This edited book of 26 chapters is divided into four parts: defining the field; mathematical problem posing in the school curriculum, mathematical problem posing in teacher education, and concluding remarks. It is not a slim book — there are 569 pages contributed by 52 authors from 16 countries, such as the U.S., Canada, Australia, Israel, Japan, Norway, Czech Republic, Singapore, Serbia, Romania, Belgium, Sweden, Italy, and the Netherlands. Despite these many and varied contributions, one gets the distinct impression that problem-posing research is still in its infancy.

Why have students pose problems, in addition to solving them? In a section titled, “Why is Problem Posing Important in School Mathematics?” the authors of Chapter 1 state that problem posing

has long been recognized as a critically important intellectual activity in scientific investigation and that the formulation of a problem is often more important than its solution. However, while problem solving has long been a fundamental activity in mathematics classrooms, problem posing has been neglected. Yet, in real life, problems must often be created or discovered by the solver. Thus, to prepare students for the future, problem posing needs to become a regular part of the mathematics curriculum.

Despite being a book by mathematics education researchers, and directed substantially at mathematics education graduate students, there are many examples of problem-posing tasks scattered throughout the book’s many chapters. Indeed, it contains a great many problem-posing tasks, and ideas for implementing them, at a variety of levels, including 3rd, 5th, and 7th grades, high school, and college. One easy technique for encouraging problem posing in a mathematics classroom is to take a given problem (with a number of parameters) and ask, in turn, “What, if not?” for each parameter. But there are other examples.

In Chapter 14, titled “An investigation of high school students’ mathematical problem posing in the United States and China,” there are examples (adapted from the literature) of free, semi-structured, and structured problem-posing situations. An example of a free problem-posing situation is the following: *There are ten girls and ten boys standing in a line. Make up as many problems as you can that use the information in some way.* An example of a semi-structured problem-solving situation is the following: *In the following picture, there is a triangle with an inscribed circle. Make up as many problems as you can that are in some way related to the picture. *An example of a structured problem-posing situation is the following: *Last night, there was a party at your cousin’s house and the doorbell rang ten times. The first time the doorbell rang only one guest arrived. Each time the doorbell rang, three more guests arrived than had arrived on the previous ring. (a) How many guests will enter on the tenth ring? Explain how you found your answer. (b) Ask as many questions as you can that are in some way related to this problem.* Results of the study include that, when US students were asked how they posed problems on the problem-posing test given in the study, they spoke in terms of “thinking outside of the box”, doing “something fun”, and about freeing their minds and letting ideas come to them, but many did not know how to explain what they had done. In contrast, many Chinese students posed problems similar to those they usually did in class. However, when asked, both U.S. and Chinese students reported that they had had little or no experience in posing mathematical problems, but thought doing so might help them learn mathematics.

Chapter 13, titled “Using problem posing as a formative assessment tool,” proposes a rubric for evaluating the complexity of problems posed by students and categorizes the anticipated solutions as being of low, moderate, and high complexity. Low complexity problem-posing tasks call heavily on recall and recognition of previously-learned concepts and typically specify what the solver is to do, such as make a calculation or solve a one-step word problem. Moderate complexity tasks involve more flexibility of thinking and choice among alternatives — the solver is expected to decide what to do, such as solve a multiple-step problem, extend a pattern, or interpret a simple argument. High complexity tasks make heavy demands on the solver, who must engage in more abstract reasoning, planning, analysis, judgment, and creative thought. Such tasks may ask a solver to use different representations to solve a problem; to describe, compare, and contrast solution methods; or to provide a mathematical justification. The assumption seems to be that the complexity of the problems posed reflects the mathematical knowledge and creativity of the problem poser.

This is just a sample of the many chapters that cover such topics as problem posing from a modeling perspective, using digital technology for problem posing, fostering creativity in mathematics classrooms, problem posing as reformulation and sense-making, middle grades preservice teachers’ mathematical problem solving and problem posing, problem posing activities in a dynamic geometry environment, and problem posing as an integral component of the mathematics classroom

Who might use this book? The editors state that the book “presents the state of the art of research in mathematical problem posing” and should be useful for graduate courses “related to mathematical problem posing and problem solving or as a foundation upon which to propose lines of inquiry into problem posing.” Clearly, as the editors indicate, the book is for mathematics education graduate students, but it could also be useful for mathematics teachers at a variety of levels who could gain ideas and tasks for classroom use.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. She remains active in mathematics education research and curriculum development.