This study investigated the role of tools in supporting students to reason about even and odd numbers. Participants included Kindergarten, Grade 1, and Grade 2 students (ages 5-8) at two schools in the USA. Students took part in a cross-sectional early algebra intervention in which they were asked to generalize, represent, justify, and reason 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…

Cognitive development of eighth-grade students, as identified by Jean Piaget, occurs during a time when many of them are transitioning between concrete operations and formal operations where the ability to think in abstract concepts becomes possible. Because of this period of transition, many eighth-grade students find difficulty in demonstrating…

This study examines the collective mathematical reasoning when students and teachers in grades 3, 4, and 5 explore fractions derived from length comparisons, in a task inspired by the Elkonin and Davydov curriculum. The analysis showed that the mathematical reasoning was mainly anchored in mathematical properties related to fractional or algebraic…

The aim of this study was to examine 6th-grade students' mathematical abstraction processes related to the concept of variable by using the teaching experiment method and to reveal their learning trajectories in the context of the RBC+C model. A teaching experiment was administered to a class of 29 middle school students for 3 weeks. Observations,…

Elementary standards include multiplication of single-digit numbers and students advance to solve complex problems and demonstrate procedural fluency in algorithms. The ability to illustrate procedural fluency in algorithms is dependent on the development of understanding and reasoning in multiplication. Development of multiplicative reasoning…

A study was conducted to explore the beliefs and practices of 49 New South Wales (NSW) primary school teachers regarding their beliefs and practices concerning the use of concrete materials in the learning and teaching of Number and Algebra. This paper reports on elements of the study regarding why and how teachers use concrete materials. Not only…

Background: Key to optimizing Computational Thinking (CT) instruction is a precise understanding of the underlying cognitive skills. Román-González et al. (2017) reported unique contributions of spatial abilities and reasoning, whereas arithmetic was not significantly related to CT. Disentangling the influence of spatial and numerical skills on CT…

Students in the elementary grades often experience difficulty setting up and solving word problems. Using an equation to represent the structure of the problem serves as an effective tool for solving word problems, but students may require specific pre-algebraic reasoning instruction about the equal sign as a relational symbol to set up and solve…

Explaining appears to dominate primary teachers' understanding of mathematical reasoning when it is not confused with problem solving. Drawing on previous literature of mathematical reasoning, we generate a view of the critical aspects of reasoning that may assist primary teachers when designing and enacting tasks to elicit and develop…

This paper examines the implementation of an instructional module on preservice elementary teachers' professional noticing of children's mathematical thinking. The module focuses on professional noticing skills through the content focus of early algebraic reasoning and uses complex video vignettes from whole class instruction in authentic…

"Modeling Mathematical Ideas" combining current research and practical strategies to build teachers and students strategic competence in problem solving.This must-have book supports teachers in understanding learning progressions that addresses conceptual guiding posts as well as students' common misconceptions in investigating and…

Looking for, recognizing, and using underlying mathematical structure is an important aspect of mathematical reasoning. We explore the use of mathematical structure in children's integer strategies by developing and exemplifying the construct of logical necessity. Students in our study used logical necessity to approach and use numbers in a…

A common approach used for introducing algebra to young adolescents is an exploration of visual patterns and expressing these patterns as functions and algebraic expressions. Past research has indicated that many adolescents experience difficulties with this approach. This paper explores teaching actions and thinking that begins to bridge many of…

Sixty-five Year 2 children with ages ranging from six to seven years participated in a teaching experiment to introduce functional thinking. The results show that young children are capable of generalising, can provide examples of relations and functions, can describe the inverse of such relationships and give valid reasons for how they found the…