This paper builds on our previous research and investigates how students' fractional competence and reasoning can provide clear evidence of non-symbolic algebraic thinking and its progressive transition towards fully generalised algebraic thinking. In a large-scale study, 470 primary students completed a written paper and pencil test. This…

One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they perceive as arbitrary without being able to justify these rules. A good algebraic understanding involves connecting the transformation rules to other characterizations of equivalence of…

Understanding of the equal sign is associated with early algebraic competence in the elementary grades and equation-solving success in middle school. Thus, it is important to find ways to build foundational understanding of the equal sign as a relational symbol. Past work promoted a conception of the equal sign as meaning "the same as".…

An understanding of the equal sign is a fundamental concept for early algebra. While literature claimed that Chinese students commonly master the relational understanding of the equal sign in the elementary school, these claims are under-researched. This study used the Mathematics Equivalence Assessment with 237 Chinese Grade 5 students. The…

The relation to operands (RO) principles describe the relation between operands and answers in arithmetic problems (e.g., the sum is always larger than its positive addends). Despite being a fundamental property of arithmetic, its empirical relation with arithmetic/algebraic problem solving has seldom been investigated. The current longitudinal…

Mathematical learning has an important role and is often prioritized in education. In K-16 education, algebra is one of the most vital mathematical content domains: it represents one of the top barriers for students pursuing a postsecondary education. Game-based learning has been effective in fostering classroom math learning environments that are…

This study investigated the role of online applets in early algebra lessons. The effect of two different types of intervention modules on developing students' early algebraic thinking abilities was compared. The first intervention module involved the use of open applets and real-life contexts (open-real). The second intervention module involved…

We describe 24 third (8-9 years old) and 24 fifth (10-11 years old) graders' generalization working with the same problem involving a function. Generalizing and representing functional relationships are considered key elements in a functional approach to early algebra. Focusing on functional relationships can provide insights into how students…

Functional thinking is an established route into algebra. However, the learning mechanisms that support the transition from arithmetic to functional thinking remain unclear. In the current study we explored children's pre-instructional intuitive reactions to functional thinking content, relying on a conceptual change perspective and using mixed…

Supporting students with mathematics learning difficulties (MLD) as they transition from arithmetic to algebra remains a challenge within the special education and mathematics education research communities. This dissertation includes 3 manuscripts focused on how fifth graders with MLD develop algebraic understandings related to equivalence and…

If chosen carefully according to teaching purposes and incorporated into the curriculum, games can enhance teaching and learning. The use of didactic games while teaching maths can support students' motivation and performance during the lesson. Therefore, educational games should be preferred in maths classes as an alternative teaching method. In…

For students to advance beyond arithmetic, they must learn how to attend to the structure of math notation. This process can be challenging due to students' left-to-right computing tendencies. Brackets are used in mathematics to indicate precedence but can also be used as superfluous cues and perceptual grouping mechanisms in instructional…

This program focused on improving mathematics content and pedagogy for fifth-grade teachers, including regular classroom teachers and exceptional education teachers, as they implemented the Common Core State Standards (CCSS) for Mathematics. Concentrating work with a single grade allowed for tightening breadth while increasing depth of content.…

Making connections within and between different aspects of mathematics is recognised as fundamental to learning mathematics with understanding. However, exactly what these connections are and how they serve the goal of learning mathematics is rarely made explicit in curriculum documents with the result that mathematics tends to be presented as a…

The transition between arithmetic and algebraic thinking is challenging for students. One notable difficulty for students is understanding algebraic symbols--pronumerals. Researchers are exploring pedagogical approaches in seeking to address this issue. The current paper is contributing to this body of literature by illustrating how an online card…