Geometry and Measurement

ECED 4251Dr. Jill Drake

Sign Up For Case Study Meetings Geometry and Measurement Chapter in Ashlock

Review Game Van Heile – Levels of Geometric Thinking Error Patterns

Last Class Mathematics/Assessment Kit Case Studies

Today’s Topics…

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The van Hiele Model of Geometric Thought

4

Make my Master Piece!

5

When is it appropriate to ask for a definition?

A definition of a concept is only possible if one knows, to some extent, the thing that is to be defined.

Pierre van Hiele

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Definition?How can you define a thing before you know what you have to define?

Most definitions are not preconceived but the finished touch of the organizing activity.

The child should not be deprived of this privilege…

Hans Freudenthal

van Hiele – Levels of Geometric Thinking Level 0: Visualization Level 1: Analysis or Descriptive Level 2: Informal Deduction or Relational Level 3: Deduction Level 4: Rigor

For specific information: See Van de Walle (2004), pp. 347

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Geometry

Level 0: Visualization (Van de Walle, 2004, p. 347)

Recognize, sort, and classify shapes based on global visual characteristics, appearances. “A square is a square because it looks like a square.” “If you turn a square and make a diamond, it’s not a

square anymore.”

Because appearance is dominant at this level, appearances can overpower properties of a shape. Page 8

Geometric Thinking…

sorting, identifying, and describing shapes manipulating physical models seeing different sizes and orientations of the same

shape as to distinguish characteristics of a shape and the features that are not relevant

building, drawing, making, putting together, and taking apart shapes

John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. (New York: Addison Wesley Longman, 2001), pp. 310-11

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Suggested Instruction

Level 1: Analysis or Descriptive Students see figures as collections of properties. They

can recognize and name properties of geometric figures, but they do not see relationships between these properties.

When describing an object, a student operating at this level might list all the properties the student knows, but not discern which properties are necessary and which are sufficient to describe the object.

(Professional Handbook for Teachers, GEOMETRY: EXPLORATIONS AND APPLICATIONS: McDougal Littell Inc., 2006, p. 4-5)

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Geometric Thinking…

It’s a rotation!

shifting from simple identification to properties, by using concrete or virtual models to define, measure, observe, and change properties

using models and/or technology to focus on defining properties, making property lists, and discussing sufficient conditions to define a shape

doing problem solving, including tasks in which properties of shapes are important components

classifying using properties of shapes.

John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. (New York: Addison Wesley Longman, 2001), pp. 310-11

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Suggested Instruction

Level 2: Informal Deduction or Relational Level

Students perceive relationships between properties and between figures. At this level, students can create meaningful definitions and give informal arguments to justify their reasoning.

Logical implications and class inclusions, such as squares being a type of rectangle, are understood. The role and significance of formal deduction, however, is not understood.

(Professional Handbook for Teachers, GEOMETRY: EXPLORATIONS AND APPLICATIONS: McDougal Littell Inc., 2006, p. 4-5)

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Geometric Thinking…

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Relational Level Example

If I know how to find the area of the rectangle, I can find the area of the triangle!

Area of triangle =

h

b

1

2h

1

2bh

doing problem solving, including tasks in which properties of shapes are important components

using models and property lists, and discussing which group of properties constitute a necessary and sufficient condition for a specific shape

using informal, deductive language ("all," "some," "none," "if-then," "what if," etc.)

investigating certain relationships among polygons to establish if the converse is also valid (e.g., "If a quadrilateral is a rectangle, it must have four right angles; if a quadrilateral has four right angles, must it also be a rectangle?")

using models and drawings (including dynamic geometry software) as tools to look for generalizations and counter-examples

making and testing hypotheses

John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. (New York: Addison Wesley Longman, 2001), pp. 310-11

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Suggested Instruction

Level 3: Deduction

Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions.

At this level, students should be able to construct proofs such as those typically found in a high school geometry class.

(Professional Handbook for Teachers, GEOMETRY: EXPLORATIONS AND APPLICATIONS: McDougal Littell Inc., 2006, p. 4-5)

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Geometric Thinking…

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Deductive Level Example

In ∆ABC, is a median.

I can prove thatArea of ∆ABM =

Area of ∆MBC.M

CB

A

BM

Level 4: Rigor

Students at this level understand the formal aspects of

deduction, such as establishing and comparing mathematical systems.

Students at this level can understand the use of indirect proof and proof by contrapositive, and can understand non-Euclidean systems.

(Professional Handbook for Teachers, GEOMETRY: EXPLORATIONS AND APPLICATIONS: McDougal Littell Inc., 2006, p. 4-5)

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Geometric Thinking…

Van Hiele: Level 0 For example: Ashlock (2006), pp. 194 -

206Diagnosing errors

Martha – Ashlock (2006), pp.194-195 Oliver – Ashlock (2006), p. 196

Suggested correction strategies Martha – Ashlock (2006), pp. 204-205 Oliver – Ashlock (2006), p. 206Page 19

Geometry

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Visualization Error

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Visualization Error

Conceptual: Unit concepts (mile, minute, penny) Unit Equivalence (12 inches = 1 foot) Place Value (whole numbers and decimals) Measurement tools (ruler measures length) Measurement Concepts (time, perimeter) Number Sense

Procedural Algorithm violations Conversion errors (gram to kilogram, lapse

time)Page 22

Measurement Errors

Work with a group of your peers to reach a consensus about…◦ Error Type: Conceptual, Procedural or Both?◦ The procedural error(s)

Ask yourselves: What exactly is this student doing to get this problem wrong?

◦ The conceptual error(s) Ask yourselves: What mathematical misunderstandings

might cause a student to make this procedural error?

Chapters 6-7 (Ashlock, 2006)

Diagnosing Errors

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Tamiko’s Case

•Describe Tamiko’s error pattern.

1. Procedural Error: 2. Conceptual Error

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Spurgeon’s Case

•Describe Spurgeon’s error pattern.

• Conceptual Strategy?

• Intermediate Strategy?

• Procedural Strategy?

Apply these conceptual understandings when diagnosing errors associated with measurement. For example: Ashlock (2006), pp. 203 - 212

Diagnosing errors Margaret – Ashlock (2006), pp. 203-204

Suggested correction strategies Margaret – Ashlock (2006), pp. 211-112

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Unit Conversions of Measurement

Martha’s Case

•Describe Martha’s error pattern.

1. Procedural Error: 2. Conceptual Error

Oliver’s Case

•Describe Oliver’s error pattern.

• Conceptual Strategy?

• Intermediate Strategy?

• Procedural Strategy?

Charlene’s Case

•Describe Charlene’s error pattern.

• Conceptual Strategy?

• Intermediate Strategy?

• Procedural Strategy?

Denny’s Case

•Describe Denny’s error pattern.

• Conceptual Strategy?

• Intermediate Strategy?

• Procedural Strategy?

Teresa’s Case

•Describe Teresa’s error pattern.

• Conceptual Strategy?

• Intermediate Strategy?

• Procedural Strategy?

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Margaret’s Case

•Describe Margaret’s error pattern.

• Conceptual Strategy?

• Intermediate Strategy?

• Procedural Strategy?

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