Understanding negative numbers can be challenging for many students, as these concepts may seem less tangible than counting objects, which are commonly represented by positive numbers. In addition, the multiplication of two negative numbers resulting in a positive might appear inconsistent and puzzling to young learners who are used to seeing the…

In this article, the authors share the potential for two types of atypical arrays (composite and partially hidden) to stimulate initial multiplicative ideas and strategies from students in a second-grade classroom. Composite arrays are defined here as nonarrays that are composed of multiple smaller complete arrays. The atypical arrays were…

The author presented a math task asking all his 3rd graders to find ways to count seven sets of eight donuts, arranged in grids of two-by-four. He hoped this task would lead students toward fluency in single-digit multiplication, albeit in a manner that was inherently personalized learning, with students choosing their own methods to reach an…

This article examines the mathematics learning experiences in proportional reasoning I created for pre-service teachers in my teacher education course in Canada. The approach aimed to enable pre-service teachers to work on and with their prior mathematics understandings, make connections that were particular to what emerged during group work, and…

The importance of multiplicative thinking in supporting students' learning of key topics and success in further mathematics is widely and clearly stated in mathematics education literature. However, multiplicative thinking is not clearly stated in curriculum documents of many countries including Australia. In the "Australian Curriculum:…

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)…

This is the first of two articles on the use of a written multiplication algorithm and the mathematics that underpins it. In this first article, the authors present a brief overview of research by mathematics educators and then provide a small selection of some of the many student work samples they have collected during their research into…

Proportionality can be considered among the most important mathematical notions in the middle school math curriculum (grades 5 to 8). It is the capstone of elementary arithmetic, number, and measurement concepts, and at the same time one of the most elementary understandings one needs for more advanced mathematics. Understanding proportionality is…

How do students cope with and make sense of polysemy in mathematics? Zazkis (1998) tackled these questions in the case of 'divisor' and 'quotient'. When requested to determine the quotient in the division of 12 by 5, some of her pre-service teachers operated in the domain of integers and argued for 2, while others adhered to rational numbers and…

During a time when students are accessing instruction virtually because of COVID-19, reflecting on the sixth of the Standards for Mathematical Practice (SMP 6) is a vital step for educators to take. Although the Common Core State Standards for Mathematics (NGA Center and CCSSO 2010) identifies SMP 6 as Attend to precision, associated expertise…

Despite the numerous benefits that stem from a coherent mathematical understanding, it is uncommon for teachers to encourage students to form a developed web of mathematical connections (Mousley, 1999). Additionally, when students explain their thinking, they frequently do not state how their reasoning connects to prior knowledge. This obscures…

Learn the most effective ways to teach elementary math, no matter how much experience you have with the subject. In this book, Fuchang Liu takes you through many common mistakes in math instruction and explains the misunderstandings behind them. He points out practices that should be avoided, helping you to adjust your lessons so that all students…

In general, units coordination refers to the relationships that students can maintain between various units when working within a numerical situation. It is critical that middle school students learn to coordinate three levels of units not only because of their importance in understanding fractions but also because of their implications for…

The Smarties-Box Challenge encourages students to apply several different mathematical capabilities and concepts--such as, estimation, multiplication, and the notion of being systematic--to solve a complex, multistep problem. To effectively engage in the Smarties-Box Challenge, students are required to demonstrate aspects of all four proficiency…

Algorithms and representations have been an important aspect of the work of mathematics, especially for understanding concepts and communicating ideas about concepts and mathematical relationships. They have played a key role in various mathematics standards documents, including the Common Core State Standards for Mathematics. However, there have…