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Transitions in Mathematics Education

Ghislaine Gueudet, et al.
Springer Open
Publication Date: 
Number of Pages: 
ICME-13 Topical Surveys
[Reviewed by
Peter T. Olszewski
, on

As the title suggests, Transitions in Mathematics Education considers various types of transitions in mathematics education. The book is a topical survey from the ICME-13 that focuses on two main topics:

  1. Conceptual changes and the learning process as a transition process.
  2. Transitions between social groups or contexts with different mathematical practices.

The book considers the learning of mathematics at all ages, from preschool to the university level and even in the workplace. The book begins with a literature review, which develops various theoretical approaches and associated methods. In Section 2 the learning process of different difficult concepts is considered. Here “pieces and processes” are viewed using the framework of Knowledge in Pieces. Section 3 looks at the double discontinuity between secondary school mathematics and university level mathematics.

Section 4 introduces the institutional perspective, viewing mathematical practices as shaped by the institution where they take place. Finally, section 5 considers two extremes: the transition between prior to school and school mathematics, and the transition between school and out of school mathematics. At the end of the book, future research ideas and directions on transition in mathematics education are presented.

In section 1.1, the book asks questions on what types of transition have been taking place and have been considered in mathematics education and considers which studies (if any) lead to the identification of areas of improvement or potential discontinuities. The notion of critical transitions is outlined by Yerushalmy (2005):

Critical transition is viewed as a learning situation that is found to involve a noticeable change of point of view. This change could become apparent as an epistemological obstacle, as a cognitive discontinuity or as a didactical gap. A transition would be identified as a necessity for entering into a different type of discourse (in terms of the language, symbols, tools, and representations involved) or more broadly as changing “lenses” used to view the concept at hand.

This quote sets the stage for discussing three types of transition: epistemological, cognitive, and discursive. The authors’ goals throughout the book are not to argue that one perspective of transition is more appropriate than the other, but rather to present to the reader these various ideas and how they can be discussed. For example, as pointed out on page 2, there have been many other types of transition studied in mathematics education that deal with the available tools teachers and students have access to: the transitions from tracers and templates to compasses for drawing circles in Geometry (Chassapis 1998) and moving to/from CAS-supported classrooms (Kendal and Stacey 2002) have been considered.

Section 2.1, Setting the Scene, focuses on one particular issue dealing with the transition in the conceptual structure and whether this structure should be considered continuous or discontinuous (CvD). After a brief history of CvD in mathematics education, section 2.2 presents the early 1980s phenomenon of “naïve conceptions.” Here, the viewpoints of Kuhn (1970), Toulmin (1972), and Bachelard (1938) are presented with the discussion of the “duck/rabbit” by Wittgenstein on pages 6–7, where the “duck/rabbit” can be seen as either one or the other but not both.

Jumping ahead to section 4.3, The Secondary-University Transition is a pivotal topic of discussion, as more and more students are having a harder time adjusting to University level learning. Here, many different views are presented, which are referenced for further reading. Larger class sizes, having more than one teacher for a subject (TAs), and getting accustomed to viewing their professors as researchers add up to an increasingly more difficult pill for students to swallow.

The final section of the book presents how we as educators should move forward with transitions. Some ideas are:

  • The need for combining approaches.
  • From gaps and obstacles to commonalities and opportunities.
  • Research results and interventions on transitions.

In reading this book, I couldn’t help but think how my own students have been having more and more difficultly adjusting to college life than in previous years. Being a former high school teacher and now a University lecturer, I know many of the challenges students face when first walking through the door of their first college classes. For many, it is too much: what they thought worked before no longer does. They are now responsible for their own learning and future. Do the high schools adequately prepare their students for this step? Do universities prepare their students for the workplace and provide them with enough discipline and motivation to be successful? Only further studies and time will tell. This book summarizes the ongoing discussion of transition, how it has changed over the last 20 years, and what we as professors can do to help students.

Peter Olszewski is a Mathematics Lecturer at Penn State Behrend, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. His research fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks He can be reached at Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.

See the table of contents in the publisher's webpage.