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Reasoning, Communication and Connections in Mathematics: Yearbook 2012, Association of Mathematics Educators

Berinderjeet Kaur and Toh Tin Lam, editors
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Annie Selden
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This fourth yearbook of the Singapore Association of Mathematics Educators contains fifteen chapters written by authors from various countries, mainly Singapore, Korea, Australia, and the U.S. The chapters stem from keynote lectures and workshops conducted during the 2011 Singapore Mathematics Teachers Conference. One of the reasons for the theme of this yearbook is that “After the curriculum review in 2006, the Singapore school mathematics framework expanded in scope to include reasoning, communication, and connections…” It may surprise U.S. readers that mathematics educators in Singapore, whose international test scores are always very high, while acknowledging that Singaporean students are very good at procedural tasks, also acknowledge that they are less good at conceptual reasoning and consider this is a matter of great concern.

The editors say that the yearbook is intended for teachers. However, it is a rather “mixed bag” in that some chapters report research studies in great detail, while others have very practical suggestions on how to modify textbooks tasks at both primary and secondary levels so they entail higher cognitive demand. One chapter describes numeracy in Australia and its importance, suggesting there might be opportunities for Singaporean teachers (of all school subjects) to promote literacy. While it is hard to see school teachers getting much out of the detailed research reports, they might well adopt some of the suggestions for modifying textbook tasks.

For example, in Chapter 4, Berinderjeet Kaur suggests that when dealing with the usual closed textbook tasks of primary school mathematics, a teacher might want to modify them by asking four “What?” questions: “What number makes sense?”, “What’s wrong?”, “What if?”, and “What’s the question, if you know the answer?”. This is illustrated with the following task: “Tickets to a concert cost $15 per adult and $8 per child. Mr. Wang bought tickets for 4 adults and 5 children. How much did he spend altogether? The suggestion is to modify this task as follows: Tickets to a concert cost ___ per adult and ___ per child. Mr. Wang paid ___ for ___ tickets. He bought tickets for ___ adults and ___ children. What numbers make sense? 4, 5, 9, $8, $15, $100. Put the numbers in the boxes where you think they fit best.

From reading various chapters, one may learn some concepts that one perhaps did not know before, as well as some information about Singaporean mathematics classrooms. For example, in Chapter 9, titled, “Understanding Classroom Talk in Secondary Three Mathematics Classes in Singapore,” which reports the kinds of teacher questions focused upon and the relationships between different kinds of questions, one learns of the prevalence, in Singaporean mathematics classes, of IRE questions in which the teacher initiates (I), students respond (R), and the teacher evaluates (E) those responses. In general, it is thought that the primary function of IRE questions is evaluative or performative, rather than exploratory and constructive. Similarly, one can learn of “closed” known-answer test questions versus “open” questions, considered genuine, authentic, and information seeking, that do not seek one right answer or a limited range of acceptable answers.

Using data from a smaller, purposeful sample of a larger data set of 625 videographed lessons and transcriptions from over 1,000 Secondary 3 students in 30 Singaporean schools, the authors of Chapter 9 concentrated on three kinds of teacher questions: performative, procedural, and conceptual. Performative talk is talk that focuses on the use of closed questions to test student knowledge and generally entails IRE sequences. Procedural talk focuses on how students complete a procedure or algorithm. Conceptual talk is evidenced by classroom exchanges that focus on such areas as clarifying meaning, exploring how to solve problems, or offering reasons and explanations. The five authors conclude that “relationships between the three forms of talk are complementary rather than orthogonal: [Singaporean] teachers do not ask one kind of question and neglect others, but, up to a point, tend to ask all three kinds of questions.”

While one can get a lot from reading various chapters, a guide to the chapters would have been useful, so one would know which chapters to read depending on one’s interests. For example, it is clear that Chapters 2 and 11 report research studies; that Chapters 3, 5, 7, and 14 give practical suggestions for teachers; and that Chapters 13 and 15 give general information on numeracy in Australia. Some chapters are easy to read, whereas some are hard to read because of the technical terms and detailed analysis of data. In addition, there is no glossary and no index, so one cannot easily look up terms, such as “SEM model,” if one does not already know their meanings.

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.

  • Reasoning, Communication and Connections in Mathematics: An Introduction (Berinderjeet KAUR and TOH Tin Lam)
  • The Epistemic Framing of Mathematical Tasks in Secondary Three Mathematics Lessons in Singapore (Ridzuan Abdul RAHIM, David HOGAN and Melvin CHAN)
  • Modifying Textbook Exercises to Incorporate Reasoning and Communication into the Primary Mathematics Classroom (Denisse R THOMPSON)
  • Some “What” Strategies that Advance Reasoning and Communication in Primary Mathematics Classrooms (Berinderjeet KAUR)
  • Reasoning and Justification in the Secondary Mathematics Classroom (Denisse R THOMPSON)
  • LOGO Project-Based Mathematics Learning for Communication, Reasoning and Connection (Hee-Chan LEW and In-Ok JANG)
  • Reasoning, Communication and Connections in A-Level Mathematics (TOH Tin Lam)
  • Visual and Spatial Reasoning: The Changing Form of Mathematics Representation and Communication (Tom LOWRIE)
  • Understanding Classroom Talk in Secondary Three Mathematics Classes in Singapore (David HOGAN, Ridzuan Abdul RAHIM, Melvin CHAN, Dennis KWEK and Philip TOWNDROW)
  • Mathematics Classroom Discourse Through Analogical Reasoning (Kyeong-Hwa LEE)
  • Students' Reasoning Errors in Writing Proof by Mathematical Induction (Alwyn Wai-Kit PANG and Jaguthsing DINDYAL)
  • Presenting Mathematics as Connected in the Secondary Classroom (LEONG Yew Hoong)
  • Numeracy: Connecting Mathematics (Barry KISSANE)
  • Making Connections Between School Mathematics and the Everyday World: The Example of Health (Marian KEMP)
  • Mathematics, Astronomy and Culture: Helping Students See Connections (Helmer ASLAKSEN)