One day a teacher writes 12 = ___ + ___ on the board. "You wrote it wrong," says one of the 1st graders. "You can't write the answer first!" exclaims another. As teachers in classrooms across the United States put the new mathematics standards into practice, they'll face countless examples like this one. When teachers…

In this paper, the dice game "Unders and Overs" is described and presented as an active learning exercise to introduce basic probability concepts. The implementation of the exercise is outlined and the resulting presentation of various probability concepts are described.

Linear programming (LP) is an application of graphing linear systems that appears in many Algebra 2 textbooks. Although not explicitly mentioned in the Common Core State Standards for Mathematics, linear programming blends seamlessly into modeling with mathematics, the fourth Standard for Mathematical Practice (CCSSI 2010, p. 7). In solving a…

This analysis of a problem that is frequently posed at professional development workshops, in print, and on the Web--the coffee-milk mixture riddle--illustrates the timeless advice of George Pólya's masterpiece on problem solving in mathematics, "How to Solve It." In his book, Pólya recommends that problems previously solved and put…

For mathematicians, definitions are the ultimate tool for reaching agreement about the nature and properties of mathematical objects. As research in school mathematics has revealed, however, mathematics learners are often reluctant, perhaps even unable, to help themselves with definitions while categorizing mathematical objects. In the research…

Box plots are frequently misinterpreted and educational attempts to correct these misinterpretations have not been successful. In this study, we used two instructional techniques that seemed powerful to change the misinterpretation of the area of the box in box plots, both separately and in combination, leading to three experimental conditions,…

This paper extends work about quantitative reasoning related to covarying quantities involved in rate of change. It reports a multiple case study of three students' reasoning about quantities involved in rate of change when working on tasks incorporating multiple representations of covarying quantities. When interpreting relationships between…

Mathematical ideas from number theory, group theory, dynamical systems, and computer science have often been used to explain card tricks. Conversely, playing cards have been often used to illustrate the mathematical concepts of probability distributions and group theory. In this paper, we describe how the 21-card trick may be used to illustrate…

The history of mathematics educational reform is replete with innovations taken up enthusiastically by early adopters without significant transfer to other classrooms. This paper explores the coupling of coding and mathematics education to create the possibility that coding may serve as a Trojan Horse for mathematics education reform. That is,…

Background: Teaching is comprised of interconnected practices. Some practices are domain neutral (DN), or independent of a content area. Examples of DN practices include emotional and instructional support and classroom organization. Others are domain specific (DS), or content dependent. Within a mathematics context, examples of DS practices…

We have developed, implemented, and assessed a teaching-learning sequence that aims to enhance students' understanding of the divergence and curl of electromagnetic fields. We designed guided-inquiry worksheets based on student difficulties we identified during semiquantitative and qualitative studies and discussions in the literature. A multiple…

Mathematical games are widely used in the primary classroom; however, not all games are equally valuable. How might teachers decide which specific games to introduce? The authors present five principles of educationally-rich games to help teachers address this issue. In so doing, they provide teachers with a usable framework to stimulate…

While 'flipped' classrooms have been heralded as a valuable tool for student learning, little is known about the process by which students learn from the out-of-class resources. This study investigates 12 undergraduate students' sense-making activities while watching videos about topics in introductory statistics. Analysis of interview data…

We explore the transition from school to university through a commognitive (Sfard 2008) analysis of twenty-two students' examination scripts from the end of year examination of a first year, year-long module on Sets, Numbers, Proofs and Probability in a UK mathematics department. Our analysis of the scripts relies on a preliminary analysis of the…

This article describes the beliefs and their transformations of members of a cohort of early-childhood, elementary and middle-level pre-service teachers (PSTs) as they professionally develop. A typological analysis of both quantitative and qualitative data collected between August 2011 and May 2013 was utilized to categorize how 40 PSTs' beliefs…