Sunday, December 23, 2007

Promises, promises

At last!

Many apologies for the delay in getting around to posting on this topic, after all of the promises I’ve made. It’s taken a while to gather all the necessary permissions from my students (who shall remain anonymous in all of the following comments), the semester’s end brought the usual concomitant tumult, meeting after meeting with administration about this matter and that added extra delay, and the last couple of weeks have been devoted both to finishing up a number of loose ends and to enjoying a relatively unstructured, somewhat work-free languor.

I did want to share my thoughts (and my students’) on the Newton v. Leibniz project, since at least one of the two reenactments proved to be a very memorable learning experience for a number of the class’s students. Though there was certainly room for improvement, I feel that my management of the project was quite effective, and many of the students put a great deal of effort into making the activity a successful one.

Below I’ll highlight some of the successes of the project: what did the students take from it? What did they learn about math, about math history, about themselves and the way in which math plays a role in their lives? What did they take from the classroom reenactment itself? On the other hand, I’ll point out some of the project’s flaws, and ways various students have suggested the project be modified the next time it’s assigned.

First, I might say a few words about the character of the reenactments.

The first section’s trial reenactment was satisfactory, but comparatively lifeless, characterized by a lack of preparedness on the part of some of the trial’s principal parties (the teams comprising Newton and Leibniz themselves, along with their counsel, and the teams comprising the scientists’ friends and colleagues). Two of the, shall we agree to say, less modest students in the section had taken on the roles of Newton and Leibniz, and it was clear that they did not always take these roles seriously. (This levity was a source of distraction for at least two of these students' colleagues.)

The second section’s reenactment was positively splendid: it was lively, heated, authentic, and
characterized not only by preparedness and seriousness, but also sincere concern for the discovery of "truth." Nearly every member of each of the principal teams prepared meticulously, and this level of preparation was clearly on exhibit in the resulting trial, in which students engaged in excited debate. Those cast as friends of one party or another fit seamlessly into their parts, attorneys on either side deliberated feverishly in reaction to the other side’s statements and questions, objections tore through the air like shrapnel. The result was an electrifying activity, and a pleasurable one: several students expressed a wish that the trial could be extended to the following class period. Said one: "the execution of the trial was a blast and I would have been willing to stay an hour after class just to continue it."

What was it that held the first section back?

"I have to admit, I was disappointed by the trial," said one student. "I understand that it’s important to have fun with projects, but there’s a different between having fun and not taking it seriously. I feel like, in general, the groups of colleagues and the historical experts came to class prepared to present information, they collaborated, and took the project seriously. I was dismayed to find that the primary groups…were relatively unprepared, and spent a great deal of class time joking with one another."

One of this students’ colleagues in the class agreed: "The arguments and the overall project could have been given a heightened sense of importance…Coming from my personal thoughts and experiences working on this project, I thought it had the potential to develop into a really intense trial, maybe not to the caliber of the To Kill a Mockingbird trial scene, but pretty close. I thought that everyone loved to argue which would ignite a heated trial."

I later asked this student for advice on making the experience more worthwhile. He gave examples of a mock trial in which he’d taken part in high school, indicating techniques his high school teacher had used to make the most of her project. He explained how in studying Beowulf his English class had put Grendel on trial, and the teacher herself played an active part in the trial by assuming the role of Grendel. "She knew her stuff about it, being an English teach," the student indicated in an e-mail to me later, "this makes it hard on the team who is unprepared." I agreed that in this way the teacher can "call out" those who are underprepared. The disadvantage of this set-up is that the pivotal role of Grendel (comparable to that of either Leibniz or Newton in our class’s rendition) is then taken away from a student, who might gain much from being cast in that part. The relative merits of either side could be debated. I very much appreciate this idea, and will definitely consider it when running the activity again.

Others were discomfited by the trial experience itself, whether or not they felt it was a success: "overall, I like the idea of a mock trial, but I found this experience to be really stressful." After listing a litany of inconveniences and discomforts experienced during the lead-up to the trial, the student admits that "despite all of this, I kind of enjoyed it." One of his fellow students agrees, clearly a lot more comfortable with the activity itself: "the actual trial was the part I found most enjoyable, because I got to get outside of the person that I normally am and really attack the other team. Being an attorney ended up being more fun than I thought it would be."

What did they take from the experience, this first section?

