How to make Maths harder

Alex and Bob were travelling together on a high-speed train when they whizzed past a field full of cattle. A told B, “There are exactly 74 cows in that field.” “What!” said B, “How did you manage to count so many heads of cattle in a split second?” “I didn’t count their heads, of course,” replied A, “I counted their hooves and divided by four!”

Parents were told not to allow their children to write numbers under each other when adding or subtracting ‘because it would be easy that way’- Josef Lauri

Parents of children in primary school know what it means to do mathematics the difficult way. They encounter it whenever they try to help their children with their maths homework, unless the school they go to has broken the mould and is using some textbooks other than the Abacus series – which is very unlikely in our centrally-imposed educational system.

Some teachers are also baffled by the Abacus system, but they have to toe the line and few dare speak out in protest.

Let me start by emphasising that, unlike what many teachers and parents seem to think, Abacus is not some new-found universally accepted way of teaching mathematics. Rather, it happens to be the trade name of a particular series of textbooks, published by a particular publisher and written by two authors who follow a particular (and marginally important, I would say) philosophy of teaching primary school mathematics.

One of the main tenets of this philosophy is the refusal to teach children efficient and quick methods (algorithms, technically speaking) to carry out calculations.

In the US, some parents were up in arms against textbooks with a similar philosophy, leading to the famed ‘Math Wars’. So the Abacus books are anything but a holy text that should be accepted on blind faith. They simply propose one method among others.

But in Malta, parents, teachers and schools have been cowered into thinking this is a method only fools do not understand. These parents and teachers have the right to be told that if children take too long to carry out a numerical routine which we could do in a couple of lines in our school days, it is not the teacher’s fault, nor the textbook’s failure, but it is a success of the system espoused by the Abacus books.

Children are deliberately not shown the most efficient way of working out a problem. I am not in favour of starting primary school maths by teaching algorithms mindlessly, but I am certainly against postponing it indefinitely. The result of doing this is that, for most primary schoolchildren, every time they encounter a problem like ‘26+37’ it is a creative challenge and they have to decide what ‘strategy’ (but no algorithms please!) to employ.

In the same way they are deliberately not encouraged to memorise, say, that 7 x 9 = 63, but to work it out as 7, 14, 21,..., a habit which many children carry with them to secondary school and find it difficult to grow out of. But then, when transforming 7 2/9 into an improper fraction, some children, especially the weaker ones, would have forgotten what they had set out to do in the first place by the time they reach 63.

And let us not think children are now not taught how to perform routines in a rigid step-by-step manner with little room for understanding, or that because they are not expected to remember the multiplication tables they are not inundated with facts. The Abacus books are littererd with facts (for example, facts of 10, facts from 10, facts to 10) and a plethora of ‘strategies’ (number line, number chart, counting up, counting down, and so forth).

And the curriculum is certainly rigid. Ask any parent who has tried to help a child with maths homework, only to be greeted with protests that that is not the method the teacher wants. I know instances when parents were told not to allow their children to write numbers under each other when adding or subtracting “because it would be easy that way”.

This method of teaching children to walk with their hands tied behind their backs will not help them walk better when their hands are freed. It will, in fact, stunt their mathematical growth because it goes precisely against the spirit of mathematics.

Alfred North Whitehead, the great logician and mathematician wrote, “Civilisation advances by extending the number of important operations which we can perform without thinking about them.”

This is exactly how mathematics has evolved through the ages. Complex problems are tackled and reduced to automatic routines so that we can use our freed creative energy to tackle harder problems. But this true nature of mathematics is being hidden from our children, at least in the first six years of their school experience – a pitiful lost opportunity.

And the result is that the techniques ‘allowed’ by Abacus are so weak that primary schoolchildren today are not even taught how to add two fractions (unless some enlightened teacher decides to break the rules). Many end up never learning how to do this because at secondary school, calculators and spreadsheets come in.

These machines should be considered beasts of burden to do things children could do themselves but which they do not bother with so that they can do something more interesting and which requires more skill and thought. In our case, however, these machines come in to make up for the lack of technical ability students inherit from their primary school days.

And this has a serious knock-on effect. For example, algebra has become notoriously difficult for many students in secondary schools. But it is only to be expected that children find algebraic manipulations difficult when they have not even mastered the numerical equivalent.

And when children find mathematics difficult they are less likely to choose science subjects, no matter whether we call them physics, chemistry and biology or some other modern-sounding titles.

Frustrated teachers and perplexed parents have for too long been brow-beaten into submission about this and other aspects of their children’s education.

As it often happens, here it might have to be the hard experience of failure (as in our recent participation in the Trends in International Mathematics and Science Study), which will convince us that the Abacus emperor is thinly clad.

Prof. Lauri is from the University’s Department of Mathematics.