Tommy's autism means that when he learns something, he learns exactly that thing that he is taught. For example, when he learns math with manipulatives, he learns to use manipulatives. He doesn't learn the theory behind the manipulatives. He doesn't learn to substitute symbols for manipulatives. And so, without the manipulatives, he has no clue what to do.
This is becoming more and more of an issue as we work on multiplication. Yes, he can now use charts which he made himself (by skip-counting) to do multiplication of single numbers through the tens tables. And he can multiply a double-digit number by a single digit number with no carrying. This is WAY more than he could do at the end of last year.
BUT - he still doesn't seem to really understand why he can do what he can do.
For example - he created his 2 times chart by putting an X on every other number. So when he sees 2X10, he simply counts ten X's, and when he's done - his finger is on the 20. He's solved the problem, and puts down the right number. But he doesn't actually know how to skip-count by twos. I know this because I've made sequencing worksheets for him - and he has a terrible time with them.
I've showed him the pattern: 0, 2, 4, 6, 8, 0. He can get that pattern and repeat it, saying twenty TWO, twenty FOUR, twenty SIX twenty EIGHT. But he still doesn't understand that 30 comes next. Instead, he says "zero."
If I hand him his chart, he reads it accurately - but again, he's just reading it, not understanding it.
I'd love to be able to say "if he can solve the problem, what difference does it make how well he understands the process?" But I'm pretty sure that it matters. These are basic, simple patterns - patterns that should be self-evident. But they're opaque to Tom.