A number of students indicated that they learned more about Leibniz than they did about Newton, given the relative obscurity of the former mathematician. Several students mentioned learning a good deal of mathematics and the history to go along with it. A few insights were more purely epistemological, dealing with the nature of mathematical knowledge and its acquisition than with the math itself: where does math come from? How does it arise? Who can lay claim to its discovery?

One student became more aware of the "social" aspect of mathematical discovery: "from this project I have learned that in math everything is a collaborative effort with [one’s] colleagues or predecessors, nothing is every truly invented on [one’s] own." Another student agreed with this assessment and reflected on the "human" provenance of mathematics: "if anything else, from this project I have taken away the important idea that a mysterious Calculus book wasn’t laid in someone’s lap. It took many years to discover and perfect." And no angels were they, the folks who cobbled calculus together. In the words of yet another student, "these [Newton’s] colleagues were not so innocent in their daily lives, and their moral drawbacks were made public, so as their future representatives we had to make Newton’s colleagues look trustworthy and knowledgeable. In all honesty I never realized such dramatic lives were led by the founders of mathematics and the sciences."

Others took from the trial more pragmatic insights, ones that might be incorporated usefully into other settings: "it was interesting to learn and expand my knowledge of Newton and Leibniz but what stuck out even more was the small details," says one student. "The seeming less significant aspect of the project was the most interesting to me. The most beneficial was the fact that this assignment involved, writing, mathematics, the overall use of language, communication, cooperation with group members. This was interesting because it was universal and very helpful in other subject areas…Thanks for incorporating more than just math into this project."

How did the second section experience the same assignment?

Overall, excitedly, and positively. I don’t believe I received a single negative comment from the second section. Not a one. The nearest thing to a negative comment came from a rather reticent student who knew herself well enough to anticipate her strengths and weaknesses: "this is not the kind of project that I like to express myself with since I’m more of a creative writing person. The poem that’s coming up will be better for me to express myself." (Indeed, she would do that; hers is one of the poems appearing in the second set of poems, in the previous post.)

On the other hand, most were thrilled by the trial experience: "with the three-second attention span of a goldfish, it really amazed me how focused I became on this project," said one. She continues: "I enjoyed working on this with my teammates, I had fun digging up facts during research, and, more than anything else, I loved picking apart the other [team’s] arguments. So many things about this project interested me, but one aspect has really stayed with me. I was inspired and charmed by the level of dedication that came out of the students, including myself, during this project…what amazed me was the dedication of the students during this project. Weeks of planning and research culminated into a wondrous thing."

Her colleagues agreed: "I think the whole approach was creative and stimulating," said one. Several indicated disbelief at first, but later conversion. Many made revelations, about math and its history, and about themselves, much like those uncovered by the first section’s students. Here’s a representative testimonial:

"When the Newton v. Leibniz project was assigned I was, in all honesty, unmotived and in no way looking forward to the trial date. The assignment somehow seemed to be kind of irrelevant and distracting to what we were actually learning in class. After the actual trial on Monday though, I cannot stress how wrong I had been…Math had seemed like one of those things that has just always been around. I never thought twice about the history behind Math because I had never had a need to do so. By doing the [trial] however, one seemingly obvious thing was made evident: someone somewhere had to actually derive these concepts and be the first to think of them. Before, I had just thought of math as math. I never took into account that there is more to it than that. Math isn’t just math; math is a conglomeration of ideas and concepts over time. A major part of math is in its history and to truly understand math, one cannot ignore its past…Math is a collection of evolved ideas over time, whose history is almost as important as its functions."

And a second:

"When I enrolled in calculus in order to fulfill my biology requirement, I expected that the semester would be full of long, grueling problems which would bring back painful memories of math classes long past. And so it has (although with a lot less stress than I had anticipated). But if you had told me at the beginning of the semester that I would be doing a project later in the semester that I would actually enjoy, I’d figure you were crazy. But I did enjoy Newton vs. Leibniz, probably for two reasons: one, there were no math computations involved; and two, I felt I could relate it to what I am interested [in] more than anything else we’ve done this semester."

