The School University Interface

1. The School University Interface Michèle Artigue LDAR, Université Paris Diderot – Paris 7 Auckland, Avril 2010

2. The School University Interface: a research perspective • An accumulation of research results, and a lot of associated experiments and innovative projects • But a diversity of approaches and changing contexts making it difficult to get a coherent and synthetic view of what we really know, and of how research results can productively guide educational action • An ambition of the ICMI Study Teaching and learning mathematics at university level, whose resulting book (Holton, 2001) is still of great interest, despite the evolution of context.

3. A diversity of perspectives Epistemological and cognitive perspectives Institutional and cultural perspectives

4. Cognitive and epistemological perspectives Discontinuities / flexibilities

5. Cognitive and epistemological perspectives: discontinuities • A diversity of approaches respectively based on: • the distinction between three mathematical worlds: Embodied, Proceptual and Formal (Tall…) • the notion of Formalizing, Unifying and Generalizing concept (FUG) (Robert, Dorier…) • the notion of epistemological obstacle (Cornu, Sierpinska, Schneider…) • Algebra / Analysis discontinuity (Legrand, Artigue…) • Not necessarily a radical change today in the secondary / university transition, but nevertheless substantial changes

6. Epistemological obstacles: the case of the limit concept • A notion due to Gaston Bachelard and imported in the didactical field by Guy Brousseau • Resistant errors do not result from a lack of knowledge but from forms of knowledge that have proved to be efficient in other contexts • The case of the limit concept, diverse categorizations but evident commonalities: • Beliefs about the nature of mathematics and the status of infinite processes • Common meaning of the notion as an inaccessible barrier or an extreme point • Abusive generalization of properties of finite processes to infinite processes (in accordance with the continuity principle stated by Leibniz) • Geometrical obstacle (when geometrical evidence opposes to analytical reasoning)

7. An example: computing the area of the sphere Why this process does not give the expected answer: 4R²? Computing the area of the sphere using a similar process to that used for computing its volume A=²R² The historical evidence (cf. Lebesgue speaking about Schwarz counter-example in La mesure des grandeurs)

8. Algebra / Analysis discontinuities From an equality symbol expressing the equivalence of expressions to an equality symbol expressing arbitrary level of closeness: A= B  >0 d(A,B)< The predominant role taken by inequalities over equalities From global perspectives to the articulation of local and global perspectives Reasoning based on equivalence / on the management of sufficient conditions, learning to loose information in a controlled way, taking into account both orders of magnitude and the local character of reasoning in Analysis

9. Cognitive and epistemological perspectives: connections and flexibility • Acknowledging the essential role played in conceptualization by connections between contexts, mathematical settings, semiotic registers, points of view… • An evolution supported by: • the increasing attention paid to the semiotic dimension of mathematical activity in educational research, • the technological evolution and its specific semiotic affordances. • Once more substantial changes at the secondary / university interface

10. The case of linear algebra (Hillel, Sierpinska, Alves Dias…) Different languages: - Geometrical - Algebraic - Abstract • Different registers: • - Graphical register • Table register • Algebraic register • - Symbolic register • Different points of view : • Cartesian • Parametric • Different modes of reasoning: • Synthetic-geometric • Analytic-arithmetic • Analytic-structural

11. Institutional and cultural perspectives

12. An institutional perspective on transition issues • Transition processes are always problematic due to: • the changes in norms and values regarding mathematical knowledge and practices from one institution to another, • the implicit character of most of these norms and values, and the way they are conveyed, • from the contextualized dimension of an important part of students’ mathematics knowledge, and the resulting issues in terms of didactical memory.

13. Institutional and cultural approaches • Diverse theoretical frameworks but commonalities: • mathematical objects emerge from human mathematical practices which are institutionally situated, • in order to understand students’ learning processes, one must first understand the institutional practices that shape these explicitly and implicitly, and the associated systems of norms and values which remain partly tacit. • The attention moves from the student to the system, and the vision of complexity changes similarly.

14. A pioneering research: Praslon’s PHD • Investigating the exact nature of the secondary-university transition in France regarding the notion of derivative and its environment • A macro-theoretical frame: the anthropological theory of didactics (ATD) (Chevallard) complemented by the affordances of existing research on Calculus and Analysis • A methodology combining the analysis of didactic material, examination texts, tests posed to students entering university math programs, regular workshops with students

15. The main results • A substantial universe around the notion of derivative already at the end of high school as attested by concept maps, but a huge increase of the landscape in the first six months at university. • Not a radical move from the proceptual to the formal world, from an intuitive and algorithmic Calculus to the approximation world of Analysis, rather an accumulation of micro-breaches less visible and not appropriately taken in charge by the institution.

