**Chapter Ten.**

**From fear of fractions to the joy of maths**

Here's a question. "What is a half of three-quarters?" I ask this question
not because it's particularly hard but because this was a question put to
the former Chief Inspector of Schools in England (1994 - 2000), Mr Chris
Woodhead on a radio show. Mr Woodhead had been complaining about falling
standards, especially in maths. This was a concern shared by many parents
and teachers. One teacher phoned the radio show, complaining that the
16-year-olds she taught were hopeless. They couldn't even solve simple
problems like "what is a half of three-quarters". The host of the show did
not ask the Chief Inspector to comment on this piece of news. Instead, she
asked him for the answer to this question. Remembering the embarrassment
an education minister had recently suffered in a similar situation (see
Chapter 8), he refused to answer. Later a spokesman for the Inspectorate
told the world that the Chief Inspector knew the answer but didn't want to
say. In any case, his speciality was English, not maths.

No doubt, the Chief Inspector did know the answer, but like many people,
lacked confidence in the answer. Why is this? Confidence comes from
knowing what you're doing and also knowing that you know what you're
doing. The way many of us are taught doesn't help us gain confidence. For
the fraction problem that may have defeated the Chief Inspector of
Schools, most of us learned (or failed to learn) a procedure that would
give the right answer. Multiply the denominators (2 x 4) to make the new
denominator, and multiply the numerators (1 x 3) to make the new
numerator. Lo and behold you have the answer 3/8. But can you be sure this
is the right answer? How can you have confidence?

One of the great beauties of mathematics - and what makes it unlike most
subjects we learn at school - is that there are many ways to find the
answer. So you will have confidence if you arrive at the same answer using
a different procedure. But few of us have learned other procedures.
Another path to confidence is to make an estimate and check that the
answer is consistent with the estimate. To do this, you need to understand
the problem. Half of something is less that the whole of it, so half of
3/4 is less than 3/4. What is more, because 1/2 of 3 is 11/2 the answer
has to be one and a half quarters, that is, more than one quarter and less
than two quarters. You could also see that 1/4 = 2/8, and that half of 2/8
is 1/8.

In England, and probably in Japan, teachers and other educational
authorities have decided which is the best method for each type of
problem, and this is what you learn. As a consequence, most of us lack
confidence in our ability to deal with fractions, so we avoid and fear
them.

It doesn't have to be like this. And it is never too late to understand
fractions or to have confidence that you do. What could be a more severe
test of confidence, and competence, than doing your calculations on TV in
front of millions of viewers willing you to succeed? This is exactly the
situation in which Rüdiger Gamm found himself. He appeared on a German TV
programme in which the audience bets whether an expert can accomplish a
difficult challenge.

Here's a question he answered very swiftly. What is 87 to the 12th power
(8712)?

[The answer, in case you have not calculated it, is
188,031,682,201,497,672,618,081.]

You couldn't do this in your head, could you? You may think this is
because you have no special talent for mathematics. But neither has Gamm.
At school, he hated maths, and was no good at it. So what happened? At the
age of 20, he found a piece of maths he could understand, and with
practice, could do. Like the champion calculator, Wim Klein, (Chapter 7:
"Good and Bad at Numbers"),
he found he could use this skill to impress. More than that, he saw the
possibility of a new path to fame and, as it turned out, to a modest
fortune. Of course, this level of achievement comes at a cost. Gamm, who
is now known as "the human calculator" and makes a living doing
calculations, works four hours a day at his new profession. He has learned
tables of squares and cubes, square roots and cube roots, like we once
learned our multiplication tables. Just as we can solve 8 times 5 almost
instantly, so Gamm can give 273 and √ 169 without calculating. But he also
knows lots of tricks and short-cuts, so when he is presented with a
problem he has many ways to solve it. He can not only check his answer by
using two different methods; he can also select the method he has found to
be the most efficient.

Most of us can work out the answer to 876 x 458 on paper. We understand
what the problem means, we know which steps are needed to do it, and we
can carry out each of the steps successfully. For example, we know that 8
x 6 is 48. But most of us could not solve the problem in our heads,
because we cannot keep in mind the whole sequence of steps and
intermediate results. Gamm has trained himself to do just this. A
scientific study of Gamm's calculating ability revealed that he uses a
part of the brain that the rest of us do not use for calculating. We are
limited by our "working memory" capacity, approximately a total of six or
seven steps. This, in my case, is about enough to multiply two two-digit
numbers such as 45 x 76, and then I often make mistakes. How then does Gamm
manage to exceed this capacity, which is usually thought to be fixed?
(Pesenti et al, 2001)

Your computer has a random access memory with a relatively small capacity,
and a hard drive with much greater capacity. However, your computer can
increase the effective capacity of RAM by temporarily offloading data not
immediately needed into dedicated "swap space" on the hard drive,
retrieving it again as required, creating a "virtual memory" with more
capacity than the RAM alone. This is similar to what all kinds of experts
seem to be able to do, but only in their domain of expertise. An expert
musician can remember the sequence of notes in a new melody much better
than a novice. An expert chess player can remember the position from a
game at a single glance. An expert waiter can remember the orders from a
table of twenty people without writing them down. All these exceed the
normal capacity of our "working memory". How is this achieved? It's
simple. You become an expert. It so happens, we are all experts in one
skill we practice every single day - language. (Butterworth 2001)

We are all experts at language. If I give you a sequence of four words
from a language unknown to you, and ask you to recall them in the
presented order, you will have great difficulty. But you would have no
difficulty remembering a list of familiar words. In fact, your capacity
for a word list is about 6 or 7, depending on the words. However, if I
make the list very much like what you practice every day, make the list
into a sentence, then you will have little trouble remembering far more
words. Even a thirty-word sentence can be no problem at all. (Wingfield &
Butterworth, 1984) This is
because of the hours of practice we have put in to understanding, and,
where necessary, remembering what people say to us. We develop very
efficient ways of coding the information so that we store it on our
internal hard disk - our long-term memory - to be retrieved as needed. No
one would be surprised if we were able to demonstrate this skill, and no
TV company would pay out a big prize for us to recall a thirty-word
sentence. What makes Gamm so unusual is that he chosen to become an expert
in what very few us have bothered with.

So could any of us become like Gamm? Most of us wouldn't want spend all
that time learning dry arithmetical facts or arcane calculation procedures.
But there is no reason to think that we couldn't if we
tried. After all, Gamm succeeded, and he seems to have been worse at maths
at school than most of us.

Some of us would be put off the whole idea of working away at mathematics,
because anything with numbers makes them anxious. As I suggested in
Chapter 8, "Home, Street, and School Mathematics", being forced at school to do tasks that one doesn't understand
is anxiety-inducing. This is especially true of maths because we are not
only forced to do these tasks, but to do them quickly. What is more,
learning maths is a process of building one concept on top of another,
and, like any kind of building, if the foundations are not secure then the
whole edifice will tumble down when too much pressure is applied. Maths
tests, even informal ones that occur as part of a lesson, can be seen as a
process of applying a stress to the building to see if it is really
secure.