## Learning Math: Geometry

# Classroom Case Studies, 6-8 Part B: Developing Geometric Reasoning (40 minutes)

## Session 10: 6-8, Part B

### In This Part

- Introducing van Hiele Levels
- Analyzing with van Hiele Levels

### Introducing van Hiele Levels

The National Council of Teachers of Mathematics (NCTM, 2000) identifies geometry as a strand in its *Principles and Standards for School Mathematics.* In grades pre-K through 12, instructional programs should enable all students to do the following:

- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
- Specify locations and describe spatial relationships using coordinate geometry and other representational systems
- Apply transformations and use symmetry to analyze mathematical situations
- Use visualization, spatial reasoning, and geometric modeling to solve problems

In grades 3-5 classrooms, students are expected to do the following:

- Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes
- Classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids
- Build and draw geometric objects
- Create and describe mental images of objects, patterns, and paths
- Use geometric models to solve problems in other areas of mathematics, such as number and measurement
- Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life

Dutch educators Pierre van Hiele and Dina van Hiele-Geldof developed a theory of five levels of geometric thought. It is just a theory, but a useful one for thinking about activities that are appropriate for your students and prepare them to move to the next level, and for designing activities for students who may be at different levels.

**Level 0: Visualization.** The objects of thought at level 0 are shapes and what they look like. Students have an overall impression of the visual characteristics of a shape, but are not explicit in their thinking. The appearance of the shape is what’s important. Students may think that a rotated square is a “diamond” and not a “square” because it looks different from their visual image of square. (*Early elementary school and, for some, late elementary school*)

**Level 1: Analysis.** The objects of thought here are “classes” of shapes rather than individual shapes. Students are able to think about, for example, what makes a rectangle a rectangle. What are the defining characteristics? They can separate that from irrelevant information like the size and the orientation. They begin to understand that if a shape belongs to a class like “square,” it has all the properties of that class (perpendicular diagonals, congruent sides, right angles, lines of symmetry, etc.). (*Late elementary school and, for some, middle school*)

**Level 2: Informal Deduction.** The objects of thought here are the properties of shapes. Students begin “if-then” thinking; for example, “If it’s a rectangle, then it has all right angles.” Students can begin to think about the minimum information necessary to define figures; for example, a quadrilateral with four congruent sides and one right angle must be a square. Observations go beyond the properties into mathematical arguments about the properties. Students can engage in an intuitive level of “proof.” (*Middle school and, for some, high school*)

**Level 3: Deduction.** The objects of thought here are the relationships among properties of geometric objects. Students can explore relationships, produce conjectures, and start to decide if the conjectures are true. The structure of axioms, definitions, theorems, etc., begins to develop. The students are able to work with abstract statements and draw conclusions based more on logic than intuition. (*This is the goal of most 10th-grade geometry courses, but many students are not developmentally ready for it.*)

**Level 4: Rigor.** The objects of thought are deductive axiomatic systems for geometry. For example, students may compare and contrast different axiomatic systems in geometry that produce our familiar Euclidean plane geometry, finite geometries, the geometry on the surface of a sphere, etc. **Note 4**

For more information on the van Hiele levels and how to work with students within each level, read the article “Geometric Thinking and Geometric Concepts” by John A. Van de Walle from *Elementary and Middle School Mathematics.*

Van de Walle, John A. (2001). Geometric Thinking and Geometric Concepts. *In Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed.* (pp. 342-349). Boston: Allyn and Bacon.

Reproduced with permission from the publisher. Copyright © 2001 by Pearson Education. All rights reserved.

### 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 SegmentIn 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. 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**

- What do you think were the key pieces of geometry content in this activity? What knowledge did you learn, solidify, or connect with better?
- 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?

**Problem B6**

What might students misunderstand or find confusing in the lesson? How could you alter the lesson or prepare them beforehand to help avoid these misunderstandings?

### Notes

**Note 4
**

If you are working with a group of colleagues, take some time to discuss your own students. Where in the van Hiele levels do you see them functioning comfortably? (There will be a range, of course, because not all students are the same.) Try to cite evidence from you classrooms: With which tasks do students find success? With which tasks do they struggle?

### Solutions

**Problem B1**

Answers will vary. Some possible responses:

- (Level 1 thinking) The students easily calculate the areas of the squares and triangles in different positions and recognize the triangles as congruent even though they are positioned differently.
- (Level 2 thinking) By summing areas to find the total and equating the areas found in two different ways, students are showing logical thinking about geometric objects.
- (Level 3 thinking) This is harder to see. The teacher is clearly trying to move them through a multi-step argument, but students may not all be aware that they are using previous results (area formulas, algebraic facts, solving equations) to prove something new.

**Problem B2**

There were many adaptations. Here are some: The proof was adapted to be one that was easier for students to follow. (It was based on the Garfield proof in this course, but was adapted to remove the need for computing with fractions.) Students worked through numerical and numeric/variable mixed problems before working with variables only. The teacher works with students as a whole class (on the proof itself) rather than asking them to work through individually or with pairs.

**Problem B3**

Answers will vary. The thinking often goes like this: “If it’s going to make a solid shape, then there must be at least three polygons meeting at a vertex. If it’s going to make a solid shape, then there must be less than a total of 360˚ around a vertex. Regular hexagons and polygons with more than six sides all have 120˚ or more at each vertex, so these shapes cannot be used.” And so on. Putting all of this together to convince yourself that only five such solids are possible constitutes level 3 thinking.

### Problem B4

Answers will vary. Some possible answers:

- Key pieces of geometry are definitions and properties of regular two- and three-dimensional figures, building polyhedra, angle relationships, and so on. We also explored Euler’s formula in Session 9, Problem A8.
- “If-then” thinking, reasoning through every possible case, and generalizing were all important parts of the activity. It was important to both know the geometry (what are the angle measures for polygons with different numbers of sides?) and to use those facts in making deductions.

**Problem B5**

Answers will vary. Students will probably gain understanding of three-dimensional figures and how they differ from polygons. They will likely gain valuable understanding and visualization skills from building and manipulating the solids and from attempting to count faces, edges, and vertices. They may not have the prerequisite knowledge of angle measures in polygons as a solid foundation. Some students may also struggle with the generalizations. If six triangles don’t work, how do we know seven triangles won’t work? Why can we eliminate polygons with seven, eight, and more sides without even trying to build them?

**Problem B6**

Answers will vary. Some ideas: Lots of experience with building generalizations in cases that are easier to check, and lots of experience with polygons will help.