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Emergent graphical shape thinking (EGST) involves interpreting or constructing a graph as dynamically generated, which is useful across science, technology, engineering, and mathematics fields. Although evidence suggests that students as young as middle school can engage in EGST with support, other research indicates most college students and U.S. teachers do not spontaneously engage in such reasoning when potentially productive. We describe a local instruction theory (LIT) to support middle school students developing EGST as part of their graphing meanings. We then present a case study to show how two students engaged with a task sequence designed with the LIT in mind to develop meanings for EGST. This article illustrates general principles researchers and educators could use to promote students' graphing meanings.

Covariational reasoning (or the coordination of two dynamically changing quantities) is central to secondary STEM subjects, but research has yet to fully explore its applicability to elementary and middle-grade levels within various STEM fields. To address this need, we selected a globally referenced STEM assessment--the Trends in International Mathematics and Science Study (TIMSS)--to investigate the extent to which covariational reasoning could be applied. Specifically, we identified the frequency of items with potential to elicit students' covariational reasoning through a content analysis of publicly released TIMSS 2011 items in Grades 4 and 8 mathematics and science. We found that approximately one-third of items in all grades and subjects had potential for covariational reasoning, and many of these items had such potential with no viable alternative strategy. Furthermore, items in every content strand and cognitive domain had potential for covariational reasoning. We interpret these findings as indicative of the salience of covariational reasoning across STEM education, and we discuss the implications of these results for research, assessment, and instruction.

Although there is much research exploring students' covariational reasoning, there is less research exploring the ways students can leverage such reasoning to coordinate more than two quantities. In this paper, we describe a system of covariational relationships as a comprehensive image of how two varying quantities, having the same attribute across different objects, each covary with respect to a third quantity and in relation to each other. We first describe relevant theoretical constructs, including reasoning covariationally to construct relationships between quantities and reasoning covariationally to compare quantities. We then present a conceptual analysis entailing three interrelated activities we conjectured could support students in reasoning covariationally to conceive of a system of covariational relationships and represent the system graphically. We provide results from two teaching experiments with four middle school students engaging in tasks designed with our conceptual analysis in mind. We highlight two different ways students reasoned covariationally compatible with our conceptual analysis. We discuss the implications of our results and provide areas for future research.

In this study, we examined graphical representations in several mathematics, science, and engineering textbooks and practitioner journals with the goal of identifying similarities and differences across these sources. To do this, we drew from the extant research on students interpreting graphs and reasoning covariationally to develop a framework to analyse different aspects of graphical representations in these sources. We present several key findings which include identifying different uses of coordinate system which are prevalent in specific sources, the prevalence of contextualized and decontextualized quantities across sources, and differences in the frequency with which certain graphing conventions are maintained. We conclude by discussing implications of this study in relation to the teaching and learning of graphical representations across these fields and directions for future research.

Critical to constructing and interpreting graphs is an individual's understanding of the underlying coordinate systems, yet coordinate systems are often overlooked or taken-for-granted in both mathematics education research and curricula. In this paper, we foreground coordinate systems and present a distinction between two uses of coordinate systems--spatial and quantitative. We describe how attending to this distinction can inform research endeavors and we provide possible alternative interpretations of findings from previous research. We close with implications for the learning and teaching of coordinate systems and graphical representations.

We were interested in exploring the extent to which advanced mathematics lecturers provide students with opportunities to play a role in considering or generating course content. To do this, we examined the questioning practices of 11 lecturers who taught advanced mathematics courses at the university level. Because we are unaware of other studies examining advanced mathematics lecturers' questioning, we first analyzed the data using an open coding scheme to categorize the types of content lecturers solicited and the opportunities they provided students to participate in generating course content. In a second round of analysis, we examined the extent to which lecturers provide students with opportunities to generate mathematical contributions and to engage in reasoning that researchers have identified as important in advanced mathematics. Our findings highlight that, although lecturers asked many questions, lecturers did not provide substantial opportunities for students to participate in generating mathematical content and reasoning. Additionally, we provide several examples of lecturers providing students with some opportunities to generate important contributions. We conclude by providing implications and areas for future research.

