This article illustrates and differentiates the unique role cognitive interviews and think-aloud interviews play in developing and validating assessments. Specifically, we describe the use of (a) cognitive interviews to gather empirical evidence to support claims about the intended construct being measured and (b) think-aloud interviews to gather…

The purpose of this study was to gain insight into 30, first year calculus students' understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up…

This paper aimed to investigate secondary school students' representations for solving geometric word problems in different clinical interviewing processes. More specifically, the focus was to understand the changes/developments in students' representations through think-aloud interviews (TAIs) and open-ended prompting interviews (OEPIs). Three…

Instruction promoting reasoning and proof should be a part of K-12 mathematics teaching. Preservice teachers (PSTs) need mathematics content instruction that helps them construct valid justifications for mathematical notions and develop positive perceptions of proof. This case study describes instruction with proof-related tasks and manipulatives…

This study investigated how a high school literacy coach provided coaching to support teachers' understanding and implementation of disciplinary literacy instruction. With a focus on collaborations between the literacy coach and teachers in the disciplines of social studies, math, and English, this article presents three case studies that…

In this study, we considered how middle school teachers understood the relationship between fractions and ratios. We used two instruments to collect data from 11 teachers and relied on Knowledge in Pieces as a lens for considering understandings teachers have and how coherent those understandings are. From our analysis, we developed three main…

Two studies were conducted to examine and compare the construct validity of scores on the Junior Metacognition Awareness Inventory (JMAI) and problem-solving interview protocols. Participants consisted of 183 middle and high school students attending a university summer program for academically talented youth. Study 1 results indicated that JMAI…

This study discusses the construction of deductive warrant derived from inductive warrant in mathematical argumentations expressed by pre-service teacher. In completing a mathematics task, a problem solver needs argumentation to determine, reveal, and support a reasonable solution. A mathematical argumentation can be analyzed by Toulmin scheme…

The Common Core Standards require students to learn content and mathematical practices, and so teachers must have content knowledge and be able to engage in practices themselves. This raises the question of how novice teachers learn to engage in mathematical practices. I investigate pre-service secondary teacher learning of mathematical practices…

This article is a report on the findings of three case studies that focused on elementary teachers' in-the-moment noticing across a month of instruction. Extending Jacobs and her colleagues' framework (Jacobs, Lamb, and Philipp, 2010), this articles categorizes the mediums (i.e., written accounts, verbal interactions, physical strategies) by which…

Considerable research suggests that standardizing curricular materials can undermine teachers' ability to facilitate learning. Nonetheless, U.S. school districts' curriculum guides have become increasingly standardized, with teachers being required to follow predetermined pacing calendars indicating which standards to teach during specific months,…

Rate of change is an important concept for students to study, and little is known about the ways in which secondary students make sense of rate of change. In this qualitative study I examined how four high school students who have not taken calculus reasoned about changing quantities when interacting with mathematical tasks involving multiple…

Research has shown that the tools provided by dynamic geometry systems (DGSs) impact students' approach to investigating open problems in Euclidean geometry. We particularly focus on cognitive processes that might be induced by certain ways of dragging in Cabri. Building on the work of Arzarello, Olivero and other researchers, we have conceived a…

Parents (n=148) and their children, ages 9 and 10, were brought together to participate in a Family Math program in Ontario. This program helped promote positive attitudes toward mathematics in parents and children and increased parents' understanding of current methods of teaching mathematics. (MKR)