Category Archives: Limits
At the Klingenstein Summer Institute this past summer, a really transformative experience that I think is going to help me bridge the gap from rookie teacher to low level intermediate teacher (I set ambitious goals), we talked a lot about identifying MISCONCEPTIONS as one of the roots of good teaching. In the math teacher group, one of the challenges that we decided would challenge ourselves with was to design an activity/lesson where the students would realize that they had a misconception on their own without me explicitly telling them. Here was my inadvertent attempt…
THE ISSUE: In a extraordinarily weak non-AP Calculus class (for now! and only compared to last year’s group, – growth mindset!!), we just spent about a week and a half talking about limits. Nothing fancy, basically just graphical and numerical limits. I had a really hard time getting going this year and do not think I did a good job at all of creating a student centered, constructivist classroom… and it showed. We had a quiz on numerical and graphical limits, and… wow. I was genuinely surprised at how low the conceptual understanding was of the main idea of limits.
THE MISCONCEPTION: Among many misconceptions, the biggest one seemed to be that the limit as x approaches a certain value is affected by the value of the function there. This mainly manifested itself with tabular limits (in which students put that the function is undefined at a point, and conclude the limit does not exist), and with open and closed endpoints on piecewise functions (closed meaning – in the teenage mind – that the limit does exist, open meaning it does not).
THE ACTIVITY: I made a four stations around the classroom and each one had 3 or 4 different functions represented either graphically or algebraically. The goal was to determine the left, right and overall limits of each function. For the tabular limits, I put a green piece of paper over the approached x-value of the function in the tables. I asked them to first determine the limits before lifting up the green piece of paper. Then lift it up, check out what the behavior of the function is like at the approached point, and see how their answers change.
The catch was that all the limits were the same. This was patently obvious with the green strip over the function values – I could see the students at each station talking and being like “What? They’re all the same…” Then, they lifted them and were like “What?!? How can that be?” The other stations were similar, except that I represented the functions graphically instead, covering up the whole x-value on the graph:
After everyone passed through all the stations, we debriefed. We talked about what changed when you lifted up the paper and what didn’t (okay, maybe this is explicitly pointing out the misconception, but only after most students realized it on their own). I asked what the point of the activity was and one student eagerly raised his hand to say, “I think the point was to show that when you lifted up the green strip, EVERYTHING about the limit changed because how the function is defined underneath.” Right, except… wait no, exactly wrong. But, from that we ended up having a good discussion and I really saw the right idea finally click in a few students heads. I really wish I had done this in the beginning instead of defining a limit more mathy-mathlike.
So the reason I decided to post about this? I got this encouraging bit of metacognition as part of a Reassessment Request:
on the table, in our last quiz i assumed that since there was an error at the limit the limit did not exist. we did the activity in class and it helped me learn that even though there might be an error or a weird number, there is still a limit if it is going to the same number from both sides
Success! Misconception realized. Exactly right young mathling. Even with the rough start, maybe this will be a great year after all.