1. Geometry and Measurement ECED 4251 Dr. Jill Drake

2.  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…

3. 3 The van Hiele Model of Geometric Thought

4. 4 Make my Master Piece!

5. 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

6. 6 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

7.  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 Page 7 Geometry

8.  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…

9.  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 Page 9 Suggested Instruction

10.  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) Page 10 Geometric Thinking…

11. It’s a rotation!

12.  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 Page 12 Suggested Instruction

13.  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) Page 13 Geometric Thinking…

14. 14 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 =

15.  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 Page 15 Suggested Instruction

16.  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) Page 16 Geometric Thinking…

17. 17 Deductive Level Example In ∆ABC, is a median. I can prove that Area of ∆ABM = Area of ∆MBC.

18.  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) Page 18 Geometric Thinking…

19.  Van Hiele: Level 0  For example: Ashlock (2006), pp. 194 - 206  Diagnosing 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. 206 Page 19 Geometry

20. Page 20 Visualization Error

21. Page 21 Visualization Error

22.  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

23. 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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25. Tamiko’s Case Describe Tamiko’s error pattern. 1.Procedural Error: 2. Conceptual Error

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27. Spurgeon’s Case Describe Spurgeon’s error pattern. Conceptual Strategy? Intermediate Strategy? Procedural Strategy?

28.  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 Page 28 Unit Conversions of Measurement

30. Martha’s Case Describe Martha’s error pattern. 1.Procedural Error: 2. Conceptual Error

32. Oliver’s Case Describe Oliver’s error pattern. Conceptual Strategy? Intermediate Strategy? Procedural Strategy?

34. Charlene’s Case Describe Charlene’s error pattern. Conceptual Strategy? Intermediate Strategy? Procedural Strategy?

36. Denny’s Case Describe Denny’s error pattern. Conceptual Strategy? Intermediate Strategy? Procedural Strategy?

38. Teresa’s Case Describe Teresa’s error pattern. Conceptual Strategy? Intermediate Strategy? Procedural Strategy?

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40. Margaret’s Case Describe Margaret’s error pattern. Conceptual Strategy? Intermediate Strategy? Procedural Strategy?

41.  Math Kits  Case Study  Questions Upcoming Considerations