Supporting students developing understandings of function has been a notoriously elusive task in mathematics education. We present Thompson and Carlson's (2017) description of a covariational meaning of function and provide an example of a student who maintains meanings compatible with this description. We use this student's activity to illustrate nuances in a covariational meaning of function and to highlight how such meanings can be powerful. We argue a student who has developed meanings compatible with a covariational meaning of function has the foundational meanings needed to understand a formal definition of function (i.e. the horse needed to pull the cart).

Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers' and students' inverse function meanings. Responding to this call, we characterize 25 pre-service teachers' inverse function meanings as inferred from our analysis of clinical interviews. After summarizing relevant research, we describe the methodology and theoretical framework we used to interpret the pre-service teachers' activities. We then present data highlighting the techniques the pre-service teachers used when responding to tasks that involved analytical and graphical representations of functions and inverse functions in both decontextualized and contextualized situations and discuss our inferences of their meanings based on their activities. We conclude with implications for the teaching and learning of inverse function and areas for future research.

GeoGebra is a free tool that has the potential to change both how and what is taught in mathematics. GeoGebra allows teachers and students to explore various mathematical ideas either through the full applet (https://www.geogebra.org/graphing) or by sharing applets via GeoGebra's Materials site (https://www.geogebra. org/materials/). It has many capabilities, including allowing students to graph functions, draw and change various geometric shapes, solve equations, and complete other algebraic procedures. Although all of these features make GeoGebra a potent tool, the full applet can be overwhelming to a novice user and can create distractions for students. Teachers should consider what tools are (and are not) important for students. This article offers suggestions for teachers on how to create GeoGebra files that maximize student engagement with mathematical ideas while minimizing potential distractions.

To engage students, many teachers wish to connect the mathematics they are teaching to other branches of mathematics or to real-world applications. The lesson presented in this article, which uses the algebraic skill of finding the equation of a line between two points and the geometric axiom that any two points define a line, does both. A historically significant coding technique relies on algebra skill and the axiom. This lesson integrates the first Standard for Mathematical Practice from the Common Core State Standards (CCSSI 2010) and many of the NCTM Process Standards--namely, Problem Solving, Reasoning and Proof, Communication, and Connections (NCTM 2000). What follows is a description of how the lesson can be presented as well as insight into adaptations that will make it a good fit for any classroom. This historically significant real-life application of a cryptographic coding technique, which incorporates first-year algebra and geometry, makes mathematics come alive in the classroom.

Can one infinity be more than another infinity? Ask students this question, and many will be puzzled; others will insist that "infinity is infinity." The question seems to pique their interest and provides an opportunity to present the beautifully simple but counterintuitive proofs concerning the size of infinity first constructed by Georg Cantor. Such a question may not appear to fit within certain specific content, but the concept of infinity emerges within many contexts throughout the secondary school curriculum, from first-year algebra to BC Calculus. Two other topics in the secondary school curriculum closely related to infinity are one-to-one and onto functions, which have clear connections to Cantor's work. A function is "one-to-one" if each output value is the image of a unique input; a function is "onto" if each element in the range is an output for some input value from the domain. Too often, students get little exposure to one-to-one and onto functions. The significance of infinity is supported by NCTM's Standards (2000), which underscore the importance of developing a deep understanding of very large numbers as well as comparing and contrasting the properties of different number systems at the secondary school level. Further, these Standards emphasize the importance of students analyzing and evaluating the mathematical thinking, arguments, and proofs of others. In this article the author suggests that a historical exploration of Cantor's proofs concerning infinite sets is one way in which teachers can develop students' understanding of different sets of numbers while allowing students to evaluate and critique the mathematical arguments of others. A bibliography is included.