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The positive relationship between students' attitudes toward mathematics and mathematics achievement is well documented. Yet there is a worsening problem of low appeal of mathematics especially at the secondary level. Therefore, in this paper we focus on three high school students who report a strong dislike of mathematics. By analyzing student surveys, interviews, and lesson observation data, we examined how some mathematical lessons improved these students' experiences (i.e., their aesthetic dimensions). We found that while student preferences varied, each student was interested in lessons that centered them as sense-makers and in which the content unfolded with suspense. Such lessons led to positive aesthetic responses such as surprise, curiosity, and satisfaction. Thus, lessons can be designed in which even students with the most negative views of mathematics can find mathematical concepts interesting. [ABSTRACT FROM AUTHOR]

Engagement in the mathematics classroom through interactions with the instructor, peers, and content are necessary for an effective learning experience. As such, it is important to understand the types of interactions that teachers utilize to engage students, especially as they have had to shift from a complete face-to-face setting to various remote modalities. Utilizing four interaction types (learner-content, learner-instructor, learner-learner, and learner-interface) this paper analyzes 35 videos of classroom instruction with the purpose of describing the interactions that take place throughout the course of the mathematics lesson. While there was not a significant difference in the type of interaction and the modality of instruction, there was a significant difference in the type of interaction enacted and the modality of instruction. [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.)

This paper reports a preliminary analysis of prospective secondary mathematics teachers’ diagrammatic reasoning and understanding of visual proofs. The data comprises responses to a task in which participants were asked to articulate how a given claim was proven through diagrams and to compose a dialogue between two students and a teacher about the proofs. The findings show that the participants’ approaches to articulate “the unspoken” of the diagrams greatly differed; their pedagogical preferences to teach the same mathematical content were also distinct. The unspoken and unarticulated, from prospective teachers’ perspective, were different from what was anticipated. [ABSTRACT FROM AUTHOR]

This study explores how students reason about congruence based on their high school learning experiences. We developed an online survey to characterize how students understand congruence and gather their recollections about high school geometry coursework. Three ways students understood congruence emerged from data analysis: pictorial, measurement-based, and transformational. Our results indicate that (1) transformational reasoning was the most productive among students’ meanings but was exhibited by less than one-fifth of participants and (2) that diverse contexts encourage critical and productive thinking about congruence. In this paper, we discuss the implications of our findings to research and teaching of congruence. [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.)

Students studying geometry at the secondary level are expected to read diagrams in different ways than those in elementary school. In this paper, we present an analysis of the changes in diagrammatic expectations by comparing the geometric diagrams found in Grade 1 U.S. textbooks with those in U.S. high school geometry textbooks. This work included developing and using a coding scheme that recognizes dimensions of reading a diagram geometrically, including the type of object represented, use of deduction, use of mental redrawing, interpretation of markings, and the necessity of the diagram. The way in which elementary and secondary students are expected to interpret diagrams was shown to change along several of these dimensions, posing potential learning barriers for students. We end our paper with a discussion of what our results mean for the learning of geometry. [ABSTRACT FROM AUTHOR]

This paper describes the role of sense of place in students' analysis of spatial data toward understanding the distribution of financial services in their city. High school students participated in a 10-session module about their city's two-tiered financial system of banks and alternative financial institutions. The paper analyzes two class sessions organized around the use of ratios, or intensive variables, to understand the distribution: an embodied distribution activity atop a large floor map and individual exploration of scalable, data-rich digital maps. Analysis investigates the role of students' sense of place as they grappled with these ratios. Students drew on their senses of place to interpret data and generate their own sets to associate in ratios. Abstract measures and data visualizations contained in the digital maps were less accessible to students in this iteration of the module. [ABSTRACT FROM AUTHOR]

This paper reports on a longitudinal study of mathematics teachers' development of a vision of teaching with technology where we document professional events and activities that point to continued evolution and devolution of those beliefs. We extend earlier work and ask participants to reflect on the experiences they have had as early career teachers, and how they have influenced their beliefs since graduation. We find that there are significant opportunities for professional learning after graduation, and recommend continued development of graduate-level coursework that is technology-dependent. We also find that the use of Desmos is particularly influential in changing beliefs about the role of technology. [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.)

