Recently, I re-started Ryan's math work. I re-looked the activities that I offered to Ryan and I tried to understand why a particular activity may or may not convey the intended lesson. Take, for example, the basic counting exercise - present some items or a picture of some items to the child and ask him to count the items. This is a typical activity and I'm sure that you see it all the time.
I had initially thought that if Ryan could count off seven items where there were seven items, that meant that he understood seven as a quantity. I was wrong. Chanting one to seven in such a situation was merely rote counting. Simply rattling off the numbers in sequence does not mean that he learns quantity. In the way that he recited one, two, three, four, to seven as he pointed to each item, he could have equally have recited do, re, mi, fa, so, la, ti, do. Or red, orange, yellow, green, blue, indigo and violet. Or something even more random - Peter, Paul, Mary, Joseph, Ryan, Richard, Rachel. You get what I mean?
When I taught Ryan to add, I would show him the process, for example, that 2+ 3 meant that we count two items, then we count three items, then we count them altogether. Ryan is very methodical - you just have to show him a process and he will follow it faithfully. As he managed to get the correct answer this way, we continued to tackle all sorts of sums this way and he could do them, no problem. So I thought he could add. But nope, I was wrong. He was just following the process without understanding its meaning. There was no understanding that he was adding two quantities together to form a bigger quantity.
When I taught him to subtract, I showed him that 5-2 meant that we took away two items from a pile of five items and then counted the remainder. He could do that too. So I thought he could subtract. But nope. Once again, I was wrong. Again, he was just applying the process.
Perhaps you think I'm being too fussy. After all, if he got the sums right, then what's the problem? Well, it's not good enough because, this way, he doesn't learn about quantity. This means that, if he could not apply the same process that he had been using, he would be stuck. He would not be able to figure out other ways to solve the problem if he did not understand the concept that we were adding quantities.
Let me give you an illustration. I first introduced numbers and counting to Ryan a few years ago and when he could count, I introduced the concept of more and less. It took him a long time to understand what I was getting at when I asked him: is 7 more than 5? To him, it was as if I was asking him, is B more than T? or is red more than blue? To him, 7 was simply the word that comes after 6, just like Tuesday comes after Monday. At that time, I didn't realise why he found it difficult, I just thought that he needed some practice - looking back, this is nonsense. It's not about practice, it's about understanding the basic concept of quantity.
Once, I asked him what is 5 minus 5? It took him a long, long time before he could answer. The question truly confounded him and, seeing him struggle with that question, I was dumbfounded myself. That's when I knew for certain that I had missed something out.
It took me some time to figure out the root of the problem. I spent a long time thinking about how we've been doing math. I referenced lots of material, especially the mathematics module in the Montessori teacher training course which I completed earlier this year. I tried to understand how children learn math and what the important concepts were. And so I realised where I went wrong.
Ever since then, I've been more conscious of how I structure our math activities. I give a bit more thought to, not just the mechanics of the activity, but I also critically analyse the effectiveness of the activity in delivering the lesson.
And so, to be safe and to plug any gaps, we are starting over with our math. The way I look at it, we are taking one step back so that we can take two steps forward later. At this point, I don't know which concepts are weak and which are strong, so I'm just going to do everything, step by step. If I see that he understands the concept already, we will skip ahead to the next one.
In the photos in this post, Ryan is working on activities similar to Montessori's spindle box and Montessori's number rods. These reinforce several concepts including quantity. I am also posting over at
HappyPlayWonder today about an activity on quantities - click
here to have a look!
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