Maths: Getting Started.

Most of the pages on this Web site are designed to enable children to learn everything on their own (although it is best if a parent can sit with them to begin with until they get the hang of things), but the early stages of maths should be taught to children when they are too young to be able to learn independently, so you will need to step in and become your child's teacher. It isn't difficult, though: all you have to do is follow the instructions here.
       I'm going to use the word "he" rather than saying "he or she" every time I refer to your child - it saves space but it's also much clearer to read. You must work with your child in short, focussed sessions, never spending so long that it bores him, and never working so intensively that it becomes a struggle. If you do it right, it becomes a game, and the rapid progress should make it fun. If he becomes bored, abandon teaching for the day immediately. You can, if necessary, use rewards such as sweets to generate interest, but if you do this you must be fair: never promise a reward for your child doing something only to demand that he must do something else in addition before he gets it. Children are very precise fairness meters and they can often explode if you don't play by the rules, even if your intentions are good, so be very carefull not to turn any of your attempts at teaching into a conflict: your aim should always be to make it fun, so be on your very best behaviour and offer lots of praise for the smallest of achievements.
       Don't be scared of the idea of teaching maths if you find it difficult yourself: the difficulties you have are almost certainly because you weren't taught maths properly, so there were gaps left in your knowledge which have never been filled. This course is designed to make sure that your child will never have to try to build any knowledge on top of an incomplete foundation, but it might also help you to fix the holes in your own understanding of maths at the same time. You may be astonished to learn that a vital early step in maths is almost never taught in schools, leaving children struggling to work out their own way of getting round the problem for themselves: some children never find a way past this barrier and they often spend the rest of their lives feeling like complete failures and believing themselves to be stupid, when in reality they have simply not been taught a key skill.

Step 1:-

The first thing you need to do is check to see if your child can count up to ten. If he doesn't find this easy you'll have to practise it with him until he can, counting things at every opportunity, though not so much as to bore him. Don't just rely on songs to teach him to count: they help a lot, but he'll have to count real objects while saying the numbers if he is's to understand what the words actually mean.
       Don't be in any rush to move beyond this first step: it is not a race. Wait until he is reasonably competent and can count objects accurately most of the time. If he isn't on top of this step, moving on to step two will be a complete waste of time.

Step 2:-

Find ten small items such as Lego bricks for your child to count. This time he will count them in groups and get the idea of adding groups together. If you have a group of four items and another group of three, he may add the three to the four by counting the whole lot from scratch even if he's already counted the first group and knows that there are four items in it. Your aim here is to get across the important idea that once you know the number of things in one group you don't need to count them again when you add a second group: you just say the number of the first group while pointing at it, and then count on up from there while counting the items in the second group, thus adding them on. Each time, you should give him a group to count, and then give him another group to add on to it.
       Your child will be ready to move on to the next step once he has stopped counting them all from scratch , but he must also consistently get the right answer for the total. If he has difficulty, you should make the second group a single item every time and only increase the size of the second group once he's got the hang of that. Don't feel that you have to use the same set of small items every time: switching to using sweets can inject a great deal of interest.
       It is also a good idea to do some counting with money at this stage. A coin with the value of two or five will behave as if it is a group of two or five items which have already been counted up, and this will help him to get his mind round the whole business of adding groups of items together more easily.

Step 3:-

Now you need to teach him how to count up to twenty so that he can add bigger numbers together. Continue with the adding as in step 2, but now you will use twenty items instead of ten. Don't move on to step 4 until he has a good grasp of those numbers over ten. Make him count money as well, and introduce a couple of coins with the value ten on them.

Step 4:-

Stop using items for counting and let your child count on his fingers instead, making sure at first that the answers aren't bigger than ten, but once he's got the hang of things you can start using some with bigger answers as well: this will lead him to notice that he doesn't have enough fingers to do this. Let him struggle with this for a while, and then move on to step 5.

Step 5:-

Read this whole section and make sure you understand it completely before you try to teach any of it to your child: I want you to understand the difficulty with the process of adding because it's far harder than it looks.
       You may not realise it, but when you add two numbers together, even small ones, you actually have to do two lots of counting at the same time, and this is the difficulty that most teachers overlook. For example, if you add 5 plus 3, you would start counting up from the 5 (as shown in red below), but in order to know when you've finished adding the 3 onto it you actually have to count up to 3 at the same time (as shown in green):-

6, 1;   7, 2;   8, 3.

