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Our paper details the ways teachers understand and navigate equity-oriented dilemmas (Berlak & Berlak, 1981) when teaching mathematical modeling and how mathematics teacher educators can support teachers' learning of culturally responsive mathematics teaching. Using Zavala and Aguirre's (in press) framework for culturally responsive mathematics teaching, we explored the ways teachers describe and frame their choices when faced with dilemmas. Findings revealed that teachers identified dilemmas with Rigor and Support most often, followed by Knowledges and Identities. Dilemmas related to Power and Participation occurred far less frequently. Implications for teacher professional development are discussed. [For the complete proceedings, see ED658295.]

In this study, we examined one experienced mathematician's class practices, with particular attention to cognitive model described in genetic decomposition. Our findings indicate that students only had limited opportunities to be familiar with the first three steps in genetic decomposition, which may potentially lead students to answer limit tasks correctly, but not necessarily having a deep conceptual understanding behind those tasks. With limited opportunities, it will be challenging for students to overcome well-known learning challenges in limit. [For the complete proceedings, see ED658295.]

This research contributes to the need to identify and expand learning environments that encourage undergraduate students to develop collaborative work skills and apply their classroom knowledge to solve real-world problems. Using qualitative methods, we examine the effects of the interaction between two teams of students when solving a Model-eliciting activity, based on the theoretical framework of the Models and Modeling Perspective. Our analysis shows that the students had two opportunities for interaction, within the team and between the modeling teams. Through these interactions, the students refined their models in three directions: mathematical knowledge, interpretation and modeling of the phenomenon, and mathematical representations. The findings of this study emphasize the significant value of Model-eliciting activities in enhancing students' collaboration and modeling abilities. [For the complete proceedings, see ED658295.]

Mathematical modeling is a process in which students investigate authentic problems and everyday situations using mathematics. In doing so, they bring their multiple mathematical knowledge bases and cultural funds of knowledge into their solution strategies. During a task called "Abuelo's Birthday", 297 students in grades 3-5 decided how to split the costs of a gift "fairly" and justified their work using early notions of ratio and proportional reasoning. We argue that these young students were successful with a complex task due to the way the realistic context connected to students' lived experiences and funds of knowledge related to home and family life. We share rich examples of how students included elements outside of the task to justify and enhance their mathematical models and conclude with implications for the importance of modeling and non-routine tasks in elementary classrooms. [For the complete proceedings, see ED658295.]

There is growing recognition that mathematical modeling can be a lever for equity in elementary mathematics classrooms. This study focuses on the impact of a professional development program focused on culturally responsive mathematical modeling on 8 kindergarten through 2nd grade teachers' practices in modeling lessons. We use a project developed observation tool to evaluate two video recorded modeling lessons from each teacher (16 total). Findings focus on patterns in the strengths and challenges in primary grade teachers' practices for teaching modeling, including how teachers' practices align with culturally responsive teaching. We discuss implications of our findings for the design and refinement of professional development. [For the complete proceedings, see ED657822.]

The purpose of this study is to gain understanding of how secondary teachers conceptualize the relationships between mathematical modeling and problem solving. Eight secondary teachers participated in semi-structured, think-aloud individual interviews. Some conceptualizations include (a) modeling and problem solving are two distinct processes; (b) modeling is a subset of, or a tool for, problem solving; (c) the two processes share similar characteristics such as a real-life scenario but are different in terms of whether a single correct answer exists; (d) the two processes are inseparable, completely enmeshed in each other, and co-dependent; and (e) problem solving is a reduced process of modeling. Teachers' conceptualizations are related to their preferred instructional sequence and the types of problems or activities they'd rather use. [For the complete proceedings, see ED657822.]

Recently, abstracted quantitative structures (AQS), a construct from quantitative reasoning, has been offered as a means to conceptualize and study mathematization during mathematical modeling. Extending this theoretical work, we provide empirical evidence that an intervention targeting participants' AQS can assist in aligning modelers' models with normatively correct models. We report on a pre/post intervention study designed to elicit alignment between symbolic forms and AQS and alignment between AQS and modeling scenarios. We used the Sørenson-Dice coefficient and cluster analysis to identify shifts in student associations of symbolic forms with modeling scenarios. [For the complete proceedings, see ED657822.]

In this report, we present cases where students constructed new quantities through operating on quantities that does not fit the definitions of existing theories on quantitative operations. As a result, we identified five quantitative operators--operators that can be used on single qualities in order to transform the quantity to a new quantity--students used while constructing mathematical models for real-world scenarios. [For the complete proceedings, see ED657822.]

