PROCEEDINGS OF THE 22ND ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2019
Oklahoma City, Oklahoma
Number of pages: 1205
Number of papers: 207
- Aaron Weinberg
- Deborah Moore-Russo
- Hortensia Soto
- Megan Wawro
2019
Oklahoma City, Oklahoma
Number of pages: 1205
Number of papers: 207
2019
Oklahoma City, Oklahoma
The SIGMAA on Research in Undergraduate Mathematics Education (RUME) community recently had to grapple with issues encountered as a result of California state law, which states that, “California must take action to avoid supporting or financing discrimination against lesbian, gay, bisexual, and transgender people” (California Assembly Bill No. 1887, 2016). In effect, this law prohibits California state-funded travel to other states which have religious freedom laws that are viewed as discriminatory to lesbian, gay, bisexual and transgender (LGBT) individuals. On June 22, 2018 Oklahoma was added to the list of states where travel was prohibited, due to a newly enacted law SB 1140. This law states that child-placement agencies will not be required to place a child in adoption or foster care in situations that “violate the agency’s written religious or moral convictions or policies” (Oklahoma Senate Bill No.1140, 2018). The enactment of SB 1140 and the prohibited state-funded travel meant that some members of the RUME community would be unable or unwilling to travel to the 2019 and 2020 RUME conferences, which were slated to be held in Oklahoma.
2019
Oklahoma City, Oklahoma
This paper reports on an instance of change in a university mathematics department which revitalized and improved their precalculus/calculus program by implementing a series of strategies, techniques, and programs which are supported by educational research. Using the Four Frames perspective for organizational culture (Bolman & Deal, 2008; Reinholz & Apkarian, 2018), we explore how the dimensions of structures, symbols, people, and power support a rich understanding of how the department’s culture supported and constrained the change initiative. We do so both generally speaking, for the entire initiative, and more in depth, regarding the development of a course coordination system. Furthermore, this case study suggests the utility of these four frames for change agents elsewhere as a tool to support the design and enactment of successful and sustainable change towards the improvement of, specifically, undergraduate mathematics education.
2019
Oklahoma City, Oklahoma
A robust conceptual understanding of function is essential for students studying calculus and higher levels of mathematics as they continue to pursue the learning of mathematics. In this study, we investigated the ways in which students in a Calculus II course understand functions by examining student engagement with a vending machine applet. Specifically, we considered how these students made sense of the univalence requirement of functions in the context of a vending machine in which a single input produces an output of two cans. We identify and discuss in detail several themes that emerged in students’ categorization of machines as functions or non-functions when encountering this two-can scenario.
2019
Oklahoma City, Oklahoma
A schema is a mental structure of concepts that are connected together and allows for the efficient functioning of director systems. Skemp (1979) discusses various qualities that this study used to look at students’ schemas. This case study focuses on a pair of Topology students and their work on a problem involving the product topology on 𝑋 × 𝑌. There were many positive qualities that the students demonstrated, but there were also difficulties with particular connections between concepts.
2019
Oklahoma City, Oklahoma
This paper is a review of research that either explicitly or implicitly examines the interplay between teacher knowledge and teaching practices sensitive to students’ mathematical thinking. I use radical constructivism as a lens to analyze how the researchers conceptualize the role of teacher knowledge in student-teacher interaction. My analysis reveals that some researchers attribute teachers’ observable actions to what knowledge teachers possess (i.e., application perspective) while some others focus on what knowledge teachers construct in-the-moment (i.e., construction perspective). I conclude the paper by discussing the potential causes and consequences of these differences as well as the affordances and limitations of each perspective.
2019
Oklahoma City, Oklahoma
The mathematics learning center (MLC) of a university may influence more aspects of a student’s life than the targeted mathematics learning. In this study we examined an MLC from the perspective of the undergraduate peer tutors employed there seeking to understand the space as a figured world. Differential use of pronouns emerged during analysis of collected stimulated recall data from the participating undergraduate mathematics peer tutors. My examination of which individuals, groups, or subgroups were included in “we” and “us” statements by the participants revealed social patterns within the MLC where both academic and non-academic behaviors indicated belonging or the potential to belong. The personal narratives of the participant tutors expanded on these ideas of coming to belong within the MLC and the implications of that belonging for their developing mathematics and STEM identities.
2019
Oklahoma City, Oklahoma
When students are coming to understand how to construct proofs, as well as how mathematicians use proofs in their work, the role of the instructor cannot be overstated. In this paper, we present an investigation into how an instructor uses her authority to empower students as legitimate proof producers and learners of mathematics. We view this empowerment and student learning through a situated lens, accounting for relationships of disciplinary authority and student agency. In our investigation, we analyzed the transcripts from three classroom episodes in an inquiry-based transition-to-proofs course. We identified instances when the instructor leveraged her institutional authority as well as her mathematics expertise authority to support students’ engagement in the dance of agency, asserting their own creative ideas as learners of mathematics while still adhering to the norms and standards of the discipline.
2019
Oklahoma City, Oklahoma
In this report we examine the performance and reasoning of span and linear independence of 126 linear algebra students who learned through a particular inquiry-oriented (IO) instructional approach compared to 129 students who did not. Students who received IO instruction outperformed Non-IO students on questions focused on span, but not on questions focused on linear independence. Our open-ended coding additionally suggested that IO students’ concept images of span and linear independence were more aligned with corresponding concept definitions than those of Non-IO students.
2019
Oklahoma City, Oklahoma
Positive and negative quantities are ubiquitous in physics, and the sign carries important and varied meanings. Unlike physics experts, novices struggle to understand the many roles signed numbers can play in physics contexts, and recent evidence shows that unresolved struggle carries over to subsequent physics courses. The mathematics education research literature documents the cognitive challenge of conceptualizing negative numbers as mathematical objects. We contribute to the growing body of research that focuses on student reasoning in a physics context about signed quantities and the role of the negative sign. This paper contributes a framework for categorizing the natures of the negative sign in physics contexts, inspired by the research into the natures of negativity in algebra. Using the framework, we analyze several published studies associated with reasoning about negativity drawn from the physics education and mathematics education research communities. We provide implications for mathematics and physics instruction and further research
2019
Oklahoma City, Oklahoma
For reasoning and proving to become a reality in mathematics classrooms, pre-service teachers (PSTs) must develop knowledge and skills for creating lessons that engage students in proof-related activities. Supporting PSTs in this process was among the goals of a capstone course: Mathematical Reasoning and Proving for Secondary Teachers. During the course, the PSTs designed and implemented in local schools four lessons that integrated within the regular secondary curriculum one of the four proof themes discussed in the course: quantification and the role of examples in proving, conditional statements, direct proof and argument evaluation, and indirect reasoning. In this paper we report on the analysis of 60 PSTs’ lesson plans in terms of opportunities for students to learn about the proof themes, pedagogical features of the lessons and cognitive demand of the proof-related tasks.
2019
Oklahoma City, Oklahoma
Measurement is a foundational concept in all STEM fields. Difficulties with measurement and converting between units of measure have been documented in medical students, chemistry students, and mathematics students at varieties of educational levels. However, less is known about why this topic is so difficult and what mental operations are entailed in mastering it. Steffe (2012) argued that students must assimilate situations with three levels of units to understand measurement conversions so we attend to students’ units coordination schemes while remaining open to other factors impacting students’ responses to measurement tasks. We found that the STEM majors in our sample who assimilated tasks with two levels of units had more difficulty with measurement tasks than those who assimilated tasks with three levels of units.
2019
Oklahoma City, Oklahoma
The creators and leaders of mathematics tutoring centers at universities make many choices about the organizational structure of their centers. Some of those choices include the location of the center, the education level of the tutors, the method of tutor training, the number of hours tutoring is available, and the way tutoring is provided (i.e. drop in or scheduled). Our group’s long-term goal is to provide research-based evidence to help faculty and administrators choose effective structures for centers. This paper documents similarities and differences between centers to provide a descriptive foundation for future hypothesis generation and testing.
2019
Oklahoma City, Oklahoma
This study raises questions about a common assumption that an advanced degree in mathematics is sufficient for teaching courses in undergraduate mathematics meaningfully. The study reports results from 24 mathematics PhD students’ solutions to a precalculus level problem requiring quantitative reasoning. We also describe the PhD students’ conceptions of what knowledge is needed to produce a meaningful solution to this task. These graduate students’ problem solving approaches and images of the reasoning abilities needed to solve the problem were classified as having either a static calculational orientation or a dynamic conceptual orientation. We share how these two orientations are exhibited in the context of teaching precalculus students. We further illustrate ways in which a teacher’s actions to support her students in conceptualizing and relating quantities led to her engaging her students in more dynamic conceptually oriented discussions.
2019
Oklahoma City, Oklahoma
This longitudinal study revisits a decade old study about the relationship between level of Graduate Teaching Assistant (GTA) instructional expertise, amount of GTA teaching experience, and academic performance of their college algebra students measured by course grades. The questions posed then remain relevant today. In the present study, college algebra grades for all students in classes taught by GTAs since the original experiment were analyzed. That is, data from twelve years (AY2006 – AY2017) and 168 sections (n = 6675) were examined. Noteworthy is the fact that success in lowering the drop rate in the treatment group held true for 15 years since the treatment was initiated. Included is a look at what has changed and what has remained the same since the original study.
