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by Matthew Leitch, 5 October 2012, expanded 3 January 2013
If your child struggles to learn number facts, such as times tables facts and number bonds, then it may be that he or she will find it easier with a better way to practise them. That's what this article is about.
You may have noticed that number facts don't sink in. Alternatively, you may have noticed that number facts seem to go in, but a few days later they've gone again. Hours of practice seem to make no difference. Progress feels agonizingly slow. If any of this is familiar, then read on.
Between the ages of about 4 years old and 10 years old most children learn number facts for mental arithmetic that include addition and subtraction of small numbers and multiplication facts from 1 × 1 up to 12 × 12. Some children learn much more than this.
It is sometimes said that maths is logical, and that you can work everything out if you need to. That's true, but you can't work it out quickly enough for that to be a workable approach. In reality, memory is vital to mathematicians, which is why they lose so much skill in just a few months of not practising.
To be specific, your child needs to be able to recall number facts within a second or two, with no working out, given problems like '5 × 7 =' and '3 + 7 =', in any order, with no other clues.
Being able to count up in 5s or chant a times table in order is useless. Furthermore, it is not enough to be able to work out times tables by adding or other strategies, or to work out addition and subtraction by counting on a ruler or with fingers. All those are temporary strategies that should be used for as short a time as possible, if at all.
If your child has to do calculations to work out simple number facts then doing bigger calculations will be exhausting and probably impossible. Your child will suffer fatigue, confusion, and failure. Mastering those number facts properly is by far the easiest and least painful way forward.
If progress is slow you may find that your child's school keeps him/her using 'scaffolding' methods, like using number lines and fingers, instead of getting on to the practical methods that are required for competence. This can lead to a child starting to believe that the number line and finger methods are the proper methods to use and that nothing else is needed. They may stop improving altogether.
Although we are used to the idea that learning multiplication facts (the 'times tables') takes place over years of schooling and requires endless grind, I have seen cases where the whole lot has been learned in a few days. Admittedly, that is not normal, but how long should it take to learn all the number facts (not just times tables), if there are no problems? Nobody seems to know.
Here's an estimate. There are 328 facts to learn, in 91 groups, (see appendix below) so if it took 2 minutes per fact (30 seconds initial study to build three chunks, and the rest for 30 tests to build speed) then that would be 656 minutes, which is just under 11 hours of concentrated study. If a student can do 10 minutes of concentrated study per school day then that means 66 days of study, which is about 13 weeks, or 3 months. A student who can focus for half an hour a day would do it in a month.
This is pure guesstimation, but surely if the number facts have not been learned within, say, a year then something is going wrong.
As a first step, cut out the activities that take up time but don't help, or actually make things worse:
DON'T chant tables or count up in fixed units (e.g. 'counting up in fives'). Children think that, because they can do this, then they know their tables and have nothing more to learn. Wrong.
DON'T use number lines, rulers, or chunking for one minute longer than is absolutely necessary. These are slow methods that schools teach to get children started and help them understand in principle what they are doing. You're only supposed to do them temporarily and could actually avoid them completely.
DON'T keep testing memory for a number fact that your child gets wrong often; it just gets them confused so stop and give them a chance to look at the answer for several seconds before trying again.
DON'T confuse being able to work out an answer with knowing the answer from memory; we want answers from memory because anything else is hard work in a bigger calculation.
There are several reasons why number facts are inherently hard to memorize. They are abstract, often unfamiliar, and there are lots of superficially similar ones. However, the most important point is that they seem arbitrary at first. Of course all the number facts can be worked out, and they are not arbitrary at all. Two plus two is always equal to four, and there is a good reason for that. However, number facts seem arbitrary unless you work them out and working them out from first principles is too slow in most practical situations. So, in practice, they feel arbitrary and so very hard to remember unless you do something to make them less arbitrary.
What your child needs to get started is ways to link questions to answers so that the answers seem less arbitrary. There are two types of link that can be used:
Very short calculation rules that can be used within 1 or 2 seconds until automatic recall develops. For example, "10 times any whole number is the whole number with an extra zero on the end".
Rules that help you pick one answer from the possibilities that pop into your mind. For example, "an even number of sixes gives an answer ending with the even number e.g. 2 × 6 = 12, 4 × 6 = 24", and "adding two small numbers gives a larger but still smallish number".
