In this note we demonstrate two instances where matrix multiplication can be easily verified. In the first setting, the matrix product appears as matrix element concatenation, and in the second, the product coincides with matrix addition. General proofs for some results are provided with a more complete description for 2×2 matrices. Suggested for…

The understanding of the relationships between pattern recognition and mathematical abilities is important for students in their learning of mathematics and mathematics teachers in the design of their teaching pedagogies. This study examines the connections between pattern recognition and mathematical abilities of students from four Singapore…

This study investigated the effect a computer intervention program has on student proficiency in the Algebra 1 End of Course (EOC) examination. This study utilized the intervention math program IXL Math, a computerized program that targets individual students' skill practice and supports students' need to accelerate mastery of Algebra 1 skills in…

As educators, it is important for us to recognize that our assessment methods affect student attitudes. If we want students to learn from their mistakes and counteract a fixed-mindset of learning, perhaps we should look at what we incentivize in the classroom. Some professors are attempting to counteract math and test anxiety, poor STEM retention,…

The design of the Cognitive Tutor Algebra I (CTA1) intelligent tutoring system assumes that students work through sections of material following a pre-specified order, and only move on from one section to the next after mastering the first section's skills. However, the software gives teachers the flexibility to override that structure, by…

Mastery-grading techniques such as mastery-based grading, specifications grading, and standards-based grading are assessment techniques professors are implementing in order to support a growth mindset of learning, decrease test anxiety, and help students develop a deeper understanding of course objectives. In this paper, we provide an analysis of…

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…

Structure sense can be interpreted as an intuitive ability towards symbolic expressions, including skills to perceive, to interpret, and to manipulate symbols in different roles. This ability shows student algebraic proficiency in dealing with various symbolic expressions and is considered important to be mastered by secondary school students for…

Girls are less likely than boys to pursue a degree or career in a mathematics-based field. Girls tend to have higher grades and similar test scores when compared with boys. However, girls' affective beliefs, including confidence, fall behind those of their male counterparts. The purpose of this study was to understand how girls perceive the…

Competency-based learning (CBL) is a grading strategy that emphasizes the mastery of content. At Middle Technical College (MTC), success rates in the college algebra class have decreased. This study will answer two questions: how will a CBL approach to learning affect student perceptions of their success and what are student perceptions of after…

The effect of metacognitive scaffolding for learning by teaching was investigated and compared against learning by being tutored. Three versions of an online learning environment for learning algebra equations were created: (1) APLUS that allows students to interactively teach a synthetic peer with a goal to have the synthetic peer pass the quiz…

This article examines how student learning is affected by the use of team-based inquiry learning, a novel pedagogy in mathematics that uses team-based learning to implement inquiry-based learning. We conducted quasi-experimental and observational studies in intermediate level mathematics courses, finding that team-based inquiry learning led to…

Despite repeated discussions and practice, algebra students continue to make variable errors, in many ways, year after year. This same problem appeared thirty years ago in a list of common errors that math teachers today would immediately recognize, many involving exponents and distribution (Marquis 1988). Similar complaints even appeared in the…

This case study highlights a high school math teacher's successful journey of taking the steps needed to move away from traditional teaching methods to innovative pedagogies supporting mastery-based teaching and learning. Mastery-based learning (also called competency-based learning) is a shift away from traditional, one-size-fits-all teaching.…

In the context of a generic harmonic oscillator, we investigated students' accuracy in determining the period, frequency, and angular frequency from mathematical and graphical representations. In a series of studies including interviews, free response tests, and multiple-choice tests developed in an iterative process, we assessed students in both…