There are many continuous quantitative dimensions in the physical world. Philosophical, psychological and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time, mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, density) and number can be…

The purpose of this analysis was to examine achievement gaps on fractions for very-low-performing students as a function of whether they receive inclusive fraction instruction or specialized fraction intervention and with the shift to Common Core State Standards (CCSS). In three randomized control trials conducted in 3 consecutive years, 203…

This paper discusses Multi-Feature Hierarchical Sequential Pattern Mining, MFH-SPAM, a novel algorithm that efficiently extracts patterns from students' learning activity sequences. This algorithm extends an existing sequential pattern mining algorithm by dynamically selecting the level of specificity for hierarchically-defined features…

In this webinar, Dr. William Schmidt of Michigan State University discussed helpful instructional tools for promoting the higher order conceptual thinking found in the Common Core Standards. The PowerPoint presentation and webinar recording are also available.

Many students or teachers ask themselves: Being given a natural number n, how many non-isomorphic groups of order n exists? The answer, generally, is not yet given. But, for certain values of the number n have answered this question. The present work gives a method to determine of all non-isomorphic groups of order 16 and gives descriptions of all…

As with previous reports of activity at the annual Institute of Mathematics Pedagogy, this does not disappoint. The tasks used to support teaching and learning in mathematics can vary in so many ways, but this variation is compounded when learners begin to access the tasks. A task that really motivates and challenges one learner is likely to…

Over the past two decades, mathematical patterns have been acknowledged as important early components of children's development of algebraic reasoning (NCTM 2000). In particular, growing patterns have attracted significant attention as a context that helps students develop an understanding of functional relationships (Lee and Freiman 2006; Moss et…

The logistic map is one of the simplest nonlinear dynamical systems that clearly exhibits the route to chaos. In this paper, we explore the evolution of the logistic map using an open-source microcontroller connected to an array of light-emitting diodes (LEDs). We divide the one-dimensional domain interval [0,1] into ten equal parts, an associate…

Symbolism is an important element within mathematical notation. Symbolism enables unambiguous communication of mathematical ideas and forms. Current mathematical symbolism is the result of much evolution and acceptance by those working in the field. When is it appropriate to challenge accepted norms and suggest alternatives or "changes"?…

This paper explores how textbooks address two central concepts in differential calculus, derivative at a point and derivative function, make the transition from one concept to the other, and establish connections between them. We analyzed how the three most widely used calculus textbooks present these two aspects of the derivative, focusing on…

We argue for decreasing the use of the word "spread" when describing the statistical idea of dispersion or variability in introductory statistics courses. In addition, we argue for increasing the use of the word "variability" as a replacement for "spread."

Students' experience in using formulas for volumes is often limited to substituting numbers into given formulas. An activity presented in this article may help students make connections between the formulas for volumes of prisms and volumes of pyramids. In addition, some interesting facts from number theory arise, demonstrating strong connections…

One of the most significant steps in learning algebra is understanding the change in the role of letters in mathematical expressions from unknowns to variables. We describe the historical development of this change in usage, starting with the ancient use of mathematical unknowns, detailing several important changes in practice that allowed for the…

This relationship is omnipresent to those who appreciate the shared attributes of these two areas of creativity. The dynamic nature of media, and further study, enable mathematicians and artists to present new and exciting manifestations of the "mathematics in art", and the "art in mathematics". The illustrative images of the relationship--that…

Motivated by the question, "What exactly about a mathematical concept should students discover, when they study it via discovery learning?", I present and demonstrate an interpretation of discovery pedagogy that attempts to address its criticism. My approach hinges on decoupling the solution process from its resultant product. Whereas theories of…