The current study examines the personal epistemology of fourth-grade elementary school teachers from Germany (n = 10) and the United States (n = 10) to gain a more nuanced understanding of teachers' beliefs about knowledge and knowing through a cross-cultural lens. Analyses of semi-structured interviews reveal similarities and differences in the…

To conceive the infinity of integers, one has to realize: (a) the unending possibility of increasing/decreasing numbers (potential infinity), (b) that the cardinality of the set of numbers is greater than that of any finite set (actual infinity), and (c) that the leap from a finite to an infinite set is itself infinite (immeasurable gap). Three…

This study examines presentations of the distributive property (DP) in two widely used U.S. elementary text series and one main Chinese text series along three dimensions: problem contexts, typical problem types within each problem context, and variability in using the DP. In general, the two U.S. texts were found to resemble each other but to…

There are infinitely many ways of talking about infinity. The assortment of discourses on learning infinity is infinite as well. When the author says "way of talking" or "discourse," she is concerned with much more than the question of how words are chosen and combined. Ways of talking are not just innocent "auxiliaries" to thinking--they shape…

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students' understanding of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The…

This article describes the development of the Precalculus Concept Assessment (PCA) instrument, a 25-item multiple-choice exam. The reasoning abilities and understandings central to precalculus and foundational for beginning calculus were identified and characterized in a series of research studies and are articulated in the PCA Taxonomy. These…

This report provides evidence of the influence of a tutorial "communication game" on fifth graders' generative understanding of the integer number line. Students matched for classroom and pretest score were randomly assigned to a tutorial (n = 19) and control group (n = 19). The tutorial group students played a 13-problem game in which student and…

In this research, we examined overall performance and gender differences in measurement skills in elementary-school students from low-income families. In Study 1, accuracy and error patterns were analyzed in a large sample of fourth-graders; in Study 2 error patterns and strategy usage were examined with a smaller sample of fourth-graders. Study 1…

The examination of alternative hypotheses can initiate students into scientific practices and equip them with scientific literacy skills that will help them participate in ongoing debates involving complex socio-scientific problems. Hypothesis testing, in which the examination of alternative hypotheses is situated, has received much attention in…

An important element of adaptive expertise involves stepping away from a routine to retool one's knowledge or environment. The current study investigated two forms of this adaptive pattern: fault-driven adaptations, which are reactions to a difficulty, and prospective adaptations, which are proactive reformulations. Graduate and undergraduate…

Son and Goldstone (2009) provided orienting activities intended to increase personal engagement, and these activities reduced participants' transfer based on concepts of signal detectability theory. I offer a theoretical interpretation that differs from theirs. First, instead of assuming that activity is contextualized in some conditions and not…

Looking away from an interlocutor's face during demanding cognitive activity can help adults and children answer challenging mental arithmetic and verbal-reasoning questions (Glenberg, Schroeder, & Robertson, 1998; Phelps, Doherty-Sneddon, & Warnock, 2006). While such "gaze aversion" (GA) is used far less by 5-year-old school children, its use…

This article reports on a case study from a design-based research project that investigated how students make sense of the disciplinary tools they are taught to use, and specifically, what personal, interpersonal, and material resources support this process. The probability topic of binomial distribution was selected due to robust documentation of…

This study investigates how collaborative knowledge construction within an inquiry learning environment can be assisted with scaffolds that aim to support students' hypothesis generation process. Sixty-six students on a university preparatory track worked collaboratively on a kinematics task. The instructional goal was to develop students'…

Three experiments are reported that investigate the extent to which an authority figure influences the level of persuasion undergraduate students and research-active mathematicians invest in mathematical arguments. We demonstrate that, in some situations, both students and researchers rate arguments as being more persuasive when they are…