Tag Archives: Stack method

The Fake Bar Model Method

Recently, I was peeping at some postings on the Facebook PSLE Parents group, and I came across the following question:

Philip had 6 times as many stickers as Rick. After Philip had given 75 stickers to Rick, he had thrice as many stickers as Rick. How many stickers did they have altogether?

Here are two solutions that caught my attention to the above primary or grade 6 word problem.

Solution contributed by Izam Marwasi
Solution by Izam Marwasi
Solution by Jenny Tan
Solution by Jenny Tan

Pseudo-Bar Model Method?

Arguably, the solution by the first problem solver offered to parents looks algebraic, to say the least. Some of you may point out that the first part uses the “unitary method,” but it’s the second part that uses algebra. Fair, I can accept this argument.

Since formal algebra, in particular the solving of algebraic equations, isn’t taught in primary or grade six, did the contributor “mistake” his solution for some form of bar model solution, although no diagram was provided? It’s not uncommon to see a number of pseudo-bar model solutions on social media or on the Websites of tuition centers, when in fact, they are algebraic, with or without any model drawings.

Many parents, secondary school teachers, or tutors, who aren’t versed with the bar model method, subconsciously use the algebraic method, with a bar model, which on closer look, reveals that the mental processes are indeed algebraic. No doubt this would create confusion in the young minds, who haven’t been exposed to formal algebra.

Does the Second Solution Pay Lip Service to Design Thinking?

What do you make of the second solution? Did you get it on first reading? Do you think an average grade five or six student would understand the logic behind the model drawing? From a pedagogical standpoint, the second solution is anything but algebraic. Although it makes use of the bar model method, I wonder what proportion of parents and their children could grasp the workings, without some frustration or struggle.

One common valid complaint by both parents and teachers is that in most assessment (or supplementary) math books that promote bar modeling, even with worked-out solutions to these oft-brain-unfriendly word problems, they’re often clueless how the problem solver knew in the first place that the bar model ought to be presented in a certain way—it’s almost as if the author knew the answer, then worked backwards to construct the model.

Indeed, as math educators, in particular, math writers, we haven’t done a good job in this area in trying to make explicit the mental processes involved in constructing the model drawings. Failure to make sense of the bar models has created more anxiety and fear in the minds of many otherwise above-average math students and their oft-kiasu parents.

Poor Presentation Isn’t an Option

Like in advanced mathematics, the poor excuse is that we shouldn’t be doing math like we’re writing essays! No one is asking the problem solver or math writer to write essays or long-winded explanations. We’re only asking them to make their logic clear: a good presentation forces them to make their thinking clearer to others, and that would help them to avoid ambiguity. Pedantry and ambiguity, no; clarity and simplicity, yes!

Clear Writing Is Clear Thinking

It’s hard work to write well, or to present one’s solution unambiguously. But that’s no excuse that we can afford to be a poor writer, and not a good thinker. As math educators or contributors, we’ve an obligation to our readers to make our presentation as clear as possible. It’s not enough to present a half-baked solution, on the basis that the emphasis in solving a math problem is to get the correct answer, and not waste the time to write grammatically correct sentences or explanations.

I Am Not a Textbook Math Author, Why Bother to Be Precise?

As teachers, we dread about grading students’ ill-written solutions, because most of us don’t want to give them a zero for an incorrect answer. However, if we’re convinced based on their argument that they do know what they’re doing, or show mathematical understanding or maturity of the concepts being tested, then we’d only minus a few marks for careless computation.

Poorly constructed or ill-presented arguments, mathematical or otherwise, don’t make us look professional. Articulating the thinking processes of our logical arguments helps us to develop our intellectual maturity; and last but not least, it makes us become a better thinker—and a better writer, too.

© Yan Kow Cheong, November 1, 2017.

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

 

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.

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.

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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.

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

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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.

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.

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.

The Dolls Problem à la Singapour

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).