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# Classroom Case Studies, K-2 Part A: Geometry as a Problem Solving Process (25 minutes)

## Session 10: K-2, Part A

The study of geometry can include both problem solving and connections to other areas of mathematics (arithmetic, algebra, etc.). Too often, classrooms focus almost exclusively on correctly identifying shapes and their properties by name. While mathematical language and clear communication are important in geometry, it is important to include other kinds of geometric problems as well so that geometry isn’t reduced to mere nomenclature. Note 2

When viewing the video segment, keep the following questions in mind:

 a. How does the teacher incorporate geometric language into the lesson without making it the focus of the lesson? What is the purpose of having students describe the shapes rather than name them? b. Where in the lesson are students learning new geometric content? What is that content? c. Where in the lesson are students solving problems and thinking mathematically? How does the problem solving relate to the geometric content? d. Thinking back to the big ideas of this course, what are some geometric ideas these students are likely to encounter through their investigation of this situation?

### Video Segment

In this video segment, second-grade students in Ms. Christiansen’s class are working to describe different figures so that their fellow students can draw them just from the description. They use vocabulary they know to describe figures, learning names for new shapes as they work.

You can find this segment on the session video approximately 22 minutes and 32 seconds after the Annenberg Media logo.

### Problem A1

Answer the questions you reflected on as you watched the video:

 a. How does the teacher incorporate geometric language into the lesson without making it the focus of the lesson? What is the purpose of having students describe the shapes rather than name them? b. Where in the lesson are students learning new geometric content? What is that content? c. Where in the lesson are students solving problems and thinking mathematically? How does the problem solving relate to the geometric content? d. Thinking back to the big ideas of this course, what are some geometric ideas these students are likely to encounter through their investigation of this situation?

### Problem A2

This lesson is not couched in a “real-world context.” Students are sorting shapes and thinking about mathematical ideas in the abstract. What are the advantages and disadvantages of this kind of lesson? Are “mathematics only” lessons important in your classroom? What purpose do they, as opposed to contextualized lessons, serve? Note 3

### Problem A3

Ms. Christiansen’s lesson is similar to one from Session 1 of this course in which you had to build designs from pattern blocks based on descriptions of those designs. Discuss the ways in which Ms. Christiansen’s lesson was similar to and different from the one in this course. What makes this more appropriate for second-grade students?

### Notes

Note 2

Before examining specific problems at this grade level, you will watch with an eye toward geometric problem solving a teacher in her classroom. The purpose in viewing the video is not to reflect on the teacher’s methods or teaching style, but to watch closely the way she brings out geometric ideas while engaging her students in a problem-solving task. Think about how the task meets students at their current level of geometric sophistication and also helps move them to the next level.

Note 3

This is a particularly good discussion to have with your colleagues. Everyone has different opinions and thoughts about the use of context in the mathematics classroom. Spend some time talking about not just what you think, but why you think it. Cite examples from your own experience instead of focusing on what you have heard others say.

### Solutions

Problem A1

 a. Answers will vary. Some ideas: The challenge of feeling a shape but not looking at it, and of describing its properties without using its name simultaneously, requires students to practice with vocabulary like “sides” and “vertices” but also keeps the lesson from seeming like a vocabulary lesson. Also, the fact that students have different roles — feeling and describing, listening and drawing — keeps the focus on communication rather than just on the words. b. We see students learn new terms like “trapezoid,” but with meaning and context and relating it to other things they know. (It’s half a hexagon.) c. Throughout the lesson, students work on solving a problem that involves discovering what the properties of a shape are. In other words, they think about what makes a square a square, a triangle a triangle, etc. d. One of the big ideas that students encounter in this lesson is the idea of classification. In this lesson, we see how students begin to develop a sense of classification of polygons based on their properties. Classification is an idea that progresses through grade levels.

Problem A2
People have very different, and often very strong, opinions about the use of context in mathematics classrooms. It is important to present students with a variety of lessons. Students can be engaged by problems that are not context-based, as well as by those with real-world connections.

Problem A3
In this lesson, students describe just one shape rather than a design made of several shapes. Also, students feel the shapes without looking and draw the shapes that are described. This gives them the opportunity to touch, draw, describe, and listen to descriptions of the shapes.