Many students share a certain amount of discomfort when encountering proofs in geometry class for the first time. The logic and reasoning process behind proof writing, however, is a vital foundation for mathematical understanding that should not be overlooked. A clearly developed argument helps students organize their thoughts and make…

Working with language-independent logic structures can help students develop both inductive and deductive reasoning skills. The Japanese publisher Nikoli (with resources available both in print and online) produces a treasure trove of language-independent logic puzzles. The Nikoli print resources are mostly in Japanese, creating the extra…

A mysterious conflict of solutions emerged when a group of tenth- and eleventh-grade students were studying a seemingly ordinary problem on combination and probability. By investigating the mysterious "conflicts" caused by multiple randomization procedures, students will gain a deeper understanding of what it means to perform a task…

The Common Core State Standards (CCSSI 2010) for Mathematical Practice have relevance even for those not in CCSS states because they describe the habits of mind that mathematicians--professionals as well as proficient school-age learners--use when doing mathematics. They provide a language to discuss aspects of mathematical practice that are of…

In the authors' attempts to incorporate problem solving into their mathematics courses, they have found that student ambition and creativity are often hampered by feelings of risk, as many students are conditioned to value a produced solution over the actual process of building one. Eliminating risk is neither possible nor desired. The challenge,…

Despite Common Core State Standards for Mathematics (CCSSI 2010) recommendations, too often students' introduction to proof consists of the study of formal axiomatic systems--for example, triangle congruence proofs--typically in an introductory geometry course with no connection back to previous work in earlier algebra courses. Van Hiele…

How do students think about an angle measure of ninety degrees? How do they think about ratios and values on the unit circle? How might angle measure be used to connect right-triangle trigonometry and circular functions? And why might asking these questions be important when introducing trigonometric functions to students? When teaching…

Confusion can arise from the subtle difference between proving a general and a particular statement, especially when general statements are presented by textbooks in ways that make them appear particular in nature. The authors discuss the implications for teaching proof in light of the current opportunities in high school geometry textbooks.

This article argues that the practices of reasoning and sense making are critical for developing students' mathematical literacy. Seven mathematics teachers collaborated throughout a school year to discuss the ideas proposed in Reasoning and Sense Making (NCTM 2009) and put them into practice through action research. Conducting action research,…

When confronted with a challenging problem, many solvers may at first think that not enough information has been provided. If, however, they suspect that the problem is solvable, this feeling typically influences them to persevere as well as monitor and reflect on their efforts. In this article the authors present four nonroutine tasks that they…

The importance of developing reasoning and justification has been highlighted in "Principles and Standards for School Mathematics" (NCTM 2000). The Common Core State Standards for Mathematics (CCSSI 2010) further reiterates the importance of reasoning and proof in several standards for mathematical practice. Students of all grades are…

Sometimes a student's unexpected solution turns a routine classroom task into a real problem, one that the teacher cannot resolve right away. Although not knowing the answer can be uncomfortable for a teacher, these moments of uncertainty are also an opportunity to model authentic problem solving. This article describes such a moment in Zahner's…

If students are going to develop reasoning and thinking skills, use their mathematical knowledge, and recognize the relevance of mathematics in their lives, they need to experience mathematics in meaningful ways. Only then will their mathematical skills be transferrable to all other parts of their lives. To promote such flexible mathematical…

The purpose of this article is to provide an image of what proof could look like in beginning algebra, a course that nearly every secondary school student encounters. The authors present an actual classroom vignette in which a rich opportunity for student reasoning arose. After analyzing the proof schemes at play, the authors provide a…

Pairs of students are looking at rows of pennies--laughing, talking, kidding one another about who is winning and who is making smart moves. The game, a simple game for two players, is One or Two?. The game begins with a player removing any one penny. Players then take turns, removing either a single penny or a pair of pennies from adjacent…