Today H. had another breakthrough with numbers. Before you can really understand the magnitude of her accomplishment, you first have to understand where she started. When she joined our family at the age of 9, she could count to 20 (by the end of our stay in China), and could write her numbers (in Arabic numerals) to 20. The problem was, that it was all done by rote, and just like 'lmnop' gets lumped into one long letter by little children singing the Alphabet Song, in H.'s mind the numbers 1-20 were one long string that you said in order and had no bearing on real life other than 2 must follow 1, and 3 must follow 2, and so on and so on. The first two years she was home we spent working with preschool number manipulatives in order to tease apart the numbers and give her some understanding of what each of them meant. We counted and counted and counted everything we could find. It was the rare moment I let her use a number without seeing the corresponding physical items which matched it.
At first it was very difficult for her to remember the name of each numeral just by looking at it (and so went spent more and more time counting objects to get to that number). I actually despaired for a while that she would never be able to recognize numbers without counting above the numeral 5. We even stopped doing math for a while because we seemed to be going backward at one point. Much deep breathing ensued.
Then a friend sent a pre-kindergarten math (Singapore Math) book that matched what H. needed at just that moment and we were off again. The break and new book were just the thing that was needed. Since that point, we reached an impass with the Singapore series and I went back to my first homeschooling love of Rod and Staff math (1st grade). And that's where we've been since the new year. H. has learned to identify and read numbers up to 100 (far past number 5 which I despaired we would never pass a year ago) and can skip count by 5's and 10's with a little help. I haven't done a huge amount with skip counting at this point, though, because I was afraid that it was lodging in the rote memory part of her brain that we had spent so much time getting out of. (Note, rote memory is not a bad thing at all if you have the concept behind the numbers you are saying. I was not convinced that this was the case for H. and that 10-20-30-40... were just a string of sounds you say together.)
In the Rod and Staff book she is working in, one of the exercises we routinely see is to have two, two-digit numbers and to have to circle either the higher one or the lower one. For instance, 36 and 26 will appear and the child is to figure out which number is the biggest (or smallest depending on directions.) This particular exercise has been a little tricky for H., but she has been managing because I have her use manipulatives for every single problem.
This is a complete aside, but I know others have commented to me about how their children from less than ideal backgrounds seem to have memory issues. Memory is a problem for H., and it sometimes shows itself in learning to read and do math because in order to do these things well, you need to hold several things in your head at once before you can put them all together. By using manipulatives, H. is able to do problems that she wouldn't be able to do otherwise because the manipulatives, besides helping to show her the problem, act as an external memory system for her. Imagine my delight when I had another pet theory confirmed by someone with letters after his name. In The Learning Brain: Memory and Brain Development in Children by Torkel Klingberg, Dr. Klingberg writes,
The education [of children with working memory deficits] also provides a range of strategies designed to make it easier for children to learn despite a poor working memory. In most cases, the method entails reducing the demands on working memory and supplying external aids to relieve them of some of the information burden. For example, the children might be given briefer instructions that don't overload their capacity (reducing demands) and notes or pictures that remind them of a sequence of actions so that they don't have to keep that information active in their working memories (external aids).
But back to H. This morning, H. was working on one of these types of exercises and I was doing something else at the moment (probably helping a little girl), and I hear TM look over and offer H. some help, by pointing out that she could count by 10's to get to the number she was trying to reach. When I got back to the table, my worst fears about H. not understanding skip counting came to pass. I took a look at what she was doing and for the number 32 she had 50 cubes stacked together in a line. I was baffled. Never before had she had difficulty of this nature on this particular activity and she suddenly couldn't figure out why that 50 before her wasn't 32.
(The Teacher of the Year award will go to anyone who can guess what she was doing before you read my explanation. I will admit I had no clue and took a very long time of watching her build her stacks to figure it out. Sometimes the only truly difficult thing about teaching is figuring out why a child is getting something wrong... there really is a very good reason 99% of the time.)
After much watching her work the problem and listening to her count (and more than one break to re-set her brain from disassociating), I finally had the A-ha! moment. By adding the counting by 10's part, it seemed to H. as though we had moved to a completely different counting system. In one system you count by 1's, in another you count by 5's, and in yet another you count by 10's, but they are separate and completely unrelated. Thus, if you are counting by 10's, you first stack up three sets of 10's to make 30, then because you need two more numbers, you need two more sets of 10 because in this land, you can only count by 10's... thus you have "counted" to 32, but end up with 50 cubes.
It is at this moment that I was incredibly thankful that I went ahead and made the, what at the time seemed a wee bit frivolous, purchase of uni-fix cubes at the last homeschooling convention. I'd never needed them before, but had bought another book which used them, so picked them up as well. If you don't know what they are, they are interlocking cubes, with sets of 10 in different colors. It is very easy to see the 10's as you count, but you can also break them into individual cubes. When I discovered the problem, we went back and did a lot of counting... counting 10 cubes, taking them apart, putting them together, adding one more, etc., etc. Then it all clicked, when we counted to 20 and I added one more block, instead of saying 30, she said 21. And there was great rejoicing. It was her own a-ha moment of realizing that instead of different and separate number systems, it was just one system used in a different way. She then went on to do the rest of the exercise correctly.
It was a huge win. I was feeling really happy until a little while later, I hear H. say, "Sorry, Mama."
"What?! What are you sorry for?"
"Sorry, Mama. The math... "
She couldn't even articulate what she was sorry about but she felt badly that she hadn't understood her math immediately. Boy, talk about a knife to a mother's heart. I carefully explained that I wasn't upset with her. It was OK that she hadn't misunderstood, but then she worked really, really hard, and then she understood and she did an amazing thing. I proud of her. I'm still not sure I have convinced her of that. When for too many years mistakes have not been allowed, it makes a child fearful to try something new or to stretch herself in any way.