The tradition of studies involving the combinatorial approach to recurring numerical sequences has accumulated a few decades of tradition, and several problems continue to attract the interest of mathematicians in several countries. This work specifically discusses the Fibonacci, Pell, and Jacobsthal sequences, focusing on Mersenne sequences. The…

Even though there is a great temptation as teachers to share what is known, many are aware of an idea called "rules that expire" (RTE) and have realized the importance of avoiding them. There is evidence that students need to understand mathematical concepts and that merely presenting rules to carry out in a procedural and disconnected…

This article presents a checklist of 10 evidence-based practices for educators to apply in mathematics instruction for students with learning disabilities. The checklist is "actionable," meaning the items on the checklist can be put into action immediately. It provides practical strategies teachers can adopt to fit their lessons…

The field of second language studies is using increasingly sophisticated methodological approaches to address a growing number of urgent, real-world problems. These methodological developments bring both new challenges and opportunities. This article briefly reviews recent ontological and methodological debates in the field, then builds on these…

As an alternative to looking solely at linear functions, a three-lesson learning progression developed for Year 6 students that incorporates triangular numbers to develop children's algebraic thinking is described and evaluated.

Humans routinely make inductive generalizations about unobserved features of objects. Previous accounts of inductive reasoning often focus on inferences about a single object or feature: accounts of causal reasoning often focus on a single object with one or more unobserved features, and accounts of property induction often focus on a single…

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…

Problem solving allows students to use what they know to achieve a goal when no solution is apparent. Traditional educational models evolved from an earlier system, based on rote memorization and designed to produce employees for industry. The workforce of tomorrow must move beyond rote learning by both applying current knowledge and using…

The note introduces sequences having M-bonacci property. Two summation formulas for sequences with M-bonacci property are derived. The formulas are generalizations of corresponding summation formulas for both M-bonacci numbers and Fibonacci numbers that have appeared previously in the literature. Applications to the Arithmetic series, "m"th "g -…

The purpose of this study was to determine the effect of systematic instruction with a concrete representation on the acquisition of an algebra skill for students with moderate developmental disabilities. Three high school students with moderate developmental disabilities participated in this study. A multiple probe across participants research…

This paper demonstrates how to design an instructional process aimed at teaching general methods of thinking, using the Landamatics theory and methodology. Landamatics is not a collection of lesson plans, but rather a general method of approaching the design of any effective course of instruction or any lesson plan. The method formulates general…

Effects of an intervention that combined self-instruction with multiple exemplar training on the generalized problem solving of five high school students with mental retardation were examined. The intervention involved preteaching self-instruction to proficiency with one exemplar before introducing multiple exemplars and embedding problem…

A metacognition model that can help understand general problem-solving deficits in learning disabled students is presented. Two components of metacognition are highlighted: executive processes and attributional beliefs. An educational package combining these components with specific strategy training is illustrated as an approach to improving…

The paper describes a pilot project in which 17 hearing impaired adolescents are participating in a program to improve thinking skills via the Instrumental Enrichment (IE) approach. IE focuses on concepts by suggesting alternative strategies by which the learner can arrive at the objective, thus allowing for opportunities to reflect on one's own…

The reasons people most often give for the failures of U.S. schools involve poverty, racial inequality, and a host of social problems. This paper argues that even if all these conditions were remedied, the schools would not produce many more people with the ability to think than they do today. Teachers, who are usually able to think, do not know…