Tuesday, October 11, 2005

Learning is stupidly simple

In November 2001 I heard an inspiring talk by Russell Bell in the Cayman Islands. Russell Bell is running a learning centre in Kingston, Jamaica, 'Helping Students Realize Their Full Potential'.

This is his main idea:

'Learning is stupidly simple. To learn one has to remove obstacles, not to "strive for higher intelligence".'

Water can't flow out of a hose if it is blocked. The same goes for learning. His book 'The Learning Process' discusses how we learn, which barriers exist, and how to remove them.

I am looking forward to a promised update of his web site.

Monday, October 10, 2005

Arithmetic without tables

When you look for patterns you may find amazing things.

2 * 6 = 12
4 * 6 = 24
6 * 6 = 36
8 * 6 = 48

Can you see a pattern?

The answer's second digit is the number we multiply six with and the first digit is half that number.

1 * 6 = 06
3 * 6 = 18
5 * 6 = 30
7 * 6 = 42
9 * 6 = 54

What is the pattern?

Let me take 7 * 6 as an example. Add 5 to 7 to get 12. 2 is the answer's second digit. Half of 7 rounded down is 3, plus the 'carry' from 12 is 4. That gives the answer's first digit.

These two patterns can be combined to a rule for multiplying any number by 6:

To each digit add half the neighbour to the right rounded down and 5 more if the digit is odd.

What is 749 * 6?

I will write this as 0749 * 6 and start from the right.

9 has no neighbour and is odd: 9 + 5 gives 4 with 1 carry.

4's neighbour is 9 and there is a carry: 4 + 4 + 1 gives 9 with no carry.

7's neighbour is 4 and is odd: 7 + 2 + 5 gives 4 with 1 carry.

0's neighbour is 7 and there is a carry: 3 + 1 gives 4.

The answer is 4494.

Study multiplication by 7 and come up with a similar rule.

These patterns were first discovered by Jakow Trachtenberg while he was a prisoner of war. He discovered that anyone who can take halfs rounded down and add small numbers can multiply.

Saturday, October 08, 2005

I hate school

Friday, October 07, 2005

The mathematical experience

What is math? A teacher's answer determines how he teaches. 'The Mathematical Experience' is a book that should widen most people's perspective.

A review from Amazon:

I was going to study history. Math? Who cared about math? Math was for those science-types. I had an image of mathematicians as bespectacled, socially-inept, hunch-shouldered gnomes who lived in universities and ventured out of their burrows for--well, maybe they didn't venture out at all.

The joke's on me. I'm a math major now. This book is one of the reasons.

I've always loved history: the march of events, the ebb and flow of cause and effect and unexpected accident. I didn't realize that math, too, had a history, an ebb and flow. If I'd ever thought about it, I would have realized that an angel didn't come down from the heavens bearing The Big Book of Math, complete with proofs. But that's what it seemed like, until I read about the almost architectural building of theorem upon theorem, idea upon idea. Math wasn't a Big Book; it evolved and grew. Grows still, I should say.

Did numbers exist? Well, of course they existed. Wait a second. What *is* a number anyway? How *does* one exist? Would they exist if there were no people?

And so I learned that math, too, has its philosophies.

Most of all, I learned that mathematicians were and are people, not gnomes in burrows who have nothing to do with the rest of the world. That math is important for more than the homework assignments that plagued my high school evening hours. That math is worth studying.

If you could convey this to heaven knows how many disgruntled and frustrated math students around the world, I wonder if they might like the subject better.

I sure did.

Thursday, October 06, 2005

The one who has the shoe on

It is important that students bring a certain ragamuffin, barefoot irreverence to their studies; they are not here to worship what is known, but to question it. - Jacob Bronowski

The IMPACT program, led by David Clarke, required pupils to give written responses fortnightly to eight questions.
  1. What was the best thing to happen in Maths during the past two weeks?
  2. Write down one new problem which you now can do.
  3. What would you most like more help with?
  4. What is the biggest worry affecting your work in Maths at the moment?
  5. Write down the most important thing you have learnt in Maths during the last two weeks.
  6. Write down one particular problem which you still find difficult.
  7. How do you feel in Maths classes at the moment? (Circle the words which apply to you.)
    A. Interested B. Relaxed C. Worried D. Successful E. Confused F. Clever G. Happy H. Bored I. Rushed J. (Write down one word of your own)
  8. How could we improve Maths classes?
Sample answers to each question:
  1. We worked hard and learnt.
  2. I can't do any problems but I can now do triangles.
  3. Fractions but the teacher thinks I know them.
  4. Sometimes I'm a bit unsure where to put the decimal point.
  5. I'm stupid in class.
  6. Algebra a bit, but because I don't understand why we don't use numbers. It would be simpler.
  7. Relaxed. Bored. I feel relaxed because I'm bored.
  8. Have less work and more learning.
Source: The Interactive Monitoring of Children's Learning of Mathematics, David Clarke, For the Learning of Mathematics, Feb 1987.

If you are a teacher, give these questions to your students. It is suggested that you ask students to write their names, but that their answers will be treated in confidence. If you are a student you are welcome to put your answers in a comment below.

It is all too seldom that students are asked how they feel about their math life. However, in some countries the students have to evaluate their school life once a year via the Internet.

