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A new classification for semantic structures of one-step word problems is proposed in this paper. The classification is based on illustrations of word problem situations in Common Core State Standards (CCSSM, 2010) and related historical studies (e.g. Weaver, 1973, 1979, 1982), as well as conceptual elaborations of embodied and grounded nature in Lakoff et al. (2000). The classification identifies two main classes: action on/change of an initial quantity and coordination/comparison of two quantities, providing a unifying characteristic of basic operations of quantities. This classification is more comprehensive and differentiated than the classification of CCSSM (2010) and Polotskaia et al. (2021), as it emphasizes conceptual demands of children's mathematics, coherence and continuity of progressions, and consistency with thinking modes and/or problem-solving strategies. [ABSTRACT FROM AUTHOR]

This case study explores theories and enactments for using whole-body movement in helping a student with visual impairment understand slope. While technology has made learning algebraic concepts easier for many students, much of it relies on sight. To accommodate the student's learning, tactile and whole-body experiences were used to give meaning to the concept of slope. The results show that the learner constructed deep, profound, and novel notions of slope. In this paper, we illustrate the way mathematical knowledge was understood and communicated through motion and language and raise theoretical questions about the relationship between the body, the environment, and mathematical ideas. We propose these pedagogies be used with all students reflective of universal design principles. [ABSTRACT FROM AUTHOR]

The development of the mathematical concept of the infinite, through the reflections that arise from personal notions and perceptions and the analysis of some ideas of Galileo and Cantor, invites us to investigate the relationship between intuition and formalization for the understanding of the said concept. This paper aims to observe and describe the shift from intuition to formalization that Omar, a first-semester undergraduate student in applied mathematics, goes through when accounting for the number of objects between two equal mirrors, one in front of the other. The results indicate that the interaction between intuition and formalism is necessary for Omar to conceive of mathematical infinity. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the relationships between adult students' mathematical micro-identities (informed by positioning theory) and their mathematics (informed by radical constructivism). I used positioning theory to conduct a detailed turn-by-turn analysis of students' micro-identities, and to inform my models of students' mathematics. Results showed how positive micro-identities can promote persistence, vulnerability, visibility, or flourishment in students' mathematics. Implications for research and practice are discussed. The latter connects to the PMENA conference theme by highlighting the type of classroom environment teachers can foster to engage their students and maximize opportunities for learning. [ABSTRACT FROM AUTHOR]

We articulate a framework for delineating student thinking in active, STEM-rich learning environments. Researchers have identified ways of reasoning that relate to specific content areas and practices within each of the STEM disciplines. However, attempts at characterizing student thinking in transdisciplinary STEM environments remains in its infancy, in need of theoretical and analytic frameworks to support this emerging research area (Li et al., 2019). This paper advances the field along both of these dimensions by articulating an analytic framework for student thinking in STEM contexts and operationalizing it through an analysis of two groups of students in elementary, informal learning situations. Our results suggest that student thinking, in these environments, is often grounded in personal experience and authority and consists of a rich mix of general and disciplinary-oriented reasoning. Implications are provided. [ABSTRACT FROM AUTHOR]

This paper describes our work to determine the naturalistic images that first-time secondsemester university calculus students possess for series convergence. We found that the students we interviewed most frequently determined whether a series converged by imagining a process of appending summands into a running total and examining whether this running total appeared to approach an asymptotic value. We provide examples and three corresponding implications of this "asymptotic running total" that informed students' actions while determining series convergence or the value of convergence. Our paper adds to the research literature by confirming students' meanings for limits reported for other topics (e.g., limit of sequence, function, Taylor series) apply to infinite series and proposing relationships between previously reported meanings for series convergence. [ABSTRACT FROM AUTHOR]

This paper introduces two theoretical constructs, open-loop covariation and closed-loop covariation, that combine covariational reasoning and causality to characterize the way that three preservice mathematics teachers conceptualize a feedback loop relationship in a mathematical task related to climate change. The study's results suggest that the preservice teachers' reasoning about feedback loop between quantities involved the ability to conceive closed-loop covariation, which in this study was characterized by two cognitive realizations: (i) the conception of simultaneous change and (ii) the recognition of circular causality. These realizations, at least for the participants, appeared to be independent from one another. The theoretical distinction between open- and closed-loop covariation could inform instructional strategies to develop students' ability to think about and model feedback loops. [ABSTRACT FROM AUTHOR]

