In his initial survey Neal said, “...my success in mathematics has not been spectacular in the past two years....” He described himself as someone that had not been competent at mathematics and had gained little understanding of the role of proofs in mathematics, and his role in creating proofs. Moreover, Neal was not alone. During the first week of class, I noted in my journal, “I am suddenly aware how much the students fear doing any math on their own.” Many of the students had been unhappy when I asked them to attempt to write a calculator program that performs long division. They found the work frustrating, but also very different from what they had done in the past.
We also note that Neal's early homework problems really show how poor his basic proof skills are. In homework 2, we saw that Neal was not careful about his language in his proofs and was quite likely to barge ahead without knowing quite what was going on. Similarly, the third problem set shows Neal's difficulty with including (and recognizing) the key steps in arguments, while also showing that Neal was competent at basic calculational work at that time. Thus, early in the course, we see evidence of a student that is mathematically weak when it comes to deeper understandings and has a negative attitude towards his own interactions with mathematics.
These two traits together are worrisome in a prospective teacher. To begin with, teachers that are only procedurally competent at mathematics do not have the deep understanding that is necessary to approach topics in multiple ways. Moreover, they are unlikely to be able to develop ideas on how to help students in specific cases. The second shortcoming is even more damaging. Teachers that do not engage in investigation of mathematical topics are unlikely to acquire what Li Ping Ma refers to as “Profound Understanding of Fundamental Mathematics (PUFM)” (Ma, 2000). In her work, she interviews Chinese and American elementary mathematics teachers. She notes that the teachers that truly exhibit PUFM are those that tackle new problems as investigations and engage in the investigation. An attitude towards mathematics that includes a belief that, “I could not afford to be creative” will make teachers unlikely to perform investigations of topics that require this sort of thinking. If one hopes to see American teachers starting classes off with problems as the Japanese teachers do in the TIMSS tapes (Stigler and Hiebert, 1999), they need to be able to engage in the problem with confidence. Moreover, as Shulman points out (1987??) teachers that lack confidence in their abilities in the content of the discipline are less likely to teach in engaging ways.
Neal echoed these arguments in the interview. When asked, “would you be more likely to deal with a student's question that you don't know the answer to? Now versus before?” his response was
Yeah, because, whereas before it seemed like oh, shove it under the rug somehow and yeah.
That is, he saw that prior to that semester, he aimed to avoid questions that he didn't know the answer to. This carried through to his understanding of an interesting problem also. Recall that his initial description of features that made a problem interesting was that it:
forms a bridge between an ordinary math concept(s) to a higher level of concept(s) in which you must use them to reach something entirely new.
That is, the problem was interesting on an external level. With this view, mathematics becomes something you transmit. Thus working against a constructivist model, and against a model promoting student engagement. Again, this view is a natural outgrowth of the attitudes and experiences pointed out above.
By the end of the semester, however, Neal is different. As his final exam shows, Neal is competent at mathematical proof. Moreover, while he was unable to give complete answers to several questions, he recognized that this was so. For example on the first problem of the final he writes:
As for the third derivation... I am empty-handed. However I see something that may be of use. ...
Thus, he recognizes the limitations of his response, but instead of trying to fool the instructor, he admits to not knowing what I am looking for, but then engages me in conversation about another interesting facet of the problem. Thus, he shows a willingness to investigate even on an exam.
We also see an attitudinal difference. At the end of the term, he states that the main aspect that makes a problem interesting is that:
there must be an inner motivation sparked inside the individual …
In the interview this is made even more explicit when he states that a good problem is:
One that starts (with) thought, mathematical thought, and interests to the individual student. ... Going along with that, there need to be motivation involved, because if there is no internal or external motivation for mathematical problem, it is pretty hard to explore, and to explore the mathematics I guess.
