Book review
Teaching Secondary School Mathematics and Statistics: Evidence-based Practice
R. Averill & R. Harvey (Eds.), NZCER Press,
Wellington, New Zealand, 2009
Volume 1: 209pp. ISBN 978-1-877398-43-8
Volume 2: 209pp. ISBN 978-1-877398-45-2
In their introductory acknowledgements to this two-volume set, editors Robin Averill and Roger Harvey recall that the initial idea behind the books was to create a suitable text for secondary mathematics initial teacher education programmes, modelling the structure around the very successful editorial efforts of Jim Neyland with Mathematics Education: A Handbook for Teachers. I expect the audience will be more numerous than that intention implied. These books have contributions from many of New Zealand’s mathematics and statistics education academics and will generate interest from anyone involved in the teaching of mathematics and statistics.
The reflective intention of the books is made clear from the beginning, with each chapter including highlighted sections of “Points to ponder” and “Focus questions”. Such a structure will provide a model for discussion within initial teacher education programmes. They also provide a suitable pause for thought for other readers.
Chapter 1 sets a fine tone for the publication. Andy Begg invites the reader to ponder “the nature of learning tasks, the nature of mathematics, the aims of education, and pedagogical strategies” (p. 11). Begg sketches some theoretical background to these ideas, sprinkles in some of his own thoughts and their development over time and leaves the reader with questions to consider. His style is engaging, eminently readable and thought provoking. He leaves readers to draw their own conclusions, or, more likely, stimulates them to explore the questions further. As is the case with all the chapters, additional reading recommendations and references will provide readers with plenty of opportunities for further reflection.
It is chapters of this nature that will appeal to a wider audience than that originally intended by the editors when they conceived this project. The questions asked of the reader are pertinent to experienced teachers reflecting on their professional purpose. The writings of Jim Neyland readily fit this category. His are chapters that encourage the muse. He draws the reader’s attention to the difference between idea and fact, and in his chapter with Derek Holton proposes that “proof, in short, is an art form” (Vol. 2, p. 170). Like Begg, Neyland ponders the nature of mathematics, causing the reader to puzzle about the nature of the mathematics they generally experience, or teach. He poses an important challenge to the belief that mathematics is based entirely on logic and reasoning.
I have my favourites. I will always pick up a reading from derek Holton with anticipation that it will be insightful, filled with common sense, challenging yet engaging and always interestingly mathematical. His contribution on problem solving doesn’t disappoint on any front. He writes a clever piece of prose around a problem that deliberately leaves the reader hanging, pondering their next move. It captures the reader in the mathematical problem-solving process, and champions the case for such experiences to be part and parcel of the mathematics classroom. It’s hard to argue against that.
In a different sense, there are other contributions on specific aspects of mathematics that will engage all teachers. In topics that reflect their recent research interests, the contributions from Chris Linsell, Nigel Calder, Sharleen Forbes and Maxine Pfannkuch, and Mike Thomas deserve a wide readership. Chris linsell provides a thorough explanation of his research into students’ strategies for solving equations. Of particular note is the observation of a correspondence between students’ numeracy strategy stage and their level of sophistication in solving equations. Nigel Calder traverses the possibilities for learning afforded by technologies that allow for multiple simultaneous representations of situations—symbolic, numeric and visual—and that are interactive and dynamic in their nature. Such technologies, he argues, allow mathematic understanding to emerge in less conventional fashion, with new opportunities for investigation
and learning. If you have missed recent advances in the development of statistical reasoning, and the use of technologies for same, then the chapter by Sharleen Forbes and Maxine Pfannkuch is an excellent starting point. It is copiously illustrated with examples suitable for the classroom and is well referenced for further exploration by teachers. Mike Thomas has long been an eloquent advocate of technology as an aid to developing students’ understanding of mathematical concepts. He delves into the skills and insights needed by teachers in order to make the best use of technology to develop students’ conceptual understanding of mathematics.
