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.

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.

I hate school

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.

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.

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?!

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.