Tag Archives: recreational math

Pi Day in Singapore

Thousands of students around the world celebrate Pi Day today, but local math students in Singapore can only dream of being part of this annual mathematical event. Singapore math students, teachers, and parents don’t (and can’t) celebrate Pi Day, as long as they officially follow the British style of writing their dates (DD/MM/YY).

What makes matters worse is that this year, Pi Day falls on the first day of the one-week school break, which makes it almost impossible for hardcore math teachers, who want to buck the calendrical trend, to get their students excited about the properties and beauties of the number Pi.

Until Singapore switches to the American style of writing dates (MM/DD/YY), which may not happen, at least during my lifetime, however, this shouldn’t prevent us from evangelizing the gospel of Pi among the local student population.

Here are seven e-gifts of the holy Pi, which I started musing about 314 minutes ago on this Pi Day.

Pi Day vs. Abacus Day

  

A 14-Month Year for Singapore ONLY!

  

Where Are You in Pi?

  

Heavenly Pi

  

The Numerology (or Pseudoscience) of Pi

  

In Remembrace of the Late Singapore PM 

  

Biblical Pi vs. Mathematical Pi

   

Happy Pi Day!

© Yan Kow Cheong, March 14, 2016.

The 12 Problems of CHRISTmaths

Vintage Christmas—Just like Baby Jesus two millennia ago!
Vintage Christmas—Just like Baby Jesus two millennia ago!

Christmas is a golden and joyful opportunity for number enthusiasts and math geeks to sharpen their creative mathematical problem-solving skills.

Here are 12 CHRISTmaths cookies that may help you shake your brain a little bit in the midst of Christmas festivities.

Warning: Refrain from forwarding this post to relatives or friends living in countries, which are intolerant of Christmas and Christianity, such as Brunei, Saudi Arabia, and Somalia, as it’s haram for “infidels” to take part in any kind of Christmas celebrations. And I assume that includes reading any on-line materials deemed un-Islamic or un-Mohammedan, which might lead believers astray from the faith.

1. Unlucky Turkeys

Estimate the number of turkeys that make their way to the supermarkets every year.

2. A Xmas Candy

Mary wanted to buy a candy that costs 25 cents. A dated vending machine would take one-cent, five-cent, and ten-cent coins in any combination. How many different ways can she use the coins to pay for the candy?

Christmastize your code!
Remember to scan your Christmas item!

3. The Dimensions of a  Cross

A square of side 25 cm has four of its corners cut off to form a cross. What is the perimeter of the cross?

4. The Number of Crossings

Two lines can cross one time, three lines three times, four lines six times, and five lines ten times. If there are 25 lines, what would be the maximum number of crossings be?

5. An Eco-Xmas

If all instances of the word “CHRISTMAS” were replaced with “XMAS,” how much ink and paper (or Xmas trees) could you save every year? How much money could be channelled back to feeding the poor and the hungry during the festive season?

XMaths Tech
© T. Gauld’s You’re all just jealous of my jetpack (2013)

6. Number of Xmas Cards

In an age of Xmas e-cards and video cards, how many Christmas greetings cards are still being sent worldwide? How many trees are being saved every festive season?

7. Does Xmas! have 25 digits?

1! = 1, 2! = 1 × 2 = 2, 5! = 1 × 2 × 3 × 4 × 5 = 120—a 3-digit number, and 10! = 1 × 2 ×⋯× 10 = 3,628,800—a 7-digit number.

(a) Without a calculator, how would you verify whether the number 25! has precisely 25 digits or not.

(b) Which positive integers n (other than the trivial case n = 1) for which n! has exactly n digits?

GST with no thanks to Father Xmas
GST (or VAT) with no thanks to Father Xmas

8. Xmas Trees

Guesstimate how big a forest would 25 million Christmas trees occupy.

9. Folding papers

Fold a single piece of paper perfectly in half, from left to right. How many creases will there be after the 25th fold, when you continue folding so that all the rectangles are folded into two halves each time?

10. Pre-Xmas Tax

Imagine Singapore were to implement a pre-Christmas tax on all kinds of Christmas marketing before the first week of December. Estimate how many extra million dollars would the Income Tax department collect every festive season.

Folding a Santa Claus
© Anonymous Folding a Santa Claus

11. A Xmas Quickie or Toughie

What is the sum of the last two digits of 1! + 2! + 3! +⋯+ 24! + 25!?

