Category Archives: Singapore Math

News and Views about the Good, the Bad, and the Ugly about Singapore Math

Numbering Our Days

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Thou shalt remember thy PIN! © 2010 Summersdale Publishers Ltd

Teach us to number our days aright, that we may gain a heart of wisdom.
— Psalm 90:12

Every second counts. Every minute counts. These sound more like clichés to many of us that few would pay attention to. Most of us live our lives as if there’ll be many more tomorrows. For the rest of us who are nearing, or have lived past, the half life, mortality is no longer a topic we can conveniently dismiss. Some try to ignore it, or pretend that age is just a number, or that they’re young at heart—they “psycho themselves” to think or speak like folks from the Positive Thinking or New Age movement.

The Three-Scores-and-Ten Lifespan

The Holy Scriptures tell us that the majority of us are approximately given a three-scores-and-ten lifespan; for a blessed minority, it’d be extended to four scores and ten—90 years. The names of two ex-political ethical leaders cross my mind: Nelson Mandela and Lee Kuan Yew.

Even with medical breakthroughs in recent decades, a look at the obituary pages in the papers every day shows that the average lifespan of a man or woman has remained fairly constant for centuries—even with women living an average of 3 to 5 years longer than men, depending on which continent of the world they live in.

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Is age a mere psychological figure? “I am seventy years young!”

The Billionth Heartbeat

For most part of human history, the billionth heartbeat has defined the length of a man’s days. Even today, that nine-zero figure remains fairly constant in a number of African or developing countries. But, thanks to medical advances and better standards of living, many in developed nations can live up to about two to three billion heartbeats.

A Satanic Alert

If you’re more of an apocalyptic type, then you’re more likely to define your mortality in terms of some multiple of the beast number. This means you’ve about 888 months during your earthly stay to live in a manner that could reduce your odds of joining folks like Idi Amin Dada, Saddam Hussein, and Adolf Hitler.

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Not all dozing folks on the bus are dead! © 2003 Summersdale Publishers Ltd

Mortality and Eternity

Show me, O Lord, my life’s end and the number of my days; let me know how fleeting is my life. —Psalm 39:4

It’s easy to count our age and the number of years we’ve lived, but it’s difficult to count the number of our remaining days—we simply can’t count from the future.

Perhaps, we need to pray the psalmist’s prayer: Teach us to number our days aright, that we may gain a heart of wisdom.

In Job 14:5, we read: Man’s days are determined; you have decreed the number of his months and have set limits he cannot exceed.

Elsewhere, Jesus told Peter: “I alone control the length of a man’s days.”

No doubt, our days are numbered, and how can we make wise use of them? How do we frame our finite days in the light of eternity? How do we break away from living unremarkably average lives? How can we plan not just for a big, meaningful day or event, but also for a big, meaningful life?

It’s high time we stop kidding ourselves: We don’t have 500 years to live. If we realize that our average lifespan of “three scores and ten” years on earth—about two to three billion heartbeats, depending on our location, position, or station in life—are insignificant in the light of eternity, we’ll value what are important: Love God and His people.

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What would you define as your “prime years”?

Math Educators, What’s Your Legacy?

With The Lord a day is like a thousand years, and a thousand years are like a day.
— 2 Peter 3:8

For us, math educators, how can we move more to the right side of the bell curve when it comes to impacting the lives of others in some areas of mathematics education? How can we say NO to living mediocre mathematical lives, although we may not presently have all the necessary tools in our mathematical toolkit to reach out to those whom we long to positively influence?

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Is fifty the new thirty-five?

A Spiritual Formula for Longevity

Let me leave you with some verses that may hold the key to seeing you live beyond the  billionth heartbeat—three verses that may be worth keeping in your heart.

Ephesians 6:2–3 

“Honor your father and mother”—which is the first commandment with a promise—”that it may go well with you and that you may enjoy long life in the earth.”

Exodus 20:12 (The 6th Commandment):

“Honor your father and your mother, so that you may live long in the land the Lord your God is giving you.”

Number your days. More than a million people die every single week. Think about it. You made it another week! You’re better off than a million folks. You can thank God that you’re alive. Make “numbering your days” a priority in your mathematical journey!

Practice

1. How many times does the heart beat in a person’s lifetime? How do the figures vary for those living in developed and developing countries?

