That doing the projects would have an effect on the students was not a big surprise to me. That it would affect how I taught content in the class was, however. This page is devoted to discussing how and why this happened.
As a traditional mathematics faculty member, I had various presuppositions of what the students knew about how mathematics is done before starting the course. Many of these presuppositions were destroyed by watching the students struggle with their research project. Upon reflection and research, I found that most of my surprises are clearly known in the literature. On the other hand, the format that these misunderstandings took, their manifestations, and their remediation all appear to be based on the projects. Below we shall look at some specific examples.
Multiple Representations: The first misunderstanding that I discovered as common among many of the projects dealt with multiple representations of the same object, and what the value of having such representations are. Dan Chazan (Chazan, 2000) points out that a problem asking students to manipulate an expression to move it from one form to another
...is the kind of problem that seems to beg students to ask “Why is this important?”
Senior mathematics education majors typically are good at such problems as they are usually asked. That is to say that given a problem asking them to simplify an expression, they are capable of completing the task quickly and easily. Of course, that the students are good at these procedures, does not imply that they understand the value of the procedures. Consider the following vignettes from the students' project work.
Vignette 1: Jill and Bill (link to project group) came in to discuss how to find the best rational approximation for the square root of 2 using a given integer denominator, without evaluating the square root of 2. For example, if the denominator is 5, we might begin by realizing that we want an integer x such that x/5 is close to the square root of 2. Consequently, we want x^2/25 to be close to 2. At this point, it seems clear that we would want x^2 to be close to 50, and we see that the best integer choice for x is 7. Moreover, we can pretty quickly deduce that this approximation must be pretty close since 7^2=49 is within 1 of 50. Thus 7/5 is the best rational approximation for the square root of 2 with denominator 5. The students had gotten stuck on this problem, when they came in to talk with me. As is typical in these types of problems, I had them begin by working through the example (in this case using denominator 7). After much discussion, the students decided they needed to find the integer closest to 7\sqrt{2} (7 multiplied by the square root of 2). Here they again got stuck. What became clear as I was asking them questions trying to get them to bring the 7 inside of the radical (so they would get \sqrt{98}, a number that is clearly close to 10), was that while they would have no trouble performing the operation, they would also never think of doing so on their own. In particular, these students had learned that 7\sqrt{2} is the “correct” form for the number \sqrt{98}, so there would be no purpose to putting it in an “incorrect” form. Mathematically, of course, there is no correct form, rather different forms give different information and are valuable at different times.
Vignette 2: On Wednesday, October 4th and before, the group working on the worm problem had written a recurrence relation to try and solve the problem. The relation was in a nice “tight” form. Namely, they had that the ratio on day d of the length the worm had crawled to the length of the band was
R[d]=W[d]/L[d]=(W[d-1]+1)/L[d-1],
where R[d] is the ratio on day d, W[d] is the length the worm has traveled on day d, and L[d] is the length of the band on day d. This formula is extremely close to the recurrence that makes the solution much more transparent. The key is to rewrite the last expression as W[d-1]/L[d-1] + 1/L[d-1] = R[d-1] + 1/L[d-1], which quickly leads to R[d] corresponding to the d-th Harmonic number. The students were not able to take this last step, however until November. Again, watching them struggle with this problem, it was clear that to them the right way of writing down a sum of two fractions with a common denominator was to simplify the equation as you would in high school algebra. Just like the previous group, they saw no benefit to having a different expression, and thus they did not look for it.
In each of these cases, the students naturally encountered a problem requiring them to have an unsimplified expression. Both times the students experienced difficulty since their understanding of simplifying answers was based on a certain rigidity of mathematics rather than based on different forms having different meanings. Notice that no textbook problem is likely to show me this misconception. Two things happened here. First, I was made aware of the misconception so that I could remediate it. Second, the problem with the misunderstanding had been contextualized for the students. Thus, on addressing this misunderstanding, I could use this contextualization, rather than discussing it in the abstract.
The domino effect was most interesting, however. Because I wanted to address the misconception for all students, I needed to further contextualize the problem in terms of the course content. In this case, I was lucky. These questions arose as I was beginning to discuss extension field theory to set up the proof that you cannot construct the cube root of 2 with straightedge and compass. One difficulty with this proof is motivating the value of the proof to the students. To motivate the field of study, I start with the question
What if we are asked to rationalize the denominator of the fraction 1/(2+\sqrt[3]{2}-2\sqrt[3]{4})?
