1. P.M van Hiele Mathematics Learning Theorist Rebecca Bonk Math 610 Fall 2009

2. Who is P.M van Hiele? Pierre Marie van Hiele, and Dina van Hiele-Geldof, were Dutch educators whose research in math education was focused on learning geometry. Their theory was presented in their doctoral dissertations in 1957.

3. What was their theory? There are five levels to describe student understanding. –Level 0: Visual –Level 1: Descriptive –Level 2: Relational –Level 3: Deductive –Level 4: Rigor American researchers typically number the levels from 1-5 so “level 0” can describe young children who cannot identify shapes. Both systems are used.

4. Level 0: Visual (or Recognition) Students can name and recognize shapes by their appearance. Identification is based on mental “prototypical” shape. Students may recognize characteristics and properties but cannot use them formally. Grades K-3

5. Visual Level Example

6. Suggestions for Instruction Examples and non-examples Tangrams Build, take apart, rearrange into different shapes Sorting, identifying, describing Vocab: visual words (“pointy”, “corner”) and correct terminology (“rectangle”, “angle”)

7. Level 1: Descriptive (or Analysis) Figures are identified by properties rather than by appearance. Connections between different shapes and their properties is not evident. Definitions are not limited to necessary and sufficient conditions (may contain too many or too few properties). Grades 4-5

8. Descriptive Level Example Find area of irregular figures “Squares are not rectangles.”

9. Suggestions for Instruction Sorting and drawing Concrete/ visual models used to define, measure, observe, and change properties Property Lists Classifying shapes using properties Vocab: sides, angles, similar, congruent, always, sometimes, never

10. Level 2: Relational (or Abstraction, or Informal Deduction) Students recognize relationships between and among properties. Students can give informal arguments for deductions and can follow formal proofs. Definitions (necessary and sufficient conditions) are meaningful, and students can handle equivalent definitions. Students need to be here by end of grade 8

11. Relational Level Example Find area of a triangle based on what you know about the area of a parallelogram Show that the sum of the angles in a triangle is 180°

12. Suggestions for Instruction Express relationships verbally Open ended tasks and problem solving Make and test hypothesis Property lists, and discussion on which properties are necessary Vocab: “if-then” statements, “what if…”, all, some, or none (ASN), converse

13. Level 3: Deduction Definitions contain only necessary conditions. Students can construct proofs. Students understand importance of definitions, axioms, and theorems. Logical reasoning is developed. High school level

14. Deductive Level Example

15. Suggestions for Instruction Formal proof Compare various proofs (ex: various ways to prove Pythagorean Theorem) Drawing and constructions Vocab: postulate, theorem, axiom

16. Level 4: Rigor Understanding of various axiomatic systems without needing a visual model Able to study non-Euclidean systems Ability to give rigorous formal proof using methods such as proof by contrapositive College level

17. Characteristics of Levels of Thinking Levels are sequential and hierarchical. Students must master one level before advancing to next level. Level advancement is based on experiences, not on age like Piaget’s theory. –Students do not automatically progress; they gain abstraction and sophistication in their thinking as a result of their experiences.

18. Characteristics of Levels of Thinking Students may reason at multiple levels or at intermediate levels. Children advance at different rates for different concepts. Each level has its own language. Terms that are considered correct at one level may be modified at another. Instruction and language at a level higher than the student may inhibit learning.

19. Phases of Learning Information Guided orientation Explication Free Orientation Integration

20. Information Phase Students state what they already know about a concept. Formal vocabulary is clarified.

21. Guided (or directed) Orientation Phase Students explore concept in a small group setting. Construct knowledge from their findings in activities. Teacher facilitates, provides hints, and asks scaffolding questions.

22. Explication Phase Students exchange ideas about their discoveries and observations regarding relationships. Misconceptions are addressed, and it is especially helpful if students can explain corrections to their fellow classmates. Correct technical language is developed and used.

23. Free Orientation Phase Students engage in more complex tasks and open ended investigations. Students begin working independently using the newly discovered relationships.

24. Integration Phase Students review and summarize what they learned. Forms basis for “Information Phase” of next level of thinking. Knowledge is available for immediate recall.

25. Where Is This In Schools? Teachers modeling correct use of terminology is essential, but often present a definition to students rather than letting them discover it. Textbook focus is on integration phase (summary) and many teachers start instruction with revealing what students should find in activity.

26. Where Is This In Schools? Activities are tiered so students first have exposure to shapes, begin sorting them visually, and then talk about properties they see. As students progress, geometry becomes more abstract. Research indicates that students may enter high school at Level 0 or 1; thus, they have great difficulty with formal proofs. Definitions progress to only contain necessary information.

27. Implications for Teaching The traditional form of teaching, by modeling and explanation, is time-efficient but not effective. Students will have a firm grasp of geometry if they are allowed to “play” with the ideas and arrive at own conclusions. Whole group discussion is important in clearing up misconceptions and presents alternate observations and understandings.

28. Implications for Teaching The more teachers know about a subject and the way students learn, the more effective they will be. –Research indicates content knowledge is low and there needs to be increased research in explaining student cognition in geometry Difficulty in assigning van Hiele level. Many students are between levels or show two levels simultaneously, and they can be at different levels for different concepts. Some researchers argue for a continuum of attainment from one level to another on a specific concept.