A presentation on the Budapest Seminars in Mathematics Education website, inspired the author to try a new approach while teaching a unit on quadrilaterals in his Geometry for K-8 Teachers course, a class primarily composed of undergraduate students interested in teaching at the elementary or middle school level. The author designed a lesson that…

This article presents the Snail problem, a relatively simple challenge about motion that offers engaging extensions involving the notion of infinity. It encourages students in grades 5-9 to connect mathematics learning to logic, history, and philosophy through analyzing the problem, making sense of quantitative relationships, and modeling with…

All students should be provided with opportunities to develop conceptual understanding prior to procedural fluency. To develop students' conceptual understanding, teachers must learn such skills as how to select, plan, and enact cognitively demanding tasks (CDT) and to evaluate evidence of student learning. Therefore, teachers need opportunities…

Although it is not a new idea, discourse continues to be a topic of discussion among teachers, teacher educators, and researchers in mathematics education. The National Council of Teachers (NCTM) and the Common Core State Standards for Mathematics (CCSSM 2010) describe mathematics classrooms as discourse communities in which whole-class…

Modeling is an effective tool to help students access mathematical concepts. Finding a math teacher who has not drawn a fraction bar or pie chart on the board would be difficult, as would finding students who have not been asked to draw models and represent numbers in different ways. In this article, the authors will discuss: (1) the properties of…

"Principles to Actions: Ensuring Mathematical Success for All" (NCTM 2014) gives teachers access to an insightful, research-informed framework that outlines ways to promote reasoning and sense making. Specifically, as students transition on their mathematical journey through middle school and beyond, their knowledge and use of…

Early experiences of reasoning while playing games of strategy are foundational for future proofs that students will be expected to build using conventionally structured arguments. But how did game playing in school occur? How can educators be sure that mathematical reasoning is going on? The authors investigated these questions to understand how…

Percentages have proven to be a challenging concept in school mathematics. At the surface, a percentage is merely a rational number, representing a ratio between a number and 100. At the conceptual core, however, a percentage is sensitive to the context, making sense with respect to a network of related quantities. In the effort of the authors to…

The phrase "interdimensional relationships" may conjure images of UFOs, alien beings, wormholes, and even mystical powers. These notions of interdimensionality were not out of bounds in the middle school learners' experiences described in this article. However, these notions do not entirely capture the mathematical relationships…

When teaching algebra concepts to middle school students, the author often hears questions that echo her own past confusion as a young student learning to write linear equations using data tables that show only input and output values. Students, expected to synthesize the relationship between these values in symbolic representation, grow…

The author wants her students to see any new mathematics--fractions, negative numbers, algebra--as logical extensions of what they already know. This article describes two students' efforts to make sense of their conflicting interpretations of 1/2 × -6, both of which were compelling and logical to them. It describes how discussion, constructing…

Mathematical modeling, a practice standard in the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010), is a process by which students develop and use mathematics as a tool to make sense of the world around them. Students investigate a real-world situation by asking mathematical questions; along the way, they need to decide how to use…

Two studies (Trautwein 2007 and Dettmers et al. 2010) show a positive correlation between high-quality homework and mathematics achievement. Students who completed their homework assignments scored better on assessments. However, these studies also showed no relationship between time spent on homework and resulting student achievement. This helped…

Science, engineering, and mathematics are fields that many students see as separate entities. But if these fields are combined with technology, they become STEM. This investigation provides a context and allows students to explore mathematics, science, and engineering within that context. It requires students to model with mathematics and find…

This article describes a project in which students investigate the question: What dihedral angles between pairs of mirrors in a kaleidoscope result in perfectly symmetric images? The unit culminates with students building their own kaleidoscopes. This content aligns with parts of the Common Core's standards for fifth grade (classify…