2. Show that most folks have an average of 888 months to live on this side of eternity.

3. Based on a two- or three-billion-heartbeat lifespan, or depending on the continent you are living in, what fraction of your lifetime have you lived? How do you plan to spend the remaining of it meaningfully?

References

Solomon, R. M. (2012). Reflections on time & eternity. Singapore: Genesis Books.

Summersdale Publishers Ltd (2011). Old is the new young. UK: Summersdale Publishers Ltd.

Fraser, B. (2010). You know you’re having a senior moment when…. West Sussex, UK: Summersdale Publishers Ltd.

Fraser, B. (2003). You know you’re getting old when…. West Sussex, UK: Summersdale Publishers Ltd.

© Yan Kow Cheong, July 27, 2013.

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A funny book with a touch of seriousness.

Problem Solving Made Difficult

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The US edition of a grade 5 Singapore math supplementary title.

Recently, while revising a grade 5 supplementary book I wrote for Marshall Cavendish, I saw that other than the answer, there was no solution or hint provided to the following question.

If Ann gave $2 to Beth, Beth would have twice as much as Ann.
If Beth gave $2 to Ann, they would have the same amount of money.
How much did each person have?

Most grade 7 Singapore math textbooks and assessment books would normally carry a few of these typical word problems, whereby students are expected to use an algebraic method to solve them. For instance, using algebra, students would form two linear equations in x and y, before solving them by the elimination, or substitution, method. A pretty standard application of solving a pair of simultaneous linear equations, by an analytic method.

However, it’s not uncommon to see these types of word problems appearing in lower-grade supplementary titles, whereby students could solve them, using the Singapore model, or bar, method; and the Sakamoto method. In other words, these grade 7 and 8 questions could be solved by grade 5 and 6 students, using a non-algebraic method.

Algebra versus Model Drawing

Conceptually speaking, I think a grade 6 or 7 student who can solve the above word problem, using a model drawing, appears to exhibit a higher level of mathematical maturity than one who simply uses two variables to represent the unknowns, before forming two simultaneous linear equations to solve them. Of course, because the numbers in this question are relatively small, it’s not surprising to catch a number of average students relying on the trial-and-error method to find the answer.

Try to solve the question, using both algebra and a model; then compare the two methods of solution. Which one do you think demands a deeper or higher level of reasoning or thinking skills?

Depicted below is a model drawing of the above grade 5 word problem.

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From the model drawing,

1 unit = 2 + 2 + 2 + 2 = 8
1 unit + 2 = 10
1 unit + 6 = 14

Ann had $10.
Beth had $14.

Generalizing the Problem

A minor change in the question, by altering the “number of times” Beth would have as much money as Ann, reveals an interesting pattern: the model drawing remains unchanged, except for the varying number of units that represent the same quantity.Here are two modified versions of the original grade 5 question.

If Ann gave $2 to Beth, Beth would have three times as much as Ann.

If Beth gave $2 to Ann, they would have the same amount of money.
How much did each person have?
Answer: Ann–$6; Beth–$10.

If Ann gave $2 to Beth, Beth would have five times as much as Ann.
If Beth gave $2 to Ann, they would have the same amount of money.
How much did each person have?
Answer: Ann–$4; Beth–$8.

From Problem Solving to Problem Posing

The two modified questions could serve as good practice for students to become skilled in model drawing, and to help them deduce numerical relationships confidently from them. Besides, they provide a good opportunity to challenge students to pose similar questions, by altering the “number of times” Beth would have as much money as Ann. Which numerical values would work, and what ones wouldn’t, in order for the model drawing to make sense, or for the question to remain solvable?

Conclusion

Let me end, by tickling you with another grade 5 question, similar to the previous three word problems.

If Ann gave $2 to Beth, Beth would have three times as much as Ann.
If Beth gave $2 to Ann, they would have twice as much money as Beth.
How much did each person have?

Answer: Ann–$4.40; Beth–$5.20.

How do you still use the model method to solve this slightly modified ratio question? Test it on your better students or colleagues! It’s slightly harder, because any obvious result isn’t easily deduced from the model drawing, as compared to the ones posed earlier on. Besides, unlike the three previous word problems whose answers are integers, this last problem has a decimal answer—it just doesn’t lend itself well to the guess-and-check strategy.

