The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge--rather than understanding of mathematical concepts and procedures--to…

Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge--rather than understanding of mathematical concepts and procedures--to…

A mathematical mnemonic is a visual cue or verbal strategy that is used to aid initial memorisation and recall of a mathematical concept or procedure. Used wisely, mathematical mnemonics can benefit students' performance and understanding. Explorations into how mathematical mnemonics work can also offer students opportunities to engage in proof…

This paper provides insight about the development of addition and subtraction strategies for nearly 22,000 Australian primary school children in 2016. The children were each assessed by their teacher using a task-based assessment interview that identified the strategies they used to mentally perform addition and subtraction, and matched these to a…

The study aimed to investigate variations in addition and subtraction fluency by observing grade three students in Norway (n = 253, M[ subscript age] = 8.38 y.) and Finland (n = 209, M[subscript age] = 9.35 y.) while controlling for their age and non-verbal reasoning. Gender differences were also examined. The focus of the study was on the…

In this study, the systematic use of variation and invariance in the teaching of mathematics is examined in accordance with two different but compatible explicit frameworks. We consider the following: (1) differences in tasks that follow on from each other can significantly change what can be learned; (2) Chinese teachers (and probably those in…

The aim of the study is to investigate the effects of using computer for developing arithmetical skills of students with mathematics learning difficulty (MLD). The study was carried out with pre-test - post-test single subject research design. The participants of the study consist of a girl and two boys who attend 3rd grade at elementary school.…

The current study examines the mental computation performance owned by students at fifth grade. This study was carried out with 118 fifth graders (11-12-year-olds) studying at 3 randomly selected primary schools that served low and middle socio-economic areas in a city of Turkey. "Mental Computation Test (MCT)" has been used to reach how…

Mathematically speaking, a "difference" is the result of a subtraction. However, when the number domain is extended from natural numbers to integers, the separation of the magnitude of a number from its value creates "different differences," where the connection to subtraction is no longer straightforward. Based on…

Measuring computational fluency, an aspect of procedural fluency, is complex. Many attempts to measure this construct have emphasised accuracy and efficiency at the expense of flexibility and appropriate strategy choice. Efforts to account for these latter constructs through assessing children's computational reasoning using structured interviews…

This article draws on some ideas explored during and after a writing workshop to develop classroom resources for the reSolve: Mathematics by Inquiry (www.resolve.edu.au) project. The project develops classroom and professional learning resources that will promote a spirit of inquiry in school mathematics from Foundation to year ten. The…

Teaching of addition, subtraction, multiplication and division in mathematics starts from the first years of primary school. The learning output for four operations (addition, subtraction, multiplication and division) affects student success at every level of mathematics education from primary to higher education. At this point errors,…

Whole number arithmetic is the foundation of higher mathematics and a core part of elementary mathematics. Awareness of pattern and underlying problem structure promote the learning of whole number arithmetic. A growing consensus has emerged on the necessity to provide students with the opportunity to engage in algebraic reasoning earlier in their…

This is the second of a two-part article that presents a theory of unit construction and coordination that underlies radical constructivist empirical studies of student learning ranging from young students' counting strategies to high school students' algebraic reasoning. In Part I, I discussed the formation of arithmetical units and composite…