1 gattegno’s mathematizing – becoming aware of dynamics of mathematics by working on imagery,...
TRANSCRIPT
1
Gattegno’s Mathematizing – becoming aware of dynamics of mathematics by working on imagery, equivalence and substitution
Els De Geest
2
‘Only awareness is educable’
3
‘Only awareness is educable’
In his book “The awareness of mathematization” (1988) he describes awareness of mathematics as the dynamics of (mathematical) relationships.
Gattegno’s concept of mathematization: becoming aware of dynamics of mathematical relationships.
4
How?
He uses mental activities to work on becoming aware of these dynamics of mathematical relationships.
5
Imagery – what is it and why would you bother?
Your thoughts?
6
Imagery – what is it and why would you bother?
There are great advantages in concentrating on mental activities which deliberately embrace
many awarenesses and which can lead to new awarenesses.
(Gattegno, 1988, p89)
7
Imagery – what is it and why would you bother?
Because images are dependent on our will, once we begin deliberately to employ them, we can very soon obtain an awareness that indeed imagery is a power of the mind, and it can yield
in a short time vast amounts of insights into fields that become almost sterile when
dynamics are removed from them (Gattegno, 1987, p35).
8
Imagery – what is it and why would you bother?
Imagery is present at will and can remain present while the mind is at work on it or on
some element within it.
(Gattegno, 1987, p35).
9
Imagery - task
Imagine a lemon…
Adapted from Sandy Dawson(1988)
10
Imagery – task: the square
Imagine a square…
11
Equivalence ~
Read the sign ~ as “equivalent to” or “is another way of saying”
(Gattegno, 1988, p22)
12
Equivalence tasks
Compliments in 100 (Gattegno, 1988, p26)
(0, 100) ~ (1, 99) ~ (2, 98) ~ (3, 97) ~ …
(10, 90) ~ (11, 89) ~ (12, 88) ~ (13, 87) ~ …
(20, 80) ~ (21, 79) ~ (22, 78) ~ …
What mathematical awarenesses are you working on?
Can you develop an image for these? Any others?
Any sense of dynamics?
13
Equivalence tasks
Additions (Gattegno, 1988, p28)
10~ 1+9 ~ 2+8 ~ …
~ 9+1 ~ 8+2 ~ …
What mathematical awarenesses are you working on?
Can you develop an image for these? Any others?
Any sense of dynamics?
14
Equivalence tasks
Subtractions (Gattegno, 1988, p35)
1 ~ 2-1 ~ 3-2 ~ 4-3 ~ …~ 98-97 ~…~ 1754-1753 ~ …
2 ~ 3-1 ~ 4-2 ~ 5-3 ~ …~ 99-97 ~…~ 1755-1753 ~ …
3 ~ …
…
What mathematical awarenesses are you working on?
Can you develop an image for these? Any others?
Any sense of dynamics?
15
Equivalence tasks
Transformations in subtraction (Gattegno, 1988, p38)
2031-784 ~ 2247-1000 (both +126)
~ 1247
2031-784 ~ 2000-753 (both -31)
~ 1247
What mathematical awarenesses are you working on?
Can you develop an image for these? Any others?
Any sense of dynamics?
16
Equivalence - tasks
Fractions (Gattegno, 1988, p83)
1 ~ 1/1 ~ 2/2 ~ 3/3 ~ … ~ n/n
What mathematical awarenesses are you working on? Can you develop an image for these? Any others? Any sense of dynamics?
What is the effect on you and your learning of using equivalence sign and not equal sign?
17
Equivalence - tasks
Fractions (Gattegno, 1988, p79)
a/b ~ 2a/2b ~ 3a/3b … Na/Nb…
What mathematical awarenesses are you working on? Can you develop an image for these? Any others? Any sense of dynamics?
What is the effect on you and your learning of using equivalence sign and not equal sign?
18
Equivalence - tasks
Fractions (Gattegno, 1988, p88)
m/n ~ m/a × a/n
~ m/a × a/b × b/n
~ m/a × a/b × b/c × c/n …
What mathematical awarenesses are you working on? Can you develop an image for these? Any others? Any sense of dynamics?
What is the effect on you and your learning of using equivalence sign and not equal sign?
19
Equivalence and substitution
"The notion of equivalence goes hand in hand with the important mathematical idea of
replacement. If two expressions are equivalent
then one may be used to replace the other at any time”
(Collis, 1975, p.17 – thanks to Ian Jones to alert me to this)
20
Substitution tasks – Old Mac Donald and his farm
Substitute what Old MacDonald had on his farm.
Are there limitations?
Any imagery?
What mathematical awarenesses are you working on here?
21
Substitution – exploring limits of properties of relationships
Which one is the odd one out: elephant, parrot, dog, cow. Why?
What mathematical awarenesses are you working on?
22
Substitution – more maths tasks
Look at the expressions:
10 5+5 13-3
√ (100) 1x10 10.00
Which one is the odd one out? Why?
What mathematical awarenesses are you working on?
23
Substitution – more maths tasks
10 ~ 1 + 9 ~ 2 + 8 ~ ? ~ ?....
~ 1 + 1 + 8 ~ …
Make up some extreme examples of your own.
What mathematical awarenesses are you working on here? Any imagery? Any dynamics?
24
Some more tasks…
1) ‘Clouding the picture’ task
2) Consider the graphs of the expression y = x2
Substitute (2x) or (x + 3) or … for x
Substitute (3y) or (y-2) or … for y.
What mathematical awarenesses are you working on? Mathematical dynamics? Any imagery?
Any other tasks that come to mind?
25
Adapted Ian Jones (2008) task
In a world in which 15+28 ~ 44, what can be
said about 15+29?
26
Looking back ….
“Only awareness is educable”
Gattegno’s concept of mathematization: becoming aware of dynamics of mathematical relationships.
G. uses mental activities to work on becoming aware of these dynamics of mathematical relationships.
Does it make more sense?
mathematics can be seen as the study of actions performed on objects.
27
Looking back ….
Because images are dependent on our will, once we begin deliberately to employ them, we can very soon obtain an awareness that indeed imagery is a power of the mind, and it can yield
in a short time vast amounts of insights into fields that become almost sterile when
dynamics are removed from them (Gattegno, 1987, p35).
28
References
Gattegno, C. (1987/2010) What we owe children. New York: Educational Solutions Worldwide. First published 1987. Reprinted 2010.
Gattegno, C. (1988) The Science of Education. Part 2B: The awareness of mathematization. New York: Educational Solutions.
Dawson, S. (1988) Words triggered by images: images triggered by words. In: John Chatley (ed) Readings in Mathematica education: mathematical images. Derby: ATM
Collis, K. F. (1975). A Study of Concrete and Formal Operations in School Mathematics: A Piagetian Viewpoint. Victoria, Australia: Australian Council for Educational Research.
Jones, Ian (2008 A diagrammatic view of the equals sign: arithmetical equivalence as a means, not an end. Research in Mathematics Education 10:2, pp151 — 165
29
Thank you
30
Imagery – task: number line
Imagine a number line…