Reasoning and problem solving are both critical acts in mathematics learning. The author has met many teachers who tell her they would love to highlight them more but feel they cannot because of the pressures of content coverage, pacing guides, and district tests. In this article, she describes a teaching intervention that highlighted problem…

Inquiry-based instruction is a student-centered approach to teaching that focuses on active learning (Barron and Darling-Hammond 2008) in which students engage with "tasks that promote reasoning and problem solving" (NCTM 2014). Specifically, such tasks encourage a variety of solution strategies and stimulate use of the NCTM Process…

Justification is a critical mathematical practice that must play a role in teaching and learning at all grade levels. Having students share their reasoning and explain how they know something is true or correct is the process of justification. The authors (a team of teachers and researchers) worked together for two years with an NSF-funded project…

Modeling using mathematics and making inferences about mathematical situations are becoming more prevalent in most fields of study. Descriptive statistics cannot be used to generalize about a population or make predictions of what can occur. Instead, inference must be used. Simulation and sampling are essential in building a foundation for…

Teachers have found that engaging students in justification can help students deepen and retain mathematical knowledge, gain a greater sense of ownership over the material, and improve communication and representation skills (Staples, Bartlo, and Thanheiser 2012). Student engagement in a justification activity can also lead to more equitable…

Developmentally, middle school students progress from being concrete thinkers and learners to abstract thinkers. Traditional middle school mathematics instruction introduces a curriculum that is intuitive and taught within a natural trajectory of the content. However, with this traditional approach, students may miss out on being exposed to…

In the elementary grades, students learn procedures to compute the four arithmetic operations on multidigit whole numbers, often by being shown a series of steps and rules. In the middle grades, students are then expected to perform these same procedures, with further twists. The Reasoning and Proof Process Standard suggests that students need to…

The combined seventh-grade and eighth-grade class began each day with a mathematical reasoning question as a warm-up activity. One day's question was: Is the product of two odd numbers always an odd number? The students took sides on the issue, and the exercise ended in frustration. Reflecting on the frustration caused by this warm-up activity,…

A math classroom is complex. Teachers navigate through a lesson using a variety of activities and tools that are designed to address students' needs and help them build conceptual understanding. Manipulatives, drawings, and other such concrete representations can help students explore mathematical ideas in physically grounded manifestations.…

Several activities that are based on the ancient method of false position, also called false assumption, are presented in this article as a way to motivate students to find the solution of literal equations in beginning algebra.

The goal of this article is to help teachers both recognize and capitalize on classroom opportunities so that students can be meaningfully engaged in mathematical proof. To do so, the authors discuss students' thoughts about proof and their abilities to construct arguments. They also offer suggestions for teachers that are intended to support…

Mathematical argumentation is an important skill. It leads to the process of proof, which is one area that students are being asked to master by the end of secondary school. Encouraging explanation and justification in math class allows this skill to develop. Explaining and justifying their ideas forces students to think deeply about mathematics…

In this article, the authors share an episode from an after-school program and focus on eight of the twenty-four urban, sixth-grade students who volunteered to work on mathematical problem solving. When students are given a supportive environment, open-ended tasks, and thoughtful teacher questioning, their reasoning can develop naturally. By…

Students explore the Pythagorean theorem and an extension using jelly beans to measure areas. Deductive thinking and algebraic notation help establish connections. (Contains 8 figures.)

This article describes an activity that incorporates 9 of the 10 National Council of Teachers of Mathematics (NCTM) Standards. Students explore a stacking garden containing multiple layers. The garden allows students to focus on finding the amount of border material needed and the amount of space available for planting. The problem balances the…