1. Van Hiele’s Learning Theory Mara Alagic

2. 2 June 2004 Levels of Geometric Thinking Precognition Level 0: Visualization/Recognition Level 1: Analysis/Descriptive Level 2: Informal Deduction Level 3: Deduction Level 4: Rigor

3. Mara Alagic3 June 2004 Learning Theory Phases Information Guided Orientation Explication Free Orientation Integration

4. Mara Alagic4 June 2004 Outline Learning theory phases Information Guided Orientation Explication Free Orientation Integration Example Theme: Properties of two-dimensional shapes Tools: Seven tangram shapes

5. Mara Alagic5 June 2004 Information Students should be encouraged to freely explore the materials; they will discover some properties & structures While students are playing, the teacher can observe and informally assess students’ thinking and language Give each student a tangram set and ask 'What can you do with these pieces?' Encourage students to share and talk about the shapes and pictures they have made Students begin to see how pieces fit together; they begin to discover the properties and relationships

6. Mara Alagic6 June 2004 Guided Orientation Activities are presented so that students’ attention is focused on particular characteristics of the shapes; Questions might be generated from the first phase: Choose two shapes. How many different shapes can you make with them? Use three shapes to make a new shape. How many ways can you make this shape? Which pieces can be made from three other pieces? Have students record what they find by tracing around the shapes, and share with others One activity will often lead to another, e.g. …

7. Mara Alagic7 June 2004 Explication Tasks and games that deliberately develop the vocabulary associated with the ideas that have been encountered so far The teacher clarifies terms the children are already using and introduces new terms Questions provide opportunities for the terms to be used: Which shapes have a right angle? How many lines of symmetry does each shape have? What is the same about all the triangles? Which shapes have parallel sides? …

8. Mara Alagic8 June 2004 Free Orientation Students engage in more challenging tasks that draw on the knowledge and skills previously developed Activities and problem-solving tasks are open- ended or can be completed in different ways The aim is to apply what they have learned and become more skillful How many ways can you make a square from some or all of the pieces? Complete classic tangram puzzles of outlines of animals Draw a tangram square (like the one illustrated above) on an 8x8 square grid, examine the pieces carefully in relation to the grid, then work out a way to enlarge all the pieces

9. Mara Alagic9 June 2004 Integration Opportunities are given for the students to pull together their new knowledge and reflect on it as a whole They should be able to express or summarize what they have learned in some way: For example, small groups of children can design charts, class books and displays that present what they have learned about the tangram shapes Small groups, pairs of students and individuals can design presentations (posters, class books, technology aided), games and lists of questions about what they have learned about the tangram shapes This will help students to logically organize their knowledge of properties of the shapes

10. Mara Alagic10 June 2004 References Van Hiele, P. M. (1959). Development and learning process. Acta Paedogogica Ultrajectina (pp. 1-31). Groningen: J. B. Wolters. A method of initiation into geometry at secondary schools. In H. Freudenthal (Ed.). Report on methods of initiation into geometry (pp.67-80). Groningen: J. B. Wolters. Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of Thinking in Geometry Among Adolescents. JRME Monograph Number 3.