Here are some good things to do:
DO tackle a small group of number facts at a time, say 3 to 12 facts in one session, depending on how quickly the learner progresses.
DO start with easy-to-remember number facts and build on them.
DO start with number facts your child already knows. For example, your child may already know that 3 + 3 = 6 from looking at a die when playing board games. Get them faster, more reliable, and more confident. If you can make those facts automatic then they can help with learning more facts.
DO focus on addition and subtraction before tackling multiplication and division.
DO focus early on the number facts that can be reached by a simple rule without relying on other number facts. This includes rules for 0 added to anything, 1 added to anything, 0 times anything, 1 times anything, anything divided by 1, 10 times anything, and 11 times 1 to 9.
DO learn about the relationship between addition facts and subtraction facts, and between multiplication facts and division facts. Then tackle the number facts in related groups such as 2 + 3 = 5, 3 + 2 = 5, 5 - 2 = 3, and 5 - 3 = 2. This involves learning about subtraction and division as soon as possible, but there are other good reasons for doing this as well.
DO use other rules that quickly give number facts based on number facts already known. For example, "to get 5 times 1 to 12, divide by two, multiply by ten, and add 5 if there was a remainder e.g. 4 × 5 ... 2 ... 20, e.g. 5 × 5 ... 2 rem 1 ... 20 + 5 ... 25."
DO give several seconds to look at and think about each number fact before trying any kind of test of memory. (See below for detailed suggestions on how to do this crucial stage.)
DO move on to testing memory as soon as a fact has been studied. The point of the tests is mainly to practise recalling the answers in response to the questions, not really to 'test' whether learning has taken place. Doing small groups of number facts each time, and using quick rules, means that you can quickly move on to lots of successful tests which promote consolidation, speed, and reliability without compromising and using unrealistic cues as you would with chanting tables in order.
DO test far beyond the point where a correct answer has been given. This is called overlearning, and overlearning helps. You want speed and automaticity.
DO test facts at random e.g. 2 + 3 = ?, then 4 - 7 = ?, then 3 + 1 = ?, and so on. That's how the facts will be needed. If you use any pattern at all your child may become dependent on that pattern.
DO mix the questions. If there is any part of the question that your child does not have to pay attention to in order to get the right answer then your test could do more harm than good. For example, if you dedicate a session to the 5 times table and ask '3 × 5 = ?' then your child only has to notice the '3' part. The rest is the same for the whole session. If you then surprise them with '3 × 10 = ?' you could get the answer '15'. If you wait a week then ask for '3 × 5 = ?' you could find your child has forgotten because all they did was learn to say '15' when you said '3 blah blah'.
DO mix operations as soon as you can. Mix addition and multiplication, mix addition and subtraction, and mix multiplication and division. This allows you to mix the questions more fully. If you find explaining division tricky, then ask 'What number multiplied by 3 makes 15?', or show '? × 3 = 15' as a step towards division number facts.
If you've been reading critically you will have noticed that there is a difficult compromise between introducing new number facts and making sure your child notices each part of the question. For example, if you introduce three new facts then your learner knows that the answer must be one of the three just introduced. They don't necessarily have to attend to all parts of the question to know which.
A certain amount of this is unavoidable, but you can minimise it by making sure all elements of the question are noticed on first study, and moving to a larger set of number facts for testing purposes as quickly as possible.
To an adult, many numbers are recognizable, special chunks with their own familiar properties. For example, think of the significance of 16, 21, 50, 7, and so on. We are particularly familiar with smaller numbers and with numbers we learned number facts about at school. Prime numbers, like 37, 41, and 59, seem less familiar and memorable.
However, to a child who has little or no knowledge of numbers they have less character and uniqueness. If you say '21 16 7 50' it just sounds like 'blah blah blah' to them.
So, take the time to build each number into a familiar unit. For example, in '3 × 5 = 15' it may be that 15 is a new number to your child but 3 and 5 are familiar. Before 15 can be linked to '3 × 5' using '=', the '15' needs to become familiar, so get him/her to take a few seconds to look at the number, say it a few times, and see where it lies on the number line. You could even think of some things that are famous about the number outside the world of mathematics, such as that 15 is the number of players in a rugby union team.
