Given a parabola in the standard form y[superscript 2] = 4ax, corresponding to three points on the parabola, such that the normals at these three points P, Q, R concur at a point M = (h, k), the equation of the circumscribing circle through the three points P, Q, and R provides a tremendous opportunity to illustrate "The Art of Algebraic…

We produce a continuum of curves all of the same length, beginning with an ellipse and ending with a cosine graph. The curves in the continuum are made by cutting and unrolling circular cones whose section is the ellipse; the initial cone is degenerate (it is the plane of the ellipse); the final cone is a circular cylinder. The curves of the…

We show that inside every triangle the locus of points satisfying a natural proportionality relationship is a parabola and go on to describe how this triangle-parabola relationship was used by Archimedes to find the area between a line and a parabola.

Given a family of "p" greater than or equal to 3 points in the plane, some three of them have the property that the smallest circle encompassing them encompasses all "p" points. Similarly, we show that for "p" greater than or equal to 3 circles, there are three of them such that the smallest circle encompassing them…

With the introduction of "Next Generation Science Standards" ("NGSS"), teachers have been called to find meaningful and engaging ways to teach science content while incorporating engineering practices and, to a lesser extent, computing and computational thinking. The task becomes even more complex when they also have to…

Dual credit courses are legislated course enrollment options available to ninth- to twelfth-grade students in the State of Texas. The dual credit program provides the opportunity for all high school students, regardless of grade level, to earn college credits while working toward a high school diploma (Houston ISD, 2018). There is no limit to the…

Much literature has presented evidence that supplemental digital resources enhance student performance with mathematics. The purpose of this study was to explore the impact of a web-adaptive digital resource, Think Through Math©, on student performance with state-mandated annual standardized mathematics assessments. This study utilized a…

Creekside High School is a high-performing public high school where students who are not White or Asian face a racialized school environment. This dissertation examines the school's initiative to detrack its ninth grade Algebra class. Detracking is the process of placing students in heterogeneous classes instead of grouping students by ability.…

Math and science textbook chapters invariably supply students with sets of problems to solve, but this widely used approach is not optimal for learning; instead, more effective learning can be achieved when many problems to solve are replaced with correct and incorrect worked examples for students to study and explain. In the present study, the…

The aim of this study is to analyze the mistakes that have been made in the group theory underlying the algebra mathematics. The 100 students taking algebra math 1 class and studying at the 2nd grade at a state university in Istanbul participated in this study. The related findings were prepared as a classical exam of 6 questions which have been…

Topic-specific didactical design research provides means not only to investigate how to learn but also what to learn, i.e., for specifying learning contents by analyzing students' comprehension processes in detail. This important characteristic of didactical design research is exemplarily shown for students' difficulties in finding symbolic…

A vectorial representation of the full sequence of events occurring during the 2D-NMR heteronuclear single-quantum correlation (HSQC) experiment is presented. The proposed vectorial representation conveys an understanding of the magnetization evolution during the HSQC pulse sequence for those who have little or no quantum mechanical background.…

This study proposes a framework for examining ways in which prospective teachers integrate mathematical knowledge acquired in advanced topics courses into explanatory knowledge for school teaching. Participants, all of whom had recently completed coursework in abstract algebra, were asked to explain concepts connected to the school mathematics…

Lesson study, a teacher-led vehicle for inquiring into teacher practice through creating, enacting, and reflecting on collaboratively designed research lessons, has been shown to improve mathematics teacher practice in the United States, such as improving knowledge about mathematics, changing teacher practice, and developing communities of…

Math and science textbook chapters invariably supply students with sets of problems to solve, but this widely used approach is not optimal for learning; instead, more effective learning can be achieved when many problems to solve are replaced with correct and incorrect worked examples for students to study and explain. In the present study, the…