MATH and MUSIC

Here are some random thoughts between mathematics and music―indeed, there are more parallelisms than thinks the brain! 

Bestsellers for publishers are: 
Music for your unborn baby! 
Problem solving for your unborn child! 

Are these mere fads? The Mozart effect and the 'Aha!' feeling 

The centuries-old debate: Why are there more mathematicians who are musicians than musicians who are mathematicians? The ratio is probably a million to one!

The-music-of-the-spheres doctrine prompted astronomers, physicists, and mathematicians to study (read) mathematics to better understand the workings of the universe.
Pythagoras' discovery that the musical notes can be expressed in integral (integers) ratios prompted him to start hypothesizing that "Everything is number!" 

Music is soothing to the ear, mind, and soul; mathematics is like a balm to the mind's eye. In this sense, music can reach out to a larger audience than math, but mathematics seems more selective in revealing its charm and beauty to mankind.

Mathematics often acts as a filter to separate the nerd from the herd; music provides a platform to bring people from different socioeconomic backgrounds together. Music unifies people, mathematics alienates them.

You don't need to know the mechanics of musical notes to appreciate music; you don't need to know abstract mathematics to appreciate the beauty and power of a mathematical theorem (or result).

The mathematics of music (technical, boring, ...) and the music of mathematics (poetic, elegant, certain, ...)

The Mozart effect and the visuospatial skills: is it just a marketing fad to tap on the fear and greed of consumers or parents? There is no concrete correlation or proof that listening to music helps one improve one's visualization skills in the long term.

Let's appreciate music for music's sake, and let's enjoy mathematics for mathematics' sake!

Mathematics is like music to our sixth sense; music is like mathematics to our spiritual sense.

Reading and doing mathematics is Greek to the lay person; speaking and singing in tongues sounds alien to the unbeliever.

God is a Great Mathematician; God ever Geometrizes! The most beautiful theorems are in His Great Book! 
God is the Perfect Musician. The psalmist sings and praises to the Lord! Music soothes the soul or spirit———―King Saul listening to David playing the flute! 

Let those who have ears hear and listen; let those who can think quantify and enumerate!

Let the angels sing! Let the angles cross!
...

© Yan Kow Cheong, April 1, 2010

The Lighter Side of Singapore Math (Part 1)

The Greeks have their Euclid’s Elements.

The Egyptians had their Rhind papyrus.

The Japanese had their sangaku.

The Chinese had their Chiu-chang Suan-shu.

What do the Singaporeans have?

The Model Method

Q: If the US math curriculum is ‘a mile wide and an inch deep,’ what about the Singapore curriculum? 

A: The Singapore curriculum is ‘a smile wide and a glitch deep.’

Some politically incorrect terms defined

Singapore: 1. A 'fine' city with the most number of restrictions and prohibitions, and a record number of supplementary (assessment) mathematics titles per hundred thousand people. 2. A city-nation with the highest number of drill-and-practice specialists per capita.

Math teacher: A complaining lot whose number has increased in the aftermath of Y2K and 9/11.

Model method: 1. An overrated and oversold approach to solving word problems. 2. Visualization attaining brand status. 3. An often-misused and abused anti-algebraic heuristic tool to tackling challenging word problems traditionally set in grades 7 to 10 to terrorize elementary grade students.

Number bond: A high-tech word to mean breaking a number into smaller numbers.

Back-to-Basics: Basic numerical skills to avoid being conned by shopkeepers, contractors, and timesharing agents.

Kumon Math franchises: The MacDonalds of Mathematics Education – the 'McDonald’s of Rote Learning.'

[Japanese] Sakamoto Math: Singapore's model method of using lines instead of bars or rectangles.

Ig Nobel Peace Prize for Singapore Prime Minister

Singapore's ban on the import of chewing gum, and others like feeding pigeons, spiting, begging in public, and failing to flush the toilet won the 1994 Ig Nobel Peace Prize for former prime Minister Lee Kuan Yew.

A Quick-and-Dirty Comparison between Singapore and USA

What appears as ‘critical or creative math questions’ for Americans are often routine questions for Singaporeans.

American textbooks offer even or odd answers, while Singapore textbooks provide almost all answers and solutions, free of charge.

Singapore may be the math envy of the US, based on its ranking at the TIMSS, but Singapore teachers long for the opportunity to be as creative as their US counterparts. Besides, Singapore teachers dream of the shorter teaching hours, and lesser dependency on grades.