When the second count gets to 3, the first count must have reached the answer, so the answer is 8. So, how can you do two lots of counting at the same time when your brain isn't designed to do such a thing? Children usually try to solve the problem by doing one lot of counting on their fingers while they do the other lot of counting in their head. If you watch the process in its most primitive form, you may see a child count out the first number on his fingers, then you see him add on the second number by putting his sixth, seventh and eighth fingers up: he is thinking "one, two, three" in his head as he does this. He may then count all the raised fingers from scratch before he can see that the answer is eight.
       This is obviously a slow method for adding and it doesn't even work if the answer's bigger than ten, so there comes a point where children have to find a new way to do adding. Unfortunately, very few children are ever taught a better method because teachers are generally blissfully unaware that there is a double counting difficulty: most children are therefore left to invent their own way of doing it and the result is that they can end up using extremely inefficient methods. Most of them do eventually get up to speed, but only once they have done so much adding over so many years that they have memorised all the correct answers - clearly it's infinitely easier for the child if he has been taught an efficient way of working out the answers instead.
       One answer to the double-counting difficulty is to count on your fingers in a different way: instead of counting up to the total on your fingers, you use as a count for the number being added on, so you would hold up three fingers if you want to add three to five, and then you count "six, seven, eight" in your head as you move your eyes in steps along the three raised fingers.This method works even for nine plus nine, but it's still slow and cannot be recommended. Even so, it may be the only method that works for some children with learning difficulties.
       A better solution is to do one of the two counts by using a different part of the brain so that the normal counting part of the brain has only one lot of counting to do. Using rhythm for the second count is a good system: the music part of the brain deals with the number being added on while the maths part of the brain counts up to the answer. Each number up to nine has its own associated pattern which must be used whenever it's added to another number. I came up with this method myself at the age of four while still in nursery school and never told anyone about it: it enabled me to add strings of numbers together rapidly and for the most part without error.
       All the patterns are based on repeating lots of four: ONE two three four ONE two three four ONE. It's better to get rid of the numbers though so that you can concentrate on the rhythm, so we'll replace them with the word "da" (which sounds marginally less silly than any alternative I can think of): DA da da da DA da da da DA. The best pattern to teach your child first is probably the one for five: DA da da da DA. Get that pattern fixed in his head by making him chant the numbers ONE two three four FIVE several times. Next, get rid of the numbers and use DA da da da DA instead. Then give him a sum to do like three plus five: he should say "three", or think it in his head, and then he should count up from there using the rhythm which he has just learned: FOUR five six sev'n EIGHT (note that he will need to learn to shorten the word "seven" as much as possible - he will do this by copying the way you do it).
       If he doesn't understand why the first number to say when applying this rhythm always has to be one up from the number he's adding five to, you may need to get those Lego bricks out again to remind him how it works in the real world: then he'll be able so see how the numbers fit to the real groups of items being added together.
       Initially you should concentrate on getting him to add five to each of the numbers from one to five, but later on you can get him to add it to anything up to fifteen, or even higher if you like. Don't move on until he finds adding five easy and gets it right most of the time: the occcasional error is just a sign of a lapse of concentration and doesn't matter much, but frequent errors indicate that there's still a problem, though boredom could be an issue if his ability slides backwards. Okay: you've read it all now, so it's time for you to go and try it out.

Step 6:-

Now teach him the nine pattern: ONE two three four FIVE six sev'n eight NINE. Make him say it several times as before, and then replace the words with "da" as before: DA da da da DA da da da DA. Then give him lots of sums to do where he has to add nine each time. Keep practising this until he finds it easy, but you'll need to show him how to say the longer numbers quickly while putting little pauses between them to keep them separate. You can tap your finger on a table while counting trough the -teen numbers (tapping: hard soft soft soft hard soft soft soft hard), making the "-een" part almost inaudible. Don't worry too much if it takes him a long time to get the hang of these bigger numbers: he'll get plenty more practice later on, so he can move on to the next step once he is reasonably good at adding nine to the numbers one to four.

Step 7:-

Now give him sums to do where he either has to add five or nine to other numbers: he has to learn to use the right pattern to add on the required number. Don't ask him to add anything other than five or nine though. Once he finds this easy, move on again.

Step 8:-

Introduce the pattern for seven now: ONE two three four FIVE six sev'n. Keep the seven short. Make your child chant through the pattern several times, then switch to the "DA da da da DA da da" version. Give him sums to do which always involve adding on seven, and once he's got the hang of it, chuck in some fives and nines as well to make sure he thinks about which pattern he should be using.