In this paper, we present the development of an investigation on the promotion of covariational reasoning in high school students (14-15 years old) in Mexico. The study consists of designing and applying a sequence of didactic activities that simulate a real situation virtually. The activities are organized through a Hypothetical Learning Trajectory supported by digital technology and elements of Cuevas-Pluvinage didactics. The activities were evaluated according to the levels of covariation proposed by Carlson and colleagues, categorizing students' achievements and difficulties for each level of understanding. The results show that the activities favor students' progress by moving from the context situation to the different representations, establishing the relationship between the variables, and identifying their functional dependence. [For the complete proceedings, see ED630210.]

This theoretical paper describes how Community-based Mathematical Modeling can advance equity and cultivate civic empathy in elementary school settings. We provide a framework for community-based mathematical modeling instruction consisting of five goals: facilitating connections, fostering engagement, promoting rigor, cultivating civic empathy, and elevating justice. We illustrate how these goals work together to advance equity and cultivate civic empathy through classroom vignettes of community-based modeling lessons. Through this theoretical synthesis, implications for community-based mathematical modeling instruction will be discussed. [For the complete proceedings, see ED630210.]

The development of modeling skills in mathematics is essential for individuals to understand, describe, control, and predict phenomena around them. This article describes the results of an investigation to find out how an activity -- based on a Models and Modeling Perspective -- stimulates the mathematics modeling skills of undergraduate students who are in the first quarter of a business degree. As a result, it was shown that the MEA [model-eliciting activity] enabled students to exhibit, develop, and refine different modeling skills, such as: identification of variables, assumptions based on the real-life context, identification of patterns, and construction of mathematical representations. [For the complete proceedings, see ED630210.]

Literature typically describes mathematization, the process of transforming a real-world situation into a mathematical model, in terms of desirable actions and behaviors students exhibit. We attended to STEM undergraduate students' quantitative reasoning as they derived equations. Analysis of the meanings they held for arithmetic operations (+, -, ·, ÷) provided insight into how participants expressed real-world relationships among entities with arithmetic relationships among values. We extend the findings from K-12 literature (e.g., using multiplication to instantiate a rate) to STEM undergraduates and found evidence of new ways of justifying the usage of arithmetic operations (e.g., using multiplication to instantiate an amount). [For the complete proceedings, see ED630210.]

Using data from teaching experiments and theories from quantitative reasoning, we built second-order accounts of students' mathematics with regards to how they conceived rate of change through operating on existing quantities. In this report, we explain three different ways STEM undergraduates structurally conceive rate of change as they constructed mathematical models for real-world scenarios. [For the complete proceedings, see ED630210.]

Existing studies have defined and assessed disciplinary literacy, mathematical literacy, and general academic language. However, there is a need to define and assess mathematics discipline-specific language (MDL), particularly for elementary school teachers. Therefore, the purpose of this study was to develop a research instrument to assess the MDL of elementary school teachers. The final instrument developed through iterative analysis included 20 items on a 4-point Likert-like scale distributed between three distinct MDL categories: technical, symbolic, and visual. Instrument validity was confirmed using Confirmatory Factor Analysis with the set of 211 video recordings and corresponding lesson plans of mathematics lessons taught by pre-service elementary school teachers enrolled in a graduate special education program. [For the complete proceedings, see ED630210.]

In this paper we address the question, how do quantitative reasoning and covariational reasoning present as students build structural conceptions of real-world situations. We use data from an exploratory teaching experiment with an undergraduate STEM major to illustrate the explanatory roles quantitative reasoning and covariational reasoning play in, (a) coordinating more than two interdependent quantities, (b) conceiving of real-world situations in more than one way, (c) constructing networks of quantitative relationships, and (d) creating a mathematical expression. We make the case that looking at mathematical model construction through the lens of quantitative reasoning and covariational reasoning may provide insights into students' mathematical decisions as they structure complex real-world scenarios. [For the complete proceedings, see ED630060.]

In this paper we present the results of an investigation related to the developing of mathematical knowledge and skills by first semester university students when solving a Model Eliciting Activity [MEA] which involves quadratic function knowledge. This was a qualitative research. The theoretical framework was Models and Modeling Perspective [MMP]. The results show that the students used their mathematical knowledge and skills related to linear and quadratic functions to describe the situation; they moved from a quantitative cycle of understanding (associated with linear and quadratic behaviors), to an algebraic cycle of understanding (associated with quadratic behaviors). [For the complete proceedings, see ED629884.]

