I never used to understand this - when I was back in school, my mother would ask me, "Do you understand the concepts?" and I would say impatiently, "yes, yes, of course." But I didn't, not really. You see, I knew all the formulae, like Area of Rectangle = Length x Breadth, Distance = Speed x Time, etc but they were just formulae to be executed, I never got the real meaning behind those formulae.

A case in point: Adeline was recounting how when she saw the sum ½ ÷ 1/4, she tried to teach her son the "inversion" method, ie flip the 1/4 and change the sign to "x", so you get ½ x 4/1 = 2. That's the way we were taught in school some 20 (ok, ok, closer to 30!!) years ago. But she couldn't explain it to her son and we didn't really GET the meaning behind the answer, until Lilian's son Brian explained, "well, 2 quarters go into half, that's why the answer is 2." Doh!

I'm confident that the Singapore education system is laying a very strong foundation in maths because having undergone the primary math curriculum for six years with Lesley-Anne, I can see how it systematically reinforces the understanding of math concepts, first in solid, tangible methods before moving into abstract forms. Maths in Singapore doesn't teach blind application of formulae to narrowly defined, topical questions. Eg. it used to be that if you came across a question with a circle, you just need to apply one of the formulae for circles and you should be able to find the answer.

These days, a question may feature a pie chart but involves your application of knowledge not just of pie charts but maybe also of circles, percentages, fractions and algebra. Which is very reflective of maths in real life - afterall nothing ever falls neatly in pre-determined categories! And that is the reason why many Singaporean parents find their kids' maths so difficult even at the lower primary levels. It's not that we're dumb or bad at maths, we were just never taught that way!

Many parents, being unable to solve their kids' maths problems (let alone teach it), immediately dismiss the subject as too difficult and call for a tutor. I would like to suggest an alternative solution that is quite straightforward. The catch is that it may not be too palatable for parents - we need to re-learn our maths. Before you groan, let me tell you why: I find that when parents embark on the learning process together with their kids, their kids often are more motivated to work. Never say you're too old - it's doable!

Let me share my experience with Lesley-Anne thus far and hopefully this will encourage you. I never attended a single math workshop and my kids never had math tuition. I was one of those stubborn and tidak apah types who only bothered to find out more about a topic if my kids didn't understand something. I naively thought, "This is primary school maths! How hard can it be??" Until Lesley-Anne hit p3 and was stumped by a question that required models in an exam paper. I knew nothing about models then. When I saw the sum, I was indignant. "Siao! Expect 9-year-olds to solve this kind of question!" I still remember complaining to Lilian when she happened to come back to Singapore for a visit. I showed her the paper expecting sympathetic horror but she said calmly, "oh, this one needs models." (Lilian, I don't know if you remember this!) I was like whaaa..tt?

From there, I looked up model drawing in assessment books and I fell in love with the ingenuity of it. It introduced me to a whole new way of visualising maths problems that I never knew existed. So that was the start. From then on, I would always keep one topic ahead of whatever Lesley-Anne was learning in school by referring to her My Pals Are Here math textbooks and understanding how it's taught (usually very simply yet brilliantly). Consistently, I found that it emphasised understanding of concepts, using meaningful drawings. New topics start from the basics and more complex ones build on topics previously taught in a logical manner.

*Qualifier: I'm referring only to mainstream maths. I never bothered with topics under the GEP maths syllabus.*

However, the textbooks cover only the straightforward and foundational stuff. Often, in assessment books or exam papers of top schools, you will find the questions are much tougher. These are the ones that usually cause much angst in parents. To be able to solve these problems, you need PRACTICE. No two ways about it, sorry. Once the net is cast open for maths questions to involve applications of multiple concepts, there are just too many computations and permutations, you cannot possibly learn ALL the possible ways of solving them by rote. The only way is to ground yourself so firmly in the principles that you are able to manipulate them in different ways to solve the problem. That's what our kids are taught. And they're made to practise. A LOT.

I haven't got it down pat. As you can tell from my less than stellar score in the Nanyang prelim paper, I still have lots to learn. But you know, I'm enjoying it. Once you realise that maths is about applying concepts to solve problems, it becomes fun. For me, it's like doing puzzles (of course I realise that's not everyone's cup of tea!) But it certainly is more interesting and effective than simply doing math by rote.

If you still need more convincing, here are some numbers for you: The Trends in International Mathematics and Science Study (TIMSS) measures the performance of math and science achievements of students around the world. In 2007, the TIMMS ranked Singapore students second in the world (just after Hong Kong) for Grade 4 (equivalent to p4) and third in the world for Grade 8 (equivalent to sec2), after Chinese Taipei and Korea. In 2003 when the TIMSS was last conducted, Singapore ranked first in both categories.

The consistently high results achieved by Singapore students in math have attracted attention in the US and several states have already adopted our math curriculum and are beginning to see results. You can read this article in the Philadelphia Inquirer and another in the LA Times.