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]

In this paper, we compare the levels of reasoning elicited during the completion of three versions of a bottle filling task: high school level matching; middle school level matching and constructed response. The goal of the tasks was to make visible secondary students covariational reasoning methods. Video of students completing the task while explaining their reasoning during one-on-one interviews were analyzed. Analysis demonstrated a wide range of reasoning when provided a matching version with a greater incidence of accuracy with students who exhibited lower levels of reasoning. Conversely, the constructed response task demonstrated higher levels of reasoning more consistently with decreased accuracy. Implications for assessment are discussed. [ABSTRACT FROM AUTHOR]

The field of mathematics teacher education has been moving collectively towards a common goal of preparing preservice mathematics teachers to teach with technology, which is explicated in the Association of Mathematics Teacher Educators’ (2017) Standards for Preparing Teachers of Mathematics. In this paper we present findings from a national survey of accredited university secondary mathematics teacher education program. The purpose of the study is to describe the current state of the ways in which programs are preparing preservice teachers to teach secondary mathematics with technology. [ABSTRACT FROM AUTHOR]

National organizations such as the Association of Mathematics Teacher Educators recommend that secondary preservice teachers (PSTs) have a strong understanding the content they will teach. Other studies (e.g. Monk, 1994) suggest that the more mathematics courses preservice teachers take after a certain point do not affect their pupils’ learning. This study aims to find evidence that participating in mathematics courses tailored for secondary PSTs as recommended by the Conference Board of Mathematical Sciences have the potential to help PSTs gain knowledge that promotes student-centered learning. This paper will share preliminary evidence from a case study that seems to provide evidence of a mathematics experience that promoted a change towards more student-centered teaching. [ABSTRACT FROM AUTHOR]

This paper reports a preliminary analysis of mathematicians’ detailed accounts on why and how advanced mathematical knowledge matters for secondary mathematics teaching, and what specific connections can be made between advanced and secondary mathematics. Drawn from interviews with 18 mathematicians from 9 universities, the findings indicate that having advanced mathematical knowledge would allow secondary mathematics teachers to 1) develop an appropriate framework of mathematics; 2) gain mathematical experiences that would enhance their students’ experience with mathematics; and 3) grow awareness of the extent of one’s mathematical knowledge. Specific examples that connect linear algebra to secondary mathematics are presented and discussed. [ABSTRACT FROM AUTHOR]

To better understand how to support high school students' engagement, advancements in research methods that provide greater understandings of malleable factors of engagement and conditions that affect students' engagement are needed. In this conceptual paper, we introduce four dilemmas that researchers need to navigate to study secondary students' engagement with mathematics: How can we concurrently capture engagement in-the-moment and at scale? What counts as a moment or experience? What sorts of experiences could be engaging? Whose perspectives on the experience should be privileged? We propose approaches for navigating these dilemmas in the context of a current research project. [ABSTRACT FROM AUTHOR]

For reasoning and proof to become a reality in mathematics classrooms, it is important to prepare teachers who have knowledge and skills to integrate reasoning and proving in their teaching. Aiming to enhance prospective secondary teachers' (PSTs) content and pedagogical knowledge related to proof, we designed and studied a capstone course Mathematical Reasoning and Proving for Secondary Teachers. This paper describes the structure of the course and illustrates how PSTs' interacted with its different components. The PSTs first strengthened their content knowledge, then developed and taught in local schools a lesson incorporating proof components. Initial data analyses show gains in PSTs' knowledge for teaching proof and dispositions towards proving, following their participation in the course. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe secondary mathematics student teachers' types of noticing while teaching. We discuss the importance of focusing on the interrelatedness of the noticing skills rather than reporting on individual skills separately. We apply the types of noticing to videos of student teachers to identify their ability to elicit and interpret student mathematical thinking in-the-moment while teaching. Results suggest that our student teachers did elicit and attend to student mathematical thinking while teaching, but how they interpreted the elicited student thinking varied. We hypothesize three reasons for why student teachers may have interpreted student mathematical thinking at a general level. [ABSTRACT FROM AUTHOR]

An important decision that professional development (PD) facilitators must make when preparing for activities with teachers is to select an appropriate tool for the intended learning goals of the PD (Sztajn, Borko, & Smith, 2017). One important and prevalent tool is artifacts of student thinking (e.g. Jacobs & Philipp, 2004). In this paper we add to the literature on artifact selection for professional development by discussing the affordances and constraints of different written artifacts of student thinking. Through a professional noticing assessment, we examine the interpretive frames (Sherin & Russ, 2014) that were invoked by 72 secondary teachers regarding 6 students' written strategies to proportional reasoning tasks. We characterize different ways teachers might make sense of different artifacts of student thinking, and discuss for what purposes PD facilitators might select particular written solutions. [ABSTRACT FROM AUTHOR]

This study explored how three high school mathematics teachers thought about using real world connections to teach mathematics. Each of these teachers expressed beliefs about mathematical objects as entities that could be seen in the real world. This belief was connected to ideas about knowing mathematics as seeing these mathematical objects. This paper elaborates this epistemology of visibility and describes some of the implications. [ABSTRACT FROM AUTHOR]

This paper outlines the curricular redesign efforts of one high school mathematics department seeking to respond to high failure rates in low level mathematics courses that are disproportionately populated by students of color. I outline the aids and challenges encountered by the department and examine the resulting course structure and its likelihood to promote more students to take advanced level mathematics courses. [ABSTRACT FROM AUTHOR]

In this paper, a model is proposed to characterize some of the ways mathematical proofs are generated and understood. The model uses the notion of ‘Key Idea’, which is defined and illustrated in the context of a task from a pre-service course for secondary school teachers. [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.)