12. An Ever-Early Xmas

Show that as one celebrates more and more Christmases (or, as one gets older and wiser), Christmas seems to come earlier every year.

Xmas Möbius Strips
Christmas Möbius Strips

References

Gould T. (2013). You’re all just jealous of my jetpack. New York: Drawn & Quarterly.

Yan, K.C. (2011). Christmaths: A creative problem solving math book. Singapore: MathPlus Publishing.

Zettwoch, D., Huizenga, J., May, T. & Weaver, R. (2013). Amazing facts… & beyond! with Leon Beyond. Minneapolis: Uncivilized Books.

A Xmas Bonus: 25 CHRISTmaths Toughies from Singapore ?? http://tinyurl.com/q9w3ne9

 

Selected Hints & Answers

2. 12 ways. Hint: Make an organized list.

3. 100 cm. 

4. 300 crossings.

5. About 30 million gallons of ink, 500 square miles of paper, and $15 trillion could be saved.

6. Hint.

7. (b) n = 22, 23, 24.

9. 225 – 1.

11. 4.

12. Hint: Why as one gets older, time appears to fly faster.

2012-12-21 23.13.28

© Yan Kow Cheong, December 25, 2015.

 

Anything Funny about Singapore Math?

Math educators, especially stressed [often self-inflicted] local teachers in Singapore, are always on the look-out for something funny or humorous to spice up their oft-boring math lessons. At least, this is the general feeling I get when I meet up with fellow teachers, who seem to be short of fertile resources; however, most are dead serious to do whatever it takes to make their teaching lessons fun and memorable.

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© Sidney Harris Sea animals are mathematical, too!

It’s often said that local Singapore math teachers are the world’s most hardworking (and arguably the world’s “most qualified” as well)—apparently, they teach the most number of hours, as compared with their peers in other countries—but for the majority of them, their drill-and-kill lessons are boring like a piece of wood. It’s as if the part of their brain responsible for creativity and fun had long been atrophied. A large number of them look like their enthusiasm for the subject have extinguished decades ago, and teaching math until their last paycheck seems like a decent job to paying the mortgages and to pampering themselves with one or two dear overseas trips every other year with their loved ones.

Indeed, Singapore math has never been known to be interesting, fun, or creative, at least this is the canned perception of thousands of local math teachers and tutors—they just want to over-prepare their students to be exam-smart and to score well. The task of educating their students to love or appreciate the beauty and power of the subject is often relegated to outsiders (enrichment and olympiad math trainers), who supposedly have more time to enrich their students with their extra-mathematical activities.

Singapore Math via Humor

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© Sidney Harris The lost art of Roman numerals

A prisoner of war in World War II, Sidney Harris is one of the few artists who seems to have got a good grasp of math and science. While school math may not be funny, math needn’t be serious for the rest of us, who may not tell the difference between mathematical writing and mathematics writing, or between ratio and proportion. Let Sidney Harris show you why a lot of things about serious math are dead funny. Mathematicians tend to take themselves very seriously, which is itself a funny thing, but S. Harris shows us through his cartoons how these symbol-minded men and women are a funny awful lot.

Angel: “I’m beginning to understand eternity, but infinity is still beyond me.”

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© Sidney Harris There is nothing new under the mathematical sun!
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© Sidney Harris Isn’t mathematics just a man-made game?
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© Sidney Harris The world’s first “mathematical plagiarizer”
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© Sidney Harris The aftermath of Pi addiction
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© Sidney Harris Maybe we’d soon spot some bunnies running around!
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© Sidney Harris Some step just needs to be accepted on faith!
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© Sidney Harris Who says mathematicians don’t need drugs?

Mathematical humor is a serious (and dangerous) business, which few want to invest their time in, because it often requires an indecent number of man- or woman-hours to put their grey matter to work in order to produce something even half-decently original or creative. The choice is yours: mediocrity or creativity?

Humorously and irreverently yours

References
Adams, D. S. (2014). Lab math. New York: Cold Spring Harbor Laboratory Press.
Harris, S. (1970). What’s so funny about science? Los Altos, Ca.: Wm. Kaufmann, Inc.

© Yan Kow Cheong, August 20, 2015.

