Studies in Algebraic & Geometric Thinking

A series of explorations in algebra and geometry with commentaries which draw upon mathematical powers, themes and pedagogical choices to illustrate the fact that any task can be seen as merely one example for a rich domain of similar tasks, and putting forward the view that a task which does not permit and encourage learners to use their own powers, to encounter core mathematical themes, is not a pedagogically effective task.

Algebraic Thinking

AS1: Up & Down Sums or
                     Why Are We Doing This, Miss?
(Inner & Outer Aspects of Tasks)

The first task begins with the following facts:
Constructs called upon include
  • Inner aspects of tasks, in addition to  the overt outer aspects and possible meta aspects
  • Watch What You Do and Say What You See as pedagogic tactics
    Exploiting diagrams when expressing generality
S1: Up & Down Sums
File Size: 164 kb
File Type: doc
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AS2: Sums to One: a study in generalisation

If someone has chosen two numbers which sum to one, which of the following two will be the larger?
    the sum of the square of the larger with the smaller
​                                     or
    the sum of the square of the smaller with the larger
Constructs called upon include
  • Dimensions of Possible Variation and Range of Permissible Change
  • Exploiting Diagrams when expressing generality
S2: sums to one
File Size: 86 kb
File Type: s2 sums to one
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S3: Marble Sharing: a study in variation

The tasks begin with a 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 problem is simplified and then re-complexified

The notions of Dimensions of Possible Variation and Range of Permissible Change are exploited both to simplify and then generalise the task.
S3: Marble Sharing
File Size: 56 kb
File Type: s3 marble sharing
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Geometric Thinking

Powers, Themes & Pedagogic Tactics

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