Many students read for pleasure, but how many, stranded in an expansive library, would willingly reach for a mathematics textbook? One reason this scenario is so inconceivable is that mathematics textbooks are organized less as exercises in storytelling and more as how-to manuals. Unfortunately many mathematics teachers instruct accordingly. This…

Can one infinity be more than another infinity? Ask students this question, and many will be puzzled; others will insist that "infinity is infinity." The question seems to pique their interest and provides an opportunity to present the beautifully simple but counterintuitive proofs concerning the size of infinity first constructed by…

When examining transformations of the plane in geometry, teachers typically have students experiment with transformations of polygons. Students are usually quick to notice patterns with ordered pairs. The Common Core State Standard, Geometry, Congruence 2 (G-CO.2), requires students to describe transformations as functions that take points in the…

"When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data," notes the Common Core State Standards Initiative (2010, p. 72). This exploration of the recursive process in the Devil and Daniel Webster problem reveals that the symbolic spreadsheet fits this bill.…

One of the most rewarding accomplishments of working with preservice secondary school mathematics teachers is helping them develop conceptually connected knowledge and see mathematics as an integrated whole rather than isolated pieces. To help students see and use the connections among various mathematical topics, the authors have paid close…

Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…

Perceiving and interpreting invariants is a complex task for a nonexpert geometry student, as various studies have shown. Nevertheless, having students work through particular kinds of activities that involve perception and interpretation of invariants and engage in discussions with classmates, guided by the teacher, can help them learn…

For students who have difficulty envisioning how the mathematics that they are learning is used outside the classroom, an adventure as field mathematicians can be enlightening. Measuring stream discharge is a field experience that allows students to engage in a hands-on (and boots-on) real-world, problem-solving activity and that integrates…

Singapore students have scored exceedingly well on international tests in mathematics. In response, there has been a desire in the United States--both at the policy level and at the school level--to emulate Singapore. Because what can be identified most easily about Singapore's school mathematics can be gleaned from curriculum documents from the…

The widespread adoption of the Common Core State Standards for Mathematics (CCSSI 2010) shows a commitment to changing mathematics teaching and learning in pursuit of increasing student achievement. CCSSM should not be viewed as just another list of content standards for publishers and assessment groups to design their products around. Many…

This article introduces an optimization task with a ready-made motivating question that may be paraphrased as follows: "Are you smarter than a Welsh corgi?" The authors present the task along with descriptions of the ways in which two groups of students approached it. These group vignettes reveal as much about the nature of calculus students'…

In recent years, the term "quantitative literacy" has become a buzzword in the mathematics community. But what does it mean, and is it something that should be incorporated into the high school mathematics classroom? In this article, the authors will define quantitative literacy (QL), discuss how teaching for QL differs from teaching a traditional…

This article presents an origami activity which can lead to rich tasks in several branches of mathematics. This activity is a good first step for preservice teachers to get to know one another, start talking to one another, and start discussing mathematics. It can also show them how mathematics teaching and learning have changed to a more…

NCTM (2000) described geometry as "a means of describing, analyzing, and understanding the world and seeing beauty in its structures" (p. 309). Dossey et al. (2002) captured the essence of this aspect of visualization by stating that geometry fosters in students an ability to "visualize and mentally manipulate geometric objects." (p. 200).…

"Reasoning and sense making should occur in every classroom every day," states "Focus in High School Mathematics: Reasoning and Sense Making" (NCTM 2009, p. 5). As this book suggests, reasoning can take many forms, including explorations and conjectures as well as explanations and justifications of student thinking. Sense making, on the other…