At the beginning of class, some of the students saw themselves as doers of mathematics, but many others did not. One of the written survey questions was
Question 42: Describe the most interesting mathematics problem you have ever worked on.
Almost half of the 13 students (Ellen didn't fill out the final survey and consequently has been left out of this account) gave answers suggesting that they had no problem that they could think of. These responses consisted of 2 blank responses (Don and Teri), 2 responses that said that they couldn't think of a particular problem (Jim and Mary), and 2 responses (Neal and Lyn) stated that they couldn't remember one. Of the two that said they couldn't remember a problem, one was Neal's (see his responses to all written questions), and the other, by Lyn, indicated some criteria for the student to rate a problem as interesting. In particular, she said:
There are several, but none that I can think of. It's those problems that through the early learning stages of math I am not able to see how things come out the way they do. Then with some instructions I am able to say "ooh," I see how.
This last suggests that Lyn does have some examples of interesting problems, but that her work with such a problem is based on knowledge transmission. That is her interaction with a problem is for someone to give her “some instructions” so she can see how to do it. It also indicates a budding aesthetic view of mathematics.
Another 4 (Tom, Ron, Jill, and Bill) of the students fell into a category of having particular “homework” style problems, or externally motivated, ranging from Jill's calculus/geometry procedural
Volumes of three dimensional objects
to Tom's more process oriented
I can't pick one. I liked the complex proofs done in abstract algebra II.
Finally, there were 3 responses (Brad, Tom, and Alan) that were indicative of students trying to solve internally motivated problems.
Brad: I have two. First, it is my own theory. It felt great that my find could actually be proven; that my theorem was true. Next , it is another project that I have looked at; proving that no odd perfect numbers exist.
Tom: I enjoyed a lot of logic problems, like ideas involving the principle of contradiction, universals, logical consequence, false premises; especially when I got to read some different authors on it.
Alan: The most interesting problem I worked on was exploring for myself the probability of getting NCAA tournament all right from start, then after 1st rnd, etc.
I placed Tome's response in this category because of the external reading, and in any case, all three of these students have chosen problems that they found appealing internally. Another feature of these problems is that they all required the students to think on their own, and the answer is less important in many of these than the process. For example, Alan comments on “exploring” the problem, and Brad picked an unsolved deep and rich mathematics problem as his second one. Additionally Tom talks of ideas and reading, rather than solving.
What these responses show is that most of the students appear to see their involvement with mathematics in the past as one of doing externally motivated problems. The last three students, however, show a more mathematical perspective on problems, one that suggests that mathematical investigation and exploration are interesting.
The post class responses to this question are less easy to analyze. As one might expect, the majority of the students (9 of them) listed the research projects they worked on in class, but few give much detail to why they found it interesting. There were a couple of responses here that are worth noting:
Alan: My most interesting mathematics problem was the project I worked on this semester because it gave me free will to think.
Don: The semester long project in math 496. It involves constructing a figure and exploring & analyzing the figure. We also explored sequences and limits in constructing the figure.
Alan's is interesting because he now sees the importance (for him) of having a “free will to think,” but since he was originally one of the upper three, this isn't particularly informative. On the other hand, Don, who had a blank response initially, makes a very interesting comment about the project involving “exploring & analyzing.” It appears from this response that Don sees his involvement and exploration as a component to the problem being interesting.
Alan then saw the whole class in this light when he said in describing the class:
Some days we came in and we would work on groups on homework, but I felt like a lot of the time it was us kind of blindly of walking through this idea of "I think this is where we want to go" and your just kind of OK we'll let them go there and wait until they run into a wall, and when they're dizzy I can help them back to the uh… 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 asked for a specific example, he mentioned the day that I sat in the corner. Thus, one has to ask what happened there. A day where the students took over the class and argued about the mathematics. For an analysis of that day, click on the above link.
Other students also felt the class was different from a typical class. When asked what made the class different, Jim said
There was more of us exploring.
However, when I look at the class, the exploring by the students became more intense at the end of the class than at the beginning, suggesting that this exploring was being created.
The above suggests that the students attitudes changed. The next question is what role the projects played. Clearly, we cannot definitively answer this question, but we can suggest a way the projects would help with this transformation, and then look at what was said concerning the projects.
