In the last post I wrote about one of my students getting pretty upset about me not telling him if he was right or not. When he told me, "I've never had a math teacher be like this," I wondered if there was an emphasis on math. Are his other teachers like this? Is it just math that he expects to be delivered in neat, compartmentalized chunks of notes, problem sets, and correct answers? This was a student who aces every test and project, so his reluctance wasn't an issue of not comprehending the statements or assignment. The discomfort he felt with not having me tell the correct answer is exactly the reason I wanted to do the project this way. I could have done the same project but hinted, and suggested, and told them if they were right. If done that way, I'd guess that they students would earn better marks on a multiple choice test about points, lines, and planes, but then they would still think that there was one correct answer. They wouldn't have that same feeling as when Ms. A and I debated if there was such a thing as two coinciding planes. If there was, would that count as an intersection?
In forcing them to actually invest, to think, to struggle, the math was getting to them. I had more kids be frustrated, throw fits, and tell me they were quitting. But they didn't. I told them to pick an answer, and move forward with it. If they got more stuck, they needed to back up and try a different answer. Or talk it out with a classmate.
It was also really eye-opening to me to see how different the process focused approach is from being product oriented. In the past I have claimed to be process oriented. My school is art focused so I hear a lot of discussion about being process oriented. Typically it goes hand in hand with a discussion about how students work at different paces and having rigid deadlines or only accepting one format of answer is stymying creative output. As a result I decided to allow students to retake tests, to emphasize that the process of learning is important, not learning by a deadline. In actuality, that is still product oriented. There is a right answer (a product) that I want them to reach, and I don't care how they get there or how long it takes, just that they get there. I also decided to allow for multiple formates on most of my projects. During reviews students can turn in videos or word problems or a review game or a set of textbook problems. Again, I felt like I was being process oriented, but there was still an outcome fixed in my mind. I wanted them to review. I didn't care how they did that, just that they did.
The execution of the project was the first time I feel like I have actually been process focused. I fought every urge to tell them "No! That's not right!" I fought (almost) every urge to hint, "Hmmm. Maybe look at number 2 again," or "Well, do you really think a plane never ends?
Ultimately, with the recap next week, there is a product. I want the students to know that if two planes intersect, they always intersect in a line. In a math class, I can't deny that there is content, some nugget of information I am hoping that they will learn. In showing a "best" answer, I hope that I am not diminishing my initial goal, for the students to learn that it's ok to struggle in math, but that they need to push forward and argue for why they think their answer is correct.
The one time I broke down was when a student had a fantastic photo of why a plane never had an edge (I'll upload it once he turns his project in). He thought the statement "Planes have an edge" was always true. I asked why and he said, "well it's impossible for something to not end." I asked if lines ended or went on forever and ever. He said, "well lines go on forever and ever," so I countered that it was possible for things to not end. He thought for a minute and said, "So if two lines cross in an x and they both go on forever and ever and you fill in the space in between the lines with like paper or more lines or something, then is that a plane? Wait! That's like the graphs we make. OH! that's called the coordinate plane isn't it? Is that why it's called the coordinate plane? Is that a plane?" I initially felt bad that I was nudging him, but after that grand epiphany I was grinning ear to ear. He was visualizing and making connections and it was beautiful. Not bad for last period on a Friday.