In accordance with views of reform-based mathematics teaching from across the globe, the authors reviewed a decade of research published in the International Group-Psychology of Mathematics Education (IG-PME) conference proceedings on reform teaching practices for mathematics pre-service teachers. Six salient practices were identified in the decade of research: lesson planning, noticing, representations, problem solving and posing, discourse and assessment. The authors found that the beneficial practices of lesson analysis, student work analysis, allowing creativity, discussion, reflection, and piloting assessments promoted reform-based practices. Further research and emphases are needed on preservice teachers' abilities to make decisions based on student thinking, understand the importance of linking multiple representations, and require explanations from students. Recommendations for broadening the research are included.
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Researchers over the years have agreed on some aspects of reform mathematics, such as that students should have a deep understanding of mathematics, which could be achieved through discourse around solving problems that have high cognitive demand. For example, in the United States (US), Ball (1993) saw reform mathematics as embodying classrooms where “students would conjecture, experiment, and make arguments; they would frame and solve problems; and they would read, write, and create things that mattered to them” (p. 374). Boaler (2002) likewise saw discourse as an important part of problem solving. While conducting research in the US, she described international ideas of reform-based mathematics as productive learning experiences, provided by “open-ended problems that encourage students to choose different methods, combine them, and discuss them with their peers” (p. 239). This is similar to the description of the reform curriculum in Greece described by Christou et al. (2002), as engaging students in “thought-provoking, original problems that involve challenging themes” (p. 257). Students were intended to be engaged in interdisciplinary, discovery activities. Chapman (2004), in her research in Canada, emphasized the importance of peer interactions as part of reform mathematics. She focused on how teachers implement discourse among students in the classroom. In Australia, reform mathematics had a similar view, with Anderson and Bobis (2005) identifying aspects of reform mathematics in Australian curricular documents of “questioning, applying strategies, communicating, reasoning and reflecting” (p. 65), which are intended to support problem solving and high order thinking. These views are also seen in the US, as Van den Heuvel-Panhuizen and Fosnot (2001) point out, a basic component of reform math is that it focuses more on processes and strategies than on answers, to provide insight into students’ thinking.
Researchers also believe that solving problems in the context of real life can be an important part of reform mathematics, as seen by Carroll (1996), in the US, who focused on the belief that a reform curriculum would use problem solving in broader mathematical contexts. Ball (1993) also included authentic tasks in her view of reform mathematics, as did Vos and Bos (2001), who described reform mathematics in the Netherlands as based on Realistic Mathematics Education, which relied on the belief that mathematics is an important part of real life, and students should create mental images to aid in problem solving. The curriculum emphasized open-ended mathematics problems set in the context of real life.
Reform mathematics has also been seen as student-centered. Sakonidis et al. (2001) summarized reforms in Greece under the umbrella of constructivism, such that the teacher provides opportunities for students to make sense of new ideas. Meanwhile, in Israel, Chissick (2002) studied teachers’ attempts to implement reform mathematics with the characteristics of open-ended tasks, use of technology, and pupil-centered teaching. Oh (2003) characterized reform teaching in South Korea by asserting that “reform-oriented teachers have been seen as ‘facilitators’ or ‘stimulators’ given that they assist students’ mathematical learning as they create a learning environment that reflects students’ ideas” (p. 406).
Anderson and Bobis (2005) included reasoning and reflecting in their reform-mathematics ideas. Reasoning was also seen in the earlier National Council of Teachers of Mathematics (NCTM, 2000) document, as part of the Process Standards of problem solving, reasoning and proof, connections, communication, and representation, as well as in Adding It Up: Helping Children Learn Mathematics (National Research Council [NRC], 2001), which identified five strands that defined mathematical proficiency: adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition. Reflection, as well as productive disposition as seen in the NRC (2001) document, were included as part of Schoenfeld’s (2006) characterization of reform mathematics as including “strategic competence, metacognitive ability (including monitoring and self-regulation), and productive beliefs and affect (or disposition)” (p. 25).