Art, science (and math) in full glory

Every year-end school hols, we try to make an excursion to either a museum or an attraction of interest. This year, we made a trip down to the Art Science Museum at Marina Bay Sands for three ongoing exhibitions - Journey to Infinity: Escher's World of Wonder, NASA - A Human Adventure, and Future World.

The Escher exhibition was the one we were really keen on. In case you don't know, M.C. Escher was an artist who combined mathematical concepts in his art, playing with perspective and architecture. One of his most well-known contributions was the artistic interpretation of tessellations to a completely new level. In Singapore, primary school kids still learn about tessellations today.

This was one of his earlier works on tessellations...

Sky and Water 1

which gradually became more complex.

Reptiles

Angels or demons?

Circle Limit IV

From 1954, Escher began working on optical illusions and concepts of infinity, which resulted in some pretty captivating and today, iconic artworks. 

Relativity

Ascending and Descending

All four of us loved the Escher exhibition, which probably says something about how OCD we are as a family 😆

As an undergraduate, I chanced upon a book of Escher's artworks and was immediately hooked. The geometry and poetry of his lines spoke to me so much that I attempted to recreate a colour version of one of his works. This was one of his early woodcuts during the period when he was consumed by the architecture of buildings:

Inside St Peter's

My not-quite-exact replica in coloured pencil.

 
Our main purpose was to visit the Escher exhibition but since there was a special all-access ticket to all three ongoing exhibitions, we decided to cover 'em all.

The NASA exhibition is good for kids and adults interested in all-things space. You get to see models of different space shuttles...

Saturn V

and gawk at space food...

 and space toilets.

You can also pay $6 to get a ride on G-Force - Astronaut Trainer Ride. It's a contraption that tilts you back and forth and spins you around. We watched for a bit to see if anyone came out of the ride walking sideways or throwing up. Nope, nobody did!

Finally, there's the Future World exhibition. In my opinion, this exhibition is great but more for younger kids (primary school and below). It takes interactive art and play to a whole new level. For example, you can draw your own a sea creature, scan it in a machine and see it swim on the digital wall. What's amazing is that the picture doesn't come out static - it moves and squirms like real sea creatures.

Something around every corner to enthrall the little ones.

The Crystal Universe with 4D technology and a heckuva lot of LED lights was quite spectacular.

I know this post might come a little late since it's nearing the end of the school hols (parents say "yay!" kids say "boo!"). However, the exhibitions are still on so you might want to book one of the upcoming weekends to take your kids.

Future World is on till 8 Jan, Escher exhibition till 26 Feb and NASA exhibition till 19 March. Special all-access pass to all three exhibitions available.

Does this spell the end of literature?

Last month, there was some debate on the drop in number of students taking Literature at 'O' levels from 16,970 in 1992 to just 3,000 in 2012, and whether this signals a decline in appreciation of the humanities among students. You can read the news article here.

While the drop in numbers sounds terribly alarming, I always hesitate to take numbers at face value. Instantly, two "BUTS" came to mind:

1) BUT: Wouldn't the decline also be due to the drop in number of kids taking 'O' levels?  Today, with the IP and other alternative educational pathways, many kids no longer take 'O' levels.

2) BUT: There are more 'O' level subjects these days, compared to 20 years ago. Wouldn't the increase in choice automatically dilute the number of kids taking any one subject (except for compulsory subjects)?

MOE's reply shed more light.  With reference to my point 2), it seems that the drop happened most drastically at the 2001 point when Combined Humanities was introduced.  This is a social studies subject with an elective of Lit, Geog or History and became a popular alternative to the other pure humanities subjects - Lit, Geog and History.

With regards to point 1) though, it's still a little hazy. MOE reveals the percentage of sec 4 cohort that took Lit in 1992 (47.9%), 2001 (21.8%) and 2012 (9%) but it's important to note that a percentage of the Sec 4 cohort not the same as a percentage of the 'O' level cohort.

A drop in 'O' level Lit does not necessarily translate into an equivalent drop in kids studying Lit at sec 4. In many IP schools, Lit is a compulsory subject for sec 4 kids as they take Language Arts (which is a fusion of English and Lit).  For the IB programme, Lit is compulsory all the way to year 6.

Typically also, even from the 1990s, Lit has been considered the more challenging humanities subject (because it requires critical thinking and language skills, not something you can mug for it), so it is usually reserved for the more academic classes (usually the top science and arts classes).  If you postulate that these kids are now likely in the IP track at secondary school, then it's no surprise that Lit has fizzled out in the 'O' level arena.

Beyond the numbers

Nevertheless, it's hard to deny that the number of students studying Lit (whether at 'O' levels or not) has declined.  Perhaps not as drastically as the numbers suggest but still undeniable... and that's sad.  I won't bother to write all about the importance of Lit since I'd already done so previously.  Besides, I came across this passionate Facebook post by Joshua Ip, which laid out all the points very convincingly.  It's a little over the top (there's a little bit of theatre in every true blue Lit student) but the points are pretty solid and more substantial than any argument I'd read in the mainstream media.  Plus it's a highly entertaining read, which reflects well on his Lit background.
 
This decline in Lit is not just about Lit alone. Looking at the bigger picture, it reflects the persistent worldview that the humanities in general are the poor cousin to the firmly footed maths and science.  After 20 years, the triple science combination is still considered the most prestigious in your sec 3 choices. In some schools, only the "top" students are offered the combination. In Lesley-Anne's school, there's no real humanities combination - you either take triple science or double science. Double maths is compulsory.

Moving on to 'A' levels, this bias continues. The 'A' level syllabus now requires you to choose a contrasting subject to create balance, ie science students have to take at least one humanities subject and arts students have to take at least one science subject.  In theory, this is great but in reality, I find the practice flawed.

If you're in the science stream, you can choose from a long list of humanities subjects, eg. 3rd language, Geog, Lit, History, Music, Art and Econs.  I always felt that Econs was such a miscategorisation.  It's nowhere remotely near a true humanities subject that encourages the broadening of worldviews or right brain thinking.  It's about statistics and trends, and the analyses are more akin to the scientific type.  That's why many science students choose Econs as their contrasting subject - yet another "pragmatic" choice.

Whereas if you're in the arts stream, you're pretty much limited to one contrasting subject - Maths.  Don't believe me? Most JCs don't offer science subjects as a contrasting subject, meaning that if you don't like maths, you're pretty much screwed.  It also sends a very clear message - the most important subject in the Singapore education system is MATHS MATHS MATHS.

