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…

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 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…

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…

Learning trajectories are built upon progressions of mathematical understandings that are typical of the general population of students. As such, they are useful frameworks for exploring how understandings of diverse learners may be similar or different from their peers, which has implications for tailoring instruction. The purpose of this…

This paper explores how a group of pre-service elementary school teachers training to become mathematics teachers for elementary schools arrived at generalizations based on patterns. Two representative problems were investigated with these preservice teachers. The focus of this study was how these preservice teachers analyze and symbolize…

Cross-curricula opportunities afforded by STEM education (Science, Technology, Engineering and Mathematics education), supports an environment where students can develop twenty-first century competencies. One approach to addressing cross-curricula opportunities in STEM education is the introduction of computer science (computer…

Empirical arguments rely on examples without necessarily addressing all cases. Students should be skeptical of empirical evidence and should seek more secure arguments for generalizations, such as those that explain why a generalization is true for all cases. Generalizing on the basis of patterns in data is an important mathematical practice;…

This paper explores strategies for teachers to work with science conceptions, both those consistent and those inconsistent with western science understanding. It emphasises the value of teachers checking their own and their students' prior understanding of concepts to be learnt. A past approach of educators has been to replace old beliefs with new…

Development in any domain is often characterized by increasingly abstract representations. Recent evidence in the domain of shape recognition provides one example; between 18 and 24 months children appear to build increasingly abstract representations of object shape [Smith, L. B. (2003). Learning to recognize objects. "Psychological…

Researchers have regarded inductive reasoning as one of the seven primary mental abilities that account for human intelligent behaviours. Researchers have also shown that inductive reasoning ability is one of the best predictors for academic performance. Modelling of inductive reasoning is therefore an important issue for providing adaptivity in…

Web-based financial commentary, in which experts routinely express market-related thought processes, is proposed as a means for college students to learn vicariously about financial markets. Undergraduate business school students from a regional university were exposed to expert market commentary from a single financial Web site for a 6-week…

Three experiments using university students explored what constitutes an event and what determines its boundaries. Results supported the hypothesis that sequences of stimuli repeating in different contexts are cut out to become cognitive entities ("things" with a beginning and an end) in their own right. Results suggest that the…