Tag Archives: visualization

Singapore Math Books on the Bar Model Method

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.

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 Method
A 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 Modeling
Another 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 Method
A 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 Method
A 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.

Hungry ghosts don’t do Singapore math

In Singapore, every year around this time, folks who believe in hungry ghosts celebrate the one-month-long “Hungry Ghost Festival” (also known as the “Seventh Month”). The Seventh Month is like an Asian equivalent of Halloween, extended to one month—just spookier.

If you think that these spiritual vagabonds encircling the island are mere fictions or imaginations of some superstitious or irrational local folks who have put their blind faith in them, you’re in for a shock. These evil spirits can drive the hell out of ghosts agnostics, including those who deny the existence of such spiritual beings.

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Hell money superstitious [or innumerate] folks can buy for a few bucks to pacify the “hungry ghosts.”

During the fearful Seventh Month, devotees would put on hold major life decisions, be it about getting married, purchasing a house, or signing a business deal. If you belong to the rational type, there’s no better time in Singapore to tie the knot (albeit there’s no guarantee that all your guests would show up on your D-Day); in fact, you can get the best deal of the year if your wedding day also happens to fall on a Friday 13—an “unlucky date” in an “unlucky month.”

Problem solving in the Seventh Month

I have no statistical data of the number of math teachers, who are hardcore Seventh Month disciples, who would play it safe, by going on some “mathematical fast” or diet during this fearful “inaupicious month.” As for the rest of us, let’s not allow fear, irrationality, or superstition to paralyze us from indulging into some creative mathematical problem solving.

Let’s see how the following “ghost” word problem may be solved using the Stack Method, a commonly used problem-solving strategy, slowing gaining popularity among math educators outside Singapore (which has often proved to be as good as, if not better than, the bar method in a number of problem-situations).

During the annual one-month-long Hungry Ghost Festival, a devotee used 1/4 and $45 of the amount in his PayHell account to buy an e-book entitled That Place Called Hades. He then donated 1/3 and $3 of the remaining amount to an on-line mortuary, whose members help to intercede for long-lost wicked souls. In the end, his PayHell account showed that he only had $55 left. How much money did he have at first?

Try solving this, using the Singapore model, or bar, method, before peeking at the quick-and-dirty stack-method solutions below.

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From the stack drawing,
2 units = 55 + 13 + 15 + 15 = 98
4 units = 2 × 98 = 196

He had $196 in his PayHell account at first.

Alternatively, we may represent the stack drawing as follows:

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From the model drawing,
2 units = 15 + 15 + 13 + 55 = 98
4 units = 2 × 98 = 196

The devotee had $196 in his account at first.

Another way of solving the “ghost question” is depicted below.

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From the stack drawing,
6u = 55 + 13 + 15 + 15 = 98
12u = 2 × 98 = 196

He had $196 in his PayHell account at first.

A prayerful exercise for the lost souls

Let me end with a “wicked problem” I initially included in Aha! Math, a recreational math title I wrote for elementary math students. My challenge to you is to solve this rate question, using the Singapore bar method; better still, what about using the stack method? Happy problem solving!

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How would you use the model, or bar, method to solve this “wicked problem”?
Reference
Yan, K. C. (2006). Aha! math! Singapore: SNP Panpac Education. 
© Yan Kow Cheong, August 28, 2013.

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A businessman won this “lucky” urn with a $488,888 bid at a recent Hungry Ghost Festival auction.

A Before-and-After Singapore Math Problem

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A Singapore math primer for grades 4–6 students, teachers, and parents

In Model Drawing for Challenging Word Problems, one of the better Singapore math primers to have been written by a non-Singaporean author for an American audience in recent years, under “Whole Numbers,” Lorraine Walker exemplified the following before-and-after problem, as we commonly call it in Singapore.

Mary had served $117, but her sister Suzanne had saved only $36. After they both earned the same amount of money washing dishes one weekend, Mary noticed she had twice as much money as Suzanne. What was the combined total they earned by doing dishes?

The solution offered is as follows:

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© 2010 Crystal Springs Books

The author shared that she did two things to make the model look much clearer:

• To add color in the “After” model;
• To slide the unit bars to the right.

This is fine if students have easy access to colored pens, and know which parts to shift, but in practice this may not always be too convenient or easy, especially if the question gets somewhat more complicated.Let me share a quick-and-dirty solution how most [elementary math] teachers and tutors in Singapore would most likely approach this before-and-after problem if they were in charge of a group of average or above-average grades 4–5 students.

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From the model drawing,

1 unit = $117 – $36 = $81
1 unit – $36 = $81 – $36 = $45

2 × $45 = $90

They earned a total of $90 by doing dishes.

Analysis of the model method

Notice that the placement of the bars matters—whether a bar representing an unknown quantity is placed before or after another bar representing a known quantity.

In our model, had we placed the [shaded] bar representing the unknown unit on the right, it would have been harder to deduce the relationship straightaway; besides, no sliding or shifting is necessary. So, placing the bar correctly helps us to figure out the relationship between the unknown unit and the known quantities easier and faster.

In general, shading and dotting the bars are preferable to coloring and sliding them, especially when the problem gets harder, with more than two conditions being involved.

The Stack Method

This word problem also lends itself very well to the Stack Method. In fact, one can argue that it may even be a better method of solution than the bar model, especially among visually inclined below-average students.

Take a look at a quick-and-dirty stack solution below, which may look similar to the bar method, but conceptually they involve different thinking processes. To a novice, it may appear that the stack method is just the bar method being depicted vertically, but it’s not. Perhaps in this question, the contrast isn’t too obvious, but for harder problems, the stack method can be seen to be more advantageous, offering a more elegant solution than the traditional bar method.

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From the stack model diagram, note that the difference $81(= $117 – $36) must stand for the extra unit belonging to Mary.

1 unit = $81
$36 + ▅ = $81
▅ = $81 – $36 = $45
2 ▅ = 2 × $45 = $90

So, they had a total of $90.

The Sakamoto Method

This before-and-after problem also lends itself pretty well to the Sakamoto method, if the students have already learned the topic on Ratio. Try it out!

Let me leave you with three practice questions I lifted up from a set of before-and-after grades 4–6 problems I plan to publish in a new title I’m currently working on, all of which encourage readers to apply both the bar and the stack methods (and the Sakamoto method, if they’re familiar with it) to solving them.

Practice

Use the model and the stack methods to solve these questions.

1. At first, Joseph had $900 and Ruth had $500. After buying the same watch, Joseph has now three times as much money as Ruth. How much did the watch cost?

2. Moses and Aaron went shopping with a total of $170. After Moses spent 3/7 of his money and Aaron spent $38, they had the same amount of money left. How much money had Aaron at first?

3. Paul and Ryan went on a holiday trip with a total of $280. After Paul had spent 4/7 of his money and Ryan had spent $52, the amount Paul had left was 1/4 of what Ryan had left. How much money did Ryan have at first?

Answers
1. $300 2. $86 3. $196

Reference
Walker, L. (2010). Model drawing for challenging word problems: Finding solutions the Singapore way. Peterborough, NH: Crystal Springs Books.

© Yan Kow Cheong, August 4, 2013.

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.

A Singapore Grade Two Tricky Question

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