So as I stated in the beginning of this post, the Singapore maths curriculum is rock solid. (Now if they can only do that for English!) I encourage you to take the time to "go back to school" and re-learn your maths. If not for your kids, then for yourself. Approach it with an open mind, I think you might surprise yourself.

## 15 comments:

Again, I have absolutely no recollection of this. How obnoxious I must have come across LOL! Thing is, I knew that models were required, I just didn't get down to doing it like you did; and I think we learnt it all wrong anyway, our only reference point were the Andrew Er workbook answers. That's why my foundation for models isn't strong at all, Brian's isn't either. Maybe I should start building it from scratch with Sean, but he's pretty resistant to models at the moment.

Nolah, you opened my eyes! Models are great for kids who have problems conceptualising what the numbers mean. For you, both your kids are so strong mathematically that they're able to visualise all the abstract numbers in their heads. I don't think they need models!

Brian still falls back on models for some types of questions, perhaps kids who are really strong in algebra can solve them algebraically, but he says he can't. So I think I should definitely build up my modelling skills as you have advised, hopefully (but highly unlikely), I'll be as adept at it as you are by the time Sean takes PSLE. I have ermmm, more than 5 years...

Ya I agree with Mon models are good for learning and explaining concepts. I think for a teacher, this is a very good tool to explain what he/she wants to say to a student. However, when it comes to exams, if mathematically strong, it is not necessary to rely on the models if the picture is in the mind. just my 2 cts la.

qx

The good news for you is that Sean may never need to take the PSLE depending on where you're located then. But anyway, Woowoowoo... I'm cheering you on. I'm sure you can do it!!

Ok, I'm taking this as an encouragement. Mon, you had no clue about models, but can dive in and do it like a pro now? I thought you were born doing models in vitrio!

So I also got chance, hehe, cos we have puh-lenty of time too!!

Ad: Of course you can!! But with your smarty pants, I suspect once you teach him, it won't take long before he's revinventing his own models and teaching you (and everyone else)!! :D

QX: In the Nanyang paper, I only used models for one measly question! So models are good for understanding concepts but during exams, the kids may not actually need to use them.

Hey hey... I like this post very much, Monica! I am not strong in Math but have taken a liking to models too. Like you, I wish we were taught that during our school days. I have...errr...hmmm.... bought a copy of P3's Fan Learning on Models to try out the problems. Call me kiasu or whatever...! Haha!

Mon: Honestly, under time pressure, I gabra zebra...never think of model diagrams. But when I had to attempt the corrections..Q46 gave me such a big headache (even with algebra) that I had to draw the model to give me an idea how to look at things. And that was the only 1 I could use model...the other one you did using model, I find algebra was faster to capture the essence as I was and still am not at the "poetry in motion" level as you.. LOL..

I seriously hope my kid can overtake me in this modelling thingy in time to come *pray hard*...my engine...rusty already la..

But it is great that you encourage parents to try because modelling really has its beauty. That said, I used to be one of those parents who think modelling is convoluted. LOL *self-slapping* Now I admit I am WRONG. It is visually CLEAR but more tedious for exams if lack in practice.

qx

Nowadays the way to draw models is just too stiff...

My niece complained that her maths teacher insisted on the following:

-Ruler for every straight line in the model

-explanation of the model underneath

-making sure every unit is equal dimensionally

That really put the kids in her class off math-they took half an hour to draw their models. It really defeats the purpose this way.

Models quite easy to draw lah, don't need to learn, just draw out the problem...

Ann: Good for you!! I heard that MOE recently published a book on the model method, probably due to demand from parents. I've not seen it myself but maybe you'll like to check it out, it's sold at Popular.

QX: You are so funny...gabra zebra... in fact, you are starting to sound like us! (me, Lilian, etc). I think you've been "hanging around" us too much LOL. The great thing about the SG math system is that there are often several ways to find the solution, since it tests understanding of concept and not application of formulae. So whatever works for you is fine. Just practise!

Veronica: That's madness lah, if Andre had to follow all those rules, he'll never get past the first question! Aiyoh, why is it there will always be people who insist on making a simple thing difficult...

Yes, I can imagine Andre agonising over his work if he was taught by that teacher...(sigh) Singapore's maths systemj is just too strict!

I know your blog is quite a while ago but the issue is still quite relevant today. I am struggling to figure out how to teach my daughter the concept of maths models. She just doesn't get it. I wonder what the teacher in school teaches but I think that is not really the problem. The problem is my daughter has a conceptualisation difficulty, I think. Are there tools, games or whatever that can help nurture this concept of maths models?

chewing: I don't know of any games or tools, sorry. Maybe it's just a question of practice until she finally sees the link? Conceptually, models are actually easier to understand than algebra as it's visual but learning how to draw and interpret the models is key. All the best to you!

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