A Singapore Grade Four Question Without Algebra

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                           2x                     x

After                        2x – 413              x – 19

Now,   2(2x – 413) = x – 19

               4x – 826 = x – 19

                        3x = 826 – 19

                        3x = 807

                          x = 807 ÷ 3 = 269

               2x – 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

Singapore at the IMO

Singapore at the IMO

In Singapore, the average ratio of female to male undergraduates at the university is three to one. Yet those who perform outstandingly are male. Few female undergraduates make it to the Honours class, and very few of them have a doctorate in the subject. In the last fifteen years, only one female Singaporean mathlete had represented Singapore at the International Mathematical Olympiad (IMO); indeed, not a universal anomaly looking at the number of over-represented male mathletes at regional and international math competitions. It almost seems that our local gifted and talented math female students lack the “math gene” to compete regionally and internationally!

Even in local publishing, only recently did we see some female writers experimenting with some school math textbooks. Traditionally, mostly male teachers or tutors have written Singapore textbooks, although increasingly and encouragingly we're seeing more female authors in the assessment (supplementary) math market. In many ways, they're still under-represented considering that up to the university level, females outperform males in most mathematics local examinations.

Singapore's Mediocre Ranking

Why do Mainland Chinese, Vietnamese and Iranian students outshine their Singaporean counterparts at the IMO? One would assume that better coaches, as compared to their counterparts in other developing countries, would provide better training to the six representatives of Singapore; yet, the prevailing situation reveals a different story.

An informal investigation shows that the Singapore Mathematical Society (SMS), the body in charge of coaching high school IMO mathletes, doesn't seem to engage the best local or foreign-born trainers to coach the IMO Singapore team. Everybody seems too busy to put the time and effort to offer their expertise to prepare the best possible IMO local team to compete with the best teams round the world. Apparently, it's not due to a lack of brainpower or finance that is the cause of Singapore's mediocre performance at the IMO every year, but because of the reluctance on the part of some professors to play their social responsibility to the nation.

Singapore – a Deprived Math-Gene Nation!

Skeptics believe that even with the best resources, our small team of talented young mathematicians may not have what it takes to win some gold medals, because the often drill-and-kill educational system and the limited choice to compete in other olympiads doesn't provide them with the conducive milieu to wanting to excel in mathematics competitions.

Local math educators reveal that local talented and gifted students would rather compete in a physics or chemistry olympiad than taking part in a mathematics competition, simply because it's easier to win a medal in these subjects than being an IMO medalist. Paradoxically, there is too much choice for our talented youngsters to compete. The time and effort needed to prepare for an IMO seems to be a key factor in attracting the best brains to represent Singapore at the IMO.

One hypothesis is that developing countries like China, Vietnam and Iran can ill-afford modern laboratories to do research; so their gifted youngsters have little choice but to focus on subjects or disciplines that don't need much equipment and resources. Singapore young scientists and mathematicians, on their part, have much opportunity to interact or work with faculty staff of tertiary institutions - one interesting point here is that faculty members from the physical and social sciences seem more willing to mentor their students than their counterparts from the mathematics department. 

Relaxing IMO Criteria

Perhaps if the IMO relaxes its citizenship criteria, Singapore may do better, because it's no secret that many of the better math students in Singapore come from neighboring countries, such as Malaysia and Indonesia, but they're not eligible to represent their host country at any math competitions. Although raised and schooled in Singapore, their nationalities hinder them from doing Singapore proud, especially when the pool of local gifted math students is infinitesimal. Many of these foreign bright students are dealt a double blow: on one hand, they can't represent their country of birth for political and racial reasons; on the other hand, as non-Singapore citizens, their brilliance prevents them from representing their host country.

Singapore's First IMO Invitation

Singapore was first invited at the IMO by host Australia in the late eighties. Since then, we have only produced few gold medalists. Singapore mathletes continue to disappoint or underperform at Olympiads, despite its relatively high standard of mathematics education, as compared to many developed and developing countries.


After all, it's hardly a coincidence that Singapore was ranked first three consecutive times at the TIMSS (Trends in International Mathematics and Science Study) - it lost its top position to Taiwan at the last TIMSS. Excellent performance at the TIMSS but average at the IMO! Nonetheless, Singapore mathletes have so far performed relatively well in other regional math competitions such as the Australian Mathematics Competition (AMC) and the Pacific Asia, where they'd collected a handful of gold medals, as compared to their other Asian counterparts.

So, what must be done to help Singapore make a significant impact at the IMO? Better coaches or better coaching methods to prepare our local mathletes? What are some of the missing ingredients, which would propel Singapore to the Top Ten in the IMO ranking?

© Yan Kow Cheong, March 1, 2010