A design experiment with 18 students in a regular seventh grade math class was conducted to investigate how to differentiate instruction for students' diverse ways of thinking during a 26-day unit on proportional reasoning. The class included students operating with three different multiplicative concepts that have been found to influence rational…

The inspection of mistakes can play a powerful role in an individual's learning process (Boaler 2015). The purpose of this article is to support the reader in selecting mistakes that can be leveraged to benefit the learning of all students. Specifically, the authors focus on "which" and "why": "which" mistakes to…

This article investigates how students in third grade learn to reason on multiplication when they first encounter that concept in the classroom context. By analysing the data from 24 classrooms focused on teaching and learning multiplication, the article aims at contributing to the research and conceptualisations about how students learn to…

The present study evaluated the effects and interaction of item set size and instructional time on students' fluency growth rates with multiplication facts over time when using an Explicit Timing intervention. The first goal was to determine an interaction between set size and instructional time across time that would influence the fluency growth…

While mathematics education is key to the post-school outcomes of students with disabilities, it has received less attention in research and practice compared with other aspects of educating this population. Skill maintenance is particularly crucial in mathematics because students build upon prior knowledge across grade levels. They also need to…

Mathematics instruction -- and interventions to support mathematics teaching -- for students with intellectual disability is important yet underexamined. This study explored a graduated instructional sequence referred to as the virtual-representational (VR) as a mathematical intervention. Researchers taught four students with disabilities what…

This investigation presents students with the challenge of determining the total number of eggs in a photograph of a stack of egg trays. The perspective of the photograph does not allow students to count all the individual eggs in the top tray but does allow them to see the number of eggs in one row and one column of the top tray. During this…

The "Making Meaning for Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The twenty-nine cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation of specific…

Inquiry-based instruction is a student-centered approach to teaching that focuses on active learning (Barron and Darling-Hammond 2008) in which students engage with "tasks that promote reasoning and problem solving" (NCTM 2014). Specifically, such tasks encourage a variety of solution strategies and stimulate use of the NCTM Process…

In order for students to move from using concrete materials to using mental strategies and from additive to multiplicative thinking, the use of arrays and visualisation is pivotal. This article describes a lesson in which students are taken through a Concrete-Representational-Abstract (CRA) approach that involves noticing structure, using…

Many general and special education teachers teach mathematics word problems by defining problems as a single operation and linking key words to specific operations. Unfortunately, teaching students to approach word problems in these ways discourages mathematical reasoning and frequently produces incorrect answers. This article lists eight common…

Many general and special education teachers across the U.S. teach word problems by defining problems as a single operation (e.g., "Today, we're working on subtraction word problems") and linking key words (e.g., more, altogether, share, twice) to specific operations (e.g., share means to divide). Unfortunately, teaching students to…

In abstract algebra, the study of concrete groups is fundamentally important to beginners. Most commonly used groups as examples are integer addition modulo n, real number addition and multiplication, permutation groups, and groups of symmetry. The last two examples are finite non-abelian groups and can be investigated with the aid of concrete…

The connections the "Next Generation Science Standards" ("NGSS") creates between science content and practices to literacy and mathematics are especially beneficial at the elementary level. In this article, the authors present a series of lessons a second-grade teacher taught within a thematic unit that integrated standards…

This paper reports on a pilot study that incorporated an alternative professional learning model that was school-based and focused an identified area of need: multiplicative thinking. The shift to multiplicative thinking can be challenging for both students and teachers due to its multifaceted nature. The study involved the delivery of six modules…