2019
Oklahoma City, Oklahoma
When trying to examine instructors’ instructional practices, specifically lecturing, qualitative studies have indicated the necessity to consider their beliefs. However, there is a dearth of quantitative belief measures specific to instructors of undergraduate mathematics courses. No one specific instrument captures the relationship between beliefs and lecturing. This paper, therefore, attempts to establish a foundation of significant factors for researchers to consider when developing belief measures to predict lecturing. We use pre-existing data from Calculus and Abstract Algebra courses to conduct factor analyses and develop composite variables. We then use multiple regression to examine composites with significant effects on time spent lecturing. Results suggest that beliefs related to a focus on skills and content, knowledge facilitation authority, expectations of student success, and the importance of particular concepts are of particular importance.
2019
Oklahoma City, Oklahoma
There are differing senses of meaning in mathematics education focusing either on mathematical understanding or on relevance. Various pedagogical practices exist in mathematics education, each with its own goals and associated challenges of implementation though the relationship between pedagogical goals and differing senses of meaning has not been explored extensively. Using pre-existing survey data on calculus instructors’ pedagogy, we used multiple regression to determine the effect of differing pedagogical decisions aligned with the “meaning of” or “meaning for” mathematics on instructors’ perceived pressure to cover course content. The results of our overall test were statistically significant. In particular, we found instructors’ focus on the meaning for mathematics had a statistically significant effect on decreasing instructors’ stress to cover material. Implications and further areas of study follow.
2019
Oklahoma City, Oklahoma
This proposal reports on a teaching experiment in which a pair of prospective secondary mathematics teachers leverage their knowledge of secondary algebra in order to develop effective understandings of the concepts of zero-divisors and the zero-product property (ZPP) in abstract algebra. A critical step in the learning trajectory involved the outright rejection of the legitimacy of zero-divisors as counterexamples to the ZPP, an activity known as monster-barring (Lakatos, 1976; Larsen & Zandieh, 2008). This monster-barring activity was then productively repurposed as a meaningful way for the students to distinguish between types of abstract algebraic structures (namely, rings that are integral domains vs. rings that are not). The examples of student activity in this teaching experiment emphasize the importance of identifying, attempting to understand, and leveraging student thinking, even when it initially appears to be counterproductive.
2019
Oklahoma City, Oklahoma
Many mathematics departments have transition to proof (TTP) courses, which prepare undergraduate students for proof-oriented mathematics. Here we discuss how common TTP textbooks treat three topics ubiquitous to such courses: logic, proof techniques and sets. We show that these texts tend to overlook the rich connections sets have to proof techniques and logic. Recent research has shown that student thinking about sets is propitious to novice students’ ability to reason about logic and construct valid arguments. We suggest several key connections TTP courses can leverage to better take advantage of their unit(s) on sets.
2019
Oklahoma City, Oklahoma
Though scholars have long called for applications and modeling to be explicitly added to classroom agenda (Niss, Blum, & Galbraith, 2007), opportunities for undergraduates to engage in modeling in the classroom remain scarce. We report a study of undergraduate STEM majors engaging in authentic, open-ended modeling tasks using differential equations through a modeling competition. In this study, we propose a logic model that captures the relationship between the advantages of mathematical modelling and mathematics self-efficacy (MSE) and investigate the extent to which a mathematical modeling intervention increased STEM majors’ Mathematics Self Efficacy (MSE).
2019
Oklahoma City, Oklahoma
This study presents findings from a series of interviews in which we observed undergraduate students’ moment-by-moment Reading of Mathematical Proof (ROMP) activity. This methodology is adapted from a validated assessment of narrative reading comprehension developed by cognitive psychologists. We demonstrate the fruitfulness of the method by describing four relatively novel phenomena that we observed in our interviews, and highlight ROMP activities that seemed to distinguish less productive and more productive readers
2019
Oklahoma City, Oklahoma
Homework accounts for the majority of undergraduate mathematics students’ interaction with the content. However, we do not know much about what students learn from homework. This paper reports on a pilot study of why professors chose particular homework problems, what they hoped students would learn from them, and whether students’ engagement with the problems reflected those outcomes. Results show students gained the desired familiarity with notation and procedures. The results also speak to how professors manage the content between what they discuss in class, homework problems, and intentional overlap between the two.
2019
Oklahoma City, Oklahoma
Recently, Ely & Ellis (2018) described a new mode of covariational reasoning—scaling-continuous reasoning—and conjectured that it might support productive student thinking in calculus. We investigate that hypothesis by analyzing how calculus students employed scaling-continuous covariational reasoning when discussing differential calculus ideas. The interviewed students who took a course based on a “local straightness” approach to calculus used scaling-continuous reasoning in their description of the derivative at a point, particularly in their imagery of zooming in on a function at a point to reveal its slope. The interviewed students who took a course based on an “informal infinitesimals” approach to calculus used scaling-continuous reasoning in their account of how zooming in on a neighborhood reveals the coordination between a bit of x (dx) and the corresponding bit of y (dy), a relationship that gives a differential equation for that curve.
2019
Oklahoma City, Oklahoma
This study explores how mathematicians view validity in the face of explicit validity issues within written mathematical arguments in the context of the Introduction to Proof (ITP) setting. An internet survey of 30 arguments was constructed leveraging common issues in validity at the ITP level, and widely distributed to research-active mathematicians in the United States. The results suggest that there is no consensus as to the effect of any single validity issue on the overall validity of an argument, lending credence to the notion that argument validity lacks a consistent set of criteria from one mathematician’s point of view to the next.
2019
Oklahoma City, Oklahoma
Research has shown that teachers and instructors’ views about the discipline they teach inform their instructional approaches. As a foundation for investigating this relationship in statistics, we explore how (or whether) beginning graduate students in statistics perceive statistics as distinct from mathematics. Using the lens of epistemology, we share findings from interviews with four, first-year graduate students who served as graduate teaching assistants (GTAs) in a statistics department. Using data collected from interviews across their first year, we constructed three models that explain how the GTAs conceived of the nature of statistics in relation to mathematics. Additionally, we identified two continua that reveal how participants came to understand the nature of doing statistics. We discuss how these models and continua form the basis of a unified statistical epistemology that has implications on their views for statistics education.
2019
Oklahoma City, Oklahoma
Linear functions of more than one variable exhibit the property that changes in the dependent variable are linear combinations of changes in the independent variables. Although multivariable calculus makes frequent use of this linearity condition, it is not known how students reason about linearity within this context. This report addresses this question by analyzing how three students incorporate linearity into their schemas for linear approximation and directional derivative. The students in this report showed a progression in their understanding from not using linearity within their reasoning to incorporating linearity into first their scheme for linear approximation and finally into their scheme for directional derivative. The results indicate that the context of linear approximation was useful for developing concepts of linearity and aiding their development of the concept of directional derivative.
2019
Oklahoma City, Oklahoma
In a culture where STEM preparation is rapidly becoming of utmost importance to the nation’s economy and educators are challenged to increase diversity and equity amongst students, quality mathematics instruction at the collegiate level is critical. Yet the majority of undergraduate mathematics teachers are not formally trained in pedagogy. This is a systemic issue, an institutionalized paradox, which originates in the mathematicians’ training grounds - mathematics PhD programs. This paper provides background on this issue and focuses on a survey of university mathematicians concerning their formal academic training and their outlooks and prioritization of pedagogical training. Attention is drawn to the disconnect between university mathematicians’ beliefs about the important role of pedagogical education in mathematics program and their resistance to promoting its implementation as a basic institutional requirement. A call for action is suggested to remedy these institutionalized systemic paradoxes.
2019
Oklahoma City, Oklahoma
To reform instruction by moving towards student-centered approaches, research has shown that faculty benefit from support and collaboration (Henderson, Beach, & Finkelstein, 2011; Speer & Wagner, 2009). In this study, we examined the ways in which a mathematician’s instruction unfolded during his participation in a faculty collaboration geared towards reforming instruction and aligning it with inquiry oriented instruction (IOI) (Kuster, Johnson, Andrews-Larson, & Keene, 2017). Results indicate the participant’s mathematics background and research interests influenced how he used student thinking in his instruction. More specifically, when mathematics content specifically aligned with the participant’s research interest he often guided students to view differential equations as he did; whereas, when the content was not aligned with his research interest, he was more open to the using his students’ thinking to drive the class forward. Implications and future research directions are discussed.
2019
Oklahoma City, Oklahoma
Advanced mathematics courses require that students possess sophisticated proving techniques. Topology is one such course in which students’ proving behaviors have not been extensively studied. In this paper, we propose that visual methods play an important role in undergraduates’ discovery of the key idea of a proof, and we describe a potential framework for students’ proving processes in a first course in undergraduate topology based on Carlson and Bloom’s (2005) problem solving framework.
2019
Oklahoma City, Oklahoma
There is evidence that instructors who are responsive to students’ thinking tend to provide more positive learning experience for students. Additionally, effective instruction relies on an instructor’s ability to respond to student thinking, which is especially relevant due to the increased attention on improving college mathematics instruction. In order to investigate instructor responsiveness to student thinking as a disposition (that guides action) and responsiveness to student thinking as an action (the enacted evidence of the underlying disposition), eight college Calculus instructors were interviewed three times over the course of one academic year. A thematic analysis of the task-based interviews indicated that instructors who exhibited a responsive disposition to their students’ thinking enact this through eliciting student thinking, reflecting on student thinking, and responding to student thinking. Further, these instructors view themselves as decision-makers, and thus feel empowered to act on their responsive disposition.
