During my afternoon time as an EA, I spent most of the class working with an 8th grader yesterday. The class was practicing solving single and multi step equations. A particular student finished his sheet of 20 problems in about 10 minutes, whereas his classmates needed closer to 30 minutes. He had a few errors and pretty consistently missed the negative sign in his final answer.
None of that is particularly interesting, but in watching him do the problems I realized he had no idea of why or what he was doing. I know this student from last year and he has an amazing ability to see, memorize, and repeat patterns (pattern-sniffing as my PCMI peeps would call it). He just looked at the numbers and knew "well you add this one away from the x to the one thats alone, then you divide by that other one." He didn't know anything about "doing the opposite" or "trying to get the x by itself" he just knew what sequence to enter into his calculator. He didn't and in fact couldn't write any of his work down. When he had made a mistake he couldn't see it or correct it and pretty much had a melt-down when I tried to explain and show him the reasons behind why he was doing each step.
Is this a problem? Is it good as long as he gets the right answer and it doesn't matter if he has a concept of what is fundamentally happening when he is punching the keys? Will it just self-correct when he gets more complex problems? I think not. From what I saw of him last year and from other struggling students, when the complexity rises, without an understanding of the base level, his ability to pattern-sniff is not longer as helpful and he is suddenly in the deep end without any floaties. There isn't a base to build off of or go back to. Similarly, when different types of problems are all jumbled together, then the pattern isn't as obvious and if the student doesn't happen to remember "oh this type of problem is done this exact way" then he is out of luck. Maybe that's part of why this brilliant boy, an others like him, does miserably on standardized tests but seems to understand everything at the time. Had I been the classroom teacher and not had the chance to spend most of the hour sitting and watching and talking, I wouldn't have known how he was solving the problems or that he really knew nothing of the basic concepts.
Bonus Story: The reason I know this kid is an excellent pattern sniffer is because he is the fastest to ever beat me at the "Block Game." I put out 15 pieces. My opponent can choose to go first or second. On each turn, we can choose to take 1, 2, or 3 pieces. The person to take the last piece loses. Over time students realize that I have a "strategy" that lets me win almost every time. If a student wins, it is typically by dumb luck. Using backward deduction will reveal that there is a set of moves that allows the first player to win 100% of the time. The second player is able to win as long as the first player doesn't know the string of moves.
Most kids take a few days and many many rounds to figure out how to win. This student figured it out in about 30 minutes. When I asked him to explain how he knew what to do he explained that he just watched me. All the other students watch the pieces, or their own plays. He knew enough to watch the person who controlled the game. From there, he memorized and analyzed every move I made to figure out what to do in each situation. He didn't know the nice "summary" of the moves or realize that there was a particular string that I was playing each time. He just knew that if my opponent pulled 3, then I pulled 3 and so on. I was very impressed by him and his ingenuity. He has become very adapted to finding and exploiting patterns, but isn't yet capable of taking the next step and coming up with a clear set of rules.
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