International Panel: Bridging Policy and Practice
A Focus on Teacher Preparation

Presentation 5: Pre-service Mathematics Teacher Education in France

Antoine Bodin
Université de Franche Comte IREM
Vivianne Durand-Guerrier
Institut Universitaire de Formation des Maítres

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Antoine Bodin

France, has a common path for future teachers. After completing a three-year degree, prospective teachers apply to the University Institute of Teacher Training (IUFM) and take the Regional Admission Test. The ministry determines how many certificates are available in each subject area, which determines the number admitted. If there are 1000 positions, 1000 are admitted. The actual number depends on how much money is available. Every available position has about six to eight applicants, so the competition is intense. Those who are admitted attend the IUFM training for two years as teachers-students.

Year one

During year one, all elementary teacher preparation is done in the IUFM. Secondary teachers are more specialized and take both subject matter courses in the University linked to the IUFM and pedagogical and didactical training in the IUFM.

Prospective elementary school teachers take the Regional Competitive Exam, and prospective secondary teachers take the National Competitive Exam (CAPES) for certification in one subject area. At the secondary level - grade 6 to 12 - it is more common for a given teacher to be specialized in one subject matter and teach only in that subject. Passing the national exam allows recruited teachers to teach anywhere in the country, but the ministry determines exactly where they should teach. A more prestigious, highly competitive examination called "Aggregation" allows prospective teachers to teach at secondary and at university level. If they pass the exam, they enter the second year in the IUFM and become civil servants paid by the ministry.

Year two

In year two, prospective teachers are recruited to teach in the schools. During this time they are both teachers and students, so they are called teacher-students. For the entire school year they assume full responsibility for teaching a class. Each teacher-student has a pedagogical advisor who serves as a mentor. Teacher-students also have another pedagogical advisor for theoretical content who is a trainer from IUFM. This person attends some of the new teacher's classes, and the new teacher attends some of the classes of this advisor. At the end of the year teacher-students take the Accreditation Exam, given under the auspices of the IUFM and the Ministry. Prospective teachers must pass this exam to earn a credential and, during this year, must also write a dissertation (called "Mémoire Professionel") and defend it in front of a board of examiners. This dissertation is written under the direction of an IUFM advisor and is part of the qualifying process devised by the ministry's inspectorate.

Viviane Durand-Guerrier

Elementary teachers receive special preparation in teaching mathematics. They do nine weeks of student teaching at three different grades. There is a period where they observe a special teacher who helps them. They try to create an appropriate classroom atmosphere for teaching a particular lesson and generate a mathematics problem to teach. Unfortunately, only one teacher in three has had the academic preparation that they need. One third of teachers are expected to become math teachers, but some do not like mathematics at all. One goal is to try to find ways to bring enjoyment back to mathematics for them.

A new program at the primary level, implemented in 2002, involves more space and geometry that the previous program. The main part of the work is to help pupils to identify important properties through problem-solving. Nearly no teacher does this in the classroom now. Passive observation or learning definitions should not substitute for this kind of work.

An example of the type of a problem solving activity in geometry that is given to prospective teachers includes the following:

Instruction

Determine all of the convex regular polyhedra
  1. Solve the problem; you may use manipulatives if necessary.
  2. Create a poster with results and justification. Show how you solved the problem and describe the difficulties you encountered, if any.
  3. Imagine a situation related to this problem for students between the ages of 7-8 and 9-10.

Frequent Answers and Questions arising from this activity include:

  1. There must be an infinite number of polyhedra, as there are for polygons.
  2. There are three polyhedra: one made of triangles, one made of squares, and one made of octagons.
  3. Is it really impossible to make a convex regular hexagon? If so, why?
  4. Some irregular polyhedra can be made of triangles and sometimes octohedra.
  5. The first investigation rarely included icosahedra (20 triangles).
  6. How can we be sure that we have found all of the possibilities?

Discussion arising from the Posters:

A theorem arises that states, "If you have six equilateral triangles with a common vertex and each shares an edge with at least one other triangle, then they necessarily give rise to a unique design."

In the real world, this works if the triangles used as manipulatives are made of a rigid material.

Consequences of this finding include:

  • No regular solid with hexagons can be constructed.
  • No regular solid with more than six triangles meeting at one vertex can be created.
  • No regular solid can be created if polygons with more than six sides are used.
  • It may be possible to construct a regular solid out of five triangles meeting at a vertex.
  • There is a need to continue with the activity, using manipulatives, in an effort to build an icosohedron. It is not obvious, but if no success is achieved, look for an explanation. Generally all groups succeed at the task.
  • Students agree that the theorem should be accepted and that at least three shapes are needed to form a vertex. As a result, they can conclude that there are exactly five regular convex solids-the Platonic Solids. Rigorous proof can be found in the "Euclidean Elements" section of Book X.

This problem concerns surface geometry, which-although it has to be taught in elementary school-is generally considered to be very difficult. Cubes and tetrahedra are used so everybody can explore the problem. You can then either make conjectures and control, or try again, or use both procedures. If you use plastic pieces, you can easily construct and deconstruct to explore the problem. Usually, nobody in the group knows an expert solution. To decide if the answer is correct or not, you need to assume a theorem that is validated during the process of solving. The problem phases clearly involve the process described by Brousseau: action, formulation, validation, and institutionalization.

This problem is interesting for several reasons. The first is that the mathematics is connected with real objects in a special way. We believe that mathematics is not a purely abstract matter. As a result, we think it is worthwhile to control the conjecture by building an object. We realize that even a simple problem may require more than "reflection" to solve. Action plays an important role. We are interested in geometry because it can be considered to be an experimental science. Euclidean geometry is a model that allows for predictions that can be tested in the real world. A wealth of geometrical knowledge that must be taught in French elementary schools involves Euclidean geometry. We believe that under certain circumstances, argumentation and proof are possible at the elementary level. Last, but not least we believe that mathematics is beautiful!

Promises and Challenges Related to the Approach in France

Seminar participants were pleased about the role that problem-solving plays in the program and with the high status, structure, and rigor of the problems used. However, there was some concern about the timing admission process. For example, in Japan, students have to decide whether or not to become teachers while they are still in high school. Also, in Brazil, the effort is to recruit prospective teachers before they complete their university degree. Participants were also concerned about the role that the exams play and about the relationship between the structure of the program and what happens in practice.

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