2019
Oklahoma City, Oklahoma
Undergraduate mathematics instructors are called by recent standards to promote prospective teachers’ learning of a transformation approach in geometry and its proofs. The novelty of this situation means it is unclear what is involved in prospective teachers’ learning of geometry from a transformation perspective, particularly if they learned geometry from an approach based on the Elements; hence undergraduate instructors may need support in this area. To begin to approach this problem, we analyze the prospective teachers’ use of the conceptual link between congruence and transformation in the context of congruence. We identify several key actions involved in using the definition of congruence in congruence proofs, and we look at ways in which several of these actions are independent of each other, hence pointing to concepts and actions that may need to be specifically addressed in instruction.
2019
Oklahoma City, Oklahoma
Calculus teaching and learning is a topic of great interest in the mathematics education research community. Specifically, the definite integral and accumulation have received quality attention in the past few years (e.g. Jones, 2015; Sealey, 2014). However, even though approximately 30% of our introductory calculus students are planning on careers in the biological and life sciences, little research exists concerning how students from these fields reason about calculus. In this study, task-based interviews were conducted with 12 undergraduate students majoring in the biological and life sciences. Students were asked to complete two similar calculus accumulation tasks, one a traditional kinematics task and the other set in the context of plant growth. Data were analyzed via open coding. Results indicate students interpreted the given information in the tasks differently, they were more likely to view the rate of change curve as representing the total accumulated quantity in the plant growth tasks.
2019
Oklahoma City, Oklahoma
The practice of providing written feedback on an undergraduate student’s proof in the form of asking questions is striking in that professors do not know whether the student attempts to answer the questions. This phenomenon leads us to investigate the reasons why professors ask questions as written feedback. We analyze the written questions of four professors teaching abstract algebra and real analysis at a medium-sized, rural, comprehensive public university in the northeast. We find that these four professors most frequently ask questions that either seek further explanation from students or address a mathematical detail within their proof. In some cases, the professors answer the questions they ask as written feedback. Overall, the professors ask questions as written feedback to encourage students’ thinking, thereby engaging students in the proof-writing process and improving the students’ proof production skills.
2019
Oklahoma City, Oklahoma
The goal of this article is to propose a way to think about the role of a multiplicative object in reasoning about formulas quantitatively and covariationally. Building off the works of others on the importance of constructing multiplicative objects when reasoning about graphical representations, I adapt their definitions to be able to include a meaningful way to discuss what it means to construct a multiplicative object with a formula. I then use the analysis of six sessions of a semester-long teaching experiment with a preservice secondary mathematics teacher to illustrate what it means not to construct and what it means to construct a multiplicative object with a formula.
2019
Oklahoma City, Oklahoma
Our field has generally reached a consensus that active learning approaches improve student success; however, there is a need to explore the ways that particular instructional approaches impact various groups of students. Here we examined the relationship between gender and student learning outcomes in one particular context – abstract algebra, taught with an Inquiry-Oriented Instructional (IOI) approach. Using hierarchical linear modeling, we analyzed content assessment data from 522 students. While the performance of IOI and non-IOI students was similar, we detected a gender performance difference (men outperforming women) in the IOI classes that was not present in the non-IOI classes. In response to these findings, we present avenues for future research on the gendered experiences of students in such classes
2019
Oklahoma City, Oklahoma
Unfortunately, students far too often have little or no intellectual need for learning the second semester calculus topic of Taylor and power series. In this study, we examine the “potential intellectual needs” (PINs) provided by commonly used textbooks. While the textbooks used different approaches, they both often lacked problems developing intellectual need, suggesting that instructors must incorporate intellectual need by themselves. To assist in this endeavor, we focus part of the paper on a discussion of including PINs for this content. We found that it may be difficult to incorporate genuine problems for first-year students through an approach based on a “family of series” meaning for Taylor/power series, but that stronger problems could be incorporated through an approach based on an “extension of linear approximation” meaning.
2019
Oklahoma City, Oklahoma
Our theory of what entails a conceptualized frame of reference is explained, along with items and rubrics designed to illuminate how teachers do or do not reason with frames of reference. We gave 551 teachers in the US and Korea frame of reference tasks, and coded the open responses with rubrics intended to rank responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. Our results show that our theoretical framework is useful in analyzing teachers’ reasoning with frames of reference, and that our items and rubrics function as useful tools in assessing teachers’ meanings for quantities within a frame of reference.
2019
Oklahoma City, Oklahoma
The purpose of this study is to analyze student understanding of isomorphism as it is taught in a university level mathematics course. We collected and studied student responses to course assignments covering the concept of isomorphism. The findings of this study support previous research that suggests student understanding of isomorphism is largely reliant on an imaged-based concept of symmetry. We found that student understanding is supported by an image-based radical constructivist approach and detail the techniques students use when first working with isomorphic mappings.
2019
Oklahoma City, Oklahoma
In this study, we present a comparative case study of two students with different epistemological frames watching the same college algebra lectures. We show that students with different epistemological frames can evaluate the same lectures in different ways, including very different evaluations of the goals and important content. Moreover, we illustrate that even when students have seemingly productive epistemological frames might give way to pragmatic decisions about earning a good grade when presented with too much information too fast. We argue that students might have productive dispositions towards mathematics, but default to a procedural orientation, and, as a result, appear indistinguishable in a class, from those who only have a procedural view of mathematics. These results illustrate how a student’s interpretation of a lecture is not inherently tied to the lecture, but rather depend on the student and her perspective on mathematics and factors in the control of the lecturer.
2019
Oklahoma City, Oklahoma
Research shows that definitions in mathematics are often not used correctly by students in mathematical proofs and problem-solving situations. By observing properties and making conjectures in non-Euclidean geometry, students can better develop their understanding of these concepts. In particular, Taxicab geometry is suggested to be introduced before other non-Euclidean geometries since it is a considerably simpler space. To further investigate this, APOS Theory is used as the framework in this analysis of responses to a real-life situation from students enrolled in a College Geometry course at a university. Through the perspective of APOS Theory, this report provides two representative illustrations of the conceptual understandings found among these students in relation to the definition of circle. By adapting and applying their knowledge of definitions from Euclidean geometry to Taxicab geometry, these students provided insight as to how Taxicab geometry concepts are assimilated into their existing understanding of concepts in geometry.
2019
Oklahoma City, Oklahoma
This study examines how the use of plausible and productive reasoning in mathematical problem solving (MPS) influences student performance on non-traditional problems. Data comes from ten individual, task-based interviews with College Algebra students. In general, students who demonstrated high use of plausible and productive reasoning had a higher percentage of correct answers on interview tasks than their peers. We propose reasons why a student may use plausible and productive reasoning and still arrive at an incorrect answer; we also consider how a student may use suboptimal reasoning and reach a correct answer.
2019
Oklahoma City, Oklahoma
This study investigates factors that influence instructors’ decisions to implement inquiry-oriented instruction. We analyzed entrance interviews with twelve Linear Algebra instructors, who participated in the Teaching Inquiry-Oriented Mathematics: Establishing Supports professional development project, to better understand the reasons why the instructors chose to shift from traditional lecturing to inquiry-oriented instructional approaches. We found three internal and three external factors that influenced the participating instructors’ choice to teach the inquiry-oriented Linear Algebra course. Implications for future research are discussed.
2019
Oklahoma City, Oklahoma
In this study we investigated how a small sample of students used variational reasoning while discussing ordinary differential equations. We found that students had flexibility in thinking of rate as an object, while simultaneously unpacking it in the same reasoning instance. We also saw that many elements of covariational reasoning and multivariational reasoning already discussed in the literature were used by the students. However, and importantly, new aspects of variational reasoning were identified in this study, including: (a) a type of variational reasoning not yet reported in the literature that we call “feedback variation” and (b) new types of objects, different from numeric-quantities, that the students covaried
2019
Oklahoma City, Oklahoma
In the United States (US) and worldwide, undergraduate mathematics instructors are increasingly aware of the value of inquiry-based instruction. We describe the intellectual origins and development of two major strands of inquiry in US higher education in mathematics, offer an explanation for apparent differences in these strands, and argue that they be united under a common vision of Inquiry-Based Mathematics Education (IBME). Central to this common vision are four pillars of IBME: student engagement in meaningful mathematics, student collaboration for sensemaking, instructor inquiry into student thinking, and equitable instructional practice to include all in rigorous mathematical learning and mathematical identity-building. We conclude by calling for a four-pronged research agenda focused on learning trajectories, transferable skills, equity, and an educational systems approach.
2019
Oklahoma City, Oklahoma
The Pathways to College Algebra curriculum aims to build concepts that cohere with the big ideas in Calculus, and initial results suggest improved readiness for Calculus by students who have taken a Pathways class. However, less is known about how Pathways might influence students’ initial understanding and reasoning about calculus concepts. Our study examines similarities and differences in how Pathways and non-Pathways students initially understand and reason about the calculus concept of the limit. Our findings suggest that Pathways students may engage a little more in quantitative reasoning and in higher covariational reasoning, and have more correct and consistent initial understandings. Further, the Pathways students were explicitly aware of how their Pathways class may have benefited their understanding of limits.
