I have recently read (and reread, as it happens) Sapiens: A Brief History of Mankind by Yuval Noah Harari. Not only is it extraordinarily well written in a style that is attractive for the nonspeci...

ABSTRACTIn this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detailing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive patterning to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tasks that were posed. We discuss implications for research and practice.

ABSTRACTInformal statistical inference has now been researched at all levels of schooling and initial tertiary study. Work in informal statistical inference is least understood in the early years, where children have had little if any exposure to data handling. A qualitative study in Australia was carried out through a series of teaching experiments with a class of five- to six-year-old children in two phases over six months. The aim of the exploratory study was to understand and support the emergence of informal statistical inference in the early years of schooling. Through activities that initially built on children’s experiences with prediction and key characteristics of informal statistical inference, the children’s actions were observed in a data-based inquiry involving prediction to identify critical relationships that then supported children in making informal statistical inferences. Implications are discussed.

In this research, we examined the relationships between gender, spatial ability, verbal-logical reasoning ability, calculus performance, and preference for visual or analytic processing. Data were collected from 150 calculus students at four high schools in two school districts. The results suggest that spatial and verbal-logical reasoning abilities are important factors of calculus performance. Preferred mode of processing was unrelated to spatial and verbal-logical reasoning abilities suggesting that cognitive abilities did not predict students’ preference for visual or analytic processing. There were no significant differences between the two sexes in cognitive abilities, preferred mode of processing, and calculus performance.

The mathematical idea of similarity is typically taught to students across the middle school years between ages 11 and 14. In this study, students' understanding of presimilarity is examined based on a set of clinical interviews of 21 students aged 12–13 years. Students were asked to scale a series of geometric figures and were found to use a variety of strategies including some that incorporated both geometric and numeric reasoning. Tasks were developed that manipulated the characteristics of figures that students were required to attend to in order to explore the boundaries of numeric reasoning and to maximize the degree to which visual reasoning could be brought to bear on the task. Contrary to the literature, student use of visual reasoning did not indicate less developed conceptions of similarity. In fact, visually-based strategies supported students as they reflected on and sought to improve wholly numeric strategies. Analysis of the interview data indicated that providing students with tasks that required them to scale more complex geometric figures improved their capability to attend to the quantifiable features of shape and to the numeric relationships between them.

A core component of informal statistical inference is the recognition that judgments based on sample data are inherently uncertain. This implies that instruction aimed at developing informal inference needs to foster basic probabilistic reasoning. In this article, we analyze and critique the now-common practice of introducing students to both “theoretical” and “experimental” probability, typically with the hope that students will come to see the latter as converging on the former as the number of observations grows. On the surface of it, this approach would seem to fit well with objectives in teaching informal inference. However, our in-depth analysis of one eighth-grader's reasoning about experimental and theoretical probabilities points to various pitfalls in this approach. We offer tentative recommendations about how some of these issues might be addressed.

For the purpose of our research we define Advanced Mathematical Knowledge (AMK) as knowledge of the subject matter acquired during undergraduate studies at colleges or universities. We examine the responses of secondary school teachers about their usage of AMK in teaching. We find that the majority of teachers focus on the purposes and advantages of their AMK for student learning, such as personal confidence, the ability to make connections, and to respond to students' questions; only a few provide content-specific examples. We conclude with a call for a more articulated relationship between AMK and mathematical knowledge for teaching.