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: (1) examining the role of cognitive conflict as it arises during validation of a model; and (2) 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. [For the complete proceedings, see ED629884.]

This study aims to explore seventh-grade students' understanding of negative integers as they engaged in mathematics history integrated model-eliciting activities in small groups. For this educational case study, we designed model-eliciting activities based on six design principles of the models-and-modeling perspective that incorporated history of negative integers. Both written data and video records of students were analyzed to elicit the facets of their models of negative integers. We found that students' thought that either daily life contexts or people's need drove the invention of negative integers. The findings also indicated students' reasoning on the evolvement of mathematics ideas by contribution of different culture, revealing the role of math history integration into the modeling process. In this sense, our study presents a unique approach in modeling literature. [For the complete proceedings, see ED630060.]

This article describes the results of an investigation based on a Models and Modeling Perspective [MMP]. We present the evolution of the models built by university students when solving a model development sequence designed to promote their learning of the exponential function. As a result, we observed that students' thinking was modified, expanded, and refined, as they developed different iterations of their models. Students' models evolved by creating, first, linear models that required direction; second, models where there was no dissociation between linear and exponential behavior; then, situated exponential models; and finally, sharable, and reusable exponential models. [For the complete proceedings, see ED630060.]

One promising approach for connecting undergraduate content coursework to secondary teaching is using teacher-created representations of practice. Using these representations effectively requires seeing teachers' use of mathematical knowledge in and for teaching (MKT). We argue that Rowland's (2013) Knowledge Quartet for MKT, in particular, the dimensions of Foundation and Contingency, is a fruitful conceptual framework for this purpose. We showcase an analytic framework derived from Rowland's work and our analysis of 85 representations of practice. These representations all featured geometry. We illustrate examples of combinations of "high" and "low" Foundation and Contingency, and show results of coding juxtaposed with performance on an instrument previously validated to measure MKT. We describe the potential for generalizing this framework to other domains, such as algebra and mathematical modeling. [For the complete proceedings, see ED630060.]

These proceedings are a written record of the research presented at the 42nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) held in Mazatlán, Mexico, virtually beginning May 27, 2021 and in-person June 2-6, 2021. The conference was originally scheduled to take place October 14-18, 2020; it was postponed, due to the COVID pandemic. In accordance with the major goals of PME-NA that include promoting international contacts and stimulating interdisciplinary research, the theme was proposed as "Entre Culturas/Across Cultures". The goal of the theme was an academic exchange that would reflect the ample diversity of ways of teaching and learning of mathematics, and of the tools and communities involved in mathematics education; as well as an exploration of how the differences in cultures imply a need to consider how research results can be taken into account in varying contexts. The papers comprise 80 Research Reports, 190 Brief Reports, 147 Poster Presentations, 10 Working Groups, 2 Research Colloquia, and Plenary and special lectures. The program also includes a special tribute to Eugenio Filloy and a tribute to the PME-NA members who were affected by the COVID-19 pandemic. [These proceedings were produced with Cinvestav (Centro de Investigación y de Estudios Avanzados del IPN), México and AMIUTEM (Asociación Mexicana de Investigadores del Uso de Tecnología en Educación Matemática, A.C.), México. Individual papers are indexed in ERIC.]

Throughout the years, Mathematical Modeling has gained international attention, not only in research but also in the development of curricula and its applications in the classroom. However, systematic literary surveys are scarce. In this paper, we present some findings from a survey of 485 international publications related to different aspects of mathematical modeling. While the number of Latin-American publications is rather small compared to the number of international publications, the vitality of the currently discussed themes in this region and their innovative perspectives testify to the international relevance of this developed work on mathematical modeling. [This paper is provided in both Spanish and English. For the complete proceedings, see ED606531.]

We analyzed 5th grade students' interactions with one computer-based modeling simulation to examine how they defined and prioritized variables in a dynamically simulated environment. Results revealed that exposure to the simulation environment helped students visualize continuous motions, interpret the different quantities present in the model, and connect how variables related to each other. Additionally, students became more precise and systematic in adjusting the variables to explore the problem, test hypotheses, and achieve desired outcomes. [For the complete proceedings, see ED606556.]