Studenting is comprised of the behaviors that students perform or exhibit in a learning situation, such as the mathematics classroom. These may or may not be intended to help themselves learn, and may or may not conform to the teacher's goals and expected actions. In this paper I report on the case of a particular student, Maria. Observations, together with excerpts of informal interviews, are used to provide a preliminary analysis using activity theory as a theoretical lens. Initial results reveal that 'good' studenting behaviors do not necessarily result in mathematics learning, nor indicate a primary motive of learning. Analysis indicates Maria's primary motive was to get a good grade and be seen as a good student. [ABSTRACT FROM AUTHOR]

Creativity in mathematics is often looked at as the exclusive domain of professional mathematicians. There have been few studies that have examined creativity in the high school classroom, particularly in its manifestation in students with exceptional mathematical abilities. One of the obstacles to the study of mathematical creativity is the ‘fuzziness’ of the construct of creativity. In this paper the notion of mathematical creativity is developed from literature in psychology. In particular the evolving systems model (Gruber & Wallace, 2000) is used to construct the case study of a ninth grade student and to assess his attempts to discover the structure of Steiner Triple Systems after working on several problems over a four-month period. The implications of this study for assessing students’ problem solving in the high school classrooms are discussed. [ABSTRACT FROM AUTHOR]

The study describes how the common representations are coordinated. I analyzed a popular geometry textbook using semiotics and a pragmatic approach to capture the variety of representations into categories and to use descriptive statistics to narrow the focus to the most common representations and coordinations. The major findings are: (1) exposing which representations are most often coordinated like written language; (2) some of the mechanisms in coordination use numbers, point names, and textbook gestures, which include color, arrows, font changes, etc. [ABSTRACT FROM AUTHOR]

A modified, bilingual Attitudes Toward Mathematics Inventory (ATMI) instrument was administered to 1,258 high school students in South Texas in an NSF-funded project on informal learning of mathematics and near peer mentoring. We explore students' survey response behaviors and examine the existence of careless and insufficient effort (CIE) responses. This is empirical research for handling the challenge of CIE responses that leads to improved survey data quality, thus eventually validating the intervention effect of the mathematical informal learning project. [ABSTRACT FROM AUTHOR]

This investigation utilized publicly available data from the High School Longitudinal Study 2009 (HSLS:09) by the National Center for Educational Statistics (NCES) to examine any latent structures among variables which may empirically support the validity and reliability of a mathematical sense of belonging (MSB) construct. Using the nationally representative survey data in the HSLS:09, the proposed study performed an exploratory factor analysis (EFA) to create a complex construct of students' MSB or mathematical belongingness. A weighted sample of over 13,000 (N=13,354) high school students was used to conduct the EFA, which was then mapped onto Mahar and colleagues' (2012) intersecting, transdisciplinary themes of belonging. Possible implications for preservice teachers, professional development for practicing teachers, as well as future research directions are discussed. [ABSTRACT FROM AUTHOR]

Ten secondary mathematics teachers were interviewed about their understanding of mathematical connections in a series of three progressively more structured individual interviews. A model for categorizing mathematical connections is presented and used as an interpretive tool. This paper reports the case of Robert, a knowledgeable and experienced mathematics teacher. He values making connections but finds articulating explicit specific mathematical connections difficult and makes connections of limited types. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

The paper presents the results of an investigation of the infinitesimal procedures for the calculation of the area of a circle depicted in 19 modern Canadian, Taiwanese, and Russian secondary schools textbooks and in their medieval counterparts. The authors conjecture that the procedures found in the textbooks are related to mathematical methods found in medieval mathematical writings, in particular, in those of Gerard of Brussels and of J. Kepler. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This paper reports on a study of how preservice secondary mathematics teachers deal with the complexity of a problem in a covariational context. We claim that the coordination of structural and operational perspectives of mathematical objects can provide affordances in dealing with complexity and covariation. ..PAT.-Conference Proceeding [ABSTRACT FROM AUTHOR]

