The objective of the study was to characterise how pre-service kindergarten teachers used a Hypothetical Learning Trajectory on length and its measurement to notice children's mathematical thinking. A total of 64 pre-service kindergarten teachers enrolled in an Early Years Education mathematics teaching course were asked to notice teaching…

Estimating and accounting for statistical uncertainty have become essential in today's information age, and crucial for cultivating a sound decision making citizenry. Engaging with statistical uncertainty early on can support the gradual development of uncertainty-related considerations that are often challenging to foster at any age. Statistical…

The purpose of this article is to present and discuss two recommended sequences of learning the areas of polygons, starting from the area of a rectangle. Exploring the algebraic derivations of the two sequences reveals that both are valid teaching progressions for introducing the area formula for various polygons. Further, it is suggested that…

Due to their abstract nature, representation of mathematical concepts through different registers favors their understanding. In the case of ''sequences and regularities'', it becomes propitious the exploration of different registers of representation in the institution of topics, such as term, order, formation law, and generating expression.…

Learning progressions detail the incremental steps that students take as they learn to master a skill. These progressions are based on developmental research about how students learn and how their thinking develops as a result of instruction. A typical progression not only describes the stages that students must master, but it also shows what…

The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional…

"That was the day I decided I was bad at math." Countless times, preservice and in-service teachers make statements such as this after sharing vivid memories of learning multiplication facts. Timed tests; public competitive games, such as Around the World; and visible displays of who has and has not mastered groups of facts still…

The findings discussed here are a small part of a larger study entitled, "Encouraging Persistence Maintaining Challenge". The paper reports five teachers' observations of the implementation of a task which was new to them. The teachers were asked to identify aspects of the task which they perceived as challenging for the Year 6 students.…

In this paper, we describe our learning progressions approach to early algebra research that involves the coordination of a curricular framework, an instructional sequence, written assessments, and levels of sophistication describing the development of students' thinking. We focus in particular on what we have learning through this approach about…

Find out how to use photographic images to support the conceptual development of proportional thinking. This paper provides insight into a sequenced activity that promotes student engagement and makes links to familiar and unfamiliar contexts.

This report summarizes recommendations from NCTM, NRC, CCSSM, NMAP, and IES to guide early numeracy instruction for elementary age students in general and special education classroom settings. We highlight common threads among general and special education research recommendations and provide a numeracy intervention curriculum model connecting…

Graphical representations (GRs) of the learning content are often used for instruction (Ainsworth, 2006). When used in learning technology, GRs can be especially useful since they allow for interactions across representations that are physically impossible, for instance by dragging and dropping symbolic statements into a chart that automatically…

This article first describes some of the basic skills and knowledge that a solid elementary school mathematics foundation requires. It then elaborates on several points germane to these practices. These are then followed with a discussion and conclude with final comments and suggestions for future research. The article sets out the five…