We combine Peirce's rule, case, and result with Toulmin's data, claim, and warrant to differentiate between deductive, inductive, abductive, and analogical reasoning within collective argumentation. In this theoretical article, we illustrate these kinds of reasoning in episodes of collective argumentation using examples from one…

Functions provide powerful tools for describing change, but research has shown that students find difficulty in using functions to create and interpret models of changing phenomena. In this study, we drew on a models and modeling perspective to design an instructional approach to develop students' abilities to describe and interpret rates of…

A hypothetical framework to characterize statistical knowledge for teaching (SKT) is described. Empirical grounding for the framework is provided by artifacts from an undergraduate course for prospective teachers that concentrated on the development of SKT. The theoretical notion of "key developmental understanding" (KDU) is used to identify…

Recent reform efforts in mathematics education have stimulated a focus on learning trajectories. At the same time, a global increase in high-stakes testing has influenced instructional practices. This study investigated how four fourth grade teachers within a school planned and enacted lessons to understand what mediated their planning and…

Context personalization refers to matching instruction to students' out-of-school interests and experiences. Belief in the benefits of matching instruction to interests is widely held in the culture of schooling; however, little research has empirically examined how interest impacts performance and learning in secondary mathematics. Here we…

The purpose of this study is to examine prospective mathematics specialists' engagement in an instructional sequence designed to elicit and develop their understandings of random processes. The study was conducted with two different sections of a probability and statistics course for K-8 teachers. Thirty-two teachers participated. Video analyses…

In this article I explore high school students' perspectives on working together in a mathematics class in which they spent a significant amount of time solving problems in small groups. The data included viewing session interviews with eight students in the class, where each student watched video clips of their own participation, explaining and…

What are the nature, forms, and roles of metaphors in mathematics instruction? We present and closely analyze three examples of idiosyncratic metaphors produced during one-to-one tutorial clinical interviews with 11-year-old participants as they attempted to use unfamiliar artifacts and procedures to reason about realistic probability problems.…

We examined children's development of strategic and conceptual knowledge for linear measurement. We conducted teaching experiments with eight students in grades 2 and 3, based on our hypothetical learning trajectory for length to check its coherence and to strengthen the domain-specific model for learning and teaching. We checked the hierarchical…

This article characterizes the development of a deep and connected body of mathematical knowledge categorized by Ball and Bass' (2003b) model of Mathematical Knowledge for Teaching (MKT), as Specialized Content Knowledge for Teaching (SCK) in algebraic reasoning and number sense. The research employed multiple cases across three years from two…

Current college admissions tests inadequately predict college success (Geiser & Studley, 2002) and provide insufficient information for students and teachers regarding college readiness at earlier stages of students' academic careers (Olson, 2006). This article reports on the creation and validation of a more informative test rooted in college…

In this article we report the influence of contextual factors on mathematics and science teachers' reasoning in risk-based decision-making. We examine previous research that presents judgments of risk as being subjectively influenced by contextual factors and other research that explores the role of context in mathematical problem-solving. Our own…

This qualitative study considers middle grades mathematics teachers' reasoning about drawn representations of fractions and decimals. We analyzed teachers' strategies based on their response to multiple-choice tasks that required analysis of drawn representations. We found that teachers' flexibility with referent units played a significant role in…

Explanations are considered to be key aids to understanding the study of mathematics, science, and other complex disciplines. This paper discusses the role of students' explanations in making sense of data and learning to reason informally about statistical inference. We closely follow students' explanations in which they utilize their experiences…

This theoretical paper relates recent interest in informal statistical inference (ISI) to the semantic theory termed inferentialism, a significant development in contemporary philosophy, which places inference at the heart of human knowing. This theory assists epistemological reflection on challenges in statistics education encountered when…