Transcript May 2004 - Extranet

The ‘reader’ and the ‘writer’ perspectives Two conceptual ways of approaching an
algebraic expression
Caroline Bardini – Université Paris 7
May 7th 2004
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.
Historical and epistemological background (2/4)
Rhetorical interpretation extended to more ‘complex’ formulae:
(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.
Algorithmical ‘translation’ of symbols suggested by Widmann and
adopted by his sucessors.
Exercise #1 (reader perspective)
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)
c)
3
1
x
+2
[2(-x+3)]2
2x  5
Historical and epistemological background (3/4)
Reader: To decipher an expression, starts with the most
‘internal’ operating signs (weekest) and progressively reconstruct the hierachy of the expression.
[5(2+x)]2
3.5 x (2+x)
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.
carried out by the reader
Whereas the starting point of the
deciphration of a symbolic expression
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
Historical and epistemological background (4/4)
Whereas the author (writer) of an expression is
guided by its ‘meaning’
Directly related to the ‘strongest’ sign, the one that structures the expression
The reader tackles the expression through the most
internal sign (‘weekest’)
Ex.1
Ex.2
Exercise #2
Goal
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
Exercise #3
In theory, both related to
the writer’s perspective
Not only the writer’s ‘will’ (expressed in natural
language) is given, but so is the algebraic expression
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 :
a2 b2
n°1 : The inverse of the square of the sum of a and b
B=
1 :
a2 + b2
n°2 : The sum of the inverses of the squares of a and b
C=
1
:
(a + b)2
n°3 : The square of the sum of the inverses of a and b
n°4 : Other(s) :
Reader/ writer perspectives
Analyse students responses
Design/analyse tasks
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)