2019
Oklahoma City, Oklahoma
Undergraduate mathematics instruction contributes to marginalization among women and racially minoritized individuals’ experiences. This report presents an analysis from a larger study that details variation in minoritized students’ perceptions of potentially marginalizing events in undergraduate mathematics instruction. With past research on undergraduate mathematics experiences largely focused on students’ post-hoc reflections and one or two race-gender intersections, this analysis extends prior work by attending to variation in students’ in-the-moment perceptions of mathematics instruction across various race-gender intersections. Findings highlight how issues of underrepresentation, stereotypes, and instructor care contributed to interpretations of instruction-related events as potentially marginalizing. The report concludes with implications for teaching practices in undergraduate mathematics that academically support and socially affirm students from historically marginalized backgrounds.
2019
Oklahoma City, Oklahoma
Students often struggle with issues of order – that is, with distinguishing between permutations and combinations – when solving counting problems. There is a need to explore potential interventions to help students conceptually understand whether “order matters” and to differentiate meaningfully between these operations. In this paper, I investigate students’ understanding of the issue of order in the context of Python computer programming. I show that some of the program commands seemed to reinforce important conceptual understandings of permutations and combinations and issues of order. I suggest that this is one example of a way in which a computational setting may facilitate mathematical learning.
2019
Oklahoma City, Oklahoma
We explore the notion of density of the set of rational numbers in the set of real numbers, as interpreted by undergraduate mathematics students. Participants’ responses to a scripting task, in which characters argue about the existence of one or infinitely many rational numbers in a real number interval, comprise the data for our study. The framework of reducing abstraction is used in explaining the participants’ mathematical behavior when coping with the task. The analysis reveals informal ideas related to density as well as unconventional understandings of density-related concepts of rational numbers and infinity
2019
Oklahoma City, Oklahoma
In this study we focus on example spaces for the concept of a function provided by prospective secondary school teachers in an undergraduate program. This is examined via responses to a scripting task – a task in which participants are presented with the beginning of a dialogue between a teacher and students, and are asked to write a script in which this dialogue is extended. The examples for functions fulfilling certain constraints provide a lens for examining the participants’ concept images of a function and the associated range of permissible change. The analysis extends previous research findings by providing refinement of students’ ideas related to functions and the concept of the function domain.
2019
Oklahoma City, Oklahoma
The symbol “𝑑𝑥” is one example of a differential, which is a calculus symbol that is found in a variety of settings and expressions. We wanted to explore how expert mathematicians think about differentials in some of these settings and expressions, in order to see what levels of consistency might appear among their views. To that end, we created an interview protocol that contained differentials in the contexts of derivatives, definite and indefinite integrals, and separable differential equations, interviewed seven mathematicians, and analyzed their responses using a form of thematic analysis. Overall, we found no instances of total agreement among all subjects, but did find several common and recurring themes, including some that were unexpected and not found in our previous studies.
2019
Oklahoma City, Oklahoma
Functions play a fundamental role both in abstract algebra and earlier courses in the mathematics curriculum. Yet little attention has been paid to how students’ understanding of function (informed by their prior experiences) supports or constrains their activity when dealing with functions in abstract algebra. In this study, we report on six abstract algebra students’ understanding of function, their function-activity in abstract algebra tasks, and the degree to which their understanding of function from prior experiences is connected to their understanding in this new setting. We conclude with two cases contrasting the activity of two students with divergent levels of connection between their function understanding and the abstract algebra setting. In general, we found that function served an important role in students’ activity and provides implications for instruction and future research.
2019
Oklahoma City, Oklahoma
Examining teaching practices in advanced mathematics is a relatively new field of scholarship despite a long history in K-12 settings. One important research area in this setting is documenting teacher noticing of students’ mathematical thinking. In this report, we extend this line of work to explore how undergraduate mathematics instructors attend to, interpret, and respond to student thinking (Jacob, Lamb, & Philipp, 2010) in abstract algebra. We surveyed 25 abstract algebra instructors with a range of experience. Overall, we found that our participants focused on student thinking to a greater degree than the elementary teachers in earlier studies. Further, their interpretations spanned two distinct foci: understanding of concepts and the formal representation system. Their proposed responses then reflected a wide span of teaching actions. This exploratory analysis unveiled a number of previously undocumented characteristics of instructor noticing at the undergraduate level which can serve to inform future research on teaching practices.
2019
Oklahoma City, Oklahoma
In an effort to improve teaching and learning in undergraduate mathematics courses and help graduate students learn how to teach, many departments across the United States have begun coordinating courses. Although coordination may provide structure and remove some variability in the classroom, there are still many decisions made in the classroom that cannot be coordinated. The purpose of this study was to examine the “uncoordinated” decisions that graduate student instructors made when enacting examples in the classroom. To examine this phenomenon, I studied the cognitive demand of the examples that graduate student instructors chose to enact and the roles that they took on while enacting high cognitive demand examples. As a result, I found that less than 27% of the examples that I observed were enacted at a high level of cognitive demand and that there were three roles (modeling, facilitating, and monitoring) that instructors took on while enacting examples.
2019
Oklahoma City, Oklahoma
Research has documented the power of play to affect learning at all ages. This research shares the kinds of mathematical student thinking elicited by incorporating the board game Catan into an inquiry-based classroom. The class was composed of 25 students, not majoring in STEM fields, who were enrolled in a freshman seminar course intended to provide an opportunity to engage in research-like inquiry. Students had played Catan in class and to engaged inquiry-oriented instruction sessions focusing on the relationships between mathematics and Catan. Student work is provided for arguments related to the value of each resource and selecting locations for initial settlements, with connections between this work and topics traditionally taught in probability classes
2019
Oklahoma City, Oklahoma
Attending to and leveraging student thinking is known to be an effective teaching practice, but little research has been done to investigate the ways in which mathematics tutors attend to student thinking. This study will use the construct of decentering and Ader & Carlson’s (2018) framework for analyzing teacher-student interactions to describe the ways in which tutors attend to student thinking in the moment. We will also provide examples of how written reflections and stimulated recall interviews can contribute to a tutor’s ability to attend to student thinking.
2019
Oklahoma City, Oklahoma
The learning of mathematics is a complex phenomenon that is influenced by both cognitive and affective factors. Little is known about the relationship between students’ affective and cognitive outcomes, as much work focuses on one or the other, but not the intersection of the two. Therefore, this study examines the relationship between students’ affective learning gains as reported on the SALG survey and their content assessment scores for differential equations courses. The goal was to determine if there was a relationship and then investigate if this relationship held for male and female students, as well as those in inquiry-oriented classes. Mixed linear models were used to examine this relationship, while simultaneously taking into account the nesting of students within instructors. Results showed there are some significant relationships between affective learning gains and content assessment scores, but these relationships are not consistent across sub-groups by gender nor instruction type.
2019
Oklahoma City, Oklahoma
In the literature on metaphor, researchers have pointed out the importance of metaphor as a tool for sense-making and have demonstrated the impact of metaphor use on cognition. In mathematics in particular, metaphor has been shown to be a valuable tool for making sense of and reasoning with mathematics. To our knowledge, there has been no research on the metaphors that professors use when communicating the nature of mathematical practice to students in advanced mathematics lectures. In this paper, we present a particular metaphor, Learning Mathematics is a Journey, that we found in a corpus of 11 advanced mathematics lectures. We describe this metaphor we found and offer some speculative analysis regarding the implications of this metaphor.
2019
Oklahoma City, Oklahoma
This paper addresses the characteristics of first time Calculus I TAs’ teaching practice of the derivative and reflection on their teaching using video-stimulated recall. Our analysis using three views of function – correspondence, variation, and covariation – shows that regardless of representations that TAs adopted, their discussion mainly addressed correspondence, and most TAs used correspondence to justify the difference quotient (DQ) in the limit definition as a function, and transition from the derivative at a point and the derivative as a function. TAs also emphasized different uses of letters as an input of the derivative in such transition from students’ point of view although they as mathematicians did not see the difference. Some TAs addressed the variational or covariational view in class and/or during reflections but in a limited way by simply acknowledging that quantities “change” without describing how they change.
2019
Oklahoma City, Oklahoma
We study how the mathematical beliefs and knowledge of peer mentors in a summer mathematics program influenced their efforts to help high school students learn to write proofs in number theory. Using Schoenfeld’s framework for understanding decision making, we analyze interviews of three undergraduate student mentors for evidence of how their views of the role of proof, norms for proof writing, and mathematical knowledge for teaching informed their pedagogical decisions. We find that each mentor developed a distinctive approach to providing feedback on student work consistent with their own values, and present evidence that the success of each approach depended on the mentor’s resources for interpreting student work.
2019
Oklahoma City, Oklahoma
Real analysis is an important course for both undergraduate and graduate students. Researching the ways students reason about challenging and abstract concepts can inform and improve instruction in real analysis. In this report, I examine two undergraduate students’ reinvention of a general metric function. To facilitate this reinvention, I conducted a 15-hour teaching experiment with undergraduate mathematics students that had completed the introductory sequence in real analysis. In this experiment, the students generalized their initial understandings of distance measurements on R to construct increasingly abstract measures of distance in various metric spaces, including sequence and function spaces. Their generalizing activity culminated in construction of a general metric function through reflected abstraction of operations relevant to distance measurement carried out in previous metric spaces. I explore the students’ generalizing activity, as well as the abstractions that supported their generalizing1.
