My three-and-a-half year old son, Divyanshu (who became Divi to help people at least pronounce his name), knows the numbers 0 through 10. Last week, I got a set of flash cards to teach him 11 – 25. As an utterly clueless father, I don’t know how other children at this age react to learning, but Divi is a very avid learner, with an attention span of five seconds.
I started teaching him trying to relate the new larger number to the smaller numbers he already knows:
Say one and one eleven.
One and two twelve.
One and three thirteen.
… and so on.
After 25, I started again from zero. In this round, he somehow figured out that if you insert ‘teen’ after a number, it becomes another number – four-teen, six-teen, seven-teen, eight-teen, and nine-teen. And when I reached 20, two and zero, he said ZERO-TEEN. However, I continued and so did his idea of the new numbers. After zero-teen came one-teen (21), two-teen (22), three-teen (23), four-teen (24) and five-teen (25). I smiled, corrected him, and restarted from 0. To my horror, he didn’t identify 10 by ‘ten’, which he knows but ‘zero-teen’. This is when I realized that he had identified a pattern, to suffix the number at units place with a teen. And this round resulted in yet hilarious ‘new’ numbers. So both 11 and 21 became one-teen, 12 and 22 became two-teen, and so on.
Divi might have developed misconceptions by identifying a pattern in the numbers, but pattern identification is what differentiates an expert from a novice. By pattern, I mean the underlying principles, meaningful arrangements, and logical reasonings.
If a chess board with some randomly placed pieces are shown to a chess expert and a non-expert for a few seconds and then asked to arrange the pieces on another board from memory, both persons fare poorly. However, when the pieces are arranged in a meaningful way, as in an actual game of chess, the expert chess player is able to arrange most of the pieces correctly, while the non-expert fares as poorly as before. The expert, in the second case, is able to identify the pattern of the pieces on the board.
Similarly, expert students are able to identify the appropriate laws of motion in word problems while novices try to plug in variable values in formulas and do not know the underlying principle on which the problem is based.
Divi’s is an oversimplified case (and resulting in a misconception) but … you get the idea.