Towards the end of the year, those same students design and build a planetary model that would help to explain all of the celestial phenomena that we experience. The teachers don't tell them what to build. They go around and conference with each group, rotating planets and flashlights and explaining, "According to your model, we would see a solar eclipse every 6 months. Is that true?" The students then are engrossed in deep conversations about their models and attempt to fix them. Eventually, when all the groups have finished building their models correctly, they use their models to answer questions such as, "It is 3pm and the moon is in the eastern sky. What time of the year is it?"
Seeing this level of amazing teaching and learning has made me question what we can do in math teaching to bring our kids to be curious, to model, and to experiment the same way that they do in these amazing science classes. Here are some elements that I think were critical to the success of these projects, and in many ways should be transferable to a math classroom:
Core element 1: A meaningful physical experience
Core element 2: Clear objectives and a way to self-assess against them
Core element 3: Time for an iterative process, conference, and reflection
Core element 4: Shared vision within the department
A meaningful physical experience: One complaint that I have received from past students is that we do too many worksheets in class. As much as I try to make learning exploratory and scaffolded, I do find it challenging to move away from worksheets. One thing that I really liked about our Precalculus experience this year was that about once per unit (as in, once per major topic), we had some activity that involved modeling and analyzing a certain motion through Logger Pro. This often came randomly, but should be far more thoughtful. Each unit should start with an investigation that links the math to a physical experience, to establish the goals of analysis, and this investigation should generate interesting questions (otherwise it's probably not a good investigation to use). Towards the end of the unit, there should be some form of follow-up where the students can create their own example and create a physical product to represent what they have learned. (For example, an end-of-unit quadratic product could be for students to create a quadratic sequence that grows visually, and to analyze its growth using a variety of algebra methods.)
Clear objectives and a way to self-assess against them: The book I previously read by Jo Boaler had underscored the importance of this. The better students understand what they should be achieving, the better they can achieve it. In the case of the science projects, the goals and objectives are clear and simply stated. The students are left questioning how to achieve that, and the methods are left open. In a math classroom, we are often eager to show kids the way, but the objectives are actually obscured. At the start of a unit, the objectives should be clearly stated. The students could, for example, keep learning journals with which they generate their own examples and explain connections in their own words. Whenever they feel that they have achieved the learning objectives, they can then submit those learning journals for review by the teacher. This is a mixture of self-assessment and teacher-supported (non-quiz) assessment. What it would provide is a clear expectation that the students are self-assessing their learning on an on-going basis.
Time for an iterative process, conference, and reflection: One thing that strikes me about these science classes is the time that those teachers take in letting their students struggle. They believe so strongly in what they do, that they will spend four or so weeks on one open-ended project. It's not all roses; some of my students claimed after the fact that they "learned nothing" from Grade 9 science, but what those students don't realize (even after the fact) is that that class was trying to teach them to think, and to think deeply. The teachers often spent the whole period just checking in on half of the class and involving in deep conferences, rather than bouncing from group to group as I have the tendency to do. Quite often, the teachers would send some of the students out of the room to work unsupervised, so that they could focus on the remaining groups. I think that this is something I will experiment with -- having longer conferences with fewer students, while providing the other students with clear expectations to struggle for a bit more on their own before asking for help.
Shared vision within the department: One thing that is extremely powerful is that the science department at our school has a similar vision. Not only do those two physical science teachers believe in teaching the way that they teach (and taking long weeks to focus on deep, meaningful assignments), but their entire department believes the same. The bio teacher teaches biology as a series of mysteries to be unfolded based on your existing knowledge. The chem teachers bring out an unknown substance, and ask the students to do whatever they can to decipher its identity. The physics teacher does exciting projects like rocket design (complete with parachute deployment), building circuits for a house, and processing sound signals. That is so important in the continuum of development of a child. If we can achieve the same coherence in our math department, it won't matter which content strands we've only skimmed over, because what we will get in the end is a confident, thinking student. I feel that holding this line is especially important in the high school age range, where the pressure of content is strong, with or without standardized exams. Believing that we can do right by the kids by re-focusing on the cognitive aspects of math is more critical in high school than ever.
Sorry, maybe you were hoping for a new resource today. I hope you enjoyed this rant nonetheless.
PS. Hello from NOLA, where I will be for about a month. What an amazing city it has been already!