Don't misunderstand me, I acknowledge that maths is important.  However, I find this obsession with maths baffling. The maths proponents tend to play up the pragmatic value of learning maths but honestly, most people will never apply the kind of maths you learn at 'A' levels.  If you want to focus on pragmatics, I would have thought it's more important to learn about human geography (urban planning!) or human biology (understand your body!).  I suspect this practice is again linked to scoring - Maths is one of those subjects with the highest percentage of distinctions at 'A' level so it's attractive to JCs looking to raise their overall scores. 

Quality not quantity

I was chatting with a teacher friend recently and he mentioned that he happened to see one of his sec 3 student's Lit essays. He was perturbed to see it littered with grammatical errors that were not corrected by the teacher.  Upon questioning, she replied "Miss W said in Lit, we're not marked on grammar, only based on the points we give."

Stab a knife through my heart now, won't ya.  Is that how we're addressing the difficulty to score in Lit, by reducing it to a factual content subject?  I don't think the teacher recognised the irony of it - that a subject supposed to test your critical thinking and arguments is awarded marks according to a marking template.  Instead of embracing the humanities, we try to turn it into something more scientific.  And when English ceases to matter in English Lit, well... I've no words.

I was so appalled by this that I checked with Lesley-Anne if it was the same at her school.  Thankfully, she shared my disbelief and said no, her Lit assignments are marked on both language and content, where original arguments are valued.  Phew.  So maybe it's just that school or that teacher.  I hope and pray that's not how Lit is marked at 'O' levels these days.

It's a sad state of affairs.  It's the belief that education in Singapore is still primarily about scoring.  There's a lot of talk about holistic education, creativity and critical thinking but the truth is, so many educators, parents and students are still stuck in the old mindset that education is about amassing facts and A's.

Lesley-Anne had this to say: "The problem is many students are still learning things in isolation, so they don't understand the relevance of what they learn beyond the narrow confines of that subject. Even English becomes relevant only to the English subject in school. So many things in the world are linked.  When we are unable to see the inter-connectedness of different issues, we miss the big picture. How then can we innovate and solve problems?"

I love how a 15-year-old can have the wisdom to see this truth where many adults cannot. I couldn't have said it better myself.

Making the reluctant case for tuition

Lesley-Anne got back her year-end exam results last week and overall, we're pretty pleased with it.  Except for Higher Chinese where she scored a B4, she received an A grade for everything else. As we all know, it's not always the case that hard work will translate into good results but this time, it really paid off for her and we're very grateful for it.

The subject we were most pleasantly surprised by though, was maths. As I'd previously blogged, Lesley-Anne had been struggling with maths this year, earning herself a C5 in E Maths and a dubious F9 in A Maths in her mid-year exams. We got her a math tutor and even though we expected to see improvement, we weren't predicting any miracles.

The results turned out to be pretty drastic. For the CA2, barely 3 months after tuition commenced, Lesley-Anne's A Maths grade jumped from F9 to C5.  For this end of year exam, she scored an A1 for E Maths and an A2 for A Maths, just one mark shy of an A1.   

To tell you the truth, I'm pretty torn by this turn of events. On one hand, of course I'm delighted by the results. On the other hand, I'm actually feeling a little perturbed. Why?  Because the results show that contrary to what her F9 grade indicated, Lesley-Anne has the capacity to learn maths. Just not from her school teacher.

This bugs me.  In my opinion, a student should be able to learn at least the rudiments of any given subject from school with the prerequisite hard work, unless he really has zero aptitude for it. I've never been a big proponent of tuition unless the child truly needs help.  However, from this experience, I found that in just 6 months, a once-a-week, two-hour tuition session made all the difference.  I can only conclude that it's not the amount of effort or the hours spent learning maths, it's the teaching method.

I'm pretty sure that Lesley-Anne would not have gotten these results if she had continued to try and learn maths purely from school. For some reason, she can't understand the way her teacher explains concepts and maths remedial didn't help since he would repeat the topics in the same way. Maybe he thought that her poor results were because she wasn't working hard enough or wasn't paying attention.

I totally get that it's more difficult to teach a classroom of kids than tutor one student but Lesley-Anne was not even remotely passing A Maths, which tells me she hadn't even grasped the basic concepts.

If it's true that our children sometimes don't do well in school because they can't understand the teachers, then this is troubling. How can our education system wean kids off tuition if it is more effective than school?  What about kids who can't afford tuition?  There is also the question of how many kids think they are stupid or do not have the aptitude to grasp a subject when maybe it's just the way it is taught that's not effective.

These are difficult issues that have always been a challenge - finding good teachers, equipping them with the right tools, making sure they have enough time and resources to do their job well, etc. Nevertheless, they need to be addressed if MOE wants to convince parents that "every school is a good school".

For now, I'll just have to be resigned to the fact Lesley-Anne's maths education is not from school but an external source.  It's not ideal but at least it's effective.

Don’t throw away the arithmetic yet

After I wrote a guest post on White Group Maths, I asked Frederick if he would like to reciprocate as I like his simple yet engaging style of writing. Happily, he agreed so here's his post. Slightly irreverent, yet relevant. For all the non maths enthusiasts, hopefully this will pique your interest in the subject!

xxxxxxxxxxxxxxxxxxxxxxx

“A new car built by my company leaves somewhere traveling at 60 mph. The rear differential locks up. The car crashes and burns with everyone trapped inside. Now, should we initiate a recall? Take the number of vehicles in the field, A, multiply by the probable rate of failure, B, multiply by the average out-of-court settlement, C. A times B times C equals X. If X is less than the cost of a recall, we don't do one”- Narrator (Fight Club, 1999)

Seriously? Using algebra on the job? Not unless one is deeply involved with actuarial work. Things would get messier if the person is an aerospace engineer we are talking about, since he has to formulate specific mathematical models to simulate efficiency of avionics systems. Or a rocket scientist who has to……… ok I shall stop, because this is the point when the average Joe would scream total irrelevance. The comforting truth is, for majority of us peace-loving, ordinary folks, the amount of mathematics which actually wriggle into our livelihoods to taunt and torment is far much lesser. In fact, almost all the knowledge acquired in school (horrors of horrors such as calculus, complex numbers, vectors et cetera) would be relegated to the junk folder of our memory banks from the very instant we graduate and step into the working world, never to be revisited again in our future endeavours.

