Teaching to support students' sense-making is challenging. It requires continuous, context-dependent decision-making about which student ideas to pursue, when, how, and why. This paper presents a single case study of an experienced teacher, Nadine, as an illustrative case in order to provide a rich description of this teacher's decisional…

Definitions are an integral aspect of mathematics. In particular, they form the backbone of deductive reasoning and facilitate precision in mathematical communication. However, when multiple non-equivalent definitions for the same term exist, their ability to serve these purposes can be called into question. While ambiguity can be productive, the…

In this paper Michael Vazquez and Dustin Webster consider the practice of deliberating about ethical case studies as a means to contribute to the professional development of educators. An ongoing debate is whether or not the study of ethical theory should be included in this practice. Vazquez and Webster argue that a popular strategy, known as the…

The present work refers to a study of 10-12-year-old students' ideas on the self- or hetero-luminosity of the Moon. To check the validity of students' reasoning on this question, we also examined their ideas on the lunar phases and how we see hetero-luminous objects. Students supported their ideas on the self- or hetero-luminosity of the Moon by…

Abstraction is essential to learning mathematics because the mathematical concepts obtained through abstraction will be more meaningful than directly receiving these concepts. This study aims to describe the pre-service teachers' abstraction in constructing relationships among quadrilaterals. This research method was explorative qualitative…

Although actual experiences of upward social mobility are historically low, many adolescents and adults express a "belief" in social mobility (e.g., that social status can change). Although a belief in upward mobility (e.g., that status can improve) can be helpful for economically disadvantaged adolescents and adults, a belief in upward…

The ideas of measurement and measurement comparisons (e.g., fractions, ratios, quotients) are introduced to students in elementary school. However, studies report that students of all ages have difficulty comparing two quantities in terms of their relative size. Students often understand fractions such as 3/7 as part-whole relationships or…

In order to learn more about student understanding of the structure of proofs, we generated a novel genre of tasks called "Proof Without Claim" (PWC). Our work can be viewed as an extension of Selden and Selden's (1995) construct of "proof framework"; while Selden and Selden discuss how the structure of a proof can be discerned…

An important consideration in the design and development of numeracy-focused coursework is ensuring that one meets students where they are with respect to both their mathematics background and their existing numeracy practices in relation to public issues. The latter consideration is especially important, given that students already think about…

Interest in the role of casuistry and casuistical questions in Kant's "Doctrine of Virtue" ("DV"), i.e. the second part of the "Metaphysics of Morals," has grown in recent years. My own position is formulated in Schuessler (2012, in German), the main thesis of which will be retained here in an updated form and with…

The purpose of this proposed quantitative, ex post facto, comparative study with a pretest-posttest design was to assess gains in overall Critical Thinking (CT) skills, and particularly in the Hypothetical-deductive Reasoning (HDR) skills, for pre-health college students tested before and after a Critical Analyses in Science course they attended…

This article reports on a qualitative investigation into students' thinking about a differential equations problem posing task; i.e. an initial value problem. Analysis of written and verbal responses to the task indicate that only four of the 34 students who participated in the study were successful in posing problems. Furthermore, only one of the…

This study identifies characteristics of polygon class learning opportunities for 8-9-year-old children during the whole-class instruction. We consider the interplay between the geometrical tasks demanding different ways of reasoning, features of children's geometrical thinking, and the teacher's moves to identify characteristics of learning…

Human decision-making is underpinned by distinct systems that differ in flexibility and associated cognitive cost. A widely accepted dichotomy distinguishes between a cheap but rigid model-free system and a flexible but costly model-based system. Typically, humans use a hybrid of both types of decision-making depending on environmental demands.…

University students are often asked to learn abstract concepts. Abstract concepts are hard to learn. Giving specific examples can help learning abstract concepts. These examples might limit understanding to the similarities between the abstract domain and particular examples. The primary purpose of this study was to test whether exposure to…