When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number…

Windmill images and shapes have a long history in geometry and can be found in problems in different mathematical contexts. In this paper, we share and discuss various problems involving windmill shapes and solutions from geometry, algebra, to elementary number theory. These problems can be used, separately or together, for students to explore…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

Identifying and addressing the knowledge gap in early numeracy is crucial, given the strong associations between early numeracy skills and later school success. The purpose of this study is to establish current viewpoints and ideas on children's numeracy development via three forms of representation: manipulative, symbolic, and static. The…

A problem sequence is presented developing the basic properties of the set of natural numbers (including associativity and commutativity of addition and multiplication, among others) from the Peano axioms, with the last portion using von Neumann's construction to provide a model satisfying these axioms. This sequence is appropriate for…

This practical book draws on Ronit Bird's teaching experience to create detailed strategies and teaching plans for students aged 9-16 who have difficulties with number. Activities and games are used to teach numeracy skills in these key areas: number components, bridging, multiplication, division and reasoning strategies. New to this edition: (1)…

In grade 3, students are expected to be able to represent a fraction on a number line by first identifying the interval from 0 to 1 as the unit and by partitioning the unit as needed on the basis of the denominator. This task extends these grade 3 fraction goals stated in the Common Core State Standards for Mathematics (NGA Center and CCSSO 2010)…

Learners intending to enter some higher education (HE) institutions in South Africa write the National Benchmark Tests (NBTs) that are expected to provide a measure of their readiness for HE. A large gap exists between the quantitative literacy competencies of many of these learners and expectations at HE level. In this article I explore the…

This proposal explores how the activity of subitizing--quickly apprehending the numerosity of a small set of items--changes with the development of number concepts. We describe how varying the orientations of items in teaching experiment sessions promoted one pre-schooler, Frank, to attend to subgroups of items and change his thinking about…

Many studies have shown the difficulties of learning and teaching the decimal number system for whole numbers. In the case of numbers bigger than one hundred, complexity is partly due to the multitude of possible relationships between units. This study was aimed to develop conditions of a resource which can help teachers to enhance their teaching…

This unlikely cast of characters, by working collaboratively in a trusting learning community, was able to identify an approach to teaching rational numbers through measuring from the everyday practices of Yup'ik Eskimo and other elders. "The beginning of everything," as named by a Yup'ik elder, provided deep insights into how practical…

This theoretical paper presents a methodology for instructional design in mathematics. It is a theoretical analysis of a proposed model for instructional design, where tasks are embedded in situations that preserve meaning with respect to particular pieces of mathematical knowledge. The model is applicable when there is an intention of teaching…

The fifth-grade curriculum presents many opportunities to use students' prior mathematical knowledge as a way to bridge new and more difficult mathematical ideas. In this article, the authors document an area tiling task given to fifth-grade students to connect aspects of area measurement covered in earlier grades to grade-level standards such as:…

Algebraic reasoning is foundational to all mathematical thinking. This is no less the case in the early years of school, where the capacity to recognise the structure of mathematical processes enables students to acquire deep conceptual understanding. It is through algebra, therefore, that students are able to explore and express mathematical…

This article introduces the formative assessment probe--a powerful tool for collecting focused, actionable information about student thinking and potential misconceptions--along with a process for targeting instruction in response to probe results. Drawing on research about common student mathematical misconceptions as well as the former work of…