What is going through this student’s brain? What would you say to the kid to help out?
Let me know if you’re having trouble reading today’s submission. If image quality continues to be a problem then I’ll experiment with typing the student work up.
We seen in the second part of 4 that the student knows to add the exponents to multiply. It’s the division that causes confusion. Perhaps because there are two possible intermediate steps (a^4/a^3, or a^2/3 + a^2/3) the student doesn’t know which to take.For 5, all there is to go on is the _lack_ of intermediate steps. The student probably doesn’t know to convert all terms to the same base first. Or perhaps there is confusion about the 1 in the exponent, usually omitted?My training is as a computer scientist, not an educator, and from this background I draw the concept of “unit tests,” essentially standards on a micro scale. Can the student multiply two exponents? Can they divide? Can they convert bases? Such tests should be run in isolation of each other. By the time we get to a problem that includes multiplication and division, or converting bases and a negative exponent, there’s not enough information to make the “diagnosis”. Is it that they don’t know how to divide, or it the composition of known things that brings confusion, or is it an arithmetic error? From a natural science perspective, try to hold as many variables constant as possible. While it’s a novel experience to analyze the taste of information presented, seeing an ordered progression of problems will help us do more than speculate.
Michael P
The thing is that, as a teacher, you’re constantly forced to make these diagnoses with imperfect information. I think that’s part of what this site is designed to help teachers get better at: making quick and accurate diagnoses based on experience and imperfect information.Also, I think that you’re totally right about how division is this student’s problem. But I’m not sure I get your diagnosis of 4. What’s the path between being confused between those two possible intermediate steps and the answer that the kid gives?
David Wees
I think that the student in question 5 basically thought to himself, “I don’t what to do with this negative on this exponent, so I’ll just ignore it.”
laurephant
5. perhaps they used a calculator. cubed is often on scientific calc. I find students do not grasp the concept of negative as division. I heard all kinds of mnemonics, cross the line change the sign or negatives always want to be in the other bunk bed, etc. None of these “stick” well except in high ability kids who are just remembering their way through math. Exponents, indices, are a abstract concept and I have found it hard to find a concrete anchor. So much science depends on grasping this concept. I have used the video http://www.youtube.com/watch?v=0fKBhvDjuy0 Powers of ten which at least explains some of this concept.