This work explores the complex cognitive processes students engage in when addressing contextual tasks requiring linear and exponential models. Grounded within Piagetian constructivism and the Knowledge in Pieces (KiP) epistemological perspective (diSessa, 1993, 2018), this empirical study in a clinical setting develops a Microgenetic Learning…

The evolution of mathematics coincided with advancements in its teaching. The 19th and 20th centuries marked a pedagogical revolution in mathematics education. This paper argues that Bruner's (1966) model, Gagné's (1985) taxonomy, innovative teaching methods emphasizing research and problem-solving, and the inclusion of data analysis topics have…

Recursive reasoning is a powerful tool used extensively in problem solving. For us, recursive reasoning includes iteration, sequences, difference equations, discrete dynamical systems, pattern identification, and mathematical induction; all of these can represent how things change, but in discrete jumps. Given the school mathematics curriculum's…

This paper presents data from the first of three iterations of teaching experiments conducted with secondary teachers. The purpose of the experiments was to investigate how teachers' combinatorial reasoning could support their development of algebraic structure, specifically structural relationships between the roots and coefficients of…

This study investigated how Japanese curricula represent functional relationships through the lenses of quantitative and covariational reasoning. Utilizing both macro and micro textbook analyses, we examined the tasks, questions, and representations in the Japanese elementary and lower secondary course of study, teachers' guide, and textbooks.…

This research aims to analyze the results of studies conducted in Turkey on reasoning skills in mathematics teaching and to reveal what kind of trend there is in this field. Within the scope of this study, databases were searched with the keywords "reasoning"(muhakeme, akil yürütme) and "reasoning skill" (Muhakeme becerisi,…

The authors define two mathematical cognitive verbs which are fundamental to the development of mathematical thinking and reasoning. They distinguish between 'describing' and 'explaining' in relation to doing mathematics, rather than using them interchangeably.

Understanding fraction as a quantity has been identified as a key developmental understanding. In this study, students in Grades 5, 8, and 11 were asked to compare the areas of two halves of the same square--a rectangle and a right triangle. Findings from this study suggest that students who understand fraction as a quantity use reasoning related…

Generalisation is a key feature of learning algebra, requiring all four proficiency strands of the Australian Curriculum: Mathematics (AC:M): Understanding, Fluency, Problem Solving and Reasoning. From a review of the literature, we propose a learning progression for algebraic generalisation consisting of five levels. Our learning progression is…

By using high-leverage models to connect student learning experiences to overarching concepts in mathematics, teachers can anchor learning in ways that allow students to make sense of content on the basis of their own prior experiences. A rectangular area model can be used as a tool for understanding problems that involve multiplicative reasoning.…

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…

Geometrical concepts play a crucial role in developing spatial thinking and reasoning. Further, curricular materials play a key role in shaping student-learning experiences in the classroom. The organisation of the content of textbooks plays a decisive role in how and when students are introduced to concepts, especially given the…

Making sense of mathematical concepts and solving mathematical problems may demand different forms of reasoning. These could be either domain-based, such as algebraic, geometric or statistical reasoning, while others are more general such as inductive/deductive reasoning. This article aims at giving visibility to a particular form of reasoning…

The mathematical proficiencies in the "Australian Curriculum: Mathematics" of understanding, problem solving, reasoning, and fluency are intended to be entwined actions that work together to build generalised understandings of mathematical concepts. A content analysis identifying the incidence of key proficiency terms (KPTs) embedded in…

This paper presents the findings of a small-scale study that investigated the issues and challenges of teaching and learning about quantitative reasoning (QR) within a project-based learning (PjBL) context. Students and teachers were surveyed and interviewed about their experiences of learning and teaching QR in that context in contrast to…