On 5 February 1999, Janey, an ex-colleague, forwarded me “Can U do a P4 maths question?”.

**A farmer has twice as many ducks as chickens. After the farmer has sold 413 ducks and 19 chickens died, he has half as many ducks as chickens. How many ducks does he have now?**

**The above primary four (or grade four) question seemed to have been originated from an educated parent or university lecturer, who lamented that he was unable to solve it without algebra. How do you find the answer without having to solve an algebraic equation?**

Before reading on, switch off your algebraic antenna for a while, and try to solve the question, using a model.

The solution given by the e-mailer is as follows:

To novice problem solvers, or to those whom the model method is a foreign heuristic, the thinking processes involved in drawing the correct model is seldom obvious. Even if you’re versed with the model or bar method, your model may be different from the one provided by the problem solver.

Different models or methods

With some effort, one can come up with no fewer than half a dozen methods of solution. Other than the Chinese

*line*and the Japanese

*Sakamoto*methods, here are three commonly known ones, the first one being a slightly modified version of the above.

Method 3: By Algebra

**Ducks**

**Chickens**

*Before*2

*x*

*x*

*After*2

*x*– 413

*x*– 19

Now, 2(2

*x*– 413) =*x*– 19 4

*x*– 826 =*x*– 19 3

*x*= 826 – 19 3

*x*= 807*x*= 807 ÷ 3 = 269

2

*x*– 413 = 2 × 269 – 413 = 125

So the number of ducks the farmer has now is 125.

**The need for multiple solutions**

In many problem solving books published locally, where the model method is the primary heuristic, few writers had provided more than one method of solution. Whether it’s due to space or time constraint from the editorial side, or because of little or no conscious effort on the part of the problem poser, to look out for alternative models, is hard to tell, especially when not many who edited or wrote those questions had had first-hand classroom experience with students who struggled with constructing those models.

It isn’t uncommon to encounter these typical questions in the Singapore Mathematics Olympiad for Primary Schools, and from the exam papers of the better local primary schools, whereby a model solution, or some intuitive method, is expected. Such problems, which would normally be solved by algebraic means at a higher level, could now be tackled using a model, when set at a lower level.

We fail to realize that these challenging questions are often constructed backwards, so that we can force the numbers to fit themselves nicely at each step of the working. No wonder even experienced primary and secondary school (elementary and middle-school) teachers are challenged to solve these word problems

*sans*algebra. Failure to be aware of this may otherwise lead many qualified teachers to feel inadequate in tackling these non-routine questions, especially when a non-algebraic solution is expected.**Filtering the nerd from the herd**

The disturbing fact about using the model method is that many schools (not just the better ones) in Singapore use these “Section C Problems” to unfairly identify and stream students into the good and bad classes⎯the A-, B- and C-band and the rest, where the students are presumably less academically inclined. So if you can’t cope with (or are allergic to) those word problems whose solutions are best solved using a model, then good luck to you! Your score on these word problems is often used to gauge how good you are in mathematics.

The worst happens when teachers themselves aren’t confident in solving these word problems. What they usually do is to leave confused students with some sketchy modeled steps, often worked out by underpaid undergrads, as hints or partial solutions to questions that are extracted from unauthorized (pirated?) exam papers.

© Yan Kow Cheong, 11 July, 2010

## 7 comments:

I posted this on Twitter as an example of the use of Singapore math models:

Girl has 4 times as much money as Boy. After she gave him $50, she had 3 times as much as he had. How much did he have?

I am a novice with models. The following is my crude attempt at a model but I know it has flaws! It's still too algebraic in nature. My sense so far is that one has to establish what the UNIT represents in the problem, then represent all other quantities in terms of that UNIT or part thereof. Working backwards seems to help me but...

Before

Girl ---- ---- ---- ----

Boy ----

After

Girl ----50 ----50 ----50

Boy ----50

Therefore the 4th ---- in Girl "Before" must equal 50 50 50 50 since she gave him 50. I could also write an equation here using the UNIT but I chose not to. Please comment on this crude attempt!!

