The problem solving performed by students in solving mathematical literacy problems shows that at the developmental stage of proportional reasoning students have different strategies, especially at the developmental stage of functional and scalar relationships. The purpose of this study is to identify and explore the developmental stages of…

A Prolog program consists of a set of facts and rules rather than imperative statements, commonly used in most other programming languages. Therefore, the Prolog language is used to encode logic, from which the inference engine deduces logical conclusions. In this article, we argue that the use of the Prolog language can be useful to help students…

Lockwood has argued that taking a set-oriented perspective is critical for successful combinatorial enumeration. To date, however, the research literature has not yet captured the cognitive processes involved in taking such a perspective. In this theoretical paper, we elaborate the constructs of spatial structuring and spatial-numerical linked…

Teachers have difficulty integrating proof in their mathematics instruction due to both narrow beliefs about proofs and limited understanding of proofs. Indirect proofs seem to be a particular cause for concern. In this exploratory study, we contribute to the research area by reporting on an empirical study of Norwegian pre-service teachers'…

Decades of research have established that solving geometry proof problems is a challenging endeavor for many students. Consequently, researchers have called for investigations that explore which aspects of proving in geometry are difficult and why this is the case. Here, results from a set of 20 interviews with students who were taught proof in…

The purpose of this study is to analyse the effects of the Interactive Learning Materials Triangle (iLMT) on the learning and knowledge retention of 12-year-old students. The iLMT is a digital version of the standard school learning materials in Serbia, and is characterized by a high degree of interactivity and immediate feedback during the…

"Productive Struggle" refers to a strategy that gives students an opportunity to increase their background mathematical knowledge. Productive Struggle helps students connect key concepts, determine how and where an error occurs, and supports students in figuring out how to use their own thinking and reasoning skills to correct an error.…

Comparison is a powerful and important way that we learn. To support teachers in the use of comparison in their instruction, the authors developed an instructional routine called compare and discuss multiple strategies (CDMS). Similar to other instructional routines, CDMS is a structured, specific, repeatable minilesson that teachers can use to…

In this article, the authors share frameworks for problem types and students' reasoning about integers. The authors found that all ways of reasoning (WoRs) were used across grade levels and that specific problem types tended to evoke particular WoRs. Specifically, students were more likely to use analogy-based reasoning on all-negatives problems…

The algebraic structure is one of the axiomatic mathematical materials that consists of definitions and theorems. Learning algebraic structure will facilitate the development of logical reasoning, hence facilitating the study of other aspects of axiomatic mathematics. Even with this, several researchers say a lack of algebraic structure sense is a…

Duncan Symons and Derek Holton discuss the different types of mathematical reasoning and what each of these might look like in the classroom. By suggesting language that can be used to describe the different methods of reasoning, they hope to provide teachers with the tools they need to better recognise and assess student reasoning.

The hammer-and-nail phenomenon highlights human tendency to approach a problem using a tool with which one is familiar instead of analyzing the problem. Pedagogical suggestions are offered to help students minimize their mathematical impulsivity, cultivate an analytic disposition, and develop conceptual understanding.

Using simulation approaches when conducting randomization tests for comparing two groups in the context of experimental studies has been promoted as a beneficial approach for supporting student learning of statistical inference. Many researchers have suggested that the data production process in simulations for the randomization test intuitively…

Mathematical proficiency is essential to supporting students' success in learning mathematics. It is the ability to use conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition in problem-solving. Cognitive independence shows students' ability to process information. Meanwhile, motivation…

In this study we investigated pre-service teachers' (PSTs) proportional reasoning and how they interpret their calculations in proportional tasks. We administered a written questionnaire to 199 PSTs and used an inductive content analysis approach for data analysis. We found that one item, in which PSTs were asked to interpret the meaning of the…