Uludag University, Bursa, Turkey

1. Democracy in Mathematics Courses Using Equivalent Definitions Dr. Menekşe Seden TAPAN Uludag University, Bursa, Turkey e-mail: tapan@uludag.edu.tr

2. DEMOCRACY and EDUCATION In a democratic society, education is inseparable from democracy education. The main purpose of education is to train people with the capacity to understand the world, to define the common good and to work towards more solidarity and tolerance between people and between peoples. How can we associate this training, this education to citizenship with mathematics teaching?

3. VERTICAL TEACHING vs. AUTOSOCIOCONSTRUCTION The vertical teaching (knowledge transfer from teacher to students) has its roots anchored very profoundly in the history of the pedagogy. The first didacticiens tried to study the behavior of the student, in particular to adapt support according to the type of teaching subject and to the categories of students while preserving this verticality.

4. VERTICAL TEACHING vs. AUTOSOCIOCONSTRUCTION But it is only very recently that didacticiens noticed that this verticality had its limits and that a change was necessary to improve the quality and the efficiency of the education. And so the integration of the democracy in education appears as an educational revolution.

5. VERTICAL TEACHING vs. AUTOSOCIOCONSTRUCTION There are no longer exclusively the archaic point of view “dominant (teacher) / dominated (students)”, but through enabling and facilitating the participation of everyone (e.g. oral participation which is a cornerstone of democracy that freedom of expression) and help the group move towards self-management and knowledge autosocioconstruction i.e. access to knowledge by himself and with others.

6. DEMOCRACY IN THE CLASSROOM We are going to look for the possibilities of integrating the democracy into a class of mathematics on the example of equivalent definitions in geometry. Before wondering about the way of integrating the democracy in the course of mathematics, it is advisable first of all to identify not the democracy but the forms of democracy. Indeed, we often have a too reducing thought towards the notion of democracy; we confuse it only in the formal democracy.

7. DIFFERENT FORMS OF DEMOCRACY • The democracy shows itself in 3 different manners. • The "formal" democracy, which legitimizes the political parties, bases on the right to vote for all, the freedom of expression and the division of the powers; • - The "fundamental" democracy bases the action of the State on the redistribution of the wealth and thus the equality of access to the education, the health and the culture;

8. DIFFERENT FORMS OF DEMOCRACY • The "substantial" democracy, which invest the social movements, militates for the fair sharing of the social production and the intensification of the civil society (BENGOA, on 1996) • So, the democracy can take various forms: direct, representative or participative (mediation and negotiation) (PARTOUNE, 1999).

9. DEMOCRACY IN THE CLASSROOM By considering its various forms, the democracy can be easily set up in a class of math thanks to the creation of work groups, by the active participation of students by the institution of a method of investigation and by the association of the usage of ICT (Web 2.0, numerical working environments etc.) with the mathematics.

10. DEMOCRACY IN THE CLASSROOM So, the learner’s freedom of expression, the status of the student passing from dominated to actor is a direct consequence of these actions which are, by definition, foundations of the formal democracy.

11. DEMOCRACY IN THE CLASSROOM The establishment of the democracy in maths courses can be made from the fact that in mathematics there is often no single truth but several truth; especially when defining an object.

12. EQUIVALENT DEFINITIONS In deed, researches in didactics show the importance of the definitions within the mathematical activity This essential place of the definitions appears from the axiomatic structure of the mathematics.

13. EQUIVALENT DEFINITIONS In any deductive science, there are primitive terms (not defined) and sentences accepted as they are, said axioms. Being given the primitive terms and the axioms, the adopted principle is " not to use an expression of the discipline on the condition that it is not determined by means of the primitive terms and by means of such expressions the meaning of which was previously explained. "

14. EQUIVALENT DEFINITIONS The sentence which determines the meaning of a term in this way is called a definition (Tarski, 1995, p. 118). The definitions and the construction of the definitions are so the base of the deductive structure of the mathematics.

15. EQUIVALENT DEFINITIONS We can distinguish two elements which are important concerning the definitions in geometry: a definition is mainly arbitrary (several alternative definitions can exist) and the conditions which form a definition must be minimal (does not contain superfluous information). So, in most of the time, there is a multitude possibility of defining a concept.

16. EQUIVALENT DEFINITIONS These two elements lean on a very important difference in the teaching of the geometry which is the difference between " to understand what is said in a definition " and " to understand how the elements of a definition are connected one to the others ".

17. EQUIVALENT DEFINITIONS It is essential that teachers become aware of the fact that there can exist several alternative definitions for the same concept. Because, if the teachers do not understand the various conventions and the consequences of each alternative definitions, they will be limited to guide the learning of their pupils.

18. EQUIVALENT DEFINITIONS • So a teacher must at least be capable of: • Recognizing the difference between a definition and a list of the properties, • Analyzing the various alternative definitions by taking into account how their elements are connected • - Excluding the superfluous properties in a definition.

19. EXAMPLE for DEMOCRACY IN THE CLASSROOM After a brief look to the importance of equivalent definition in the axiomatic structure of mathematics, lets return to the question of integrating democracy in maths courses. An example of the possibility of the integration of the democracy in mathematics teaching is described in the following.

20. EXAMPLE for DEMOCRACY IN THE CLASSROOM In a mathematics course, the teacher asks the students gathered in small groups to write a definition of the rectangle. A representative of every group is going to present the definition of the rectangle so written. The student so speaks either that on his behalf but in the name of the group ( representative democracy).

21. EXAMPLE for DEMOCRACY IN THE CLASSROOM Indeed, let’s take the case where three groups arrive to three different definitions for the rectangle:

22. EXAMPLE for DEMOCRACY IN THE CLASSROOM Definition 1: Rectangle is a quadrangle with the set of opposite sides equals and with a right angle Definition 2: Rectangle is a quadrangle with the set of opposite sides parallels and with a right angle Definition 3: Rectangle is a parallelogram with a right angle Although different, all three of these definitions are true.

23. EXAMPLE for DEMOCRACY IN THE CLASSROOM However the acceptance of the definition of a group pass at first by the listening of the other one while relying on the person delegated for the representation. The learner thus learns to analyze an assertion (information) and to criticize it. This process develops capacities to build up a self-opinion through the dialogue with the others, to learn to use various means to express clearly an opinion or a request (direct democracy).

24. EXAMPLE for DEMOCRACY IN THE CLASSROOM After the phase of negotiation and mediation (participative democracy), students can arrive to the established fact that all three definitions are correct without authoritarian intervention of the teacher.

25. DEMOCRACY IN THE CLASSROOM The progress described shows that in mathematics, contrary to the common ideas, there is not only a single truth. Thus the mathematics course is a privilege and convenient place for the education to the notion of tolerance, respect for the thought of others, the capacity to listen to others with a critical and open glance at the same time. This so-called tolerance requires the non existence of prejudgments (PARTOUNE, 2005). In contrast, the classroom must maintain a laïc character to not confuse with the notion of tolerance.

26. TO YOU…… Write at least two definitions for the parallelogram… The parallelogram… is a rectangle without right angle…