The movement of groundwater in underground aquifers is an ideal physical example of many important themes in mathematical modeling, ranging from general principles (like Occam's Razor) to specific techniques (such as geometry, linear equations, and the calculus). This article gives a self-contained introduction to groundwater modeling with…

Students analyze a photograph to solve mathematical questions related to the images captured in the photograph. This month, photographs of a massive sculpture near the National Mall in Washington, D.C., provide opportunities for counting problems, generalizing from a pattern, and fitting a quadratic to data.

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.

There are at least five different equivalent definitions of the absolute value concept. In instances where the task is an equation or inequality with only one or two absolute value expressions, it is a worthy educational experience for learners to solve the task using each one of the definitions. On the other hand, if more than two absolute value…

From a young age children feel the need to identify two-dimensional geometric figures (shapes) and three-dimensional geometric figures (solids). The National Council of Teachers of Mathematics indicates the importance of being able to identify and name various geometric figures by kindergarten age. One of the objectives of this study was to learn…

The goal of the Rural Math Excel Partnership (RMEP) project, a development project funded by the U.S. Department of Education Investing in Innovation (i3) grant program, was to develop a model of shared responsibility among families, teachers, and communities in rural areas as collective support for student success in and preparation for advanced…

It might be said that for most occupations there is now less of a need for mathematics than there was say fifty years ago. But, the author argues, geometry, probability, and statistics constitute essential knowledge for everyone. Maybe not the geometry of Euclid, but certainly geometrical ways of thinking that might enable us to describe the world…

This discussion paper put forwards variation as a theme to structure mathematical experience and mathematics pedagogy. Patterns of variation from Marton's Theory of Variation are understood and developed as types of variation interaction that enhance mathematical understanding. An idea of a discernment unit comprising mutually supporting variation…

We relate the factorization of an integer N in two ways as N = xy = wz with x + y = w - z to the inscribed and escribed circles of a Pythagorean triangle.

Perceiving and interpreting invariants is a complex task for a nonexpert geometry student, as various studies have shown. Nevertheless, having students work through particular kinds of activities that involve perception and interpretation of invariants and engage in discussions with classmates, guided by the teacher, can help them learn…

Rhythm is a fundamental dimension of human nature at both biological and social levels. However, existing research literature has not sufficiently investigated its role in mathematical cognition and behavior. The purpose of this article is to bring the concept of "incarnate rhythm" into current discourses in the field of mathematical learning and…

With the adoption of the Common Core State Standards for Mathematics (CCSSM), many teachers are changing their classroom structure from teacher-directed to student-centered. When the author began designing and using problem-based tasks she saw a drastic improvement in student engagement and problem-solving skills. The author describes the Cake…

In this work we examined the content of approximately 950 mathematics questions used in measuring achievement among 4th, 5th and 6th graders in the US in an attempt to determine how the current standardized examinations, which were developed to reflect new national curricular mandates, compared to the old measures used for the same purpose. Using…

This report will determine to what degree the AP Physics 1 and 2 and AP Calculus AB and BC frameworks are aligned with the Trends in International Mathematics and Science Study (TIMSS) Advanced Physics and Mathematics frameworks. This will enable an exploration of any differences in content coverage and levels of complexity, and will set the stage…

The Partnership for Assessment of Readiness for College and Careers (PARCC) is a state-led consortium working to develop next-generation assessments that more accurately, compared to previous assessments, measure student progress toward college and career readiness. The PARCC assessments include both English Language Arts/Literacy (ELA/L) and…