Source: *Journal of Mathematics Teacher Education**, 14(1):*49–66. (February, 2011)

(Reviewed by the Portal Team)

The purpose of this article is to provide evidence that teachers’ observations of students’ mathematical activity in research project on students’ development of mathematical ideas can provide rich opportunities for teachers to learn about students’ mathematical reasoning.

The following research questions guided the analysis of the teachers’ observational reports: (1) What evidence is there, if any, of teachers attending to aspects of students’ mathematical reasoning,

(2) What evidence is there, if any, of teachers learning about students’ mathematical reasoning?

Research setting

Nine mathematics teachers and 24 sixth-grade students participated in the IML project, which took place in a middle school, located in an urban, low-income, and minority community in the United States. The teacher interns consisted of two elementary school teachers, four middle school mathematics teachers, and three mathematics teacher coaches.

The teachers were able to see that students can naturally come up with problem-solving strategies, create a language to describe their mathematical thinking, adopt justification as part of their mathematical practice, develop valid ideas about what counts as an acceptable mathematical justification, and come up with mathematical discoveries.

The observations also helped the teachers realize that giving students the opportunity to explore ideas and decide on the validity of their mathematical arguments can help students develop into powerful and autonomous mathematical thinkers, and encouraging students to explain their mathematical thinking is a powerful way of promoting students’ ability to engage in mathematical justification.

The analysis suggests that three main factors enhanced the teachers’ learning of students’ mathematical reasoning in the IML project.

First, the IML was a research project on students’ development of mathematical ideas.

In the IML project, the students’ ways of reasoning and how they constructed their knowledge were the results of the research, not of preconceived goals.

The results were several opportunities for teachers to observe reasoning- rich students’ mathematical activities some of which have been documented elsewhere (Maher et al. 2007; Weber et al. 2008). This certainly enhanced the teachers’ learning about students’ reasoning.

Second, the IML was an after-school project. This means that there was not a fixed curriculum and there were no typical pressures of regular classrooms such as having to cover large amounts of material and preparing students for standardized tests.

Finally, in their role as interns, the teachers worked closely with researchers in the IML project.

In particular, researchers contributed observations of students’ mathematical reasoning that helped the teachers attend to particular aspects of students’ mathematical reasoning.

Researchers also promoted teachers’ collective discussion of their observations, which helped the teachers articulate their insights into students’ mathematical reasoning. Such actions from researchers certainly helped promote teachers’ learning in the project.

The results of this study are significant, particularly with regard to future research projects.

It is particularly significant that the teachers were impressed by the students’ mathematical reasoning.

Therefore, this study suggests that teachers’ observations of students’ mathematical activity in IML-type settings might help teachers develop an understanding of mathematics that is effective for teaching.

Finally, another area for future research regards ways of incorporating the benefits of IML-type after-school programs in teacher education and professional development. One way is to run such programs and encourage teacher participation in them. Another way is to involve teachers in the study of videotapes of students’ mathematical activities from such programs.