Share with us how your students or colleagues fare on this last question. Remember: No algebra allowed!

© Yan Kow Cheong, July 12, 2013.

Singapore math authors-millionaires

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Mr. Chow’s new revised grade 7 textbook—a US edition is also available, which competes with an equivalent title in the “Math in Focus” series.

It’s an open secret that two of the well-paid math authors in Singapore are Dr. Fong Ho Kheong and Mr. Chow Wai Keung—two non-Singaporeans who have made it to the Millionaire Dollar Club. Also on the Forbes’ Singapore Math List are local folks like Dr. Y. H. Leong, Andrew Er, Fabian Ng, and Lee-Ann Goh, albeit their names are most likely alien to those outside Singapore.

Obscure writing, obscene royalties

A talking point in the local mathematical community is that both millionaires-authors “can’t write”—their titles are notoriously heavily edited or ghostwritten by editors. For instance, there is a decade-long local joke that over a hundred editors have their “editorial footprints” on Dr. Fong’s dozen odd titles.

Form or substance

As for Mr. Victor Chow, his critics remarked that his series of no-frills Discovering Maths titles—apparently a canned version of his ill-written books, which have been poorly received in Hong Kong—is ironically (or miraculously?) doing pretty well in Singapore, in spite of the fact that the competitors’ authors have been household names in math education for decades—many of whom are still teaching teachers.

Many attributed the decent or successful adoption of the Discovering Mathematics series in local schools, primarily because of better sales and marketing strategies by the publisher, as compared to those used by its competitors—form has allegedly triumphed over substance, thanks to lateral (and often shady) marketing.

Interestingly, that many in academia and in local publishing circles subscribe to the above views or rumors, whether because they’re jealous and envious of their “obscene” royalties, is understandable. Apparently, they rationalized that Dr. Fong’s and Mr. Chow’s “below-average writing skills” didn’t match their deserved earnings.

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Dr. Fong co-authored latest grade 1 textbook, based on the new Singapore syllabus.

A mix of jealousy and envy and …

Having had the opportunity to speak with some of Dr. Fong’s ex-colleagues, and those who know him personally, it sounds to me that jealousy and envy feature high in discrediting him for “earning so much,” as they feel that they “can lecture better” and “have written more quality research papers” than him.

The argument is that writing textbooks (even successful ones) are for second-rate math educators and mathematicians—unspokenly, first-rate math folks write papers and speak at conferences; second- and third-rate folks write textbooks, or become consultants of these textbooks.

What is seldom talked about is that a number of these so-called seasoned lecturers feel marginalized or “blacklisted” by local publishers for not approaching them—many are still waiting for publishers to line up outside their offices to beg them to write for them. As a result, it’s not surprising that a number of them condescendingly blame local publishers and editors for choosing second-rate writers to author the school textbooks.

Dr. Fong—Singapore’s math popularizer

What we seldom hear, though, is that albeit Dr. Fong might arguably be a “boring presenter or lecturer,” as remarked by his critics, he nevertheless had the guts to promote his books in public, unlike his fellow ex-colleagues who think that it’s a “degrading job” to become a salesperson in promoting their titles at math conferences. Today, who’s having the last laugh to the bank?

In fact, it’s probably not an exaggeration to say that other than Fabian Ng and one or two ghostwriters, it’s Dr. Fong who helped popularize the Singapore model method and the problem-solving strategies locally, through his supplementary math books and public talks in the nineties, written for both students and parents. Yes, long before the Andrew Er’s and Yeap Ban Har’s books were spotted in the local market.

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An assessment title of yesteryear—a forerunner of Dr. Fong & Company’s titles.

When risky wasn’t the new safe yet

At the other end of the wealth distribution curve, we’ve dozens of local math writers who wouldn’t dare to being a full-time author, simply because they’re more likely to end up begging than earning enough royalty to pay their bills. Unless you’re a shrewd textbook author-entrepreneur like Dr. Fong, the rest of us write more for our egos than expect any financial rewards, albeit few would admit it.

Negative royalties

I’ve also heard of local math authors who had earned “negative royalties,” which means they owed the publisher instead—they had sold zero copies, and dozens of free copies were given, as part of some book promotion or launch.

Math can make you rich!

