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alexpgpI was not "the sharpest tack in the box," as the expression goes, back in elementary school.
In my defense, however, I think it's fair to say that—if my role could be described as that of a "tack"—I had no idea I was supposed to be "sharp." Furthermore, I would have had no idea how to achieve sharpness had I been aware there was a need to do so. But I digress, and the metaphor has taken enough punishment…
As do most, I did figure out, early on, that right answers were better than wrong answers, and so I did what I could to come up with right answers as often as possible. Sometimes, however, the task was beyond me, as was the case in learning simple arithmetic.
Now, somewhere along the line, I had managed to learn to count, forwards and backwards, and for better or worse, I would entertain myself at home by seeing how high I could count (in my head) during the commercials that interrupted my enjoyment of afternoon cartoons on television. (There was something about the pattern of counting that appealed to me, and as a side-effect, I developed an ability to count very quickly.)
Obviously, I had a native understanding that adding one to a number made it the next larger number, but I didn't really recognize what I was doing as "addition." So when it came to learning to add a number other than one to some number—say 6 and 5—oh, I was an endless source of wrong answers (or shrugged shoulders), as I had no idea how to do it without counting and using my fingers.
I was reprimanded for using this approach and told that it was "wrong." Apparently, the answer was supposed to appear inside my head, but how that worked wasn't exactly clear. The teacher proceeded to drill me using flash cards, which merely sped up the rate at which I provided wrong answers. I recall I wasn't bothered so much by how I kept coming up with wrong answers as much as by how the other kids in the class apparently knew the answers (and without hesitation), not to mention the fact that they laughed at me.
As it turned out, it was good that I had no idea of the "right" way of going about addressing the problem, which apparently involved simply repeating the information over and over until it somehow got stuck in your head. (To this day, that kind of rote learning simply has no effect on me.) This left me in a state of blissful ignorance, free to figure out a way to get the right answer on my own. The result wasn't particularly pretty, but it worked.
For each number between 2 and 9, I created a mental "constellation." For the numeral 2, I visualized two stars, one at the top of the curve of the 2, and the other in the middle of the horizontal part at the bottom. In my visualized numeral 5, two stars were located at the ends of the top horizontal and the other three lay equidistant along the loop of the 5. The numeral 9 had three stars along the top of the loop, another three along the bottom of the loop, and the final three along the bottom, "hook" portion of the number.
Basically, I had—without knowing it—created a mnemonic device that broke every digit down into the corresponding number of stars, and since I could count like a demon by ones, all I had to do was add one as I mentally visited each star of a number's "constellation." So when I was asked, say, "What's six plus seven?" I would start with the six, visualize the seven, and instead of using my fingers, I'd visualize the constellation for seven, and as my "mind's eye" landed on each star, I'd mentally count sev-ate-ni-ten-lev-twel- and announce "Thirteen!" as the answer.
There were a couple of times (especially when the number to be added was a 9) when my answer came late enough to prompt my teacher to ask if I was still counting by ones in my head when doing the addition. By that time, I knew that if I answered in the affirmative, that it'd be... the wrong answer.
And so I gave... the right, if untrue, answer. Soon after, however, that answer became the truth, for while I still used my constellations, I had begun to count by twos and threes as easily as I had at the beginning of my journey, by ones.
I was young and, as they say, just learning to get by. I didn't fully grasp addition—at least, not in the way my teacher had wanted—until some time in junior high school.
This week's entry is a two-parter! View the second part, about a 'design flaw', here.
In my defense, however, I think it's fair to say that—if my role could be described as that of a "tack"—I had no idea I was supposed to be "sharp." Furthermore, I would have had no idea how to achieve sharpness had I been aware there was a need to do so. But I digress, and the metaphor has taken enough punishment…
As do most, I did figure out, early on, that right answers were better than wrong answers, and so I did what I could to come up with right answers as often as possible. Sometimes, however, the task was beyond me, as was the case in learning simple arithmetic.
Now, somewhere along the line, I had managed to learn to count, forwards and backwards, and for better or worse, I would entertain myself at home by seeing how high I could count (in my head) during the commercials that interrupted my enjoyment of afternoon cartoons on television. (There was something about the pattern of counting that appealed to me, and as a side-effect, I developed an ability to count very quickly.)
Obviously, I had a native understanding that adding one to a number made it the next larger number, but I didn't really recognize what I was doing as "addition." So when it came to learning to add a number other than one to some number—say 6 and 5—oh, I was an endless source of wrong answers (or shrugged shoulders), as I had no idea how to do it without counting and using my fingers.