This paper describes the APOS-Slope framework, a tool for studying students’ understanding of slope. The paper outlines how this framework merges the theoretical lens of APOS with past work describing multiple conceptualizations of slope. Descriptions of student understanding of slope at each stage of the framework are provided, and implications of the framework for future research are detailed [ABSTRACT FROM AUTHOR]

This study examined the role of student generated drawings to offload cognitive demands of a mathematical problem. We used Unit Transformation Graphs to compare students' thought processes when they had to solve the problem mentally, and when they were allowed to use pen and paper. The results indicated that the possibility to rely on drawings helped the participants to free up working memory resources and complete a cognitively demanding fractional task. [ABSTRACT FROM AUTHOR]

This paper frames children's mathematics as mathematics. Specifically, it draws upon our knowledge of children's mathematics and applies it to understanding the prime number theorem. Elementary school arithmetic emphasizes two principal operations: addition and multiplication. Through their units coordination activity, children construct two mathematical worlds: an additive world and a multiplicative world. Understanding how children might map between their additive and multiplicative worlds provides insights into the prime number theorem. It also helps us appreciate the power of children's mathematics, constructed through the coordination of their own mental actions. [ABSTRACT FROM AUTHOR]

In this paper, I introduce a mode of reasoning, which I call spatial proximity reasoning, that entails coordinating the (co)variation of an object's degree of proximity (i.e., closeness or nearness) to other objects (i.e., reference objects) in a dynamic real-world situation. I compare and contrast spatial proximity reasoning with quantitative (co)variational reasoning. I also present the implications of spatial proximity reasoning on students' graphing activities in oneand two-dimensional spaces. [ABSTRACT FROM AUTHOR]

In this paper we explore how college students across different courses appeared to interpret the meaning of parentheses or brackets in the context of algebraic syntax. This work was influenced by theories of computational vs structural thinking, and also considered the extent to which students' definitions, computational work, and explanations appeared to be consistent with specific normative definitions of parentheses. In analyzing student work, several categories of students' conceptions emerged, which may be helpful in diagnosing which conceptions may be more productive or problematic as students progress through algebra. For students who appear to conceptualize parentheses as a cue to non-normative procedures, several categories of procedures were found, which could have implications for instruction. [ABSTRACT FROM AUTHOR]

In this paper, we describe a framework for characterizing students' graphical reasoning, focusing on providing an empirically-based list of students' graphical resources. The graphical forms framework builds on the knowledge-in-pieces perspective of cognitive structure to describe the intuitive ideas, called "graphical forms", that are activated and used to interpret and construct graphs. In this study, we expand on the current knowledge base related to the specific graphical forms used by students. Based on data involving pairs of students interpreting and constructing graphs we present a list of empirically documented graphical forms and organize them according to similarity. We end with implications regarding graphical forms' utility in understanding how students construct graphical meanings and how instructors can support students in graphical reasoning. [ABSTRACT FROM AUTHOR]

In this paper, I propose a new construct named analytic equation sense to conceptually model a desired way of reasoning that involves students’ algebraic manipulations and use of equivalent expressions. Building from the analysis of two existing models in the field, I argue for the need for a new model and use empirical evidence to explain the new model. [ABSTRACT FROM AUTHOR]

Students often struggle to make sense of integer concepts because they contradict their whole number understanding. In this paper, I unfold how four fifth graders interpret a story conflict and its resolution within different interactive versions of a mathematics storybook, which was designed to highlight the contradictory ideas between the absolute and linear values of integers in the context of temperature. Over three sessions of reading and retelling, students paid more attention to the differences between positive and negative numbers’ order and value and more often referred to the mathematical conflict and its resolution. This study informs the effectiveness of introducing the concept of integers by presenting conflict and illuminates ways that students can be supported through the resolution process. [ABSTRACT FROM AUTHOR]

Copyright of Conference Papers -- Psychology of Mathematics & Education of North America is the property of Psychology of Mathematics & Education of North America and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

All humans have bodies. Our bodies profoundly affect our perspectives and understandings of the world. Today's schools neglect the body as a resource for mathematical thinking and learning. Recently this changed and researchers are putting the body back into the equation. Recent research explores changes in the modality and scale of mathematical activity, putting the power of paper and pencil into whole-body coordinated movements in open spaces. This study designs and studies learning in new forms of mathematical activity that involve multiple, moving bodies. This paper reports on the first phase of a two phase project in which we use interviews with experts in a variety of disciplinary fields, asking people to find and make sense of emergent mathematics in choreographed, ensemble performances from the opening ceremony of the 2016 Rio Olympic Games, to design and study experimental mathematics learning. [ABSTRACT FROM AUTHOR]

