Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 10, Part B:
Developing Geometric Reasoning

In This Part: Introducing van Hiele Levels | Analyzing with van Hiele Levels

 In this course, we have primarily worked across levels 2-4. You may feel that the activities we've done are not appropriate for the level of your students, and you're probably right. The goal for this session is for you to think about problems and activities that are at your students' level, and how to help them prepare for the next level of thinking. In grades 6-8, students should be working comfortably at level 1. Ideally, they will have begun working on drawing logical conclusions and "if-then" thinking characteristic of level 2, but not all students may be comfortable with that kind of task. During middle school, students should be prepared for work at the van Hiele level 3. This means reasoning through more complicated mathematical arguments, leading into some early proofs.

 Video Segment In this clip from Ms. Weber's eighth-grade class, the teacher leads the students as a whole class through a proof of the Pythagorean theorem. Students have already reviewed the statement of the theorem, and they have worked through some numerical examples like the one the teacher works through in general. Note 5 If you are using a VCR, you can find this segment on the session video approximately 16 minutes and 0 seconds after the Annenberg Media logo.

Problem B1

Where in the video do you see evidence of the following?

 • (Level 1 thinking) Students thinking about classes of shapes rather than the individual shapes. Do students seem concerned with orientation or size of the figures? • (Level 2 thinking) "If-then" reasoning and making geometric arguments • (Level 3 thinking) Students working more abstractly, drawing conclusions based on logic more than on intuition

 Problem B2 Ms. Weber's lesson was based on a lesson from Session 6 of this course. Discuss the ways in which Ms. Weber's lesson was similar to and different from the one in this course. What adaptations did she make and why?

 Problem B3 In Session 9, you worked on the problem of building the five Platonic solids and then arguing from the construction that only five such solids were possible. Recall your own experience in this activity as an adult mathematics learner. During the activity, when did you have to use level 2 thinking? (How did you know when to stop building with triangles and move on to other figures? How did you convince yourself that no other Platonic solids were possible?) What about level 3 thinking?

Problem B4

 a. What do you think were the key pieces of geometry content in this activity? What knowledge did you learn, solidify, or connect with better? b. What do you think were the key thinking and reasoning skills in this activity? How did the reasoning and geometric content tie together?

 Problem B5 Now think about students in grades 6-8 and how this Platonic solids activity might work with them. What must students know and be comfortable with to get the most out of this activity? What are potential stumbling blocks for them?