Monthly Archives: July 2014

How and when school children learn the times table

7times8equals56
This post was motivated by a long public discussion of learning the times table at UK schools.

Dropping maths at the age of 16

It seems that a public acceptance of being bad in maths may not be considered as a great shame, even by a finance minister, read BBC magazine: UK Chancellor refused to answer a “times table” question. How could that happen?

One retired maths teacher has told us that many students do not continue studying maths and English at the age of 16-18. Here is an update from BBC education: a long-term possibility might be to require all would-be teachers to study maths to 18.

This long term plan can be compared with the long existing tradition in other countries where maths and native language/literature are compulsory for all students up to the university level.

Why are UK students really lucky?

Here is the most popular comment on BBC: “I’m studying a Masters in Physics at University and STILL don’t know my times tables by heart, I got an A* at GCSE Maths and an A at A-Level.”

Does it imply that UK students may not know the times table? It is probably better not to reflect on conclusions that overseas students can make about UK qualifications and universities.

Here is our reply: you are really lucky to study in the UK, because without the times table you wouldn’t progress to a secondary school in a different country.

In many other educational systems, if children start school at the age of 6-7, then they finish learning the times table by the end of their 1st class at the age of 7-8.

Why is learning the times table so hard?

As with all other obstacles, learning mathematics becomes unnecessarily hard when pre-requisite concepts are missed. The necessary earlier step is the simpler addition table that has the sum i+j in the intersection of the i-th row and j-th column.

+ 1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11
3 4 5 6 7 8 9 10 11 12
4 5 6 7 8 9 10 11 12 13
5 6 7 8 9 10 11 12 13 14
6 7 8 9 10 11 12 13 14 15
7 8 9 10 11 12 13 14 15 16
8 9 10 11 12 13 14 15 16 17
9 10 11 12 13 14 15 16 17 18

Notice how the table above is beautifully symmetric like many great results in mathematics. When students learn the times table not by rote, but by actually trying to compute products in their head, they naturally start adding smaller numbers.

For instance, to find \(7\times 8\), one can sequentially compute the following partial sums: \(8+8=16\), \(\;16+8=24\), \(\; 24+8=32\), \(\; 32+8=40\), \(\; 40+8=48\), \(\; 48+8=56\). These sums require the addition table above, which should be studied before the times table.

So we get \(7\times 8=56\) after completing 6 easy additions. If we are already good at multiplying only by 2,  faster way is to do 3 multiplications: \(7\times 2=14\), \(\; 14\times 2=28\), \(\; 28\times 2=56\).

Can the times table be learned only by rote?

Let us compare the times table with another basic skill of walking. Children usually start learning to walk earlier than learning the times table.

Toddlers fall (fail) many times before they become confident walkers. After a first fall a baby looks at their parent to understand how to react: if the parent looks afraid then the baby starts crying, if the parent smiles then the baby also smiles.

So almost all children successfully learn to walk, not by rote, but by actually trying to walk, despite numerous falls or even small injuries. If a kid can not walk by the age of 3, it is usually considered as a physical disability.

A criterion of success in learning

In our opinion, the same powerful principle works for learning any other basic skill. So we wish our readers to follow the excellent example of smart babies, namely more practice and consider every failure (or a fall) as one more step in a right direction.

There is even the internal criterion of success without any comparison with your peers. If people can walk, they walk easily and without making any conscious efforts. Similarly, multiplying single-digit numbers should be as effortless and enjoyable as walking or breathing.

  • Riddle 17: can the square \(n^2\) of an integer have the last digit 8?
  • How to submit: to write your full answer, submit a comment.
  • Hint: the required knowledge is (a small bit of) the times table.
  • Warning: give an example or justify that there is no such number.
  • Prize: free 1-year access to one of our interactive web tutorials.
  • Restriction: only the first correct public answer will be rewarded.

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