Sorry for the title of this post, but I really wanted to link to this video.
Anyway, what’s going through this students head? What would you say to the student when going over the problem?
(Also, remember to drop into the comments of this post and leave some feedback about the site.)
5 replies on ““The limit does (not) exist””
My guess here is that the absolute value sign is just another grouping symbol to this student. Not an especially important one either, I would guess. If you are confused by how to treat it (meaning that you don’t remember or never understood the piecewise definition) then simply treating it as x -1 is fine. Alternately, to give the student more credit for understanding, it is possible that the student was thinking only of the right hand limit in which case the absolute value sign can be effectively ignored.
I suspect there is also no sense that the three problems are related. The difference between left and right hand seems to mean nothing to the student in the third problem.
I guess what I would work on is trying to encourage this student to find quick ways to confirm their answer. In this case, look up, and mental mental math after factoring. I am not really concerned about the initial error, just the lack of initiative. I also find it extremely difficult to get students into this mode of pausing to think about their solution.
Sometimes I wonder if we ask them to do too many things (which leads to them not pausing to think because they feel they have to rush on to the next problem). I think Lola Swint’s point that the student doesn’t see the three problems as related really gets to the heart of it. What can we do to change that? Asking them on the exam to explain the relationship between the problems seems like it’s just supporting their bad habit rather than heading them toward a better one.
The really sad part to me is that they even had the beautiful graph and still didn’t see it as an answer to the last part. Maybe this is just a case of “if the algebra and the graph or table of numbers disagree, the algebra must be right” — it’s more abstract, it’s symbolic, so it’s good.
To fix the problem that the student sees no relation between these three problems, couldn’t we just call the whole thing 1 with parts a, b, and c?
Their factoring is great on the bottom, minus the |x-1| being factored out. They clearly understand that they can factor out a removable discontinuity, but what they don’t understand is that their (agreed, beautiful),graph has already given them the answer.
I think this student just needs a little more confidence in their test taking ability. They’re second-guessing themselves where they don’t need to.