Guessing, for Pólya, is an important way of getting an initial handle on a mathematical problem. An argument can be made to place guessing in any one of the first three steps of the four-step approach to problem solving as described in "How to Solve It" (Pólya 1945). It could be a part of understanding the problem, devising a plan, or…

According to the National Center for Education Statistics (2016), the number of public school students in the United States who were English learners (ELs) was estimated at 4.5 million. However, in 2005, the national standardized testing scores (Fry 2007) show that about 51 percent of eighth-grade EL students were behind other students in reading…

Since 1950, the Prisoner's Dilemma has intrigued economists and amused fans of mathematics. It presents a situation in which two players acting to their own advantage do not do as well together as two players whose actions oppose their individual interests--hence, the dilemma. Variations of the Prisoner's Dilemma have appeared in diverse…

Students often have the misconception that mathematics involves reading the problem, staring at the page, and then writing down an answer. With more than twenty years of experience working with students in eighth-grade math through college calculus, the author has been increasingly interested in how she can foster problem-solving skills in her…

Rich mathematical modeling activities are crucial to giving students agency and making mathematics meaningful. Proportional reasoning and transitional algebraic reasoning are the primary topics in the prealgebra curriculum, so a need exists for meaningful modeling activities using proportional reasoning in addition to geometric modeling. In…

The task shared in this article provides geometry students with opportunities to recall and use basic geometry vocabulary, extend their knowledge of area relationships, and create area formulas. It is characterized by reasoning and sense making (NCTM 2009) and the "Construct viable arguments and critique the reasoning of others"…

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…

The Common Core State Standards for Mathematical Practice (CCSSI 2010) elevate the ways of thinking used to create mathematical results to the same level of importance as the results themselves. These mathematical habits of mind (MHoM), or "specialized ways of approaching mathematical problems and thinking about mathematical concepts that…

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…

In October 2014, an interesting new app became freely available in the Apple Store®. It is called PhotoMath®: (1) Hold a phone or tablet up to a mathematics problem in a textbook; (2) The app shows the answer to the problem; and (3) Select Steps to see the step-by-step process for getting from the problem to the solution. The app works for simple…

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…

In this article, Ceyanes, Lockwood, and Gill use "The Geometer's Sketchpad" in a sequence of algebra lessons to introduce quadratic functions as the product of two linear functions. The lessons provide an enriched look at the creation of parabolas and allow students to experience the foundational idea that all functions are…

With the acceptance by many states of the Common Core State Standards for Mathematics, new emphasis is being placed on students' ability to engage in mathematical practices such as understanding problems (including word problems), reading and critiquing arguments, and making explicit use of definitions (CCSSI 2010). Engaging students in…

Sometimes, in the teaching and learning of mathematics, open-ended problems posed by teachers or students can lead to a fuller understanding of mathematical concepts--a depth of understanding that no one could have anticipated. Interesting solutions and ideas emerged unexpectedly when the authors asked prospective and in-service teachers an "old"…

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…