Learning trajectories (LTs) can inform teaching by contributing a variety of pedagogical and content-related insights and strategies. They support sequencing topic introduction and development. Because ideas evolve gradually and often rely on a careful introduction of new representations, operations, cases of numbers, structures, and definitions,…

To learn mathematics, historically students had no choice but to memorize fundamental facts and apply memorized algorithms. Since 1995 in the US, all states have adopted standards to govern K-12 mathematics instruction, and in most, standards have de-emphasized memorization and emphasized reasoning based on concepts. This change assumed the brain…

Responsive teaching is a highly effective strategy that promotes student learning. In math classrooms, teachers might funnel students towards a normative answer or focus students to reflect on their own thinking, deepening their understanding of math concepts. When teachers focus, they treat students' contributions as resources for collective…

I present my perspective on the design process in this article, arguing for a focus on student learning and "slow design" that stems from knowledge of mathematics and their support system in the learning process. I have a question about the design process academization and task design research direction. Numerous examples from my work at…

This research was designed to investigate the conceptualisations and thinking strategies Indigenous Australian students use in counting tasks. Eighteen Aboriginal students, in years 1 to 11 at a remote community school, were interviewed using standard counting tasks and a "counting" task that involved fetching "maku" (witchetty…

The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our…

This study considers the evolving influence of variation and expectation on the development of school students' appreciation of distribution as displayed in their construction of graphical representations of data sets. Three interview protocols are employed, presenting different contexts within which 109 students, ranging in age from 6 to 15…

The concept of learning progressions offers one promising approach to developing the knowledge needed to define the "track" that students may be on, or should be on Learning progressions can inform teachers about what to expect from their students. They provide an empirical basis for choices about when to teach what to whom Learning…

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 book provides a landscape of learning that helps teachers recognize, support, and celebrate their students' capacity to structure their worlds algebraically. It identifies the models, contexts, and landmarks that facilitate algebraic thinking in young students and provides insightful and practical methods for teachers, math supervisors, and…

How do composing and decomposing numbers connect with the properties of addition? Focus on the ideas that you need to thoroughly understand in order to teach with confidence. The mathematical content of this book focuses on essential knowledge for teachers about numbers and number systems. It is organized around one big idea and supported by…

Informal inferential reasoning has shown some promise in developing students' deeper understanding of statistical processes. This paper presents a framework to think about three key principles of informal inference--generalizations "beyond the data," probabilistic language, and data as evidence. The authors use primary school classroom…

This paper examines the extent to which mathematical knowledge, and its related pedagogy, is inextricably linked to the tools--physical, virtual, cultural--in which it is expressed. Our goal is to focus on a few exemplars of computational tools, and to describe with some illustrative examples, how mathematical meanings are shaped by their use. We…

How do you refute the erroneous claim that all ratios are fractions? This book goes beyond a simple introduction to ratios, proportions, and proportional reasoning. It will help broaden and deepen your mathematical understanding of one of the most challenging topics for students--and teachers--to grasp. It will help you engage your students,…

This mixed-methods study investigates the effects of student attitudes and behaviours on the outcomes of learning mathematics with computer tools. A computer tool was used to help students develop the mathematical concept of function. In the whole sample (N = 521), student attitudes could account for a 3.4 point difference in test scores between…