Math and math education : A vision of its evolution

1. Math and math education : A vision of its evolution Ricardo Cantoral and Rosa María Farfán TA-C, ICME 10 Cinvestav IPN - Mexico

2. Aims and focus, TA-C • How do new developments in Mathematics influence the teaching of mathematics? • How are teachers in mathematics trained in Mathematics? • How can mathematicians and educators collaborate to construct better curricula and improve teaching methods? (Final programme of ICME – 10, p. 103)

3. Looking towards the future • Testimonies : Some examples of attempts. • Plans : Constructing web sites, cooperative events, getting users of mathematics to testify about their etc. • New ideas : Understand the complex relationship between Mathematics and mathematics education and construct a vision of its evolution.

4. How math influence to teaching math Mathematics Teaching ofmathematics

6. We are using some references: • Cantoral, R. and Farfán, R. (2003). Mathematics education: a vision of its evolution. Educational Studies in Mathematics53 (3). • Cantoral, R. et Farfán, R. (2004). Sur la sensibilité a le contradiction en mathématiques et l’origine de l’analyse complexe. Recherches en Didactique des Mathématiques24 (3).

7. We are using some reference: • Cantoral, R. e Ferrari, M. (2004). Uno studio socioepistemologico della previsione. La matematica e la sua didattica 2. • de Guzmán, M. El papel del matemático frente a los problemas de la educación matemática. Spain: Complutense de Madrid, 1993. • Holton, D (ed.). The teaching and learning of mathematics at university level. The Netherlands: Kluwer Academic Publishers, 2001.

8. the fourth International Congress of Mathematicians… Rome, 1908 • The Congress, recognizing the importance of a comparative study on the methods and plans of teaching mathematics at secondary schools, charges Professors F. Klein, G. Greenhill, and Henri Fehr to constitute and International Commission to study these questions and to present a report to the next Congress (Lehto, 1998, p. 13)

9. A Call for ChangeMathematical Association of America • On the mathematical preparation of teachers… • “The mathematical experiences recommended for teachers at the K-4 level require that mathematics departments offer courses specifically designed for these audience.” (Leitzel, 1991, p. 11)

10. So, we prefer talk about … • The dialogue of mathematics education research “with other scientific communities in particular the mathematics research community”…as Sierpinska and Kilpatrick said in 1998; it was one of the issues raised at the outset of recent ICMI Study on research in mathematics education. (Hodgson, B. 2001, p. 516).

11. … are connected … Research in Mathematics Education Mathematics instruction Research in Mathematics

12. Aims and focus, TA-C • How do new developments in Mathematics Education and Mathematics influence the teaching of mathematics? • How are teachers in mathematics trained in Mathematics Educationand Mathematics? • How can one create community between mathematicians, mathematics educators and mathematics teachers, to construct curricula, textbooks and improve mathematics teaching?

13. From teaching to research … • Collatz’ conjecture  n  IN ,  k  IN   k (n) = 1 • Infinitesimal models for teaching calculus Let be i a formal expression, i is an positive infinitesimal if  x  IR+ it follows 0 < i < x

14. Our problematic… • we will deal with shall be those relating to the evolution of the study of educational phenomena that take place when mathematical knowledge, socially produced outside of school environments, is introduced into the educational system, forcing it to undergo a series of modifications that directly affects both its structure and functionality.

15. This process of incorporating highly specialized knowledge into the educational system creates a series of non-trivial theoretical and practical problems, which require methodological approaches and suitable theorists … will allow us to understand the mechanisms for the adaptation of mathematical and scientific knowledge into practice both for teachers as well as students.

16. We shall present a serie of examples that demonstrate the evolution at different times, which we have called: • “didactics” without students • “didactics” without school • “didactics” without sociocultural settings and • “didactics” without … “didactics as pedagogical approaches ”

17. Didactics without students • The classic problematic in mathematics education was the designing of presentations with mathematical content for schools considered more accessible to students and teachers than those so-called “traditional presentations”. It was assumed that a presentation better adapted to schools and their employees could only be created by means of reflection by mathematics professionals.

