A report on the state of numeracy of peoples of 24 selected countries shows Britons came 21st and British young people have about the same numerical skills as their retiring grand parents. I don’t know about you but as a teacher of secondary mathematics, I am shocked not.
What normally follows news and reports like this is the government setting up a committee to look into the report and come up with recommendations to reverse the trend. Hundreds of thousands of pounds, if not millions, would be spent on this mission to save the good people of the United Kingdom from innumeracy. This is not unlike the joke about America’s NASA spending millions of dollars to design pens that will write in space, where lack of gravity makes conventional pens useless as the ink don’t drip into the writing tips. Although NASA succeeded in creating this non-gravity dependent pen at a huge cost, the Russians achieved the same purpose of being able to write in space (the bottom line) by using pencils!
Just like the punch line of this joke, I will suggest that the government take a different approach and save tax payers’ money. I have a very cheap alternative based on my professional diagnosis of British secondary school students that I teach on daily basis: make it mandatory for children to learn the 1 to 12 multiplication or times table before their 10th birthday! Perhaps, not mandatory, as we live in a civilised and free society where it is not fashionable to force people to do anything. The DfE can perhaps encourage parents to help make their children learn the times’ table before they are ten.
You might be wondering why learning the times table is considered to be so important in tackling innumeracy and the answer is found in why British young people are so poor at numeracy: most don’t know the times table! I know that this sounds redundant but please bear with me and read on. Being numerate means being able to add, subtract, multiply and divide numbers. The extent of our numerical ability is measured by factors like: (i) how quickly we can carry out calculations and (ii) the complexity of numbers and operations that we are capable of calculating.
Speed of calculation is perhaps the most important reason to learn the times table. While it would take a 12 year old child who knows the times table about 2 seconds to find the answer to “7 times 8”, it could take a peer who doesn’t more than 30 seconds to do the same. And the latter is likely to get the answer wrong! This leads to frustration in the youngsters making them conclude that they are poor at mathematics or, even worse, that the subject is incomprehensible. At the very early stages, it makes sense for kids to count their fingers when adding small numbers. This strategy, however, needs to be “weaned off” as soon as possible. Try reading this piece by spelling every word and then putting them together to form words as you did when you were first learning to read! You might notice how laborious it is. Worse still, you can’t really comprehend what you are reading because too much time elapsed between the words! What you experienced here is similar to what children who count fingers go through when carrying out calculations or learning new numerical concepts in maths lessons.
The easiest of the 4 numerical operations is addition. Multiplication is repeated addition. Subtraction is backward addition while division is repeated backward addition. In other words, if you can add numbers, you can also multiply, subtract and divide them. How quickly and accurately we do these is what varies among individuals. Individuals who know the times table have one advantage: they have a bank of ready made answers stored in their memory that they can quickly recall when needed. You could say that they have a sort of phone ‘contact list’. When they need to call a friend, they don’t have to put the numbers together; they just look into their ‘contacts list’ and voila! They get the number quickly and they never mix digits up as many of us did before the golden age of mobile phones. The question: 72 divide by 8 would take a 12 year old who knows the times table about 3 seconds while a peer who doesn’t could take more than 30 seconds and still get the answer wrong. Students who don’t know the times table by heart will struggle with division, fractions, powers (squares, cubes etc) and roots (square roots, cube roots etc). It does not get easier as they grow older and have to learn more complex concepts, according to the age based curriculum.
Many British young people do not know the times table by heart due to the education system’s discouragement of learning by rote. Learning by understanding is encouraged and personally, I find it to be a much better way. We don’t need children to regurgitate facts, we want them to be able to interpret and analyse these facts and give their own judgement on what they have learned and how to apply them in different circumstances. I spent a lot of time memorising different formulae for my mathematics, physics and chemistry final secondary examinations in my native country. On hindsight, I think too much time was spent memorising the formulae at the expense of exploring their meanings and applications. The only students who succeeded were those who through excruciating mental labour were able to memorise the formulae and learn how to interpret and apply them. There is the possibility that students who might have been better at interpreting and application could have been screened off on the account of not been able to memorise the formulae.
