This article models how to plan and facilitate implementing the same task in two sixth-grade classrooms with two different learning goals using the Five Practices structure.

Getting the right answers in maths is only half the problem. Understanding why what you're doing works is the part that often stumps students and teachers alike. The essential guide for mathematics teachers and those training to teach, "Yes, but Why?" answers all your questions, and sheds light on the hidden connections between…

Fractions are an imperative skill for students to master to achieve success in future mathematics concepts and classes. Yet many students, especially those experiencing math difficulties and/or characteristics of emotional and behavioral disorders, continue to exit elementary school without a concrete foundation of fractions skills required to…

The purpose of the study is to investigate pre-service elementary teachers' noticing expertise of students' mathematical thinking within the context of fractions. For this purpose, 32 pre-service elementary teachers were asked to examine students' answers and ways of thinking given in written excerpts related to fractions. The data was mainly…

For students to develop adequate conceptual understanding about fractions, they require experiences that afford them opportunities to think flexibly about multiple interpretations of fractions. This article presents a sequence of lessons designed to offer such experiences for students in Years 3 and Year 4. The lessons adopt a challenging tasks…

The complexity of understanding unit fractions is often underappreciated in instruction. We introduce a continuum of children's understanding of unit fractions to explore this complexity and to help teachers make sense of children's strategies and recognize milestones in the development of unit-fraction understanding. Suggestions for developing…

This article highlights the importance of re-defining units unconventionally with fractions. We provide three examples of tasks Grade 5 students engaged in; we highlight the creative ways that they flexibly re-defined units and engaged deeply with fractions. We offer suggestions for supporting unconventional units for promoting conceptual…

The purpose of this article is to share how one teacher implemented exploratory writing within a mathematics lesson to support her third graders in making sense of fractions and communicating their mathematical ideas. The author defines exploratory writing, then describes the teacher's lesson while also presenting a general overview about how to…

This article presents activities focused on developing some essential Year 1-3 fraction understandings (i.e., joining/composing, decomposing/ partitioning) while aligning these activities to the Australian Curriculum content strands and content descriptions. Additionally, the authors describe the materials/resources needed, the key vocabulary…

The book synergizes research on number across two disciplines--mathematics education and psychology. The underlying problem the book addresses is how the brain constructs number. The opening chapter frames the problem in terms of children's activity, including mental and physical actions. Subsequent chapters are organized into sections that…

Productive struggle is an essential part of mathematics instruction that promotes learning with deep understanding. A video scenario is used to provide a glimpse of productive struggle in action and to showcase its characteristics for both students and teachers. Suggestions for supporting productive struggle are provided.

In this article, the author describes how to help develop students' conceptual understanding of fractions through the tactical card-based game Nearest to One. Nearest to One promotes number-sense based strategies for comparing fractions, such as benchmarking (e.g., Is the fraction greater or less than one-half?) and building to the next whole,…

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)…

Our middle school mathematics department used lesson study to investigate how to introduce fractions division to our sixth-grade students. We highlight our learnings during the Study and Plan phases, describe our observations during the lesson, and provide tips for educators interested in using lesson study to study their own content.

This document includes instructional tips on: (1) Building on students' informal understanding of sharing and proportionality to develop initial fraction concepts; (2) Helping students recognize that fractions are numbers that expand the number system beyond whole numbers, and using number lines to teach this and other fraction concepts; (3)…