Here he is making clear the need for an inner motivation. Moreover, when asked how he sees this influencing his teaching, he states:
I think a big idea that we were shown through the course was that we should bounce ideas off of each other. A good way of exploring mathematics a good way of learning mathematics is by bouncing ideas off of somebody, because otherwise it seems you can get stuck and don't know where to go and give up. Something like that. Back to the idea of oh mathematics if you can't solve it within five minutes, then you can't do it. So it challenged that idea. So one thing I definitely want to apply as a teacher is to try to instill the idea with kids, with students, encourage them to bounce ideas off of each other. So in creating lessons, in creating discussions, group work when bringing up challenging ideas, see what kind of questions they have. Trying to spark an interest with the students themselves, but letting them do the exploration so they can create a sense of ownership - perhaps.
Reading this, we see that Neal's intentions are to engage students and get them to take ownership of mathematics, certainly something that the curriculum reform movement is striving for. Continuing along this line, however, Neal suggests that he doesn't think the students should have complete control over the content, as he talks about getting students to link a new concept with a previous one.
The last change we note here, is the change in Neal's confidence. When approaching mathematical material early in the course, Neal found it difficult to get started. In responding to the question: “Do you feel more confident in attacking problems at this point?” Neal stated:
I do. You probably want more than a two word answer…. I think part of the answer to that question goes back to what we just talked about as far as not being afraid to ask questions in mathematics. Not being afraid to say, well, OK why are we why are we looking at this, where does it come from, and I think there is a motivation factor in there. Once you know why, you can go ahead and maybe develop some confidence in attacking things.
So his confidence has improved, and one of the things he notes is the importance of this confidence in allowing him to ask questions.
Above we have argued that by the end of the course Neal had gone through a major transformation. This raises the question: What factors led to this change? In analyzing this question, of course, we need to recognize that this course was not the only influence on Neal's attitudes that term. For example, he took a teacher education class at MSU that discussed philosophy of mathematics issues at the same time as the capstone class. On the other hand, it is worthwhile to try and nail down some of the factors.
I believe that the project, and Neal's experience in particular, was a major influence, but I also think that more was happening. Listening to Neal's words on the issue, a common theme is shown in the following:
One thing that was good about the project was that we were able to take the whole semester to examine it.
I think for me as a student struggling with mathematics coming into the course, even though this is my seventh higher level mathematics proof class, was that I needed to have some time to examine all the things that were possible. In other words, I guess, where I got grown accustomed to doing problem sets a couple days before they are due instead of look at the problems OK if I don't know how to do it, I don't need to do it right this instant, I can sit on it and then it might come to me the following day, so that extra time, I thought, was important in developing ideas that were essential to what a mathematician does, not even a mathematician, but someone… the exploration process requires that time frame. It is essential for that.
That is, he doesn't name the project as the influence for a change of attitude, but he consistently refers to it as an example. This shows that Neal views the project as an example of a good activity, but does not believe it to be the cause for a change. On the other hand, he does attribute this to the whole class.
You know, piggyback on each other one definitely complemented the other (the class and the project), but I think it was more so the project group helping to complement the class work.
One should be careful, however, to point out Neal's use of the phrase “project group”. He clearly sees the group as an important component to the transformation. I also see the group as an important component. In particular, as we have seen earlier, a strong group member has the potential of destroying confidence in other members, rather than making them stronger.
One other factor that Neal mentioned was his motivation and interest in the course, which he attributed to my showing them that I cared about their ability to teach mathematics. In particular:
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.
I would expand on this slightly. What seemed incredibly important to me on this part was the conversations we would have in class. Alan commented in the group interview that
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.
The point is that the students felt in control of the class. They felt that they were engaging in the mathematics. What surprised me was that I also felt in control of the class. We never strayed far from anything I had planned from the first week of the term, although the path to get where I was going seldom seemed to be how I wanted to do it.
The last issue to examine is Neal's learning of the content in the class. Running down the five process outcomes we have:
Neal's final exam analysis of the quadratic equation derivations does a very nice job of linking his learning to the high school. In particular, he links his understandings of the processes of mathematics to the classroom in his statement
...these things are useful to talk about in the classroom - why we do these operations...
In the project, however, this was less apparent. I think that his work on homework sets 9 and 10, however, also provides evidence for the success of his ability to look down at the curriculum and see how advanced mathematics is reflected there. In particular, his responses on why the complex numbers and matrices appear in the standards.