Step 9:-

Now it's time to teach him how to do all the others. Give him sums to do where he has to add on any number from one to nine and get him to work out the right pattern for himself, but you should always use the same number several times in a row to reinforce the pattern he has just learned, and then you should mix in some numbers with patterns he already knows, further reinforcing them in his head. To work out each pattern he should chant through it using the number names, then switch to using "da", and then apply that pattern to the actual sum. You'll need to help him do this, and some children may need a lot more help and practice for each new number pattern than others, but don't worry if it doesn't come easily because it says nothing about his intelligence: bigger brains often take longer to program than small ones.
       Keep working on this with him from time to time, several times a day for a few minutes if it doesn't bore him, and he will soon get to the point where he knows straight away which pattern to use to add each number. Once he becomes sufficiently good at it, you should encourage him to do the counting silently in his head. Don't be in any hurry to move on to the next step if he is not fully on top of this one: it's better to build on a foundation that's a little stronger than necessary than a foundation that's too weak.

Step 10:-

Teach your child to count up to thirty, if you haven't already done so.

Step 11:-

Teach your child to count up to a hundred in tens.

Step 12:-

Teach your child to count up to a hundred in ones. Once he is capable of this, you can give him sums to add where the first number is anything from one to a hundred, or more: the sky's the limit. And find a way to celebrate: he has achieved a significant milestone.

Step 13:-

In preparation for subtraction you need to make sure your child can count downwards as well as up. Start off by counting down with him from ten to zero, or "blastoff" if you can find a way to make it more fun (such as a rocket propelled by water and pressurised air). Don't move on until he can do this on his own with ease.

Step 14:-

Now you are ready to introduce subtraction. If your child doesn't know what taking away means, you'll have to get out twenty Lego bricks to demonstrate what it's all about. Taking away using the rhythm patterns is just like adding, except you count down instead of up. Give him some take-away sums where the number being taken away can be anything up to nine and the number it's to be taken away from must be either bigger or the same size. Don't make the numbers bigger than ten to start with.
       Sometimes you might want to make the number to take away bigger than the one it's to be taken away from: he'll get to zero before the pattern runs out. Tell him it's impossible to take more things away that there are to start with and use the bricks to prove it. (He doesn't need to know anything about negative numbers for now - they're horribly abstract because they don't actually exist in the real world).

Step 15:-

Teach your child to count down from twenty to zero, and then do some more subtraction taking numbers up to nine away from numbers as big as twenty.

Step 16:-

Give him sums to do which can be either adding or taking away: he needs to learn to think about which type of sum it is every time so that he counts in the right direction.

Step 17:-

Teach him to count down from thirty to zero.

Step 18:-

Teach him to count down from a hundred to zero in tens.

Step 19:-

Teach him to count down from a hundred to zero in ones. This is the biggest counting challenge of all: it gives a child a considerable sence of achievement when they are able to do it. Another celebration is due, for this is another big milestone.

Step 20:-

Your child is nearly ready to tackle big sums on paper, but before he does so you need to check a couple of things. Firstly, can he read numbers? If not, you could write them out on cards (or find numbers in some other form such as fridge magnets or on wooden blocks - you'll almost certainly have something suitable in the house already) and test him with them, giving him a chance each time to tell you which number it is before you tell him. If he makes no progress with this, he may have eyesight problems so you may need to seek the advice of a specialist.
       The second thing to check is that he understands written numbers of more than one digit. If he doesn't yet, then you may need to write out lot of numbers of two, three and four digits and use them like the cards (or objects) with single-digit numbers on them, again teaching him to read them out correctly. He needs to learn to count the digits to determine whether to say "thousand" or "hundred" after the first digit he reads out, and you may need to spend some time with him practising this.

Steps 21 and upwards:-

The groundwork is complete and your child is ready to do some serious adding, but we have to move on to the next page for this, and the course changes radically in style because it is aimed at children who have been taught how to read by another part of this Web site. If your child can't read yet, you should switch attention to that now and put the maths on the back burner for a time because reading is far more important, though you must keep doing a little maths to maintain what he has already learned - making sure you supply a little pocket money is a good idea as it will give him something to count and the motivation to do it (and you don't have to give him the same amount every time, though it would be very unfair to reduce it). When you return to the maths course, there's absolutely no reason for you to stop helping your child if you want to, but do give him a reasonable chance to learn each step on his own before you step in: learning to teach himself is a valuable skill which is never encouraged in schools (for an obvious reason).