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.
– a proof due to Zolotarev and its connection to dealing cards.
A list of 240 and counting published proofs of the theorema aureum.
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.
A slick solution to the broken stick problem or otherwise known as the uncooked spaghetti problem
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.
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.
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)
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.
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.
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
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.”
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.
“The man who has no more problems to solve, is out of the game.”
- Elbert Hubbard (1856 – 1915), American Writer and Philosopher.
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.
A well written note by Frenkel that attempts to explain Deligne’s work that won him this year’s Abel Prize.
Another report on how students don’t learn as well when they attempt to multi-task.
With the recent korean wave, we get to see lots of Korean pop, tv, movie stars in the media. But I found that I can never tell them apart. Especially, the ladies, tall, thin, beautiful but too homogenous. (Of course, korea is well known to be a cosmetic surgery heaven, but let’s not go there.)
Here’s finally some no-nonsense maths to prove the point. A great post analyzing the 2013 contestants of the Miss Korea competiton. In a nutshell, there are 6 eigenfaces – great word – and all the rest of the faces can be reconstructed by some linear combinations of these 6.
Apparently a couple of influential papers — by two world famous economists Reinhart and Rogoff from Harvard — that were quoted by politicians to shape policies contain serious errors that invalidated their claims. They made the common mistake of not selecting all the entries in the column when averaging, and reported a negative value when in fact the average was positive.
More details can be found here. There was no byline for the post, but the blog is by David Bailey and Jon Borwein, respected mathematicians.
So should we stop using excel because it has tremendous potential for disasters such as this? Yes, if you ascribe to the “everyone is an idiot” philosophy, and then you make everyone do all the calculations and data entry manually thereby increasing the chance of more errors.
Let’s end with a joke. “Why should doctors, nurses, economists, judges, auditors, accountants learn mathematics?”
“To prevent idiots from getting into these professions.”
… it is no true wisdom that you offer your disciples, but only its semblance, for by telling them of many things without teaching them you will make them seem to know much, while for the most part they know nothing …
This quote caught my attention for it seems to make a case of “telling” vs “teaching”. Of course it is somewhat taken out of context. Socrates was actually commenting on the technological advancement (back then) known to us as writing or books. And many people nowadays use the passage as a critique of the internet. A more complete transcript may be found here.
that human being can understand? By that I mean not those formal logic nonsense. Tim Gowers is doing an interesting experiment to get readers to judge proofs of exercises in metric space theory. The three proofs are supposed to be written by an undergraduate, a phd student and a computer. I only looked at the first problem which is quite straightforward: prove that the intersection of two open metric spaces is open.
Honestly, I can’t tell which one is the computer. The main problem is one does not know how the program was written. Here’s my analysis of the solutions. The first two proofs are very similar in the sense that they both use open balls. (The third did not.) The writing is also much clearer with good choice of notations. One thing that is clear is that the first two writers already knew what is the key to the proof – take the minimum of the the radii of the two open balls. The third proof, on the other hand, seemed to be using the various conditions to force the choice. Perhaps that makes it the computer, assuming that the programmer do not actually instruct the computer to reorganize and rewrite the proof. If the first two were humans, I still cannot tell which is which. The second proof looks to be more carefully written, but that does not actually imply the level of mathematical sophistication. Or does it?