The empirical data in this study are from a series of two lessons on measurement implemented in seven classes with 119 six-year-old students in Sweden. Both problem solving and problem posing were shown to be important in early mathematics when students in this study worked on one problem-solving task and one problem-posing task on measurement. As…

Seeds of Algebraic Thinking comes from the Knowledge in Pieces (KiP) perspective of learning. KiP is a systems approach to learning that stems from the constructivist idea that people learn by building on prior knowledge. As people experience the world, they acquire small, sub-conceptual knowledge elements. When people engage in a particular…

We offer an analysis of calculus assessment items that highlights ways to evaluate students' application of important meanings and support their engagement in generative ways of reasoning. Our central aim is to identify characteristics of items that require students to apply their understanding of key ideas. We coordinate this analysis of…

The COVID-19 pandemic has affected student learning; according to the National Assessment of Educational Progress (2022), scores for students in the United States have shown a decline in mathematics. . One way educators can strengthen students' current mathematical knowledge and bridge to new content is through Number Talks. Number Talks are brief…

The hammer-and-nail phenomenon highlights human tendency to approach a problem using a tool with which one is familiar instead of analyzing the problem. Pedagogical suggestions are offered to help students minimize their mathematical impulsivity, cultivate an analytic disposition, and develop conceptual understanding.

Problem solving has been the focus of the Common Core State Standards for Mathematical Practice. Helping students acquire critical-thinking and problem-solving skills has become the primary goal of mathematics education across all grade levels. However, research has found that many students struggle with word problems because of poor text…

This article explains how explorations into the quadratic formula can offer students opportunities to learn about the structure of algebraic expressions. In this article, the author leverages the graphical interpretation of the quadratic formula and describes an activity in which students derive the quadratic formula by quantifying the symmetry of…

Quadratic modeling problems are commonplace in high school mathematics courses; they typically situate quadratic patterns of change and their corresponding parabolic graph within real-world contexts. Traditional approaches to this type of problem lend themselves to making connections across different representations (e.g., Garofalo and Trinter…

Skills in creativity are needed to meet the needs of today's organizations, and design thinking is a process that one can learn to become more creative. Yet the diminishing exposure to and pursuit of humanities courses, which have traditionally developed these skills, has put pressure on business schools to fill the gap. This experiential learning…

This article examines a common proportional reasoning problem used in schools, often referred to as the Orange Juice task. The authors show how these six strategies described by Nikula (e.g., Unitizing, Norming, etc.) and one additional strategy can be used to either solve or make progress in the Orange Juice task. The article presents work from…

Students often use incoherent strategies in their problem solving involving force and motion, as revealed, e.g. when they are asked to draw force diagrams for amusement rides involving circular motion, whether in horizontal or vertical planes. Depending on the questions asked, assignments involving circular motion can reveal different types of…

This book provides prospective and practicing teachers with research insights into the mathematical difficulties of students with learning disabilities and classroom practices that address these difficulties. This linkage between research and practice celebrates teachers as learners of their own students' mathematical thinking, thus contributing…

Mathematics has been taught throughout history without much more than a straightedge, a compass, and an abacus. So there is little question that all the major concepts of mathematics can be delivered without the aid of modern technology. But clearly, just the time-saving aspect of using technology can greatly enhance the art of teaching. In this…

Mathematical modeling, a key strand in mathematics, engages students in rich, authentic, exciting, and culturally relevant problems and connects abstract mathematics to the surrounding world. In this, article, the authors describe a modeling activity that can be used when teaching linear equations. Modeling problems, in general, are typically high…

Numerous studies have shown that mathematical knowledge is situational, meaning that students' abilities to transfer and apply skills depends on the conditions in which they were learned (Barab 1999; Boaler 2002, 2016). Given that the real world seldom presents problems that are confined to a single discipline (Barab 1999), it is imperative that…