Tag Archives: Singapore model method

Stack Modeling as Mathematical Art

Gain that competitive edge, by being a  creative Singapore math educator and problem solver!
Gain that competitive edge, by being a creative Singapore math educator and problem solver! Title available on App Store and Google play.

One Singapore’s problem-solving strategy that is gaining currency among more and more local teachers in Singapore is the Stack Model Method, which has proved to be conceptually more advantageous—a more intuitive and creative strategy—than the bar model method. On a lighter note, let’s look at a dozen benefits one could derive should one fearlessly embrace this visualization problem-solving strategy to solve word problems.

1. As a Form of Therapy

Like bar modeling, getting involved in stack modeling may act as a form of visual therapy, especially among visual learners, and for those who need a diagram or model to make sense of a problem-situation. Indeed, a model drawing is often worth more than a dozen lines of algebraic symbols.

2. A Possible Cure to Dementia

Like Sudoku and crossword puzzles, practicing the science and art of stack modeling may help arrest one’s schizophrenia or dementia, particularly those who fear that their grey matter might play tricks on them in their golden years.

3. Prevention of Visual or Spatial Atrophy

For folks wishing to enhance their visualization skills, stack modeling could potentially turn their worry of short-term visual apathy and long-term visual atrophy into aha! moments of advanced visual literacy.

4. A Disruptive Methodology and Pedagogy

When most Singapore coaches and teachers are no longer excited or thrilled about the Singapore’s model method, what they need is a more powerful and intuitive problem-solving strategy like the stack model method to give them that competitive edge over their peers, all of whom are involved in the business of Singapore math—from training and coaching to consulting and ghostwriting.

 

Age Problems 3-4
An age-related problem from “The Stack Model Method (Grades 3-4)

 

5. A Platform for Creative Thinking in Mathematics

Getting acquainted to the stack model method would not only help one to hone one’s visualization skills, but it’ll also refine one’s problem-solving and creative thinking skills. Being mindful that competing stack models could be designed to figure out the answer, the challenge is to come up with the most elegant stack model that could vow even the mathophobics!

6. Look-See Proofs for Kids

Stalk modeling could help remove any “mathematical cataract” from one’s mind’s eye to better “see” how the parts relate to the whole. The way stack models are drawn (up-and-down and sideways) often allows one to see numerical relationships that would otherwise be difficult to visualize if bar models were used instead.

7. The Beauty and Power of Model Diagrams

Even those who are agnostic to the Singapore math curriculum, a “Stack Modeling” lesson could help enliven the beauty and power of model diagrams in creative problem solving. The stack model method could act as a catalyst to “seeing” the connection between parts and whole—normally, the same result would be tediously or boringly derived by analytic or algebraic means, understood only by students in higher grades.

8. A Simple but Not Simplistic Strategy

Like Trial and Error, or Guess and Check, the stack model method shows that Draw a Diagram is a simple, but not simplistic, problem-solving strategy. The stack model reinforces the idea that often “less is more.” The simplicity of a stack model can reveal much hidden information that is often lost in an algebraic argument.

9. A Branded Problem-Solving Strategy

For math educators who might think that Singapore math, or the bar model method, in particular, is a mere fad in mathematics education, the stack model method further disproves that myth. Like bar modeling, stack modeling shows that a simple problem-solving strategy like the “draw a diagram” has what it takes to attaining brand status, especially when we consider the types of challenging word problems that lend themselves to both bar and stack models, and which could also be assigned to a younger audience.

10. Stack Modeling as a Creative Art

To the novice problem solver, stack modeling is a science; to the seasoned problem solver, stack modeling is an art— the challenge is to come up with more than one stack model to arrive at the answer. Remember: Not all stack models are created equal!

 

Before-After 3-4
A solution page from “The Stack Model Method (Grades 3-4)

 

11. Earn as You Learn

If you are a mathepreneur, you can easily steal the ideas in The Stack Model Method: An Intuitive and Creative Approach to Solving Word Problems to write a more expensive Singapore math book on the subject. There are dozens of ethically challenged ghost writers and cash-strapped undergrads from China, India, and the Philippines, who can help you professionally plagiarize any types of editable contents! You earn as you learn! Of course, you need to mail them your copy, or buy a new copy for them to do the “creative work” at a fractional cost! Make sure you don’t get caught, though!

12. Green Math à la Singapour

Ecologically speaking, stack modeling, which generally uses less space than bar modeling, could help math educators save millions of ink and square miles of paper [aka trees]. In economic terms, millions of dollars could be saved by the right choice of model drawing. In other words, stack modeling could act as a catalyst to help one play one’s part in reducing one’s carbon footprints!

From Bar to Stack Modeling

With a bit of imagination, I bet you could come up with another dozen benefits of stack modeling. The stack model method is no longer an option, nor should it be treated as a mere visualization strategy to be discussed only during an enrichment math lesson.

The stack method is going to be a problem-solving strategy of choice, as more math educators worldwide invest the time to learn and apply it to solve non-routine questions in elementary math. Be among the first creative problem solvers to embrace the stack model method, as you gain that methodological or pedagogical edge over your fellow math educators!

References

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

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

© Yan Kow Cheong, January 10, 2015.

 

Differences-Gap 5-6
A screenshot from “The Stack Model Method (Grades 5-6)” without the Thought Process

 

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.

A Grade 5 Bicycles-and-Tricycles Problem

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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.

20140220-180550.jpg

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.

The Singapore Excess-and-Shortage Problem

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

20131026-231338.jpg

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.

20131027-213637.jpg

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

3 units + 16  = 5 units – 6

20131026-231524.jpg

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.

20131026-231534.jpg

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.

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

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.

Picture

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.

Picture

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:

Picture

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.

Picture

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!

Picture

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.

Picture

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:

Picture

© 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.

Picture

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.

Picture

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.

Problem Solving Made Difficult

Picture

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.

Picture

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.

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?

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

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