Perhaps the most surprising thing that happened in this class was that students started pushing me to discuss deeper mathematical topics. In every course I teach, I will mention important problems that drove the mathematical discovery. Usually one or two students might express some interest in knowing the answer to the question, but particularly in the mathematics education capstone course, it is unusual for students to ask me about the proofs of related theorems. This term, however, the students stopped me at least three times to ask about proofs and deeper explanations of such topics. The three that come to mind are that the students asked me to outline the proof of Gödel's incompleteness theorem, the transcendentality of 2^{\sqrt{2}}, and about infinitesimal arithmetic. In addition, while I always plan to discuss cardinality and the proof that the set of integers has the same cardinality as the set of rational numbers, this topic was covered about a month in advance of when I would typically cover it. The issue arose out of an open conversation about commutativity that led to discussions of infinity. (For my diary entry click here.) Most years, if I have these student driven discussions in the capstone class, few students seem to be paying attention. More often the conversations happen in my office. What made this class surprising to me was the depth of the questions, that they took place during class, and that the questions were raised by many different students.
The students also felt that this class was different from their typical experience. Alan said of the course:
So that's what it felt like to me, like we as a class decided where to take, where to take the next step in terms of what was going to come up the next day, we kind of decided, OK this is where we think were going to go and sometimes you'd be like "OK, no don't go there cause it's going to take us a whole class period and it's really bad," and other times you would let us go that way and we learned a lot from that.
When the students were asked to elaborate on whether this was like other classes they had taken, overwhelming they said this class was different. In comparing his behavior in the class to other classes, Ron said
I think externally I acted the same, I speak up in other classes also. But internally I definitely felt more comfortable when I did speak up in this class.
This idea that the class was more comfortable was common throughout the discussion, although there was great disagreement among the students as to why they felt more comfortable with some attributing it to the other students, some to me, and some to the subject matter. I suspect all three are responsible in part, although in comparing that term's class to those from other terms, the two major difference that I have noted is that I spent more time in class discussing topics that were not even necessarily tangentially related to the class. Alan noted in the interview that
One of the things I wanted to mention about that last question is that um I felt comfortable asking you questions that had nothing to do with the class, and I think that was a really big thing, there were just so many things that I never knew and I always wish I knew and I always wish I could ask them, but I never asked them. I guess I felt comfortable asking you stuff that I knew had nothing to do with what we were going to cover that day but I knew that you would answer them and not look down on me that way.
Thus the comfort level extended to the level that the students could talk mathematics in class knowing it didn't have to be directly related to class, and from my perspective, this comfort carried over in to asking questions that they may have felt were unrelated, even though from my more advanced perspective they were.
Related to this, during the interview the students got into a discussion of “sloppiness” in the class. The conversation started when John remarked on the projects that:
I think it (having the projects due in three weeks) definitely would have cut some of the sloppiness out.
This idea of sloppiness intrigued many of us. John went on to elucidate his meaning of sloppiness as
I always thought the projects and the class were at their best when they were directed, when everyone was interested, and when our conversation was directed towards trying to come to an understanding of the problem and that wasn't always the case and when things got sloppy I'm using sloppy in a pejorative sense a bad sense, the class to me was less fun.
Jim agreed with John somewhat in saying:
I kind of agree with him too, I thought like there were some times where yes we had conversations and stuff but I thought like it got out of hand and sometimes we would be at 3:40 and we didn't get anything accomplished, whereas like "God I wish I would have understood this better."
At this point, I was curious what Jim meant by sloppy. Asking both students to elucidate, they said:
John: I would say sloppy that we weren't talking about mathematics.
Jim: I would say the opposite. I would say that we felt like, well only because I felt like I would come to class I had been doing my homework and I didn't understand something and today you are supposed to be going of er that specific topic and we would get off tangent and talk about other mathematical stuff which is cool, but I felt like this was supposed to be the day that I was going to understand this stuff.
John, a top student in many of his math classes, considered sloppiness to be when the class wasn't talking about mathematics, whereas Jim, a weaker student included in the definition of sloppiness any conversation that was not directly tied to the course material. The distinction between John and Jim's views are key here. John had taken more math courses than any other student in the class and was probably the most mathematical of any of the students in the course. Thus, he saw the proper discussion of this mathematics course to be mathematics in any form, and only saw problems with the discussions when they strayed off of mathematical topics. (An example of what I believe John considered a non-mathematical discussion is that one day we discussed tracking for 10 minutes in class.) Jim, on the other hand, saw the times we discussed mathematics that was not germane to the homework assignments as sloppy. This latter view seems to be shared by many of the students in the class, at least in the sense that in most math classes they only feel comfortable asking questions that are clearly directly related to the material being covered. Now relate this to Alan's quote up above: I guess I felt comfortable asking you stuff that I knew had nothing to do with what we were going to cover that day... While it was certainly true that the questions on Gödel's theorem, countability, etc. were not ones that I had planned for us to cover that day, they weren't questions that were foreign to the class in my opinion, but to the students they sometimes were.