That said, there are a few basic math skills definitely worth preserving throughout adulthood, chief amongst them is arithmetic (which includes mental calculation to a certain extent). What then, as one may question, is the value of being able to count competently and efficiently? Herein I shall share five benefits you would be rewarded with in daily living:

1. Producing fairly accurate ascertainment of costs on the spot

Say there is a 50 dollar note resting snugly within your wallet, and you have decided to give yourself a nice treat. This restaurant with a particularly unique ambience and good food is offering a classy buffet spread at $44.90, exclusive of the additional 10% service charge and 7% GST. Will you end up washing the dishes for a whole day in their kitchen because you didn’t have sufficient funds to make payment? Rounding up the cost to $45, and doing a quick mental check that the 10% service charge is equivalent to $4.50 (which therefore gives a sub-total of approximately 50 bucks without factoring in the other 7% surcharge), you would have concluded it is wise to give this place a miss for the moment.

2. Prudent budgeting

This absolutely goes hand in hand with prudent spending. At some point in our lives, the temptation to shell out cash to get that big-ticket item is just excruciatingly overwhelming. Should you succumb to the devil or postpone the purchase? Or perhaps clench your teeth and walk away forever? A little diligent math done beforehand coupled with a rational, disciplined mind-set would help prevent penalisation in the form of a deeper shade of red on your personal balance sheet. And all it takes is a typically straightforward process of adding and subtracting of numbers, with some interest rates thrown in as multiplication factors if you are looking at a bank loan for financing purposes. It is indeed as easy as it sounds.

3. Reconciling facts with statistics

Statistics can be augmented to present a fictional truth, however blunt, raw facts never lie. Think: an ailing man is considering undergoing a major surgical procedure, and an unscrupulous hospital cites success rates of 75 percent. An independent study by a non-profit medical foundation demonstrates that in a recent batch of 100 patients who were wheeled into operating theatres worldwide, 60 came out with white sheets draped over them. If his tingling mathematical sense kicked in to warn him of this existing discrepancy, he would think twice (probably thrice) before putting his signature on the consent form. That moment of pause could have well given him greater clarity in planning his next course of action. Moral of the story: be discerning by doing your own math homework, as this saves you the potential agony of being deceived by conniving, slippery salesmen. And it might save your life too.

4. Impressing that girl

Neil Strauss completely forgot about penning this down in his book “The Game”, so here is the tip: you can in fact impress the hot girl of your dreams with your superior mental calculation skills. Say she is totalling up the impending bill for her cart of groceries in one corner of the supermarket using her fingers, and you swoop in to give her the correct grand figure. Chances are she can’t take her eyes off you, at least for a while. Tried and tested by yours sincerely, and works like a charm.

5. Keeping Dementia at bay

While a cure currently still eludes scientists and doctors, studies have shown that maintaining cognitive fitness goes some way in delaying the progression of symptoms, thereby enhancing the overall quality of life. Hence, start entertaining yourself with KENKEN puzzles, and reduce the odds of being ravaged by old-age Alzheimer’s.

Frederick Koh is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com

Do we really need to learn calculus in school?

I received a request from Frederick Koh, a JC maths tutor asking me to contribute a guest post on his blog, White Group Mathematics. My first reaction was maths? What do I know about maths?

He then clarified that I didn't actually have to write something on maths, just anything on education but the obsessive personality that I am thought a maths-related post would be more appropriate.

Objectively, I think I have enough of a logical brain to tackle basic maths. I coached both my kids in primary school maths and there's something about the puzzle-solving aspects of primary school maths that appeal to me. My personal experience with maths was generally positive too - I sailed through maths in primary school and E Maths at 'O' levels.

However, this led me to erroneously believe that I could manage Maths C at 'A' levels even though I hadn't studied Add Maths at 'O' levels. Boy, was I wrong. I sat in oblivion throughout the 2 years, despite my classmates' best efforts to help. The tutor practically gave up on me and skipped past me whenever he asked my classmates a question. In the end, by a miracle, I achieved a C grade, to which the tutor reacted epically - "You?? You got a C???"

There's just something about the strings of random numbers that baffle my mostly right brain. Sometimes, they're attached to letters, other times, with funny incomprehensible squiggles and notations. Worst still, sometimes the numbers don't even line up in the same row (miniature ones written above or below other numbers like they're an afterthought). It's like Morse code in an alien language.

I have a PRC friend and back in China, he was in the gifted programme for maths. He shared how in China, there are only gifted programmes for maths and science - such is the emphasis on what is commonly perceived as the pragmatic and "superior" subjects.

"What about China's great legacy of literature and the arts?" I asked. He explained that the government felt China's heritage in the arts caused the country's economic decline so they are now over-compensating. In fact, he told me that all the current China leaders have backgrounds in maths or science.

He feels that this complete neglect of the humanities is a great disservice to Chinese kids. He cites his own example where he wishes that he had learned more soft skills like communications, people management and so on so he can better function at work. He says up to today, he has yet to apply any of the calculus he'd studied (even though he once worked as an engineer).

As someone who's always been in the humanities underdog camp, it isn't hard for me to sympathise with him. But I got to thinking, there are actually two different controversies at work here.

The more obvious one is clearly the maths/science-humanities struggle. Which is more important? It would be easy for me to side with the humanities but my answer is both, and I'm not just trying to be politically correct. The maths and sciences set the foundation for logical thinking and deduction, there's no denying the importance of this. It's not enough just to be able to give customers the correct change or work out how much that bag of apples at the supermarket costs. That's why even though Lesley-Anne doesn't have the aptitude for maths, I tell her that having a good foundation of maths is important.

As for the humanities, well, it fosters critical thinking and deeper reflection into the intangibles, into human behaviour. Much in this world is different shades of grey, not black-and-white. Solutions often can't be calculated via a fixed formula or measured on a quantitative scale, and the humanities teach us how to wrap our minds around the fuzzy and give it meaning. In this connected world where people brashly push forth their arguments and opinions, it's more important than ever not to blindly believe what you read and to question everything with a critical mind. (Yes, including my post).

But beyond the superficial maths/science vs humanities conundrum lies another conflict in my friend's statement. He felt that the years he spent learning calculus would have been better spent learning skills like communication because he could use it in real life. It's this intrinsic belief that in education, what you learn must be usable to be considered useful.

Granted, if you're training to be a mechanic, I sure hope whatever you learn will be useful enough to enable you to fix my car. But this idea that education has to be practical is essentially another left brain argument. To me, it dilutes the value of education because it reduces education to yet another commodity.

It means that if you don't intend to draw, there's no point in learning art. By that same token, learning Chinese is needed only if we intend to do future business with China. I've always thought this moot point as I suspect all the enterprising Chinese entrepreneurs are mastering English as we speak, for the same pragmatic reasons.