Computational activity is increasingly relevant in education and society, and researchers have investigated its role in students' mathematical thinking and activity. More work is needed within mathematics education to explore ways in which computational activity might afford development of mathematical practices. In this paper, we specifically examine the generalizing activity of undergraduate students who solved combinatorial problems in the context of Python programming. We demonstrate instances of generalizing in terms of Ellis et al.'s (2017) framework, and we argue that some opportunities were facilitated and supported by the computational setting in which the students worked. [ABSTRACT FROM AUTHOR]

This paper characterizes the way three preservice mathematics teachers (PSTs) understand and quantify the rate of change as they model the link between carbon dioxide (CO2) pollution and global warming. I also discuss what PSTs learned about the concept of forcing by CO2, a key metric of global warming. The PSTs completed a mathematical task during an individual, task-based interview. The study revealed three levels of understanding of the rate of change in relation to quantitative operations (comparison versus coordination), graphing activity (pointwise versus smooth and continuous), and concavity (discovering versus anticipating). Depending on their level of understanding, PSTs could imagine the rate of change changing discretely or continuously with respect to an independent variable. PSTs also learn four central ideas regarding the forcing by CO2 as a result of working on the task. [ABSTRACT FROM AUTHOR]

In this theoretical paper we compare the Piagetian perspective on knowledge construction to mathematical model construction, with the aim to understand how mathematical modeling enables learning of mathematics and learning of science, as is often claimed. We do this by examining data through two lenses:(i) examining the role of cognitive conflict as it arises during validation of a model and (ii) viewing model validation as a reflection on activity-effect relationship. We explain why we chose to look deeply into model validation specifically, present examples for each lens, and consider implications. [ABSTRACT FROM AUTHOR]

Researchers have identified students' difficulties reasoning about inverse functions. Through our review of this literature, three meanings stand out: a formal, 'undoing', and quantitative meaning. Using these meanings as a guide, we analyzed student work collected from a lesson on the topic of inverse functions taught by an experienced high school mathematics teacher, using a novel task. In analyzing the data, we noticed tensions between students' understanding of the context and of inverse function as treated in curricula. In this paper, we illustrate these tensions and describe potential implications for students' productive construction of the meanings of inverse function. [ABSTRACT FROM AUTHOR]

I propose a model of student-teacher interaction with attention to teachers’ construction and enactment of knowledge of students’ mathematics. I adopt the theoretical perspective of radical constructivism to discuss how it enables us to simultaneously account for individual teacher cognition and the dynamics of student-teacher interaction. Specifically, I characterize a mechanism by which student-teacher interaction is generative to teachers’ mathematical learning at a cognitive level. I close the paper by discussing the implications for research on teacher decentering and teacher knowledge, and the potential methodological challenges. [ABSTRACT FROM AUTHOR]

Systems of equations are an important topic in school mathematics. However, there is limited research examining productive ways of supporting students’ understandings of systems of equations. In this paper, we first present a conceptual analysis of potential ways students may leverage their quantitative and covariational reasoning to graph systems of relationships. We then describe results from a design experiment in which we examined the potential of supporting middle-grades students in reasoning in ways compatible with this conceptual analysis. We highlight two different ways of reasoning students engaged in as they compared the relative magnitudes of two quantities with respect to a third quantity and leveraged this reasoning to graphically represent two relationships on the same coordinate system. We draw implications from these results for the teaching and learning of systems of equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the teachers’ reconstruction of a complex concept, that of limit, in a scientific debate/cooperative learning environment. Throughout the implementation of a set of proper designed activities (in one semester), first, participants were encouraged to elicit their intuitive ideas related to potential infinity and, second, when teachers (students from a graduate program) presented their tasks to the whole class, we tried to promote confrontations (cognitive conflicts) with the teachers’ ideas about potential and actual infinity in a scientific debate. Later we asked for individual work as an individual reflection and at the end of all the sessions, we interviewed each participant to test and document what he or she had retained. We present on this paper a general description about the complete experiment and a specific case. [ABSTRACT FROM AUTHOR]

The study reported on in this paper is an interview study conducted with 20 7th and 8th grade students whose purpose was to understand the generalizations they could make about non-linear meanings of multiplication (NLMM) and non-linear growth (NLG) in the context of solving combinatorics problems. The paper identifies productive challenges for the students, and thus fruitful areas where the students could generalize their reasoning about NLMM and NLG. [ABSTRACT FROM AUTHOR]

