What is the reason for this reflection?

You should have developed some basic geoboard area techniques in the hands-on section. These are the foundation of a substantial curriculum that makes connections between algebra and geometry. You are asked to comment on your experiences with this tool.

In the reading, you have seen a critique of the traditional Algebra 1 course. Please share your reactions to this critique.

Asynchronous communication lets you build a community of content inquirers that fits our hectic schedules.

Share your insights and your problems with them and the Field Expert. (One Field Expert will be reading the posts and representing your curriculum.)

1. Problems 1, 10, 15, and 17 on pp. 36-37 are wide open, in that there is not a single correct answer. What are the pros and cons of problems of this type?
2. Problems 8 and 13, on the other hand, provide strong hints towards effective and accessible techniques for area. What are the pros and cons of problems of this type?
3. The remaining problems are traditional one-correct-answer problems. What are the pros and cons of problems of this type?
4. Why do the authors of the material discourage the teaching of area formulas at this point? Do you agree?
5. Does the lesson provide access to the weaker students? How? Does it provide challenge to the stronger students? How?
6. Which parts of the lesson are most likely to enhance discourse? How can you, as the teacher, intervene to promote discussion and reflection?
7. How could the lesson be improved to work better with your students?

Read the comments of others and make two of your own in the Algebra discussion area. Look for the Tools: Algebra Activity 2 thread.

What is the reason for this second reflection?

In the first reflection, you probably thought mostly about the activity itself. In this round, you will react to others' comments, and try to step back and look at the big picture: how can a geometric activity about geoboard area contribute to learning algebra?

One hour to post your ideas.

One hour to comment on postings and other reflections.

1. Algebra is best learned in context. Instead of only working with abstract x's and y's, students need to explore domains where they can be comfortable and self-confident that lend themselves to the development of algebraic insights. Pattern block perimeter and geoboard area are two such domains.

2. Mathematical microworlds like geoboards allow most students to collect interesting data without the messiness of "real world" errors and complexities. These concrete contexts are ideal ramps for learning basic algebra. (See for example pp. 74-75 #1-12 for problems that are definitely geared to developing algebraic symbol sense, while being anchored in the geometric world of the geoboard.)

3. How do the geoboard activities undermine or support the following beliefs?
   
   • The only way to learn math is from teacher explanations.
   • In math, there's nothing to discuss: either you get it, or you don't.
   • Math is about memorizing formulas.
   • Area is when you multiply, perimeter is when you add.
   • First you learn the theory, then you apply it.
   • To do math well, you need to explain your ideas in writing.

Read the comments of others and make two of your own in the Algebra discussion area. Look for the Tools: Algebra Activity 2 thread.

If you want to explore a more substantial geoboard area problem, take a look at pp. 157-158 of the handout, where students are challenged to come up with a formula for area that is based on the number of pegs enclosed by the rubber band. (This is known as Pick's Formula.)

Algebra: Themes, Tools, Concepts also uses the geoboard as an arena for the exploration of similar figures and slope. This is not in your handout, but if you have access to the book, see pp. 115-116, 293-294, 403-405. Or create your own geoboard lesson on this subject!

If you pursue any of these extensions, consider sharing what you find out with others in the discussion area.