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…

This article provides conceptual ideas, data, and exercises, for integrating original sources of recent, state of the art, world-class life science research in the undergraduate mathematics curriculum and classroom. To this end, this article shows how one of the main goals of calculus in the life sciences, fitting parameters to data and assessing…

Evaluations of professional development programmes often seek to represent definitive outcomes, with phenomena posited as discrete, bounded and independent entities, and researchers positioned as external actors. An alternative is to understand the complexity of these relationships as entanglements by applying Baradian concepts. The value of this…

Ensuring students master their basic facts is a shared priority among teachers, parents, and administrators. As a result, the number of available resources for basic facts instruction, from concrete to virtual, is vast and continues to grow. Basic fact fluency has always been of interest to elementary school teachers and is particularly relevant…

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,…

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…

Internationally, schools with small numbers of children and constrained funding are not able to group the children in classes based on their age. Rather, the children are combined into larger 'mixed-age' classes. Schools and teachers manage this situation in different ways, for example by teaching the class as two or more separate age-based groups…

Math fact fluency involves the quick, accurate retrieval of basic arithmetic combinations and the ability to use this fact knowledge efficiently. Math fact retrieval is typically considered fluent when performed accurately within 2 to 3 seconds, and "efficiency" refers to students' ability to apply fact knowledge to more complex…

In engineering sciences like mechanical or automotive engineering, pre-knowledge in mathematics is required to follow deeply specific engineering lectures. Mathematical skills are essential preconditions to successfully complete engineering lectures. It is necessary that these preconditions are identified for each individual student so that…

Like many math educators, I have spent much of my career bound to traditional methods of instruction and assessment. In recent years, motivated by a growing understanding that such approaches may not result in equitable or inclusive classroom environments, my teaching philosophy has shifted radically. In this article, I describe how I transformed…

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.…

Here, we report on the development of a theoretical framework for mathematics teaching that looks aside from common traditional/reform dichotomies. Attention is focused on managing the amount of new information students must attend to, while structuring that information in a manner that allows discernment of key features and continuous extension…

The learning of "basic facts"--single-digit combinations for addition, subtraction, multiplication, and division--has long been a focus of elementary school mathematics. Many people remember completing endless worksheets, timed tests, and flash card drills as they attempted to "master" their basic facts as children. However,…

In the authors' examination of various instructional programs, they observed that most provide all the necessary data to solve proportion problems, employ compatible numbers that are usually unrealistic, present numbers (data) in the order in which they are to be manipulated, discuss contexts that cannot be easily replicated, and present…

For decades, classroom instruction was dominated by lectures, textbooks and note-taking. And for decades, students have often been bored and unengaged. They took notes and did what they had to do to memorize lessons just long enough to pass an exam. But short-term memorization does not help much in preparing students for college, advanced training…