Although students' invented strategies typically prove to be meaningful and effective in improving the students' mathematical understanding, much remains unexplored in the current literature. This study examined, through a teaching-scenario task, the nature of 80 preservice teachers' reasoning and responses to students' informal and formal…

Current curriculum initiatives (e.g., National Governors Association Center for Best Practices and Council of Chief State School Officers 2010) advocate that models be used in the mathematics classroom. However, despite their apparent promise, there comes a point when models break, a point in the mathematical problem space where the model cannot,…

This article contributes to the research literature concerning prospective elementary teachers' mathematical thinking and learning with a focus on flexibility. We present a case study of a prospective elementary teachers' development of flexibility in mental addition and subtraction during a Number and Operations course. Building upon the…

There is a call for enabling students to use a range of efficient mental and written strategies when solving addition and subtraction problems. To do so, students should recognise numerical structures and be able to change a problem to an equivalent problem. The purpose of this article is to suggest an activity to facilitate such understanding in…

Algorithms and representations have been an important aspect of the work of mathematics, especially for understanding concepts and communicating ideas about concepts and mathematical relationships. They have played a key role in various mathematics standards documents, including the Common Core State Standards for Mathematics. However, there have…

The transition from whole numbers to integers involves challenges for both students and teachers. Leadership in mathematics education calls for an ability to translate depth of understanding into effective teaching methods, and this landscape includes alternative treatments of familiar topics. Noting the multiple meanings associated with the…

This article discusses a triangle of numbers that are related to the derangement numbers. These numbers satisfy a Pascal-like recurrence relation with subtraction instead of addition. We describe how they relate to numbers studied by other authors and use them to generalize Euler's famous recurrence relation for the derangement numbers.

The data used for the qualitative analysis reported here were generated as part of a larger study to understand and characterize teacher practice related to engaging students in algorithmic thinking associated with the fraction operations of addition, subtraction, multiplication and division. This paper presents ways in which teachers used…

This study explores the extent to which a teacher elicited students' mathematical reasoning through the use of challenging tasks and the role her knowledge played in doing so. I characterised the teacher's knowledge in terms of a "local theory" of instruction, a form of pedagogical content knowledge that involves an empirically tested set of…

Textbooks, like many other resources teachers have at hand, are meant to be an aid for instruction; however there is little research with textbooks or on their potential to develop metacognitive knowledge. Metacognitive knowledge has received substantial attention in the literature, in particular for its relationship with problem-solving in…

This book is about playing with mathematics, teaching: (1) How to multiply and divide large numbers mentally; (2) Making addition and subtraction easy; (3) Tricks for understanding fractions and decimals; and (4) How to quickly check answers for each calculation. Described as a foolproof way to do multiplication, division, factoring estimating,…

The purpose of this curriculum guide is to assist Training Specialists in teaching Adult Basic Education/General Educational Development (ABE/GED) students basic skills in language arts and mathematics. The guide begins with an introduction (purpose, overview, adult learner characteristics, teacher type), curriculum policies, curriculum structure,…