The probability certain random quadratics have real roots

Early in high school algebra, quadratics chosen as examples by teachers and textbooks alike tend to have integer coefficients and to factorise over the integers. This can give the misleading impression that such quadratics are the norm. As students progress into calculus and begin regularly seeing quadratics that are not as ‘nice’, we hope they become disabused of this notion. Indeed, even if the coefficients of the quadratic are integers, the probability that the quadratic factorises over the integers tends to zero as the range from which the integers are drawn grows (see [1]). But what if we ask about a behaviour less restrictive than factorising, say merely having real roots? This is the problem that concerns this Article.

An Investigation of Success Factors in a High School Algebra Intervention Program

This quantitative study allowed investigation of a high school Algebra intervention program through examination of potential relationships among teachers’ beliefs about teaching and learning, teachers’ instructional styles, students’ academic self-concept in mathematics, and students’ mathematics achievement. Existing research focused on individual components used in this study on the elementary level; thus, leaving a gap in understanding of how factors related to the success of high school students. Results may provide information to teachers and administrators regarding relationships among factors shown to impact student achievement in mathematics, and provide evaluation of an Algebra intervention program at the secondary level. The researcher utilized the National Council of Teachers of Mathematics Teaching and Learning Beliefs Questionnaire to identify beliefs about teaching and learning mathematics and the Reformed Teaching Observation Protocol (RTOP) to determine how closely instruction in Algebra classrooms aligned with constructivist practices. Students were given the Academic Self-Description Questionnaire II (ASDQII) as pre-and-post measures of academic self-concept in relation to mathematics; then, data were checked for relationships to achievement, measured by common semester final exams. Pearson Product Moment Correlation Coefficient analysis determined significant relationships existed between RTOP scores and the ASDQII prompts: I am hopeless when it comes to mathematics; work in mathematics is easy for me; and I get good marks in mathematics classes. Significant relationships were defined between Teaching Style and both Productive and Unproductive Teacher Beliefs, as well as between students’ academic self-concept in mathematics and overall mastery of Algebra content, measured by scores on common semester finals.

Algebra Instruction for Students with Learning Disabilities in the Era of Common Core

The Common Core State Standards in Mathematics (CCSSM) were released more than 10 years ago. This set of standards outlines the mathematics that all students should know and be able to do to prepare them for post-secondary education and employment. Students with learning disabilities (LD) continue to underperform in relation to their peers without disabilities in secondary mathematics. As high school Algebra I is a required course for the majority of students, research-based instructional practices should be utilized to support students with LD in Algebra I. This article summarizes recent research on instructional practices for teaching algebra content that aligns to the CCSSM. Specifically, three types of instructional practices have been found to promote progress in the high school algebra content: (a) concrete-representational-abstract integration, (b) virtual manipulative instruction, and (c) gestures and diagrams.

A Heart-Centered Stance: Receptivity to Algebra Teachers’ and Students’ Multidimensional Experiences

The algebra classroom in urban public high schools in the United States is a complex space, ripe with many challenges and opportunities. In this paper I introduce the notion of a heart-centered stance for the teacher and the educator, and a method of engaging in creative expression for reflection and introspection toward individual change in the rich context of the high school algebra classroom. My evolving relationships with two high school algebra teachers, observations of their classrooms, as well as my own self-study and professional growth, are incorporated into this paper as I introduce and exemplify two tenets of a heart-centered stance: multidimensionality of experience and receptivity to relatedness. This study suggests the possibility of using creative artistic expression and a self-study approach to support the transformation of educators’ perspectives toward research, creative activities, and outreach that are receptive to the mathematical experiences of teachers and students in our local communities.

Beyond Rise Over Run: A Learning Trajectory for Slope

This article presents the results of a design-based research study related to slope that took place in a high school algebra 1 classroom. In the study, students explored situations related to making predictions. As students engaged with these situations, they reinvented and made meaningful multiple subconstructs of slope. I present the findings in the form of a learning trajectory for slope, which describes how learning happened over time, the principles that guide the design of activities that support this learning, and the rationale for how the activities support learning.

Asked & Answered

The Asked & Answered department shares excerpts from discussion threads on the online MyNCTM community. In this issue, featured threads highlight responses to members' questions related to mathematical depth in preschool, spiral review in the upper elementary grades, ideas for differentiation in middle school, and projects for high school algebra.

Why Math?

An entry-level high school algebra project increases student engagement as they explore reasons to learn mathematics.