Well, it’s about over. Over the next two days, my students will submit their project finals for grading. I have been intimately involved in developing each project as a consultant, so I already know the work they have done very well. Some, inevitably, have slacked off and will pay for that — but most have engaged pretty deeply and the products in most cases are better than I expected, which is very exciting and inspiring.
When I started this process, I had no idea what to expect. I feared that students couldn’t tie the concrete system they built to the abstract model they had to create. After I wrote the project specifications, I feared they couldn’t learn to do what I required of them — to be honest, I was not sure how to do all of it myself. Most importantly, I feared that we would emerge from the process with a rather convincing feeling that the project had not been a good learning experience. I did not want to go back to the drawing board on this one, but I knew that was a possible outcome.
I am very pleased with the outcome of the process. I expect some really nice work and I look forward to showing it off to my students early in the year next year to get them fired up to take on this very same project (with some changes) in its second iteration. We have all learned a lot.
It has been providential that I chose at the last moment to have all of my classes do this project. I modified the requirements so that my Pre-Calculus students would have to draw on their new knowledge of trigonometric functions — knowledge my Algebra 2 students don’t yet have — and so that my Calculus students would have to combine their differential calculus skills with linear algebra and polynomial skills they developed two years (or more) before. As a result, I have a well-defined sequence of projects, and it is clear just by viewing the graphs of the models which class each group represents. This gives the students a visual sense of how their mathematical power grows as they move through the advanced math sequence. I could not have devised a better teaser for those students who wonder just what awaits them in the advanced courses.
Another providential development was my requirement that students match their graphs to photographs of the course they created for their ball bearings (for my Algebra 2 and Pre- Calculus groups). This drove several of them to the web in search of graphing utilities, where, independently, they discovered and began mastering Desmos. We also took time to learn how to graph piecewise functions on the TI-8x series calculators we use in class. The students also had to learn how to integrate the Desmos product — a downloadable pdf file containing an exportable graphic — into a presentation application so that they could combine the graph and the photograph to make a new graphic. Many chose PowerPoint, but others took the time to discover how to do this using Prezi, Glogster, or other applications. One group projected their Desmos graphic on my Interwrite board (we turn our homework in on baked clay tablets in cuneiform, too) and traced it on a printout of their photograph. Low-tech? Sure — but the product meets specifications; the solution is ingenious in its own way.
“I love it when a plan comes together.”
The diverse methods and the determination that spread among the groups in most of my classes were a joy to see. The final satisfaction began to grow inside me as my more accomplished students, leaders in their groups, began to express personal pride not only in the product but in the simple fact that they managed to do the project in just a few weeks with no exemplars or specific experience. It is very satisfying for me to see and hear hard-working and capable students sit back after finishing a tough project and say, to anyone in particular, “I feel really good about doing this.”
Riding a Snowball?
I am already very excited about next year: we’ll do this project for the first semester final in all of my classes, and the projects I receive over the next two days will be “Exhibit A” for the argument “You can do this.” At the same time, I’m going to make the Algebra 2 and Pre-Calculus projects more demanding. The Calculus project has yet to be done in its entirety, so I will not adapt it yet. The Pre-Calculus survivors who take it on next year will have plenty of experience and confidence from this year to push them along.
“You gotta try this”
Teachers, I think we should all give this a shot. Try a project. Write up a crazy project — one that forces students to look for a connection between a concrete system and the abstractions that populate their advanced math courses — and play along. Don’t worry if parts of it seem impossible: encourage them to solve the problem, and take part in the process as a consultant. The one rule I had was this: I did not spread the word on any essential discovery until one group had made that discovery independently. Desmos is very, very popular in my classes now, but I told them not a word about it until one group discovered and mastered it on their own. The technique for overlaying the Desmos product on a photograph using PowerPoint is something all of my students understand now, but I did not guide any group to it until one group discovered it. The Prezi technique is still beyond me; I’ll have to learn that this summer.
Did they learn math? Oh, yeah.
That’s all very well, you might say, but what about the math? Well, that wasn’t as easy to pull off. We began with the simple process of turning a relative maximum or minimum and a point between that extremum and the next into a partial parabola; this mimicked the smooth curves that defined the course of the ball bearing through our vinyl tubing. This was the technique my Algebra 2 students chose to use: they had to solve ten to seventeen three-variable systems of three equations to generate their piecewise functions. They did not discover until very late that they could have used matrices and their graphing calculators to make the algebra much quicker and less prone to minor errors. That’s right: they solved each system using substitution and elimination. Had they discovered this earlier, several groups would have done much better, but, frankly, the Algebra 2 classes suffered not only for lack of mathematical knowledge but for maturity, and this will probably prove to be a benefit to them as they face projects in the future. They’ll be wiser for it.