And a third:

"Upon the receiving the details of the assignment, I would be lying if I say that I didn’t let out a few groans. First of all, it was a group project. Group projects have seemed to be the epitome of evil to me for the past several years of my life. I kept flashing back to previous group projects in high school where it seemed that I pulled most of the weight of the project, whether or not it was intentional…Second of all, the assignment seemed a bit silly. I had only briefly heard about Leibniz and I felt as if there was no point to argue the Leibniz side of the dispute. Obviously, I was a bit wrong…From this project, I have changed. I have changed my outlook on the idea of group projects. Everyone in my group was amazing, and they all did their part. I realized that I needed to stop being so paranoid and pessimistic [about] others in my groups. I actually like group projects a little bit, now, and it is nice to work and talk with others who I would have otherwise possibly never talked to."

Though this last student had a change of heart about working on a project as one member of a team, others had different experiences: "this was one of the most interesting projects I have ever participated in and not in a bad way. At the beginning of the project I was very [skeptical] and did not know what to [expect]. Even though I had a rough start, in the end I thoroughly enjoyed myself. I learned a lot about myself and what role I play in groups. This may be cliche, however, I truly did learn about myself and realized that I do not like working in a group at all, unless I can pick the members in my group and that does not always turn out great. I realize this is a big problem because for the most of my remaining life I will be expected to work in groups."

As in the first section, some students had much to say about how the project had lent them insight into the process through which mathematical knowledge is constructed: "it’s [mathematical discovery is] a constant building of ideas on top of each other, almost like a pyramid with each layer built being built upon a previous layer. Newton of Leibniz could not have done their work without the work of mathematicians such as Isaac Barrow, Barrow couldn’t have done his work without the influence of scientists before him." As this student indicates, this "influence" causes difficulties when it comes time to ascribe credit: "The idea of creating new concepts also made my group and myself question what it means to actually invent something…Whoever created it first gets the credit, right? Well, then I started thinking, the more complex something is the more parts there are, the more parts there are the more other people are involved in the process."

Assigning credit is made more difficult by the fact that real people were involved, a fact brought to the fore by the students’ research during the project. "We often think of scientists as being ‘morally superior’ to politicians in that they are interested in obtaining the facts," says one student, who continues, "while I am sure this is true to an extent, scientists want fame and success just as much as politicians want power…In researching Newton vs. Leibniz, I was reminded that scientists are only human." This student was able to relate his experience in his section’s trial to his leisure reading: "It’s reinvigorated my interest in reading the last two books of the Baroque Cycle [a series of historical fiction novels], the last of which specifically deals with the calculus controversy. I can’t wait to see how Neal Stephenson portrays Newton: embittered, jealous, or sincere in that an injustice was done? In either case, I know it will be an entertaining read."

Others were able to make similar connections to their interests outside of class. Our friend with the three-second "goldfish" attention span hopes to recycle the courtroom concept for her own classes once she begins her career as a teacher: "as impressed as I was with the level of participation during the trial, I began to formulate a way I could use the idea of a trial in my own classroom. After reading a little of Change of Basis, I realized that this was not a strict model and could be applied across different disciplines. It was reassuring to know that while the process and student involvement stayed with me, it also stayed with my professor before me."

Perhaps the most ecstatic comments came from a student who had an epiphany concerning his own approach to mastery of mathematics: "for me it is much easier to understand mathematics when I say what I am writing down, in verbal form in my head," he explains. "I have always done this and before I had no idea why. Now that I have reasoning and have explored the subject, I have a better understanding of how to teach myself and be a good student of Calculus and the broader field of Mathematics. As a person that does not make breakthroughs like that on a regular basis, I was floored to learn more of how I learn. All due to this project!…If I were you I would keep assigning this project to Calc I classes."

Never fear, I believe I shall.

While the first section’s experience of the reenactment of Newton v. Leibniz offered mixed results, I was overwhelmed by the energy and alacrity with which the second section embraced the project, and gratified by their positive feedback. In all honesty the project was a bitch of a row to hoe, but it’s clear to me that the harvest was a rich enough one to warrant sowing these seeds again. Made wiser by the suggestions my students have offered me, and simply by the experience itself, I’m sure the next installment will be more successful than the last. I think I can safely say I’ve put together a significant learning experience with this trial.

And with this post, I’ve fulfilled a promise I’ve now made for several weeks.

I hope to post again soon with comments on my upcoming courses (choose-your-own-adventure testing in Graph Theory), on my teaching-related Winter Break leisure reading (Jonathan Kozol’s vitriolic The night is dark and I am far from home), and so forth.

Now, however, the night is indeed dark, and I should soon away to bed.

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