Tag Archives: problem-solving strategy

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.