Using a mixed methods approach, we explore a relationship between students' graph reasoning and graph selection via a fully online assessment. Our population includes 673 students enrolled in college algebra, an introductory undergraduate mathematics course, across four U.S. postsecondary institutions. The assessment is accessible on computers,…

Reversibility thinking carried out mentally in mathematical operations has an important role in the process of understanding concepts as it involves developing a thinking process from beginning to end and from end to beginning. This qualitative research aims to describe students' reversible thinking processes in solving algebra problems,…

Mathematics is a collection of cognitive products that have unique characteristics from other scientific disciplines. As cognitive products, mathematics, and thought processes are two things that cannot be separated. Although in the literature there have been many approaches proposed to support the analysis of students' thinking processes, not…

Algebraic thinking is the ability to generalize about numbers and calculations, find concepts from patterns and functions and form ideas using symbols. It is important to know the student's algebraic thinking process, by knowing the student's thinking process one can find out the location of student difficulties and the causes of these…

Researchers from multiple disciplines have found that fractions and algebra knowledge are linked. One major strand of research has identified children's "units coordination" structures as crucial for success with fractions and algebra via multiplicative reasoning, whereas a second strand of research points to "magnitude…

Researchers from multiple disciplines have found that fractions and algebra knowledge are linked. One major strand of research has identified children's "units coordination" structures as crucial for success with fractions and algebra via multiplicative reasoning, whereas a second strand of research points to "magnitude…

This paper presents results of a study aimed at describing and discussing evidence/features of advanced algebraic thinking processes. To achieve these objectives, we analysed the written production of students enrolled in engineering courses working on mathematical modelling tasks related to differential equations. Our guiding question was as…

The design principles of cognitive load theory and learning by analogy has independently contributed to our understanding why an instruction will or will not work. In an experimental study involving 97 Year 9 Australian students conducted in regular classrooms, we evaluated the effect of the unguided problem-solving approach, worked examples…

In this paper, we report the analysis of thought processes used by Pre-Service Teachers' (PSTs') through clinical interviews as they solved an algebra task involving a linear pattern. The PST's were asked about a mathematical model they had constructed to describe a pattern problem. Our analysis suggests that conflict factors arise due to…

This paper discusses findings from an ongoing study investigating mental mechanisms involved in the conceptualisation of linear transformations from the perspective of Action (A), Process (P), Object (O), and Schema (S) (APOS) theory. Data reported in this paper came from 44 first-year linear algebra students' responses on a task regarding the…

This study examined college calculus instructors' preferences in solving two calculus tasks to examine college calculus instructors' use of important cognitive roots in understanding derivatives of function. Our results showed that only one instructor consistently uses cognitive roots while other instructors either focus on algebraic methods or…

Research on dyscalculia has focused almost exclusively on elementary-aged students' deficits in speed and accuracy in arithmetic calculation. This case study expands our understanding of dyscalculia by documenting how one college student with dyscalculia understood algebra during a one-on-one design experiment. A detailed case study of 19 video…

In this study, it is aimed to determine the algebraic thinking habits of two eighth grade school students through the answers they gave in the process of solving mathematical problems. The algebraic habits of mind (ZCA) theoretical framework developed by Driscoll (1999) was used to reveal these thinking habits. The research design of this study is…

The importance of mathematical connections in teaching mathematics is broadly acknowledged in the literature. Nevertheless, more research is needed to clarify how different types of mathematical connections manifest in classroom interactions and how they can support student learning. This article uses the commognitive framework to analyse 15…

Over the last several centuries, mathematicians have developed sophisticated symbol systems to represent ideas often imperceptible to their five senses. Although conventional definitions exist for these notations, individuals attribute their personalized meanings to these symbols during their mathematical activities. In some instances, students…