Dr. Fong and Mr. Chow both show that you needn’t be the best writer in town, not even a decent one, but if you work hard and smart, and ignore your critics; and if you’ve faith that your publisher has a good sales and marketing strategy, it’s possible to make a decent living in math education.

And what’s even more amazing is that both are foreign-born writers, who have seized the opportunity to make it big in Singapore, when the majority, some of whom are no doubt smarter and better than them, have let their intellectual or mathematical pride and arrogance prevent them from contributing more to raising the standard of mathematics education in Singapore.

© Yan Kow Cheong, June 30, 2013.

Postscript: The author (@Zero_Math and @MathPlus) is a self-professed zeronaire, who is “infinitely jealous and envious” of these authors-millionaires, who have shown us that with hard work (and some luck by the side) “one can get rich with math,” infinitesimal as the chances may be.

The legitimacy of the bar method

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“In Step Maths” grades 1-6 used to be a popular series among local schools—a far more user-friendly series than the “My Pals Are Here” and ‘Math in Focus” series.

During this haziest and most polluted week in Singapore, while looking out for some teaching tips in some dated teaching guides, I came across the following grade 3 Singapore math question, which looks more like a grade 5 question to me:

A number represented by the letter B, divided by 6 and then added to 6, gives the same answer as when the same number B is divided by 9 and then added to 9. What is the number B?

How would you solve it, using the Singapore model, or bar, method? Give it a try before peeping at the solution below, which is the one given in the guide. Would you accept the teacher’s guide’s solution as one that effectively uses the power of the bar model in arriving at the answer?

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A bar-modeled solution to the above grade 3 word problem.

Is there an abuse or misuse of the bar method?

Personally, I’m not too comfortable with the given solution, as I feel it lacks some legitimacy in the effective use of the bar method in arriving at the answer. What do you think? Do you sense a misuse or abuse of the visualization strategy? How would you use the bar method, or any non-algebraic method, in solving this question? Share your thoughts with us on whether the bar method has legitimately been applied to solve this grade 3 word problem.

Reference
Gunasingham, V. (2004). In Step Maths Teacher’s Guide 3A. Singapore: SNP Panpac Pte Ltd.

© Yan Kow Cheong, June 21, 2013.

Is Singapore math frickin’ hard?

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There is a millennium myth that Singapore math is hard or tough—that only geeks from some remote parts of Asia (or from some red little dot on the world map) should do it. The mass media (and math educators, too) have implicitly mythologized that those who are presently struggling with school math, should avoid Singapore math, in whatever form it’s being presented, totally. A grave mistake, indeed!

Is Singapore math a mere fad?

After reading dozens of tweets and blog posts on the pluses and minuses of Singapore math, it sounds as if Singapore math has a certain mystique around it—some kind of foreign math bestowed by some creatures from outer space to terrorize those who wished math were an optional subject in elementary school.

Not to say, folks who totally reject Singapore math on the basis that it’s just another fad in math education, or another marketing gimmick to promote an allegedly “better foreign curriculum” to math educators. They believe that “back-to-basics math,” whatever that phrase means to them, with all its memorizing and drill-and-kill exercises, is a necessary mathematical evil to get kids to learn arithmetic.

Singapore math OR/AND Everyday Math

Singapore math? Sure, no problem! It’s no big deal!

Well, it’s a big deal for traditional publishers, which may lose tons of money if more states and schools continue to embrace this “foreign brand” of math education.

Poor writing and teaching from a number of us could have indirectly contributed to the white lie that if you can’t cope with Everyday Math or Saxon Math, or whatever math textbook your school or state is currently using, Singapore math is worse! You might as well forget about it!

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One of the first few Singapore supplementary titles to promote the model method in the mid-nineties.

The bar method as a powerful problem-solving strategy

Objectively speaking, Singapore math doesn’t come close to most pedagogical insights or creative ideas featured in journals and periodicals published by the MAA and the NTCM. Personally, I must admit that I become a better teacher, writer, and editor, thanks to these first-class publications. The Singapore model method, although a key component of the Singapore math curriculum, is just one of the problem-solving strategies we use every day, as part of our problem-solving toolkit.

On the other hand, for vested interests on the part of some publishers, little has been done to promote other problem-solving strategies, such as the Stack Method and the Sakamoto method, which are as powerful, if not more elegant, than the bar method—in fact, more and more local students and teachers are using them as they see their advantages over the model method in a number of problem situations.