16. An accumulation of micro-breaches • An accumulation of micro-breaches: • an increasing speed in the introduction of new objects • a greater diversity of tasks • much more autonomy given in the solving process • a new balance between the particular and the general, the tool and object dimensions of mathematical concepts • objects more controlled by definitions, results more systematically proved, and proofs which are no longer “the cherry on the cake” but take the status of mathematical methods • Resulting in a didactic gap students are asked to fill by themselves essentially

17. A task in the didactic gap • Let us consider the periodic function f withperiod 1 defined by f(x)=x.(1-x) on [0, 1[ (the graph on [-3,3] is given) • Q1 : Is this function continuous? Differentiable? • Q2 : The notion of symmetric derivative is formally introduced and students are asked to compute the derivatives and symmetric derivatives of f, if they exist, at points ½, ¼ and 0, and to compare these. • Q3 : Students have to say if the following three conjectures are true or false and justify their answers: • Every even function defined on IR has a symmetric derivative at 0. • Every even function defined on IR has a derivative at 0. • If a function defined on IR has a derivative for x=a, it has also a symmetric derivative, and the two are equal.

19. What do the students do? • Q1 : 1/3 of the students do not see the problem, 1/4 answer that f does not have a derivative at 0. • Q1 : There is no connection between the algebraic and the graphical registers • Q2 : Students are not afraid by the formal definition of the symmetric derivative and are able to exploit it correctly in simple cases, but they fall into the trap (point 0), even those who had correctly answered Q1 • Q3 : Few answers. The absolute value function is the favoured counter-example Students who don’t refuse to work on definitions, on general statements, but are poorly equipped for such a mathematical work

20. Some more recent contributions • Berge’s thesis on the evolution of institutional relationships to the notion of completeness of IR along university courses, and their impact on students’ personal relationships: each course functions as a specific and isolated institution • Research on the management of quantifiers in university textbooks and courses: an evident lack of coherence in practices (Arsac, Durand-Guerrier & Chellougui ) • Bosch, Fonseca, Gascón research on limits: the existing gap between the rigid, punctual and praxis based high school praxelogies on the one hand and the regional and theory based praxeologies of the university • And also research on students’ forms of personal work (Castela, Lithner), on university teachers’ expectations and assessment modes (Gueudet), and practices (Nardi)

21. From understanding to action • Didactic research shows evident regularities and coherent patterns BUT • Only local solutions, and experiments and innovations indeed reflect: • the diversity of epistemological choices and didactical approaches • the diversity of institutional means and institutional constraints • the diversity of educational contexts and cultures • the diversity of the visions developed about technology and its potential role

22. Ten years ago: the ICMI Study Already, many interesting projects: • The « Analysis-project » at the University of Warwick • The « Scientific Debate » at the University of Grenoble I • Teaching through projects at the University of Roskilde • The « Active/Interactive Classroom » at Duke University

23. An evident diversity but also common trends in successful experiences • The design of long terms projects in an iterative way and the attention given at making these ecologically viable. • Projects sensitive to the mathematical culture of students entering tertiary education and trying to make sense of it. • Projects trying to make students feel responsible and develop their autonomy. • Projects trying to develop assessment methods in accordance with their foundational principles. • Projects which try more and more to rely on existing research for better understanding students’ behaviour and difficulties and design appropriate didactic strategies.

24. And also, as a leitmotiv are stressed the following points • The human cost of the enterprise. • The fact that it cannot be achieved in a sustained way by isolated persons but requires team work. • The necessity of a clear institutional support, and of some institutional acknowledgment for those who strongly engage in such projects. • The fact that the ordinary training of university mathematicians poorly prepare them to the difficult problems they have more and more to face.

25. And, since that time… • More and more actions directed towards high school students and their teachers. • Actions aiming at the professional development of university teachers(Jaworski, Nardi…). • The increasing role given to technology for supporting communication and exchanges between students, teachers and students, promoting collaborative work, accompanying students’ personal work, and extending the range of educational resources at the disposal of students and teachers.