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

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

4. Overall view (2/3) 3. Deepen the theoretical and philosophical framework. Articulation between philosophy and mathematics Mathematical notation G.G. Granger ( ‘94) J. Vuillemin ( ‘62) C. Babbage (1821) D. André ( ‘09)F. Cajori ( ‘28) M. Dascal ( ‘78) Retrospective and epistemological study about constitution of symbolic language M. Serfati (1997) 4. Collate educational research to epistemological work Resonances between epistemology and didactics. Highlight some students errors / another point of vue of students behaviors. ? On the influence of signs in Mathematical Reasoning 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

5. Overall view (3/3) 5. Epistemology – An essential tool in the experimental part of the work Designing tasksAnalysing students responsesA.I. ( x -1) x ( y -1) « 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» Year 8 (13 yrs) Year 10 (15 yrs) « (…) 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 ». 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. « 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. »

6. 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) Duval Treatment/conversion Arzarello Algebraic sense/ Contextualised sense representation Drouhard Connotation/interpretation Let a, b, c the medians of a triangle. The intersecting point of a and b is the same as the one of b and c. 4’ and ‘8/2’ have the same denotation but express different ways of conceiving the same number. (1901) The expressions ‘Melbourne’ and ‘the australian capital’ have different senses but denote the same city.

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

8. V- Sense and denotation and Drouhard’s work Interpretationof an algebraic expression X within a given framework is every object that ‘corresponds’ to the denotation of X within this framework. 3x+7 Graphical framework : line y = 3x+7 Arithmetical: the writing of a number congru to 7 modulo 3 Connotation is the subjective perception someone has of an algebraic expression (a+b) 2 = a 2 + b 2 Students ignore that denotation keeps unchanged under algebraic transformations a = 2, b =3 2513 « We don’t get the same value? That isn’t surprising! We haven’t done the same thing! » sense over denotation Primacy of sense over denotation connotation over denotation Primacy of connotation over denotation

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