Students Identifying Misconceptions (Instead of Me)

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.

Posted on September 29, 2011, in Calculus, Limits, Misconceptions. Bookmark the permalink. 11 Comments.

  1. Hihi… You gave me an idea for my next assessment. I bought scratch off stickers (scratch off, like lotto tickets). I think I’ve got my kids to avoid that misconception (I do it with something called the “enchanted limit window” which is… well, whatever, stupid).

    But anyway: I am going to have a two part question where (1) I ask students to determine the limit with the middle number covered by the scratch off sticker (like your green sheet of paper). And then (2) they scratch off the sticker, and see if they want to revise their answer.

    LOVE IT. I’m curious to see how they’ll do. Plus I’ve been dying to use these scratch off stickers.

  2. reason number like 5049845 why you rock. such a cool idea with the scratch stickers. where did you get those??? i am actually very curious about the enchanted limit window… green strip of paper = not that sticky of an idea.

    • i just ordered them online (type “scratch off stickers” or “scratch off labels” and a thousand sites will pop up). my kids take the assessment tomorrow! if it wasn’t for your green strip idea, i wouldn’t have had this idea. YAAY for synergy.

      i don’t think the enchanted limit window actually is “sticky” — definitely not any stickier than your green strips.. i just made a smartboard version of the window/shudders a la this video at around 2:30 (… and then “erased” the shudders. so i’d show them just one point and ask “what’s the limit” and after like 2 slides they’re like “WE DON’T KNOW.” So I’d erase both shudders and ask “what’s the limit?” and they get it. Or if we’re doing right handed limits, I’d just erase the right hand… Whatever, it’s not that exciting. I just like calling it “enchanted limit window.”

  3. I swear, you Klingons rock so much it makes my head spin. Thanks for all these amazing ideas!

    – Elizabeth (aka @cheesemonkeysf on Twitter)

  4. Nikki Pinakidis

    Would you be willing to send attachments of the tables/graphs you used for this activity? Looks awesome!

    • I’m so sorry, I’m between computers right now and am moving across the country, so I don’t have the file handy unfortunately. the pictures above show about half of them though. i did a mix of limit does exist, limit does not exist and then a range of things going on with the point (point DNE, exists somewhere else, exists where it should be). i’ll have it in about 2 or 3 weeks when i get access to my hard drive, so I can send it then if I remember – if I don’t just leave another comment!

  5. I introduced limits today in AP Calculus AB. It was a mess. I’ve never been happy with my students intuitive or formal understanding of limits. There are some times where lack of understanding is related to student effort, etc., but this time it’s 100% on me. Terrible approach.

    At any rate, this being the first August where I’ve been aware of the math blog amazingness spread all over the Internet, I hopped on to Sam Shah’s virtual file cabinet and ended up here. I love the activity. I was sad to see the handouts won’t be available for a couple of weeks, since I want to use this activity tomorrow, so I made some of my own (stealing super-heavily from what you shared in the photos, particularly for the tabular limits).

    If I had more time I would have made a second station with tables, but it is what it is for now.

    Thanks for sharing a great activity! I’m excited to see how my students respond tomorrow in class.

    For anyone interested in my plagiarized version of the handouts, here’s what I made:

    Note: I don’t have green paper—chuckle, chuckle—so I went with another approach to hiding the limits. I’ll be curious to see if what I did for my own convenience helps, hurts, or is neutral in its affect on their understanding/success with the activity.

    • Biggest limit tip I have: defining a limit with the phrase “what SHOULD be there”. It’s really helpful in distinguishing if a limit does or does not exist (if there isn’t something that should be there it doesn’t exist) and also getting them to ignore what’s happening at that point (doesn’t matter what’s there matters what should be there). Can’t wait to check out your update! (On a phone now)

  6. Thanks so much! I am just starting to “relearn” how to teach math and these blogs help me provide a more meaningful experience for my students!

  7. Thank you so much for this activity and discussion. Today was the first day I did limits in our last trimester only non-AP calc, and I feel some students really struggled with understanding. I especially love your comment Bowman to think of the limit as what SHOULD be there.

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