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…

"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…

Offering secondary math educators an innovative holistic and process-orientated approach for implementing nonroutine problems into their curriculum, this book defines and establishes practical strategies to develop students' problem-solving skills. The text focuses on the process skills necessary to solve nonroutine problems in mathematics and…

Students' difficulties understanding the meaning of logarithms could stem in part from differences between teachers' and students' views of them. The purpose of this article is to unpack some specialized content knowledge for teaching logarithms. The author discusses the history of logarithms to show why they can be understood as repeated…

Probabilistic reasoning underpins much of middle school students' future work in data analysis and inferential statistics. Unfortunately for many middle school students, probabilistic reasoning is not intuitive. One specific area in which students seem to struggle is determining the probability of compound events (Moritz and Watson 2000). Research…

Mathematical modeling, a key strand in mathematics, engages students in rich, authentic, exciting, and culturally relevant problems and connects abstract mathematics to the surrounding world. In this, article, the authors describe a modeling activity that can be used when teaching linear equations. Modeling problems, in general, are typically high…

This article demonstrates how to consider differentiating instruction for diverse learners while maintaining the cognitive demand of a mathematics task. The authors present scenarios involving hypothetical cases of students in inclusive classrooms who engaged in productive struggle within the differentiated task. The authors specifically focus on…

This article explores some of the common mathematical difficulties that 11- to 16-year-old students experience with respect to their learning of physics. The definition of "understanding" expressed in the article is in the sense of transferability of mathematical skills from topic to topic within physics as well as between the separate…

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…

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…

Jill Cheeseman and Deborah Michels let us eavesdrop on a young child's attempt to solve a four-digit subtraction. In the process they describe the use of an empty number line as a tool for describing the calculation process. If you haven't tried using an empty number line, then the authors outline some compelling reasons for using empty number…

Mastering algebra is important for future math and postsecondary success. Educators will find practical recommendations for how to improve algebra instruction in the What Works Clearinghouse (WWC) practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students". The methods and examples included in…

This article investigates teachers' attitudes toward English Language Learners (ELLs) at three middle schools in Texas, New York, and Pennsylvania. These schools have been critically impacted by the presence of ELLs, whether due to large numbers, or to smaller, concentrated populations in schools that traditionally have not served ELLs. This…

What is the relationship between ACT scores and success in college? For decades, admissions policies in colleges and universities across the country have required applicants to submit scores from a college entrance exam, most typically the ACT (American College Testing) or SAT (Scholastic Aptitude Test). This requirement suggests that high school…

This paper summarizes difficulties that chemistry students at all levels commonly exhibit when translating, manipulating, and interpreting mathematical expressions that contain logarithms, and offers approaches that the authors have found useful to help students overcome such difficulties. The online supplement provides problem sets created by the…