Mathematical modeling is a high-leverage topic, critical STEM education and civic engagement. This study investigates culturally responsive, school and community-based approaches that support mathematical modeling with elementary students. Specifically, we analyzed two modeling lessons in one fifth grade classroom, with a focus on how students drew upon their experiences and funds of knowledge as they engaged in the mathematical modeling process. Our findings illustrate different ways that students' experiences and situational knowledge informed and guided their modeling activity, including the quantities that they deemed relevant, and how they interpreted and refined their solutions. We also attend to the teacher's role in supporting such connections. Implications for mathematics teacher educators and research are included. [For the complete proceedings, see ED606556.]

This working group will engage PMENA members to better understanding the nature of mathematical modeling in the early grades while considering the student perspective and recognizing the importance of teachers knowing their students and the contexts that are meaningful to their students. We will investigate how PK-6 teachers demonstrate the interdisciplinary nature of mathematical modeling, the diversity of mathematical approaches taken by student modelers, and the multiple pathways the teacher can use to elicit students' mathematical thinking. We will explore how mathematical modeling bridges equity and social community in teaching and learning mathematics for all students. Exemplar tasks that emphasized local contexts and tapped into students' funds of knowledge and student artifacts will be shared to illustrate the child's perspective and the developmental progression. These topics will facilitate group discussions exploring the learning progression for mathematical modeling thinking and habits of mind that can develop for emergent mathematical modelers from an early grade. Finally, based on the interests of the participants, we will devote work time to finding synergistic collaborative topics to pursue for future research and practice. [For the complete proceedings, see ED606556.]

Formulating a mathematical model is a dynamic process, and attending to changes in students' models as they engage in modeling tasks is critical for informing pedagogical interventions. In this report, we coordinate constructs from literature on mathematical modeling, quantitative and covariational reasoning, and semiotics to characterize changes in mathematical models. We illustrate the application of these constructs using data from an undergraduate solving a modeling task. [For the complete proceedings, see ED606556.]

I discuss how three preservice mathematics teachers' (PSTs') covariational reasoning supported the mathematization of a simple energy balance model (EBM) for global warming, and how such mathematization shaped PSTs' understanding of the link CO2 pollution and global warming. I use Thompson & Carlson's (2017) levels of covariational reasoning and Thompson, Carlson, Byerley, and Hatfield (2014) descriptions of understanding and meaning to inform the discussion of results. The PST completed the EBM Task during an individual, task-based interview. The analysis revealed that Chunky Continuous Covariation level supports the mathematization of the EBM in terms of a covariation's rapidity of change. The analysis also revealed that particular mathematizations resulted in particular meanings for radiative equilibrium, which in turn have implications for understanding the link between CO2 pollution and global warming.

This article is a report on a teacher study group that focused on three elementary teachers' perceptions of mathematical modeling in contrast to typical mathematics instruction. Through the theoretical lens of figured worlds, I discuss how mathematics instruction was conceptualized across the classrooms in terms of artifacts, discourse, and identity. I then highlight, through four themes, how mathematical modeling challenged the ways in which both the teachers and students understood what it means to know and do mathematics. Findings suggest that the practice of mathematical modeling allowed for access, empowerment, and real world connections that were typically not present in classroom instruction. In addition, it challenged student positioning in the classroom in terms of who was framed as capable of doing mathematics. [For complete proceedings, see ED581294.]

In this research we discuss relating the description of certain harmonic phenomena, with models that represent them, with the goal of promoting the understanding of the period, the frequency, and the amplitude of the trigonometric functions, particularly referring to sinusoidal functions; in this way the frontier between sinusoidal functions and one of its areas of application to physics is preserved. This is an ongoing research study where digital technology like a movement sensor is employed in order to generate graphics related to sinusoidal functions, in which the movement features followed by harmonic phenomena are identified. It is important to mention the use of an approach where it is possible to transit alternatively from the analysis of the phenomena to the mathematical model. It is conjectured that the way it approaches sinusoidal functions could help assign meaning to the model f(x) =Acos(?x+[phi]). [Written in both Spanish and English. For the complete proceedings, see ED583608.]

Multiple representations, multiple modalities, and technology can be helpful in the understanding of mathematical concepts when used in an appropriate manner (Shah & Freedman, 2003; Goldman, 2003), but this alone does not account for the student benefits of creating and using dynamic models over teacher generated graphs to construct connections between representations. By uncovering the dynamic nature of mathematics, calculus becomes more transparent as relationships and patterns emerge. The struggle to understand becomes worthwhile and rewarding for students as they create and observe the action of a dynamic mathematical object. This study shows an improvement in attitude, and academic achievement when students develop dynamic mathematical object to understand calculus and poses new questions to explore. [For the complete proceedings, see ED583608.]