Check out an inexpensive (but risky) way to make a Singapore math lesson less boring: The Use of Humor in Mathematics. The author would be glad to visit local schools and tuition centers to conduct in-service three-hour math courses for fellow primary and secondary math teachers, who long to bring some humor to their everyday mathematical classrooms—as part of their annual 100 hours professional upgrading. Please use his e-mail coordinates on the Contact page.

The Numerology about Mr. Lee Kuan Yew

In the aftermath of the death of Singapore’s founding father, Mr. Lee Kuan Yew (1923–2015), a number of numerological tidbits (or numerical curiosities, to put it mildly) floated on social media, which got a number of apparently self-professed innumerates pretty excited. Here are three such postings that I saw in my Facebook feed and on WhatsApp.

RIP: Lee Kuan Yew (1923–2015)

A numerological message that was circulated among WhatsApp users in Singapore
(© Unknown)—A numerological message that was shared among Singapore WhatsApp users

The WhatsApp message gives the impression that it was the works of some “pseudo-mathematician,” but it could very well have been the digital footprints of a “mathematical crank” or an amateur-numerologist, who wanted to tickle mathophobics with such numerical coincidences.

Did Singapore’s numerologists (or pseudo-mathematicians) fail to point out some of the following numerological absurdities?

The digital root of Mr. Lee’s birth year is 1 + 9 + 2 + 3 = 15, which stands for the last two digits of the year he experienced his last heartbeat.

The pollution index for that week was in an unhealthy range, and the average PSI for the six-day mourning period was about 91.

Or, were there exactly 91 priests on vigil at an undisclosed Roman Catholic Church, who were interceding for Mr. Lee to ensure that his heavenly destination is 100% secured, although his manifold deeds to the nation inarguably exceeds the number of his political faux pas, especially vis-à-vis his political enemies or opponents?

Or, did 91 senior monks and nuns (or did I mistake them for disciples of Shintoism?) resort to “synchronized chanting” to ensure that the highest level of enlightenment be bestowed on the late Mr. Lee, who might be reincarnated as a future Buddha for his numerous selfish deeds towards his oft-ungrateful and unappreciative fellow citizens?

And did any police personnel verify whether there were 91,000 odd mourners in black attire on that Black Sunday, not to say, 91 VIPs or Heads of States who attended the eulogy, depending on one’s definition of a VIP?

 

The Numerology of the Old Guard

One Facebook numerological factoid that circulated in the first post-LKY week was the following:

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Singapore’s political fathers who outlived the biblical three-scores-and-ten lifespan

At face value, these nonagenarians had their blessed lives prolonged up to “four scores and ten and one” years. Sounds like their good earthly or political deeds were good karma for their longetivity? Are they the recipients of the following success equation?

Sacrifice + Service + Incorruptibility + Risk  = Political Success + Longevity

Observe that simply taking the difference between the birth year and the death year of Mr. S Rajaratnam suggests that he died at the age of 91; however, if we look closely at the month dates (Feb. 25, 1915 – Feb. 22, 2006), he was still 90 years old, when he passed away. The same argument goes for Dr. Toh Chin Chye (Dec. 10, 1921 – Feb. 3, 2012), who wasn’t yet 91, when he died. So, always take the pseudoscience of numerology with a grain of salt. As with fengshui charlatans, a degree of skepticism towards numerologists of all sizes and shapes isn’t an option—wear your critical-thinking cap when meeting, or reading about, these paranormal folks!

 

Fortune via Misfortune—From 4D to 5C

(© Unknown) Punters used combinations of the digits related to Mr. Lee death date to lure Lady Luck.

To rational non-punters or non-gamblers, betting on someone’s death date, whether he or she was poor or rich on this side of eternity, seems like an extreme case of bad taste,  or simply showing zero respect for the deceased and their family members. However, in superstitious circles, that practice isn’t uncommon among mathematically challenged or superstitious punters, who think that bad luck paranormally translates into good omen, if they bet on the digits derived from the death date or age of a recently deceased person.

In fact, during the nation’s six-day mourning period for its founder, besides the long queues of those who wanted to pay their last respects to Mr. Lee at the Parliament House, another common sight islandwide were meters-long lines of 4D or TOTO punters, who wanted to cash in on the “lucky digits” to retire prematurely, hoping to lay hold of the traditional 5Cs (cash, car, condo, credit cardcountry club), coveted by hundreds of thousands of materialistic Singaporeans.

