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S.M.A.R.T. Circle Overview

Andy Liu
Publication Date: 
Number of Pages: 
Springer Texts in Education
[Reviewed by
Woong Lim
, on

Joint review of S.M.A.R.T. Circle Overview, S.M.A.R.T. Circle Projects, and S.M.A.R.T. Circle Minicourses, all by Andy Liu

Math Circles are informal mathematical learning communities, often led by mathematicians, that aim to engage teachers or students in mathematical inquiries through solving challenging and interesting problems. The problems are typically selected from outside the school curriculum and the topics are connected to advanced mathematics. There are hundreds of math circles in the U.S., meeting the intellectual and social needs of gifted and curious mathematics students (in a math students’ circle) as well as teachers (in a math teachers’ circle). These three books on the S.M.A.R.T. Circle by Andy Liu, a professor emeritus at the University of Alberta, offer both problems (with detailed solutions) and his own account of running a math students’ circle in Canada. Here “S.M.A.R.T” stands for Saturday Mathematical Activities, Recreations, and Tutorials.

The book comes in three volumes. The first volume, S.M.A.R.T. Circle Overview, provides three sample projects (problems with rich contexts, such as stories with characters) and a sample mini-course (mathematical explorations centered around a specific topic with some degree of coherence). The volume also contains (1) many past problems from highly competitive math contests, sorted out chronologically or by mathematical topics, (2) the learning materials (games and short problems) he used for running math camps for junior high students, and (3) his collected project ideas appropriate for math fairs.

The other two volumes are a more detailed collection of projects (volume II) and mini-courses (volume III). The topics of S.M.A.R.T. Circle Projects range from familiar ones (to me) such as The Towers of Hanoi or Pascal’s Triangle to newer, more novel projects, such as The Coach’s Dilemma or Tetris Number Theory. I was not familiar with the latter two topics and shared the activities with a group of middle school teachers, who seemed to enjoy the mathematical ideas and said they would introduce them to their gifted students. S.M.A.R.T. Circle Mini-courses offers five topics in geometry and five random, but quite exciting, additional topics.

With the understanding that this book is not meant to be “covered” or “taught” with a short span of time for the contest prep course, I believe the teachers who work with gifted and talented students should appreciate the richness of the problems, along with related contexts and discussions detailed in the book. All three volumes should serve as an amazing source of materials for teachers working with motivated junior high students who enjoy enrichment materials and extra challenges outside school mathematics. I can also imagine this book can serve those (mathematicians or master teachers) who facilitate math circles and are looking for new materials and ideas for social events. I recommend that the leaders of math circles who are at the beginning of their programs read S.M.A.R.T. Circle Projects and move on to the project and mini-course of their interest, and that those who have enough experience with math circles consider using the problems and discussions in the other volumes, especially when the table of contents looks interesting. Of course, the materials are not necessarily new, since the author is sharing published materials and old problems. Nonetheless, just looking at the content and the organization of the book may provide great insight into designing an effective curriculum for fostering mathematical inquiry, grit, and a sense of community for exceptional and highly motivated junior high students.

Woong Lim ( is an Assistant Professor of Mathematics Education at University of New Mexico. His research interests include mathematics teacher education; and discourse, language, and equity in the mathematics classroom.

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