Education is not the same as training, it has to have a higher purpose. That's been one of the main criticisms of the Singapore education system, that it doesn't educate individuals, it trains them. Beyond learning how to read and write, count apples and how gravity works, education should enable us to be more thinking versions of ourselves.

At every level, we should have this, to different degrees. At primary school, it could be as simple as asking questions about a science experiment. At secondary school, critical analysis of a social issue. At JC and university? Well, why not calculus?

Right upfront, I'll say the only thing I know about Calculus is that it's the name of a character in the Tintin comics. I looked at the universal authority on all things aka wikipedia and here's what it says: Calculus (Latin: a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals and infinite series.

Ahhhh... nope. Catch no ball.

But despite my ignorance about calculus, I'm pretty certain that calculus does make you think deeper about certain concepts of maths. And as long as something you learn in school helps you exercise your brain in thinking deeper and more laterally, chances are, it's valuable. You just don't know it. The same way that many people think literature isn't useful cos nobody spouts poetry at work in real life. But literature helps you read between the lines, analyse human behaviour and appreciate the subtleties of the written word, all of which is important in life.

So I wouldn't write off calculus just yet. (Do I hear Frederick heave a sigh of relief?) As I've always advised parents, when your child has to choose his or her subject combinations, instead of saying, "take the most useful one" (which is only marginally better than "take the subjects you can score in" or "take the subjects that can earn you the most money"), tell them to take the subjects they're passionate about. Passion ignites learning and from there, they will have a better chance of extracting value from it and becoming more thinking individuals.

That's what education is all about.


This post was published on White Group Mathematics here.

Revisiting maths models

I haven't written a maths post in ages, since Lesley-Anne finished her PSLE, actually!

However, this year, I found that Andre frequently had difficulty with problem sums that featured two different variables in numbers, eg. number of coins vs value of coins. Somehow, he couldn't grasp the method that is commonly taught in school and used in assessment books. I guess it doesn't help that these sorts of sums tend to look very complicated and that gives him a mental block.

After some experimentation, I found a way to teach him using the model method and I'm happy to say, it really works! A similar question came out in his mid-year exam and he could solve it. Ironically, the teacher didn't understand the method and put a question mark next to it, which annoyed me a little. As long as he could arrive at the answer, I thought it shouldn't matter that he didn't use her method.

Anyway, I thought I'd share it here, for the benefit of those kids who might face the same difficulty. This probably works with kids who are visual learners.

1. In a coin pouch, there are 12 fifty-cent coins more than twenty-cent coins. If the total value of the fifty-cent coins is $21 more than the total value of the twenty-cent coins, find

a) the number of fifty-cent coins in the pouch
b) the total value of coins in the pouch

As a starting point for the model, assume the number of the coins for each denomination are the SAME and draw your model based on the VALUE of the coins, This is easy cos in terms of value, 50cts will always be 5 parts and 20cts will always be 2 parts.

Next, you draw in the value of the extra coins, in this case the 12 fifty-cent coins (always, always remind them that they're looking at VALUE, not number. This is critical!)

So you find the value of 12 fifty-cent coins, ie 12 x 0.50ct = $6 and add that to the model.

Now, you're told the value of the fifty-cent coins is $21 more than the twenty-cent coins. This is represented by the portion as drawn here.

Clearly, $21 - $6 = $15 -> 3 parts, so 1 part -> $15 ÷ 3 = $5

To find the number of fifty-cent coins simply take the total value and divide it by 0.50

$5 x 2 + $21 =$31
$31 ÷ 0.50 = 62

Answer: a) There are 62 fifty-cent coins in the pouch.

Finding total value of coins in the pouch is also a cinch, just find the value of the twenty-cent coins and add it to $31.

2 x $5 = $10
$10 + $31 = $41

Answer: b) The total value of coins in the pouch is $41.


This type of question can also be in forms other than money, eg. number of animals vs number of legs, or in this next example, number of vehicles vs number of wheels.

2. In a carpark, there are motorcycles and cars. 5/7 of the wheels are the wheels of the cars. There are 12 more cars than motorcycles. How many wheels are there in the carpark?

Similar to Question 1, first assume the number of both types of vehicles are the same and draw the model based on the number of WHEELS (4 wheels per car, vs 2 wheels per motorcycle).

Next, add in the additional wheels for 12 cars, which is 12 x 4 = 48.

Now, the question states that 5/7 of the wheels are the wheels of cars, meaning 2/7 of the wheels are the wheels of the motorcycles.

Looking at the model, there are already 2 parts to the motorcycles vs 4 parts to the cars, therefore 48 has to be equivalent to 1 part.

So total number of wheels is 48 x 7 = 336.

Answer: There are 336 wheels in the carpark.

For those of you new to my blog, I'm a firm believer of using models for maths as they've helped my kids, who are both visual learners (and algebraically-challenged), tremendously. If you're interested, you can visit some of my old posts on how to use math models (click on the 'Mathematics' label under the Blog Contents column on the right).

The girl who is humanities-bound

If there ever was any doubt that Lesley-Anne is right brain-oriented, this removed it once and for all.


This is the result of the Higher Ability Selection Test administered by the Australian Council for Educational Research that Lesley-Anne sat for late last year. The school made all the students who entered via DSA sit for the test, to see if it can replace the current General Ability Test (GAT) for DSA applicants. I suppose they want to assess if the results are consistent with those of the GAT.

The test comprises a few comprehension passages, a series of math questions and a writing test. Since there are two English components and only one math component, it seems to me the test is slightly skewed in favour of those good in language. That definitely contributed towards Lesley-Anne's total score (not that we're complaining).

Anyway, this test sort of came at the right time. Since Lesley-Anne is is in sec 2, she'll have to choose her stream for sec 3 and 4 by the end of this year. The results confirm our belief that she'll fare much better going down the humanities route instead of the ever popular triple-science stream.

Sec 1 maths project on GST Offset Package

This is EOY (End of Year exams) season for secondary school kids and Lesley-Anne has been busy hitting the books. From what I see on Facebook, students have been complaining incessantly but really, they have it good compared to primary school.

For one, their school holidays will start much earlier - Lesley-Anne's begin about 2 weeks before Andre's and I'm dreading the inevitable whining from him. Secondly, exams are just one component in the calculation of the final grade. In Lesley-Anne's school, the grade takes into account numerous mini tests, assignments and projects done throughout the year, so it really is about being consistent. It also takes the pressure off the final exams somewhat.

