Caroline Bardini – Université Paris 7 April 16th 2004

1. Establishing bonds between epistemology and mathematics education - towards a better understanding of one’s relationship to algebraic symbolism Caroline Bardini – Université Paris 7 April 16th 2004

2. René Magritte, The human condition I (1933) I - The core • Frailty with which some students carry out algebraic tasks as factorizing. A task accomplished blindly, where some « unquestionable » rules take place, transforming remaining meaningless algebraic expressions. • How do students percieve the expressions they manipulate? How do they perceive its different constituent? What is hidden, for students, behind a symbol?

3. II- Overall view (1/3) 1. Mathematics education research within the « early algebra » field Different uses of history of algebra Modelling the different stages through which mathematical concepts are built Designing /analysing teaching sequences Sfard Kieran Harper Radford 2. Mathematics education research more specifically related to the study of algebraic symbols Tinged with semiotics and rely on sense and denotation (Frege-1892) philosophical concepts Duval Drouhard Arzarello

4. ? Stacey & MacGregor. TRIANGLE problem The perimeter of this triangle is 44 cm. Write an algebraic equation and work out x. Students: x =30 2x cm x cm 14 cm Overall view (2/3) 3. Deepen the theoretical and philosophical framework. Articulation between philosophy and mathematics J. Vuillemin ( ‘62) Retrospective and epistemological study about constitution of symbolic language G.G. Granger ( ‘94) Mathematical notation C. Babbage (1821) M. Dascal ( ‘78) F. Cajori ( ‘28) D. André ( ‘09) M. Serfati (1997) On the influence of signs in Mathematical Reasoning 4. Collate educational research to epistemological work Resonances between epistemology and didactics. Highlight some students errors / another point of vue of students behaviors.

5. Overall view (3/3) 5. Epistemology – An essential tool in the experimental part of the work Designing tasks Analysing students responses A.I. « (…) by doing the amount of hazelnuts on width-1 x amount of hazelnuts on lenght-1 = amount of chocolat chips contained in a chocolat bar ». (x -1) x (y -1) Year 8 (13 yrs) Year 10 (15 yrs)    « We can know the amount of chocolat chips by substracting one to the width of the chocolat bar and by doing the same with the lenght and then multiplying them  together» « You just have to take the amount of hazelnuts in length minus one hazelnuts for the edge and multiply by the amount of hazelnuts in width minus one for the edge. » Intermediate description between rhetorics and symbolic that takes into account not only the presence of symbols but also the order by which the different operators appear.

6. Duval Arzarello Drouhard Algebraic sense/ Contextualised sense Connotation/ interpretation Treatment/conversion III – Sense and denotation Sense and denotation – Frege (1892) Sign: every way used to designate something, having the same status as a name (word, various caracters, signs). Sense: the way (mean) through which the sign is given to us Denotation: what the sign designates (= object itself) Let a, b, c the medians of a triangle. The intersectingpoint of a and b is the same as the one of b and c. (1901) The expressions ‘Melbourne’ and ‘the australian capital’ have different senses but denote the same city. 4’ and ‘8/2’ have the same denotation but express different ways of conceiving the same number. representation

7. Transforming expressions originally set up in natural language into objects belonging to the same field. Decimals and fractions: two different representation systems. Students know how to add decimal and fractions (= treatments), can’t shift from one representation to another (=conversion) 0,25+0,25 = 0,5 ¼ + ¼ = ½ Frege – substituting words/group of words with same senses, different senses, same denotation, etc. IV- Sense and denotation and Duval’s work Eg.: Calculating = substituting new expressions to given expressions within the same s.system (i.e. writing numbers). Changing representations of an object within the same semiotic system. Internal transformation Treatment Duval Eg.: Formulating a problem given in natural language into equations. Transpose the representation of an information/ object/ situation into a different system. External transformation. Conversion Sense, denotation and Treatment Sense, denotation and Conversion 0,25 and ¼ : different senses, one denotation. Eg: if 0,25 and ¼ are not seen as refering to the same object, one cannot be thought as the substituent of the other. Therefore conversion cannot be conceived Different treatment procedures Distinguishing sense from denotation is essential to conversion. Sense comands the treatment procedure.

8. Graphical framework : line y = 3x+7 3x+7 Arithmetical: the writing of a number congru to 7 modulo 3 V- Sense and denotation and Drouhard’s work Interpretation of an algebraic expression X within a given framework is every object that ‘corresponds’ to the denotation of X within this framework. Connotation is the subjective perception someone has of an algebraic expression (a+b)2 = a2 + b2 a = 2, b =3 25 13 Students ignore that denotation keeps unchanged under algebraic transformations « We don’t get the same value? That isn’t surprising! We haven’t done the same thing! » Primacy of sense over denotation Primacy of connotation over denotation

9. n(n+1) (x+5) = x n2+n x2+x+1 = 0 mapping A={0,2,6,...} denotation n(n+1) n2 + n algebraic sense product of two consecutive nbrs. Rectangle area contextualised sense VI - Sense and denotation and Arzarello’s work Algebraic sense: the way by which the expressions are given, trhough different rules. Contextualised sense: Formulas express different thoughts, with respect to the different contexts they are used. Elementary number theory : product of 2 consectutive nbrs n(n+1) Geometry: Surface of a rectangle which sides are n and n+1