1. The ‘reader’ and the ‘writer’ perspectives - Two conceptual ways of approaching an algebraic expression Caroline Bardini – Université Paris 7 May 7 th 2004

2. Historical and epistemological background (1/4) 1526 Widmann Add the number 30 to the number 3 Substract the number 17 from the number 4 1608 Clavius 1 – 7 From the value of the unknown, deduct the number 7 Author’s intention: Provide the reader the symbolical depiction of an elementary instruction A rule describing the execution of an operation (eg. substraction) where two quantities (numerically given or not) are involved.

3. Rhetorical interpretation extended to more ‘complex’ formulae: Historical and epistemological background (2/4) (2+x) x 3.5 Add the integer represented by the symbol ‘2’ to the unknown number which symbol is ‘x’. Then multiply the prior result to the number which symbol is ‘3.5’. One should interpret (‘well formed’) assembled symbols as the execution of compound instructions, that is as a sequence of elementary instructions, followed in a very precise order. Reader Algorithmical ‘translation’ of symbols suggested by Widmann and adopted by his sucessors.

4. The following instructions - take a number x - multiply it by 2 - substract 5 from the result - take the square root out of the result - add 3 to the result constitute an algorithm by the end of which we obtain the formulae: Find out algorithms leading to each of the following expressions : a) [5(2+x)] 2 b) +2 c) [2(-x+3)] 2 Exercise #1 (reader perspective)

5. Historical and epistemological background (3/4) Reader: To decipher an expression, starts with the most ‘internal’ operating signs ( weekest ) and progressively re- construct the hierachy of the expression. Synthetical approach [5(2+x)] 2 Add the number represented by the sign ‘2’ to the number which sign is x. Multiply the result by the number represented by the sign ‘5’. Square the last result. Whereas the starting point of the deciphration of a symbolic expression carried out by the reader is that of interpret the most internal operational signs (e.g. sum) the major will of the author (writer) is to represent, by the means of symbols, a square. Underpinning sign 3.5 x (2+x)

6. Whereas the author (writer) of an expression is guided by its ‘meaning’ Directly related to the ‘strongest’ sign, the one that structures the expression Analytical approach Historical and epistemological background (4/4) The reader tackles the expression through the most internal sign (‘ weekest ’) Ex.1Ex.2 Synthetical approach

7. Exercise #2 Pupil supposed to play the role of the writer of an expression, i.e. translate symbolically the author ‘will’ expressed in natural language. Translate the following sentences onto algebraic expressions: a) The double of the square of a b) The sum of the square of 5 and the double of a c) The difference between 3 and the product of 7 by x d) The square of the sum of 7 and x e) The ratio of the sum of 3 and a and the difference between b and 8 Goal

8. Exercise #3 Not only the writer’s ‘will’ (expressed in natural language) is given, but so is the algebraic expression In theory, both related to the writer’s perspective Link each of the mathematical expressions listed bellow to the sentence you think describes it best. a and b are two non-zero numbers. (Fill Others…) A= 1 + 1 :n°1 : The inverse of the square of the sum of a and b a 2 b 2 B= 1 :n°2 : The sum of the inverses of the squares of a and b a 2 + b 2 C= 1 :n°3 : The square of the sum of the inverses of a and b (a + b) 2 n°4 : Other(s) :

9. Reader/ writer perspectives Design/analyse tasks Analyse students responses Coming next… « substract one from the width, substract one from the lenght and then multiply them together » « Take the width minus one and multiply it by the lenght minus one » (w-1) x (l-1)