Origin of Singapore Math

SINGAPORE MATH = China Math ∩ Russia Math ∩ Israel Math ∩ US Math ∩ ...

What's STEM for Singapore?

Not Science, Technology, Engineering, and Mathematics, but Strategies, Tuitions, Examinations, and Marks!

© Yan Kow Cheong, April 1, 2010

Bad Algebra and the Double Negative

In algebra, two negatives make a positive; in English, two negatives cancel each other – or are equivalent to an affirmative.

Expressions like 'The middle class didn’t get nothing from the government' or 'Nobody told me nothing' are regarded as illiterate in current English, and they’d be avoided. Yet, arguably, using two negatives (like not or no or never) helps make the negation stronger:

Math editors cannot not write.

Math writers cannot not plagiarize.

In 18th century England, to standardize the English language, language experts believed that multiple negatives were illogical. Using algebra as an analogy, they claimed that two negatives make a positive, and disallowed speaking styles like 'She don’t know nothing' or 'He don’t never look at me.' The argument was neither good logic nor good algebra; in algebra, -x plus -x equals -2x, not 2x. The two negatives equal two negatives. So, double negatives are no-no's.

Logical or not, the rule has been a success, and double negatives like 'I don’t know nothing about it' tend to label a speaker as unrefined. People with a little knowledge pounce on a double negative as self-righteously as they decry a split infinitive. As a result, one seldom sees expressions involving not and clear negatives like nothing or nobody in writing.

“You ain’t seen nothin’ yet!” Ronald Reagan US President

Basically, double negatives aren’t wrong, and some languages use repeated negatives to strengthen the effect. The logic had since changed, and a sequence of negatives came to be regarded as self-canceling instead of reinforcing each other.

Just like there’s almost an exception to every rule, a double negative can be a positive statement – it can be useful as a way of emphasizing a point, as in this sentence:

The managing editor was not unaware of the plagiarism.

At the least, a double negative is acceptable when it’s used as a kind of figure of speech with intentional canceling effect, as in It has not gone unnoticed (= It most certainly has been noticed). In fact, a double negative is an effective way to make a mildly positive statement. For example, instead of undermining the creativity of an editor, a critic may write that she 'is not untalented.'

Here are some double negatives which are actually positive statements:

Double negative: He doesn’t dislike her.
Positive meaning: He likes her (Well, sort of …).

Double negative: We cannot pretend and do nothing.
Positive meaning: We must do something.

Neither-Nor

The double negative, neither-nor, defies the so-called logic that two negatives cancel each other. We don’t say: 'Neither substitution or elimination can solve the simultaneous linear equations,' avoiding a double negative. In standard English we use two negatives, with no idea that they cancel each other: 'Neither substitution nor elimination can solve the simultaneous linear equations.'

Beware of words like barely, hardly and scarcely, 

which have negative meaning.

We’re also less consistent with negatives like hardly or scarcely. It’s uncommon to hear

I can’t hardly recall the formula.

I don’t hardly think I should.

Standard usage requires cutting one of the negatives:

I can hardly recall the formula.

I hardly think I should or I don’t think I should.

When but has a negative meaning, there is still more confusion, even though standard usage dictates a single negative:

 I can’t help but imagine the worst. 

I can’t help imagining the worst. 

I don’t doubt but that she means well. 

I don’t doubt that she means well.

That double negative is still popular today although language gurus urge us not to use it. If we say that 'Saddam is not entirely dictatorial,' we leave very little for the positive, only whatever small amount of non-dictatorship between entirely and not entirely. If we say that Hitler is 'not half bad,' we may be allowing at least 50 percent on the positive side. If I say that Mao’s views on democracy are 'hardly without prejudice,' I am certainly accusing him of some prejudice, but the negative approach seems to make feel safer from libel suits.

Although the double negative is considered ungrammatical in good writing, however, as we saw earlier, it can serve as an effective way to make a positive statement. At the worst, we may be judged as being uncivilized if we misuse the double negative – we may not impress positively on the reader or listener.

Let me end by saying that it’s not unthinkable that in a not uneducated move in a not too distant future, no small number of editors will be revealing their not unhidden talents to the not unchallenging task of freeing non-uneditable manuscripts of double negatives.

Non-negatively yours

Practice

Which of the following are incorrect?

(a) I wouldn’t be surprised if she didn’t show up unannounced.

(b) The men unloosened their ties.