BY the way, your post is extremely significant and helpful to me as an American educator. Your insights and understanding of math pedagogy are profound. Here's the irony. Several US school districts implemented Singapore Math materials over the past 10 years without realizing how much training is required for teachers. You're suggesting that even Singapore teachers occasionally have difficulty in explaining the model method! I would really appreciate it if you would contact me at dmarain@gmail.com for further discussion...

Dave Marain

Hi Dave

Thanks for taking time to comment on the Singapore model method in solving arithmetic word problems.

Your solution to the boy-girl problem looks fine to me. BTW, you may be interested to look at another non-algebraic method of solution I'll be e-mailing to you shortly. And your concern as to whether your solution may appear quasi- or pseudo-algebraic isn't uncommon even among problem solvers who are versed with the model or bar method.

In many ways and instances, the model heuristic is nothing but algebra in disguise. Instead of using the variable x, we use a unit, part or line to represent some unknown quantity. It's probably not a white lie to say that in a number of modeled solutions, the model method dresses up as some pre-algebraic heuristic in some visual attire. Ironically, this is where the power of the model method lies: It allows us to use visualization to solve higher-order word problems, which would normally be solved by algebraic means. In other words, the model method enables us to solve [including challenging] word problems at a lower grade, when traditionally they're only assigned to upper-grade students, using algebra to solve them.

You raised the point about equipping teachers to be proficient in using the model method. Well, believe it or not, albeit the model method has been around for about two decades now, there has never been any formal in-service training for elementary school teachers to learn it the effective way. The heuristic was popularized by assessment writers-entrepreneurs in the late 90s, and through some talks and workshops conducted by speakers not affiliated with the Ministry of Education, SIngapore. In fact, it was only last year that the MOE decided to publish a "monograph" on the model method - apparently to be given as some kind of gift or souvenir to visiting foreign educators, unfamiliar with the heuristic.

Most of us learned the model method the hard way, by starting to solve some challenging word problems in assessment titles, either as teachers who have little or no choice but to teach it in schools, or as parents who try to help our children with this visual heuristic. In fact, only in recent years that some lecturers from the National Institute of Education (NIE) had started conducting some interviews and surveys to assess how students used the model method in solving word problems on certain topics. You may wish to look at some papers by Dr. Ng Swee Fong and her co-researchers from the NIE.

Thanks to Singapore's enviable ranking in the TIMSS, and to the free marketing by American homeschoolers, the model method is experiencing a second life in our local mathematical landscape. just when we thought we've had enough of it! The math hype here in the last decade has been on problem-solving heuristics (guess & check, draw a table, ...) and thinking strategies, which American math educators seemingly are no longer interested in.

Meanwhile, you may wish to read an article on the model method I wrote for "The Mathematics Educator" many years ago, by clicking at http://tinyurl.com/yf9kv34

Mathematically yours

Kow-Cheong Yan (@MathPlus) - who is keen to co-write with math educators on the Singapore model method.

I posted a different strategy that is more pictorial in nature based upon how Jacob's Elementary Algebra introduces problem solving and my own oddball way of doing things. Thanks for the link to your article--I'm passing it on!

The link to my solution is: http://aut2bhomeincarolina.blogspot.com/2011/05/pictorial-ways-of-solving-algebraic.html

Hi Steve and Tammy

Thanks for offering a different visual approach to solving the chickens-ducks problem.

Another method of solution is never too many to share among our mathematical brethren, especially when it comes to promoting creative thinking in mathematics.

Kow-Cheong

One of my readers asked a question about the first statement: "A farmer has twice as many ducks as chickens." Shouldn't that be past tense? "A farmer HAD twice as many ducks as chickens."

Indeed, the entire question should be in "past tense." I printed the question verbatim, as it's e-mailed to me a decade-odd ago; besides, I'm not the poser nor the copyrights holder of this Singapore grade four question. I intentionally posted it with all its "grammatical warts."

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