2019
Oklahoma City, Oklahoma
The aim of this research was to investigate the nature of difficulties with algebra in calculus problems from the perspective of students. We employed Skemp’s (1979) theory to analyze two calculus students’ difficulties with algebra in an interview setting. Our findings indicate that although these students were aware of their challenges with algebra, they struggled to resolve those issues in the context of calculus. Likewise, both seem to struggle in different ways with algebra outside the context of calculus. Implications for teaching based on our current research will be provided.
2019
Oklahoma City, Oklahoma
Although different instructional models for teaching mathematics have arisen over the past decades, lecturing continues to be the preferred approach of abstract algebra instructors. We identified facilitating and constraining factors of instructional change by analyzing thirteen instructor interviews. Factors were further classified as internal or external; as related to factors of community, sources, curriculum, procedures, empowerment and feelings; and as institutional, networking or change management. Additionally, different levels of resistance or support were identified for each factor. Some results of our analyses include finding that supportive faculty chairs and colleagues strongly facilitate attempts at instructional change while departments open to change serve as a moderately supportive external factor towards instructional change. Student resistance constitutes the most frequent constraining factor that instructors face.
2019
Oklahoma City, Oklahoma
This study uses thematic analysis to examine the conceptual metaphors used by two abstract algebra teachers to describe the concepts of isomorphism and homomorphism, both in interviews outside instruction and during class. These metaphors are compared to the metaphors used by their students to describe these concepts. While the two instructors utilized similar metaphors for isomorphism, they did not share metaphors for homomorphism. Further, when looking from interviews to instruction, there was again more alignment with isomorphism than with homomorphism, with metaphors used to discuss homomorphism during the interviews being less present during instruction than those used to discuss isomorphism. The students in these two classes appeared to incorporate the instructors’ metaphors to varying degrees.
2019
Oklahoma City, Oklahoma
The transition to proof is difficult for students – what developments do students show while learning how to prove? I present a short-term longitudinal, qualitative analysis of 11 undergraduates taking a transition to proof course, of the developments seen in their proving. Within this, I follow one student whose proof reasoning grew but whose performance on the proof construction tasks declined. Investigating this single-subject case serves as an example of the complicated interplay between development and performance. It serves as a reminder for how attending to performance does not account for students’ thinking and vice versa.
2019
Oklahoma City, Oklahoma
Inquiry-Oriented Instruction (IOI) holds promise for providing equitable learning opportunities for men and women. We consider two abstract algebra instructors whose women exhibited different learning outcomes. We explore how this disparity in the women’s achievement might be related to differences in these instructors’ implementation of IOI, specifically regarding how they elicited and evaluated student contributions. We used Reinholz and Shah’s (2018) EQUIP observation tool to investigate differences between students’ participation opportunities in each class. We found a significant difference in the instructors’ number of interactions with men and women during class discussion. We also found significant differences in the instructors’ methods for soliciting and responding to student contributions in class discussion. A discussion of these differences in instructional practices, as well as implications for future work, is provided.
2019
Oklahoma City, Oklahoma
Research on student understanding of eigentheory in linear algebra has expanded recently, yet few studies address student understanding of the Characteristic Equation (CE). In this study, we explore students’ conceptual and procedural knowledge of deriving and using the CE. Consulting Star’s (2005) characterization of deep and superficial conceptual and procedural knowledge, we developed the Conceptual and Procedural Knowledge framework for classifying the quality of students’ conceptual and procedural knowledge of both deriving and using the CE along a continuum. Most of our students exhibited deeper conceptual and procedural knowledge of using the CE than of deriving the CE. Furthermore, most students demonstrated deeper procedural knowledge than conceptual knowledge of deriving the CE. Examples of student work are provided, and implications for instruction and future research are discussed.
2019
Oklahoma City, Oklahoma
In this paper, we share findings around four prospective secondary mathematics teachers’ attention to varying curricula while planning an algebra lesson. We specifically address how their attention interacted with their interpretations of and responses to the curriculum materials via idea sequences, and further we study how the format of the curriculum materials plays a role in influencing these interactions. We discuss the result that sequences of interpretations and responses are always initiated by attention for PSTs, which itself is influenced by curriculum elements and format. We end with a discussion of implications around the need for curriculum use practices in teacher education and professional development.
2019
Oklahoma City, Oklahoma
In this study of mathematics teaching, we explore how to measure inquiry-oriented practices of college mathematics instructors. We offer a conceptualization of inquiry-oriented instruction organized by the instructional triangle (Cohen, Raudenbush, & Ball, 2003) and introduce an instrument developed to explore the extent to which elements of inquiry-oriented instruction are present in the teaching of university mathematics courses. This scale has been developed to explore what practices instructors currently use and eventually investigate the relationship between beliefs and practice. We show how we have operationalized inquiry-based instruction as self-report items and report preliminary findings that indicate our scales are performing well. We show that some inquiry-oriented practices are significantly more present in upper-division courses than lower-division courses. This suggests that at least some components of inquiry-oriented instruction are not reducible to individual differences (whether the instructor is an inquiry-based instructor), but also dependent in the context of instruction.
2019
Oklahoma City, Oklahoma
Mathematics faculty spend considerable time scoring and providing feedback on student-generated proofs. Yet there is very little research on the feedback professors provide on proofs during the grading process. In this paper, we discuss a coding scheme developed for categorizing the feedback professors write on student-generated proofs in abstract algebra and real analysis. We then explore the types of annotations that professors make on student proof attempts. The results show that professors generously use annotations (like checkmarks) as informal grading tools or to signify things they have read when grading, most feedback focuses on a particular part of the proof that is no more than a few lines, and the majority of feedback does not convey why the feedback was given.
2019
Oklahoma City, Oklahoma
Using Thompson and Carlson’s (2017) definition of a variable and the results of teaching sessions with two preservice secondary mathematics students, I describe the role of quantitative and covariational reasoning in constructing a formula with variables to describe a relationship between covarying quantities in a dynamic geometric context—the Parallelogram Problem. I report that although each student reasoned with a dynamic situation, their symbolic representations of that situation did not necessarily entail variables. I conclude that providing students with dynamic situations with which to construct formulas provides them opportunities to construct formulas with variables representing covariational relationships between quantities.
2019
Oklahoma City, Oklahoma
In this paper, we analyzed a mathematician’s daily teaching journals of a 5-day series of teaching episodes on eigenvalues and eigenvectors in a first-year linear algebra course. We employed Tall’s (2013) three world model, in conjunction with Tall and Vinner’s (1981) concept images and concept definitions, to follow the mathematician and instructor’s movements between Tall’s worlds. The study showed that the instructor strived to build concept images that, while perhaps mirroring his own concept images, did not resonate with the students.
2019
Oklahoma City, Oklahoma
In this study, we describe how a funded professional development program for collegiate mathematics faculty impacted their teaching philosophy. As a result of this program, participants felt an obligation to attend to students’ needs as part of student-centered learning, found rich tasks useful to connect students’ prior knowledge with new content, recognized the value of creating a sense of community in the classroom, and established a community among themselves. This newfound community was especially valuable for adjunct faculty. The participants expressed how they shifted and aligned their teaching beliefs with teaching practices, appreciated evidence-based teaching techniques, taught with intention, and realized how both they and their students had more fun in the classroom. Furthermore, the participants came to comprehend how these practical and philosophical transformations fostered equitable teaching practices in the mathematics classroom. The structure of this program may serve as a model for future professional development programs.
2019
Oklahoma City, Oklahoma
In this paper, we illustrate and discuss two undergraduate students’ reasoning about quantities’ magnitudes. One student identified regularities regarding the relationship between two quantities by focusing on successive amounts of change of one quantity (i.e., a pattern) while the other attended to relative amounts of changes in both quantities (i.e., a relationship). We illustrate that although reasoning about amounts of change is useful for making sense of the rate of change in quantities, reasoning about relative changes in identifying a relationship between quantities’ magnitudes is likely more productive in developing the concept of rate of change
2019
Oklahoma City, Oklahoma
In this report, I examine the use of intuition by a mathematically experienced adult on the autism spectrum given a paradoxical mathematical problem involving infinity. I compare both his level of use of intuition and the importance he places on it against results from students in the general population. Interview results combined with previous data suggest that students on the autism spectrum are less likely to use approaches based in intuition, place less importance on intuitive ideas compared to other explanations, and may also have different views of the nature of intuition. Analysis of possible reasons for showing these differences and implications for teaching and further autism-related research are presented.
2019
Oklahoma City, Oklahoma
The undergraduate preparation of pre-service teachers requires attention to the factors that enable and constrain their application of mathematical knowledge to positive effect in the classroom. In this paper, we examine how the instructional decisions of three in-service secondary mathematics teachers were influenced by individually consistent patterns of such mediating factors. Acknowledging that such factors might be content-dependent, we focus on teachers’ instruction of exponential functions, a topic foundational to both secondary and collegiate mathematics. We developed models of each teacher’s implicit learning theory, professional identity, values and goals for students’ learning, and beliefs about the nature of mathematics and about what constitutes genuine mathematical engagement. We illustrate these results by summarizing our analyses of a selection of mediating factors for each teacher. We conclude with a discussion of the implications of our findings for the preparation of pre-service mathematics teachers at the undergraduate level.
