The purpose of the study is to evaluate how prospective mathematics teachers (PMTs) modify tasks to facilitate students' learning of pattern generalization through the use of their mathematical knowledge for teaching. Case study, which is a type of qualitative research method, was used to determine the mathematical characteristics that PMTs use…

Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…

What better way to interest students in mathematics than using a Super Bowl® commercial? A Prudential® insurance commercial aired during the Super Bowl in 2013 was the impetus for this lesson (see it here on You-Tube™: http://www.youtube.com/ watch?v=IsNiKGMSHUQ). In the commercial, four hundred people were polled on "How Old Is the Oldest…

Students often view mathematics as a set of unrelated facts and procedures and fail to make the connections between and among related topics. One role of a teacher is to help students understand that mathematics is an interrelated discipline. Another role is to assist students in the scaffolding of their knowledge so that they can make connections…

What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…

This article presents a new approach to generalizing the definition of means. By this approach we easily obtain generalized means which are quite different from standard arithmetic, geometric and harmonic means.

The aim of this study was to investigate the effectiveness, generalizability, and the permanency of the instruction with the touch math technique. Direct instruction was used to the instruction of the basic summation skills of the students with mild intellectual disabilities. A multiple probe design across the subjects was used in this study. The…

Developing meaning for symbolic representations is an important part of the middle school mathematics experience for all students. Teachers must not see simply writing symbolic representations as the end goal, but focus on the internal meaning that students ascribe to these symbols. Too often, students lack a deep understanding of the algebraic…

The NCTM calls for the use of rich tasks that encourage students to apply their own reasoning to problem situations. When students work through algebraic generalization tasks, their reasoning often elicits a variety of strategies (Lannin 2003; Stacey 1989; Swafford and Langrall 2000). Challenges for teachers include facilitating student awareness…

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…

Discusses several mathematics problems involving the generalization of rules from patterns. Cautions against using conjectures as proof that a generalization will always hold true. (MKR)

This note contains material to be presented to students in a first course in differential equations immediately after they have completed studying first-order differential equations and their applications. The purpose of presenting this material is four-fold: to review definitions studied previously; to provide a historical context which cites the…

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…

Algebraic thinking in the middle grades involves constructing generalizations, moving beyond the focus of specific calculations in elementary school. This creates exciting opportunities to examine the validity of general arguments. Creating rich mathematical tasks and asking questions related to justification encourages students to examine what…

Discusses the process of generalization. Illustrates the process by generalizing the classic problem of how a farmer can get a fox, a goose, and a bag of corn across a river in a boat that is large enough only for him and one of the three items. (MDH)