The body of research examining students' graphing understandings across STEM fields indicates students are not developing productive meanings for graphs. We conjecture such failings may, in part, be explainable by features of students' use of coordinate systems and graphing activity that are under examined. In this theoretical report, we present a conceptual analysis of different ways students may reason about graphs and coordinate systems. Specifically, we describe two different uses of coordinate systems--spatial and quantitative--students might leverage and two ways of reasoning--static and emergent--students might engage in as they construct or interpret graphs. We characterize how a student may engage in each kind of reasoning in each use of coordinate system. We intend this paper to serve as a theoretical lens for future empirical studies examining students' developing graphing understandings. [ABSTRACT FROM AUTHOR]

This paper discusses a student coordinating and comparing amounts of change in two covarying quantities' magnitudes (as opposed to numerical values). We describe not only his coherent system of mental actions involved in coordinating magnitudes, but also how this system affords his reasoning about graphical concavity and a quotient between magnitudes. [ABSTRACT FROM AUTHOR]

This study investigates one Calculus student's meanings for quantifiers in Calculus statements involving multiple quantifiers. The student was asked in a two-hour long clinical interview to evaluate and interpret the Intermediate Value Theorem (IVT) and three other statements whose logical structure was similar to the IVT except for the order of both the quantifiers and their attached variables. Four different meanings for variables attached to universal and existential quantifiers emerged from his responses at various moments of the interview. In this paper we detail these four meanings with empirical evidence and discuss implications of our findings to research and teaching of quantified variables in Calculus statements. [ABSTRACT FROM AUTHOR]

This paper presents selected findings from an assessment of university students' moment-bymoment reading of mathematical proof. This method, adapted from an assessment of narrative reading validated by psychologists, yields novel insights into the strategies students use to construct meaning for the equations in a proof text. In particular, we present evidence that novice readers constructed meanings for the equations using mathematical practices familiar from non-proof oriented courses - substituting and solving for variables - while more experienced readers drew upon practices native to proof-oriented mathematics - inferring properties of quantities. We refer to these as computational and inferential orientations, respectively. We interpret this mismatch of practices in terms of Systemic Functional Linguistic's notion of textual metafunction and briefly discuss its implications for proof-oriented instruction. [ABSTRACT FROM AUTHOR]

In this paper, we rely on Epistemic Network Analysis to examine the connections between knowledge resources to understand the coherence of teachers' knowledge resources used in solving proportional reasoning tasks. Our findings show that teachers who scored lower on the LMT tended to use more knowledge resources and to coordinate knowledge resources in less systematic ways than those teachers who scored higher on the LMT. We also noted a shift from strategies suggest additive reasoning to strategies that focus on the multiplicative nature of proportions. Implications for teacher development and for further research are included. [ABSTRACT FROM AUTHOR]

The importance of doing formal axiomatic proofs and students' difficulties with proof have been well documented in research. This paper focuses on formal proofs that use triangle congruence postulates, which students create in high school geometry. Examining student work with proofs in an clinical interview setting, we analyzed students' transition from planning their proofs to writing their formal proofs using a two-column format. We found that much of the reasoning students described in their planning did not match the reasoning they used in their written proofs. Using a lens of spatial/logical structuring, we illustrate this finding by providing an illustrative example in which a student's planning does not match the deductions that she wrote in her formal proof. [ABSTRACT FROM AUTHOR]

We explored 105 second and fifth graders' performance on integer subtraction problems before and after analyzing different contrasting worked examples involving integers. The students, as part of a larger study, completed a pretest, were randomly assigned to intervention groups -- which differed in the problems they compared -- and participated in two small-group sessions, one whole-class lesson on integer subtraction, and a posttest. The students made progress in solving integer problems from pretest to posttest. In this paper, we focus on students who provided only positive or zero answers on their pretest. The trends in their posttest answers show important differences among the intervention groups concerning their use of number order and interpretations of operations in integer arithmetic. [ABSTRACT FROM AUTHOR]

