We automate and randomize the building of linear systems with a parameter, appropriate for assigning to students. When the parameter takes on a specific value, the system has no solutions. When the parameter takes on a different value, the system has an infinite number of solutions.

This paper initiates a teaching sequence that focuses on building up equivalent definitions to the standard ones for the limit concept in Real Analysis. It comprises two parts: The first provides a classroom assignment where students, guided by Analysis lecturers, are led to develop an alternative definition to the [epsilon] - [delta] one for…

Even when using mixed-ability grouping--grouping students so that students with a range of mathematical thinking work together--there is potential to position some students as more capable than others. In this article, the authors describe two approaches to mixed-ability grouping, providing examples of how teachers using those approaches describe…

As students have struggled to use the "chip model" (i.e., red and yellow chips representing positive and negative numbers) to model integer addition and subtraction and have found it confusing, the authors developed a series of activities based on adding and removing opposite objects to and from a boat to better help students in this…

Virtual reality (VR) has gained prominence and a growing prevalence. However, parental attitudes toward using VR for educational purposes (i.e., educational VR (EVR)) are under-researched. This study investigates parents' perceptions and experiences regarding the use of EVR, focussing on mathematics learning. Using the technology acceptance model…

Reasoning is the process of thinking used to make claims and reach conclusions. It is among the first skills that should be taught to students in mathematics lessons and contributes to the creation and development of mathematical foundations. This study identifies evolution and trends in research on mathematical reasoning over the last four…

In the primary grades, engaging in counting collections provides students with regular opportunities to make sense of numbers in dynamic ways, build social skills, and communicate about mathematics. This article aims to guide elementary educators through a team's process of planning and implementing counting collections in kindergarten. Counting…

As new technological developments continue to change the educational landscape, it is not an exception in the area of proof and proving. This classroom note introduces the use of one of the trending proofs assistants -- the Lean theorem prover. We first provide a technical account of Lean, then exemplify Lean proofs in propositional logic, number…

This paper is a practical how-to guide to help you start using active learning or to have greater success and more fun with it. We categorize active learning techniques as Think, Pair, Share, Composite, Group, Move, or Lead and discuss how to implement activities in each category, along with advice on creating engaging, effective, and equitable…

Mathematical games can be used to engage learners, promote fluency and deepen conceptual understanding. When considering the level of engagement of students, sometimes a small change can have a large impact. Something as simple as switching dice for cards in a game and, importantly, introducing a 'non-replacement' mechanism to the gameplay, can…

In this paper, we present a qualitative case study on an online graduate program for practicing teachers and educators that explores contemporary mathematics and its integration into K to 12 education. Data for the study comprised students' work in the courses, student feedback, and notes from instructor debriefings for two cohorts of the program.…

Schools have been using various approaches to address the pandemic-related struggles that students are experiencing with mathematics. There is an overwhelming consensus among both educators and researchers that we need to adopt acceleration rather than remediation as a tool to counteract the challenges that students currently face. Acceleration is…

Every application of integration by parts can be done with a tabular method. The trick is to identify and consider each new integral in the table before deciding how to proceed. This paper supplements a classic introduction to integration by parts with a particular tabular method called Row Integration by Parts (RIP). Approaches to tabular methods…

Typically, vocabulary instruction is associated as part of reading instruction. However, vocabulary instruction is also a necessary component of mathematics instruction. In fact, state and national standards (e.g., Common Core State Standards) require elementary students to communicate about mathematics to provide clear explanations, construct…

The mathematical modeling process is also a powerful way to teach students the meaning of data. In mathematical modeling, students are collecting data to answer a question. Data collection is the means to an end and not the end in itself. The author was looking for an engaging way to introduce his third-grade mathematics class to the skill of…