The spiral project paper takes the form of a report of activities. The students discuss various things they did to investigate the problem. As Lyn said during the interview
Like should we continue with this, should we continue with this? I thought it was funny, exploringly. You could kind of read it into anything you wanted it to be. And Dr. Bennett let us, you allowed us to go ahead and diverge, like from the original thing. Like if you thought of something else you wanted to prove, you said, "go ahead and do it," and I liked that idea.
Part of letting them explore led them to have a lot to report in the exploration stage, and they report on trying to find exact lengths (and formulas) for each successive construction, whether there was any time that the construction returned to exactly the same ray that it started with, the role that error plays in calculator calculations, and whether the figure has a limiting bound on the radius.
Rather than stating a theorem explicitly as one might in a mathematics paper, the students made their conclusions in a less formal way (even though the proof(s) was relatively formal). For example,
In our explorations, we found that the sequences of triangles were not bounded. This led us to ask the question, "What needs to change in order to find a sequence of triangles that is bounded?"
As is clear from the above, they also stated their questions in this same way. The main proof (p.4) in the paper was well written, but it could have been slightly improved at one point where they were not quite as clear about why delta had to be positive.
While the more conversational style may have detracted slightly from the paper, the students made up for this by spending two and a half pages relating the work to the high school currriculum. The following paragraph is indicative of the work the students do in this portion of the paper:
More influential in this problem was the ability to construct the picture, which would not be easily done without some practice in the construction of numbers using only straight edge and compass. Also, we needed ideas like Pythagorean Theorem, and formula of the Area of a triangle to find the sides of each successive triangle, which are concepts often introduced in Algebra or earlier. Since our recursion had 1 square root in it, we were forced to understand how to construct a side length that is a square root. This could have many uses in a high school geometry class for this project would be an opportunity for students to apply the construction tools they have learned to a real problem that may not have an answer (We certainly found no concrete all ending answer and we are college seniors). The idea that there is still math to be proven, even at the level that high school students can understand, is not an idea that is stressed enough in high schools. Most high school students think that math is just memorizing what has already been discovered and that there really is no more math to discover, a misconception we wish we could wipe away. Math is supposed to be a creative endeavor, in which students can explore the world around them. The idea that students with some simple construction tools could at least begin to explore this topic, is an interesting topic for teachers to explore.
Above, there is evidence for multiple depths of the students understanding of how to relate the project to the high school curriculum. Moreover, there is the beginning of a sample of how the students might go through and transform content knowledge to pedagogical content knowledge as they teach. Namely to work through solving a deeper problem and reflect back on how they solved it. In particular, notice the thread of the argument. First they mention what they did (before the paragraph), then they analyze the dfficulty, next they list the big ideas needed to complete this difficult problem, and then they start relating it to the geometry curriculum. At this point, however, the relationship they draw is towards a meta-concept in mathematics, namely, how to help students maintain a belief in mathematicas as a creative endeavor and the role of exploration.
Not all of their analysis, however, leads to meta-mathematical concepts. They also see relationships in the use of recurrences, using calculators as problem solving tools (as opposed to calculation machines), dealing with error, to polar coordinates, etc. This breadth of topics and ideas was where the paper excelled.
To see the entire project paper click here.