This article deals with a short historical introduction to determinants with applications to the theory of equations, geometry, multiple integrals, differential equations and linear algebra. Included are some properties of determinants with proofs, eigenvalues, eigenvectors and characteristic equations with examples of applications to simple…

Computational estimation is an important skill necessary for students' mathematical development. Students who can estimate well for computations rely on an understanding of many mathematical topics, including a strong number sense, which facilitates understanding the mathematical operations and contextual evidence within a problem. In turn, good…

The study of function transformations helps students understand the function concept which is a basic and main concept in mathematics, but this study is problematic to school students as well as college students, especially when transformations are performed on non-basic functions. The current research tried to facilitate grade 9 students'…

The purpose of this research was to explore to what extent probing learners' errors on simplifying algebraic fractions would help them to reduce those errors. The study assumed that students' errors and misconceptions persist because they handle mathematical concepts as if they were isolated chunks of knowledge. A sample of 15 grade 10 girls took…

Many rules taught in mathematics classrooms "expire" when students develop knowledge that is more sophisticated, such as using new number systems. For example, in elementary grades, students are sometimes taught that "addition makes bigger" or "subtraction makes smaller" when learning to compute with whole numbers,…

This paper reports on the reflective practice of a group of nine secondary mathematics pre-service teachers. The pre-service teachers participated in a year-long, school-based professional experience program which focussed on observing, co-teaching and reflecting on a series of problem-solving lessons in two junior secondary school mathematics…

At a time when teacher training is being moved to school-based programmes it is important to engage in a research-informed dialogue about creating more distinctive, and cost-effective 21st century models of teacher training. Three years ago I began feasibility field testing the Lesson Observation On-line (Evidence Portfolio) Platform [LOOP]…

This study addressed the opportunities to learn (OTL) modeling in algebra provided to secondary mathematics pre-service teachers (PSTs). To investigate these OTL, we interviewed five instructors of required mathematics and mathematics education courses that had the potential to include opportunities for PSTs to learn algebra at three universities.…

For centuries, the basic operations of school mathematics have been identified as addition, subtraction, multiplication, and division. Notably, these operations are "basic," not because they are foundational to mathematics knowledge, but because they were vital to a newly industrialized and market-driven economy several hundred years…

For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…

Over the past two decades, mathematics education reformers have increasingly called for new approaches to assessment. This call urges mathematics teachers to merge their assessment with pedagogy, to employ assessments that engage students in the process of "doing mathematics," and to exploit assessment to support learning. This paper…

I advance a theoretically and empirically-grounded case for designing for and learning from failure, and instantiate it in a learning design called Productive Failure (PF). I describe the key mechanisms and the design principles of PF. The PF learning design comprises a generation and exploration phase followed by a consolidation and knowledge…

Partnerships between researchers and teachers are central to stimulating advancements in a linking research and practice agenda. This paper addresses two key aims. First, research on supporting students' representational fluency in technology-rich algebra learning environments is used to illustrate a linking research and practice agenda. Three…

Do your students think a triangle can be constructed from any three given line segments? Do they believe that a transformation affects only the pre-image--not the whole plane? Do they understand that examples--no matter how many they find--cannot prove a conjecture but one counterexample is sufficient to disprove it? "What tasks can you…

This review-specific protocol guides the review of research that informs the What Works Clearinghouse (WWC) intervention reports in the Secondary Mathematics topic area. The review-specific protocol is used in conjunction with the "WWC Procedures and Standards Handbook (version 3.0)." This review focuses on mathematics interventions that…