The groups final project was exceptionally good. Students were required to give an electronic version of their projects. The project paper itself is about 20 pages long. For the full work click here.
As with many projects, the students decided to write their paper as a report of what they did. A paper reporting on the investigation is probably what I receive on about 75% of the projects. The other project papers tend to look more like traditional papers. In the "worm project" an indicative paragraph of this style is the following:
Initially when we read this problem, our reaction was that it did not make sense that the worm would ever make it to the end since the band always seemed to be increased by more than the distance the worm crawled. However, we felt that in order to get any understanding of what was really going on, we needed to generate some data to interpret. Thus, we began by making a table of the day, the length of the band at the beginning of the day, the length of the band at the end of the day, the location of the worm at the beginning of each day, the location of the worm after crawling his inch each day, and the location of the worm at the end of each day after he moved with the stretching.
However, even in these papers, the students recognize the need to follow more traditional standards, and in particular, they are expected to state and prove theorems. One such early example is:
Theorem: If the worm does not move with the stretching of the band, the worm will never get beyond of the way to the end.
Proof:
Taking the limit of as d approaches infinity gives us the limit as d goes to infinity of . This limit is equal to infinity over infinity, and hence, by L'Hôpital's Rule, it is equal to the limit as d goes to infinity of 1/10. Thus, the limit of as d approaches infinity is 1/10. So, the worm will only make it one tenth of the way to the end of the band if it does not move when the band stretches.
While the proof of the theorem is not self-contained, it does make sense in the context of the paper. They also provide one of the main answers to the problem (that of whether or not the worm makes it to the end) later in the paper:
Theorem: The worm will reach the end of the band assuming (1) the worm has no length, (2) the elastic is fixed at the end where the worm starts, (3) the band stretches uniformly, (4) the ten inches are added after the worm crawls his one inch for the day, and (5) the rope never goes back to its previous length.
While I have omitted the proof here, I would like to point out the after note the students included after the proof:
Note: At this point it is also important to note that this proof shows us why it is necessary that the worm moves some too when the band stretches. If the worm's location is not adjusted in the same proportion as the band stretches, we do not get the cancellation in the equation for R[d] that was essential for the divergence of R[d].
Thus, the students recognized the importance of using the proof to explain one of their earlier questions, namely: "Why did it matter that the worm moved with the band?"
After summarizing everything they had done in solving the problem, the students then discussed applications of the problem to the high school curriculum. They began this with:
Now, after working through the worm problem, we can reflect on some of the math skills that high school students possess that we have used to solve the problem. The first skill that we used in solving the worm
problem was our ability to make a table of data and derive equations from that data. This skill is something that on a more basic level than in the problem can be done in a high school classroom. For example, students may be presented with a table of data similar to the following:
Day 1 2 3 4 5 6 7 8 9 10 Progress 10% 100% From this table, students may be asked to fill in the missing data and write an equation such as % = 10d. A similar problem could also consist of making their own table and writing an equation for the data in the table when simply presented with a word problem.
While this particular insight is not particularly deep, it gives some of the flavor of the work. Other links to the high school curriculum that the students found were finding equations bounding another equation, estimating the area under the curve y=1/x, and using spreadsheets.
Since the solution of the word problem relies heavily on recognizing the harmonic series, the student group decided to look for other instances of the harmonic series showing up in different ways. In particular, they discussed the role of harmonics in music in some detail, before concluding with:
Series such as the harmonic series occur in many natural settings. The key to solving the worm problem was finding whether the series for R[d] converged or diverged. At first glance, one may not expect the worm to make it to the end just as when first looking at the harmonic series it is not obvious that it will diverge. Although the use of series is central to solving the problem, there are many other math skills along the way that we could present in a high school classroom. Using these skills and technology, high school students could gain a general understanding of the problem.
Working on a challenging problem with vague direction gave us a chance to see what mathematical research is like. There were many directions we could have gone in with this problem, and often we had to decide what it was we were trying to solve or what we were interested in knowing. We also had to learn that we can not expect answers overnight. Sometimes the path one takes in solving a problem leads to a dead end, but without testing the options the answer will never be found.
These last two paragraphs summarize two of the main things that I want the students to get out of this class, that one can bring higher mathematics to the high school classroom, and a flavor of what doing mathematics is like.