This lesson is a follow-up to the Universcale Project on distance comparisons. The lesson is aimed at giving students a chance to help each other and collaborate on their work. I believe that students thrive when they get a chance to share their work an explain their ideas.
I start class by handing back their projects. I have given each student a grade and detailed feedback. If they made an error the comments will help them fix their mistake. If they did really well, I give them extension questions to push their thinking. Students use this time to review their work and help each other look back at the mathematics in the project. After about 10 minutes, I ask one partnership from each table to move their location. They bring their project with them.
Once the tables are all mixed up. I ask each partnership to share the objects they chose and the types of comparisons they made. The other partnership should be able to find the answers to the given comparisons. Then, they swap roles.
So the cycle is something like this:
1) Switch tables
2) Read your question to the other partnership
3) Give them the measurements they ask for
4) Give the parentership 3 minutes to solve your problem and show their work.
5) Take 2 minutes to compare answers and help each other with any misconceptions.
6) Have the other partnership read their question and repeat steps 3,4 and 5.
After students solve the problems at the table they visit, I give them sticky notes. The partnerships switch reports and write comments to the author. We often discuss the types of comments that are appropriate for this exercise. It is fine if students want to compliment the author and write stuff like, "this was great!" However, they also need to make comments about the mathematics or reasoning of the author. Readers can help by offering alternate algorithms or more efficient ways to calculate with scientific notation. Readers can also ask questions if there is something they don't understand. I give them about 5 minutes to write their comments and then ask everyone to switch back.
In this next 5-minute chunk, students have an opportunity to respond to the questions or comments they received. If they answered a question, they give their response to me and hand it over to the student who had the question. If they can't answer a question, we solve it as a class.
Sometimes groups are done very quickly with these steps. I make sure to always have a few fun and challenging problems on the board. For example, I might display some really great comparisons from other classes on the board.
At this point we talk as a class about what students noticed in the work they read. I know that time is tight in teaching, but this type of conversation is critical. Students need a chance to reflect on the structure of the work they saw. They will notice how great some of the work is and want to model that great work on the next project. This conversation gives students ideas on how to improve their work. They see what others are doing and start thinking, "hey that's awesome, I should do that." This is what collaboration is all about.
I finish class by sharing some of the challenging situations I encountered while grading these projects. There are always such rich misconceptions and wonderfully challenging problems in these projects. I use this time to share what I notices. I often take photos of projects and share on the projector. It is especially valuable to pull up work and say, "what went wrong here?" These types of questions make students active members in the dialogue around mathematics. I believe that this is what engaging students in Mathematical Practice 3 is all about.