Developing Thinking In Mathematics

I would like to point you in the direction of three books that I return to over and over again to improve the way I teach mathematics.

______

Developing Thinking in Algebra

by John Mason (Author), Alan Graham (Author), Sue Johnston-Wilder (Author)

(ISBN 1-4129-1171-0)

Developing Thinking in Statistics

by Alan Graham

(ISBN 1-4129-1167-2) 

Developing Thinking in Geometry

by Sue Johnston-Wilder (Editor), John Mason (Editor)

(ISBN 1-4129-1169-9)

________

Developing Thinking in Algebra emphasises generality:

“Algebraic thinking (particularly the recognition and articulation of generality) is within the reach of all learners, and vital if they are to participate fully in society.” Introduction page ix.

“A lesson without learners having the opportunity to express a generality is not a mathematics lesson.” Introduction page ix.

I dip into this book frequently and never come away disappointed. The book broadens our understanding of generalisation and especially generalisation in mathematics. It is a great source of ideas for a lesson and or gives me ideas on how to improve my pedagogy.

Developing Thinking in Statistics stress the four main areas of statistics:

• Describing
• Comparing
• Interrelating
• Uncertainty

As I teach statistics I come back to the same question over and over again “Which of the four main areas of statistics is relevant to the current situation.” This gives me perspective and allows me to flip between the details of a particular problem and the big picture in an instance. The author stresses the importance of why we study statistics as opposed to the mechanics of the subject. He cites Paul Cobb (1999):

“It is already apparent that debates about public policy issues tend to involve reasoning with data. In this discourse, policy decisions are justified by presenting arguments based on the analysis of data. In many respects, this discourse is increasingly becoming the language of power in the public area. Inability to participate in this discourse results in the de facto disentrancement that spawns the capacity for thinking critically about the world in which they live.”

Developing Thinking in Geometry:

This is my favourite of the three books. This book stresses invariance in the midst of change. The three internal angles of a planar triangle always add up to 180° regardless of the shape of the triangle. When constraints are imposed on a system, e.g. a scalene triangle, to isosceles and then equilateral they produce an increasing number of interesting properties. I always use the question “What changes and what remains the same?” when teaching transformations. This generally produces a productive learning environment. he computer software packages like Cabri or Geometer’s Sketchpad are ideal in bringing this subject to life in a very dynamic way. Both are available from .

List of References:

Mason, J. Graham, A. Johnston-Wilder, S. (2005). Developing Thinking in Algebra. London, Thousand Oaks, California CA, New Dehli. Paul Chapman Publishing, A Sage Publication Company.

Graham, A. (2006). Developing Thinking in Statistics. London, Thousand Oaks, California CA, New Dehli. Paul Chapman Publishing, A Sage Publication Company.

Cobb, P. (1999). Individual and Collective Mathematical Development: The Case of Statistical Data Analysis. Mathematical Thinking and Learning Volume 1, Issue 1, Pages 5-44.

Johnston-Wilder, S. Mason, J. (2005). Developing Thinking in Geometry. London, Thousand Oaks, California CA, New Dehli. Paul Chapman Publishing, A Sage Publication Company.