(\sqrt[3]{2} means the cube root of 2 and \sqrt[3]{4} means the cube root of 4). The value of this question being that we can solve this question with our graphing calculators and representing the number in the denominator as a three by three matrix over the rational numbers. (See chapter 3 of the text.) At exactly this point in the class, I then stopped and pointed out the value of having multiple representations, and then I brought up both of the project groups and how they had run into this problem of needing to recognize the value in having different ways to present the numbers.
At the end of the spring term, Neal allowed me to take a copy of his class notes, and the value of this presentation is apparent in that his notes included the comment
* There's value at writing things in different ways.
A second example of this sort of interaction was found over the role of definition. Yakel and Cobb have discussed the socio-cultural norms of a classroom, and how these norms influence student work in mathematics. One particular issue involved in this is the role of definitions in mathematics. Mathematical presentation usually runs: definition, theorem, proof. However, this is the sanitized version of the mathematical process that runs much more characteristically as conjecture, proof, definition, theorem. (In fact, the process is far more complicated if one follows Lakatos's perspective here (Lakatos, 1976).) Since mathematics is rarely presented this way, ( such presentation appear to decrease the material that can be covered, and sometimes the understanding of the content that can be achieved) students often do not understand the role of definition in mathematics. One particular manifestation of this is that they do not see it as their role to create definitions (or in the language of Yakel and Cobb, it is not within the socio-cultural norms of the classroom). Consider the following vignette:
Vignette 3: The Euclidean Algorithm Project Group had reached a certain stage in their project where they had conjectured that the smallest pair of numbers needed was always a pair of consecutive Fibonacci numbers. By the fourth week they had come up with an induction proof of this, except that they had no definition of smallest pair. In their weekly project update, they said:
A more specific question is how do you define a pair of numbers being smaller than another pair? Is it defined by the distance from the origin? Is it defined by the lower of the two? The higher of the two? That is obviously an integral part of proving the smallest pair that takes n steps for the Euclidean algorithm.
They recognized how crucial the definition is, but they don't recognize that making the definition is within their purview. A week and a half later the project group came in to see me (the only time during the term that the group came to me as a whole) and discuss this exact problem. My diary entry (on October 25) included the following discussion
They (the Euclidean Algorithm group) seem very strongly to want to avoid making up their own definition. This is really an issue of control and mathematical belief. Ron was able to explain how the proof that the Fibonacci numbers give the minimal pairs should work, but he couldn't see how to do it because he didn't have the right definition for least. I told them that part of mathematics was making up the "right" definition. That is the definition that leads to the most interesting mathematics.
As in the previous case, I needed to bring this discussion of the role of definition in mathematics to the class and contextualize it for the whole class while making use of the roles it took on in the project groups (the Approximations group also had trouble committing to a definition of best). In this case, I delayed discussion on it for a while. At the time of the discussion with the Euclidean algorithm group, we were working through the solution of the cubic by radicals. When we reached the general cubic requiring the use of complex numbers to solve, I was able to pull out the historical development of the complex numbers (which I always discuss) and point out that the definition of the complex numbers arose when Bombelli was trying to find real roots to cubic equations, and not because somebody wanted to create a root to a quadratic equation with no real roots. (Or at least only tangentially so, since Bombelli needed to work with the square root of -121 along the way. His idea was to allow for this placeholder symbol that squared to -1, and the only purpose to it was to allow him to find real roots to the cubic.) At this point, it comes up that the definition of the complex numbers arose because it was what was needed to make a solution work. Again, in Neal's notes we see this arise in the comment
definitions - go through changes because we define what is useful
And again, as I brought this up, I was able to tie it to the project groups that had encountered this difficulty.
The above examples were only two of many where I needed to bring contextual meaning to mathematical misunderstandings in the midst of the content being delivered. Moreover, once we had this issue in context once, the class would then spiral back to it on several occasions. Thus, the value of multiple expressions for the same object was discussed the day that we classified the Pythagorean triples by looking at intersections of lines having rational slopes and going through the point (-1,0) with the unit circle. Here we used an unusual formulation of a line, (slope, x-intercept). This led to a brief discussion on the different information that different formulas for lines bring up.
Similarly, when Dedekind cuts are introduced, the class encountered the questions of the role of definition in mathematics again. But in all of the cases here, the projects enabled a contextualization outside of the content of the class, and then my job was to contextualize these mathematical understandings inside the context of the class. I believe that this combination was extremely valuable, and greatly improved student understanding of the importance of these ideas.
Back to Content and Project connections page.