In elementary mathematics instruction, teachers often compromise access and cognitive demand in pursuit of one or the other. I explored how five different co-teaching models impact students' access to cognitively demanding mathematical learning opportunities (MLOs) in elementary classrooms. Through ethnographic lesson observations, artifact…

No matter what path students pursue in postsecondary education and in life, they need to be able to problem-solve and think critically. The study of mathematics teaches skills and habits of mind that are an essential part of the formula for students to succeed in further studies and in their careers. Requiring all students, regardless of the…

The purpose of this article is to share how one teacher implemented exploratory writing within a mathematics lesson to support her third graders in making sense of fractions and communicating their mathematical ideas. The author defines exploratory writing, then describes the teacher's lesson while also presenting a general overview about how to…

Cognitively demanding tasks provide important opportunities for students to develop an understanding of mathematics; however, they are challenging to launch and implement. The authors designed a secondary methods unit on launching tasks. Participants in the study were enrolled in five different methods courses. Using a noticing framework, findings…

The most challenging parts of teaching mathematics are engaging students and helping them understand the connections between mathematics concepts. In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the fourth-grade level through visualization, play, and…

"Mathematics Teaching On Target" is a guidebook for improving mathematics teaching, based on the Teaching for Robust Understanding (TRU) Framework and its five dimensions -- The Mathematics, Cognitive Demand, Equitable Access, Agency, Ownership, and Identity, and Formative Assessment. You'll be guided to refine your classroom activities…

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…

The hammer-and-nail phenomenon highlights human tendency to approach a problem using a tool with which one is familiar instead of analyzing the problem. Pedagogical suggestions are offered to help students minimize their mathematical impulsivity, cultivate an analytic disposition, and develop conceptual understanding.

This unique volume surveys recent research on spatial visualization in mathematics in the fields of cognitive psychology and mathematics education. The general topic of spatial skill and mathematics has a long research tradition, but has been gaining attention in recent years, although much of this research happens in disconnected subfields. This…

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…

Young children naturally engage in mathematics as they play and explore. Although many early childhood educators are aware that children are frequently interacting with mathematical ideas, a certain level of knowledge and awareness is required to identify and highlight the mathematics that children are engaging in and to create opportunities for…

Early years mathematics classrooms can be colourful, exciting, and challenging places of learning. Andrea McDonough and fellow teachers have noticed that some students make good decisions about using materials to assist their problem solving, but this is not always the case. These experiences lead her to ask the following questions: (1) Are…

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,…

Often, students who solve fraction tasks respond in ways that indicate inadequate conceptual grounding of unit fractions. Many elementary school curricula use folding, partitioning, shading, and naming parts of various wholes to develop children's understanding of unit and then nonunit fractions (e.g., coloring three of four parts of a pizza and…

Teachers across all grade levels agree that problem solving and reasoning are areas of weakness in students. Assessments among U.S. students indicate that these weaknesses persist (NCTM 2014) in spite of repeated calls that date back more than thirty years for increased problem solving, reasoning, and sense making in our schools. The NCTM is…