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Fabien Ng in his heyday was a household name in mathematics education in Singapore—it looks like he’s since almost disappeared from the local publishing scene.

Don’t throw out the mathematical baby yet!

You may not wish to adopt the Singapore math curriculum, or even part of it; but at least consider the model and stack methods, not to say, the Sakamoto method, as part of your arsenal of problem-solving strategies (or heuristics, as we call them here). Don’t let traditional publishers (or “math editors” with a limited repertoire of problem-solving strategies) prevent you from acquiring new mathematical tools to improve your mathematical problem-solving skills.

In Singapore, the model, or bar, method is formally taught until grade six to solve a number of word problems, because from grade seven onwards, we want the students to switch over to algebra. However, this doesn’t mean that we’d totally ban the use of the model method in higher grades, because in a number of cases, the model or stack method often offers a more elegant or intuitive method of solution than its algebraic counterpart.

© Yan Kow Cheong, May 27, 2013.

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A booklet comprising of PSLE (grade 6) past exam papers, which cost me only 88 cents in the eighties.

A Singapore Grade Two Tricky Question

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A classic elementary math problem that folks from a number of professions, from psychologists to professors to priests like to ask is the following:

A bat and a ball cost $1.10 in total. 
The bat costs a dollar more than the ball. 
How much does the ball cost?

For novice problem solvers, the immediate, intuitive answer is 10 cents. Yet the correct response is 5 cents. Why is that so?

If the ball is 10 cents, then the bar has to cost $1.10, which totals $1.20. Why do most of us jump to the wrong conclusion—that the ball costs 10 cents?

We should expect few students to bother checking whether the intuitive answer of 10 cents could possibly be wrong. Research by Professor Shane Frederick (2005) finds that this is the most popular answer even among bright college students, be they from MIT or Harvard.

A few years ago, I included a similar question for a grade 2 supplementary title, as it was in vogue in some local text papers. See the question below.

Recently, while revising the book, I saw that the model drawing had been somewhat modified by the editor. Although a model drawing would likely help a grade 2 child to better visualize what is happening, however, a better shading, or the use of a dotted line, would have made the model easier to understand. Can you improve the model drawing?

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Try to solve the above question in a different way, using the same model drawing.

Interestingly, I find out that even after warning students of the danger of simply accepting the obvious answer, or reminding them of the importance of checking their answer, variations of the above question do not seem to help them improve their scores. I recently tickled my Fan page readers with the following mathematical trickie.

Two cousins together are 11. 
One is 10 years older than the other. 
Find out how old both of them are.

Let me end with this Cognitive Reflection Test (CTR), which is made up of tricky questions whose answers tend to trap the unwary, and which may be suitably given to problem solvers in lower grades.

1. If it takes 5 machines 5 minutes to make 5 bearings, how long would it take 100 machines to make 100 bearings?

2. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

3. A frog is climbing up a wall which is 12 m high. Every day, it climbs up 3 m but slips down 2 m. How many days will it take the frog to first reach the top of the wall?

4. A cyclist traveled from P to Q at 20 km/h, and went back at 10 km/h. What is his average speed for the entire journey?

5. It costs $5 to cut a log into 6 pieces. How much will it cost to cut the log into 12 pieces?

Expected incorrect answers

1. 100 minutes. 2. 24 days. 3. 12 days. 4. 15 km/h 5. $10

Correct answers

References

Donnelly, R. (2013). The art of thinking clearly. UK: Sceptre.

Yan, K. C. (2012). Mathematical quickies & trickies. Singapore: MathPlus Publishing.

Postscript: What’s your CTR score? Here is something to ponder about: Those with a high CTR score are often atheists; those with low CTR results tend to believe in God or some deity.

© Yan Kow Cheong, May 7, 2013.

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For practice on the Singapore model method, this title may help—visit Singaporemath.com

Can Math Be Sexy?

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One thing is almost certain is that if Danica McKellar, actress and mathematician-turned-math-author, were to write a book on Singapore math, my bet is that it would unlikely be approved by the Singapore’s Ministry of Education (MOE), although few would deny that her title would probably be a terrific draw among local students, even in conservative or puritan Singapore—it may even end up being the first math book on a Singapore bestseller’s list.