One of the more interesting maths projects that Lesley-Anne had to do this year was to create a scenario to show the application of the GST Offset Package in a Singaporean family. The students could be as creative as they wanted but all workings had to be shown and they were limited to two pages. They were scored on creativity, presentation and accuracy of calculation.

Lesley-Anne decided to draw a cartoon and named her protagonist Andre (who's apparently a math whiz!) Notice also that she drew the parents as hopeless with their finances. Hmm...
Gst Offset

I thought it was fun and apparently her teacher thought so too cos she was one of the two or three in her class who received 15/15 for the project. Maths is still Lesley-Anne's weaker subject so it helps to have this project pull her grade up. She claims Andre brought her luck. Maybe he will feature in more of her assignments!

Maths mummy retires

I realise I haven't had a maths post in a long time which is kind of ironic since at one point, I think my blog was primarily about maths.

There's a simple explanation for this: maths was Lesley-Anne's stumbling block and I had to be actively involved to guide her through the subject so that she would be sufficiently prepared for PSLE. Now that the PSLE is over, I'm off the hook!

This is not to imply that Lesley-Anne has magically grown some mathematical DNA overnight. It's just that the transfer from primary to secondary school seems to have sparked a switch in my brain - primary school: Mummy On, secondary school: Mummy Off. I don't know what it is but I feel like at secondary school, the child should be mature enough to learn on her own and make sure she meets her deadlines. Mummy's role is now purely functional, ie see that she gets three square meals a day (well, one is in school so that's two) and that she gets enough sleep.

This mental switch is quite strange - it impacts me not just intellectually but affectively. I mean, Lesley-Anne came home with a 6/10 for her maths quiz and I was quite unperturbed, in fact nonchalant. My response was, "Do you know where you went wrong? You gao tim (handle) yourself, ok?" In other words, her not getting great results doesn't even stress me out anymore. Somewhere in the illogical region of my brain (maybe the part that controls food), I assume that she'll find a way to manage.

Maybe there's another reason for it and that is, I couldn't help even if I wanted to. If I thought primary school maths was tough, at least I could get away with drawing cutesy models and working out problem sums with narratives like "Ali and Ahmad have 25 balloons between them". How fun is that?

In secondary school, maths is serious business. Here are two samples of the no-nonsense questions from one of Lesley-Anne's math quizzes:

1) Given that 450 = 2 x 3² x 5² and 2625 = 3 x 5³ x 7, find

• the largest common factor of 450 and 2625
• the least common multiple of 450 and 2625, leaving your answer in index notation
• the smallest positive integer n, if 450n is a perfect cube
• the smallest positive interger m, if 450m is a multiple of 2625

2) Find a three-digit number A such that A + 17 is divisible by 15, A + 77 is divisible by 25, and A + 107 is divisible by 35.

Imagine a big, neon question mark above my head. What happened to all those fictitious names, the transferring of women and men into Halls A and B, and time taken to walk from playground to library? Sorry, this is secondary school. No more Ahmad and Ali. Didn't you hear? Maths is about NUMBERS. And PS, I have no burning desire to know the answers to the questions above so please don't email me.

After each topic, the teacher gives a quiz to test that the students understand what they have been taught. Those who score less than 6/10 have to attend maths remedial for that topic. I think it's quite a good system as the remedial sessions zoom in on specific topics rather than on specific individuals. After all, some kids may have difficulty only in selected topics.

And while we're on the subject of maths, Lesley-Anne called home sometime in January, informing me that she had been selected for Maths Olympiad training. She couldn't stop giggling as she told me. I asked her several times, "Sure or not???" She attributes it to her superior tikam skills on the MCQ diagnostic test ("Hmm... so many Cs, I think I'll colour a B...")

I let her go for the two-hour weekly training sessions, figuring hey, free maths tuition, why not? She's since been having thoughts about dropping out as she claims she has no idea what the teacher is talking about.

Bring back Ali and Ahmad, I say!

The PSLE maths conundrum

Last PSLE paper today! Woohoo!!

By now, some of you may have heard of the ruckus over the PSLE maths paper on Thursday. In short, the paper was a killer. How tough was it? Let's see - a boy who's a Math Olympiad platinum winner cited the paper as Math Olympiad standard. Others have said it was even harder than some of the top schools' prelim papers.

Lesley-Anne came home teary-eyed as she couldn't solve three 4-mark questions and she barely finished the paper, no time to check at all. From what I heard over the grapevine, tears were flowing in PSLE exam halls all over Singapore. Lilian and I were predicting that complaint letters by panicky parents would rapidly find their way into the Straits Times' forum page. Surprisingly, this has not happened yet, although Today newspaper has already written a piece on it here.

I'm not one of those parents, in case you're wondering. The reason is that I have no wish to become one of those kiasu, whiny parents who complain whenever things don't go their way. Besides, I don't think it will make an iota of a difference (what, you expect MOE to hand you an A* because you complained?)

I have no issue with difficult PSLE papers. I think generally, it evens itself out because the PSLE T-score is moderated, meaning that each child is assessed not on the actual score on his exam paper but how he performs in relation to the rest of the cohort. The better you perform compared to your p6 peers, the higher your T-score will be, regardless of your actual mark. So a difficult paper is actually advantageous for kids adept in that subject because they will be better able to differentiate themselves from the rest of the pack.

Having said that, I think the challenge is to be able to pitch the level of the paper appropriately. In the case of this year's maths paper, I think it was unrealistically high. Of course this is just my gut feel since I haven't seen the paper, I'm going by comments I've read in forums and anecdotal accounts.

In most years, the PSLE maths paper contains a couple of very challenging questions which I assume is to suss out the truly bright mathematical talents. If so, then the majority of students are not expected to know how to solve these, which is fine. However, this year, the few challenging questions were so difficult that they stumped many of even the top tier maths whizzes, eg. those who participate in the Math Olympiad and those in GEP. If this is true, then instead of being able to identify the top 5%, maybe you end up identifying, say the top 0.5%. The top 5% then gets bunched up with perhaps the top 20% because all of them couldn't solve the same questions. (All numbers are arbitrary).

My question then is, how is this useful in any way? The handful of maths prodigies has most likely already been identified, through their past performance. What is the value of having a measurement tool that can differentiate only the top 0.5% followed by the next large category of 20%? Not much, in my opinion.