Numbers are not the only chunks a child needs to notice and get familiar with. Each 'question' is different and the student needs to recognize each one. For example, '2 + 3' is different from '2 × 3' and each needs to become familiar and recognizable. So, take a few seconds to look at each of the three elements of each question, then get used to seeing them together and recognizing that familiar chunk. When the question and the answer are familiar it is time to learn to put them together.
Understand that a fact like '4 × 7 = 28' is a lot to take in when all you know is '4', '×', '7', '=', '2', and '8'. To be precise, it's 6 things to take in and that's too much in one step. However, combining '4', '×', and '7' to make '4 × 7' is an easier step, combining '2' and '8' to make '28' isn't hard either, and finally putting '4 × 7' together with '=' and '28' to make '4 × 7 = 28' is just lumping together three elements.
These details, though they seem trivial, all need careful, patient attention. By studying the details before the first test of memory, and whenever a recall mistake occurs, your child can make better progress.
Here are some rules you can use for addition and subtraction facts:
0 + X = X; X - 0 = X
Need X + 1, then count on one from X.
A + B = B + A
Need A - B? What added to B makes A? (E.g. Need 6 - 4. What added to 4 makes 6? It's 2.)
Know A + B and need A + (B + 1)? It's just one more. (E.g. if you know 2 + 2 = 4 and need 2 + 3, then it's 4 + 1, i.e. 5.)
Know A + B and need (A + 1) + (B - 1)? It's the same. (E.g. if you know 2 = 2 = 4 and need 3 + 1, then it's also 4.)
Here are some rules you can use for multiplication and division facts:
0 × X = 0; 0 ÷ X = 0
1 × X = X; X ÷ 1 = X
An even number of sixes ends with the even number and the first digit is half that, up to 8 × 6. Specifically, 2 × 6 = 12, 4 × 6 = 24, 6 × 6 = 36, 8 × 6 = 48.
10 × X = X0
11 × X = XX
9 × X = (X-1) (9-(X-1)) (E.g. 9 × 4 = has first digit 3 and second digit 6.)
Need A ÷ B? What times B makes A? (E.g. Need 45 ÷ 9. What times 9 makes 45? It's 5.)
Know A × B and need A × (B + 1)? It's A × B plus A. (This works best when the final addition does not require carrying a digit. For example, 3 × 4 via 2 × 4 requires doing 8 + 4, which involves carrying a digit to reach 12. In contrast, 4 × 4 derived from 3 × 4 just needs 12 + 4, which is easier.)
Know A × B and need A × (2 × B)? It's twice A × B. (This works best when the addition at the end does not require carrying a digit. For example, it works well for 4 × 6 used to reach 8 × 6. Twice 24 is obviously 48. In contrast, it's harder to use 3 × 6 to get 6 × 6 because twice 18 requires carrying to get to 36.)
However, with all but the simplest of rules (e.g. times by 1 or 10) the real goal is to be able to recall the answers immediately with no working out. So, be especially careful not to mistake memory for working out with a rule, and keep on pushing for speed until memory takes over and is doing the work.
Maths Accelerator is a simple web page program, free for anyone to use, that provides tests following these principles. Don't be fooled by the lack of 'fun' graphics or the detailed performance statistics. Children love being able to see how they are improving and, just like in a computer game, they study those stats intently. Even young children seem to be more motivated by progress than by shooting frogs, or any other irrelevant sugar coating.
One important limitation of Maths Accelerator is that it does not provide ideal support for the very earliest stages of learning number facts. However, as soon as you know enough to tackle one of the tests it is an excellent way to practise.
I hope this helps you and your children. Learning number facts is another of those big hurdles for young students.
First, the add and subtract facts needed for primary school arithmetic. Adding or subtracting 0 or 1 can be done by very simple rules, so no number facts are included in the list for them. This leaves 128 number facts in 36 groups.
Next, the multiply and divide facts needed for primary school arithmetic. Multiplying by 0, 1, or 10 can be done by a simple rule, so no number facts are included in the list for them. This leaves 200 number facts in 55 groups.
About the author: Matthew Leitch has been studying the applied psychology of learning and memory since about 1979 and holds a BSc in psychology from University College London. Until recently he worked as a consultant in risk management and systems for a leading professional services firm.
Contact the author at: firstname.lastname@example.org
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