Mastering "Same" and "Different" ??!??

How can a child who is mastering double digit multiplication, who can describe earth's biomes with accuracy, who can read a full-length novel with fluency - NOT grasp the concepts of "same" and "different?"

I am grappling with this question this week - in part because I simply can't figure out whether the problem is conceptual or semantic. Obviously, Tom can tell the difference between, say, a peanut butter sandwich and a cat. He can even tell you, when asked, the differences between summer and winter, oceans and lakes, and so forth.

He knows that birds belong to the same group - and that the bird group is different from the reptile group.

When I place two equations in front of him - say 7 +2 and 2 +7 - he can handily tell you that each adds up to 9.

BUT!

When I ask him - "So - do 7+2 and 2+7 add up to the same thing?" He looks at me confused and says "I don't know... um... no."

"But - they both add up to nine."

"Yes..."

"And nine is the same thing as nine, right?"

"I don't know..."

"Look, Tom, here's 2 + 7 using the cuisinaire rods. And here's 7 +2. Can you place them one on top of the other?" (He does - and they are identical in length.)

"Are they the same length?"

"I don't know... no."

"Tom! Look at them! They are exactly the same length!"

"They are???"

"OK, I tell you what, show me two rods that are different." (He pulls out a blue and a red rod.)

"These are different."

"Yes, they are. Now show me two the same." (He pulls out two reds.)

"These are the same."

"Right. So... are these two rods (7+2) the SAME length as those two rods (2+7)?"

"Ummm...."

OK, folks, is Tom just playing dumb? Am I using the wrong words? Or is he truly not grasping what looks to me to be the most basic of concepts? SOMEthing is going haywire here!!

Problems with Patterns

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.

Math: What School Did Wrong!

One of our biggest public school frustrations had been in the area of math. We were absolutely certain that Tom could and should be moving forward much more quickly - but the teachers either wouldn't or couldn't do so. TouchMath had been a helpful tool for teaching some calculation (especially basic addition and subtraction), becaue it had him count "touchpoints" on each number and thus add and subtract without having to use his fingers.

But he had been doing double digit adding and subracting with and without carrying/borrowing for two solid years!

Finally, at the very end of fourth grade, his teacher started using our Touchmath materials with the whole class to work on "skip counting" (counting by 2s, 3s, etc.) as a prelude to multiplication. TouchMath also using skip counting by 5s and 10s to teach money and time concepts, so he had those sets of numbers pretty well memorized. But why wasn't he doing multiplication? Fractions? Measurement? Decimals? I was determined to push him forward.

I started out using some math sheets I generated and/or printed from sites like softschools.com and enchanted learning - and they worked well for certain types of problems. I quickly saw that he could do simple word problems (Joe has 6 apples. He gets two more. How many does he have in all?) without any prompting or visual tools (though he never had come home with word problems from school). And basic fractions were no problem at all: he could identify and even create representations of 1/3, 1/2, 2/3, etc.

But he was still having terrible problems with basic addition subtraction - because he'd forget all about carrying/borrowing. He didn't seem to grasp bigger/smaller beyond the number 10. And when I asked him to count by two's, he could do so only up to number 26. Then he pooped out.

Within a few weeks, I figured out the problem.

It seems that, in teaching Tom skip counting, his teachers used a chart and had him memorize 2, 4, 6, 8, etc. But they neglected to TELL him about the pattern he was forming. As a result, he could count by twos to 26 - but had no idea what came next. I used a number chart and a pencil, and we went through saying skip, 2 (put an X on the 2), skip, 4 (put an X on the four). We did the same thing for threes and fours and fives. He has NO trouble using the charts to multiply up to 100!

I ran into the same problem with bigger/smaller. He seemed to be guessing about bigger/smaller when the numbers got bigger than 10 - and I finally realized that no one had given him rules for deciding relative size of symbolic numbers (as opposed to piles of objects). I explained more digits means a higher number. If there are the same number of digits, compare the digits on the left. If they are the same, go on to the next pair. When you find a pair that don't match, compare them. The number with the highest digit is the biggest number.

He got it.

In short: being autistic, he didn't "see" patterns just because they were repeated. He needed to have the patterns explained. But once they were explained, he whizzed forward!

The down side of all this is that I am having to create my own worksheets at odd hours to let him practice all of this. But I'm hoping that, within the next couple of months, I'll be able to return to computer-generated worksheets - and even get online with Tom (so far he's not really very excited about computer games, but I think I can get him going...).