(c) We will go irregardless of the weather.
(d) Not for nothing was Mathematics called 'the Queen of Sciences.'
(e) I am a friend of him who is an enemy of my enemy.
(f) Don’t give me no wrong answers.

(g) She doesn’t need no tuitions.

(h) We didn’t write nothing.

(i) I didn’t know hardly anybody who didn’t want me to edit.

References

Allen, Robert (2005). How to Write Better English. Penguin Books.

Dolainski, Stephen (2004). Grammar Traps: A Handbook of the 20 Most Common Grammar Mistakes And How o Avoid Them. Toluca Lake, CA: Paragraph Publishers.

Gorrell, Robert (1994). Watch Your Language!: Mother Tongue and Her Wayward Children. Reno, Nevada: University of Nevada Press.

© Yan Kow Cheong, March 24 2010.

Mathematics Year in Singapore

Mathematics Year in Singapore

In February 2009, I e-mailed the following text, to be considered for publication in the Forum Pages of The Straits Times. 

The year 2008 has been declared “Mathematics Year” in Germany. In fact, since 2000, Germany has dedicated each year to one particular science. Riding on the good standing and ranking of Singapore’s performance in mathematics globally, such as its triple firsts in “Trends in International Mathematics and Science Study” (TIMSS), its decent gold tally at mathematics competitions in recent years, and the State of California’s official endorsement of Singapore textbooks, I hope the Ministry of Education and its partners would seriously consider emulating its German counterparts, by celebrating Singapore’s “Mathematics Year” in 2010 to coincide with the 2010 Youth Olympic Games.

Besides providing ample opportunity for both locals and foreign guests to witness the beauty, power and utility of the often-disliked subject in school, celebrating Mathematics Year 2010 would also enhance Singapore’s image as being more than just a haven of assessment maths titles and a paradise for maths tutors.  

Like hosting the Youth Olympics, the experiences gained in planning and organising this first Asian “Mathematics Year” would not only strengthen Singapore’s position on the Mathematical World Map, but also prepare her

to help others in future as they too celebrate their year-long events.

So, let’s make it a national effort to humanise the multi-faceted aspects of mathematics (mathematics as an art, a language, a science, and so on) and to promote the “Speaking Mathematics” campaign – that speaking the language of science and of technology holds the key to maintaining Singapore’s competitive edge in the region.

Just as most of us have the capacity to enjoy music and the arts, everyone has some capacity for mathematics appreciation. Mathematics as a way of thinking (and of life) is not reserved for a blessed few, but is accessible to

all those who are curious in exploring the common human potential.

Think of the Internet, iPod, digital camera, black hole, MRI, and GPS, just to cite a few by-products from harnessing the power of mathematics. And declaring the Year of Mathematics 2010 in Singapore is a first step to enhancing our quantitative literacy. So, let’s count on mathematics for Mathematics Counts!

Yan Kow Cheong

353 words

I received a pre-formulated reply from a Malay lady that since they receive over a hundred readers' e-mails every day, they suggested that I channeled mine to the appropriate ministries for consideration instead. 

I'd have probabilistically received a few more of those standard replies had I decided to contact the relevant ministries. Next, I decided to e-mail it to the morning free paper, Today. This time round, the disappointment was greater for I didn't even receive an acknowledgement that they ever received, much less read, my e-mail.

Without losing hope for a future Mathematics Year, let's look at five lessons we could have learned had we celebrated the Year of Mathematics 2010 in Singapore.        
                                                                                    
1. Mathematics is many-sided.

Math is more than “learning to calculate” or "solving challenging word problems." Math is an art, a science, a language, a tree, a religion, a way of thinking, or a way of life.

2. Math is hard and difficult.

Math is only easy for math teachers and for those who know little math. It's not hard, it's damn hard! If it weren't so, it wouldn’t appeal much to the nerds. 

3. Don’t oversell or overvalue it.

There’s no guarantee that people will know more math at the end of the term or year. If people can appreciate or value its power as a language of science and of technology, it's a bonus for math educators.

4. People matter more than proofs.

Although the subject is abstract, the gist of a powerful or beautiful mathematical result can be shared to an educated or numerate public if there are more mathematically inclined media people to share the good mathematical news: they can write about new mathematics by talking to those in the ivory towers.  

5. Make it an island-wide effort.

Invite people from all walks of life to share how they use math in their daily life — how math plays an important part in the success of their careers and lives, or how the advancement of technology (web, e-learning, wireless communications, space travel, etc.) is dependent on the mastery of advanced mathematics. 