2019
Oklahoma City, Oklahoma
The purpose of this study is to investigate multivariable calculus students’ communication of vectors by examining how their responses on a mindmap assignment change over time. A mindmap is a visual network of connected and related concepts often with one image or topic centrally located. Through this open-ended instrument, we conduct a qualitative analysis to explore the connections students make between different aspects and multiple representations of vectors.
2019
Oklahoma City, Oklahoma
Calculus is often an essential milestone during a students’ time in college and can be especially impactful for students wishing to major in in a math or science field. Given its relative importance, the ways in which calculus courses are delivered can have a lasting impact on a student’s trajectory and relationship with mathematics. In this study we document the ways in which three calculus course variations at the same University operate to promote different mathematics identities for students. Drawing on the Holland et. al.’s (1998) framework of figured worlds we showcase the ways in which these course variations act as if they are different calculus worlds that constitute socially organized and produced realms of being. We highlight the ways in which these figured worlds position or fail to position students with the opportunity to refigure themselves and others as learners and doers of mathematics.
2019
Oklahoma City, Oklahoma
Previous research has illuminated and defined meanings and understandings that students demonstrate when reasoning about graphical images. This study used verbal and physical cues to classify students’ reasoning as either static or emergent thinking. Eye-tracking software provided further insight into precisely what students were attending to when reasoning about these graphical images. Eye-tracking results, such as eye movements, switches between depictions of relevant quantities, and total time spent on attending to attributes of the graph depicting quantities, were used to uncover patterns that emerged within groups of students that exhibited similar in-the-moment meanings and understandings. Results indicate that eye-tracking data supports previously defined verbal and physical indicators of students’ ways of reasoning, and can document a change in attention for participants whose ways of reasoning over the course of a task change.
2019
Oklahoma City, Oklahoma
Eigentheory is an important concept for modeling quantum mechanical systems. The focus of the research presented is physics students’ reasoning about eigenvectors and eigenvalues as they transition from linear algebra into quantum mechanics. Interviews were conducted at the beginning of the semester with 17 students at two different universities’ during the first week of a quantum mechanics course. Interview responses were analyzed using a Resources (Hammer, 2000) framework, which allowed us to characterize nuances in how students understand various aspects of an eigentheory problem. We share three subthemes of results to illustrate this: interpreting the equations graphically, interpreting the equals sign, and determining solutions.
2019
Oklahoma City, Oklahoma
This study investigates instructor perceptions of their teaching, as well as their students’ learning, obstacles encountered, and methods of implementation from the use of Primary Source Projects (PSPs). PSPs are curricular modules designed to teach core mathematical topics from primary historical sources rather than from standard textbooks. In essence, they are a form ofinquiry-based-learning that incorporates the history of mathematics through original sources. We provide an overview of results from two semesters of implementation reports and surveys administered at the beginning and end of the semester by instructors who implemented PSPs in their undergraduate mathematics class.
2019
Oklahoma City, Oklahoma
We utilize two analyses to confirm a multidimensional framework for analyzing contributions to classroom discourse, previous analysis using the framework and analysis of instances of sociomathematical norm negotiation juxtaposed with it (Cobb & Yackel, 1996). The framework considers social, epistemic, and argumentative activities exhibited in talk-turns during whole class discussion. In this study we show that collective and individual development occurred in an inquiry-oriented differential equations course and discuss patterns in ways learning partners participated in whole class discussions during sociomathematical norm negotiation.
2019
Oklahoma City, Oklahoma
Several studies have shown that student-centered instruction can help improve student success and persistence in STEM-related fields (e.g., Freeman et al., 2014). Despite this, institutional change can be difficult to enact. Accordingly, it is important to understand how departments both initiate and sustain meaningful change. For this paper we use interview data collected in Spring 2017 to examine how institutional and departmental factors affected reform efforts at two different institutions. In particular, we compare how two universities’ leadership and commitment to educational innovation contribute to the initiation, implementation, and sustainability of active learning in the undergraduate calculus sequence (Precalculus through Calculus 2).
2019
Oklahoma City, Oklahoma
In this study we use latent class analysis, distractor analysis, and qualitative analysis of cognitive interviews of student responses to questions on an algebra concept inventory, in order to generate theories about how students’ selections of specific answer choices may reflect different stages or types of algebraic conceptual understanding. Our analysis reveals three groups of students in elementary algebra courses, which we label as “mostly random guessing”, “some procedural fluency with key misconceptions”, and “procedural fluency with emergent conceptual understanding”. Student responses also revealed high rates of misconceptions that stem from misuse or misunderstanding of procedures, and whose prevalence often correlates with higher levels of procedural fluency.
2019
Oklahoma City, Oklahoma
Recent research in mathematics education has uncovered a host of teaching behaviors that are commonly enacted by instructors of advanced mathematics courses. While these descriptive accounts of math teaching are useful, little investigation has been conducted into the reasons for why these practices are so prevalent. In this study, we interviewed seven mathematicians about regularities that have been observed in the literature on the teaching of advanced mathematics. In this report we discuss whether mathematicians view these findings as accurate (they often did), whether they thought these regularities were productive or problematic teaching practices, and why mathematicians engaged in these teaching practices. We discuss how these themes may elucidate the practices of instructors, and later propose implications of the methods of the present study for changing how advanced math courses are taught.
2019
Oklahoma City, Oklahoma
In this study, two universities created and implemented a student-centered graduate student instructor observation protocol (GSIOP) and a post-observational Red-Yellow-Green feedback structure (RYG feedback). The GSIOP and RYG feedback was used with novice graduate student instructors (GSIs) by experienced GSIs through a peer-mentorship program. Ten trained mentor GSIs completed 50 sets of three observations of novice GSIs. Analyzing 151 GSIOPs and 151 RYG feedback meetings longitudinally provided insight to identify what types of feedback informed and influenced GSIOP scores. After qualitatively coding feedback along multiple dimensions, we found certain forms of feedback were more influential for GSI development than others with respect to change in GSIOP score. Our results indicate contextually-specific feedback leads to more observed changes and improvement across multiple observations than decontextualized feedback.
2019
Oklahoma City, Oklahoma
This paper focuses on how a teacher’s image of student thinking influences the meanings she conveyed to students. I observed a calculus teacher’s lessons and interviewed the teacher and her students. By exploring the data, I see the extent to which the teacher attention to student thinking has an impact on (1) the ways she expressed her meanings during instruction, (2) the ways she interpreted students’ understandings that they expressed, (3) the ways she decided to adjust instructional actions. My analyses suggest that teachers need to think about how students might understand their instructional actions so that they can better convey what they intend to their students.
2019
Oklahoma City, Oklahoma
Inquiry-oriented instruction (IOI) has been shown to increase students’ cognitive outcomes, but the relationship between students’ cognitive and affective outcomes in IOI remains unclear. Furthermore, students’ perceptions of the usefulness of and their success in an academic setting are related to their engagement and achievement. The present mixed methods study seeks to understand students’ perceptions of IOI and whether those perceptions are related to their achievement. Results of qualitative analysis reveal four themes related to developing conceptual understanding, connecting ideas, feelings of helplessness, and a preference for traditional instruction. Results of a mixed linear model show positive or neutral perceptions of IOI are related to higher achievement. The relationship between these results is discussed, and is framed in motivational theory
2019
Oklahoma City, Oklahoma
Undergraduate math tutoring is an important venue for student learning, yet little empirical work has been done to study tutoring interactions and few theories specifically address tutoring interactions. Drawing upon literature from problem solving, peer learning, and mathematics teaching, this report proposes a schema for Mathematical Knowledge for Tutors (MKTu). The proposed framework expands Ball’s (2008) Mathematical Knowledge for Teaching by adding dimensions of affect and self-regulation. This additional depth reflects the individualism, immediacy, and interactivity which are unique to the tutoring setting where problem solving and mentoring take place between an advanced undergraduate tutor and an undergraduate student.
2019
Oklahoma City, Oklahoma
At the 21st Annual Conference on Research in Undergraduate Mathematics Education, Ed Dubinsky highlighted the disparity between what the research community knows and what is actually used by practicing instructors. One of the heaviest burdens on instructors is the continual assessment of student understanding as it develops. This theoretical paper proposes to address this practical issue by describing how to dynamically construct multiple-choice items that assess student knowledge as it progresses throughout a course. By utilizing Automated Item Generation in conjunction with already-published results or any theoretical foundation that describes how students may develop understanding of a concept, the research community can develop and disseminate theoretically grounded and easy-to-use assessments that can track student understanding over the course of a semester.
2019
Oklahoma City, Oklahoma
The use of the Cartesian Coordinate system (CCS) pervades secondary and tertiary mathematics curriculum, as the dominant convention for displaying graphs of functions. The CCS in two dimensions may be framed as a conceptual blend of two number lines and a Euclidean plane (Lakoff & Núñez, 2000). Within the concept of a number line is a conceptual metaphor uniting numerical values with points on a line. While such a description of the CCS may describe a shared understanding of the convention among the mathematics community, it may not account for the ways in which individual students interpret graphs presented in the CCS. Other theories, such as David et al.’s (2017) constructs of value-thinking and location-thinking, have been proposed to account for students’ graphical interpretations. In this paper, I outline these two ways of framing conceptions of graphs, the uses of each framework, and their relation to each other.
