Melissa Boston, Duquesne University; Jonathan Bostic, Bowling Green State University; Kristin Lesseig, Washington State University—Vancouver; Milan Sherman, Drake University

In this article, we provide information to assist mathematics teacher educators in selecting classroom observation tools. We review three classroom observation tools: (1) the Reform-Oriented Teaching Observation Protocol (RTOP); (2) the Instructional Quality Assessment (IQA) in Mathematics; and (3) the Mathematical Quality of Instruction (MQI). We begin by describing each tool and providing examples of research studies or program evaluations using each tool. We then look across tools to identify each tool’s specific focus, and we discuss how the features of each tool (and the protocol for its use) might serve as affordances or constraints in relation to the goals, purposes, and resources of a specific investigation. We close the article with suggestions for how each tool might be used by mathematics teacher educators to support teachers’ learning and instructional change.

Anne K. Morris, University of Delaware; James Hiebert, University of Delaware

Two studies were conducted to identify the conditions under which instructors teaching the same mathematics teacher preparation course would continuously improve their shared instructional products (lesson plans for class sessions) using small amounts of data on preservice teacher performance. Findings indicated that when lesson-level student performance data were simply collected, by course section, the instructors could make important changes to the lessons but did not often do so. However, when the instructors were encouraged to compare data across semesters, they generated hypotheses that guided instructional improvements, which then were tested through multiple cycles. The cycles of hypothesis testing helped instructors clarify the goals for improvement, use the performance data to test whether changes were actually improvements, and reduce their tolerance for marginal student performance.

Margaret S. Smith, Editor, Mathematics Teacher Educator

Reflections on the first three years of *Mathematics Teacher Educator.*

Gwyneth Hughes, Boise State University; Jonathan Brendefur, Boise State University; Michele Carney, Boise State University,

As the focus of mathematics education moves from memorization toward reasoning and problem solving, professional development for in-service teachers must model these activities while simultaneously increasing participants’ mathematical knowledge. We examine a representative task from a mathematics professional development course that uses rational number operation as an opportunity for problem solving and modeling. Transcripts exemplify the growth teachers make in deeply understanding the content—division of fractions—while engaging in guided reinvention and classroom discourse. We propose 4 interconnected qualities of this task that allow participants to engage in and reflect on the process of guided reinvention: (1) authentic context with multiple solution methods, including visual; (2) cognitive dissonance; (3) deep engagement; and (4) impact on mathematical knowledge for teaching.

Jodi Fasteen, Portland State University, Oregon; Kathleen Melhuish, Portland State University, Oregon; Eva Thanheiser, Portland State University, Oregon

Prior research has shown that preservice teachers (PSTs) are able to demonstrate procedural fluency with whole number rules and operations, but struggle to explain why these procedures work. Alternate bases provide a context for building conceptual understanding for overly routine rules. In this study, we analyze how PSTs are able to make sense of multiplication by 10five in base five. PSTs’ mathematical activity shifted from a procedurally based concatenated digits approach to an explanation based on the structure of the place value number system.

Laura Bofferding, Purdue University; Melissa Kemmerle, Stanford University

This article presents the results of an exploratory study detailing 4 teacher candidates’ initial implementations of a number string protocol in which they presented sequences of related problems to 3rd graders. We detail how the teacher candidates were taught the components of the protocol in their methods course and describe the math-talk (student-participation) levels that occurred during their 1st number string experience with their students. We coded the lesson transcripts for math-talk levels, which range from teacher-led to student-driven, and provide examples of the number strings and excerpts from the teacher candidates’ reflections to illustrate our results. Results indicate that number strings are a supportive structure for beginning teachers as they facilitate math talk.