Wednesday, October 05, 2005

Under my nose

Many complain that math is of little use. 'I've never had to divide two fractions after I left school,' is a common sigh. I have three comments to this.

1. I've never had to divide two fractions either.
2. Mathematical problem solving skills I have used on a daily basis.
3. I have found math in unexpected places.

A bit more on 3.

Today, I had to rename these files on my computer:

I wanted to rename them in order of date like this:

But, when I tried to rename one of them, this monster appeared:

A math problem is born: How few files is it enough to rename to get the job done?

I can rename jecengine06.mdb to jecengine10.mdb without any problems, but I can't rename jecengine04.mdb to jegengine09.mdb without first renaming the existing jecengine09.mdb. I would like to rename the existing 09 to 07, but I can't since 07 already exists.

You might think, this is silly, who cares how few renames have to be done, just do it any way you like, it won't take long anyway. I agree! The practical value in this case is close to nil, but, like Mount Everest, the problem has been discovered and has to be conquered. Why? Because it is a challenge, it is fun, and it feels good to create something.

So, tell me, how few renames do you need?!

Tuesday, October 04, 2005

Proof by authority

Aristotle and Dennis Lindley

There are several ways to prove that you are right. One is by reasoning, another is by authority. News media these days prefer the latter. Here is a recent example.

Professor Dennis Victor Lindley, 80, well-known statistician from University College London, is a prankster or in search of fame. He just launched this formula:

M is the ideal age to get married. Y is the age you start looking for a partner. X is the age you intend to stop looking. e is 2.718281828..., an irrational constant only second to pi in fame.

An example. If you intend to look for the right one from 18 to 50 you should get married at 30, since 18 + (50 - 18)/2.718281828 = 29.8.

The idea is very simple, for every year you plan to look for a partner wait about 0.37 years, or four and a half months. In the example above, 32 years searching time gives 32 x 0.37 = 11.8 waiting time. So, if you start at 18, get married at 29.8.

Does the professor believe in his formula. I would hope not. Does he justify it? If he does, it is not reported in the media.

Is it a joke or a demonstration of stupidity? I go for the first. The use of e is the clue. The precision given by it and that it should crop up in a formula of this kind.

I don't think the professor's prank does mathematics any good. People in general have a distorted view of the subject already and the formula makes it worse, not better.

Mathematical thinking can clarify things as mathematician John Allen Paulos demonstrates in his column Who's Counting and books.

Monday, October 03, 2005

Mathematicians talking

Michael Atiyah and Isadore Singer interviewed by Martin Raussen and Christian Skau in May last year.

When mathematicians talk about how they work and why they work, I listen. Basically because I like to have supported my ideas about what problem solving is and feels like. This week I read an interview (pdf) with Michael Atiyah and Isadore Singer.

Some highlights from the interview:

Atiyah: A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently.

Atiyah: Almost all mathematics originally arose from external reality, even numbers and counting. At some point, mathematics then turned to ask internal questions, e.g., the theory of prime numbers, which is not directly related to experience but evolved out of it. There are parts of mathematics about which the human mind asks internal questions just out of curiosity.

Singer: I find it disconcerting speaking to some of my young colleagues, because they have absorbed, reorganized, and simplified a great deal of known material into a new language, much of which I don’t understand. Often I’ll finally say, “Oh; is that all you meant?” Their new conceptual framework allows them to encompass succinctly considerably more than I can express with mine. Though impressed with the progress, I must confess impatience because it takes me so long to understand what is really being said.

Atiyah: My fundamental approach to doing research is always to ask questions. You ask “Why is this true?” when there is something mysterious or if a proof seems very complicated. I used to say— as a kind of joke—that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture—it may be a beautiful result but with terrible proofs—you spend your time trying to find better ones; you do not listen to the lecture. It is all about asking questions—you simply have to have an inquisitive mind! Out of ten questions, nine will lead nowhere, and one leads to something productive. You constantly have to be inquisitive and be prepared to go in any direction. If you go in new directions, then you have to learn new material.

Singer: ... when I try out my ideas, I’m wrong 99% of the time. I learn from that and from studying the ideas, techniques, and procedures of successful methods. My stubbornness wastes lots of time and energy. But on the rare occasion when my internal sense of mathematics is right, I’ve done something different.

Singer: I love to play tennis, and I try to do so two to three times a week. That refreshes me, and I think that it has helped me work hard in mathematics all these years.

Atiyah: I believe that if you do mathematics, you need a good relaxation that is not intellectual—being outside in the open air, climbing a mountain, working in your garden. But you actually do mathematics meanwhile. While you go for a long walk in the hills or you work in your garden, the ideas can still carry on. My wife complains, because when I walk she knows I am thinking of mathematics.

Other sources for listening to mathematicians talking about what they do is the book 'Mathematical People - Profiles and Interviews' from 1985, with a continuation in 1990 called 'More Mathematical People: Contemporary Conversations.' (ISBN 0817631917 and 0120482509). There is a similar book about computer programmers called 'Programmers at work: interviews with 19 programmers who shaped the computer industry' (ISBN 1556152116).

A sigh: when will we get MathConversations, podcasts where mathematicians talk about their work as in the interview above? ITConversations already exist for the computer people. Click here for an interview with the creator of php. The language that helps you bring this blog.

Saturday, October 01, 2005

Knowledge is power

More Dilbert cartoons.