Drawing from sources in gesture studies, cognitive science, the anthropology of religion and art/architecture history, this article explores cultural, bodily and cosmological resonances carried (unintentionally) by mathematical graphs on Cartesian coordinates. Concepts of asymmetric bodily spaces, grids, orthogonality, mapping and sacred spaces…

Because fractions and percentages can be difficult for children to grasp, connecting them whenever possible is beneficial. Linking them can foster representational fluency as children simultaneously see the part-whole relationship expressed numerically (as a fraction and as a percentage) and visually (as a pie chart). NCTM advocates these…

Composition of functions is one of the five big ideas identified in NCTM's "Developing Essential Understanding of Functions, Grades 9-12" (Cooney, Beckmann, and Lloyd 2010). Through multiple representations (another big idea) and the use of The Geometer's Sketchpad[R] (GSP), students can directly manipulate variables and thus see dynamic visual…

Assuming that textbooks give literary expression to cultural and ideological values of a nation or group, we propose the analysis of chemistry textbooks used in Brazilian universities throughout the twentieth century. We analyzed iconographic and textual aspects of 31 textbooks which had significant diffusion in the context of Brazilian…

When hoping to initiate or sustain students' interest in mathematics teachers should always consider relevance, relevance to students' lives and in the middle and later years of instruction in high school and university, accessibility. A topic such as the mathematics behind networks science, more specifically scale-free graphs, is up-to-date,…

Functions are one of the most important and powerful tools in mathematics because they allow the symbolic, visual, and verbal representation of relationships between variables. The power of functions, as well as the numerous real-world uses of functions, make them an important part of the development of algebraic reasoning in the middle grades.…

This article is essentially devoted to a brief historical introduction to Euler's formula for polyhedra, topology, theory of graphs and networks with many examples from the real-world. Celebrated Konigsberg seven-bridge problem and some of the basic properties of graphs and networks for some understanding of the macroscopic behaviour of real…

The book "If the World Were a Village," by David J. Smith, is the context for analysing and creating graphs of the world's demographic information. Students examine numerical information regarding the more than six billion world inhabitants by imagining the world's population as 100 people.

The activities described here allow students to explore the concept of diffusion with the use of common equipment such as computers, webcams and analysis software. The procedure includes taking a series of digital pictures of a container of water with a webcam as a dye slowly diffuses. At known time points, measurements of the pixel densities…

Graphs are important for supporting critical thinking and scientific argumentation because students can use them to reason, make judgments and decisions, and solve problems like a scientist (Connery 2007). Yet teaching students how to use math to actually think critically continues to be difficult for teachers. This article describes two…

Students frequently have difficulty determining whether a given real-life situation is best modeled as a linear relationship or as an exponential relationship. One root of such difficulty is the lack of deep understanding of the very concept of "rate of change." The authors will provide a lesson that allows students to reveal their misconceptions…

Data from a cough challenge study are displayed with dot charts to demonstrate the importance of graphs in understanding data, principles of graph construction and visual perception. The data are available for use in the classroom.

It is common for both algebra and calculus instructors to use power functions of various degrees as well as exponential functions to examine and compare rates of growth. This can be done on a chalkboard, with a graphing calculator, or with a spreadsheet. Instructors often are careful to connect the symbolic and graphical (and occasionally the…

Today, beginning algebra in the high school setting is associated more with remediation than pride. Students enroll by mandate and attend under duress. Class rosters in this "graveyard" course, as it is often referred to, include sophomores and juniors who are attempting the course for the second or third time. Even the ninth graders…

Entropy analyses as a function of the extent of reaction are presented for a number of physicochemical processes, including vaporization of a liquid, dimerization of nitrogen dioxide, and the autoionization of water. Graphs of the total entropy change versus the extent of reaction give a visual representation of chemical equilibrium and the second…