The paper deals with the question of the practicability and the effectiveness of different approaches to foster students' mathematical modelling competencies. Within the modelling project ERMO (Acquirement of modelling competencies) a holistic and an atomistic approach of mathematical modelling were compared in order to find out which approach is more effective in fostering the students' modelling competencies. The results of modelling tests with three measurement points show that both approaches foster students' modelling competencies, but both approaches have strengths and weaknesses. The data indicates that the holistic approach is more effectively for students with weaker performance in mathematics. [For the complete proceedings, see ED597799.]

The paper describes the history of how learning trajectories (LTs) were associated with the Common Core State Standards for Mathematics (CCSS-M) and discusses the degree to which the two correspond faithfully. It reports on a website, www.turnonccmath.com, which organizes the K-8 standards into 18 LTs describing the development of big ideas over time, informed by empirical studies of learners. The paper illustrates how descriptors for each LT identify: (1) conceptual principles, (2) strategies representations, and misconceptions, (3) meaningful distinctions and multiple models, (4) coherent structure, and (5) bridging standards. The design principles for the website are illustrated describing how the CCSS-M are related to a learning trajectory on division and multiplication. [For the complete proceedings, see ED584829.]

Given the Common Core's dual emphases on mathematical modeling, there is a need to understand modeling as a practice and content standard to develop students' mathematical modeling skills. This study of 12 students from differing levels of mathematics instruction and English Language proficiency includes analysis of their modeling with mathematics and a focus on their transitions through a mathematical modeling cycle. Findings suggest that students were engaging in critical processes that support mathematical modeling. We posit that conventional word problems can augment the benefits of using mathematical modeling tasks and can help educators explore a process-oriented approach to mathematical modeling. [For the complete proceedings, see ED583989.]

This study investigates the extent to which pre-service elementary teachers (PETs) use their realworld knowledge to solve problems for which the result of the arithmetic operation is problematic, if one takes into consideration the reality of the context. A paper-and-pencil test was administered to 566 PETs enrolled in mathematics content courses. The test included 8 experimental items and 4 buffer items. The findings for a sample of 68 PETs are not very encouraging. The total number of realistic responses varied from 5 to 58 (out of 68 possible for each problem). [For the complete proceedings, see ED585874.]

The focus of this report is on the process of resolution of a task; specifically, on the validation of the mathematical model proposed by a group of students and the numerical result that is constructed within this model. Habermas' construct of rational behavior is used to describe validity conditions that emerge and are used by the students as means for validation. We take a classroom episode from a design experiment to examine how the emergence of these conditions points to a socially constituted mathematical epistemology in the secondary school mathematics classroom, to shared and tacit principles of the didactic contract concerning the knowledge there, and to non-mathematical references that are taken for granted. [For the complete proceedings, see ED597799.]

The study described herein is part of a larger, inclusive research study exploring the effects of model-eliciting activities (MEAs) of "real-life" situations on the development of students' mathematical creativity. This part aim at revealing students' cognitive abilities that are involved in the creative modeling processes using a qualitative analytical method. The participants were mathematically talented students, members of the "Kidumatica" math club. The data include videotapes, classroom observation and modeling products. Three core categories--appropriateness, 'mathematical resourcefulness' and inventiveness--of students' cognitive creative abilities are identified, defined and illustrated. These findings may give a better understanding of the larger concept of mathematical creativity. [For the complete proceedings, see ED597799.]

This study utilized an innovative data analysis approach to examine how engineering undergraduates engaged in mathematical modelling. Individual modelling routes were constructed via modelling activity diagrams and were used to critically examine the theoretical framework. Implications for the theoretical model are offered along with implications for future research. [For the complete proceedings, see ED597799.]

The research-project Co[superscript 2]CA investigates the influence of teacher-trainings on in-service teachers' expertise. Within a teacher-training-study 27 in-service teachers have been trained in selected ideas about teaching, having an exemplary focus on central aspects of formative assessment in competency-oriented mathematics. At the end of the teacher-trainings the teachers' pedagogical content knowledge as crucial aspect of teachers' expertise has been evaluated by using tests being sensitive to the teacher-trainings. Results of this evaluation point out: Within teacher-trainings pedagogical content knowledge can be conveyed to in-service teachers successfully. [For the complete proceedings, see ED597799.]

