I grew up in India. We had in school what were called "mental math" tests. A teacher would rapidfire at us 50 questions in quick succession, and we, ready with our pencils and paper, would have to quickly jot down the answer, and whoever did the most correct answers got the highest marks. The questions went like this:
Teacher, reading out:
19 4s are ... (2-3 seconds pause)
1000 - 564 (2-3 seconds pause)
Does november have 30 or 31 days, yes or no (2-3 seconds pause) ...
... and so on.
So, obviously, the only way you could do well in these tests was practice, practice, practice. And my grandfather (blinded by cataract, he would hold his transistor radio, and walk up and down listening to the news all day) took responsibility. Every morning before school was what we called "Tables time": Times Table Time!
I would sit and sing away the tables, from 2 to 20, and then there would be the harder practice: how you could use the tables and place value to quickly figure out higher numbers multiplying each other without resort to pen and paper.
I started school at 3 years of age (in those days there were no rules on being too young or old, there was a test, if you passed, they let you sit in class :)). And this game continued in simple form till about the time I was in Year 2, and then became a lot harder and more pleasurable, because the nature of my grandfather's questions changed to: "Is this number X prime, or not".
So, of course, armed with this experience, as a parent I introduced the times tables to my son when he was 3. He loves his (ever-growing) set of hot wheels cars, and I thought this is perfect. So, the 2s, 5s, and 10s happened quickly (little cars arranged in rows of 2s, then skip count verbally, then the same with 5s, and then 10s, maybe a post on this later with pictures). Worked like a charm. Before he went to pre-school, he was chiming away at questions like, if there are 3 wires and 2 birds on each wire, how many birds altogether?
By the time he was starting formal school, kindergarten at 5 years of age, (the age for starting school in Australia), he knew his tables from 2 to 10, maybe not the 8 or 9, but if he did not "remember" he knew how to work it out, and he knew the importance of "remembering" the answer, when say we are walking on the road, mum asks a question, and I have to quickly work out the answer.
And later, he did not even need to memorize or learn by externally provided parental support, because he began to figure out the number patterns. For example, here are multiples of 11: I did not make him learn it, he was not counting by groups of 11, he simply figured out the pattern of progression, especially after 110, all by himself. He knows he is counting by groups of 11, but that is not the "algorithm" used, the algorithm emerged from number play and pattern recognition. And to be honest, I realize, even today, the delight of this. To date, I discover a new sequence that will make me go, WOW! (sequences and series post later). And my hope is that it all starts here, in elementary school math.
So, to me, maths is as much about discovery and play and conceptual understanding, as it is about practice and making memory stronger. Only when you dribble a ball for hours, you master a game. Only when you run everyday or go to gym everyday, do you master fitness. Only when you run your fingers again and again on the piano, do you master playing it. Only when you sing repetitive patterns of notes again and again (the term in Indian classical music is called "alankars"), do you master melody. Only when you put in practice, do you master anything.
And later, he did not even need to memorize or learn by externally provided parental support, because he began to figure out the number patterns. For example, here are multiples of 11: I did not make him learn it, he was not counting by groups of 11, he simply figured out the pattern of progression, especially after 110, all by himself. He knows he is counting by groups of 11, but that is not the "algorithm" used, the algorithm emerged from number play and pattern recognition. And to be honest, I realize, even today, the delight of this. To date, I discover a new sequence that will make me go, WOW! (sequences and series post later). And my hope is that it all starts here, in elementary school math.
So, to me, maths is as much about discovery and play and conceptual understanding, as it is about practice and making memory stronger. Only when you dribble a ball for hours, you master a game. Only when you run everyday or go to gym everyday, do you master fitness. Only when you run your fingers again and again on the piano, do you master playing it. Only when you sing repetitive patterns of notes again and again (the term in Indian classical music is called "alankars"), do you master melody. Only when you put in practice, do you master anything.
By the way, I completely agree, that every child will have their own pace, and practice without conceptual understanding is not a good idea at all. But, in working with my child, I found that the reverse is also true: conceptual understanding without practice will never take root and flower in the brain. And sometimes, like my post yesterday, pattern based understanding comes not only from learning the formal rule or concept or schema or prototype and its application (deduction), it comes from doing some pattern over and over again, and then discovering the general rule or schema from the examples (induction). In other words, deductive learning is one way, inductive the other, (there is a third, but maybe later on that), and mastering something comes from both the deductive AND the inductive ways.
Finally, I found a couple of days ago, that this seems to be the case not only in math, but also in language (see specially his answer to the last question by the interviewer): Steven Pinker's Sense of Style.
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