mathematical powers,

mathematical themes

and

pedagogical constructs

for informing the learning, doing and teaching of mathematics.

For more information about these than is in the individual studies, see below.

Each study demonstrates how a single mathematical task can be seen as an example selected from a vast space or domain of related tasks.

or

Why Are We Doing This, Miss?

(Inner & Outer Aspects of Tasks)

The tasks begin with an invitation to generalise the following
observations

Constructs called upon include

Inner aspects of tasks, in addition to the overt outer aspects

Watch What You Do and Say What You See as pedagogic tactics

Exploiting diagrams when expressing generality

Download pdf file Up & Down Sums last update August 8 2006.

# Study
2: Sums To One:

a study in generalisation

The tasks begin with the question of whether, if someone has chosen two
numbers which sum to one,

which will be the larger:

the sum of the square of the larger and the smaller

the sum of the square of the smaller and the larger.

Constructs called upon include

Dimensions of Possible Variation and Range of Permissible Change

Exploiting Diagrams when expressing generality

Download pdf file Sums To One last update July 11 2006.

# Study
3: Marble Sharing

The tasks begin with a simple, traditional sharing problem:

The notions of Dimensions of Possible Variation and Range of Permissible Change are exploited both to simplify and then generalise the task.

Download pdf file Marble Sharing last update July 11 2006.

# Powers,Themes
and Constructs

For the moment, see

Mason, J. & Johnston-Wilder, S. (2006). Designing and Using Mathematical Tasks. St. Albans: Tarquin.

Mason, J. with Johnston-Wilder, S. & Graham, A. (2005). Developing Thinking in Algebra. London: Sage (Paul Chapman).

Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematics Education, London: RoutledgeFalmer.

# Mathematical
Powers

# Mathematical Themes

# Pedagogical Constructs

Inner aspects of tasks, in addition to the overt outer aspects

Watch What You Do and Say What You See as pedagogic tactics

Exploiting diagrams when expressing generality

Download pdf file Up & Down Sums last update August 8 2006.

a study in generalisation

which will be the larger:

the sum of the square of the larger and the smaller

the sum of the square of the smaller and the larger.

Constructs called upon include

Dimensions of Possible Variation and Range of Permissible Change

Exploiting Diagrams when expressing generality

Download pdf file Sums To One last update July 11 2006.

If Anne gives John one of her marbles, she will then have one more than
twice as many marbles as John then has. However, if instead,
John gives Anne one of his marbles, he will have one more than a third
as many marbles as Anne then has. How many marbles have they
each currently?

The notions of Dimensions of Possible Variation and Range of Permissible Change are exploited both to simplify and then generalise the task.

Download pdf file Marble Sharing last update July 11 2006.

Mason, J. & Johnston-Wilder, S. (2006). Designing and Using Mathematical Tasks. St. Albans: Tarquin.

Mason, J. with Johnston-Wilder, S. & Graham, A. (2005). Developing Thinking in Algebra. London: Sage (Paul Chapman).

Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematics Education, London: RoutledgeFalmer.

Comments and questions gratefully received

j.h.mason @
open.ac.uk [spaces are there to impede
spam-generating search engines]