Linear algebra is best done with block matrices. As evidence in support of this thesis, we present numerous examples suitable for classroom presentation.

Education leaders in Virginia use early access to Algebra I as one method to increase students' likelihood of graduating from high school prepared for college and careers. This study used administrative data to calculate descriptive statistics on Algebra I outcomes and attainment of college preparatory diplomas among students by their grade 5 math…

To increase opportunities for students to take more advanced math courses in high school, many school districts enroll grade 8 students in Algebra I, a gateway course for advanced math. But students who take Algebra I in grade 8 and skip other math courses, such as grade 8 general math, might miss opportunities to develop the foundational…

This article is the second in a series of activities that discusses some interesting relationships with triangles. Brian Sherman shows how to determine the altitudes of a triangle, its area, and the lengths of its circumradius and inradius. [For the first activity in the series, see EJ1259412.]

When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number…

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector--WiiGraph--with 11-year-old children. In the selected episodes, the children used this motion detector to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their…

Within a study of student reasoning in abstract algebra, we encountered the claim "division and multiplication are the same operation." What might prompt a student to make this claim? What kind of influence might believing it have on their mathematical development? We explored the philosophical roots of "sameness" claims to…

Standards-based grading (SBG) is an assessment/grading system that attaches scores to the mastery level of specific learning objectives. This methodology is gaining momentum in the K-12 environment but has been largely overlooked at the post-secondary level. In this paper, I will discuss the implementation of this practice in a College Algebra…

In this paper we share a classroom implementation of a task about basis in linear algebra, which was originally developed for research about the topic. The task asks students to construct an everyday situation that captures the definition of basis, and then to critique it mathematically. Using this task, the original research study uncovered…

The purpose of this note is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from two rectangular sheets and then discuss some of the mathematical questions that arise in the context of geometry and algebra.

We propose a divisibility criterion for elements of a generic Unique Factorization Domain. As a consequence, we obtain a general divisibility criterion for polynomials over Unique Factorization Domains. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school…

A complex web of factors can influence whether students participate in computer science (CS) during high school. In order to increase participation in CS for all students, there is a need to better understand who is currently participating and what factors might be hindering participation. This study utilized a large-scale, student-level dataset…

The mathematical method of linear equations involving the coefficients of the equation of a chemical reaction can be used to balance the equation of nearly every chemical reaction. For its application, only the rules of simple algebra need be known. Hence, the method could even be used by students of an elementary chemistry course.

In this paper, we discuss the utilization of an innovative learning tool, the Sphero BOLT robot, in a 10-day algebra-based mathematics education course for graduate students. Students created routes for their BOLTs to travel and determined ways to measure the distance, rate, and time of their robots' movements. The student prompt, sample student…

This article details a high school classroom activity in which algebra tiles are used to help students gain a conceptual understanding of completing the square. The activity utilises the eight elements of effective mathematics education suggested in the National Council of Teacher of Mathematics' (NCTM's) "Principles to action: Ensuring…