ABSTRACT: In this research paper, it is intended to (a) check the hypothesis that all the misconceptions with regard to the mathematical concept of set originate in the tacit influence of the ‘collection model’ and (b) determine what happens as an effect of time and more formal education of sets between the tacit model and the formal properties. Several misconceptions have been determined among both elementary and secondary school students due to the collection model in their mind. Even after learning the formal properties of a set in the mathematical sense, the students are still influenced by this model. Our analysis revealed that from Grade 7 to Grade 10 the number of misconceptions students have tend to decrease. [ABSTRACT FROM AUTHOR]

There is a popular myth that mathematical creativity is a trait possessed by a small group of people. This is perpetuated by the lack of understanding of what influences mathematical creativity, especially in the classroom. In particular, professional mathematicians often name aesthetic experiences as a critical aspect of their mathematical creativity. Therefore, this study examines aesthetic dimensions of episodes of student creativity from the high school classroom. Several types of aesthetic experiences arose across the episodes. Students were motivated to take creative action by experiences of mystery or discomfort. Furthermore, in some instances, taking a creative action was accompanied and propelled by a sense of craziness or satisfaction. [ABSTRACT FROM AUTHOR]

The purpose of this study is to create and evaluate an analytical framework for examining instructional materials. In particular, this study focuses on the creation of a framework intended to analyze the extent to which instructional materials promote quantitative literacy among learners. This study investigates current literature attempting to define quantitative literacy, provides an overview of theoretical frameworks designed to analyze levels of quantitative literacy among students, offers examples of relevant textbook analysis frameworks, and provides a brief overview of the impact of instructional materials on learning in general. A framework intended to analyze quantitative literacy promotion among various instructional materials is developed based on relevant empirical and theoretical research related to the aforementioned areas of study. One elementary, middle, and high school program will be evaluated using the proposed framework. [ABSTRACT FROM AUTHOR]

This research report is a part of my Ph.D. thesis on which I have worked for one and a half year. My work is indeed preliminary: to some extend, it sketches my future work this spring. [ABSTRACT FROM AUTHOR]

’And you just change it to two square root five.’ Some time after a student said this, I asked her classmates, ‘What does that mean when she says ‘just’?’ This exchange was part of a semester-long conversation in which I aimed to raise the students’ awareness of their language practice in mathematics class. When the students and I disagreed over how to use the word ‘just’, we became aware of connections between our use of the word and the way we take mathematics procedures for granted. Morgan (1998), as a result of her extensive study of secondary school mathematics writing, identifies the need for students to become more aware of their language practice. Discourse analysis has tremendous potential for providing insight into mathematics and its classroom practice. Like Morgan, I want both mathematics students and educators to benefit from increased language awareness. [ABSTRACT FROM AUTHOR]

Irrational Numbers: Dimensions of Knowledge Student: Is pi the only irrational number? Teacher: No, because there is also root-two. Mathematically, of course, it is a correct answer. Pi cannot be the only irrational number if there exists another one. However, we note in this reply a missed opportunity, perhaps due to the teacher’s limited understanding of the concept, to extend the discussion of irrational numbers. This exchange, observed by one of the authors of this proposal, inspired a study on prospective secondary teachers’ (PST’s) understanding of irrationality, part of which is reported here. The purpose of the study was to provide an account of PST’s understandings and misunderstandings of irrational numbers, to interpret how the understanding of irrationality is acquired, and to explain how and why difficulties occur. [ABSTRACT FROM AUTHOR]

This study evolved out of a commitment to improve mathematics teaching and learning in an alternative high school and to begin to reverse the cycle of educational failure for students labeled ‘at-risk.’ Although research in teacher preparation has explored the ways in which preservice teachers learn to teach mathematics, few studies have focused on how teaching in an alternative high school interacts with and complicates this process. Paralleling the need to improve the preparation of preservice teachers to work with at-risk and other marginalized students is the need for more effective mathematics education programs for at-risk students. The purpose of this study was to investigate the influences of reform-based methodologies and materials on preservice teachers’ instructional strategies and the mathematical development of at-risk students in an alternative high school. The analysis suggests that, in addition to influencing the preservice teachers’ instructional practices and constructs of student knowledge and student learning, the reform-based methodologies and materials had a positive impact on student knowledge, learning, and attitudes. By examining the experiences of preservice teachers and students in an alternative high school, the study discussed here offers a model for the improvement of mathematics education for at-risk students. [ABSTRACT FROM AUTHOR]

