Learning and Doing Mathematics (2nd Edition).

Morony, Will

Learning and Doing Mathematics (2nd Edition)

John Mason

Tarquin

1988 (first edition); 1999 (second edition)

87 pp.

Soft cover

1-85853-049-0

[ILLUSTRATION OMITTED]

Two things initially brought this small book to my attention when I saw the cover. First was the title. "Learning and Doing Mathematics" is very much a theme in the context of the Australian Curriculum: Mathematics (AuC:M). The four Proficiencies--Problem solving, Reasoning, Understanding and Fluency--are identified as being about "doing" mathematics. They are central to the philosophy of the AuC:M, and the way it is written. Incorporating the Proficiencies in classroom programs is a major challenge in the implementation of the AuC:M.

The author's name was the second. John Mason, from the UK, is well-known to many involved in school mathematics through numerous visits and presentations. Indeed, he has given Keynote 1 and other talks at I several of AAMT's bien-1 nial conferences. He has also co-authored several books that are print versions of his thinking and suggestions for teachers of mathematics.

When I actually opened the book I was stunned to see that it is the second edition of a book originally written in 1988. Many people reading this review were students at school at the time--some may not even have been born! Can a book written 24 years ago be relevant now?

Mason argues that 'specialising' and 'generalising' are at the heart of doing mathematics." As a result, the first three sections are dedicated to these two process, first individually and then in combination. He follows with sections on "Convincing Yourself and Others" and "When is an Argument Valid?". Just from those titles the connection to at least two of the Proficiencies--Problem solving and Reasoning--seems pretty clear and, therefore, the contents likely to be useful to Australian teachers of mathematics today.

Just a listing of the titles of the sections gives no inkling of the what the book is about. Mason is upfront in stating his belief that "the essence of learning is doing." Teachers of mathematics will nod their heads at that, seeing it as how they approach their work in the classroom. However, Mason is true to his word--the text expects readers to do mathematics in order to learn. In practice, this means that at least once or twice on every page there is a mathematical challenge for the reader followed by "Try it now" or "Do so now" as part of what can only be described as a conversation with the reader.

The challenges cover a range of number, algebra and geometry areas--many are non-trivial (for me, at least!). Things like

   Generalise the statement "Given any rectangle, there is another
   rectangle with the same perimeter but smaller area"

   What is the smallest number with exactly 12 divisors? How do you
   know it is the smallest?

The whole point is for readers to do much more than merely read about solving problems in mathematics--they are actually doing it, and learning from their experiences.

What will teachers get from 'working through' this book? First of all there is the fun and stimulation of actually doing some challenging mathematics and thinking about it. As a result teachers' deeper understanding of problem solving can only be helpful in the context of helping students develop the Proficiencies in the classroom.

So is this 24 year old book useful to today's teachers? For me the answer is an unequivocal yes--but only if they are prepared to put in the effort and grapple with the ideas it contains. One of the downsides of its age, however, is that the text is slightly fuzzy, and the cover illustration is bitmapped. This suggests that the new version is a digital image of a hard copy book as there are no digital originals as there would be today.

Reviewed by Will Morony, AAMT Inc.

<wmorony@aamt.edu.au>