“Gods” make Comeback at Toyota

is the headline of this bloomberg piece that reports the strange phenomena that human beings are replacing robots in the manufacturing lines. The point is that by automating some of these production process, much of the skill learnt by the human worker is lost. Without these knowledge, the humans cannot innovate and improve on the current process.

I see a close parallel between this and how students are losing their mathematical/computational ability because they are doing all their calculations on a calculator. Sure, if all the mathematics that you will do in future is to add some numbers on your bills, then by all means do that. But if you hope to develop a more mathematical/algebraic thinking then the hours of doing addition and multiplication sums are not something you should dispense with.

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Come Together

is not the beatles song but the title of episode 16 of a tv program called discover science. I happened to catch it on local tv and the combination of quirky Japanese humour and the clever introduction to mathematics and science caught my attention. A preview of the episode is available at the above link but you need to watch the whole episode to find the link to parabolas.

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Mathematics and poetry

A nice article by Ornes on slate and a paper by Glaz.

My favourites has to be the Fermat Last Theorem Poetry Challenge which I have read about previously elsewhere. Here’s one entry by
E. Howe, H. Lenstra, D. Moulton:
“My butter, garcon, is writ large in!”
a diner was heard to be chargin’.
“I HAD to write there,”
exclaimed waiter Pierre,
“I couldn’t find room in the margarine.”

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Maths & Fashion

That headline caught my eye and earned the newspaper article a more detailed look. It’s about a sub brand from Issey Miyake called 132.5, which apparently stands for 1 piece of material which forms a 3d dress but yet can be folded flat into 2d. The meaning of the .5 wasn’t that clear to me.

Fashion’s not my thing but this is certainly cool enough to merit a link.

http://www.isseymiyake.com/en/brands/132_5.html

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Advice to Young Mathematicians

The Princeton Companion to Mathematics contains a short section of sound advice to young mathematicans by Sir Micheal Atiyah and others. It is certainly worth reading.
The section can be downloaded from here.

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Quadratic reciprocity

– a proof due to Zolotarev and its connection to dealing cards.

A list of 240 and counting published proofs of the theorema aureum.

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Of word problems and place values

If you google “Abbott & Costello 7 x 13 = 28″, you can get a funny sketch of how to show 7 x 13 = 28 by dividing, multiplying and adding. There are a couple of versions, and in one of them, Abbott ask Costello

A: “did you ever go to school stupid?”
C: “Yes sir, and I came out the same way!”

Another classic one goes like this

A captain owns 26 sheep and 10 goats. How old is the captain?

According to the book Making Sense of Word Problems by Verschaffel, Greer and de Corte, children from different parts of the world have actually offered answers to the question.

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The broken stick problem

A slick solution to the broken stick problem or otherwise known as the uncooked spaghetti problem

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That’s an emphatic NO!

all 49% of it.

Seen via math-fail.com
I was curious about the source and google pointed me here.

To restore our sanity, here’s a neat problem from the 2012 AMC 8:
Let R be a set of 9 distinct integers. Six of the elements are 2, 3, 4, 6, 9 and 14. What is the number of possible values of the median of R.

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Polynomial Puzzle and Lill’s Method

I have in mind a polynomial with nonnegative integer coefficients. Can you determine completely the polynomial by just asking two questions?

I forgot where I saw this from but it was interesting enough to spur a few lunch time discussions with colleagues. I finally saw a reference in Dan Kalman’s book “Polynomia and Related Realms.” It was asked by I. B. Keene on the College Mathematics Journal, vol 36 (2005) page 100 and answered on page 159.

2005? I suspect the problem is much older than that.

Kalman’s book is a gem. In the first few pages alone, I learned about the Horner form of a polynomial and the fascinating Lill’s method for geometric visualization of the real roots of a polynomial.

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Mathematical Induction

Apparently the French terminology for mathematical induction is “raisonnement par recurrence” or reasoning by recurrence. That makes so much more sense.