This study examines how joint exploration is established and maintained among students and the teacher in secondary mathematics classrooms. We use the theoretical perspective of positioning to conceptualize joint exploration as involving the negotiation and coordination among participants to position students with epistemic agency and authority. Using a constant comparative method, we use classroom video data of two episodes containing joint exploration and closely analyze the shifts in epistemic positioning within them. We find that shifts in epistemic positioning, especially when students position one another with epistemic authority, help to support continued joint exploration. We also find that the teacher can play an important role in decentering themselves as the epistemic authority. [ABSTRACT FROM AUTHOR]

Qualitative research explores the complexity of teaching and learning, but there are barriers to using qualitative findings to inform practice/policy. We see a critical need to integrate findings across qualitative studies to develop synthesized, actionable recommendations for mathematics teacher educators. Using a meta-aggregation of 11 qualitative studies, we investigated how teacher educators’ mathematical knowledge for teaching can inform their support of formative assessment practices in secondary mathematics classrooms. Synthesized findings led to nine lines of action that we provide as recommendations for mathematics teacher educators. [ABSTRACT FROM AUTHOR]

In this article I examine a new course offering at an urban high school, Climate Justice Algebra, to determine if the course is Culturally Relevant Pedagogy (CRP). I examine the tensions between academic success (grade level mathematics learning) and climate justice learning found during two tasks administered to the class. I consider the antagonism between covering grade level mathematics standards and engaging students in Culturally Relevant tasks when melding the two practices together. [ABSTRACT FROM AUTHOR]

Thompson (2015) proposed that a productive mathematical meaning is one that prepares students for future learning and lends coherence to extant meanings. We propose that a productive mathematical meaning related to expanding algebraic expressions and factoring polynomials revolves around developing an understanding of structural relationships between (binomial) factors and polynomials. We briefly outline a combinatorial approach to support a productive mathematical meaning for expanding and factoring relationships. We then report on our analysis of four Algebra 1 and Algebra 2 textbooks as we investigated their approaches to these topics. Our findings suggest that significant work on curricular development is needed to support productive mathematical meanings in this domain. [ABSTRACT FROM AUTHOR]

This study reviews how slope is developed in expository materials across a seven-textbook series. Slope development is analyzed using a framework of five slope components to describe which components are used and connected, and by investigating accompanying levels of covariational reasoning. Findings suggest that the series describes slope from multiple components, and this development is grounded in various levels of covariational reasoning. While many connections were found between components, occurrences of both visual and nonvisual approaches within components were not prevalent. Suggestions include building connections between Behavior Indicator and Determining Property components through descriptions of covariation as well as more connections to the Steepness component. [ABSTRACT FROM AUTHOR]

The use of questioning is an effective strategy for orchestrating collective argumentation. However, teachers with minimal experience facilitating argumentation may conceive of effective support as providing little to no verbal input in the argumentation. In this study, we analyzed one teacher's analysis and critique of her support for collective argumentation during her first three years of teaching. We argue that learning to analyze her support for collective argumentation enriched the teacher's understanding of questioning. More specifically, by explicitly identifying how her questions elicited components of arguments from students, the teacher re-evaluated her questions, focusing on purpose rather than form. Implications from this study draw connections between learning to facilitate argumentation and the dilemma of telling that teachers encounter when trying to teach mathematics in ways that honor students' thinking and sense-making. [ABSTRACT FROM AUTHOR]

Teachers often find it hard to balance between justice to the students' input and leading the class towards the decided goal. We focus on how the teacher orchestrates the balance between whole class student authority and accountability to the discipline. In the case a student presents the result from group work but at some point needs help. The teacher de-personalises the discussion and directs the class' attention to the subject and not to individual students. Thereby, the class is treated as a community with a shared authority. By the end, collective learning has taken place. [ABSTRACT FROM AUTHOR]

The purpose of this research study is to characterize secondary teachers' orientations toward mathematics engagement. Results indicated that these 16 high school mathematics teachers tended to emphasize a cognitive dimension for engagement in their orientations, usually intertwined with an additional dimension (affective, social, or behavioral). Understanding teachers' thinking about engagement is a critical step toward helping teachers improve their practice to support their students' engagement in mathematics learning. [ABSTRACT FROM AUTHOR]

This case-study sought to describe instruction that supported students' mathematical learning (ML) and social-emotional learning (SEL). Transcripts from audio-recorded lesson observations and teacher interviews, field notes, and written teacher reflections were collected to answer two research questions: (1) In what ways does a high school mathematics teacher support ML and SEL during instruction? (2) How does the teacher characterize her attempts to provide social-emotional and mathematical supports during instruction? Preliminary findings suggest that strategies for revising ideas and handling errors support students ML and SEL. [ABSTRACT FROM AUTHOR]