One of the major ideas seen in some of the other students' surveys was that these connections were important. For Neal, however, it is less clear how much he met this objective. In the project, however, they do discuss this when they talk of:
Throughout the semester, we discussed many different possibilities of where to start our proof. Among these were ideas such as: what to do with summations, exponents, and factorials; proof by contradiction (week seven); analogous proofs to those of the Taylor and Maclaurin series; induction of a series (proven later); one-to-one and onto mapping (also proven later); and the extreme case of enumerating the permutations for cards 7 through 10
I would say that Neal was more aware of connections between subjects at the end of the course, however, based on discussions with him.
Certainly Neal and all of the students saw some of the questions since we discussed solving equations, proving the impossibility of the trisection of the general angle, the history behind proving that p and e are irrational, and the history behind Dedekind's definition of the real numbers.
I think this is the goal that was most explicitly met. Consider the quote from Neal upon being asked: “ Did the project excite you at any point, or did it bore you at any point, can you describe some of that?”
Yeah, I don't. I wouldn't say that it bored me at any point, and I would say that it definitely excited me at some points. It was totally a sense of ownership. Ownership and accomplishment the sense of accomplishment at the end for sure. And as you're going through… let's see … we did the project, the project was on cards, ten cards, and when we started collecting our data, it was fun. It was fun, seriously, the only part that wasn't I think, was that ok, after you do about 200 trials when you know you are each going to do 500 hand trials, it was tedious. You knew you had to do it, but it was still OK. Beyond that point when we were actually looking at the mathematics more, the sense of trying to figure out where you are, where you need to go, how to get there, it was challenging and that challenge proved to be an accomplishment, which was a good feeling as a struggling mathematics student.
Neal felt he had successfully experienced such research, and based on their project, I would concur.
Again referring to Neal's interview, when asked about what sort of questions he might ask as a teacher, he said:
So one thing I definitely want to apply as a teacher is to try to instill the idea with kids, with students, encourage them to bounce ideas off of each other. SO in creating lessons, in creating discussions, group work when bringing up challenging ideas, see what kind of questions they have. Trying to spark an interest with the students themselves, but letting them do the exploration so they can create a sense of ownership - perhaps.
Certainly, he has begun to see the value in asking richer mathematical questions.
In terms of the content goals:
Neal shows that he has learned this in his ninth homework where he says:
... in each parallel we're building a system of numbers. We are exhausting a list of every possible number in different ways that we can think of them...
Thus, he sees different ways of looking at the numbers. In terms of recognizing the strengths and weaknesses, there is less evidence for that.
I think that one of the flaws of the final exam is that it did not do a good job of measuring this outcome. I did ask Neal this question in the oral portion, and he showed a much more thorough understanding then than during the midterm.
This is shown pretty clearly in the final exam on problem 4, although he takes the formulaic way out. In his oral portion, however, he did show that he understood the geometric picture leading to this solution. Moreover, this appears in pages 1 and 2 of homework #9.
The final exam problem 1 addressed this issue. He certainly did so in terms of the two methods he presented. That he missed using translations as part of the work, argues that he might have done better on this portion.
This was shown in the work on Dedekind cuts, and it his argument in problem set 10
What we mean when we look from whole numbers to decimals & fractions to irrationals is that we're filling out the number line - (BUT WE DON'T ACTUALLY SAY THAT!!!)
In context, he was pointing out the need to fill out the number line, which is precisely the key for the need of Dedekind cuts. While Neal did not mention in his homework or final exam the difficulty that students and teachers have with .9999...=1, he was certainly one of the students to discuss this issue at various times.
This objective was probably not met. The second problem on the final exam was to analyze this issue, and there was one homework set that briefly touched on it. However, the goal was downplayed in the class.
Certainly discussed, and Neal performed adequately, if not spectacularly on that homework set.
His group did so, although as said in the grading, they might have directed it more to the high school curriculum.
In summary, Neal fulfilled many of the objectives of the course. Perhaps the most impressive work Neal did, however, does not represent itself well in this portfolio. He did spectacularly well on the oral final portion, and his work on the oral final was a major influence on his final grade.