Even the very best pupils can be handicapped by calculators

Imaginative thinking is exactly what is tested in Oxford University’s maths entrance examinations. Many sixth form pupils cannot perform well in these examinations because they have not developed their mathematical intelligence. The reason for this is simply because they have relied on the use of their calculator throughout their secondary maths education. Therefore, they have not developed strategies for manipulating numbers, for rearranging them to make the calculation tractable, instead hey have relied on their calculating machine to crunch the numbers for them. Today these exams are testing for something that was unremarkable 40 years ago, before the ubiquity of the handheld calculator. University mathematics is about imagination, and imagination is what puppet maths helps develop. Rather than pupils simply learning routines for manipulating numbers by rote, puppet maths puts these algorithms in an imaginative context.
But this is not just for the pupil who wishes to go onto advanced study, every pupil can benefit from the use of their imagination when doing sums. It’s what makes an otherwise dry subject interesting, indeed fun.

Calculators for evaluating fractions

The use of the calculator stands in the way of understanding in maths. In the days before calculators became common the divide symbol was practically unused, everyone wrote a division as a fraction. This is vital if pupils are to understand what the calculation is doing. Use of the calculator hides the method and process of division from the pupil, denying them understanding of what they’re doing. In my teaching I had a number of able pupils give me an answer of “4.99” to a particular question I had set. The right answer was “5”. This led to much complaint when I marked them wrong. Had the pupils written the problem as a fraction, they would have been able to cancel out numbers top and bottom. This would have left them with the correct answer without having to do any calculation at all. They could have solved the problem with a few strokes of their pen, in less time than it took them to type the numbers into their calculator, and without rounding errors.

Fractions continued

Understanding of fractions depends upon the way the pupil thinks about them. If children write their fractions on two lines, then that is how they think of a fraction, as two numbers, and they begin to appreciate a relationship between them. If they are using a calculator to divide on fraction by another, then they do not think of fractions as being a complete representation of a quantity, a number comprising of a top part (numerator) and a bottom part (denominator), they think of a fraction as being an uncompleted operation which can be accomplished on a calculator, whose answer is a single number. This diminishes their appreciation of mathematics and their ability to use mathematics.

More on fractions

The next problem with fractions is that pupils are taught the words “numerator” and “denominator” for the top and the bottom of the fraction, which they learn dutifully, but which mean nothing to them. They do not realise what the numbers signify. They do not realise that the word “numerator” means “how many”; they don’t realise that “denominator” means “what sort of thing” (as in denomination relating to currency or religion). This lack of understanding hampers the pupil as the rules they are expected to learn for manipulating the fraction are both opaque and arbitrary. Once they understand the meaning of these two numbers then the reason why they have to jump through the hoops they do to manipulate the fraction becomes apparent, and much more understandable.

Writing fractions

The line in a fraction should not be sloped.

Often when learning maths pupils have difficulty with fractions. This starts with the way they write the fraction on the page. Copying their parents or other adults, they will draw the line between the numerator and the denominator as a slanting slash

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rather than as a flat, horizontal line and when asked to write it with the horizontal line, they are reluctant to do so. The slash looks so much more stylish, suave and sophisticated. They cannot immediately see why they should use the horizontal version.

But it is very important. It is absolutely vital to be able to see at a glance which number is at the top and which is at the bottom.

Multiplication Songs

Have you noticed how, after watching a Disney film, young children are able to pick up and remember the words to the songs that are in those films? They appear to do so effortlessly. Now contrast that with learning the times tables. How boring was that, coupled with the fear of getting a fact wrong. The solution would seem to be to put the times tables to music.
Many attempts have been made at this. Most of the results have been abysmally bad. Yes the times tables are set to music, but the music doesn't pass the "Old Grey Whistle Test". (Composers in New York in the 1920s, where it was hot in the summer and who worked with the windows open, could tell if they had a hit, if the tramps on the street who overheard them composing whistled their tunes). If the music is tuneless and unmemorable, then it adds nothing to the memorability of the words, the times tables.
Putting the times tables to music is a challenging task, because the lyricist is constrained in the words he can use, seven sevens is forty nine whether or not the music has room for 8 syllables or not. The lyricist can change "Paris in May" for "Paris in April" because the tune requires the extra syllable, but such a change with the times tables would make a nonsense of the entire effort.
We have put the times tables to music, music that is memorable and tuneful, and we haven't found it done better elsewhere. We'd love our arrangement to become as widely used as the tune that is used to teach infants the alphabet, or indeed "Happy Birthday". Your children could be among the first to learn their times tables this way if you study maths with our puppets at "Maths Puppets".

Algorithms

When learning maths, children have to remember the routines (mathematical algorithms) that they are taught. This is a feat of memory for many. It is much easier for them if they have stories to prompt them. Instead of remembering an abstract set of rules, they can remember a story played out by puppets, a much more attractive option. This demonstrates the added value achieved by teaching maths using puppets.