Going by her math titles (Kiss my math; Math doesn’t sulk; and Hot algebra exposed!), which are primarily targeted at an American or liberal audience, should the TV personality be tempted to write a math book for a local audience, it would be a miracle if her manuscript got pass the first round of Singapore’s MOE’s censorship board—not before making most MOE curriculum specialists flush of embarrassment.

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Would you buy this math title for your son or daughter?

Infidel Math

Indeed, the Taliban and the Ayatollahs wouldn’t approve Danica McKellar’s irreverent titles, not to say, her irreligious style of writing, but what about “moderate” nations like Singapore, which unspokenly or secretly longs to be portrayed as a conservative society with “high moral standards”? McKellar’s math books are anything but boring! In fact, she thinks that math can be easy, relevant, and even glamorous. By adding a little glamour and humor to the teaching of mathematics, it looks like the math advocate has, to a large extent, demonstrated that math can actually be accessible to young girls—and young boys, too!

 

A Math Role Model for Girls (and Boys)

In math education circles, it’s not surprising that Danica McKellar is regarded by many [open-minded] parents and math educators as a terrific role model [for girls and young women]. She teaches the value of confidence that comes from feeling [math-]smart. Her supporters think that her books should be required reading for every math-anxious school girl! The message seems to be that physical beauty and quantitative literacy need’t be mutually exclusive.

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Would you give away this irreverent guide about pre-algebra to your neighbor’s son?

From Boyfriend to Babysitting

A random look at some chapters of Math Doesn’t Sulk needs no explanation why teenage girls wouldn’t want to give math a serious try, or a second chance, especially if they want to appear smart and beautiful, to their boyfriends. Would you put down a math title with these chapter headings?

 

Chapter 1: How to Make a Killing on eBay (Prime Numbers and Prime Factorization)

Chapter 2: Do You Still Have a Crush on Him? (Finding the Greatest Common Factor)

Chapter 7: Is Your Sister Trying to Cheat You Out of Your Fair Share? (Comparing Fractions)

Chapter 11: Why Calculators Would Make Terrible Boyfriends (Converting Fractions and Mixed Numbers to Decimals)

Chapter 12: How to Entertain Yourself while Babysitting a Devil Child (Converting Decimals to Fractions) 

Math Doesn’t Sulk also comes with a math horoscope, math personality quizzes, and real-life testimonials. What else more can one expect from a math book? In fact, Danica herself exemplifies her own life from being a terrified middle-school math student to a confident actress, and more.

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X-rated algebra with a dose of irreverence and humor

Girls get curves: Geometry takes shape

I’m currently looking at a copy of McKellar’s latest publication, Girls get curves; personally, I think it’s the most useful of all her publications so far, as it covers a number of middle- and high-school topics, such as congruency, similarity, and proofs, which are relevant to my teaching and writing. My wild guess is that her next title would be one on Calculus and Trigonometry!
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Learn some proofs with Danica as your personal tutor and coach!

Don’t judge a book by its irreverent title!

Suspend your judgement for a while, even if McKellar’s irreverent titles make you feel a bit squeamish or uneasy. Who knows? You may end up learning a thing or two about some long-forgotten, or decades-old avoided, math.

 

Personally, the style of writing of these pop math books is enough to arouse my interest, leave aside the math, which somewhat lacks rigor, as compared to the standard expected of Singapore math students at the same grades. For example, in Singapore,  geometric proof and trigonometry are formally covered at grade 9 or 10. However, McKellar’s  informal and conversational writing style could help us loosen our often-stiff mathematics writing, which has traditionally plagued most Singapore-published boring textbooks, as they go through the “rigorous” (or tedious?) process of MOE’s standards of quality and morality.

References

Ho, S. T., Khor, N. H. & Yan, K. C. (2013). Additional Maths 360. Singapore: Marshall Cavendish Education.

McKellar, D. (2012). Girls get curves: Geometry takes shape. New York: Hudson Street Press.

McKellar, D. (2011). Hot X: Algebra exposed! New York: Plume.

McKellar, D. (2008). Kiss my math: Showing pre-algebra who’s boss. New York: Hudson Street Press.

McKelkar, D. (2007). Math doesn’t suck: How to survive middle school math without losing your mind or breaking a nail. New York: Plume.

