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This yearbook, produced by the Singapore Association of Mathematics Educators (SAME), contains fifteen chapters written by authors from various Singapore, Malaysia, Korea, Australia, Japan, and the U.K. The chapters stem from invited lectures presented during the 2011 conference organized by the Singapore Mathematical Society and SAME. The subject was selected because metacognition is one of five components of the Singapore school mathematics curriculum framework for problem solving. One of that framework’s principles for mathematics teaching states that “Teaching should build on students’ knowledge, take cognizance of students’ interests and experiences; and engage them in active and reflective learning.”
In Chapter 5 one finds that although local studies have shown that Singapore secondary students are aware of some aspects of metacognitive reflection, such as the importance of monitoring their thinking during problem solving, “there is no evidence that this general level of awareness is helpful [to them], especially when they have to solve non-routine or challenging problems. (p. 97). Indeed, most “tend to use traditional learning strategies such as memorise formulae, pay attention, and complete assigned homework.” (p. 98). Thus, perhaps the SAME conference organizers felt a need for focusing the conference on nurturing reflective learners of mathematics.
What seems clear is that reflective thinking about mathematics does not come naturally to most students, that teachers need well designed tasks that promote reflection, and that “any task for which an easily-obtained answer is the desired endpoint is unlikely to lead to reflection on its own”. In addition, there needs to be “good teaching and deliberative talk to promote reflection that brings new, higher level perspectives.” (p. 156). To illustrate this, Chapter 8 discusses six routine calculation tasks (usable with primary students), along with pedagogic prompts that can generate a special kind of reflection that “involves noticing patterns, similarities and differences in repetitive work that has just been carried out.” (p. 164). For example, the task, “Think of a number. Multiply it by 10. Add 6. Divide by 2. Add 2. Divide by one more than your original number. The answer is ..” can engender discussion with 4^{th} graders about why this is so and bring up the inverse relation between multiplication and division, but “nothing will happen at all unless the teacher invites students to compare their starting points,” (pp.158–9).
Not only do students need to reflect on their mathematics learning, inservice and preservice mathematics teachers need to reflect on their teaching. Chapter 6 reports a study in which Chinese primary teachers were shown a video of an excellent mathematics class and asked suitable questions, such as “Given a chance, would you teach the lesson differently?”, in order to provoke them to reflect on their own teaching. Chapter 7 reports the results of activities to promote reflection that were included in an Australian pre-service primary teacher education unit. One activity “required students to choose an aspect of mathematics that they had recently been engaged in learning, and to reflect on which learning activities were helpful to them, which were not helpful and why that was the case.” The other “required students to reflect on their experiences using technology, such as Internet resources, computer software or calculators to learn some mathematics.” (p. 130). The students reported personal histories and emotions, as well as such reflections as having previously thought that there was only one “right way” to do mathematics.
Perhaps the most interesting chapter came from the plenary address by a National University of Singapore medical professor, who has developed an interest in neurocognition. He spoke of the importance of the myelin sheaths around the axons of nerves (in the brain) that form pathways — the “greater the number of layers [in the sheath], the greater is the intensity and speed of transmission of the nerve impulses along those pathways. “ (p. 19). He considered “the idea of learning as developing new neural pathways.” (p. 22). I got the impression, without his actually saying so, that he was conjecturing that reflection (on mathematics or other topics) can be useful for increasing the number of layers in relevant parts of a person’s myelin sheath.
School teachers at all levels will get some idea of tasks that can be used to induce reflection in students’ mathematical thinking. However, Chapter 11 is devoted to reflection when advanced Korean high school students were learning polar coordinates that could be useful to any calculus teacher. I found this to be an interesting book, though of course with some chapters more informative than others.
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.