 

Number Theory over Numerology

Fengshui in the Gym
(© BBC) Chinese numerology in the gym? Or, is it just a mild form of deification of a political figure?

Instead of promoting a numerological or pseudoscientific gospel based on Mr. Kuan Yew’s death date or age, which only helps to propagate superstition and pseudoscience, perhaps a “mathematically healthy” exercise would be to leverage on the D-day to teach our students and their parents some basic numerical properties—for example, conducting a recreational math session on “Number Theory 101” for secondary  1–4 (or grades 7–10) students might prove more meaningful or fruitful than dabbling in some numerological prestidigitation, or unhealthy divination.

A Search for Patterns

91  is the product of two primes: 91 = 7 × 13

91 = 1² + 2² + 3² + 4² + 5² + 6²

91 is also the sum of three squares: 1² + 3² + 9²

Are there other ways of writing the number 91 as a sum of squares?

91 = 33 + 43

 

Non-Numerological Questions to Promote Problem-Solving Skills

Let’s look at an “inauspicious number” of elementary- and middle-school (primary 5–secondary 4) math questions, which could help promote numeracy rather than numerology among students and teachers.

1. Sum of Integers

Show that the number 91 may be represented as the sum of consecutive whole numbers. In how many ways can this be done?

2. The Recurring Decimal

What fraction represents the recurring decimal 0.919191…?

3. Palindromic in Base n

For what base(s) will the decimal number 91 be a palindromic number (a number that reads the same when its digits are reversed)? For example, 91 = 101013.

4. The Billion Heartbeat

Does a 91-year-lifespan last less or more than a billion heartbeats?

5. Day of the Week

Mr. Lee Kuan Yew (September 16, 1923–March 23, 2015) died on a Monday. Using the 28-year cycle of the Gregorian calendar, which day of the week was he born?

6. One Equation, Two Variables

If x and y are integers, how many solutions does the equation x² – y² = 91 have?

7. Singapore’s New Orchid

A new orchid—Singapore’s national flower—had been named after Mr. Lee: Aranda Lee Kuan YewUsing the code A = x, B = x + 1, C = x + 2, …, , does there exist an integer x such that ARANDA sums up to 91? In other words, does there exist a numerological system such that  A + R + A + N + D + A = 91?

8. Singapore’s Coin Goes Octal

Singapore's "lucky" octagonal one-dollar coin
The alleged involvement of Mr. Lee in Singapore’s “lucky” octagonal one-dollar coin

There is an apocryphal story that had circulated for many years linking Mr. Lee Kuan Yew with Singapore’s octagonal one-dollar coin. A high-ranking monk had apparently told Mr. Lee that Singapore’s fortune would continue to rise only if Singaporeans were to carry a bagua—the eight-sided fengshui symbol. That prediction allegedly prompted the Monetary Authority of Singapore to issue the octagonal shape of the nation’s one-dollar coin.

That rumor was later put to rest by no other than self-declared agnostic Mr. Lee himself in one of his books, Hard Truths. He remarked that he had zero faith in horoscopes, much less the pseudoscience of fengshui.

What is the sum of the interior angles of the Singapore’s eight-sided coin?

9. Show that the largest number k for which the decimal expansion of 2k does not contain the digit 1 is 91.

© Yan Kow Cheong, April 26, 2015.

Resurrection isn't an option in Singapore!
Resurrection isn’t an option in Singapore!

Selected Answers/Hints

1. One example is 91 = 1 + 2 + 3 +⋯+ 13.
2. 91/99.
5. Mr. Lee was born on a Sunday.
6. Hint: Show that x² – y² = 91 has 8 integer solutions.
9. Hint: Use a computer to verify the result.

Life’s Simple Mathematical Pleasures

Stealing the idea from Nancy Vu’s “Just Little Things,” I tried to reflect on some mathematical things that give us delight and pleasure as we go about our hectic 24/7/365 lives.

Here are some of my personal mathematical pleasures.

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It’s now your turn to reflect on some aha! mathematical moments  to share with the rest of us.

References

Wu, N. (2013). Just little things: A celebration of life’s simple pleasures. New York: A Perigee Book.

JustLittleThings.net

© Yan Kow Cheong, November 10, 2013.

Mathematical Fiction Is Not Optional

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“The Parrot’s Theorem” (translated from “Le Théorème du Perroquet”) was an instant bestseller in France when it was published in 1998.