18. Didactics withoutstudents • … textbooks and educational materials were produced, which systematically failed to take into consideration other factors such as those of a cognitive or emotional nature or those relating to the sociocultural issues of knowledge. Instead, they sought to produce that which the school ought to use, without carrying out an in-depth study of school culture.

19. Traditional methods…  r2 b  h

20. A Didactics without students • Example: Students were offered various learning activities in order to estimate the value of a given area, such as the area covered by the following

21. 3 cm. 6 cm. Didactics without students • The introduction of a cover made up of elements, of which the area is known, was proposed. For example, a rectangle with sides of 3 by 6 cm. • Then if the area sought is denoted as A cm2, it thus fulfills the relationship 0  A  18. A

22. Didactics without students A 4  A  18 A more refined approximation

23. Didactics without students A 4  A  18 A more refined approximation a1 A b1 a1a2 A b2b1 a1a2a3 A b3b2b1 a1a2a3a4 A b4b3b2b1

24. Didactics without students During this procedure, the student is not in charge of the learning process, but only of its execution. The new question was: how the people learn mathematics?

25. Didacticswithoutstudents • … it can be seen that, mathematically, the limit of the sequences an and bnis, in both cases, A, so the approximation process would lead, by a kind of educational sensualism, to students being convinced that such limit exists and that their conceptions of the area and what its representation through approximations

26. Didactics without school • In the 1980s, an action program was presented at the ICME – 4 around which our discipline gradually developed. It was based on approaches such as that of Professor Freudenthal who presented questions such: • How do people learn? • How can we learn to observe learning processes?

27. Didactics without school • … this led to a new paradigm of research that modified its purpose and method of study. This has led to a cognitive approach to investigation with the systematic observation and description of the achievements of students and various learning experiences.

28. Didactics without school One of the classic examples in research on teaching and learning of Calculus, consisted of explore the answers for two questions on a single sheet given to students finishing their high school diploma or starting university, which would lead to contradictory mathematical answers without this contradiction being noticed by the students:

29. Didactics without school a) Comparethe numbers 1 and 0.999 Regular answer 0.999...  1 b) Calculate the sum of the series Regular answer 1

30. where is (x) > 0?

31. where is (x) > 0?

32. + + + where is (x) > 0?

33. where is (x) > 0? ?

34. Didactics without sociocultural settings In a research project, we reported that sought to express the concept of convergence of infinite series, making use of new educational approaches among university professors, in order to find the association of the notion of convergence with the scientific study of conduction of heat.

35. Didactics in school, but without the school sociocultural settings The phenomenon of heat conduction was an issue dealt with both by Rational Mechanics and Mathematical Analysis during the eighteenth century and for which, at that time, no definitive answer was found.

36. Didactics in school, but without the school sociocultural settings Definition (Marsden, 1974, p. 47) A serie infinite ∑ Xkwhere Xk Rn, converges to x  Rn If the sequence of partial sums Converge to x and write Convergence of sequences (opcit, p. 44) Teo. A sequence Xk en Rn converges to x Rn if for every > 0 exists N  k  N implies  Xk- x < 

37. Didactics in school, but without the school sociocultural settings

38. Temperatures on AB line, at time t0 T Temp t = t0 x A B Radial Position

39. Didactics in sociocultural settings The prediction idea as a fundamental tool for understanding variation.Newton’s binomial expression is first written as : and not as This notion of prediction is socially constructed from the daily experiences of individuals, since in certain situations we need to know the value that a magnitude will acquire with time.

40. Didactics in sociocultural settings In our opinion, these findings favor the discussion and preparation of proposals for teaching that deal with what should be taught and not only, as has been customary, with how it should be taught. In summary, the purpose of our research is to study that which is socioepistemological in mathematical knowledge and includes the primary intuitions of the student in order to redesign the scholar mathematical discourse.

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42. Cantoral, R. (2000). El futuro del cálculo infinitesimal. ICME 8 Sevilla. Mexico: GEI. Cantoral, R. (1997). An example of the Sociological Point of View in Math Education: The Case of Analytical Functions at the University Level. Principal speaker, Conference on Research in Mathematics Education. MSU, USA. Cantoral, R. and Farfán, R. (1998). Pensamiento y lenguaje variacional en la introducción al análisis. Épsilon 42, 353 – 369.

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