Copyright of Conference Papers -- Psychology of Mathematics & Education of North America is the property of Psychology of Mathematics & Education of North America and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The following paper examines the cognitive processes that occur in the course of student interactions during collaborative learning activities. We find that conceptual blending and distributed cognition are useful models for interpreting these interactions using the cooperative group itself as the cognitive unit of analysis. Conceptual blending is rarely characterized in the literature as a group phenomenon. This paper advances that notion and describes how these theories can be integrated to provide a framework for understanding the cognitive processes of small groups in a classroom setting. ..PAT.-Conference Proceeding [ABSTRACT FROM AUTHOR]

Recent mathematics standards (NCTM, 2000) place central importance on the development of mathematical reasoning. Reasoning involves more than dealing with individual concepts and focusing on conventions, skills, and techniques that produce answers to "type problems." Reasoning involves creating a rich cognitive map of content along with habits of mind that help connect and make sense of mathematics. Many areas of mathematics have recently been framed by describing the nature of applicable mathematical reasoning; for example, geometric reasoning, proportional reasoning, algebraic reasoning, and statistical reasoning. While there are similarities across all types of reasoning (e.g., deductive, inductive, and intuitive reasoning), there are also content-specific reasoning processes. As an example, statistical reasoning begins with a process of data analysis - an iterative process that includes asking a question, collecting data, analyzing the data, and forming and communicating conclusions. Learning statistics therefore includes two central components: the concepts and conventions of the discipline itself (Konold, 2002) and the ability to apply this knowledge in a broader, more flexible, and skeptical manner when analyzing, interpreting and/or communicating results obtained by statistical analysis. Garfield (2002) concurs when she defines statistical reasoning as "the way people reason with statistical ideas and make sense of statistical information. This involves making interpretations based on sets of data, graphical representations, and statistical summaries" (p. 1). In this paper, we build on the research literature and our current work to argue that this kind of statistical reasoning is facilitated by using tasks that are designed to engage students in comparing distributions of two or more data sets (e.g., Russell, Schifter & Bastable, 2002; Author, in press, 2002; Author, in press). [ABSTRACT FROM AUTHOR]

This paper analyzes the questions students ask in two very different middle school mathematics classrooms. Both classrooms were taught by experienced and respected mathematics teachers, but the instruction was very different between the two classrooms. Mr. Cordoba characterized his instruction as student-centered, project-based, and focused on understanding. Mr. Ezzo used a more traditional approach. Classroom instruction was videotaped, student questions were highlighted and categorized, and instructional moves were analyzed. Results show that authority distribution in the classroom is connected to the types of questions students ask in mathematics lessons. [ABSTRACT FROM AUTHOR]

In this paper we frame our observations in enactivism, specifically problem posing, to propose the notion of problem drift as a method to analyze the curriculum generating actions of small group learning systems in relation to teacher interventions intended to trigger specific content goals. Teacher attentiveness to problem drift is suggested to be valuable in advancing the content goals in small group work. [ABSTRACT FROM AUTHOR]

Generalization is a critical aspect of doing mathematics, with policy makers recommending that it be a central component of mathematics instruction at all levels. This recommendation poses serious challenges, however, given researchers consistently identifying students' difficulties in creating and expressing normative mathematical generalizations. We address these challenges by introducing a comprehensive framework characterizing students' generalizing, the Relating-Forming-Extending framework. Based on individual interviews with 90 students, we identify three major forms of generalizing and address relationships between forms of abstraction and forms of generalization. This paper presents the generalization framework and discusses the ways in which different forms of generalizing can play out in activity. [ABSTRACT FROM AUTHOR]

This paper discusses a pre-service secondary teacher's mental actions in the context of her partitioning activity and reasoning about amounts of change (i.e., coordinating the amounts of change of one quantity for changes of another) to structure dynamic situations. We illustrate ways of thinking that necessitated that she carried out partitioning activity, and thus had difficulty representing actions when reasoning with changes on graphs and situations. [ABSTRACT FROM AUTHOR]

Copyright of Conference Papers -- Psychology of Mathematics & Education of North America is the property of Psychology of Mathematics & Education of North America and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We investigate thirty-three second and fifth-grade students' solution strategies on integer addition problems before and after analyzing contrasting cases with integer addition and participating in a lesson on integers. The students took a pretest, participated in two small group sessions and a short lesson, and took a posttest. Even though the results reveal significant gains for both grades from pretest to posttest, second graders gained significantly higher than fifth graders. In this paper, we explore students' treatment of the negative sign and describe this gain difference. [ABSTRACT FROM AUTHOR]

