A one-degree angle would certainly be difficult to draw accurately, especially without the aid of other tools like a protractor. However, mentally making a one-degree angle in visualized imagination is not only possible, but it is important for students' emergent understandings of angular measure. When students begin learning about length,…

This article draws from Hiroko Kawaguchi Warshauer's conceptualisation of productive struggle, which capitalises on experiences that elicit struggle, that is, those that require students to 'expend intellectual effort' and that are productive, in the sense that they seek to 'advance thinking and deepen' understanding of mathematical concepts and…

One characteristic of high-quality mathematics teaching is supporting students in engaging in tasks of high cognitive demand. In this paper, we explore relationships between two elementary teachers' efforts to integrate computational thinking (CT) practices--abstraction, debugging, and decomposition--into their mathematics instruction and their…

This study examines Gazeta Matematica's (GM) influence on mathematics education in Romania from 1895-1945. Analyzing 13 articles from GM's archives, this research explores discussions on mathematics methodology and didactic issues and highlights Ion Ionescu's contributions. Findings reveal tensions between idealized rigorous pedagogy and practical…

One of the main tendencies of mathematical development in 19th and 20th centuries seem to be on rigor and formalization. Rigor and formalization took place on axiomatic basis leading to more abstraction. Euclidean type of an axiomatic model became a model of mathematics even for constructively developed analysis. Even though rigor and axiomatic…

Prospective elementary teacher (PSET) education programs vary greatly in the courses and course sequences employed to prepare their students. This article explores potential tradeoffs that arise for mathematics teacher educators, PSETs, and their future students due to the choices PSET education programs make regarding their design. Specifically,…

This article is the next in our series of papers that was presented at the 27th Biennial AAMT Conference in Brisbane earlier this year. In this article, Michael Minas uses a case study approach to explore how enabling prompts can be used most effectively to support teaching with challenging tasks.

Results in Mathematics Education, from different theoretical perspectives, highlight students' difficulties with graphs, both in interpreting given ones and in producing their own graphs. In this paper, we take an affective lens to understand both a student's relationship with mathematical graphs and a teacher's role when a task with a graph is…

Context personalization is an instructional design principle where tasks are presented to students in the context of their interest areas like sports, music, or video games. Personalization allows for understanding of domain principles to be grounded in concrete and familiar experiences. By making connections to prior knowledge, personalization…

Conventional coordinate systems are often considered representational tools for reasoning about mathematical concepts. However, researchers have shown that students experience persistent difficulties as they engage in graphing activity. Using examples from research and textbooks, we present a framework based on a conceptual analysis of the use of…

Problem solving has long been a priority in mathematics education, and the first Common Core mathematical practice (SMP1) focuses on this priority through the language of "Make sense of problems and persevere in solving them." We present findings from a survey about how prospective elementary teachers' (PTs) make sense of potential…

How people see the world, even how they research it, is influenced by beliefs. Some beliefs are conscious and the result of research, or at least amenable to research. Others are largely invisible. They may feel like "common knowledge" (though myth, not knowledge), unrecognized premises that are part of the surrounding culture. As we…

In this review, we attempt to integrate two crucial aspects of numerical development: learning the magnitudes of individual numbers and learning arithmetic. Numerical magnitude development involves gaining increasingly precise knowledge of increasing ranges and types of numbers: from non-symbolic to small symbolic numbers, from smaller to larger…

The degree of element interactivity determines the complexity and therefore the intrinsic cognitive load of linear equations. The unpacking of linear equations at the level of operational and relational lines allows the classification of linear equations in a hierarchical level of complexity. Mapping similar operational and relational lines across…

Mathematical discussions in which students compare alternative solutions to a problem can be powerful modes for students to engage and refine their misconceptions into conceptual understanding, as well as to develop understanding of the mathematics underlying common algorithms. At the same time, these discussions are challenging to lead…