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# Classroom Case Studies, 3-5 Part A: Geometry as a Problem-Solving Process (25 minutes)

## Session 10: 3-5, 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 following video segment, keep the following questions in mind:

1. How does the teacher incorporate geometric language into the lesson without making it the focus of the lesson?
2. Where in the lesson are students learning new geometric content? What is that content?
3. Where in the lesson are students solving problems and thinking mathematically? How does the problem solving relate to the geometric content?
4. 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, fifth-grade students in Ms. Kurchian’s class are working to sort different figures into Venn diagrams. They discuss whether shapes fit certain criteria given by the labels on their diagrams and when shapes can fit multiple criteria.

If you are using a VCR, you can find this segment on the session video approximately 9 minutes and 38 seconds after the Annenberg Media logo.

### Problem A1

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

1. How does the teacher incorporate geometric language into the lesson without making it the focus of the lesson?
2. Where in the lesson are students learning new geometric content? What is that content?
3. Where in the lesson are students solving problems and thinking mathematically?
4. How does the problem solving relate to the geometric content?
5. 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. Kurchian’s lesson was based on a lesson from this course in Session 3. Discuss the ways Ms. Kurchian’s lesson is similar to and different from the one from Session 3 of this course. What adaptations did she make and why?

### 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 who has also taken the course. 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.

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.

### Problem A1

1. Answers will vary. Some ideas: The lesson is really one of analysis, but students must know and use geometric vocabulary to work with the labels and to communicate with their group about the problems.
2. It may not be clear what from the lesson is new to the students, but it seems that some of the logical relationships, and possibly some of the vocabulary, might be.
3. Answers will vary. Throughout the lesson, students are thinking analytically to solve problems such as placing the polygons into appropriate Venn diagrams and answering questions about which properties are shared by certain groups of polygons and which ones are not.
4. One of the geometric ideas of the course that students encounter in this lesson is the idea of classification. In this lesson, we see how students extend their knowledge of classification of polygons by thinking about categories of shapes and analyzing them in terms of properties that they share. Classification also progresses through grade levels and here we see how the class develops from the more basic ideas and vocabulary towards more complex levels of thinking and problem solving.

### 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

There were lots of adaptations — the labels and shapes were selected to be more familiar to students in the class. The students used just two loops in the Venn diagram rather than three loops. The activity is more carefully structured in terms of different groups working with different pairs of labels.