Issues relating to using Te reo Māori for the teaching of secondary school mathematics are eloquently laid open by Tamsin Meaney and Uenuku Fairhall. The continuing evolution of Māori terms for mathematical concepts is well captured. They state that “the language, as a treasure, needs to be seen as a dynamic, adaptable communication medium” (Vol. 1, p. 147), just like any other living language, and illustrate action in the Māori community to develop the language to capture mathematical concepts with integrity. They also express a wish for pedagogical practices to be enacted that foster a community of mathematical learning for Māori in Te reo Māori. In a related chapter, Averill et al. challenge the view that mathematics is culture-free, and maintain that “traditional educational practice has not reflected our bicultural heritage” (Vol. 2, p. 27). They then go about describing strategies to address this imbalance.
Gifted and talented students are the topic of discussion by Brenda Bicknell and derek Holton. Advice is offered for the identification of these students, appealing to the work of Krutetski and Gardner. A discussion on the relative merits of acceleration and enrichment approaches to the gifted is followed by a thorough overview relating to the development of appropriately challenging programmes.
Most chapters in the collection are clearly crafted with an inservice-teacher audience in mind, while still holding appeal to experienced teachers. Jane McChesney theorises about the learning of mathematics using constructivist and sociocultural perspectives. Readily accessible resources that would be familiar to many teachers make this an approachable beginner’s guide to an exploration of learning theories. Robin Averill introduces student voice in a refreshing insight to the classroom that would serve well to prepare inservice teachers whose memories of the classroom may have dimmed over time. Her prompts with respect to the caring teacher would stimulate a valuable discussion for those teachers intent only on content delivery. Roger Harvey’s chapter is a compilation of ideas from his own practice and recent mathematical education research about the teaching of the division of fractions. Steve French challenges the image of a maths classroom that simply advocates for single solution strategies and the memorising of facts. Such an environment is unlikely to promote the creative, critical and metacognitive thinking espoused in the mathematics curriculum. A range of practical strategies and engaging problems puts some flesh on his vision of a thinking classroom. There are chapters offering advice on the division of fractions, the solution of algebra word problems, mathematical investigations and probability; and there’s a practical modelling chapter with plenty of warnings about the use of safety glasses. Issues for students for whom English is an additional language get an airing from Bishton, Gleeson and Tait. Devices suggested to support these students include three-level guides, split-information activities and a cline continuum. The books are completed by a couple of chapters containing a mix of perspectives from beginning teachers and advice for the starting years of a teaching career. Such advice may help normal people realise they are not on their own when the reality of teaching hits home at the start of a career. It can be a culture shock—it’s an intense occupation.
As often occurs in an edited collection of chapters from a great variety of authors, there is an uneven nature to the collection as a whole. Don’t expect a consistent philosophical point of view. Some authors are thankfully explicit, while for others you need to read between their lines. That may add to the book’s ability to act as stimulus material for inservice teachers.
At times the opportunity afforded by focus questions is wasted by making them either rhetorical or emotive. Within Volume 1 repetition occurs— John Mason’s comments on the generalising needed in a maths lesson occur on pages 15 and 33; the problems on pages 31 and 51 do not differ in substance. A few errors might confuse the reader (Vol. 1, p. 42, p. 90); bits of text sometimes detract from the general impact: Volume 2, page 142, last paragraph reads like a late addition after an initial review. The sentence relating to the probability of marriage dangles with scant meaning: “the probability that a person would be married before the age of 25 is not 50 percent because we need more information.” I have no idea what the author was trying to convey by this remark. Perhaps a tighter editorial oversight could have limited this.
This is a surprisingly easy read. This reflects the sound reasoning in many of the articles, and an effort on the part of the authors to resist educational jargon. I recommend it as reading for reflection on teaching practice by anyone involved in the teaching of mathematics. You are unlikely to agree with every author; and therein lies an opportunity for further reflection and discussion with colleagues.
Kevin Hannah
Team Leader
UC Education Plus
University of Canterbury