Although tests and assessments--such as those used to maintain a physician's Board certification--are often viewed merely as tools for decision-making about one's performance level, strong evidence now indicates that the experience of being tested is a powerful learning experience in its own right: The act of retrieving targeted information from…

Generative learning strategies are intended to improve students' learning by prompting them to actively make sense of the material to be learned. But are they effective for all students? This review provides an overview of six popular generative learning strategies: concept mapping, explaining, predicting, questioning, testing, and drawing. Its…

This article explores three processes involved in attending to evidence of students' thinking, one of the Mathematics Teaching Practices found in "Principles to Actions: Ensuring Mathematical Success for All." These processes, explored during a classroom activity on proportional relationships, are discussed in this article, another…

Many well-intended instructors use Socratic or leveled questioning to facilitate the discussion of an assigned reading. While this engages a few students, most can opt to remain silent. The seven step strategy described in this article provides an alternative to classroom silence and engages all students. Students discuss a single reading as they…

We share ideas for preparing for and enacting high-cognitive demand tasks in ways that support students in articulating and justifying their ideas. We offer strategies for developing and posing several types of purposeful questions: (1) eliciting thinking, (2) generating ideas, (3) clarifying explanations, and (4) justifying claims.

Masterful teachers don't just ask a lot of questions; they ask questions in a purposeful way. In this article, Costa and Kallick describe five strategies that can help teachers become more purposeful in designing and posing questions. One strategy is to plan questions that elicit student thinking at various cognitive levels, from simple recall of…

The purpose of this paper is to account for the advancement of questioning in mathematics education. Four sections focus on: (1) Importance of Questioning in Education (a historical perspective of questioning from the 1960s through the 1990s); (2) Cognitive Classification of Questions (Costa's Levels of Thinking, Webb's Depth of Knowledge, and…

"Reader, say something smart. Right now. Share a deep insight or a subtle point. Quick. No? OK (with obvious disappointment), I'll come back to you later. Anybody else?" We've all experienced this in school, the author notes--the teacher giving up, concluding that we weren't going to say something smart in the allotted…

In this article, the authors present a tool, the Cognitive Rigor Matrix (CRM; Hess et al. 2009), as a means to analyze and reflect on the type of questions posed by mathematics teachers. This tool is intended to promote and develop higher-order thinking and inquiry through the use of purposeful questions and mathematical tasks. The authors…

The authors have seen an inspiring phenomenon in certain classrooms--students thinking through their answers to teacher questions, responding thoughtfully to other students' answers, even self-correcting or providing more information after they've answered a teacher aloud. The strategy behind these student actions is a skillful use of "wait…

Through a series of six guided classroom discoveries, students create, via targeted questions, a definition for deciding when two sets have the same cardinality. The program begins by developing basic facts about cardinalities of finite sets. Extending two of these facts to infinite sets yields two statements on comparing infinite cardinalities…

If a teacher asked their students what thinking looks like, what would they say? Would they just look at the teacher quizzically? The question is challenging because thinking is largely an invisible endeavor, and developing thoughtful students can be a daunting task. However, the job of mathematics teachers is to develop students who think about…

The authors believe teaching for understanding begins with the development of a few essential orientations. Teachers must have an appreciation for student centered mathematics teaching, valuing an approach that builds on student thinking. In addition, teachers must appreciate the complexity of students' mathematical thinking and ideas. Once these…

Open-ended questions, as discussed in this article, are questions that can be solved or explained in a variety of ways, that focus on conceptual aspects of mathematics, and that have the potential to expose students' understanding and misconceptions. When working with teachers who are using open-ended questions with their students for the…

Findings derived from self-reported, structured survey questionnaires are commonly used in evaluation and applied research to inform policy-making and program decisions. Although there are a variety of issues related to the quality of survey evidence (e.g., sampling precision), the validity of response processes--how respondents process thoughts…