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Interactions around unexpected, incorrect, or dis-preferred responses can be powerful sites of learning for both teachers and students. The information that teachers uncover through probing student thinking can then guide their pedagogical response. We report on a study of prospective teachers' skills and capabilities around a particular problem of practice: eliciting student thinking when a student has an incorrect answer. In this case, if the student's thinking is sufficiently probed, the student is able to recognize the mistake and revise their work. Focusing on prospective teachers at the beginning of a teacher preparation program, we illustrate how knowledge of this kind of eliciting skill can be gathered through the use of a live teaching simulation. Our findings reveal that these prospective teachers were more fluent with eliciting the student's process than the student's conceptual understanding. Further, they focused more on eliciting the revised method and/or solution than asking about why the mistake was made. We consider the findings in terms of the skills brought by prospective teachers that could be built upon in teacher education, skills that need to be learned, and skills that need to be unlearned.

This paper compares lessons on linear equations from the same curriculum materials taught by two teachers of different levels of mathematical knowledge for teaching (MKT). The analysis indicates that the mathematical quality of instruction in these two classrooms appears to be a function of differences in MKT. Although the two teachers were teaching from the same curriculum materials, the teacher with higher MKT had more complete and concise ways to describe key concepts, had multiple ways to represent ideas about linear equations, could move nimbly among different mathematical expressions of linear relationships, and gave students a larger role in articulating the mathematical ideas of the lesson. However, curriculum materials seem to have moderated what would otherwise have been larger disparities in the quality of instruction between the two teachers. The lower-MKT teacher made minor mathematical errors, stayed on topic, and defined concepts in reasonably accurate ways when he followed the curriculum materials closely. (Contains 2 notes and 4 figures.)

This study illuminates claims that teachers' mathematical knowledge plays an important role in their teaching of this subject matter. In particular, we focus on teachers' mathematical knowledge for teaching (MKT), which includes both the mathematical knowledge that is common to individuals working in diverse professions and the mathematical knowledge that is specialized to teaching. We use a series of five case studies and associated quantitative data to detail how MKT is associated with the mathematical quality of instruction. Although there is a significant, strong, and positive association between levels of MKT and the mathematical quality of instruction, we also find that there are a number of important factors that mediate this relationship, either supporting or hindering teachers' use of knowledge in practice. (Contains 5 tables, 9 figures and 3 footnotes.)

In this paper, the authors complete the summative stage of the validity argument approach, then use their experiences to reflect on the validity argument as a method. They begin by evaluating the inferences and assumptions of the interpretive argument for the MKT measures. Then they examine both the form and the structure of the interpretive argument for the MKT measures with an eye to generalizations that can be made to other efforts to construct interpretive arguments. Finally they attempt to draw some reasonable generalizations and conclusions concerning the summative stage.

This paper provides a summary of the authors' attempts to uncover links between their measures, classroom mathematics instruction, and student learning. This paper also provides evidence regarding one central critique of their measures: that multiple-choice assessments cannot validly represent the knowledge, skills, and judgment involved in actual teaching practice. To the extent that there is a relationship between the test domain and practice domain, it strengthens their claim that they can measure knowledge for teaching in this format. In this article the authors describe the aims, methods, and preliminary findings from a large-scale statistical study of teacher knowledge and student achievement and a videotape study of teachers' classroom mathematics instruction. If a teacher knowledge/student achievement relationship is found in the former, the latter will help identify the mechanisms through which this relationship is realized. Finally, as a means for informing the debate about why test validation is often better conceptualized than practiced, they consider the cost and obstacles to this type of work. (Contains 2 tables, 1 figure and 1 footnote.)

The article completes the summative stage of the validity argument approach. The interferences and assumptions of the interpretive argument for the Mathematical Knowledge for Teaching (MKT) measures is evaluated. The form and the structure of the interpretive argument for the MKT measures is examined with an eye to generalizations that can be made to other efforts to construct interpretive arguments.

This article discusses the relationship of measure scores to classroom teaching and student learning. It discusses the formulated assumption as well as several observable inferences about the relationship of the measures of teachers' mathematical knowledge for teaching to their actual classroom mathematical work.