It's certainly possible to do higher level maths without knowing tables, and I've had a few dyslexic pupils go on to study maths at university, even without knowing tables instantly. The difference is that maths just made sense to them, easily - their calculations were very fast, but more importantly, there were already able to follow the lessons without getting hung up on the tables, because they just understood how numbers worked. And if that's the case, it's fine.
The difference is in more average or struggling pupils, who would have been able to cope OK with secondary level maths if they had more of the basics as automatic skills. They still have a slight fear of maths, a feeling that it's all a bit mysterious, and adding several extra layers of calculation on top every time they want to solve an algebraic equation, for example, just makes that whole process of algebra seem incredibly mysterious and complex. You want them to be able to follow the point of the lesson - how to set out a simple equation, for example, and how to work backwards in little steps - but that is a whole lot easier if you can simply divide something and get the answer quickly, and follow through with the logic of the lesson, rather than having to side step and work out the answer.
So it's not rote learning in and of itself that is desireable. And certainly not at the expense of understanding - if a child doesn't really understand what multiplication is (or how it relates to division), then that is a first priority, before any worries about instant recall. But once the concepts are understood, then instant recall does make later lessons easier. It's not that it's important in and of itself, but it's more that having them learned instantly can stop later lessons seeming more complicated and convincing a child that they don't understand fractions or algebra or anything anyway, and giving up. If the child is quite mathematical already, then knowing the tables instantly isn't necessarily going to hinder them, and there are some children who simply can't end up learning them automatically - and certainly they need to know ways of counting on and calculating them as well, for times when instant recall isn't possible or appropriate. But for most children, it is useful to encourage instant recall just to make their lives easier at secondary school.
Another aspect of rote learning that is sometimes neglected is just the way that it makes things utterly automatic in the end, to the point where remembering how to do it is not an issue. It is boring, but in some ways, that can be beneficial. I had to do pages of multiplication problems at primary school as homework, long after I knew how to do it, and I certainly understood it fine. So the homework was boring, and there are other things I would have rather done. But this produced a couple of good effects: one, I was so used to it by then that I could just dash off the answers whilst thinking that I wanted to be done with it all - it had become a totally automatic procedure that could then be a tool later in life without it being any kind of an issue or needing any thought, and impossible to forget. Nowadays, I have GCSE pupils who still see a 2x2 multiplication as a 'maths problem' that will come up on an exam, rather than a tool that they use to do maths problems. Secondly, being bored with rote work meant that I wanted to get through it as fast as possible, and thus discovered shortcuts, which were actually quite valuable insights into how numbers works.
That would be my soap box, instead of rote learning of tables. Knowing that if you know 2x3 is 6, then you also know what 6 divided by 2, and 6 divided by 3 are, is such a help and my kids don't seem to be taught this. (Division is a black art to them.)
Oh, absolutely. This is something that they really don't stress enough - how all the facts are related, and what division actually is. It's a huge help later for doing fractions, algebra, etc., and even for things like calculator papers where they need to do division, and it would be quite a simple problem on a calculator, but the children don't realise that what they need to do is divide, because it's rarely been taught in those words before. They have the same problem with subtraction, having spent most of primary doing mental maths, where subtraction involves counting up from the smaller number to the higher number - which is a great strategy, and I'm all in favour of children using it mentally - but they don't seem to realise that it is subtraction, so when they come to show their work later on, they often don't know how to show that that is the operation they did. They can add up two angles of a triangle to get 135 degrees, for example. They can mentally get from 135 to 180, and say that the missing angle must be 45 degrees, fine. But what they can't do is write down 180-135 = 45 to show how they did it. In their minds, they were adding, and they are at a loss of what to write. Some of them can write 135+45 = 180, which is probably enough to get them the marks, but I don't think that actually shows quite the same point - it's proving something after the fact (after you've got the answer) rather than showing how you got the answer from the numbers that you were given. And I think that's an important concept in 'showing your work'. (But then, maths is a bit of a passion of mine!!).
Dyslexiateach - are you able to give examples of some of the non-rote learning methods you use?
Many of the strategies given above are useful, like the fingers trick for the 9s, or all the doubling and halving calculations - all worth a child understanding, because it does make the whole number system clearer to them if they do. So if a child can't learn them instantly, I would certainly encourage those sorts of strategies next, because it also helps with division. If you can do things like 3x4 is 12, so 6x4 must be 24 because it's twice as much,, it's not a big jump to being able to do 24 divided by 6 by thinking of it as two twelves, which are each made up of two sixes. That's already half way to doing factorising! And when you get the hang of things like that, it's easier to do things like 72 divided by 6 by cutting it in half to 36, and then using your times tables to do 36/6, and then doubling the answer. And endless other variations - you soon see numbers as combinations of other groups of numbers, which is really important.
