1. Meaningful geometry education
Michiel Doorman
En dan misschien nog met een slotzin als: "Common sense reasoning is used to make mathematics more meaningful for students, and with the intention to extend their common sense reasoning with new mathematical means.“
En HF: “A structure of mathematics is no guide to structuring curriculum”
Uit: Hans Freudenthal, H. (1968). Didactical Principles in Mathematics Instruction. North-Holland Mathematical Library, 34,

2. Developments in the past decades in the Netherlands

3. Example activities from 1976 and 2002 [activity 1]
Have a look at the example geometry activities from Dutch textbooks
What are the activities for the students?
What could be the aim of each of the activities?
Can you see a difference between 1976 and with respect to “meaningful”?

4. What happened between 1976 and 2002?
Experiments were carried out (at FI) during the 1970s and 1980s through working with teachers and students in classrooms
Identify themes and problems that result in meaningful mathematical activities
Find empirical support for another approach to geometry

5. This ambition was not new
Dieudonne (1959) drew attention to the out-dated geometry program: “A bas Euclide”
Complaint: teaching approaches overemphasized the axiomatic method, which did not meet the needs of a technological society nor used modern mathematical and scientific languages
New directions: transformation geometry, linear algebra, vector spaces, …

6. Potential starting points
Friedrich Fröbel (1782–1852) used blocks, mosaics and other educational toys for practical activities in an early phase
Maria Montessori (1870–1952) also promoted attention for spatial orientation with playing as a context for geometrical explorations
Dieke van Hiele-Geldof & Pierre van Hiele (both PhD in 1957) investigated the potential of introductory activities with concrete materials followed by level raising
Tatyana Ehrenfest ( ) developed an introductory course with geometrical phenomena (3D) as a starting point

7. Potential starting points: Van Hiele’s
As an example, the cube:
Do not define a cube, but present students different kinds of solid and wire cubes. Discuss similarities and differences between these objects
Students become familiar with the objects and with concepts such as edge and right angle (e.g. defined by folding)
During subsequent analyses of the objects, other characteristics, patterns and symmetries are identified and relationships are constructed
This process passes different levels of understanding, labelled as
visualization (ground level)
analysis
informal deduction
generalisation, construction of a system of relationships
deduction, and rigor

8. Potential starting points: Tatyana Ehrenfest (a quote)
“The road from chaos to system (…), is not shown … For students ‘Geometry’ becomes a game with abstract objects that are isolated of all concreteness … Instead of operating with concepts – which can be acquired through the act of abstraction of one’s own living experiences – students have to work with names and drawings which do often not refer to anything they know”

9. The road from chaos to system
Mathematics educators often take the system as starting point (Freudenthal: anti-didactic inversion)
Risks
Mathematics as an isolated discipline
The system is meaningless for students
Geometry education should support students in ‘Grasping Space’

10. In search of an alternative road
Activity 2: try to solve the tasks

11. Examples Which image comes from which camera?
Taking position in space (2D -> 3D)

12. The 180o line in Films
Axis of action; spatial continuity; the 180 degrees system.

13. Examples
Can he see any rabbits?
Top view and lines of sight

14. Examples
What is higher? The tower or the bridge?

15. Examples
Shadows from sun and lamp

16. Examples (Goddijn, 1980)
Shadows from sun and lamp

17. Examples (Goddijn, 1980)
Shadows from sun and lamp

18. Experiences
Geometry can be connected to phenomena that students can experience in everyday life
Students can be considered as active and creative investigators and explainers of problems
Geometrical reasoning is supported by ‘situation’ models (lines of sight, top and side views)

19. Reflection on the examples
In the 2002 textbook, the topic of reasoning with lines and angles has changed into using 3D contexts
The 1976 textbook starts with explaining the coordinate system, while in the 2002 textbook this occurs much later in the context of other orientation systems
An important role for phenomena that create the need for drawing top views or side views, with lines that explain what might be going on
Students are involved in inquiring situations and developing geometrical means to organize and investigate the space around them

20. Deduction? [activity 3]

22. Deduction? Through local organization
The path from exploration in context to finding an underpinning argumentation

23. Conclusions
Realistic geometry education gradually emerged from an approach building on experiences with concrete objects and grasping space
Characterized by:
Starting in space with inquiry-based tasks in 3D contexts that elicit geometrical reasoning
Ending with geometry as a deductive system (for all students?)

24. Take home messages
Freudenthal: “A structure of mathematics is no guide for structuring curriculum” (HF, 1968)
Geometry curriculum: Develop inquiring minds that see and can explore and explain geometrical phenomena/situations
Next steps: Geometry also supports other math-topics, e.g. algebra Euclid maybe not an example of structuring a curriculum, but…
En HF: “A structure of mathematics is no guide to structuring curriculum”
Uit: Hans Freudenthal, H. (1968). Didactical Principles in Mathematics Instruction. North-Holland Mathematical Library, 34,

25. Meaningful geometry education
Thank you