Once Asher regained his fluidity and felt comfortable exploring numbers and what they represented, we moved on to two other basic concepts: grouping (or subtraction and addition) and multiplication (with division). Both of these processes really contain the aspects of their counterpart within them (as in addition and subtraction) so seperating the functions did not make sense to me.
Asher already has a good sense of writing out numbers, including place values and what they represent (at least as far as ones, tens, hundreds, thousands, ten thousands and hundred thousands), so we did not review this. He was not very fast at regrouping, and to be efficient and comfortable with mental arithmatic, this is a necessary skill. I find using mathematically accurate manipulatives to be most helpful in practising this. I bought ours here for $35, and I would not try to teach math without them. They have been in use since they arrived four years ago.
I started him with ten units (single cubes). He played with them for a while, dividing them and handling them. Then I took all ten of them in my hands and hid them. I opened one hand and he had to tell me how many cubes were in each hand (3 out, 7 in; 5 out, 5 in, etc). If he shouted the numbers faster than I did, he won a token (a luckybean). Once he collected ten tokens, he won a square of dark chocolate. We only played this game for three days, and he mastered groupings up to ten.
We are now practicing groupings up to one hundred. I use nine rods of ten and then ten cubes. Now I divide them up under two cloths, and the game continues. Next week, we will likely be ready for numbers up to one thousand.
As Asher was handling the cubes, he started building with them. This led to a natural exploration of squaring and cubing numbers. When he held a single cube, it led him to think about dimesions. All of this easily transitioned into discussing multiples and divisions.
Again, the rods, squares and cube provided an easy visual for multiplication. But here is where you can get really tricky! A huge variety of resources and games exist for exploring and practicing multiplication tables. RIght now, we are still exploring and conceptualizing, not memorizing, so here are the materials I am using:
1. A Waldorf style math board (base ten). Asher and I hammered ten nails into a scrap of plywood in a circle, drew the numbers on in Sharpie, and found a scrap of yarn to do this. There is a good video illustrating how to use this tool here. However, I had one major issue with it as it does not explain why you can reverse the direction when doing the complimentary times table. Asher and I discussed this on our own, though, and then it all made sense to him.
2. We did some skip counting (ie-- counting by twos, eights, etc). However, I find this more helpful at the memorization, not conceptualization, stage as it is very easy for Asher to forget that multiplying and dividing also represent groups of things (as in 2x4 means two groups of four).
3. We have all of these multi-pies from the toymaker printed out. (Image from thetoymaker.com). They are fabulous! They show the numbers' relations to one another in groupings, so they can be used to both mulitply and divide. Plus they are pretty and free.
As we move into memorization, I am sure I will add more tips and tricks here. One thing I do try and do is to recite any math facts both backwards and forwards, and in compliments (learn both multiplication and division at the same time so that 3x7 is 21, and 21 divided by 3 is 7, etc).