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| Geometry in Practice by Matt Mayberry, Math Teacher Spaulding High School, Barre, VT Matt Mayberry is a second-year math teacher at Spaulding High School in Barre, Vermont. Matt also works as a business consultant, specializing in executive training for Fortune 500 companies. Prior to consulting, he worked as an experimental physicist in fusion energy research. Matt has also shared a Math service-learning project involving students in investigating school "failure" rates, along with an insightful reflection on his experiences  Redesigning the Front EntranceI taught two sections of geometry to sophomores last year, and I was interested in having them do an architectural project to apply their geometry skills and learn about design. Having had some luck getting a project idea from my principal for my algebra class, I asked her if she had any suggestions for an architecture project. She immediately responded with the idea of redesigning the front entrance to the school along with the front lobby. She felt it was unattractive (the building resembles a factory—a metaphor for education?). The space around the entrance and front lobby did not foster the sense of school pride that she envisioned. Once again we had a client and a project! There was clearly a great opportunity for student involvement here, but at the same time, we didn’t want to raise the hopes of students that funding would be available in the near future for a major renovation of the building. We therefore agreed that the project would be presented as a “what if” exercise to give students a chance to explore possibilities that could someday lead to improvements if the budget climate improved. Professional Support Makes a Difference John Pandolfo, Kevin Beard and I taught the same full year geometry course last year. We all agreed to allocate about 10% of our total class time to the project. While none of us was an architect by training, all of us had worked on substantial design projects in industry before becoming teachers, so we had some first-hand experience with the design process. We consulted a local architect as well as Dan Lajeunesse, the teacher of the Pre-Engineering Graphics course at the Barre Tech Center to get suggestions on how to proceed. We then structured the project in two phases: 1) needs assessment, and 2) conceptual design. Students would work in teams of three or four and would ultimately be responsible for producing a set of scale drawings and a 3-dimensional model of their design. The project would culminate with design presentations by each team.  To give students a sense of the architectural design process up front, we invited a local architect to speak to our classes and show them examples of drawings and models. Students also did some research about architecture on the web. The needs assessment phase of the project consisted of a variety of activities. Students took photographs of the lobby and front entrance area with digital cameras (see photos on the next page). They also conducted brief interviews to determine the opinions of students and faculty members about what they liked or didn’t like about the front entranceway and what they thought could be improved about it. During Christmas vacation, students were assigned to go on a “photo tour” of the local Barre area and photograph the inside and outside of some of the historically interesting buildings to gain a sense of the architectural heritage of the area. These activities were designed to get students thinking about both the aesthetics and the practical side of their designs. Taking Knowledge for Granted In one early assignment we asked our students to measure the current dimensions of the lobby and front entranceway using tape measures. I was surprised when John, the geometry teacher, suggested that we might need to teach the students how to do this. (Don’t high school students know how to use a tape measure?!) Sure enough, his instincts were right. When we asked students to measure the dimensions of the tables in our classroom, we discovered that many didn’t know how to read the scale properly (inches or cm?, decimals or fractions?). Here were some basic life skills that had never been covered in school! Later in the conceptual design phase of the project, students were asked to create scale drawings and 3-d models fabricated out of foam board. The activity of designing to scale provided a further opportunity to reinforce some basic math skills. For example, coming up with scale factors required students to convert between fractions into decimals. Producing scale drawings also required students to use proportions to find the correct length of objects on their scale drawings. These relatively simple concepts had been covered more than once in their previous classes, but in the “heat of battle” we found that students weren’t always able to apply what they had already “learned” reliably. These examples point out the difference between book learning and applying knowledge in real-world situations.  Experiencing the “Real” WorldThe design process is fascinating to observe in the classroom. It is a highly complex, chaotic process, quite different from the methodical step-by-step problem solving approach we often teach in math class. We watched groups struggle to combine their diverse ideas into a coherent theme. Some groups had difficulty getting along and being productive, while others seemed to click naturally. Most groups had to tone down their ambitious designs when they realized they were too complicated or couldn’t be completed in the allotted time. Many groups procrastinated and then panicked at the end. All of these were all-too-familiar patterns to those of us with industrial project management experience. Occasionally there was one of those “aha” moments where everything seemed to come together for a group and they would become excited and energized. There were also individual breakthroughs. One of these resulted in a very unique design concept. One day after school—yes, students came in after school to work on their models!—some students were working on their 3-D foam board models. I was showing one student how you could make a curved wall by making parallel slits part way through the foam board and then bending it. Another student saw this and suddenly realized that she could design a round lobby area. With great enthusiasm and focus, and after a couple hours of work of exacting work with her Exact-o knife, she created a very cool model of a round lobby. This model is shown in the photo on the next page. Peer Critique For their final presentations, we asked each team to show their drawings, write a summary of their design, and display their 3-D foam board models. We developed a rubric that the students used to rate each other on their oral and poster presentations. Students from the Mr. Lajeunesse’s Pre-engineering Graphics classes also attended these classes and served as the “crit” panel, providing additional feedback to our students about their designs. Providing feedback to each other was something that students were initially uncomfortable doing. Some students withheld suggestions for improvement because they didn’t want to appear critical of their peers. Others (especially those on our crit panel) launched into criticism without pointing out any positive aspects of the design. One thing that seemed to help both situations was encouraging students to describe all the things they liked about a design before offering any suggestions for improvement. This is yet another example of an essential skill in the workplace—being able to give constructive feedback to a colleague—that a CSL project can bring to the classroom. A Team Counts What surprised me about the presentations was how some students who were under-performing in geometry suddenly excelled when presenting an idea they were enthusiastic about to an audience. They radiated an energy I hadn’t seen before when dealing with congruency conjectures. Other students (some of whom were excelling in geometry and who had done excellent work with their designs) were clearly uncomfortable presenting in front of the class. They stared at the ground, fidgeted, and mumbled their way through their talks. As I watched the students, I couldn’t help but reflect on my own professional experience and the importance of giving a good presentation. No matter how great your work was on a project, what counted in the end was your presentation of your results. This was a lesson that was never taught to me, and I only learned it later in life after several painful experiences. Perhaps the most important lesson for students from this project was about making decisions as a team. Coming to consensus about the details of the redesigned front entranceway was not always easy, but most teams learned how to manage the process fairly well. In a final journal assignment, we asked students what they did well and what they might do differently in their next project. Here is a sampling of their responses:
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