Upon further pushing during the group interview, the students suggested that what made for their comfort level was my enthusiasm and that I “never blew off any question.” Again, these may be important traits, but in my experience for me they are not sufficient to create the comfort level the students had. Neal, however, had another suggestion, namely:
I think some of it came from the first day or the first week, the introduction of you and yourself and we could see you as a person. Because of the personal things you did tell us, I mean you didn't tell us your whole life story, but being able to relate.
The details of the introduction were certainly deeper than many years, so if this is it, why did it work? I would suggest the following hypothesis:
Students in mathematics classes with low confidence feel that the questions they are allowed to ask need to be directly tied to the material covered. However, these students may not be as aware of what is related to the material (particularly in an abstract class), and thus they are less likely to ask non-procedural questions.
Let us look at what might support this hypothesis, and what does not. Following Neal's comment above the discussion continued:
John: His son plays soccer.
Jill: His dad tie on the first day of the semester.
John: You laughed more than any other math professor I have ever had.
Jim: When I was referring to you (Dr. Bennett) earlier as the god of mathematics I meant that as far as if there was a problem I had, I expected you would be able to solve it, I mean one way or the other, I mean and ultimately I would come to visit with you and I would end up solving it just by talking with you about it. But just by knowing that things are going to be OK, all I have to do is talk to Dr. Bennett and things are going to be OK I mean I came to you one day, he helped me with my 481 homework. I had no clue...
Certainly Neal's comment about seeing me as a person is borne out. Moreover, none of the students contradicted this statement. But the importance of allowing less mathematical discussions is not so clear. However, look at Ron's comment:
There would be some questions that I thought was simple in you know we know what the pecking order of the class was, and when someone on the bottom of the pecking order would ask a question you certainly wouldn't take it as a silly question.
Thus, he believed that I wouldn't take questions as “silly,” and this fits nicely with his comment that he “felt more comfortable when he did speak up (in this class versus other mathematics classes).” Along the same line, Lyn pointed out that when they asked questions,
We knew you were going to take the time out and put the objective aside.
So again that the questions came from a comfort level in the class and that I would “put the objective aside.” However, I never felt that I put the objective aside, merely that I was getting to a deeper objective, that the students might not see.
One should ask how I reconcile this last remark about never putting the objective aside with the statement about the students seeing me as a person. I don't entirely, except to say that the first day of every class I spend about 10 minutes introducing myself to the students and cueing them into facts I consider important. That particular term, my son playing soccer was important because it affected when I would be in my office. Otherwise, more of the personal information had to do with describing my personal experience with doing mathematics and with mathematics teaching. Finally, my view of teaching as a personal experience means that when I reflect on teaching issues, these reflections are necessarily personal.
Interested in getting a better understanding of the role of sloppiness in the conversations, I asked Neal in the one-on-one interview (held about 2 weeks after the group interview) to think more about the sloppiness or fluff. His response:
As an education student in a higher level mathematics class, that was if not a key ingredient, one of the most important key ingredients of why I loved this course because we were able to, we had a sense, … Dr. Bennett gave us a sense that he cared about our ability to teach, our ability to teach mathematics, not just to learn mathematics on our own. And so that was a different twist on a higher level math class that never had before seemingly, which is a problem, I think, of the setup of the coursework that we're required as mathematics education students. And so that was an intrinsic motivational factor. So when we had discussions about tracking and some of the other "fluff" things, lectures I guess (CB: John fluff) John fluff yeah right. In a sense those are two different kinds of fluff, and I think they reach out to different kinds of people. The Gödel's theorem fluff reached out to John, the tracking fluff reached out to myself and some of the other math education students.
Again we see this distinction between definitions of sloppiness. Neal falls closer to Jim's definition, but he didn't mind the “mathematical fluff,” and upon my mentioning that it wasn't just John that asked about the “Gödel's Theorem fluff,” Neal allowed that was true, and his one feeling was that the key was that students trusted me and that much of that had to do with directing topics towards the high school curriculum. In any case, he felt that the “teaching fluff” helped create an environment where students felt freer to ask mathematical questions. I think one other key statement here is that the “fluff” that was not mathematical being talked about was teaching oriented. Thus, when I mention this discussion of fluff, I am not suggesting that talking about the baseball game last night is likely to create a situation where the students will ask more mathematical questions, but rather that opening up to questions that are tangential to the course in my eyes (like tracking), can help to create an environment in the classroom that makes it more likely for the students to ask questions.
Certainly other issues affect conversations, and addressing only one is unlikely to dramatically change conversations. Moreover, the students interactions among themselves are extremely important to this endeavor. The projects certainly caused some students (like Neal) to be more confident, something that also suggests a greater openness to asking mathematical questions.
By no means do I feel that I have heavily supported the hypothesis above, but I think it is one that bears looking at. Namely, I suspect that the willingness in a mathematics class to talk about things only tangentially related to mathematics can paradoxically lead to more student led questions about mathematics. One should be careful here, however as I do not mean to imply that it would be worthwhile to simply start a conversation about sports or something else in general in the classroom, but rather that encouraging student questions about related topics that interest them may be a vehicle to improving the nature of the student faculty interaction in a content driven class.