Schoenfeld (1980) suggests that students experiences with mathematics are quite limited. The typical undergraduate student has never been given a problem that encourages deep analytical thinking. Rather, the problems usually come from a section of the book, and the students mostly can follow methodologies from that section, or use theorems from that section to solve the problem. Thus, the typical student has had little experience with actually solving rich problems on their own. Moreover, when problems of this nature are presented in mathematics classes, they are often presented as examples of using a particular trick. For example, in some discrete mathematics classes, the derangement problem (which the 10 cards problem is based on) is done in an advanced section on inclusion-exclusion as a problem that this technique solves. Consequently, students can become convinced that mathematics is a collection of techniques, but techniques that exist before problems, as opposed to techniques developed to solve problems.
In the teaching world, we see more example of this with American schools. In the TIMSS study, one of the American mathematicians commented that in American classes “I have trouble finding the mathematics; I just see interactions between students and teachers” (p.26 Stigler and Hiebert, 1999). What was missing was an understanding that mathematics is much more than the procedures. If this is symptomatic of the teaching, it follows that the students would learn these attitudes.
The role that the projects play is that of counterexample. Namely, the research projects force the students to work on a rich mathematical problem. They force the student to become involved in the problem of developing algorithms and arguments. They give the student ownership of mathematics, and they allow for the students to confront a different view of mathematics, one that more closely resembles the mathematical researcher's view. Schoenfeld (1980) argues that for most mathematicians this happens in graduate school as they encounter their own research problem. The projects thus play the role of a research problem for these students. The role of the ownership in the transformation of attitudes is then crucial, as it allows them to recognize in a clear-cut way that they are capable, and consequently that there is hope for them to participate in mathematics.
This last paragraph suggests a way in which the projects might help change attitudes, but it doesn't show that they do. In truth, it would be hard to design an experiment that would definitively show this since one can hardly control any of the other variables in students' learning. Thus our argument will spring from what the students said.
The first question is did the students view the projects as a way of seeing what doing mathematics is like. Alan said that the class and the projects were both about how you do mathematics, so he clearly did so. Mary, disagreed with Alan in terms of the class being about how mathematics is done, but she did say
the whole idea of the problem (the research project) the whole time was so that we could see what it was like to actually do mathematics, or something. That's what I got from it.
So not only did she see the project this way, she also felt that she gained some idea about how you do mathematics. Jill, who belonged to a group that did not function well, said of the projects
...and I felt that when we did the project, what Alan was saying has a lot of validation, because I hadn't had a math class that made you not know where you were going.
Again we see that doing the project was somehow very different for her than what she had seen in previous classes. Thus, we have validation that the projects were somehow new to the students (at least from their perspective). But what sorts of things made the projects different? The design of the projects intended to cause the students to look across various mathematical fields, something I suspected that students had rarely done. Brad said of their project that
Well just in general, we had to use, as I am sure everyone did, many different aspects of math from a bunch of different classes and I have never had that in any other math class before, and that's what we need as future math teachers because we need to show to our students, you know, this relates to everything, you're gonna have to remember this, use what you know, … yeah, use what you know to discover what you don't know, and I really found that in ours.
This suggests that he saw the project as being different simply because it brought together different fields. This also validates Schoenfeld's argument that students are used to being shown (either intentionally or unintentionally) where to look for the answer. Thus, this appears to support that the projects played a counterexample role for the students.
One other aspect to all of this is the role of time. It appeared to me that allowing the students to work unfettered on the problems for most of the semester, encouraged them to try a variety of different approaches. Upon being asked how the projects might have been different if they had been due in three weeks, John said:
I think it definitely would have cut some of the sloppiness out. Also probably less likely to go beyond the scope of the project and go on to the second question that was part of the problem, one of the few problems we had with our project is that we came up with a bound and sort of stopped there we sort of bogged down once we had what we thought was a pretty good bound, and a lot of that is because we didn't meet very many times after that after we came up with our bound it became like "let's write it up" and I think that probably would have been a lot more widespread.
While he used sloppiness in a pejorative manner (as he said later), other students were less clear. In particular, the following conversation followed John's comment:
Alan: I guess for me, I guess for me, for our group, that was the best part of the project - the sloppiness.
John: That you could be sloppy?
Lyn: Yeah, we were able to sit down and throw ideas out it's like okay we can try this or try that we didn't feel like we had to prove this, this is our only choice let's stick to this one idea.
Lyn and Alan are then arguing that part of what was good about the projects (and new for them) was that they had free range to make choices. This free range promotes ownership of the mathematics, and a feeling that the mathematics is their work. One sees similar arguments arising out of Brad's and Neal's discussion of their project. Thus, again, at least in many student's cases, and in particular the cases of Lyn, Don, Neal, and Jim, (three of whom gave no answer to the most interesting problem they had ever worked on in the class, and all of whom seemed to gain a changed attitude about their relationship to mathematics.
Return to Discussion of changes of course wrought
by projects.