Reflecting to Improve Teaching
Written by: Dr. Christine von Renesse
The following video clip was filmed in a Calculus 2 class for math majors (and some other majors) at Westfield State University in Spring 2013. We used some materials from Discovering the Art of Mathematics: The Infinite to work with series. We also used materials from Mairead Greene about Koch's snowflake.
While transcribing the video I noticed that my story of what happened in my interaction with Loghan was different from the actual exchange that you see in the video. This was really interesting to me, because it allows me to see myself more clearly and hopefully grow as a teacher.
Calculus 2 Conversation
Caption: This video clip was taken in a calculus 2 class in spring 2013. You can see Prof. Christine von Renesse in a mathematical conversation with a student, trying to use inquiry-based questioning techniques. While you watch this video try to understand the student's thinking and find questions that you would have used instead during this conversation.
This was my story
You can see in the clip how the student Loghan is trying to get me to tell her if she "is right" (and how to solve the problem) while I am trying to make her explain her reasoning and think deeper herself. Notice how I "revoice" her statements and try to ask clarifying questions.
In the end of the clip I thought I understood her conjecture but unfortunately she is incorrect. You can see how I am struggling to reflect her conjecture back to her without giving away that it is not correct. I am doing this because I want my students to revisit their thinking later when they have seen more examples of series. I believe that these kinds of "incorrect" conjectures are a necessary and important part of doing mathematics, as long as they keep an open mind and change their thinking once they have more information. Eventually students need to be able to prove conjectures or at least "make sense" of them.
I am still not sure how convinced Loghan is of her conjecture or if she was just "making a guess" in the hope that it would make me give her an answer? This video clip is from the beginning of the semester and so the students are still getting used to IBL and the fact that I will not evaluate their reasoning.
Now let's look at what actually happened
I included my immediate reflections into the following transcription to show you what I learned from this experience.
Loghan: When it (the investigation) asked: "Which one is finite and which one is infinite" does that mean – like – that this is infinite… because it doesn't have - like a… ? (trailing off…pointing to the Wolfram Alpha App on her phone which shows the values of some partial sums.)
CvR: What do you think, when you look at this one (points to the series);
do you think that if it keeps going will it be infinite or will it have a erratic jumping around behavior? (I meant the sum or series when I said "it". Since distinguishing these two is the main problem for the students, this is a bad mistake on my end. I want my students to be precise so I have to work on this as well.)
Loghan: Infinite? (Clearly guessing)
CvR: And how do you know? (This is a good clarifying question.)
Loghan: Because it is greater than one, right? (Clearly hoping for a confirmation)
CvR: And what does it mean, that it is greater than one? (Now I wonder what she means by "it". I should have asked her about that at this point. Before I know what she is talking about it makes no sense to ask if it is bigger than one or not. Not a great question.)
Loghan: That the number is gonna get continuously bigger, isn't it? It's just like a decimal. It doesn't get greater than one, so eventually it is just going to stop. (So she is probably thinking about the sum in the first sentence, seeing the decimals of the partial sums. I assume she is talking about the sequence (2/3)^n in the second sentence, noticing that it gets smaller? I should have asked her to show me what she is looking at on her phone. I assumed I know what she is doing, but now I realize that I can not understand her thinking.)
CvR: Aha? (My version of giving wait time and stalling a response)
Loghan: Maybe? (looks unsure)
CvR: Well, it won't stop. It will get smaller, but it never actually be zero, right? Or it get's larger but never hits one? (I think I am getting confused myself here between the sequence and the sum because I am so upset that Loghan thinks something is going to stop. I mean the sequence in the first two sentences. Then I am talking about a sequence (maybe a partial sums sequence) that get's larger. I really should not have talked about something getting larger and never hitting one…)
Loghan: So its never gonna be larger than one, which means it will be finite? It stops, basically close to one? (Is she saying that because of my confused statement about the decimal never getting to one? Oh boy…)
CvR: What stops close to one, the sum?
Loghan: The sum (looking more confident). The sum doesn't get greater than one which means its finite.
CvR: So you are saying: Whenever a number, say 2/3 in your example, or ¼ on the board, is less than one, the sum will always be less than one.
Loghan: Yes. (Looking happy)
CvR: Ahm, that is a conjecture we have to look at – if it is true or not. We don't know yet, right? Because we haven't found a way yet to find out an exact answer for these series. So I like that as a conjecture. I think working through the next packet will help you figure out the exact number… (Notice how I speak faster, because I am getting nervous? I have to handle this incorrect conjecture, keeping it open. I think that was ok)
Loghan: Ok. (Keeps working). (I am not sure what she will do next, will she just ignore this investigation and move on to the next packet? That's not what I wanted. She still is supposed to answer the question. Just the conjecture should be evaluated after the next packet. I think I left her in a space where she could not continue effectively…)
What did I learn from this short transcription?
I have to listen more closely to what a student is saying to me instead of assuming that they just have one of the typical misconceptions. I need evidence for my thinking before I ask clarifying (and hence leading) questions. It is interesting to see how much I lead Loghan (in the wrong direction) by making an actual statement: "Well, it won't stop. It will get smaller, but it never actually be zero, right? Or it get's larger but never hits one?" These look like questions but they are really statements.
What I could have said instead:
- What is getting continuously bigger, can you show me?
- Can I see what you are looking at on your phone? What did you compute?
- What do you mean, it will stop, can you show me what stops?
- Why are you comparing values to 1? What is special about 1?
I learned that watching yourself on videos and transcribing the dialogs are powerful tools for reflecting on your teaching. It gives us a true "reality check".