Transcript May 2004 - Extranet

The ‘reader’ and the ‘writer’ perspectives
–A tool for analysing students’ relationship to
algebraic expressions -
Caroline Bardini – Université Paris 7
May 21st 2004
Reader/ writer perspective – Historical background
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
Task #1
Algorithmical ‘translation’ of symbols
The author/ writer of an expression is guided by its ‘meaning’.
structur
e
Translate symbolically the author ‘will’ expressed
Task #2
in natural language
Task #1
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
Task #2
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
Reader/ writer perspective – Sliding the focus (1/2)
[5(2+x)]2
reader
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.
One should interpret assembled symbols as the execution of compound
instructions, that is as a sequence of elementary instructions, followed in a
very precise order.
Shift
Reader
To decipher an expression, starts with the
most
‘internal’
operating
signs
(‘weekests’) and progressively reconstructs the hierachy of the expression.
Writer
Guided by its ‘meaning’
Directly related to the sign that structures
the expression (the ‘strongest’ )
Focus on the operational signs involved in an expression and
their “status”, depending on the perspective adopted.
Reader/ writer perspective – Sliding the focus (2/2)
Looking at the order when one considers the operational signs
involved in an expression.
Descartes (1637) : z2
az+bb
5 (cross)
4(
1 .a + 1. a.a + b.b
4
2
)
2 (point)
3 (cross)
1 2 5 41 2 1 3 1
2 (point)
1 (point)
1 (line)
1 (point)
1
a
4
a
a
b
b
1 (line)
1
2
Analysing students responses (1/ 2)
Task
Student year 8
Completing the
writer perspective
Re-constructing an algebraic expression from
its description given in natural language
“The sum of the third of the square of a and b”
“ It’s the sum…of the third… then the sum.
So two expressions put…
Divided by 3 with a plus in the middle.
Then there is a squared and b squared ”
1
+
a2 + b2
Signifiant and syntax of 3 3
2
the sign
a2 linked together
b2
3
+
3
3
Step-by step builds the combinatory tree of the expression
4
Link each of the mathematical expressions listed bellow to the sentence you think
describes it best. If you chose « other(s) » for one mathematical expression,
write down your own sentence to describe it in the provided rectangle.
a and b are two non-zero numbers.
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) :
Analysing students responses (2/ 2)
(w-1) x (l-1)
« substract one from the width,
substract one from the lenght and
then multiply them together »
Operational/structural framework
« Take the width minus one
and multiply it by the lenght
minus one »
operational
2 (cross)
Reader/writer framework
Reader
1 (line)
w
1
1 (line)
1
l
Close to description of the formula, linear
Take into account not only the presence of
symbols in a description, but also the order
by which the different operators appear.
« (…) by doing width-1 x lenght-1
= amount of choc chips in a bar».