For alignment with state and NCTM math standards, visit www.mathdoesntsuck.com/standards

© Yan Kow Cheong, April 20, 2013.

The Dolls Problem à la Singapour

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Following a request from a Linkedln friend to provide a solution that makes use of the Singapore model method to the question below—I couldn’t trace the origin of this word problem—here’s a quick-and-dirty sketch of a five-model-drawing solution.

Jazmine buys and sells antique dolls on the Internet. Yesterday, she focused on dolls from the Civil War period. She began the day by selling one-fourth of her dolls from that period. Then she sold six more. Just before lunch she sold one-fourth of the remaining Civil War dolls. After lunch, she bought some Civil War dolls, increasing her collection by one-sixth. Then she bought some more, doubling her collection. Just before she quit for the day, she sold two thirds of her Civil War dolls. After all that, she had fourteen of these dolls left. How many dolls did Jazmine have before she began trading yesterday?

It wouldn’t be surprising that this kind of brain-unfriendly word problem, set in a test or exam, might give some un-mathophobic grade five or six students sweaty palms, or goose pimples, if they started feeling clueless after attempting to solve it for some five odd minutes!

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A quick-and-dirty solution that makes use of the model method.

Using the “work backwards” strategy repeatedly, the model drawings show that Jazmine had 40 dolls before she began trading yesterday.

If you’re an “algebraic freak,” by all means, use algebra to check your answer—I decided to give the algebraic approach a miss this time round.

Disproportionate parts or units

Notice that I’ve loosely used “units” and “parts” alternately to represent each model drawing. And I’ve also used each unit, or part, in a rather disproportionate manner, as compared to textbooks’ modeled solutions, which generally depict the bars (or rectangles) proportionately, based on their respective numerical values—which is secondary to the reasoning or thinking processes.

The above dolls problem is similar to a question I discussed in an earlier post, except that the present one is slightly harder; otherwise, it adopts the same problem-solving strategies for  its solution.

Sakamoto and Stack Methods

My next task is to check whether the Stack method or the Sakamoto method to the above word problem is conceptually “friendlier” than the model method. Are there intuitive or elegant solutions other than the one that embraces the bar method? Meanwhile, please send us your solution(s) to the dolls problem.

© Yan Kow Cheong, March 27, 2013.

Postscript: Although math was my favorite subject in school, I don’t recall solving questions similar to the above word problem. I doubt if I would be able to solve it when I was in grade five or six. It looks like this present younger generation has been given the shorter end of the mathematical stick—worse, if math happens not to be their cup of tea! It’s no surprise that strangers, young and old, angrily tell me of their negative mathematical experiences in school—how they disliked math (and their math teachers).

17 Theomatical Haikus for Math Educators

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Composing some Theomatics-related haikus may prove therapeutic for stressful math educators, who are prone to overusing their left part of the brain. Why not let these 17-syllabled verses reactivate some atrophied part of your grey matter? Who knows? This right-brained activity may indirectly help rekindle your mathematical creativity!

The Trinity

The True Living God
Ever Three and ever One
A mystery, indeed!

Christ And Mathematics Education (C.A.M.E)

Come to worship Him
Christ and math education
Join ACMS

Theomatics, Anyone?

Teach math Christianly 
As an act of true worship
It sure honors Him

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Two Conference Proceedings of the Association of Christians in the Mathematical Sciences (ACMS)

Pi in the Sky

The Biblical pi
Is a rational number
From the Book of Kings

π = 3.14 and John 3:16

So close, yet so far
Rational and eternal
The union is null.

The X-tian Pi

The true Christian life
Is like the contextual pi
Constantly changing

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The molten sea as described in 1 Kings 7:23. © The Golem Press

CHRISTmaths

Learning math with God
It’s time to take up your cross
To shake up your brain

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facebook.com/CHRISTmaths

A Divine Paradox

100 percent man
And 100 percent God, too
What a paradox!

God’s Nature

The nature of God
1 + 1 + 1 (mod 2)
Same, yet different

The Most Quoted Verse

What’s John 3:16?
God’s numerical message
Of His Love for us

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A treat for the mind, the eyes, and the spirit

 

Singapore Math

The Model Method
A blessed strategy
For problem solvers

TIMSS & PISA

A little red dot
Has blessed math educators
From all walks of life

Our Servant Math

In one God we serve
May His Spirit guide us
To make math serve us

The Mathematical Book of Life

Conjectures and Proofs
The Great Mathematician
Will bless you with both

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2 Kings 3:16 Calligraphy by Andrzej Kot (Poland)

What’s Prayer?

