Ohio Journal of School Mathematics

Gabriel’s Horn and the Painter's Paradox in Perspective

Richard Kaufman — Tue, 02 May 2023 00:00:00 +0000

Gabriel’s Horn is usually discussed as the painter’s paradox. The horn can hold a finite volume of paint, but its inner surface area is infinite and, therefore, cannot be painted. This may seem counterintuitive at first. In this paper, we provide the following perspective: Any finite volume consists of an infinite number of area layers, which amounts to an infinite surface area. This is shown using an example of a “mathematical” ice cube which melts into an infinitely thin film of infinite surface area. Students can appreciate this before they encounter calculus, which is normally used to establish the painter’s paradox. So, we show a perspective that is accessible to a wider range of students, and which is also applicable to all volumes besides just Gabriel’s Horn.

A Laboratory for Secondary Math

Steven P. Meiring — Tue, 02 May 2023 00:00:00 +0000

Students should have opportunities to learn in representative formats for which they will eventually have to apply their knowledge. That is the premise of various curricular innovations over the past 30 years, including authentic learning, problem solving, rich problems/tasks, collaborative learning, modeling, numerical/quantitative literacy, technology, cooperative projects, or STEM-related learning. It also includes tasks related to traditional learning presented in an investigate and discovery format. This article argues for “a place” in the curriculum wherein students and teachers assume a virtual laboratory approach, at times, for learning across the secondary level as an organizational feature, and as an incentive for publishers to provide the activities and tools required to support application of learning for each secondary subject.

Examining Textbooks to Support Prospective Teachers’ Pedagogical Content Knowledge of Geometry

Wed, 02 Aug 2023 00:00:00 +0000

This study describes how textbooks can support prospective teachers’ pedagogical content knowledge in geometry. Relational content analysis methodology guided the collection of the study’s data from five textbooks used in geometry content courses for elementary mathematics teachers to identify pedagogical content knowledge elements and examine the differences and similarities between books. The results suggest that connecting children’s ideas and work to the teacher’s role in the classroom provides opportunities to develop intertwined knowledge of geometry content, students, and teaching. Using standards and standardized test questions in the books can help develop curricular knowledge of geometry contents in K–5 classrooms.

Building a Foundational Understanding of Systems of Equations

Mark D. Hogue, Miranda Spengler, Rachel Hughes, Caitlyn Bell — Wed, 02 Aug 2023 00:00:00 +0000

The teaching of systems of equations has often been reduced to a series of procedures and steps, often boiling off the context that is inherent with each individual system. Delivering learning opportunities around systems requires teachers to present students with opportunities to engage with the Standards of Mathematical Practice, where students reason, model, and persevere. In this article, classroom teachers are provided with ideas for practice that center around the use of problems to introduce the various methods for solving simultaneous equations.

Delving Deeper: Squares within Squares and Cubes within Cubes

Ruby Schwan, Sarah Daye — Wed, 02 Aug 2023 00:00:00 +0000

This article examines the steps to find out how many squares can be drawn within a 100 by 100 array of points. Strategies of pattern analysis and organization are discussed, and the problem is then extended to include a spatial awareness component. The writers solve these fascinating problems step-by-step and then explain how they could be implemented in the high school Geometry classroom.

Unpacking the Ambiguous Case to Develop Conceptual Knowledge and Representational Competence

Kyle Schultz — Thu, 03 Aug 2023 00:00:00 +0000

The ambiguous case is a trigonometry topic for which high school students are often told “stay away from angle-side-side.” In many cases, however, these students do not get the opportunity to explore the underlying mathematical context that serves as the basis of this warning. After a briefly presenting an overview of the ambiguous case, the author describes a mathematical activity using simple homemade manipulatives to support secondary and post-secondary students’ work to unpack and understand the ambiguous case. It then discusses how this activity can support teachers’ understanding of representational competence and recommends general practices supporting students' purposeful and effective use of mathematical representations.

The Power of Basic Math: Preparing Students for a World Beyond the Classroom

Reuben Herrle, Jennifer Flory Edwards, Michael Todd Edwards — Thu, 03 Aug 2023 00:00:00 +0000

In this short op-ed, the authors make the argument for the use of real-world contexts to promote student interest and engagement at all levels of instruction.