Adolescent and children's concepts of multiplication and fractions have been linked to differences in the number of levels of units they coordinate. In this paper, we discuss relationships between adult students' conceptual structures for coordinating units and their pre-calculus understandings. We conducted interviews and calculus readiness…

The benefits of using Microsoft Excel built-in functions in an undergraduate Multistage Separations course are analyzed based on 15 years of experience teaching mass transfer in Chemical Engineering programs at both the Bachelor's and Master's levels. Eight reasons for using spreadsheets in a Mass Transfer virtual course are given. Through the…

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of…

The purpose of this study is to compare the differences of four essential aspects (i.e., representational forms, contextual features, cognitive demand levels, and response types) embedded in mathematical problems within the topics of numbers and operations in mathematics textbooks used in Finland, Indonesia, Malaysia, Singapore, and Taiwan. This…

This article reports on an exploration of how second-graders can learn mathematics through programming. We started from the theory that a suitably designed programming language can serve children as a language for expressing and experimenting with mathematical ideas and processes in order to do mathematics and thereby, with appropriate tasks and…

I examined the relationship between teachers' identification of mathematical points in written lessons and students' mathematical learning opportunities. Lessons in teachers' guides and classroom instruction were analyzed for written mathematical points and those articulated by teachers during instruction. Teachers who appropriately identified…

Understanding how environments influence learning requires attending not only to what is present but also to what is absent. In the context of mathematics learning, this means attending not only to problems that children encounter frequently in textbooks but also to ones that appear rarely. We present research in this article showing that students…

This article explores some entertaining examples of mathematical thinking in working with the alphabet. They involve thinking linearly, sequentially, in order, and logically, and require problem solving--mathematical thinking. The article discusses creatively exploring the alphabet through alphabet books and creating alternative alphabet lists.

Understanding how environments influence learning requires attending not only to what is present but also to what is absent. In the context of mathematics learning, this means attending not only to problems that children encounter frequently in textbooks but also to ones that appear rarely. We present research in this article showing that students…

The development of mathematical reasoning is a key proficiency for mathematics within the Australian Curriculum. However, reasoning can be difficult for teachers to assess, particularly with pen and paper tests. In this study, interview tasks were designed across three curriculum areas at three different levels to assess student reasoning through…

It may sound strange, but in today's mathematics classroom, teachers are encouraged to have students struggle. This struggle, however, is not what might come to mind upon first hearing the word. The struggle I refer to here is what researchers have termed "productive struggle," which is present when students must persevere in solving…

In this dissertation, I explore ways to support secondary school students' meaningful understanding of quadratic functions. Specifically, I investigate how students co-developed representational fluency (RF) and functional thinking (FT), when they gained meaningful understanding of quadratic functions. I also characterize students' co-emergence of…

The purpose of this study was to examine the psychometric properties of the Precalculus Concept Assessment (PCA), a 25-item multiple-choice instrument designed to assess student reasoning abilities and understanding of foundational calculus concepts (Carlson et al., 2010). When this study was conducted, the extant research on the PCA and the PCA…

Differential and integral calculus textbooks are widely used as the main resource for teaching. They appear in a variety of forms and adopt various approaches to present the content. In this paper, we turn our attention to one chapter of a calculus textbook and our focus is on the introduction of the derivative concept. With the purpose of…

The standard technique for solving equations with radicals is to square both sides of the equation as many times as necessary to eliminate all radicals. Because the procedure violates logical equivalence, it results in extraneous solutions that do not satisfy the original equation, making it necessary to check all solutions against the original…