This paper studies grade 7 (age 13) students' expressed understanding about the equal sign/notion of equivalence in order to investigate what aspects of the concept that could be seen as an established knowledge at lower secondary school/middle school. Using items from different instruments and combining these to a new one that covered a broad…

In learning mathematics in the 21st century and after the COVID-19 pandemic, a multimodal role in the process of drawing conclusions involving mathematical symbols or signs is very much needed. This process is called multimodal semiotic reasoning. This research aims to study the literature on multimodal semiotic reasoning research articles. Until…

The teacher's use of representations is a crucial aspect of instructional quality in mathematics education, given their pivotal role in facilitating mathematics learning. However, in our international research community, perspectives on what constitutes high-quality use of representations may vary. This cross-cultural study aims to explore whether…

Generalization is one of the core aspects of algebraic thinking, and research literature points to the importance of objectification when generalizing mathematical discourses. To increase the field's understanding of the objectification process in various algebra discourses, our study analyses Grade 6 teachers' use of words, symbols, routines and…

Research has shown there are algebra concepts elementary teachers can introduce that help prepare elementary students for the eventual transition to algebra (e.g., the relational interpretation of the equal sign). "Early algebra" refers to the use of informal approaches to introduce such concepts to elementary students. "Strip…

This qualitative case study analyzed how first year university mathematics students used words, images, and symbols to convey the same mathematical ideas. The study was located within the interpretivist paradigm and took naturalistic methodology. Twenty-six first year students were purposefully selected to participate in the study. Data was…

In high school geometry, proving theorems and applying them to geometry problems is an expectation for students. An essential part of most geometry proofs is the diagram because it not only helps encapsulate the claim being proved but can also be a tool in reasoning or communicating an argument. This interview-based study investigated how high…

Research in the area of equivalence and the equal sign dates back decades, demonstrating students often possess misconceptions concerning the meaning of equivalence and the equal sign. Students often understand the equal sign to mean the answer comes next (i.e., an operational definition) instead of the same as (i.e., a relational definition).…

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…

A language is a system of communication, consisting of a set of sounds or written symbols that enable people to communicate. In chemistry, a particular language is required in order to represent the phenomenological world by means of symbols. Choosing the right words and knowing the precise definitions for chemical concepts is needed for avoiding…

In this study, we investigated how secondary students interpret algebraic expressions that contain literal symbols to stand for variables. We hypothesized that the natural number bias (i.e., the tendency to over-rely on knowledge and experiences based on natural numbers) would affect students to think that the literal symbols stand for natural…

Although previous research has demonstrated that the acquisition of mathematical skills requires support from multiple cognitive abilities, the associations between cognitive precursors in different domains and mathematics at different hierarchical levels among primary school children are not well understood. This study explores the cognitive…

Researchers have demonstrated a much-needed shift in pedagogical practices to incorporate literacy strategies. This dissertation provides additional empirical evidence to expand the body of research by utilizing a comparative readability analysis that examines the academic language of a popular calculus textbook. A sample from the published…

Theoretical yield and extent of reaction (degree of completion) are important concepts in stoichiometry and in chemical kinetics. We provide expressions describing theoretical yield (Y) and maximum extent of reaction (fractional yield, f[subscript max]) for five examples of commonly encountered reversible chemical reactions: A [equilibrium sign]…

This study investigated how students reason about the partial derivative and the directional derivative of a multivariable function at a given point, using different graphical representations for the function in the problem statement. Questions were formulated to be as isomorphic as possible in both mathematics and physics contexts and were given…