Raising expectations is nothing new. Every iteration of standards elevates the expectations for what students should know and be able to do. The Common Core State Standards for Mathematics (CCSSM) is no exception, with standards for content and practice that move beyond memorization of traditional algorithms to "make sense of problems and…

In Indonesia, proportion is being taught formally in Grade 5 (10-11 years old). However, the existing learning approach does not support the development of the students' proportional reasoning. The way to teach proportion by giving cross multiplication is not meaningful for the students. They just memorize the procedure without understanding how…

Gain a new perspective on the sharing of erroneous solutions in classroom discussions. Based on their research in grades four and six, the authors reveal how student-to-student correction of errors promotes mathematical reasoning and understanding. Tips for teachers include strategies for using students' errors to encourage reasoning during…

This article will describe a writing assignment designed for use in a liberal arts college whose mission stresses effective written communication both within and across disciplines. In this assignment, students write three separate solutions to the same mathematics problem: one for a mathematical peer, a second for a contemporary that does not…

Large-scale numeracy assessments are intended to facilitate the improvement of educational outcomes; however, it is not clear exactly how this is to be achieved. To move towards the goal of numeracy for all, it is necessary to systematically address issues that are known to be difficult, pervasive and persistent. This paper includes an analysis of…

There has been a controversial debate if individuals solve fraction comparison tasks componentially by comparing the numerators and denominators, or holistically by considering the numerical magnitudes of both fractions. Recent research suggested that expert mathematicians predominantly use componential strategies for fraction pairs with common…

Proportional reasoning is an important cornerstone in children's mathematical development. This sort of reasoning has been shown to develop across the early years of schooling (ages 8 to 10) through the middle years (ages 11-14). In the early years, children tend to use additive reasoning to generate solutions to problems, while later comparisons…

In this article, we present an integer sequence approach to solve the classic water jugs problem. The solution steps can be obtained easily by additions and subtractions only, which is suitable for manual calculation or programming by computer. This approach can be introduced to secondary and undergraduate students, and also to teachers and…

Mastering algebra is important for future math and postsecondary success. Educators will find practical recommendations for how to improve algebra instruction in the What Works Clearinghouse (WWC) practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students". The methods and examples included in…

The aim of this paper is to present and discuss some of the evidence regarding the resources that students use when they establish relationships between a contextual situation and an ordinary differential equation (ODE). We present research results obtained from work by seven students in a graduate level course in mathematics education, where they…

Teachers who participated in a recent summer institute focused on implementing the Common Core State Standards for Mathematics (CCSSM). Throughout the summer institute, teachers had many in-depth discussions about the meanings they attributed to the Standards for Mathematical Practice (SMPs). In this article, the authors share some of the…

This study explores the "language repair strategies" (aimed at repairing communication problems) of two bilingual speakers during mathematics word problem tutoring sessions. Bilingual repair was shown to gradually shift from a linguistic to an epistemic focus during problem solving (i.e., communication became more conceptually focused…

In teaching proportion, the teachers usually give the students a formal formula, such as cross-multiplication, without providing them the meaning of proportionality. Thus, the students just apply it in order to solve proportional problems. However, it does not connote that they understand what proportionality means. In this paper, the authors…

Effective teaching practice for improving student achievement in mathematics is a critical area for instructional design. Further, results from international assessments of mathematics achievement have indicated that students in Korea typically earned test scores higher then international averages. The purpose of this study was to investigate the…

Combinatorial enumeration has a variety of important applications, but there is much evidence indicating that students struggle with solving counting problems. The roots of such difficulty, as well as ways to mitigate such difficulty, have not yet been thoroughly studied. In this paper, one particular aspect of students' counting activity is…