We taught a group of experienced in-service primary school mathematics teachers the notion of mathematical model, in order to foster the interdisciplinary mathematics teaching in primary school. In particular, we developed an exercise in which they were supposed to construct a mathematical model on the basis of primary school mathematics. We found out that the formal mathematical knowledge needed to perform the exercise was not sufficient to successfully cope with it. The main factor that influences the ability of the teachers to cope with this type of activity is the depth of their mathematical knowledge which we identify with a person's mathematical insight. [For the complete proceedings, see ED597799.]

Proof is a central concept in mathematics education, yet mathematics educators have failed to reach a consensus on how proof should be conceptualized. I advocate defining proof as a clustered concept, in the sense of Lakoff (1987). I contend that this offers a better account of mathematicians' practice with respect to proof than previous accounts that attempted to define a proof as an argument possessing an essential property, such as being convincing or deductive. I also argue that it leads to useful pedagogical consequences. [For the complete proceedings, see ED597799.]

This study examined how mathematical modeling activities within a collaborative group impact on students' perceived 'value' of mathematics. With a unified framework of Makiguchi's theory of 'value', mathematical disposition, and identity, the study identified the elements of the value--beauty, gains, and social good--with the observable evidences of mathematical disposition and identity. A total of 60 college students participated in 'Lifestyle' mathematical modeling project. Both qualitative and quantitative methods were used for data collection and analysis. The result from a paired-samples t-test showed the significant changes in students' mathematical disposition. The results from the analysis of students' written responses and interview data described how the context of the modeling tasks and the collaborative group interplayed with students' perceived value. [For the complete proceedings, see ED597799.]

In this proposal, I introduce a method for modeling the dynamics of a sixth-grade student's accommodation of his fractions scheme to include a disembedding operation (Steffe & Olive, 2010). I will describe a three-part approach consisting of a constructivist teaching experiment, retrospective analysis, and stochastic modeling of the student's activity across ecologies within the teaching experiment (cf. Steffe & Thompson, 2000). As disembedding requires coordination of two levels of units--a disembedded part and an unadulterated whole (Steffe, 2001)--the results also serve as an initial model for describing the dynamics of an individual's units coordinating within and across contexts. [For the complete proceedings, see ED584443.]

We present five strategies that mathematics majors can use to improve their proof comprehension. We argue these strategies are effective by presenting qualitative excerpts illustrating the ways in which the employment of these strategies helped four undergraduate students understand the proofs they were reading. Furthermore, we present results of a survey in which the majority of 83 mathematicians indicated they would like students to use these proof reading strategies. Finally, we argue that these strategies are underused by presenting results from another part of the survey, in which the majority of 175 mathematics majors claimed not to employ these strategies. [For the complete proceedings, see ED584443.]

We unpack the notion of mathematics teacher educators modeling teaching in mathematics teacher education courses. Specifically, we investigated what practicing teachers gained from mathematics teacher educators' modeling by examining: (1) What do mathematics teacher educators believe they model about effective instructional practice? (2) What do practicing teachers notice about the mathematics teacher educator's pedagogy and identify as effective mathematics teaching? (3) In what ways do these perspectives align in mathematics courses for practicing teachers? The results provide insight into what and how teachers learn from engaging in inquiry-oriented teaching. We discuss the implications for the education of mathematics teachers. [For the complete proceedings, see ED585874.]

The theme of the 38th meeting of the International Group for the Psychology of Mathematics Education (PME 38) and the 36th meeting of the North American Chapter of the Psychology of Mathematics Education (PME-NA 36) was "Mathematics Education at the Edge." Academically, the theme provides opportunities to highlight and examine mathematics education research that is: (1) breaking new ground or on the cutting edge of innovative research and research methodologies; and (2) exploring issues with groups that are often positioned at the edge or periphery of educational research, such as social justice, peace education, equity, and Indigenous education. Geographically, the theme "Mathematics Education at the Edge" describes the very place of the conference setting, Vancouver, a city situated at the edge of Canada on the Pacific Ocean and Coast Mountain Range. The papers in the six volumes of these proceedings are organized according to the type of presentation. Volume 1 contains the presentations of the plenary speakers, Research Forum activities, Discussion Group activities, Working Session activities and the National Presentation of mathematics education in Canada. Volumes 2-5 contain the Research Reports of the conference, while Volume 6 consists of the Short Oral and Poster Presentations. The organization of PME 2014 is a collaborative effort involving teams of colleagues at the University of British Columbia and Simon Fraser University.