The purpose of this paper is to describe the processes utilised to develop an online learning module within the Opening Real Science (ORS) project--"Modelling the present: Predicting the future." The module was realised through an interdisciplinary collaboration, among mathematicians, scientists and mathematics and science educators that…

Mathematical "funds of knowledge" (Civil 2007) that students bring from their community and home lives can offer entry points for concepts such as functions and rate of change, which are important "big ideas" across all grade levels (NCTM 2000). The Common Core State Standards (CCSSI 2010) ask high school students to…

We present the quotient rule version of integration by parts and demonstrate its use.

The NCTM Algebra Standard suggests that students use technology to explore the effects of varying the parameters in y = ax2 + bx + c. This article discusses an extension of this task that incorporates dynamic geometry software to engage students in generating, testing, and proving mathematical conjectures.

This is the fourth paper in our Light-Emitting Diodes series. The series aims to create a systematic library of LED-based materials and to provide readers with the description of experiments and the pedagogical treatment that would help their students construct, test, and apply physics concepts and mathematical relations. The first paper provided…

One of the challenges in learning fractions is understanding how and why a fraction can have multiple interpretations. As presented in one textbook, a fraction is "a symbol, such as 2/3, 5/1, or 8/5, used to name a part of a whole, a part of a set, a location on a number line, or a division of whole numbers" (Charles et al. 2012, p.…

A model solution to a proof question on an examination is explored and subjected to a detailed analysis in terms of Toulmin's scheme of argumentation. In doing so, the ways in which the scheme has been variously used in the mathematics education and philosophical literature are contrasted. The analysis raises a number of issues concerning the…

This work is part of a research project that aims to enhance engineering students' learning of how to apply mathematics in modelling activities of real-world situations, through the construction (design and programming) of videogames. We want also for students to relate their mathematical knowledge with other disciplines (e.g., physics, computer…

To learn mathematical practices, students need opportunities to engage in them. But simply providing such opportunities may not be sufficient to support all students. Simultaneously, explicitly teaching mathematical practices could be problematic if instruction becomes prescriptive. This study investigated how teachers might make mathematical…

There is a paucity of evidence supporting the use of curriculum-based mathematics measures (M-CBMs) at the middle school level, which makes data-based decisions challenging for school professionals. The purpose of this study was to examine the relationships among three existing M-CBM indices: (a) basic facts, (b) concepts/application, and (c)…

In this article, Clay and Rhee use the mathematics topic of circles and the lines that intersect them to introduce the idea of looking at the single mathematical idea of relationships--in this case, between angles and arcs--across a group of problems. They introduce the mathematics that underlies these relationships, beginning with the questions…

We report on an empirical study grounded in our sustained implementation over ten years of a sequence of three-term undergraduate core mathematics courses centred on microworlds. The survey study investigates students' views on 15 competencies potentially developed as they, individually or in pairs, create 12 Exploratory Objects, i.e.,…

It is surprising to students to learn that a natural combination of simple functions, the function sin(1/x), exhibits behaviour that is a great challenge to visualize. When x is large the function is relatively easy to draw; as x gets smaller the function begins to behave in an increasingly wild manner. The sin(1/x) function can serve as one of…

This study tested the hypothesis that intuitions about the effect of operations, e.g., "addition makes bigger" and "division makes smaller", are still present in educated adults, even after years of instruction. To establish the intuitive character, we applied a reaction time methodology, grounded in dual process theories of reasoning. Educated…

Having a deeper understanding of division derived from multiple models is of great importance for teachers and students. For example, students will benefit from a greater understanding of division contexts as they study long division, fractions, and division of fractions. The purpose of this article is to build on teachers' and students'…