Inarguably, the benefits to celebrating the Year of Mathematics 2010 would have been manifold to tens of thousands of citizens and residents, to the community, and to the country. What a missed opportunity to coincide "Singapore's first Mathematics Year" with the first Youth Olympics 2010!

© Yan Kow Cheong, March 18, 2010

13 Ways to Spot a Math Nerd

Thirteen Ways to Spot a Math Nerd 


1. They are aroused at the sight and sound of numbers. 


2. They wait for the bus and say, "I bet three buses will come all at one go."

3. They have irregular-sized pizza delivered to the mathematics or computer science department. 


4. Their idea of rebellion is to go to the whiteboard and compute everything in binary, or do calculations in Roman numerals. 


5. They sleepwalk uttering obscenities at Greek symbols, or murmuring in Latin.


6. They send e-mails to each other even though their desks are less than two meters away.


7. They refer to their friends, fiends or foes by their e-mail coordinates.


8. They explain to their five-year-old neighbors the difference between number and numeral, minus and negative, or zero and nothing.


9. They see a woman's boobs and think of a smooth polynomial curve, with its pair of minimum points.


10. They insist that "1 + 1 = 2" is not true until they use the axioms (undefined terms) to prove it.


11. They say, "I'll see you there at 7PM plus or minus 10 minutes."


12. They use insults like "You’re two standard deviations from the norm, you symbol-minded dummy!"


13. They prefer doing math to dating, dining or having sex.
© Yan Kow Cheong, March 17, 2010 

Immoral Algebra

Immoral ALGEBRA

Here is an online algebra question, forwarded to me almost every year, which has been circulating the globe many times:

Maya is 21 years older than her son Raju.

In 6 years from now Maya will be 5 times as old as Raju.

Question: Where's Maya's husband?

My reply to the sender was: My limited algebraic knowledge leads me to infer that the "normal-child-to-be" was some one year away to see the world (Answer: x = –1). Challenge: Try solving this problem using the Singapore model method.

(

So I can only conclude that the "conservative couple" was in some "missionary position" then.

My sender's forwarded answer was: 

Current age of Raju is -0.75 ie -9 months ie he is 

being conceived therefore husband is on top of wife

And my reply following the e-mailer's answer was:

 

I bet is that if we've a few of these CRE8TIVE word problems in our traditionally sterile MOE-approved textbooks, more students would be excited to reading algebra until they're old enough to have kids of their own. Students would be begging teachers and tutors for more algebra and more word problems – two bêtes-noires of school mathematics. Who knows? This may turn out to be an inexpensive cure to addressing our declining birthrate! When most students' favorite four-letter word would then be MATH MATH MATH ... and MORE MORE MORE!

Stripping Off Your Maths

The above online algebra word problem reminds me of a math workshop organized by the Singapore Science Centre in late eighties, when the speaker-and-author, Mr Ang Tok Woon, shared with fellow teachers that a good metaphor could help weak or uninterested students recall mathematical concepts or formulas rather effectively.

He used the idea of ‘stripping off’ some unknowns when teaching the topic on "Subject of Formula" to a group of academically challenged [ITE or SAF mature] students. For instance, given E = mc², ‘making c the subject of formula’ means expressing c in terms of E and m. And he was surprised that years later when some of his ex-students met him, they reminded him that they had forgotten everything else he taught them except for that part on stripping off.

Sexing Up the “Sixth Factor”

Only a lack of imagination and creativity would deprive hundreds of thousands of pubescent teenagers from experiencing the joys (and pains) of algebra.

As we explore how best to incorporate the sixth factor — the ‘context knowledge’ — as suggested by Assoc. Prof. Wong Khoon Yoong from the National Institute of Education, Singapore, to extend the present two-decades-old Singapore Mathematics Curriculum Framework comprising of the five factors (Skills, Attitudes, Concepts, Processes and Metacognition), diagrammatically placed along the sides of a pentagon, I think posing ‘real-world problems’ with a recreational or cultural flavor would go a long way to arousing the algebraic interests of our students who might otherwise give algebra a miss, once it becomes an optional topic. 

Indeed, we need a fertile dosage of algebra to debunk the myth that the common language of mathematics is difficult and boring. Algebra, or any topic in mathematics, for that matter, is amoral and neutral. Mathematics can only be boring as long as we’ve boring teachers and writers (and editors, math consultants and math specialists) — or all five — churning out unexciting textbooks. However, matters would have been a notch above boring if we had had fewer boring reviewers and curriculum specialists.