My main concern, however, is the impact on kids who are in the middle of the bell curve (which would be the majority). Again going by what I've heard, the rest of the paper was no piece of cake either, with many questions requiring a lot of time and thinking, even for Paper 1, which is traditionally more straight forward. This meant that many kids were not able to finish the papers. I know exams are also a test of time management but to what degree? I mean, is someone necessarily a better technician because he can assemble a gadget in 20 minutes versus one who can do it in 25 minutes? What are we testing here?

The purpose of any exam, especially a national one, should be to test understanding of the concepts and the ability to apply them, not to trip the kids up. I'm not suggesting you have a super simple paper that everyone can sail through but if you have a paper where a good proportion of the kids (who have gone through the national education system) is unable to perform satisfactorily in, I think it puts a question mark not on the students but on the system. Something is wrong - either the kids were not taught adequately or the exam not set correctly.

The casualties of course, are the kids. You can argue that since the T-score is moderated, kids shouldn't be too unduely distressed over an overly difficult paper. To that, I say we sometimes forget that these are 12-year-olds. Give them a break. Many of these kids have been slogging for the PSLE all year, diligently doing paper after paper, sacrificing tv and other pleasures for the hope of performing well at the PSLE. Being unable to answer the questions or even finish a paper sends them this message: "Your effort was not good enough." Demoralising is an understatement. Even if they do manage to attain a reasonable T-score, it's like a back-handed compliment - they just didn't do as badly as others.

Lesley-Anne was very disappointed with her performance in the paper after putting in so much work, although I acknowledge that having DSA does relieve the anxiety somewhat (thank God!) I hope that for the sake of the sanity of future parents and kids, the national exam policies and practices can be reviewed and adjusted appropriately. Not too much to ask, surely?

Going back to school for maths

I was prompted to write another math post after all the strong reactions to my math challenge. First let me state, there are many areas of our education system that I'm critical about but one thing I've never doubted is the quality of Singapore's maths education. What I love about it is it really teaches understanding of concepts.

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.

Answers to problems sums in NY p6 prelim paper

I hope some of you attempted the Math Challenge I issued yesterday! Lilian posted Brian's workings for some of the problem sums in the Nanyang paper. Since she has asked for my workings, here they are, at least the ones that I got right. The sums where I erred, please refer to her post for the solutions. Do note however that my aim is to try and explain how the answers are derived, so they are text-heavy. In an actual exam, the kids just need to show the numbers.

The hardest questions were in Section C, from Q42 onwards, so here they are.

Q42 (a): The figure shows a rectangular field ABCD. Mike walked from A to B to C to D and he covered a distance of 57m. Sandy walked from B to C to D to A and she covered a distance of 48m. What is the perimeter of the field?

A: I used algebra, B = breadth, L = length. Based on Sandy's distance, 2B + L = 48 (equation 1). Based on Mike's distance, 2L + B = 57 (equation 2).

Using equation 1, L = 48 - 2B. Substitute this into equation 2, you'll get:
2 (48 - 2B) + B = 57
96 - 4B + B = 57
4B - B = 96 - 57
3B = 39
B = 39 ÷ 3 = 13

Mike's distance is just short of one breadth from the perimeter of the field, so 57 + 13 = 70

Answer: a) The perimeter of the field is 70m.

(b): Some construction work was undertaken on the same field and a semi-circular part was removed from it. What is the perimeter of the field after the construction? (Take ╥ = 22/7)
Note: I got this one wrong due to a careless mistake, but here's the correct answer.

A: First, find the perimeter of the semi-circle, which is ½ (╥d)
½ (22/7 x 10.5) = ½ (321/7) = 16.5

The two remaining bits of the breadth are B - 10.5 = 13 - 10.5 = 2.5

The perimeter of the field is Mike's distance + perimeter of semi-circle + remaining bits of B
57 + 16.5 + 2.5 = 76

Answer: b) The perimeter of the field after construction is 76m.

Q43: got this wrong, refer to Lilian's post.

Q44: The tickets for a show are priced at $10 and $5. The number of ten-dollar tickets available is 1½ times the number of five-dollar tickets. 5 out of 6 ten-dollar tickets and all the five-dollar tickets were sold. The ticket sales amounted to $5600. How much more would have been collected if all the tickets were sold?

A: This is the only sum in the entire paper where I used a model (bah, so much for being a model mum!) First, note that there are two different variables here, the number of tickets and the value of the tickets. There are 1½ the number of ten-dollar tickets vs five-dollar tickets available, ie for every five-dollar ticket ($5), there is 1½ ten-dollar ticket ($15). In terms of value, the ten-dollar tickets are three times that of the five-dollar tickets. I drew a model for the value of the tickets available (right pic).

Next, we know 1/6 of the ten-dollar tickets were not sold, so I cut the ten-dollar ticket portion into six portions. Since I halve each of the 3 portions for the ten-dollar ticket, I have to do the same for the five-dollar tickets. (If you're not clear about the rules for math models, refer to my earlier post.) The shaded part is the part that is unsold.

Since ticket sales amounted to $5,600, $5,600 is represented by the 7 unshaded parts.
1 unknown part = 5600 ÷ 7 = 800 (which is equivalent to the shaded part)

Answer: $800 more would have been collected.

Q45 A rectangular tank measuring 60cm by 35cm by 40cm is half-filled with water. If Tap A is turned on, it will take 6 min to fill the remaining half of the tank to its brim. Tap B drains water from the tank at a rate of 12 litres per min. How long will it take for the tank to be filled to 1/8 of its capacity if both taps are turned on at the same time?

A: First, find how much water is in the tank, which is total volume of tank divided by 2.
(60 x 35 x 40) ÷ 2 = 84,000 ÷ 2 = 42,000 cm³ or 42 litres

The rate of Tap A is 42 ÷ 6 = 7 litres/min

The rate of Tap B is -12 litres/min (take note of the minus, since Tap B drains water, not adds water).

This means that every minute both taps are on, the tank loses 5 litres (7-12)

We need to find out how much time it takes for the tank to reach 1/8 of its capacity (which is 42 ÷ 4 = 10.5). The amount of water it needs to lose is 42 - 10.5 = 31.5.

- 5 litres takes 1 min
- 31.5 litres takes 31.5/5 = 6.3 mins

Answer: It takes 6.3 mins for the tank to be filled to 1/8 of its capacity.

Q46: got this wrong, refer to Lilian's post.

Q47. At a school carnival, there were 520 more girls than boys. 1/8 of the girls and 20% of the boys left the carnival. In the end, there were 488 more girls than boys. (a) Did more girls or boys leave the carnival? How many more?

