Studies in
Algebraic Thinking

John Mason

Last update August 8 2006

Outline of the Studies

Studies in Algebra is a series of explorations in school algebra drawing upon and illustrating the use of
    mathematical powers,
    mathematical themes
    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.

Study 1: Up & Down Sums
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:

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.

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

Comments and questions gratefully received
j.h.mason @
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