Sylvia Nasar’s A Beautiful Mind and G. H. Hardy’s A Mathematician Apology are two nonfiction mathematical classics for both mathematicians and mathematics educators. Lesser known are the mathematical novels which often feature characters whose speciality is number theory, also known as higher arithmetic, and elevated as math’s purest abstract branch.

Mathematics à la Tom Clancy

Two novels that revolve around famous unsolved problems in mathematics are Philibert Schogt’s The Wild Numbers and Apostolis Doxiadis’s Uncle Petros & Goldbach’s Conjecture.

If you’re looking for math, women, sex, and back-stabbing, The Wild Numbers is a math melodrama unlikely to disappoint.

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Winner of the New South Wales Premier’s Prize

 

Fictional math

Who are these mathematical fiction books targeted? Math and science teachers? Educated laypersons? Pure mathematicians may like to read them, yet at the same time they may complain that the mathematics discussed in these books is anything but rigorous.

These books seldom fail to convey the following subtle messages:

• The thin line between mathematical genius and madness.

• The search for mathematical truth at all costs, and the heavy price of finding it.

• The arrogance and pride of pure mathematicians who look down on their peers, most of whom work as applied mathematicians and research scientists.

• The relatively high divorce rate among first-rate mathematicians as compared to their peers in other disciplines.

• Mathematics is apparently a young’s man game; one has past one’s prime if one hasn’t written one’s best paper by the age of 40.

• Mathematicians are from Mars; math educators are from Venus.

• Pure mathematicians (or number theorists) are first-rate mathematicians; applied mathematicians are second- or third-rate mathematicians. To the left of the “mathematical intelligence” bell curve are math educators from schools of education, and high-school math teachers.

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“Reality Conditions” is collection of 16 short stories, which is ideal for leisure reading—it’s suitable for promoting quantitative literacy, or it’d serve as the basis for a creative course on “Mathematics in Fiction.”

The joy of reading mathematics

Let’s rekindle the joy of appreciating mathematics for mathematics’s sake. Let’s welcome poetry, design thinking, and creativity, whatever ingredient that may help to draw the community into recognizing and appreciating the language of science and of technology. These “pure-math-for-poets” titles have a place in our mathematics curriculum, as they could help promote the humanistic element of mathematics.

Here are ten titles you may wish to introduce to your students, as part of a mathematics appreciation or enrichment course.

The New York Times Book of Mathematics

The Best Writing on Mathematics 2010 

Clifton Fadiman’s Fantasia Mathematica

Clifton Fadiman’s The Mathematical Magpie

Don DeLillo’s Ratner’s Star

Edwin A. Abbott’s Flatland

Hiroshi Yuki’s Math Girls

John Green’s An Abundance of Katherines

Philip J. Davis’s The Thread: A Mathematical Yarn

Thomas Pynchon’s Gravity’s Rainbow

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Juvenile fiction—A child prodigy and his friend tried to create a mathematical formula to explain his love relationships.

References

Green, J. (2006). An abundance of Katherines. New York: Dutton Books.

Guedj, D. (2000). The parrot’s theorem. London: Orion Books Ltd.

Kolata, G. & Hoffman, P. (eds.) (2013). The New York Times book of mathematics: More than 100 years of writing by the numbers. New York: Sterling.

Hiroshi, Y. (2011). Math girls. Austin, Texas: Bento Books.

Pitici, M. (ed.) (2011). The best writing on mathematics 2010. Princeton, New Jersey: Princeton University Press.

Wallace, D. F. (2012). Both flesh and not: Essays. New York: Little, Brown and Company.

Woolfe, S. (1996). Leaning towards infinity: A novel. NSW, Australia: Random House Australia Pty Ltd.

© Yan Kow Cheong, September 12, 2013.

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Murderous math that doesn’t kill!

Can Math Be Sexy?

One thing is almost certain is that if Danica McKellar, actress and mathematician-turned-math-author, were to write a book on Singapore math, my bet is that it would unlikely be approved by the Singapore’s Ministry of Education (MOE), although few would deny that her title would probably be a terrific draw among local students, even in conservative or puritan Singapore—it may even end up being the first math book on a Singapore bestseller’s list.