2019
Oklahoma City, Oklahoma
This theoretical paper introduces a field-base hypothesis, according to which the intensity and type of an intellectual need that students can experience for checking their solution to a problem might be related to the epistemological status of methods that they employed for solving the problem. The hypothesis emerged from the analysis of a final exam in a first-year course where 421 students worked on four problems in linear algebra. In one of them, 33 students provided evidence of checking their solutions, all of which appeared as educated guesses. No written evidence of checks was indicated in the deductive solutions, in which the students utilized algorithms, procedures, and theorems that were introduced to them in the course. Thus, it might be proposed that problem-solving methods with a low epistemological status (e.g., educated guesses) may instigate the need for checking a solution as a means to compensate for their status.
2019
Oklahoma City, Oklahoma
This theoretical report addresses the challenge and promise of improving prospective secondary mathematics teachers’ experiences in undergraduate mathematics courses through tasks embedded in pedagogical contexts. The objective of this approach, used by multiple nationally-funded projects, is to enhance the development of teachers’ MKT. We report on the construction of a framework for observing and analyzing the development of teachers’ MKT. This framework is the result of integrating several existing frameworks and analyzing a sample of prospective secondary teachers’ responses to tasks embedded in pedagogical contexts. We discuss the methods used to build this framework, the strengths and weaknesses of the framework, and the potential of the framework for informing future work in curriculum design and implementation.
2019
Oklahoma City, Oklahoma
In the mathematics education literature on proof, there is a longstanding conversation about proofs that only convince versus proofs that explain. In this theoretical report, we aim to extend both of those ideas by exploring proofs in the domain of combinatorics. As an example of an affordance of the combinatorial setting, we explore proofs of binomial identities, which offer novel insights into current distinctions and ideas in the literature about the nature of proof. We demonstrate examples of proofs that can be explanatory or convincing (or both), depending on how a person understands the claim being made (which we refer to as their preferred semantic representation system). We conclude with points of discussion and potential implications.
2019
Oklahoma City, Oklahoma
While there has been research on students’ understanding of the meaning of the equals sign, there has yet to be a thorough discussion in math education on a strong meaning of the equals sign. This paper discusses the philosophical and logical literature on the identity relation and reviews the math education research community’s attempt to characterize a productive meaning for the equals sign.
2019
Oklahoma City, Oklahoma
This paper discusses the conceptual basis for differentiating an equation, an essential aspect of implicit differentiation. We explain that implicit differentiation is more than merely the procedure of differentiating an equation and carefully provide a conceptual analysis of what is entailed in understanding the legitimacy of this procedure. This conceptual analysis provides a basis for discussion of the literature, as well as empirical justification for the importance of this topic.
2019
Oklahoma City, Oklahoma
Evidence to date that active, student-centered learning in mathematics classrooms contributes to desired student outcomes has now accumulated to compelling levels. However, promoting and supporting widespread use of alternative practices is challenging, even amongst practitioners open to such changes. One contributing factor is the fact that a majority of instructional change efforts focus on only a small portion of the instructional system, while true transformation requires systemic reform. Successful institutional change initiatives have been shown to involve common features: they involve ongoing interventions, align with individuals’ beliefs, and work within the existing landscapes of institutional values. Here we propose a theory to support instructional change in undergraduate mathematics by adding a new dimension – instructor peer observation– to an existing model for institutional change (the CACAO model), thereby aligning with evidence regarding what promotes effective change. An exemplar is given to illustrate how this theory might be realized in practice.
2019
Oklahoma City, Oklahoma
A disciplinary literacy perspective suggests that the goal of instruction in any discipline is to apprentice students into increasing participation in the disciplinary community. In this paper we explore four distinct types of mathematical texts and the critical role each plays in mathematical meaning-making. We argue that understanding the nature and uses of mathematical text types moves undergraduate students closer to the goal of approximating/engaging in mathematical practices, resulting in greater access to powerful mathematics.
2019
Oklahoma City, Oklahoma
Prospective secondary mathematics teachers frequently take as many (or more) mathematics courses from a mathematics department as they do methods courses from an education department. Sadly, however, prospective secondary teachers frequently view their mathematical experiences in such courses as unrelated to their future teaching (e.g., Zazkis & Leikin, 2010). Yet there is some optimism that having instructors alter their instructional approaches in such mathematics courses can enhance such experiences to be a positive part of their preparation for teaching. This theoretical report elaborates on four points of connection to secondary teaching that can be made in undergraduate mathematics courses, illustrated via examples from abstract algebra, and organized along a spectrum of intended implications on secondary teaching. The purpose is to provide a theoretical bridging between instructional approaches in undergraduate mathematics and aspects of secondary teaching practice.
2019
Oklahoma City, Oklahoma
Self-explanation is a reading strategy in which readers explain a text to themselves as they encounter new information. Hodds, Alcock, and Inglis (2014) reported proof comprehension gains on students who had been trained to self-explain, when compared to students who had not received this training. We report a multiple case study in which we interviewed undergraduate students in introductory and advanced proof-based courses, to examine their understanding of self-explanation training and their use of this strategy throughout one semester. Preliminary findings indicate that self-explanation made students examine each line of the proof more deliberately, because they knew they would have to hold themselves accountable for figuring out how to explain each line of the proof. However, some students reported almost never using the technique, either because they prioritized the proof techniques demonstrated by their professors, or because they only felt the need to do so with particularly difficult proofs
2019
Oklahoma City, Oklahoma
In this report we present findings from a preliminary investigation aimed at describing models of course coordination systems currently in place within university precalculus and single variable calculus courses. Hierarchical cluster analysis was used on national survey data to identify homogeneous clusters of courses based on the intended use of uniform course elements across sections. The analysis revealed eleven clusters of courses, nested within five larger groups. We briefly describe each of the eleven clusters in terms of the uniform course elements and the five larger groups in terms of the clusters nested within them. We then characterize these groups with respect to department type (Masters- versus PhD-granting), course level (Precalculus, Calculus 1, and Calculus 2), regularity of instructor meetings, and type of course coordinator.
2019
Oklahoma City, Oklahoma
The concept of rate constants is important for developing a deep understanding of chemical kinetics, an area of chemistry that models the rates of reactions. Reaction rates are modeled mathematically, typically using an equation called a “rate law”. One of the terms in this equation, the rate constant, embodies important variables that affect rate, such as temperature-dependence, Our primary research focus in this work is investigating the question: How do students reason about rate constants in chemical kinetics? Preliminary analysis reveals that students often conflate ideas from chemical kinetics and equilibrium, such as rate constants and equilibrium constants. Furthermore, students demonstrated varying levels of sophistication regarding the distinction and relationship between rate and rate constants. Finally, students conveyed different ideas about the mathematical nature of the rate constant quantity.
2019
Oklahoma City, Oklahoma
Non-Euclidean geometries are commonly used in college geometry courses to highlight aspects of Euclidean geometry. Scholars have theorized that working in non-Euclidean geometries requires thinking at the highest van Hiele level of geometric thinking, which was developed by investigating students’ learning of Euclidean geometry, but few have pursued this empirically. This empirical study seeks to develop levels of geometric thinking for students in Taxicab geometry, which is the non-Euclidean geometry that is closest in structure to Euclidean geometry. Students in a college geometry course that included prospective secondary teachers were audio-recorded in group discussions as they completed tasks about congruence and transformations in taxicab geometry, and their written work was collected. Portraits of participants’ thinking about Taxicab geometry were developed, leading to a proposed structure for the levels of geometric thinking for Taxicab geometry.
2019
Oklahoma City, Oklahoma
In this presentation I will explore the various ways in which a pair of students used representations of the group concept to explore the structure of symmetry groups. During a series of teaching experiments, a pair of mathematics education graduate students were asked to develop an algorithm for classifying chemically important point groups beginning with an investigation of a few ball and stick models of molecules. The progress the students made through the use of each symbolization of the group concept is framed with the Realistic Mathematics Education design heuristic of emergent models.
2019
Oklahoma City, Oklahoma
The construct of mathematical knowledge for teaching (MKT) has transformed research and practice regarding the mathematical preparation of future teachers. However, the measures used to assess MKT are largely written tasks, which may fail to adequately represent the nature of content knowledge as it is used in instructional decision making. This preliminary report shares initial findings into one measure of MKT in practice – mathematical errors made during planning and enactment of mathematics instruction. We analyzed lesson plans and classroom video from prospective secondary mathematics teachers (PSTs)’ supervised field experiences in college algebra course. We found that there tended be more errors related to understanding of functions (especially logarithmic), but relatively few errors happened overall during instruction. Of the errors made during planning, the majority of these errors were issues of mathematical precision. Implications for the mathematical preparation of secondary PSTs, as well as research on MKT in practice, are discussed.
2019
Oklahoma City, Oklahoma
In this brief research report, we address the recent calls to improve undergraduate mathematics instruction through our investigation of an instructor’s teaching of derivative in a Calculus course. Considering his efforts to modify the presentation of derivative in the textbook as attempts to improve his teaching as a result of his engagement with primary historical sources, we analyze his teaching to identify the changes in his practice by using Speer, Smith, and Horvath’s (2010) framework for “teaching practice.” With our analysis of instructor interviews and video-recordings of classroom sessions, we observe that Leonhard Euler’s use of differentials in defining derivative had responded to his pedagogical concerns, and had convincing power as a method, which, in turn, led him to make significant changes in how he selects and sequences content for his teaching.