However, that still isn't instant recall, and if the child can learn them instantly, it is worth it. For many children, rapid fire practice in random order will be enough, just constantly going over them with flashcards, etc, so that they have to do the problems in the order that they come up in, without time to calculate or without secretly finding them problems in order (which children do on written tests if you aren't watching them, even if the problems are actually supposed to be in a random order!).
Many dyslexic children have problems with sequencing, which can make it difficult to use the counting on methods, and also with phonological skills, which makes rote memory hard. It's a lot to have to remember "six eights are forty-eight; seven eights are fifty six", and get all the words in the right place and right order without getting totally mixed up.
So it can help to remove as much of the verbal aspect as possible, and use other senses instead. This is not how it's usually done at school, and there's not a lot out there already, so I had to come up with my own methods. One that I created was a card game that used the positions on the clock to act as spatial cues - the child puts the answers to the table at the position on the clock that matches (e.g., in the 3x table, 3 goes at one o'clock, 6 goes at two o'clock, 9 goes at three o'clock, etc). After playing the game lots of times, the children were able to associate the card with the position on the clock, and this meant that they only had one verbal aspect to remember - the answer - and the other part they could do visio-spatially. For example, they would be able to picture that the '21' card goes at 7 o'clock, so if they need to know 3x7, they just have to imagine what card went at the 7 o'clock position, or to do 21 divided by 3, they think 'where did the 21 card go?'. I find it very effective for reinforcing the idea that division and multiplication are intertwined, because what the children are really practising is division. There are a lot more details on my website, but as I also created a game for iphones that does this automatically (children got bored with having to shuffle and deal the cards all the time, and enjoyed the multisensory aspect of the touch screen, so I decided to make it easier for them!) and the website has the game for sale, I am not sure that I am allowed to mention it here. I got told off for advertising. but it's quite hard to explain how to use visual methods without mentioning it because it's much easier to explain when you see the pictures, it being a visual method and all!! Also there isn't a lot else out there for non-verbal ways, so it's hard to not mention it if people want to know what I use. But it's totally easy to create the game yourself from looking at the pictures/rules on the site, as that's what I did for years. It's the classic game of clock patience, adapted for tables, and I originally just made it with decks of real cards, with stickers and numbers glued on and laminated. The website is [[http://www.timestableclock.com/ here] and if you look at the 'about' page, it explains how to play and how you could create the game yourself. I find that all the colour coding is also good for chidlren with strong visual memories, because it shows things like 7x3 and 3x7 being the same thing, as they are always shown in the same combinations of colours.
So that is my most useful method for children who are strong visual learners.
A second method that I use a lot for children who don't have good rote memories is a rhyming scheme. It was from a site where you could buy the cards/stories, and then I adapated and elaborated them to make them easier to remember and distinguish. I think it's called 'Times tables rhyme'. There are characters for each digit - I only bother with 2-9, as I think 1x, 10x, 11x and 12x are either easy enough or not needed. The characters all interact with each other, and there's a little story about it that ends with a catch-phrase that rhymes with the answer to the problem. For example, character 6 and character 7 meet up at the gym, and one says to the other 'I didn't know you were sporty too', which rhymes with 42, so 6x7 is 42. The advantages of the system are again that it allows that instant access to a fact without having to count through the table, which is what a lot of dyslexics find hard. And although it sounds like a lot of memory work for each fact, it's a much more multisensory way of memorising, so the children have more cues to use. I get them to draw the characters, draw the stories, act them out, put on voices, etc. The disadvantages are that some of the stories sound very similar to each other and it's not obvious why it should be these two particular characters in a situation and not others. I've fixed that by elaborating the stories and giving more links to specifics about the character, personality etc so that it's more inevitable why it should be those two that are meeting at the gym, say. The second problem is that the rhymes can be hard to do for children with phonological difficulties, not least because they are not perfect rhymes to begin with, and you have to stretch your imagination a little on some of them. With enough practice, many children can get over that though. The third problem is that it is quite memory intensive, and the child has to really throw themselves into the system, because at first, learning all the characters, and then a whole list of phrases, and then the rhymes to them, seems like a lot. And the phrases have to be learned exactly - if instead of learning "plenty of fun" (twenty one), the child remembers "lots of fun", it's not going to rhyme!! However, I have used it successfully with many children, and a lot of them really enjoy it. It does take continual practice to move from having to go through the whole story, to just the phrase, to eventually knowing them instantly. Many children never get to the instant recall stage, but at least this method means that they dont have the error-prone counting through the table, but can go straight to the fact that they need.
Those are the main non-verbal methods that I use. A third method I use, that is still verbal rote memory, but at least encourages the idea of multiplication and division being opposites, is triangle flash cards. You put the three numbers of the fact (e.g., 4, 6, and 24) in each corner of a triangular shaped card, and cover up one number with your thumb, and the child has to say what the other number is. It shows how 4x6 is related to 24/6 and 24/4, and also that 4x6 is no different to 6x4. If you are going to use flash cards and rote memory, then I find that can be a useful way. You can also buy triangular flash cards already printed.
hope that helps.