The NCTM research committee made a recent, urgent call for mathematics education researchers to "examine and deeply reflect on our research practices through an equity lens." With this in mind, we use this paper to reflect on the ways in which Steffe's work has contributed to three facets of equity. We also suggest opportunities for researchers working within this framework to deepen their commitments to issues of equity. [ABSTRACT FROM AUTHOR]

This paper is about the threshold between gesture and touch in mathematical activity, focusing on the role of multitouch technology. Drawing on the work of gesture theorist Jürgen Streek, we discuss the notion of indexical gestures, in the context of young children's mathematical explorations with a multitouch iPad environment designed to promote counting on and with the fingers. [ABSTRACT FROM AUTHOR]

Computer scientists have reported on "computational thinking," which Aho (2012) defines as thought process involved in formulating problems so their solutions can be represented as computational steps and algorithms" (p. 832). We wanted to investigate such thinking in mathematics. To study this, we interviewed five mathematicians about the role of computation their work, and the notion of "algorithmic thinking" developed from these interviews. We foun mathematicians value this thinking, and in this paper we define algorithmic thinking and prese number of applications and affordances of such thinking. [ABSTRACT FROM AUTHOR]

In this theoretical paper, I consider reversibility as a defining characteristic of mathematics. Inverse pairs of formalized operations, such as multiplication and division, provide obvious examples of this reversibility. However, there are exceptions, such as multiplying by 0. If we are to follow Piaget's lead in defining mathematics as the science of reversible mental actions, such exceptions must be examined. We consider the case of multiplying by 0 by adopting Davydov's model of multiplication as a transformation of units and by investigating the underlying mental actions. Results of this investigation have implications for breaking down the barriers between various domains of mathematics. [ABSTRACT FROM AUTHOR]

This paper reports findings from a study that explored the effect of a secondary mathematics teacher's level of attention to quantitative reasoning on the quality and coherence of his instruction of angle measure. I analyzed 37 videos of an experienced teacher's instruction of trigonometric functions to characterize the extent to which the teacher attended to supporting students in reasoning quantitatively, and to examine the consequences of this attention (or lack thereof) on the quality and coherence of the meanings the teacher's instruction supported. My analysis revealed that the incoherencies in the teacher's instruction were occasioned by his inattention to quantitative reasoning. [ABSTRACT FROM AUTHOR]

This paper reports on a study of College Algebra students that examined their understanding of polynomial functions of degree three or more. The study included interviews with students as they completed mathematical problem tasks, enabling a focus on the students' on-going cognitive actions. The results summarize how the students' existing conceptual structures guide their actions to produce and make connections among polynomial function graphs. [ABSTRACT FROM AUTHOR]

In this study I investigate Saldanha and Thompson's (1998) claim that conceptualizing a coordinate pair in the Cartesian coordinate system as a multiplicative object, a way to unite two quantities' values, supports students in conceptualizing graphs as emergent representations of how two quantities' values change together. I presented three university precalculus students with an animated task showing varying values of two quantities along the axes and asked each student to sketch a graph of how the two quantities changed together. In this paper I document the difficulty students encountered when they did not conceptualize a coordinate pair as a multiplicative object. I address why the convention of "over x and up y" inhibits students from constructing a coordinate pair as a multiplicative object and I provide recommendations for supporting students in constructing coordinate pairs as multiplicative objects. [ABSTRACT FROM AUTHOR]

This paper considers what students attend to as they first encounter R³ coordinate axes and are asked to graph y = 3. Graphs are critical representations in single and multivariable calculus, yet findings from research indicate that students struggle with graphing functions of more than one variable. We found that some students thought y = 3 in R³ would be a line, while others thought it would be a plane. In creating their graphs, students attended to equidistance, parallelism, specific points, and the role of x and z. Students' use of these ideas was often generalised from thinking about the graphs of y = b equations in R². A key finding is that the students who thought the graph was a plane always attended to the z variable as free. [ABSTRACT FROM AUTHOR]

This paper explores the nature of students' quantitative reasoning and conceptions of functions supporting their ability to symbolize quadratic function rules, and the meanings students make of these rules. We analyzed middle school students' problem solving activity during a small group teaching experiment (n=6) emphasizing quadratic growth through covarying quantities. Results indicate four modes of reasoning supportive of students' symbolization of quadratic function rules: (a) correspondence, (b) variation and correspondence, (c) covariation, (d) flexible covariation and correspondence. We discuss implications for research on learning vis-à-vis students' representational fluency, as well as design principles to support quantitative reasoning. [ABSTRACT FROM AUTHOR]