For students taking higher level mathematics courses, the transition from computational to proof-based courses such as analysis and algebra not only introduces a new format of writing and communication, but also a new level of abstraction. This study examines the affordances of one particular tool to aid students in this transition: a proof…

Several studies indicate that exploring mathematical ideas by using more than one approach to prove the same statement is an important matter in mathematics education. In this work, we have collected a few different methods of proving the multinomial theorem. The goal is to help further the understanding of this theorem for those who may not be…

Learning standards such as the "Common Core State Standards for Mathematics" [CCSSM] (NGA Center & CCSSO, 2010) advocate that we develop students' algebraic thinking "beginning in kindergarten." Such a tall order requires innovative approaches that re-imagine what teaching and learning mathematics means for the elementary…

Detailed mathematical reasoning abilities can help students to understand higher mathematical abilities, including proving, problem-solving, and critical thinking. However, based on surveys and research, students' mathematical reasoning abilities are low and require significant attention. Therefore, this study aims to gain a detailed picture of…

The number line estimation task is frequently used to measure children's numerical magnitude understanding. It is unclear whether the resulting straight, horizontal, left-to-right-oriented estimate patterns indicate task constraints or children's intuitive number--space mapping. Three- to six-year-old children (N = 72, M[subscript age] = 4.89, 56%…

This paper presents the results of a study aimed at diagnosing how newly enrolled mathematics majors reason from a mathematical standpoint when they are required to perform inductive and deductive processes and validate or refute claims of a general nature. The theoretical framework underpinning this research comprises studies and investigations…

The aim of this pilot study was to examine the immediate and long-term effects of an intervention program that aims to improve preschoolers' (N = 36, M = 4.49 years, SD = 0.35) early numeracy skills by combining the learning of numerical relational skills via story reading with fundamental motor skill practice. The intervention program was piloted…

Accumulating evidence suggests that there is a spontaneous preference for numerical, compared to non-numerical (e.g., cumulative surface area), information. However, given a paucity of research on the perception of non-numerical magnitudes, it is unclear whether this preference reflects a specific bias towards number, or a general bias towards the…

Learning to calculate the area of composite shapes is an important application of area measurement but evidence suggests that many middle school students struggle to calculate the area of even simple composite shapes. In this article, I report the findings of a classroom design study conducted to investigate the collective development of…

Efforts to increase the number of young people in England studying mathematics post-16 have historically focused on participation in standalone mathematics qualifications. However, following the recent A-level reforms, many advanced level students are engaging with some form of mathematics through the mathematical content now formally embedded…

Determining preservice teachers' competence in proportional reasoning is a very complex task, and existing research methods do not provide effective tools in doing that. This study describes an interview structure and application of it in determining preservice teachers' competence in proportional reasoning. The interview structure consists of…

This research examines the relationship between prerequisite algebra knowledge and graduate corporate finance course performance. Standardized math questions designed for an undergraduate introductory finance course also predict quantitative performance metrics among 128 business graduate students. Results demonstrate that prerequisite math…

Calculus education research on integration is coalescing around the theme that teaching integration based on quantitative reasoning is crucial for robust understanding of and usage of integrals. This paper contributes to this International Journal for Research in Undergraduate Education special issue on the teaching and learning of definite…

Learning-support system is an umbrella term that we use for digital resources that assign students with mathematical questions and give automatic feedback on the inserted answers. Transitioning between questions and feedback is characteristic to students' work with such systems. We apply the commognitive framework to explore the role of…

This study explores students' use of mathematical objects in each problem-solving phase based on an onto-semiotic perspective. The subjects of this research are students who solve problems in different ways but all with the correct result. The first student uses organizational data by applying the concept of permutations. In contrast, the second…