When illogic reigns
When 1 + 1 is not 2
When naught can be one

One Life, Two Unions

One plus one is one.
Then, one plus one turns to 3.
One plus one turns nought.

Primes & Priests

Both are hard to find.
They hold the key to success.
Have faith in them both!

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Calligraphy by Hermann Zapf (Germany)

 

Reference

Yan, K. C. (2011). Mathematical haikus for Kiasus. www.singaporemathplus.com

© Yan Kow Cheong, May 18, 2013.

Algebra or Model Method

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On December 19, 2012, in her Confession from a Homeschool Mom: Singapore Math Stumped Me Today, Monise L. Seward, blogged that her 6th grader woke her up to ask her for help on a word problem in her Singapore Math book. The nonroutine question she shared was the following:

Mrs. Pappas had some apples. She sold 1/3 of the apples plus 5 more on the first day. She sold 1/3 of the remaining apples plus 5 more on the second day. She had 125 apples left in the end. How many apples did Mrs. Pappas have in the beginning?

If you use algebra to solve this problem, it’s unlikely that the working will arouse any excitement; in fact, you may find this method of solution to be somewhat uninteresting or boring. Yes, algebra does religiously solve the problem, but the solution is anything but elegant. Moreover, most average grade five or six students wouldn’t have acquired the maturity to solve it algebraically.

An algebraic check

Besides working backwards to check the answer, after solving the question, using the Singapore model method, I also checked it out by algebra.Looking at the symbolic clutter, I guessed then that even our Singapore grades 7 and 8 average students would likely be challenged to solve this problem by algebraic means.If we stick to working with only one variable, then we may end up with an unappetizing equation such as the following to solve.

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A mum’s solution

Without a good working knowledge with the Singapore model method, we can expect most teachers and parents (who still remember their school math) to solve the question in a way similar to the one sketched and tweeted by my Pinterest and Twitter friend at PragmaticMom.com, as shown below:

2/3x – 5 = 125

2/3x = 125 + 5

2/3x = 130

x = (130 x 3)/2

x = 195

You add the 5 because you have to subtract the 5 apples from the 2/3 calculation because they were added in additionally. Then you do it again …

2/3x – 5 = 195

2/3 x = 195 + 5

2/3 x = 200

x = (200 x 3)/2

x = 300

No algebra, please!

Pretend for a while that algebra is an alien language to you! And you can only use a non-algebraic explanation to communicate your solution to a ten-year-old child! How would you go about doing it? Can you think of some intuitive methods?

After coming across this grade six question via @pragmaticmom, I tweeted a quick-and-dirty solution to the above problem, using the model, or bar, method. Give it a try first, before comparing yours with mine.

Sakamoto math

I also remarked that we could also solve this question by the Sakamoto method, which I assumed most of you in the United States might not be familiar with; so I’ll skip presenting the Japanese method of solution here.

Working backwards

Had the above grade five or six question involved the fraction 1/2 instead of 1/3, then using the “work backwards” strategy, via a flow chart, would have yielded an equally decent method of solution as the model approach.

Mrs. Pappas had some apples. She sold 1/2 of the apples plus 5 more on the first day. She sold 1/2 of the remaining apples plus 5 more on the second day. She had 125 apples left in the end. How many apples did Mrs. Pappas have in the beginning?

See a sketch of a common method of solution below.

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Two bonus questions

Let me leave you with two grade 3/4 problems that lend themselves easily to the model method.1. A shop owner sold 2 more iPads than half the number of iPads in his stock. He then sold 2 fewer iPads than half of the remaining iPads. If he was left with 28 iPads, how many iPads did he have in his stock in the beginning?

2. Sarah used $8 to buy a book. She then used half of the remaining money to buy a bag. Lastly, she spent $1 more than half of what she had left on a meal. In the end, she had only $5 left. How much money had Sarah at first?
Answers: 1. 108 iPads; 2. $32

© Yan Kow Cheong, March 13, 2013.