# 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
and
pedagogical constructs
for informing the learning, doing and teaching of mathematics.

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     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
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

# Study 3: Marble Sharing

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.

# 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.