A: If we put aside the 520 more girls at first, we have an equal number of boys and girls. So we convert 1/8 and 20% (1/5) into a common denominator to be able to compare the number of boys and girls who left the carnival, which is:

Girls - 5/40 + 1/8 x 520 = 5/40 + 65
Boys - 8/40

So the number of boys and girls still at the carnival is:

Girls - 35/40 + 520 - 65 = 35/40 + 455
Boys - 32/40

This means there were 3/40 + 455 more girls than boys still at the carnival, which is equivalent to 488.
3/40 + 455 = 488
3/40 = 488 - 455 = 33
1/40 = 33/3 = 11

Since 1/40 = 11, the number of girls who left the carnival is 5 x 11 + 65 = 120
The number of boys who left the carnival is 8 x 11 = 88
120 - 88 = 32

Answer: a) 32 more girls left the carnival than boys. (Note: This is the roundabout way! Lilian's is way more direct but I couldn't work it out at that time).

(b) How many children were there at the carnival in the end?

The number of children still at the carnival is:
35/40 + 455 (girls) + 32/40 (boys)
= 67/40 + 455
= 67 x 11 + 455 = 1192

Answer: b) There were 1,192 children at the carnival in the end.

Q48: Couldn't do this, refer to Lilian's post.

The Math Challenge to parents

When I heard the news that the swine flu has become pandemic, I turned to Kenneth and asked, "If it spreads and schools have to close, will that mean there's no PSLE? Then all the kids this year can have a free pass to sec. 1!"

See how deluded I've become? Well, rather than wait for that pipedream to turn into reality, here's a better suggestion: prepare your kids for the eventuality. And by prepare, I don't necessarily mean teach. I mean prepare them psychologically and emotionally and most importantly, support them.

The best form of empathy is to walk in the footsteps of the other person and that's what Lilian and I did. Well, Lilian did it first. She actually printed out the Nanyang Primary School p6 prelim math paper 2007 and DID the paper. Then she urged me to do the same, which I did, based on allocated time and everything.

Guess how we performed? I amassed a dismal score of 73/100. Even more mortifying, I made four careless mistakes (the thing I'm always nagging Lesley-Anne about!!) which cost me 8 valuable marks. (I had to mention this because I'd like to think that I could have scored over 80/100 which wouldn't be quite so embarrassing.) I'll let Lilian decide whether she wants to reveal her score but here's a hint: she performed even worse than I did LOL (sorry Lilian!!) Her son Brian, on the other hand, managed to score 82/100 and mind you, this is without having ever attended formal math lessons on most of the p6 topics! Brilliant lah, Lilian. Luckily he's the one sitting for the exam, not you, hehe.

I learnt plenty from this exercise, mostly that when you have a time crunch, it's extraordinarily difficult to think clearly and it's all too easy to make careless mistakes, especially when you have very complex questions with multiple steps. I didn't have time to check - I had the grand total of 3 minutes left for the last problem sum which didn't matter anyway because there was no way I could have solved it. 5 marks out the window.

It is a very stressful and draining process, even more so for our kids because they know how much is at stake. So today, I'm issuing a Math Challenge, especially to parents whose kids are taking the PSLE this year - walk in the footsteps of your child. Since Lilian and I took the Nanyang Primary School p6 prelim math paper 2007, this can be a benchmark (plus I'm curious to know how other parents will fare!) You can print out the paper here. Give yourself 2 hrs 15 mins, no breaks. I'm even ok with giving you a handicap since some of you might protest that you're rusty when it comes to maths formulas - you're allowed to refer to your kids' p6 math textbooks.

Nanyang Primary School is one of the local primary schools known to have the toughest maths papers, so don't be discouraged if you can't do some of the sums. But I guarantee you that after that exercise, you will gain a newfound respect for what our 12-year-olds have to go through. And when you understand better how it feels to be them, I'm sure you'll be better able to guide and coach them in their education journey.

Are boys really better at math and girls at languages?

There's a common belief that boys are better at math than girls. Based on my personal experience with Lesley-Anne and Andre, I had accepted this old adage without question.

Well, guess what? A study by a team of scientists conducted on SAT and math scores from 7 million students in the US found that this belief is actually fiction, not fact. Whether they compared average performance, the scores of gifted children or students' ability to solve complex math problems, girls measured up to boys.

So why does this misconception persist? According to University of Wisconsin-Madison psychology professor Janet Hyde, the study’s leader, cultural beliefs like this are “incredibly influential.” “Because if your mom or your teacher thinks you can’t do math, that can have a big impact on your math self concept.”

It makes sense. Sometimes when a friend of mine says her daughter is struggling with math, I spout this glib reply, "Girls lah, they're not as good in math." So slap me now because apparently, not only am I wrong, I'm reinforcing the stereotype and adding to the self-fulfilling prophecy of these young girls - "I can't do this math paper because I'm a girl, I'm not as good in math. I'm not as good in math, so I won't be able to do this math paper." Doh!

I think it has partly to do with our eagerness to embrace the left brain-right brain theory, ie boys are more inclined towards left-brain work (logical, analytical) and girls are more right-brained (creative). That's why along with the boys-are-better-at-math belief, we have the girls-are-better-at-languages theory.

But you know what? The part about girls being better at languages IS true, at least in the early years. Researchers have found that this is due to the way words are processed. Girls use the language decoding portion of their brains, enabling them to decipher abstract information. Boys however, rely more on vision and hearing to process information.

Doug Burman, an aneuroscientist at Northwestern University’s Developmental Cognitive Neuroscience Laboratory in Evanston explained, “For girls, it doesn’t matter whether you are reading or hearing the words, the information gets converted into an abstract meaning, an abstract thought. For boys, the research suggests it’s really going to be very important whether they’re hearing or reading words. That is going to determine how well they’re processing the language.” These results explain why girls consistently score higher than boys at language tests. But this advantage may taper off after secondary school.

While these findings warrant a separate post on catering to the different learning styles of boys and girls, for now, the lesson here is not to be so quick to dismiss your kids' learning difficulties as biological. Everyone is different. I guess I should have realised this from my own experience - I don't fall clearly under the right brian or left brain style. I'm reasonably good at math but suck at science. People tell me my English is pretty "powderful" but I couldn't learn Chinese to save my life.

So what does this mean? It means that human beings are so complex that we can't compartmentalise them into neat boxes. Even if something holds true for the vast majority, it may not hold true for us. We all have our unique strengths and weaknesses that are largely tied to our interests. If our kids are struggling in certain subjects, let's not be so quick to believe that these are beyond their abilities and instead, try to stir up interest in those subjects. Evidence has shown again and again that the kids who tend to do well eventually are not necessarily those who have the best talents but those who keep persevering.

