This article describes "Build It!--The Rectangle Game" task that uses the context of a game to develop mathematical generalizations based on strategy. The underlying mathematics in this game-based task is for students to discover factors and prime and composite numbers through 100. The playful use of "The Rectangle Game"…

When we "think like a computer scientist," we are able to systematically solve problems in different fields, create software applications that support various needs, and design artefacts that model complex systems. Abstraction is a soft skill embedded in all those endeavours, being a main cornerstone of computational thinking. Our…

Fragile understanding is where new learning begins. Students' understanding of new concepts is often shaky at first, when they have only had limited experiences with or single viewpoints on an idea. This is not inherently bad. Despite teachers' best efforts, students' tenuous grasp of mathematics concepts often falters with time or when presented…

Inherent in the Common Core's Standard for Mathematical Practice to "look for and express regularity in repeated reasoning" (SMP 8) is the idea that students engage in this practice by generalizing (NGA Center and CCSSO 2010). In mathematics, generalizing involves "lifting" and communicating about ideas at a level where the…

The very nature of algebra concerns the generalization of patterns (Lee 1996). Patterning activities that are geometric in nature can serve as powerful contexts that engage students in algebraic thinking and visually support them in constructing a variety of generalizations and justifications (e.g., Healy and Hoyles 1999; Lannin 2005). In this…

There is a need to teach the pivotal skill of mathematical problem solving to students with severe disabilities, moving beyond basic skills like computation to higher level thinking skills. Problem solving is emphasized as a Standard for Mathematical Practice in the Common Core State Standards across grade levels. This article describes a…

In many mathematics classrooms, the teacher provides "worked examples" to demonstrate how students should perform certain algorithms or processes. Some students find it difficult to generalize from the examples that teachers provide and cannot apply what they have learned in new situations (Watson and Mason 2002). Instead, teachers might…

Paula Worth presents in this article a means of challenging students' tendency to generalise even when they know that they should not. How can teachers encourage their students to say something meaningful about the past while avoiding making unwarranted generalisations? Worth takes teachers through the process of planning her own enquiry designed…

Although discussions about inductive reasoning can be traced back thousands of years (Fitelson 2011), the implementation of the Standards for Mathematical Practice (SMP) within the Common Core State Standards (CCSSI 2010) is generating renewed attention to how students learn mathematics. The third SMP, "Construct viable arguments and critique…

Elizabeth Carr writes here about a new scheme of work she developed to teach students about diversity in Victorian society. When dealing with a concept such as diversity, it can be easy for students to slip into stereotypes based on simplistic understandings of the experiences of people at the time. To resolve this, Carr argues, teachers need to…

A functional class refers to a circumstance in which responding is controlled by features of stimuli that are common to all the class members. We argue that behavior with respect to conceptual stimuli entails more than discrimination among classes and generalization within classes. We suggest that an analysis of substitution of stimulus functions…

This article focuses on the ways in which four eighth-grade girls, each with varying levels of algebraic understanding, share ideas, debate, and gradually move toward generalizations inherent in the "Painted Cube" problem. The intent of this article is to examine how students move to progressive formalization and to provide insights into the ways…

Many students with disabilities continue to struggle with understanding what mathematics means. They memorize basic facts or step-by-step mathematical procedures without understanding the underlying concepts related to the problems. Thus, instruction designed to help students understand the meaning of the mathematics that they are learning in…

This paper describes the methods and techniques called Conceptual Analysis (CA), a rigorous procedure to generate (without involuntary omissions and repetitions) knowledge bases for the development of knowledge-based systems. An introduction is given of CA and how it can be used to produce knowledge bases. A discussion is presented on what is…

During the 1960s and early 1970s two educational movements were the evolution of the middle school and the structures of disciplines approach to social studies. These movements involved school organization, curriculum development, and methodologies incorporating the inquiry or conceptual approach. Learners had trouble with the conceptual approach,…