Methods of Mathematical Proof and Logic in Mathematics Courses at the Faculty of Education – Sana'a University

This study aimed to identify the extent to which mathematical proof and logic methods are achieved in mathematics courses for student-teachers at the University of Sana'a. To achieve this objective, the descriptive and analytical method was used. Checklists for analyzing mathematical content according to methods of mathematical proof and logic were developed, and a questionnaire was used to verify the validity of the checklists. The tools were validated by a jury of experts, and the degree of agreement was (98.1%) (85.82%). The tools were applied to a sample of mathematics courses, including mathematical analysis, real analysis and abstract algebra (1), (2). The findings revealed that the most frequent proof methods found in mathematical courses were proof by deduction and transgression (65.86%) of the total methods, followed by proof by mathematical induction (11.75%), and the least frequent was proof by contradiction (9.61%). The courses did not include method of evaluative, critical and reversed proof. The direct method of proof was (82.93%), whereas the occurrence of the indirect proof method was (17.07%). The course content also did not include method of evaluative, critical and reversed proof, and there were statistically significant differences at (0.01), between the weights of the methods of mathematical proof and logic, which were included in the current courses, and the weights that should be included.

Teacher collaboration and instructional program coherence across the transition to high school

This study explores to what extent principals perceive instructional program coherence (IPC) and vertical collaboration occur in high schools and their mid-level feeder schools. It also measures IPC and its five components' association with different measures of student achievement. The study took place in the state of Missouri, and 312 principals of high schools with separate feeder middle schools were invited to participate. Electronic and paper surveys as well as public data were collected and analyzed using descriptive statistics, bivariate correlations, and linear regression. The findings show that principals perceive their school's IPC to be fairly strong, but their vertical collaboration needs improvement. The findings also indicate that only particular IPC components such as curriculum, instruction positively influenced graduation rates, academic climate positively influenced the EOC Algebra 1 scores, and vertical collaboration positively associated with higher average freshman GPA's. A major limitation to this study was the low response rate.

Roller Coasters and Quadratic Equations

Students explore the relationship between thrill-seeking rides and their algebra 1 curriculum in an informative, engaging, student-centered lesson that was designed using a project-based-instruction (PBI) framework.

Foundations to Algebraic Mastery

Realizing that Algebra 1 is a gatekeeper to not only higher mathematics but STEM careers in general, it is imperative that our students master the content matter. Our Nation’s report card shows we are not progressing in this area. To assist in algebraic mastery, this paper describes and provides concrete examples of four research-based pedagogical elements that can aid in student success: (a) basic skill development, (b) computational ease, (c) step-by-step scaffolding, and (d) the extensive use of the Explain-Practice-Assess (EPA) Strategy. Basic skill development assures that all students begin with the requisite background, providing equal opportunity for success, which can promote student engagement. By eliminating unnecessary computational complexity, students are more likely to participate and persevere in problem-solving. The step-by-step scaffolding meets the students where they are and incrementally brings them to mastery, with new material taught in digestible bites. The EPA strategy provides a mean to move students through a topic at an appropriate pace—not moved too quickly; students are given the time necessary to conceptually understand the concepts taught. The four elements described herein serve as a guide to help Algebra I teachers attain success for all students.

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.