### Teaching geometry, round 2.

Sep. 19th, 2012 03:48 pm**starmadeshadow**

Back in the saddle, for almost three weeks now. Something of a different class, when compared with the group I taught over the summer. Since about half of the 10 students during the summer were from different campuses, there were big differences in the basic knowledge of the class, so we spent a fair amount of time making sure we were all on the same page. This semester I have 18 students who have all been through the prerequisite course on this campus, so there is much more I can take as given. They're already pretty good at teacher's solutions, and are a lot less skittish about getting up in front of the board than the summer crowd.

I've been wanting to incorporate my reading about complex instruction, but so far have not got there. However, when working in groups I've asked the students to be aware of the group dynamics - is there someone who is monopolising? someone who's not saying anything? I also asked them to try to see that neither of these situations occurs to much. I've noticed some good cases of a student drawing another student out. Maybe next week I'll formalise the group work a little more - if I can find the time and energy!

Monday was the first quiz. I decided to make a bit more of a formal exercise of the grading, so once the students were in groups I explained that they need to put together a model solution for their question, with a grading rubric, then grade the questions. I also asked them to categorize each error as arithmetic (or careless) or conceptual, and if conceptual, what concept was misunderstood, and how. Each group will hand in a report summarizing their findings, as well as noting any overall conceptual error they noticed across the class, and any particularly good solutions (very clear, or solving the problem in a novel way). They seemed to take to it all with gusto, and the groups were busily discussing their grading when I came into the classroom today.

Last week we spent one of the classes in the computer lab, working through a basic geogebra tutorial. There were quite a few comments of 'This is fun!', which was great. The quiz questions which they handed in on Friday were generally of a high standard, with a few really outstanding, interesting problems. I overheard a few discussions of possible quiz questions today - it seems like there's a bit of competition going for who can come up with the best one!

Today I talked a little about the van Hiele model of geometric understanding - it's the pedagogical model on which the Common Core standards are based, and provides a useful framework for teaching geometry. The basic idea is the geometric concepts and abilities are developed sequentially, and it's necessary to master each level before moving to the next. The levels are, roughly,

Level 0: Visual. Recognizing basic shapes by appearance, without attention to parts, attributes or properties (so might identify any figure with 3 vertices as a triangle, even if the sides aren't straight, but might not identify a triangle as such if it is in a funny orientation, or is extremely long and thin).

Level 1: Analysis. Recognizing and naming the properties of geometric figures, but not understanding relationships between different types of shapes (so will not identify a square as a type of rectangle.)

Level 2: Abstract: Able to logically order properties, create meaningful definitions and make informal arguments (so can deduce that a square is a parallelogram, and also a rectangle).

Level 3: Students can construct proofs, understand the roles of axioms and derinitions, and know the meaning of necessary and sufficient conditions. High school geometry.

Level 4: Rigor. Students understand the formal aspects of deduction, can compare mathematical systems, understand non-Euclidean systems, and use indirect proof and proof by contradiction.

As I mentioned each level, we looked at the corresponding questions on the van Hiele geometry test, which give 5 questions for assessing understanding at each level. I think maybe I didn't give the students enough time to read the questions, although we were able to see some practical examples of what is expected at each level.

Having said all that, I used scroobius's awesome elfling as an example of someone who seems to have been born knowing Level 1 concepts.

I've been wanting to incorporate my reading about complex instruction, but so far have not got there. However, when working in groups I've asked the students to be aware of the group dynamics - is there someone who is monopolising? someone who's not saying anything? I also asked them to try to see that neither of these situations occurs to much. I've noticed some good cases of a student drawing another student out. Maybe next week I'll formalise the group work a little more - if I can find the time and energy!

Monday was the first quiz. I decided to make a bit more of a formal exercise of the grading, so once the students were in groups I explained that they need to put together a model solution for their question, with a grading rubric, then grade the questions. I also asked them to categorize each error as arithmetic (or careless) or conceptual, and if conceptual, what concept was misunderstood, and how. Each group will hand in a report summarizing their findings, as well as noting any overall conceptual error they noticed across the class, and any particularly good solutions (very clear, or solving the problem in a novel way). They seemed to take to it all with gusto, and the groups were busily discussing their grading when I came into the classroom today.

Last week we spent one of the classes in the computer lab, working through a basic geogebra tutorial. There were quite a few comments of 'This is fun!', which was great. The quiz questions which they handed in on Friday were generally of a high standard, with a few really outstanding, interesting problems. I overheard a few discussions of possible quiz questions today - it seems like there's a bit of competition going for who can come up with the best one!

Today I talked a little about the van Hiele model of geometric understanding - it's the pedagogical model on which the Common Core standards are based, and provides a useful framework for teaching geometry. The basic idea is the geometric concepts and abilities are developed sequentially, and it's necessary to master each level before moving to the next. The levels are, roughly,

Level 0: Visual. Recognizing basic shapes by appearance, without attention to parts, attributes or properties (so might identify any figure with 3 vertices as a triangle, even if the sides aren't straight, but might not identify a triangle as such if it is in a funny orientation, or is extremely long and thin).

Level 1: Analysis. Recognizing and naming the properties of geometric figures, but not understanding relationships between different types of shapes (so will not identify a square as a type of rectangle.)

Level 2: Abstract: Able to logically order properties, create meaningful definitions and make informal arguments (so can deduce that a square is a parallelogram, and also a rectangle).

Level 3: Students can construct proofs, understand the roles of axioms and derinitions, and know the meaning of necessary and sufficient conditions. High school geometry.

Level 4: Rigor. Students understand the formal aspects of deduction, can compare mathematical systems, understand non-Euclidean systems, and use indirect proof and proof by contradiction.

As I mentioned each level, we looked at the corresponding questions on the van Hiele geometry test, which give 5 questions for assessing understanding at each level. I think maybe I didn't give the students enough time to read the questions, although we were able to see some practical examples of what is expected at each level.

Having said all that, I used scroobius's awesome elfling as an example of someone who seems to have been born knowing Level 1 concepts.