P.M van Hiele Mathematics Learning Theorist

1. P.M van HieleMathematics 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 Show that the sum of the angles in a triangle is 180° Find area of a triangle based on what you know about the area of a parallelogram

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.