Going by her math titles (Kiss my mathMath doesn’t sulk; and Hot algebra exposed!), which are primarily targeted at an American or liberal audience, should the TV personality be tempted to write a math book for a local audience, it would be a miracle if her manuscript got pass the first round of Singapore’s MOE’s censorship board—not before making most MOE curriculum specialists flush of embarrassment.

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Would you buy this math title for your son or daughter?

Infidel Math

Indeed, the Taliban and the Ayatollahs wouldn’t approve Danica McKellar’s irreverent titles, not to say, her irreligious style of writing, but what about “moderate” nations like Singapore, which unspokenly or secretly longs to be portrayed as a conservative society with “high moral standards”? McKellar’s math books are anything but boring! In fact, she thinks that math can be easy, relevant, and even glamorous. By adding a little glamour and humor to the teaching of mathematics, it looks like the math advocate has, to a large extent, demonstrated that math can actually be accessible to young girls—and young boys, too!

 

A Math Role Model for Girls (and Boys)

In math education circles, it’s not surprising that Danica McKellar is regarded by many [open-minded] parents and math educators as a terrific role model [for girls and young women]. She teaches the value of confidence that comes from feeling [math-]smart. Her supporters think that her books should be required reading for every math-anxious school girl! The message seems to be that physical beauty and quantitative literacy need’t be mutually exclusive.

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Would you give away this irreverent guide about pre-algebra to your neighbor’s son?

From Boyfriend to Babysitting

A random look at some chapters of Math Doesn’t Sulk needs no explanation why teenage girls wouldn’t want to give math a serious try, or a second chance, especially if they want to appear smart and beautiful, to their boyfriends. Would you put down a math title with these chapter headings?

 

Chapter 1: How to Make a Killing on eBay (Prime Numbers and Prime Factorization)

Chapter 2: Do You Still Have a Crush on Him? (Finding the Greatest Common Factor)

Chapter 7: Is Your Sister Trying to Cheat You Out of Your Fair Share? (Comparing Fractions)

Chapter 11: Why Calculators Would Make Terrible Boyfriends (Converting Fractions and Mixed Numbers to Decimals)

Chapter 12: How to Entertain Yourself while Babysitting a Devil Child (Converting Decimals to Fractions) 

Math Doesn’t Sulk also comes with a math horoscope, math personality quizzes, and real-life testimonials. What else more can one expect from a math book? In fact, Danica herself exemplifies her own life from being a terrified middle-school math student to a confident actress, and more.

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X-rated algebra with a dose of irreverence and humor

Girls get curves: Geometry takes shape

I’m currently looking at a copy of McKellar’s latest publication, Girls get curves; personally, I think it’s the most useful of all her publications so far, as it covers a number of middle- and high-school topics, such as congruency, similarity, and proofs, which are relevant to my teaching and writing. My wild guess is that her next title would be one on Calculus and Trigonometry!
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Learn some proofs with Danica as your personal tutor and coach!

Don’t judge a book by its irreverent title!

Suspend your judgement for a while, even if McKellar’s irreverent titles make you feel a bit squeamish or uneasy. Who knows? You may end up learning a thing or two about some long-forgotten, or decades-old avoided, math.

 

Personally, the style of writing of these pop math books is enough to arouse my interest, leave aside the math, which somewhat lacks rigor, as compared to the standard expected of Singapore math students at the same grades. For example, in Singapore,  geometric proof and trigonometry are formally covered at grade 9 or 10. However, McKellar’s  informal and conversational writing style could help us loosen our often-stiff mathematics writing, which has traditionally plagued most Singapore-published boring textbooks, as they go through the “rigorous” (or tedious?) process of MOE’s standards of quality and morality.

References

Ho, S. T., Khor, N. H. & Yan, K. C. (2013). Additional Maths 360. Singapore: Marshall Cavendish Education.

McKellar, D. (2012). Girls get curves: Geometry takes shape. New York: Hudson Street Press.

McKellar, D. (2011). Hot X: Algebra exposed! New York: Plume.

McKellar, D. (2008). Kiss my math: Showing pre-algebra who’s boss. New York: Hudson Street Press.

McKelkar, D. (2007). Math doesn’t suck: How to survive middle school math without losing your mind or breaking a nail. New York: Plume.

For alignment with state and NCTM math standards, visit www.mathdoesntsuck.com/standards

© Yan Kow Cheong, April 20, 2013.