Pre-counting skills: before a child can start counting, he needs to understand that there is such a thing as quantity, and means to represent it, first by showing and implicitly comparing smaller or larger groups of actual objects (or their representation), and then by using numbers. This is where the Doman method, embodied in a convenient format by the Brillkids 'Little Math' program, is really useful to develop this concept. This program shows quantities of objects (or people, symbols, etc) and makes the association with the number in fast sequence... so little by little, your child understands, in part at the subconscious level, that certain quantities can be represented by certain numbers. While using the program, it is a good idea to read stories that include counting and count objects in real life in front of your child: from 0-5 (0 is a very important concept, and I think it is good to introduce it early); 6-10; 11-20. Once your child's general vocabulary 'explodes' (sometime between 18 and 22 months), you can start encouraging counting by rote, so that he can have the vocabulary to practice 'one-to-one' object correspondence (in other words, counting actual objects on his own).
Counting skills: Actual counting is 'one to one' correspondence, i.e. pointing to objects sequentially and counting, without missing any object or counting any object twice. At first, this is going to take a while (perhaps several months), but practice makes perfect. This is where cute little puzzles, such as the eeBoo Counting Puzzles, come in to teach 'one to one' correspondence for up to 20. At the same time, you can go beyond the number 20 for rote counting (Joey could count to at least 100 by the time he was 19 months old). Once your child is comfortable counting to high numbers, other two important skills to develop (we did that toward the end of Joey's first year and through the first half of his second year) are counting backward (important for subtractions) and skip counting (by 2, 5, 10, 25), which is important for multiplication, and for counting money.
Addition: once your child has mastered 'one to one' correspondence for numbers up to 20, you can start teaching the concept of addition, which is basically the understanding that when you add two numbers, the result is the sum of the two (it seems obvious doesn't it..? It is, but not to a one year old). For this, I have found that again, eeboo type puzzles or Eeboo Math Addition Flash Cards or 'Addition Match-Ups' from Lakeshore Learning were very helpful because they show additions in a visual manner which also helps reinforce 'one to one' correspondence. Once you feel your child understands the basic concept, you can move on to having him figure out additions by counting on his fingers and some automation with additions which result in 10, then in 20.
Addition by counting on fingers: For X+Y, the way to do it is to have your child start at X, then use his fingers to add Y. For instance, 5+2 can be solved by starting at 5, raising two fingers to count: '6', '7'. If you think about it, no one needs to add more than 9 at a time. And when you do need to add bigger numbers, then it is time to write out additions, which is the next step. And by the way, your child does not need to be able to really flex his fingers to do this (this is hard for children under three years of age), he can do it with fingers extended as long as he pays sufficient attention. Regarding automation (memorization of simple additions), the tools we used for that were the 'Fact Family House', the 'Addition machine' (make sure you only allow you child to 'click' to check his answer, not before he gives an answer), 'Beat the Clock! Addition Facts to 18 Practice Board' (all three from Lakeshore Learning) and the Math Slam. The bottom line is to practice a lot and in a lot of different ways.
Solving more sophisticated additions: Once it is fairly painless for your child to do simple additions (which result in 20 or less), and assuming you have had him practice writing numbers (see the second year), it really will not be very hard to teach him to write out additions, first with no regrouping, and then eventually (it happened to us in the fourth year, when Joey was a young three) with regrouping.
Subtractions: once your child is doing additions, teaching subtraction is exactly the same process, except it will go much, much faster. He will use his fingers to count backward (also a very important skill). A hurdle will be to teach him to 'borrow', particularly when borrowing not just from the tens, but also hundreds and beyond in the same subtraction, but as always, it is a matter of practice.
