Purpose: It is crucial for teachers to understand what students already know and what they are thinking, before conducting their lessons, because identification of the cause is the first step of remedy. This study analyses students' conceptions that underlie the errors they make in algebraic expressions under two categories of conceptual…

The Klein bottle and hairy ball theorem are important concepts in advanced mathematics and they are both examples of the Poincaré-Hopf theorem. Complex theories such as these usually remain unrealized in the minds of mathematicians. In this collaborative work between a mathematician and a designer, we introduce the concept of the hairy Klein…

Preservice mathematics teachers are subject to multiple influences when they make in-the-moment instructional decisions in classroom settings. Little is known about which influences make a significant contribution to decisions regarding the mathematical content knowledge (MCK) that preservice teachers decide to enact. The process by which…

Machine learning (ML) methods for causal inference have gained popularity due to their flexibility to predict the outcome model and the propensity score. In this article, we provide a within-group approach for ML-based causal inference methods in order to robustly estimate average treatment effects in multilevel studies when there is cluster-level…

The retention of foundational knowledge is crucial in learning and teaching mathematics. However, a significant part of university students do not achieve long-term knowledge and problem-solving skills. A possible tool to increase further retention is testing, the strategic use of retrieval to enhance memory. In this study, the effect of a special…

This paper investigates the expectations that a group of preservice secondary mathematics teachers had for how students would approach four particular algebra problems. Their responses to a set of open-ended items were content analysed. Findings show that the teachers ranged in the expectations that they held for their students. Some teachers…

Mathematical learning is supported by caring teaching, which is complex, political work particularly for white teachers in racially diverse classrooms. To investigate white teachers' understandings of care and its influence on mathematics instruction, I conducted a six-month qualitative study in two racially diverse ninth-grade Algebra 1…

In this paper we studied the personal mathematical work of 14 students in initial teacher training, in their first year of studies at a public university in Chile, based on two tasks on complex numbers. These tasks were set in a Computer Aided Assessment System (CAA) based on Moodle and Wiris and it was proposed to solve them using different…

We conducted a descriptive exploratory study in which we analyzed 313 sixth to eighth grade students' answers to a word problem, accompanied by diagrams, involving generalization in an algebraic functional context. In this research, we jointly addressed two objectives: (a) to determine the strategies deployed by students to generalize and (b) to…

In this article we analyze how students reason about linear combinations across multiple digital environments. We present the work of three groups of undergraduate students in the Southeast United States (US) who were considered ready to take linear algebra. The students played the game "Vector Unknown," reflected upon aspects of their…

Research about the optimal methodologies for guiding the training of primary school instructors in the instruction of early algebra remains in the process of determination. This manuscript delineates the methodology involved in developing and validating an assessment tool to evaluate the mathematical acumen of prospective primary educators in the…

Developing structure sense is an important part of learning algebra. We investigated learners' structure sense of algebraic expressions involving brackets. This led us to propose the constructs "surface structure" sense and "systemic structure" sense. Using a random sample of 58 Grade 10 learners scoring above 40% in a test, we…

Because the failure rate for college algebra is large and sometimes up to 50%, college algebra serves as a gatekeeper to STEM fields and degree completion. Therefore, it is critical to examine students who are at risk of dropping, withdrawing, and failing college algebra help-seeking behavior. The purpose of this mixed methods study was to focus…

Online credit recovery (OCR) courses are the most common means through which students retake courses required for high school graduation. Yet a growing body of research has raised concerns regarding student learning in these courses, with low quality assessments posited as one contributing factor. To address this concern, we reviewed every…

Online collaborative learning (OCL) based on the community of inquiry (CoI) framework is often implemented in the form of an online discussion activity for teaching subjects that require learners to revise their understandings and invent ways to solve problems. However, implementing OCL is challenging due to the difficulty of fostering…