This study aims at elaborating a well-established theoretical framework that distinguishes three modes of thinking in linear algebra: the analytic-arithmetic, the synthetic-geometric, and the analytic-structural mode. It describes and analyzes the bundle of signs produced by an engineering student during an interview, where she was asked to recall…

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and…

Although there have been a huge number of attempts to improve school algebra teaching, several countries are still struggling to improve students algebraic skills. In this study, we focus on the specific case of Sweden where students for several decades have had major problems mastering algebra. In order to get a better understanding of the…

This article reports on an empirical study that investigates the way students' performance in solving arithmetical tasks may be related to their performance in solving algebraic tasks. The sample consisted of 203 Grade 6 students. The arithmetical tasks involved arithmetical expressions with known quantities, whereas the algebraic tasks involved…

To understand the ways that manipulatives might support changes in students' reasoning about algebraic generalizations, a constructivist teaching experiment was conducted with two sixth-grade students. The students interpreted numerical situations with units of one and could construct units of units in mental activity. Initially, the students'…

This manuscript contributes to research on how algebraic thinking about operations and properties can develop, and relevant forms of curricular activity. The key question asked is: how can students' early algebraic activity be fostered through focusing on relations involving operations and properties? We adopt Radford's three aspects of algebraic…

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…

We investigated understanding of the linear independence concept based on the type and nature of connections displayed in seven non-mathematics majors' interview responses to a set of open-ended questions. Through a qualitative analysis, we identified six categories of frequently displayed connections. There were also recognizable differences in…

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…

The mathematics education literature indicates a consensus regarding the importance of developing algebraic thinking in elementary school mathematics. However, the approaches used to implement this concept vary around the world. This study examines how a popular standards-based elementary mathematics textbook series in China provides opportunities…

Students' difficulties in algebra are generally caused by the use of algebraic notation, the meaning of letters, as well as the types of relationships or methods used. Therefore, interpreting letters in algebra is one of the critical points in the transition from arithmetic to algebra. However, many students have misconceptions in interpreting…

In this paper, we describe an experiment in using history to work on problem-solving and the relationship between arithmetic and algebra. The students involved attended the first year of the Italian upper secondary school (grade 9). The original sources we used are problems from Italian treatises on arithmetic and algebra that appeared in the…

This study explored the mathematical learning experiences of adult students enrolled in a community college developmental mathematics course during the Fall 2023 semester. The study's main objective was to identify the types of experiences that help adult students learn foundational mathematics. Another goal of the study was to provide empirical…

The aim of the exploratory method research centered on the presence of mathematical tools in STEM through three main questions: Is mathematics an essential tool in the field of STEM? Can mathematics complete projects solely with mathematical and digital tools? Does understanding mathematical modeling affect STEM teaching? A better understanding of…

The paper presents some reflections and activities developed by researchers and teachers involved in teacher education programs on cultural transposition. The construct of cultural transposition is presented as a condition for decentralizing the didactic practice of a specific cultural context through contact with other didactic practices of…