Addition: once your child has mastered 'one to one' correspondence for numbers up to 20, you can start teaching the concept of addition, which is basically the understanding that when you add two numbers, the result is the sum of the two (it seems obvious doesn't it..? It is, but not to a one year old). For this, I have found that again, eeboo type puzzles or Eeboo Math Addition Flash Cards or 'Addition Match-Ups' from Lakeshore Learning were very helpful because they show additions in a visual manner which also helps reinforce 'one to one' correspondence. Once you feel your child understands the basic concept, you can move on to having him figure out additions by counting on his fingers and some automation with additions which result in 10, then in 20.
Addition by counting on fingers: For X+Y, the way to do it is to have your child start at X, then use his fingers to add Y. For instance, 5+2 can be solved by starting at 5, raising two fingers to count: '6', '7'. If you think about it, no one needs to add more than 9 at a time. And when you do need to add bigger numbers, then it is time to write out additions, which is the next step. And by the way, your child does not need to be able to really flex his fingers to do this (this is hard for children under three years of age), he can do it with fingers extended as long as he pays sufficient attention. Regarding automation (memorization of simple additions), the tools we used for that were the 'Fact Family House', the 'Addition machine' (make sure you only allow you child to 'click' to check his answer, not before he gives an answer), 'Beat the Clock! Addition Facts to 18 Practice Board' (all three from Lakeshore Learning) and the Math Slam. The bottom line is to practice a lot and in a lot of different ways.
Solving more sophisticated additions: Once it is fairly painless for your child to do simple additions (which result in 20 or less), and assuming you have had him practice writing numbers (see the second year), it really will not be very hard to teach him to write out additions, first with no regrouping, and then eventually (it happened to us in the fourth year, when Joey was a young three) with regrouping.
Subtractions: once your child is doing additions, teaching subtraction is exactly the same process, except it will go much, much faster. He will use his fingers to count backward (also a very important skill). A hurdle will be to teach him to 'borrow', particularly when borrowing not just from the tens, but also hundreds and beyond in the same subtraction, but as always, it is a matter of practice.
Multiplications: important pre-emergent skills for multiplication include skip counting. While you teach addition, it is also important to point out additions of identical digit and say "2+2 is the same thing as 2X2" as often as possible. When the time came to teach him the concept of multiplication, we used a manipulable called 'Learn-To-Multiply Magnetic Center' (Lakeshore Learning) to really drive home the idea of 'X groups of Y' is 'X multiplied by Y'. Don't be afraid to slow down and spend some time on this. Once Joey had a solid understanding of the concepts, for the purpose of memorizing the multiplication tables, I recited the multiplication tables instead of a lullaby (I do two rows and he does two, then the following night, he does the same two I did the night before, and I do two new ones). He loves it. At first, before he had memorized any of them, I reminded him that the next result was always the previous result + the multiplier (so if 2X2=4 then 2X3=4+2), which not only increased his understanding of how it worked, but also got him unstuck when he forgot one. At three and a half, Joey knows his multiplication tables. To seal the deal, we will use the multiplication machine and Beat the Clock! Multiplication Facts Practice Board (both from Lakeshore Learning), as we have for additions and subtractions. Soon (not yet), we will teach him to write out multiplications with multiple digits. I will then update this post.
October 2013 update: Joey has started doing complex multiplications with the John Hopkins University Center for Talented Youth program (accelerated grade 3 math). We could have taught him earlier, but decided he would get to it soon enough with that program, and there he is.
Divisions: we are just getting started with that. The way we are presenting the concept is to say 'if you have X objects, and you want to divide them into Y equal parts, then how many objects would you have in each part?'. I really want to solidify this concept before jumping to 'how to solve divisions', which entails asking the question: 'What do you multiply X by to get Z?' because I think it bypasses a real understanding of the concept (although we have made exceptions). I will update this post when we are further along with divisions.
By the way, as I mentioned in my 'Best Stuff' page and on the 3 TO 4 YEARS OLD: OUR PROGRAM page, we love the FlashMaster: Handheld computer for mastering multiplication tables that makes flashcards obsolete to practice all these skills.
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