Recommendations for achieving generalized instructional outcomes often overlook the capacity for generative learning for most verbally competent humans. Four children (ages 5-8) participated in this project. In Study 1, we provided decontextualized discrete trial teaching to establish arbitrary relations between colors, pictures of characters, and…

This study investigated how three fifth-grade students' understanding of fraction and decimal magnitude evolved over the course of a five-week teaching experiment. Students participated in teaching and learning sessions focused on developing concepts of fraction and decimal magnitude. The following questions guided this study: (1) How do fifth…

According to an aptitude-treatment interaction experiment (Lehmann et al., "Metacognition and Learning, 11," 89-105, 2016, N = 47, published in "Metacognition and Learning"), perceptually disfluent texts facilitated retention and comprehension performance (but not transfer performance) only for learners with higher working…

This paper presents data from the first of three iterations of teaching experiments conducted with secondary teachers. The purpose of the experiments was to investigate how teachers' combinatorial reasoning could support their development of algebraic structure, specifically structural relationships between the roots and coefficients of…

Tangent lines are often first introduced to students in geometry during the study of circles. The topic may be repeatedly reintroduced to students in different contexts throughout their schooling, and often each reintroduction is accompanied by a new, nonequivalent definition of tangent lines. In calculus, tangent lines are again reintroduced to…

In this study, hypothetical samples of students' work on a task involving pattern generalizations were used to examine the characteristics of the ways in which 154 elementary prospective teachers (PSTs) paid attention to students' work in mathematics. The analysis included what the PSTs attended to, their interpretations, and their suggestions for…

This study explored whether a daytime nap aids children's acquisition of letter-sound knowledge, which is a fundamental component for learning to read. Thirty-two preschool children in Sydney, Australia (M[subscript age] = 4 years;3 months) were taught letter-sound mappings in two sessions: one followed by a nap and the other by a wakeful period.…

This review addresses a lacuna in the literature concerning interdisciplinary teaching by analyzing peer-reviewed articles that discuss empirical evidence of interdisciplinary teaching practices. The article reports on the wide array of purposes, approaches and designs of interdisciplinary teaching and learning found in the review, but an…

This paper will elaborate five levels of algebraic generalisation based on an analysis of students' responses to Reframing Mathematical Futures II (RMFII) tasks designed to assess algebraic reasoning. The five levels of algebraic generalisation will be elaborated and illustrated using selected tasks from the RMFII study. The five levels will be…

A great deal of research on second language (L2) input has provided useful linguistic evidence for the development of generalizations about and hypotheses on the usage-based nature of L2 knowledge. However, despite the fact that classrooms are one of the most ubiquitous sites of L2 learning, we still know very little about the linguistic quality…

Learning progressions have become an important construct in educational research, in part because of their ability to inform the design of coherent standards, curricula, assessments, and instruction. In this paper, I discuss how a learning progressions approach has guided our development of an early algebra innovation for the elementary grades and…

Generalisation is a skill that enables learners to acquire knowledge in general, and mathematical knowledge in particular. It is a core aspect of algebraic thinking and, in particular, of functional thinking, as a type of algebraic thinking. Introducing primary school children to functional thinking fosters their ability to generalise, explain and…

Conjectures are a key component of mathematical inquiry, a process in which the students raise conjectures, refute or dismiss some of them, and formulate additional ones. Taking a design-based research approach, we formulated a design principle for personal feedback in supporting the iterative process of conjecturing. We empirically explored the…

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and…

New challenges in the development of teaching methods lead to a large number of new tools, methods, and approaches to teaching. The structure and functions of a class as a basic social group in education is being radically transformed, becoming more and more virtual especially in COVID-19/post-COVID period. In this regard, this study proposes a…