Every student has mathematical strengths beyond knowing basic facts, solving problems quickly, or showing work clearly. In this article, the author presents Smiles as an "on-ramp" task that supports students working together by unveiling and leveraging mathematical strengths. Nielsen describes on-ramp mathematics tasks as scaffolds that…

Generalizing, conjecturing, representing, justifying, and refuting are integral parts of algebraic thinking and mathematical thinking in general (Lannin et al., 2011). The activity described in this article makes a case for generalizing as an overall mindset for any introductory algebra or geometry class by illustrating how generalization problems…

Duncan Symons and Derek Holton discuss the different types of mathematical reasoning and what each of these might look like in the classroom. By suggesting language that can be used to describe the different methods of reasoning, they hope to provide teachers with the tools they need to better recognise and assess student reasoning.

This paper studies how four primary-school in-service teachers develop the mathematical practices of conjecturing and proving. From the consideration of professional development as the legitimate peripheral participation in communities of practice, these teachers' mathematical practices have been characterised by using a theoretical framework…

As part of the "Connecting Mathematics Learning through Spatial Reasoning" project, a Spatial Reasoning Mathematics Program (SRMP) intervention was implemented with one cohort of 30 students in grades 3 through 4. The SRMP embedded transformation skills in learning sequences comprising repeating and growing patterns, 2D and 3D…

The present study is a part of design research in local instructional theory in a pre-algebraic lesson using the Realistic Mathematics Education (RME) approach. The article will focus on recommendations for the type of pre-algebra class that supports elementary school students' algebraic thinking. As design research study, it followed the three…

Learning progressions (LPs) describe the development of domain-specific knowledge, skills, and understanding. Each level of an LP characterizes a phase of student thinking en route to a target performance. The rationale behind LP development is to provide road maps that can be used to guide student thinking from one level to the next. The validity…

Problem Statement: Gaining reasoning skills in early years affects the formal proving skills in the following years, thus it is quite significant. The acquiring of this skill is only possible with the approaches that the teachers used in the process. At this point, the problem to be researched in terms of making proofs is seen in how middle school…

This article describes how in early mathematics learning, young children are often asked to recognize and describe visual patterns in their environment--perhaps on their clothing, a toy, or the carpet; around a picture frame; or in the playground equipment. Exploring patterns in the early years is seen as an important introduction to algebraic…

The importance of developing reasoning and justification has been highlighted in "Principles and Standards for School Mathematics" (NCTM 2000). The Common Core State Standards for Mathematics (CCSSI 2010) further reiterates the importance of reasoning and proof in several standards for mathematical practice. Students of all grades are…

Mathematics curricula in the early years emphasize exploring patterns as an important stepping stone to algebraic thinking. Key expectations for young children are for them to recognize and describe visual patterns in the environment. When children are asked to identify patterns, what criteria do they apply to make a decision and on what do they…

Initial exposure to algebraic thinking involves the critical leap from working with numbers to thinking with variables. The transition to thinking mathematically using variables has many layers, and for all students an abstraction that is clear in one setting may be opaque in another. Geometric counting and the resulting algebraic patterns provide…

Most future teachers are familiar with number patterns that represent an arithmetic sequence, and most are able to determine the general representation of the "n"th number in the pattern. However, when they are given a visual representation instead of the numbers in the pattern, it is not always easy for them to make the connection between the…

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…

The habit of looking for patterns, the skills to find them, and the expectation that patterns have explanations is an essential mathematical habit of mind for young children (Goldenberg, Shteingold, & Feurzeig 2003, 23). Work with patterns leads to the ability to form generalizations, the bedrock of algebraic thinking, and teachers must nurture…