According to David Reid and Christine Knipping*

“Mathematical induction” is based on deductive reasoning, not inductive reasoning. The confusing terminology comes about because reasoning by recurrence makes use of specific cases, and as we saw above inductive reasoning is sometimes defined as reasoning beginning from specific cases. Reasoning by recurrence, however, also makes use of a general rule …”

Although, I have doubts whether the typical local student has any clear idea about what is deduction and what is induction.

*Proof in Mathematics Education, Research, Learning and Teaching. Sense Publishers, The Netherlands, 2010. (page 99)

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Ranking Journals

has always been a task destined for controversy. Most academics have an implicit ranking of journals in their heads. When considering where to publish, the first consideration is usually appropriateness of the journal, followed by publishing in as good a journal as their paper is worthy of. Because of inherent differences in opinions, an explicit listing of journals is always subject to criticism. On the other hand, administrators want an explicit list so that they can rank academics according on a scorecard.

The only explicit and open list that I know of is the one published by the Australian Mathematics Society. My former university also had an internal distribution only list for the purpose of work review. The European Society for Research in Mathematics Education (ERME) has recently announced a ranking (of 17 journals) of mathematics education journal.

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A combinatorialist worth his salt

Just as every analyst automatically interchanges the order of integration, and every combinatorialist worth his salt double counts at the drop of a hat, Erdös was constantly on the lookout for opportunities to apply the probabilistic method. [emphasis mine]

Taken from Bollabas’ article on Paul Erdös. A few months late but still, happy 100th birthday!

Source: Bollabas, B. To Prove and Conjecture: Paul Erdos and His Mathematics,
The American Mathematical Monthly, Vol. 105, No. 3 (Mar., 1998), pp. 209-237.

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The beauty of number theory

The realization that she is posing a problem, or working on one, that Zeno or Euler could have posed might stimulate in a student an aesthetic motivation much more profound than the purported motivation factor associated with the “real-life” problems populating so many textbooks and curricula. (p.80-81)

Source: Sinclair, N. (2006). For the beauty of number theory. In R. Zazkis and S. Campbell (Eds). Number Theory in Mathematics Education: Perspectives and prospects.Lawrence Erlbaum Associates, Inc.

A very nice java applet called the colour calculator that is written by Sinclair can be accessed here

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I have trouble finding the mathematics

Two unrelated incidents. First, I attended a talk by Sergiy Klymchuk who talked about misconceptions in mathematics, especially in calculus. He also pointed out an example of a disastrous error in a paper published in a highly rated mathematics education journal. The exact reference was not given but with a little clever searching I found the passage.

The second, also from a lecture, which quoted this observation from the book The Teaching Gap (page 25-26). Mind you, this is really from the horse’s mouth since lecturer was the one who was quoted in the book.

“I believe I can summarize the main differences among the teaching styles of the three countries.” Everyone perked up at this, and here is what he had to say:
“In Japanese lessons, there is the mathematics on one hand, and the students on the other. The students engage with the mathematics, and the teacher mediates the relationship between the two.
In Germany, there is the mathematics as well, but the teacher owns the mathematics and parcels it out to students as he sees fit, giving facts and explanations at just the right time.
In U.S. lessons, there are the students and there is the teacher. I have trouble find the mathematics; I just see interaction between students and teachers.”

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42

What is the answer to the life, the universe and everything?

Using the simplest encryption scheme, that of setting a to 1, b to 2, etc.

math = 13+1+20+8. That cannot be a simple coincidence.

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Do you have a problem?

“The man who has no more problems to solve, is out of the game.”

- Elbert Hubbard (1856 – 1915), American Writer and Philosopher.

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Of vulgar mechanicks

A Vulgar Mechanick can practice what he has been taught or seen done, but if he is in an error he knows not how to find it out and correct it, and if you put him out of his road he is at a stand. Whereas he that is able to reason nimbly and judiciously about figure, force, and motion, is never at rest till he gets over every rub.

Issac Newton – 1694

In this day and age, the above quote is definitely offensive, but nevertheless still holds true. Too many students are only interested — some even demand — in learning by rote.

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2013 Abel Prize

A well written note by Frenkel that attempts to explain Deligne’s work that won him this year’s Abel Prize.

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Multitasking with studying

Another report on how students don’t learn as well when they attempt to multi-task.

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