Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with…

Generalisation is a skill that enables learners to acquire knowledge in general, and mathematical knowledge in particular. It is a core aspect of algebraic thinking and, in particular, of functional thinking, as a type of algebraic thinking. Introducing primary school children to functional thinking fosters their ability to generalise, explain and…

Background & aims: In recent years, there has been an increased interest in analyzing the mathematical performance of students with learning difficulties in order to provide them with teaching methods adapted to their needs. In particular, the importance of studying the type of informal strategy that students use when solving problems has been…

Learning trajectories are built upon progressions of mathematical understandings that are typical of the general population of students. As such, they are useful frameworks for exploring how understandings of diverse learners may be similar or different from their peers, which has implications for tailoring instruction. The purpose of this…

We conducted a comprehensive synthesis of computer-assisted instruction (CAI) research that aimed to facilitate the learning of mathematical word problem solving involving elementary and secondary school students with learning disabilities or difficulties in mathematics. We examined a total of 13 studies under the four instructional categories:…

Many schools use computer-based testing to measure students' progress for end-of-the-year and statewide assessments. There is little research to support whether computer-based testing accurately reflects student progress, particularly among students with learning, performance, and generalization difficulties. This article summarizes an…

In this paper, we describe our learning progressions approach to early algebra research that involves the coordination of a curricular framework, an instructional sequence, written assessments, and levels of sophistication describing the development of students' thinking. We focus in particular on what we have learning through this approach about…

This paper explored variation of strategy use in pattern generalization across different generalization types and across grade level. A test was designed to assess students' strategy use in pattern generalization tasks. The test was given to a sample of 1232 students from grades 4 to 11 from five schools in Lebanon. The findings of this study…

Individual differences in emotion, cognitions, and task choice following achievement failure are found among four- to seven-year-olds. However, neither performance deterioration during failure nor generalization after failure--aspects of the helpless pattern in 10-year-olds--have been reliably demonstrated in this age group. In the present study,…

A yearlong experimental study showed positive effects of a professional development project that involved 19 urban elementary schools, 180 teachers, and 3735 students from one of the lowest performing school districts in California. Algebraic reasoning as generalized arithmetic and the study of relations was used as the centerpiece for work with…

This guest editorial describes ways teachers can use guessing games about an unknown item in a "mystery box" to help children improve their abilities to listen to others, recall information, ask purposeful questions, classify items by class, make inferences, synthesize information, and draw conclusions. The author presents information…

This study examined the effects of providing explicit metacognitive instruction prior to cognitive strategy instruction on students' with learning disabilities ability to generalize a vocabulary strategy learned in a special education setting to a general education setting. The subjects were eight elementary school students with learning…

This study examined the effect of using an outdoor nature investigation program on the ability of third-grade students to transfer knowledge. Specifically, the study determined whether the program enhanced students' ability to transfer declarative (facts and concepts), procedural (process skills), and schematic (experience) knowledge to a set of…

Examined the problem-solving strategies of above average students (n=42) in grades 2-4 on problems involving forgotten or new material while practicing arithmetic with a computer. Identified the different problem-solving strategies used, sorted them into categories, and illustrated them with examples from students' protocols. Made suggestions for…