1 Gattegno’s Mathematizing – becoming aware of dynamics of mathematics by working on imagery,...

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Gattegno’s Mathematizing – becoming aware of dynamics of mathematics by working on imagery, equivalence and substitution

Els De Geest

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‘Only awareness is educable’

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‘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.

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How?

He uses mental activities to work on becoming aware of these dynamics of mathematical relationships.

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Imagery – what is it and why would you bother?

Your thoughts?

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There are great advantages in concentrating on mental activities which deliberately embrace

many awarenesses and which can lead to new awarenesses.

(Gattegno, 1988, p89)

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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).

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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).

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Imagery - task

Imagine a lemon…

Adapted from Sandy Dawson(1988)

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Imagery – task: the square

Imagine a square…

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Equivalence ~

Read the sign ~ as “equivalent to” or “is another way of saying”

(Gattegno, 1988, p22)

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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?

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Equivalence tasks

Additions (Gattegno, 1988, p28)

10~ 1+9 ~ 2+8 ~ …

~ 9+1 ~ 8+2 ~ …




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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 ~ …

…




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Equivalence tasks

Transformations in subtraction (Gattegno, 1988, p38)

2031-784 ~ 2247-1000 (both +126)

~ 1247

2031-784 ~ 2000-753 (both -31)

~ 1247




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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?

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Equivalence - tasks


a/b ~ 2a/2b ~ 3a/3b … Na/Nb…



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Equivalence - tasks


m/n ~ m/a × a/n

~ m/a × a/b × b/n

~ m/a × a/b × b/c × c/n …



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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)

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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?

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Substitution – exploring limits of properties of relationships

Which one is the odd one out: elephant, parrot, dog, cow. Why?


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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?


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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?

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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?

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Adapted Ian Jones (2008) task

In a world in which 15+28 ~ 44, what can be

said about 15+29?

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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.

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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).

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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

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Thank you

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Imagery – task: number line

Imagine a number line…

1 Gattegno’s Mathematizing – becoming aware of dynamics of mathematics by working on imagery,...

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