether. The bicycles have 35 more wheels than the tricycles. How many bicycles and tricycles are there?

Go ahead and give it a try. What do you discover? Do you make any headway? In solving the bicycles-and-tricycles question, I find that there are no fewer of half a dozen methods or strategies, which could be introduced to elementary school students, three of which lend themselves easily to the model, or bar, method, excluding the Sakamoto method.

© Yan Kow Cheong, March 4, 2014.

Some Advice for Singapore Mathletes

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

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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.

Happy problem solving!

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© Yan Kow Cheong, Feb. 7, 2014.

Life’s Simple Mathematical Pleasures

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Stealing the idea from Nancy Vu’s “Just Little Things,” I tried to reflect on some mathematical things that give us delight and pleasure as we go about our hectic 24/7/365 lives.

Here are some of my personal mathematical pleasures.

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It’s now your turn to reflect on some aha! mathematical moments  to share with the rest of us.

References

Wu, N. (2013). Just little things: A celebration of life’s simple pleasures. New York: A Perigee Book.

JustLittleThings.net

© Yan Kow Cheong, November 10, 2013.

The Singapore Excess-and-Shortage Problem

Problem Solving, Sakamoto method, Singapore Math, Singapore Model Method, Stack Method, , , , , , , ,

In Singapore, in grades four and five, there is one type of word problems that seldom fail to appear in most local problem-solving math books and school test papers, but almost inexistent in local textbooks and workbooks. This is another proof that most Singapore math textbooks ill-prepare local students to tackle non-routine questions, which are often used to filter the nerd from the herd, or at least stream the “better students” into the A-band classes.

Here are two examples of these “excess-and-shortage word problems.”

Some oranges are to be shared among a group of children. If each child gets 3 oranges, there will be 2 oranges left. If each child gets 4 oranges, there will be a shortage of 2 oranges. How many children are there in the group?

A math book costs $9 and a science book costs $7. If Steve spends all his money in the science books, he still has $6 left. However, if he buys the same number of math books, he needs another $8 more.
(a) How many books is Steve buying?
(b) How much money does he have?

A Numerical Recipe

Depicted below is a page from a grade 3/4 olympiad math book. It seems that the author preferred to give a quick-and-easy numerical recipe to solving these types of excess-and-shortage problems—it’s probably more convenient and less time-consuming to do so than to give a didactic exposition how one could logically or intuitively solve these questions with insight.

A page from Terry Chew’s “Maths Olympiad” (2007).

Strictly speaking, it’s incorrect to categorize these questions under the main heading of “Excess-and-Shortage Problems,” because it’s not uncommon to have situations, when the conditions may involve two cases of shortage, or two instances of excess.

In other words, these incorrectly called “excess-and-shortage” questions are made up of three types:
・Both conditions lead to an excess.
・Both conditions lead to a lack or shortage.
・One condition leads to an excess, the other to a shortage.

One Problem, Three [Non-Algebraic] Methods of Solution

Let’s consider one of these excess-and-shortage word problems, looking at how it would normally be solved by elementary math students, who have no training in formal algebra.

Jerry bought some candies for his students. If he gave each student 3 candies, he would have 16 candies left. If he gave each student 5 candies, he would be short of 6 candies.
(a) How many students are there?
(b) How many candies did Jerry buy?

If the above question were posed as a grade 7 math problem in Singapore, most students would solve it by algebra. However, in lower grades, a model (or intuitive) method is often presented. A survey of Singapore math assessment titles and test papers reveals that there are no fewer than half a dozen problem-solving strategies currently being used by teachers, tutors, and parents. Let’s look at three common methods of solution.

Method 1

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Difference in the number of candies = 5 – 3 = 2

The 16 extra candies are distributed among 16 ÷ 2 = 8 students, and the needed 6 candies among another 6 ÷ 2 = 3 students.

Total number of students = 8 + 3 = 11

(a) There are 11 students.

(b) Number of candies = 3 × 11 + 16 = 49 or  5 × 11 –  6 = 49

Jerry bought 49 candies.

Method 2

Let 1 unit represent the number of students.

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Since the number of candies remains the same in both cases, we have

3 units + 16  = 5 units – 6

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From the model,
2 units = 16 + 6 = 22
1 unit = 22 ÷ 2 = 11
3 units + 16 = 3 × 11 + 16 = 49

(a) There are 11 students.
(b) Jerry bought 49 candies.

Method 2 is similar to the Sakamoto method. Do you see why?

Method 3

The difference in the number of candies is 5 – 3 = 2.

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The extra 16 candies and the needed 6 candies give a total of 16 + 6 = 22 candies, which are then distributed, so that all students each received 2 extra candies.

The number of students is 22 ÷ 2 = 11.

The number of candies is 11 × 3 + 16 = 49, or 11 × 5 – 6 = 49.

Similar, Yet Different

Feedback from teachers, tutors, and parents suggests that even above-average students are often confused and challenged by the variety of these so-called shortage-and-excess problems, not including word problems that are set at a contest level. This is one main reason why a formulaic recipe may often do more harm than good in instilling confidence in students’ mathematical problem-solving skills.

Here are two grade 4 examples with a twist:

When a carton of apples were packed into bags of 4, there would be 3 apples left over. When the same number of apples were packed into bags of 6, there would still be 3 apples left over. What could be the least number of apples in the carton? (15)

Rose had some money to buy some plastic files. If she bought 12 files, she would need $8 more. If she bought 9 files, she would be left with $5. How much money did Rose have? ($44)

Conclusion

Exposing students of mixed abilities to various types of these excess-and-shortage word problems, and to different methods of solution, will help them gain confidence in, and sharpen, their problem-solving skills. Moreover, promoting non-algebraic (or intuitive) methods also allows these non-routine questions to be set in lower grades, whereby a diagram, or a model drawing, often lends itself easily to the solution.

References

Chew, T. (2008). Maths olympiad: Unleash the maths olympian in you — Intermediate (Pr 4 & 5, 10 – 12 years old). Singapore: Singapore Asian Publications.

Chew, T. (2007). Maths olympiad: Unleash the maths olympian in you — Beginner (Pr 3 & 4, 9 – 10 years old). Singapore: Singapore Asian Publications.

Yan, K. C. (2011). Primary mathematics challenging word problems. Singapore: Marshall Cavendish Education.

© Singapore Math, October 27, 2013.

PMCWP4-2See Worked Example 2 on page 8; try questions 7-8 on page 12.