Some Nudity May Promote Numeracy 

Another episode proves that infusing some perceived elements of decent nudity – art for mathematics’ sake – to enhance numeracy (quantitative literacy, as it is commonly known in the US) may not be a far-fetched idea. Years ago I was attending a two-day teachers’ course entitled Andragogical Principles, an in-service program for those teaching working adults or mature students, when the lecturer confided to us that by letting his students write about their [wild?] fantasies, they actually improved their language proficiencies. In fact, I wouldn’t be surprised that some of them are now working as freelance writers or editors for some adult publications!

Indeed, numeracy spiced with some exciting part of human biology may act as a catalyst (‘mathematical aphrodisiac’) to attracting even mathematically dull students to sign up for some extra-mathematical activities to enhance their quantitative literacy. As mathematics educators, we critically need creative methods to promote our mathematical trade; otherwise, we’ll end up losing potential candidates with a mind for the abstract to major in other disciplines that often promise more tangible rewards.

MATH$ I$ MONEY

Like guerrilla marketers, we need to go all out to promote and sell the idea that MATHEMATIC$ I$ MONEY! Some born with the ‘mathematical gene’ definitely stand a high probability of becoming millionaires, by cracking some unsolved mathematical nuts – for example, a few million-dollar prizes are on offer by the Clay Mathematics Institute to whoever can find the solution to some fiendishly difficult mathematical puzzles. For the majority of us, we can at least hope to secure some decent jobs thanks to the logical, analytical and deductive skills acquired through years of toying around with notions and notations.

If the oldest profession on earth continues to thrive because of the money factor, there is no reason why it cannot share a common denominator with the most disliked subject in school. With some financial carrots, more people would be motivated to speak the language of science and technology. Like sex, mathematics is money — and it can be enjoyable, too!

MATHESAURUS, Anyone?

For a few years now, I've secretly been writing mathematical words and terms, to be categorized under a so-called MATHESAURUS. I wanted to christen the title A Politically Correct Mathematics Glossary or A Quick-and-Dirty Mathesaurus Dictionary.
 
Offhand, here's a sample of my irreverent definitions:

• Abacus A primitive calculator that questionably originates from China. The Japanese version, called the Soroban, claims to be able to perform square and cube roots, as well as differentiation and integration. Unlike the calculator, its batteries last a lifetime. 

• Actuary One who misuses or abuses statistics to justify the high premiums charged by one's insurance company.

• AMC Abbreviation for “Australian Mathematical Competitions.” The world’s popular competition, AMC caters for those mathletes who do not qualify for the IMO, but hope to be medalists, nonetheless. See IMO.

• Boolean Algebra The Arithmetic of “Or” and “And.”

• Coordinates A way of describing a location on a grid. For instance, the author's e-mail coordinates are: kcyan.mathplus@gmail.com Some common daily use of coordinates are: (a) A cab driver looking for (5, 8), 5th Avenue and 8th Street; (b) Finding a seat in a theatre, say L25 (Row L, Number 25), and the like.

• Mathophobia Anonymous A gathering of people who pride themselves of their mathematical ineptitude. The members meet clandestinely to avoid being overcharged and conned by unscrupulous agents. 

• Quadratic Equation A misnomer for naming an equation of degree 2; should be renamed “biratic equation.”

• Trial-and-Error Method A hit-and-miss approach. The mathematical equivalent for looking for a word in the dictionary.

• World Wide Web (WWW) Comprises virtual cities of bits; the world of 1s and 0s. A city of bits. A communications medium that encourages gossip, freedom of speech, and gambling; sale of pink pills, yellow literature, blue movies, and so on.

• NCTM’s Standards An ideal list of guidelines and recommendations for mathematical excellence in K-12 by the National Council of Teachers of Mathematics, to be implemented in a non-ideal classroom.

• One-to-one correspondence The most primitive and also the most sophisticated method of counting. Used by farmers to count sheep, and by mathematicians to study different sizes of infinities and to come up with some visual (look-see) proofs.

Would you, readers, be keen to co-author with me the above irreverent guide to relevant mathematical terms? Any takers (friends, fiends or foes) who would be excited by such a general math-lite title? Please direct them to me. 

I couldn't be more serious about this funny title. I think it's going to be exciting to co-write and publish such a MATHESAURUS.

Irreverently yours
K C Yan 

© Yan Kow Cheong, March 11, 2010