Van Hiele’s Learning Theory

1. Van Hiele’s Learning Theory Mara Alagic

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

3. Learning Theory Phases • Information • Guided Orientation • Explication • Free Orientation • Integration Mara Alagic

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

5. 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 Mara Alagic

6. 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. … Mara Alagic

7. 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? … Mara Alagic

8. 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 Mara Alagic

9. 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 Mara Alagic

10. 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. Mara Alagic