reflections after visiting two other math classrooms on rosebud

Today I was out of the classroom for "district growth" purposes. I went to a nearby country school and observed a 6-8 grade math classroom from 8:30 until 11:15, and then observed an 8th grade math classroom from 12:20 until 1:30. It was interesting to see the different dynamics in the rooms. I feel like my classroom is more like the pure 8th grade classroom structure wise, but I learned equally as much about teaching from the multi-grade room.

Multi-grade had lots of little workshop stations and a high energy level. I would classify the teacher as more "excited" about math rather than "serious" about it, which I think is good. I have drifted more towards the serious drill sergeant side, and that is not conducive to students being excited learners. The multi-grade room had a large focus on discovering concepts through activities such as measuring the circumference of different objects around the gym and using that to help them find diameter, cutting apart a piece of paper to see exponential growth, and relating the area of a triangle to the area of a rectangle in order to observe where the 1/2 comes from in the formula for the area of a triangle. I feel that this "discovery learning" is very valuable and leads to a more intuitive understanding of the material, thus increasing retention of knowledge. However, I wonder about the time consuming nature of learning by discovering...It seems to me that students should get lots of practice working with different kinds of problems in order that they might be able to solve problems quickly and efficiently. Not only does discovery learning take a long time to convey the intended concept, but also students don't get very fast at preforming certain tasks at a high level of accuracy. They need to get enough practice so that they don't have to re-discover a concept over and over and over again. Discovery learning is critical thinking at its best; I do LOVE that. Also, students doing discovery learning seemed to be highly invested in their work and were curious learners. They helped one another learn and asked each other math questions.

The pure 8th grade class was one day ahead of the multi-grade. They did the lesson using cutting of paper to discover exponential growth yesterday, which appeared to have been an introduction to a new unit. In this class, the teacher showed a picture and a video of a roller coaster in order to get at the idea of slope.  They then used a table they had made the previous day to plot some points on a graph, which turned out to be an exponential curve. The class related this curve back to the shape of the roller coaster, and then made posters of what they had already graphed in their notebooks (?). After working on the posters(but not completing them-they will get a few minutes at the beginning of class tomorrow to finish), they came back together as a class and the students came up with an equation that represented the tables that they used to make their graphs. The students came up with an idea for the equation that was outside of what the teacher had planned, but he was able to adopt his plan and use the totally valid equation students came up with.

Things that I gleaned from today:

1. It is important for students to have feedback on their work. Having a station in the room where students can go to check their answer is a way for students to practice their work correctly, and not just practice doing a process wrong over and over again because they were not told it was wrong. 
2. Having students observe their own class and give feedback on what is going well and what needs to be improved is a way that they can take ownership for whether classroom behavior contributing to or distracting from student learning. 
3. Writing in math notebooks, expressing how they are feeling about math or school in general, is a constructive and mutually beneficial way for me to get to know my students. 
4. Students need variety during their day. Their minds need to think about different applications of math over our 80 minute blocks in order to stay interested and hard-working for such a long period of time.
5. Students should feel proud of their notebooks and feel that they are a tool that they know how to use. Notebooks can be used as a resource during tests. 
6. Lowering your voice and whispering works better for getting classroom attention than does raising your voice. 
7. When you are asking for students to raise their hands to volunteer information, it seems to encourage them if you raise your own hand.