The Maths Wars, and the best way to teach children their Times Tables

Many people believe that if anyone actually did invent the notion of “Times Tables” it was probably Pythagoras (570 to 495 BC). Indeed, in many languages what in English is called the Times Tables is called the Table of Pythagorus.

But the best we can say for Pythagoras is that he may have re-invented the tables, because mathematical tables containing multiplications can be found dating from 1900 BC in Babylonian clay tablets.

It is clear, however, that similar thinking to that which inspired Pythagoras was going on at the same time in China where the “nine-nine” song or the nine-nine table was produced. When first written down, the table started with 9 x 9 and worked backwards – probably because nine was one of the sacred numbers in China.

Indeed, in 2002 Chinese archaeologists unearthed the earliest artefact of the nine-nine table that has been discovered - a written wooden script from around 480BC on which was written: "four eight thirty two, five eight forty, six eight forty eight."

Victorius of Aquitaine also has a claim to fame in the history of times tables because in 493AD he wrote a 98 column multiplication table using Roman numerals (without, of course, any zeros) giving the product of every number from 2 to 50, and some numbers from 1000 to the fraction 1/144. To make the table usable the rows appeared in blocks, descending at first by 100, then by tens, and so forth.

These approaches led to the traditional rote method of learning multiplication based on memorisation of columns in the table, in a form like
1 2 3
4 5 6
7 8 9
For the four outside numbers one moves round in a clockwise direction, so looking at the seven times table one starts with 7. Then one moves up to the 4, and thinks of the next number after 7 that ends with a 4 – the answer is 14. After that it is 21.

After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.

A similar table is used for the multiples of 2 4 6 and 8.

Of course, we wouldn’t find many of these original multiplication systems to be of much interest in teaching today – not least because the Mesopotamians used a sexagesimal (base 60) number system, which used multiples from one to 20, and then up by 10s to 60. (They also treated division as multiplication by the reciprocal).

The scholarship that led to the discovery and deciphering of the Mesopotamian system (including the understanding of the fact that a number of the tablets that have survived actually contain mathematical errors) is, however, some way away from more recent times where discussion about times tables and their use has led to what has become known as the “Maths Wars” (although to be precise we should use the American expression, “Math Wars”).

At the heart of the debate is not so much the maths itself but the explicit question of how children can best be taught these skills, and whether one should use fixed, step-by-step procedures for solving math problems (such as teaching times tables in a way that means they can be recalled at a moment’s notice) or a more enquiry-based approach involving real-world problems that help develop number sense, reasoning, and problem-solving skills.

In the enquiry-based approach the idea is that an understanding of times tables follows from the enquiry. In the reverse view the understanding arises after things like the tables are learned.

In the end the debate in America seemed to resolve itself into the view that both approaches work. And the simple fact is also that both approaches don’t work. Much depends on the way in which the approach is used.

Beyond doubt the research shows that if you teach using a method you believe in, the chances are the children will understand and be able to use that method. If you are pushed into using a method that appears to you to be wrong, the chances are that learning will be reduced. Belief in an approach and good teaching are intrinsically linked.

The Math Wars in the US arose after 1989, when the National Council of Teachers of Mathematics in the United States undertook a review of maths teaching and developed the notion that all students should learn higher-order thinking skills. Their view was that they should reduce the emphasis on rote memorization, such as multiplication tables.

Those who followed the new approach worked with children who used drawings and manipulated numbers, who became more involved in cutting and pasting, gathering data and playing conceptual games. That doesn’t mean that times tables weren’t learned – but the central focus was elsewhere. Those holding the reverse view emphasised times tables and the like, but that did not exclude data gathering, cutting and pasting, etc, etc.

And, in fact, successful maths teaching emerged from both directions. There is no clear data that shows that across a large number of teachers, starting from one end rather than the other gives a benefit.

Based on these findings it was generally agreed that pupils must first develop computational skills before they can understand concepts of mathematics. But equally it was agreed that these skills should be memorised and practised using time-tested traditional methods until they become automatic.

Of course, not everyone agreed that time is better spent practising skills rather than in investigations inventing alternatives. It all depends on the what the teacher feels works best with these particular young people.

So we might all agree that calculator use can be a good thing after number sense has developed and basic skills have been mastered. Just as we might agree that approaches which lack explanations of methods make it difficult to help with homework.

But perhaps one of the most important differences between a discovery system of teaching maths and a system based around such concepts as time tables is the fact that discovery methods are almost by definition somewhat haphazard in their approach, whereas a system based around (for example) learning tables is highly structured.

In fact some research suggests that to a certain degree all children benefit from both, but that some children have a propensity to learn in a structured way while others benefit the most from learning through their own experiments.

Thus we now have two thoughts to consider. One is that most teachers work best when using a system that the teacher believes in. The other is that some children find it easier to learn through one method and some prefer another.

Interestingly, in America the National Mathematics Advisory Panel, created by George Bush in 2008, called for a halt to all extreme positions and for a use of both approaches in teaching maths. The panel examined the scientific evidence related to the teaching and learning of mathematics, and said, "All-encompassing recommendations that instruction should be entirely 'student centered' or 'teacher directed' are not supported by research. If such recommendations exist, they should be rescinded. If they are being considered, they should be avoided. High-quality research does not support the exclusive use of either approach."

Of course that didn’t end the debate – indeed it has intensified and even become the subject of a long running cartoon – the Weapons of Math Destruction”.

Which leads to the fact that one way or another you will want to teach multiplication. Which is why we have produced the Photocopiable Times Tables resources, accompanied by the Funky Times Table CD. For more details go to www.topical-resources.co.uk/numeracy