British students do not have this problem as all the complex formulae in the subject curriculum are provided in the examinations. The examinations test ability to interpret and apply these formulae but not the ability to remember them. I prefer this as a teacher because being able to interpret and apply the formulae is much more important than being able to recall them. Interestingly, those who can interpret and apply are also (mostly) the ones who can recall but that is not the point here. The students at least have one less thing to worry about: forgetting the formulae or worse (and more likely) mixing up the variables. Unlike the system of rote learning, we can be sure that our best students are those who have excelled in application of the concepts and not just those who are best at remembering them. This is particularly so because the concepts and formulae are always in reference materials and while it is good to know them by heart, it is not important to do so in order to use them in real life applications.
While students who combine the ability to memorise formulae with interpretation and application will usually beat those who have the need to look at reference materials, especially in quiz competitions and similar settings, winners of these quiz competitions don’t end up as Nobel Laureates. We may compare them with winners of Spelling Bees competitions; they don’t always become literary giants!
As I have tried to show above, I am not in favour of learning by rote but the problem being highlighted in this piece is not on the high end of the intellectual spectrum of our students, it is on the lower end. The damning report that brought about this piece is based on general numeracy among young people in the participating countries and not about how well the whiz kids compete against each other. The solution need not be some complex or exotic policy but a review of how British kids are doing in their maths classes. We could start by asking secondary maths teachers of their experiences in teaching the subject and the abilities of our children as they enter Key Stage 3 (junior secondary). Far too many come in without any knowledge of the times table. Students in countries with some rote learning would have learned the 12 times table in primary school. They would therefore appear to be better than those who don’t in general numeracy.
The bigger impact of knowledge of times table is in the ability to comprehend other numerical concepts. First of all, students who know the times table would not fall into low maths sets upon entrance into secondary schools. In some cases, knowing and being able to apply the times table for division, for example, would get the child into top set in Year 7! This is because most of what is needed to get the sort of attainment level needed to be in a top set is around multiplication and division. Knowledge of the times table is by no means a silver bullet and it does not guarantee that a student will be good in maths but it helps more than anything else at junior secondary level.
When new concepts are being taught, knowledge of the prerequisites is assumed and the teacher just gets on with what needs to be learned. In teaching area of rectangles and squares for instance, the teacher would assume (and correctly so) that the students can multiply numbers. S/he would also expect the students to be able to find missing sides when areas are given by dividing or finding square roots. In teaching long multiplication and division, knowledge of the times table would again be assumed. The student who lacks this knowledge would be lost and frustrated in the class. Things would be happening too quickly for him as it takes far too long to count fingers (including the likelihood of getting the answer wrong by plus or minus 1).
The concept being taught might not be out of his scope of understanding initially but as the exercises comes in fast and furious, he loses the connection and frustration sets in. Conclusion: maths is too hard! This student would now be apathetic to the subject and a vicious cycle begins. Often times, students who come into secondary school without knowledge of times table (usually on Levels 3 and below) would end up making little to no progress at all the way through secondary school. I know of several instances, on the other hand, where such students learned the times table and quickly became high achievers.
Children shouldn’t be counting fingers beyond early primary school. Mental addition and subtraction should be encourage as early as possible and the ultimate mental activity for children under 10 is random recitation of the times table. From ‘1 times 1’ to ’12 times 12’ there are exactly 78 unique answers out of 144, because two numbers yield the same answer when multiplied irrespective of the arrangement. Take away the 1, 10 and 11 times tables which are ridiculously easy to remember, and we have roughly 40 unique multiplication answers to be memorised. There are no tricks here but plain memorisation. Parents can download multiplication sheets off the Internet where there are countless colourful ones. To encourage the children, parents can play assorted multiplication games with them. Games could vary from speed based (e.g. who can recite the 7 times table the quickest) to random recollection (e.g. 2 players quiz each other randomly) and division games when the times tables have been memorised.
With dedication, any 10 year old can learn up to 12 times table within a month; they learn to play more complex games in less time! Considering the good that this basic knowledge do to our children’s education, I hope that the government encourages knowledge of the times table and that parents, more importantly, see it as an absolute skill that they must help their children acquire in order for them to not only cope but compete in the real world. The competition starts much earlier than most people think.