This article describes the results of an investigation based on a Models and Modeling Perspective [MMP]. We present the evolution of the models built by university students when solving a model development sequence designed to promote their learning of the exponential function. As a result, we observed that students' thinking was modified,…

Students' construction, coordination, and abstraction of units underlie success across multiple mathematics domains. Structures for coordinating units underscore notions of numbers as composite units (e.g., five is a unit of five and five units of one). In this working group, we seek to facilitate collaboration amongst researchers and educators…

Researchers, who have studied the understanding of the density property in the set of decimal numbers, have shown that the student uses the property of the discrete of natural numbers to solve tasks related to density. So, a restructuring of concepts is necessary, that is, a conceptual change from "the discrete" to "the dense".…

Number talks, a popular mathematics teaching routine in the United States, may offer supports for beginning teachers (BTs) to engage in ambitious instruction. BTs' enactments of number talks, however, are varied, and there are few empirical studies that explore how BTs' enactment of number talks could be more (or less) ambitious. This paper draws…

Prompting people to estimate climate change numbers before showing them the true value can shift learners' attitudes and conceptions. Yet, interventions created for such a learning experience are not easily accessible to the general public. The purpose of this preregistered study was to address this research gap by developing and testing an openly…

The number line is a model that is used to measure, count, order and even operate, which requires a symbolic interpretation. Then, we investigate the conceptions of 72 in service secondary teachers, when they manage the model of the number line associated with order, spatial location and the relative position between numbers and marks. In a…

We offer this synthesized framework as a tool to reveal mathematical activity in a non-formal makerspace. In particular, we connect research at different grain sizes to illustrate and explain how mathematics plays a crucial, if often implicit, role in this activity. We begin with describing the Approximate Number System and the Ratio-Processing…

This study aims to explore seventh-grade students' understanding of negative integers as they engaged in mathematics history integrated model-eliciting activities in small groups. For this educational case study, we designed model-eliciting activities based on six design principles of the models-and-modeling perspective that incorporated history…

Fractions are one of the most difficult areas of mathematics for all students and especially for students with learning disabilities (LD). An incomplete understanding of fractions during the elementary and middle school may be why some students view fractions as "really small" or "less than 1" compared to others who view them…

This proposal explores how the activity of subitizing--quickly apprehending the numerosity of a small set of items--changes with the development of number concepts. We describe how varying the orientations of items in teaching experiment sessions promoted one pre-schooler, Frank, to attend to subgroups of items and change his thinking about…

Being able to effectively interpret students' thinking and respond effectively in the moment are important skills that preservice teachers (PSTs) need to learn. This study zooms in on 14 PSTs' planning, teaching, and reflections involved in number string lessons, and investigates to what extent PSTs anticipate their students' strategies and…

The number sequences describe a hierarchy of students' concepts of number. This research uses two defining cognitive structures of the number sequences--units coordination and the splitting operation--to model middle-grades students' abilities to write linear equations representing the multiplicative relationship between two unknowns. Results…

The present study extends recent advances developing and applying measures of mathematical content knowledge for teaching. Recent research has demonstrated that the Diagnosing Teachers' Multiplicative Reasoning Fractions survey provides information about distinct but related components necessary for reasoning in terms of quantities when solving…

Horizon Content Knowledge (HCK, Ball, Thames & Phelps, 2008), knowledge that teachers hold about the interconnectedness of mathematics, has been recognized as an integral component of Mathematics Knowledge for Teaching, yet little is known about how or where it is taught in the preservice teacher (PST) curriculum or how researchers should…

This unlikely cast of characters, by working collaboratively in a trusting learning community, was able to identify an approach to teaching rational numbers through measuring from the everyday practices of Yup'ik Eskimo and other elders. "The beginning of everything," as named by a Yup'ik elder, provided deep insights into how practical…