For the Pre-Calculus students, the mathematical turning point in the project came when they realized they could abandon the parabola approach for trigonometric functions. These functions make their graphs look much “smoother” than the Algebra 2 graphs, though there are still obvious points where even those who do not understand calculus can see that the rates of change at intersection points are unequal.
My Calculus students were undermanned — three pairs of students — and lacked time because of the traditional expectations of seniors in their last semester, including early finals, graduation, and other additional duties. I exempted them from the graph-photograph overlay requirement, but the requirement that their graphs be differentiable (and therefore perfectly smooth) proved a stiff challenge. Our solution this year was to alternate quadratic and cubic graphs, which created systems of four equations in four unknowns that had solutions. One quirk in this process is that the numbers involved at the conclusion of the course, which we measured over twenty feet (610 cm) in centimeters, were so large that the TI-8x series calculators we use calculated that the resulting matrices were singular (the determinants were too close to zero). We had to find high-powered on-line linear algebra calculators to do the final calculations … but it worked out.
What did I learn? I learned we can trust this to work.
What, then, did I learn about project-based learning from this experience? I learned that it’s nice to hit on some intuitive blessings in your first run. I don’t believe in luck, so I am properly grateful for grace: the providential decisions and inclusions in the project plan brought out exactly the kinds of experiences that project-based learning should produce. Students had to do independent research without knowing that a solution even exists, and they had to improvise using the tools they discovered to rig a solution. In the process, they had to apply their mathematical skills to connect the abstract world of mathematical models to a concrete system — in this case, one of their own devising. I know from this experience that students can indeed learn new knowledge this way, and the activity, when led by capable students and guided by even a very inexperienced teacher of this kind of learning generates much academic leverage.
The students I taught this semester are leaving my class with confidence and knowledge that applies far beyond my classroom. They know they can seek, find, and master new resources. They understand how to learn to create unfamiliar forms of presentation from these new resources. Finally, they know they can build a bridge between the abstract and the concrete using these tools and their communicative talents. I’m not just looking forward to seeing what these students will create in their other classes — I’m looking forward to seeing them do all this for me again soon.
In closing …
A final reflection on the year fits here: I don’t do this job for the kids. I love them. I can’t work without them. Nevertheless, I do this for me. I do this because I want to be the best I can be at it, and I can’t get better without them. This attitude suits the activity well, because high school students particularly are looking for what is exceptional. Their emotional makeup and the society surrounding them drive them to seek out and emulate extremes — this is what drives so many adults to frustration when dealing with them. If teachers are serious about arresting that urge to shock and harnessing that reckless energy for positive purposes, then we’ve got to stand out as exemplars of ability and of teaching. We have to be authentically cool — our ideas have to work out and our ability has to be clear — to gain the authority and trust we need to help our students. Experienced teachers know the drill here: you can’t get away with faking it in front of a group of adolescents, because they’ll find you out and, once they do, you’ll never win them over.
If this project had flopped, I’d have faced a long road back to credibility with my students. That was the source of my fear. Why did I fear? Because I just don’t know how blessed I am and I don’t appreciate the effect I have on my students. I don’t know about the other teachers out there, but this year has opened my eyes to the impact I have. The little pitchers have big ears and big eyes. We have to give them an authentically good show. Did they know I was taking a risk? I think they did. Did they want it to work? For themselves, they wanted to get good grades and, as they realized that I intended to show others their work, they wanted to put their names on good products. Did they want it for me? Who cares? I teach students who want “it” — now I have to make the most of that both for me and for them. I am already very excited about pushing them more and discovering what other wonderful things we can do together, because this has gone better than I ever expected.
I have a lot to learn about this before I’m really in control of it. This project, for example, allowed students to fall back on methods they had learned in more traditional instruction. When I’m truly good at this, I’ll be able to use projects to drive students to learn new techniques and strategies. I’ll be striving for that kind of subtlety as I learn more, because once that’s in place, I’ll have a model for truly authentic life-long learning. In general, however, I am now sold: project-based learning is good for my students and is definitely going to be part of my quest to be the best teacher I can be for the foreseeable future.
Ora et labora ut in omnibus glorificetur Deus.