I was reprimanded for using this approach and told that it was "wrong." Apparently, the answer was supposed to appear inside my head, but how that worked wasn't exactly clear. The teacher proceeded to drill me using flash cards, which merely sped up the rate at which I provided wrong answers. I recall I wasn't bothered so much by how I kept coming up with wrong answers as much as by how the other kids in the class apparently knew the answers (and without hesitation), not to mention the fact that they laughed at me.
As it turned out, it was good that I had no idea of the "right" way of going about addressing the problem, which apparently involved simply repeating the information over and over until it somehow got stuck in your head. (To this day, that kind of rote learning simply has no effect on me.) This left me in a state of blissful ignorance, free to figure out a way to get the right answer on my own. The result wasn't particularly pretty, but it worked.
For each number between 2 and 9, I created a mental "constellation." For the numeral 2, I visualized two stars, one at the top of the curve of the 2, and the other in the middle of the horizontal part at the bottom. In my visualized numeral 5, two stars were located at the ends of the top horizontal and the other three lay equidistant along the loop of the 5. The numeral 9 had three stars along the top of the loop, another three along the bottom of the loop, and the final three along the bottom, "hook" portion of the number.
Basically, I had—without knowing it—created a mnemonic device that broke every digit down into the corresponding number of stars, and since I could count like a demon by ones, all I had to do was add one as I mentally visited each star of a number's "constellation." So when I was asked, say, "What's six plus seven?" I would start with the six, visualize the seven, and instead of using my fingers, I'd visualize the constellation for seven, and as my "mind's eye" landed on each star, I'd mentally count sev-ate-ni-ten-lev-twel- and announce "Thirteen!" as the answer.
There were a couple of times (especially when the number to be added was a 9) when my answer came late enough to prompt my teacher to ask if I was still counting by ones in my head when doing the addition. By that time, I knew that if I answered in the affirmative, that it'd be... the wrong answer.
And so I gave... the right, if untrue, answer. Soon after, however, that answer became the truth, for while I still used my constellations, I had begun to count by twos and threes as easily as I had at the beginning of my journey, by ones.
I was young and, as they say, just learning to get by. I didn't fully grasp addition—at least, not in the way my teacher had wanted—until some time in junior high school.
This week's entry is a two-parter! View the second part, about a 'design flaw', here.
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no subject
Date: 2013-07-16 04:54 am (UTC)no subject
Date: 2013-07-19 04:46 am (UTC)no subject
Date: 2013-07-17 07:01 pm (UTC)no subject
Date: 2013-07-19 04:51 am (UTC)Thanks for reading!
no subject
Date: 2013-07-18 07:58 pm (UTC)I wonder if your teacher would be boggled that you later went into rocket science!
My stumbling block was division, in the 3rd grade. I had no problem with multiplication, but somehow the visuals and sequence for long division just did not come easily. But I wound up with a math minor in college, and now work in computer science, so those early grades don't show everything after all. ;)
no subject
Date: 2013-07-19 05:01 am (UTC)In practice, what you end up with is a distribution, and if things are planned right, there'll be some kind of special programs in place designed for the folks at the front and back ends of that distribution.
I don't think "late bloomers" surprise teachers all that much, as I believe the phenomenon is not all that rare. On the other hand, there are societies that place a lot of emphasis on performance in the early grades (e.g., Japan, Germany)—in effect determining one's academic future by around 7th grade—which may allow better utilization of educational resources, but completely ignores the "late bloomer" phenomenon.
Thanks for the comment.
no subject
Date: 2013-07-19 01:25 am (UTC)no subject
Date: 2013-07-19 05:02 am (UTC)no subject
Date: 2013-07-19 01:30 am (UTC)I had enormous difficulty with addition and subtraction in elementary school. Perversely enough, multiplication and division were easy for me, and I picked them up super quickly well before I was able to add reliably without counting on my fingers. For a long time, I would fake-add by doing a monstrous hybrid of multiplication and +1 or -1 depending on what set of numbers I was playing with. It worked out pretty well for most of the time I had to do any math.
no subject
Date: 2013-07-19 05:03 am (UTC)Thanks for the kind words.
no subject
Date: 2013-07-19 10:17 pm (UTC)no subject
Date: 2013-07-20 10:43 am (UTC)I seem to recall this was even before I was aware of the concept of "constellation."
no subject
Date: 2013-07-20 12:43 am (UTC)no subject
Date: 2013-07-20 10:49 am (UTC)I think what teachers really need to focus on is to make the material they teach compelling, although that may be as difficult as tailoring approaches to learning.
Standardization only makes the situation worse, as it requires conformance to fundamentally arbitrary norms.
Thanks for the comment.