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## Learning Math: Geometry

# Classroom Case Studies, K-2 Part B: Developing Geometric Reasoning (40 minutes)

## Session 10: K-2

### 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 pre-K-2 classrooms, students are expected to do the following:

- Recognize, name, build, draw, compare, and sort two- and three-dimensional shapes
- Describe attributes and parts of two- and three-dimensional shapes
- Investigate and predict the results of putting together and taking apart two- and three-dimensional shapes
- Recognize and apply slides, flips, and turns
- Recognize and create shapes that have symmetry
- Create mental images of geometric shapes using spatial memory and spatial visualization
- Recognize and represent shapes from different perspectives

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 which 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 of the publisher. Copyright © 2001 by Pearson Education. All rights reserved.

Principles and Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 2000). Standards on Geometry: Grades K-2, 41, 96.

Reproduced with permission from the publisher. Copyright © 2001 by the National Council of Teachers of Mathematics. 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.

Students in pre-K-2 generally fall at level 0 (visualization). This level describes students who reason about shapes primarily on the basis of visual considerations of the whole without explicit regard to the properties of the components (Burger and Shaughnessy, 1986). One goal of the schooling of these students is to move them to level 1 (analysis), where they can informally analyze component parts and attributes. In the primary grades, students build the foundation for understanding shapes, both two- and three-dimensional. They learn what shapes look like, the features that distinguish shapes from one another, and ways to describe shapes.

*Navigating through Geometry in Prekindergarten – Grade 2,* p. 9

Reston, VA: National Council of Teachers of Mathematics, 2001

## Video SegmentWatch this clip from Ms. Christiansen’s class again, and think about how both the lesson and the teacher are encouraging students to move to that next level of geometric reasoning. You can find this segment on the session video approximately 22 minutes and 32 seconds after the Annenberg Media logo. |

** **

**Problem B1**

Where in the video do you see evidence of the following?

- (Level 0 thinking) Students thinking about particular shapes and not their properties
- (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?

**Problem B2**

In Session 3, you worked on the problem of finding hidden polygons. Recall your own experience in this activity as an adult mathematics learner. During the activity, when did you have to use level 1 thinking? (How did you think about properties of figures to help you find them?)

**Problem B3
**

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

b. |
What do you think were the key thinking and reasoning skills in this activity? How did the reasoning and geometric content tie together? |

**Problem B4**

Now think about K-2 students and how this hidden polygon 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 B5**

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?

Findell, Carol R.; Small, Marian; Cavanagh, Mary; Dacey, Linda; Greenes, Carole E.; and Sheffield, Linda Jensen. Navigating through Geometry in Prekindergarten-Grade 2. (Reston, VA: National Council of Teachers of Mathematics, 2001), 9.

Reproduced with permission from the publisher. Copyright © 2001 by the National Council of Teachers of Mathematics. All rights reserved.

### 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?

**Note 5
**

Again, remember that the focus of the video case study is not to examine teaching practice, but to focus on the students and their thought processes.

### Solutions

**Problem B1**

Answers will vary. Some possible responses:

- (Level 0 thinking) The students’ natural response was to name the particular shape they were holding rather than to focus on properties. Also, they relate the shapes to others they know, thinking of a trapezoid as half a hexagon.
- (Level 1 thinking) The activity forces students to feel a shape, turning it around in their hands, and learn that it is the same shape no matter what its position or orientation. They are trying to focus on properties and how they can determine the class of shape from those properties.

**Problem B2**

Answers will vary. Some examples of level 1 thinking required by the activity include the identification of non-standard kinds of polygons (concave, asymmetric, etc.). Also, level 1 thinking requires knowing if you’ve already counted one of the shapes, even if you find it a different way or list the vertices in a different order.

**Problem B3**

Answers will vary. Some possible answers:

a. |
Key pieces of geometry are names, properties (number of sides and vertices), and naming of polygons. In addition, you need to be flexible in your thinking about polygons, recognizing irregular and concave polygons in addition to more regular and familiar ones. |

b. |
Reasoning includes algorithmically finding every polygon, determining if you have found duplicates, and determining when you are done. |

**Problem B4**

Answers will vary. Students will probably get a broader view of polygons, including familiarity with irregular and differently oriented polygons. They may struggle with the naming of polygons and with finding all of them in the more complicated figures.

**Problem B5
**

Answers will vary. Some ideas: Alter the activity to have less of a focus on the naming of vertices; include opportunities to draw, color, or cut out the shapes. If students have seen lots of examples of irregular polygons and have had opportunities to cut up and put polygons together to form others, they will have more success with an activity like hidden polygons.