The magic of squares

For many kids, maths is about abstract numbers. Even though they may be able to execute formulas and work out the solutions to problems, they often don't get the concept behind these numbers. That's the reason I love maths models - they enable you to have a very concrete visualisation of what the numbers represent, which really helps understanding.

Lilian's boys have an extraordinary ability to grasp mathematical concepts, fueled by her very imaginative way of teaching (and obvious love of maths herself!) In one of her older posts, she wrote about her method of explaining squares, which I thought was absolutely brilliant.

This method helps you find the answer to large squares without doing long multiplication, but really, that's not the point. (If you want to find an answer fast, use a calculator). It helps you see very clearly how squares work and how the answer is derived.

She starts off with the basic premise of a 10 x 10 square, which in pictorial form, is basically 10 columns x 10 rows (right pic). Most kids will know 10 x 10 = 100 (which is also the area of the square).




Now, say you want to find out what is 15 x 15. Visually, what this means is that you add another 5 columns and 5 rows to your basic 10 square (right pic).

And don't forget that little square in the right bottom corner which consists of 5 columns and 5 rows. Your final 15 x 15 square will look like this (bottom pic):

Now we just need to add up all the different areas of the square. We already know that the basic 10 square = 100. Each of the additional 5 rows/columns is 5 x 10 = 50. That corner bit is 5 x 5 = 25 (bottom pic). Add all of that up and you get 225. Therefore, 15 x 15 = 225.

You can do this all the way from 11 x 11 to 20 x 20, after which you use the 20 x 20 basic square. You can read more details on how Lilian got her 6-year-old Sean to work it out here.

Just last week, I came across another intriguing pattern on squares, thanks to Adeline's precocious son. He discovered while doing some multiplication, that when you multiply any two numbers that are two apart, the answer is always one less than the square of that middle number. (Ok, ok! I know that sounds very confusing!) Let me show you what I mean:

4 x 6 = 24
5 x 5 = 25

5 x 7 = 35
6 x 6 = 36

6 x 8 = 48
7 x 7 = 49

See the pattern? This is true no matter how large the number. For instance,

246 x 248 = 61,008
247 x 247 = 61,009

(I used the calculator lah, what did you think??) There's a very simple explanation to the pattern, which I will attempt to show visually.

Here is a basic 8 square - 8 columns x 8 rows (right pic).









To change it into a 7 x 9 rectangle, you essentially take one row and move it to a column.

You'll find that you have an extra square (bottom pic) because the number in the column will always be one more than the number of rows (which has been reduced from the original by one).

See? It's so simple I don't know why I never saw the pattern before. And it took a 6-year-old to point it out :P

Once again, seeing this pattern probably won't help your kids do their sums any quicker but I believe it facilitates understanding of how squares work.

More maths problems by request

I've received a couple of maths problems on my blog. While I'm flattered that you have such faith in my maths abilities, I can't promise that I'll entertain all future requests (or be able to solve them all!) since this is not a dedicated maths blog. Do click on my mathematics label to see the workings of other maths problems I've done.

But for now, these are the two I received. Thanks for your patience!

By wo wen tian:

Hi, how to solve this problem without using Algebra? At first, two shops A & B had a total of 1,040 sacks of rice. After Shop A has cleared 3/4 of its stock and Shop B has cleared 3/5 of its stock. Shop B now has 52 more sacks of rice than Shop A. How many sacks of rice does each shop have at first?

It's quite easy to solve this problem using models, you just need to work backwards. First, you draw the model for the final scenario, ie Shop B has 52 more sacks of rice than Shop A (right pic).

Then you add in the initial stock of rice. Shop A now has only 1/4 of its original stock so you need to draw in another 3 units (to make up the 3/4). Shaded parts depict stock of rice that was cleared.

Shop B now has only 2/5 of its original stock, meaning 1 unit + 52 sacks = 2/5 of original stock. To add in the original 3/5, you need to draw another 1 unit + 52 sacks (2/5) and ½ unit + 26 sacks (1/5).

Remember one of my rules for models is every unknown unit has to be equal in value, so since you have a half unit, you need to cut every other unknown unit into half (including those for Shop A - just slice horizontally across). Now, you can see (below) that you have 13 units + 52 + 52 + 26 and all that is equivalent to 1,040 sacks of rice.

13 units = 1,040 - 52 - 52 - 26 = 910
1 unit = 910 ÷ 13 = 70

Shop A is 8 units, so 8 x 70 = 560
Shop B is 5 units + 52 + 52 + 26 = 5 x 70 + 130 = 480

Answer: Shop A has 560 sacks of rice at first and Shop B has 480.


Help needed said...

There are 600 children in Team A and 30% of them are boys. There are 400 children in Team B and 60% of them are boys. After some children are transferred from Team B to Team A , 40% of the children in Team A and 60% of the children in Team B are boys. How many children are transferred from Team B to Team A?

This question is tricky - I couldn't solve it using models, I used a combination of ratio, percentage and algebra. First, we are given the number of children in each team, so we can work out how many boys and girls there were in each team originally.

600 x 30% = 180, so Team A originally had 180 boys and 420 girls (600 - 180)
400 x 60% = 240, so Team B originally had 240 boys and 160 girls (400 - 240)

Next, we use ratio. The ratio of girls to boys was:

At first: Team A - 7 : 3 Team B - 4 : 6
After : Team A - 6 : 4 Team B - 4 : 6

Notice that the ratio of girls to boys for Team B remained the same, even after the transfer. This means that the proportion of girls and boys transferred out of Team B was also 4 : 6 (or 2 : 3), ie 2 units of girls and 3 units of boys were transferred out.

Using algebra, the number of children in Team A after the transfer can be expressed as:

(Girls) 420 + 2 units = 60%
(Boys) 180 + 3 units = 40%

The lowest common multiple of 60 and 40 is 120, so convert both equations to = 120 and you can combine both equations.

2 (420 + 2 units) = 3 (180 + 3 units)
840 + 4 units = 540 + 9 units
9 units - 4 units = 840 - 540
5 units = 300

Remember a total of 5 units of children were transferred out of Team B (2 units of girls and 3 units of boys), so you don't even need to find out how many children is represented by 1 unit.

Answer: 300 children were transferred from Team B to Team A.

If anyone can solve this problem using models, do let me know.