To understand how the students’ viewed the workshop course, and what they felt was important in it, we conducted one-on-one interviews with five students taking the course this year and discussed the course at length in the focus group with five students from a range of classes (two taking the course this year, only one of whom was in the in-depth interviews). In addition, we collected student surveys asking the students to reflect on each portion of the course. In each portion, we asked students to reflect on the workshop sequence (MATH 190/191). The reflection was handled differently in each. In the interviews, students were directly asked about the workshop class, to describe it, to discuss the problem solving, to discuss the community, to discuss transfer, and to discuss general learning in all classes. In the focus group, we asked students to make a top three list of what they would say about the class to incoming freshman. On the class reflection items, we asked students to reflect on different aspects of the class.
In the focus group, students focused on three main aspects: habits of mind, community, and information about careers. The students lumped many aspects together in the habits of mind section, including problem solving, mathematical writing (although student C suggested that she didn’t think of it as important as problem solving), clarity, conciseness, etc. These habits of mind are linked closely to what the students see transferring beyond the classroom. Under community, they included getting to know and work with both the other students and the professors.
The first and most forceful response of the students was community. All five students in the focus group joined in after student L said, that getting connected was the most important thing. Words and phrases that were used by students to describe community included connected (L), collaboration (C & W), do it in groups (W), cooperation (M), meet the professors (J), and camaraderie (W). Moreover, students consistently agreed with each other as they listed new ways that the community was being described. Then, when discussing habits of mind, W comments that, “you can learn other styles of thinking by collaborating with a group.”
This idea of the learning being facilitated by the community among the students comes up in the interviews also. For example, student AD comments, “it is just a group of friends feels like to me, who do problem solving pretty much, we present different ideas people present their different approaches to solving problems.” At another point she comments how they learned a lot from each other. Of the other four students, one explicitly said it was peer teaching, and of the other three, two discussed the community learning as solving problems together. The one student not discussing the notion of community learning or problem solving tested into our remedial mathematics class in the fall term, and it was clear from much of his work that he saw the class in a more traditional light.
Looking deeper at student comments along this line, there are three recurring themes:
All of the students made comments about how the problem solving sessions involved the students presenting to each other and how they solved the problem as a class. In particular, both ES and CL commented on how they would build solutions off of each other. This specifically arose in class when CL presented an incorrect solution to a problem that provided the key idea to ES so that he could solve the problem. Students saw the process of moving to peer teaching and solving as a class as something that evolved over time. For example, AD said, “I think a lot of it (the community) is getting more comfortable with each other.” But she also felt that the class “taught me a lot about community… accepting the fact that I am not the smartest… I feel I have so much to learn from the people around me now.” AF (an honors student) comments how “listening to other people’s thought processes that are different from my own unlocks things I don’t think about,” showing a discovery of how to learn from others, something he suggested was a new idea to him in a math class. Again, ES and CL make similar statements, but AD sees the idea of presenting as a time to get information from classmates, and states, “you are standing in front of your classmates and trying to get their information.” Thus, even he sees some peer teaching in the class, despite denying it later.
The role of the instructor also comes through pretty clearly. ES stated
You (the professor) let us solve it by ourselves so we learn how we can work together. Usually you just sit in the back of the classroom and look at us, unless we go wrong, then you let us know, but as long as we are on the right track, you pretty much let us go on as far as we can
This is essentially what all students said with the exception of AM. Other students termed the professor a “guide,” “moderator,” or “facilitator.” Again the notion that the main role of the professor was to ask questions when something incorrect was presented, however, several students suggested one other important role of the professor was to help the students talk to each other. Moreover the professor “sort of taught us that just because the answer might not be right, doesn't mean it's not useful information.” These notions were reflected in the focus group also, with one student pointing out multiple times that an important thing they learned in the workshop course was that it taught her that “you can learn from your mistakes as well as build on incomplete work.” This is an important lesson, as Polya points out that guesses are important, and even wrong guesses teach you something.
The third theme surprised us. Almost all the students in the interviews alluded to a perceived difference between the “honors” students and the other students in the class. AD, CL, and ES, all mentioned this in a somewhat negative fashion, suggesting that early on in the class, they saw the existence of these students as problematic, namely that the honors students would be doing most of the problem solving. However, all three of these students suggested that a key time for them in the class was when they were able to solve something the honors students didn’t get. The one honors student interviewed (AD) suggested similarly that he didn’t expect to be learning from the other students, but was surprised at how much he could learn from them. The fifth student did not make any such comments as was consistent with his view of the class. He rather looked at the class as one in which students worked toward the answer, but the professor still made the final calls.
The third theme raises two questions. First, do students in most math classes feel unsafe to make suggestions if they are not the honors students? And second, what features of the workshop help to alleviate these fears assuming they have them? Based on the number of responses by students about the difference between the “honors” students and the non-honors students, it certainly appears that most students did have this fear. Even AD and ES, both of whom talked of how in high school they were more likely to be the “top” students, noted concerns about the honors students. Indeed, clearly one of the “honors” students (that was not interviewed) essentially boasted in his portfolio how that when he put in the effort, he wrote quite well and it was his lack of effort that led to poor write ups [1] . For the most part, however, we lack evidence in this study as to whether this is truly the feeling in most mathematics classes. We have better evidence, however, as to what makes at least some students feel more comfortable, the 15-minute problems.
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[1] This by the way was not as true as he believed based on final write-ups in the portfolio he believed to be very good.