2019
Oklahoma City, Oklahoma
Business majors represent a significant proportion of the population of students enrolled in calculus at the college level. However, there is a lack in research literature that tackles the teaching and learning of business applications at this level. This pilot study represents the beginning phases of a project that aims to investigate business students’ reasoning through tasks pertaining to marginal analysis (derivatives in a business context), accumulation functions and Riemann sums. A preliminary analysis of interviews with two pairs of students is presented, with an emphasis on their thought process while answering questions related to cost, revenue and profit functions as well as their marginal counterparts. The context-based activities were designed with a realistic mathematics education perspective, motivated by guided reinvention.
2019
Oklahoma City, Oklahoma
We presented introduction to proof students from five different US universities with multiply quantified statements to assess and interpret. The survey was designed to allow us to compare the influence of syntax, semantics, and pragmatics in student interpretation. We analyzed the ways students interpreted the statements both before and after instruction. Current analysis suggests that students became more sensitive to syntax (reversing quantifier order) after instruction and became better able to construct a semantically odd construal (e.g. the distance between two points is equal to multiple numbers). Our analysis of pragmatics suggests that students were more likely before instruction to construct a relevant construal, but we did not find evidence that truth-value influenced students’ interpretation of the given claims.
2019
Oklahoma City, Oklahoma
I report on a pilot study for an explanatory multi-method (Creswell & Plano Clark, 2011) research project that examines how undergraduate students in mathematics courses make use of online resources in order to assist with their studies. A survey of 42 students in a diverse undergraduate institution along with 4 semi-structured follow-up interviews were used to collect preliminary data on how these undergraduates make use of the internet as well as to test the data collection protocol. Initial findings suggest that students make use of online resources (beyond those assigned by the instructor) extensively and to a greater extent than in other subject areas. I also report on which resources are being used by students and find evidence of two distinct ways in which these resources are being employed. Questions will be posed about how an expanded follow-up study can best be of service to the mathematics education research community.
2019
Oklahoma City, Oklahoma
Recently, Adiredja and Andrews-Larson (2017) challenged the field to consider and recognize the political and contextual nature of teaching and learning postsecondary mathematics education including its power dynamics and social discourses. In this preliminary report, we discuss the early stages of a classroom teaching experiment to bridge research and practice by bringing social justice topics into a differential equations course. Our iterative research process consists of using theory that informs our instructional design and theory that informs our classroom analysis. Here we discuss preliminary results from the classroom analysis through Gutiérrez’s (2009, 2013) four dimensions of equity. Preliminary results show that identity and power emerge from student portfolios after engaging in a climate change problem but more work is necessary in our instructional design to draw out those dimensions more explicitly.
2019
Oklahoma City, Oklahoma
Mathematics Graduate Teaching Assistants (MGTAs) are both current and future teachers of college mathematics, but there is limited research investigating their growth as teachers. To create better professional development for training MGTAs, we first need to understand how they learn to teach. This study aims to identify why MGTAs change their teaching practices and what factors influence their development as teachers. Survey, group interview, and individual interview data from seven MGTAs at a doctoral-granting university were analyzed deductively using complexity science as a framework.
2019
Oklahoma City, Oklahoma
This study describes a service learning-based mathematics course for non-math majors at a private liberal arts university in the Midwest. Thirty-six undergraduate students participated in the course and developed lesson plans from the content taught in class. Students then taught the lessons to third graders at a local public elementary school. Undergraduates wrote self-reflections that were collected after the service and analyzed. Data reveal students felt an increase in value and more confident learning mathematical concepts because of its real-world application in the community. We conclude that including a service learning component in teaching mathematics is valuable. Service learning can help students understand mathematics beyond numbers and equations and see its importance in societal reform.
2019
Oklahoma City, Oklahoma
In many countries, concerns have been raised regarding the lack of participation of students in mathematics at the university level due to a dearth of skilled professionals to meet the needs of an increasingly technological, and thus mathematical, world. In this paper, we report on a study in which we are comparing first and final year undergraduate students’ experiences in mathematics departments. We focus on students’ conceptions of the supports and challenges that they experience in mathematics departments, using a multimodal data collection method, photovoice. We will share findings from this ongoing research project focusing on comparisons between first and final year students’ perceptions of their learning environment. The knowledge that will be gained from this research is crucial in understanding students’ lived experiences and thus making suggestions to address university mathematics pipeline issues
2019
Oklahoma City, Oklahoma
In this preliminary report, we present data and preliminary findings on the instruction that follows active learning activities designed to introduce first-semester calculus students to the definite integral. We are particularly interested in the two to three days of class that follow these group work activities to see how instructors leverage the content of the activities to summarize and build the structure of the Riemann sum and definite integral. Video data of five instructors has been collected, and we present preliminary analysis focused on the ways in which one of the instructors introduced the definite integral as a sum of products.
2019
Oklahoma City, Oklahoma
Compelling evidence supports the benefits of active learning environments in undergraduate mathematics. Research shows that such environments can benefit all students, and especially benefit students who have been traditionally underrepresented in mathematics. To move beyond the general idea “inquiry supports equity,” we provide an analysis of two inquiry-oriented classrooms to highlight the ways in which equitable participation may or may not be present, particularly in terms of gender. We found some evidence of equitable participation in one of the classrooms, while the other was dominated by men in the class. These early findings suggest that more research is required to uncover the ways in which inquiry-oriented environments may or may not be equitable.
2019
Oklahoma City, Oklahoma
Six collegiate mathematics instructors, who had all previously won teaching awards, were interviewed about their beliefs on learning. Differences between the beliefs of PhD and non-PhD mathematicians were evident, perhaps connected to the student population with which each worked. Furthermore, the four PhD mathematicians all held very different beliefs about learning and modelled their teaching accordingly. Additionally, each of the four had created at least one teaching analogy for himself (climbing instructor/spark, showman/coach, Sherpa, facilitator) that spoke to the role he saw himself in within the classroom.
2019
Oklahoma City, Oklahoma
We describe a process to characterize the questions asked by instructors and students in community college algebra courses. The goal is to measure the quality of mathematical questions that can speak to the level of student cognitive engagement with mathematics and to connect that quality with student outcomes in the course. As a first step, we explore the relation between frequency of different types of questions and other variables collected in the project. We seek to engage the audience in discussing the affordances and limitations of this work for assessing quality of instruction in connection to students’ performance.
2019
Oklahoma City, Oklahoma
This preliminary report examines students’ interpretations of free variables in linear algebra. In linear algebra, students build understandings of concepts, such as a (in)consistent system of equations, a linearly independent set of vectors, and a subspace. All these concepts will be the foundation for students’ future learning in various fields. Therefore, it is crucial to investigate the notion of free variables as it is one of the constructs underlying work with each of these concepts. Here, I analyzed 110 linear algebra students’ written assessments from three different classes using grounded theory (Strauss & Corbin, 1994). The analysis shows that students use free variables as a conceptual tool to answer questions given in different problem settings. This paper reports categories of students’ interpretations of free variables and explores what the free variables mean to students in each category.
2019
Oklahoma City, Oklahoma
The concept of mathematical maturity is one that, for some, elicits clear meanings and perhaps illustrations of ideal mathematical students. Mathematicians have been reported to use this term in various ways, yet there is no clear or empirically based description of mathematical maturity at this time. This proposal explores existing descriptions of mathematical maturity as well as descriptions of the related concepts of mathematical intuition and mathematical beliefs. This proposal reports preliminary findings from interviews with mathematicians investigating their understandings of mathematical maturity. Preliminary results include three main components of mathematical maturity: ways of thinking about mathematics, mathematical intuition, and comfort with and competence in mathematics.
2019
Oklahoma City, Oklahoma
Answering the call of Francis Su (Su, 2017) that "math is for human flourishing" and a challenge by Rochelle Gutiérrez to rehumanize math (Gutiérrez, 2018), I changed assessments in two university calculus II courses. The traditional way to change assessments is to change the questions, either by type or by content. Instead, I focused on changing/rehumanizing the structure of exams to include small group discussions between students for part of the exams. This assessment change, along with a consistently enacted classroom mission statement, produced higher exam scores and improved student engagement. Through surveys, focus groups and interview data, students also reported feeling they had a deeper understanding of concepts, as well as a humane and positive math experience in a math class they thought was very difficult.
2019
Oklahoma City, Oklahoma
The results we report are a product of the first iteration of a design-based study that uses a game, Vector Unknown, to support students in learning about vector equations in both algebraic and geometric contexts. While playing the game, students employed various numeric and geometric strategies that reflect differing levels of mathematical sophistication. Additionally, results indicate that students developed connections between the algebraic and geometric contexts during gameplay. The game’s design was a collaborative effort between mathematics educators and computer scientists and was based on a framework that integrates inquiry-oriented instruction and inquiry-based learning (IO/IBL), game-based learning (GBL), and realistic mathematics education (RME).
2019
Oklahoma City, Oklahoma
In this study, we investigate the beliefs of undergraduate mathematics tutors. Thirty-three tutors completed surveys and twenty-five participated in interviews to assess their attitudes towards mathematics and their beliefs about the roles of a tutor and instructor. Our analysis provides examples of orientations, goals, and resources that were expressed by tutors in surveys and interviews. Tutors in this study viewed their role as distinct and supplementary to that of a teacher. The orientations, goals, and resources identified in this study provide a foundation for future studies that explain and predict tutor decision making. Although tutors are not content experts, they offer a valuable perspective that is different than that of the instructor.
2019
Oklahoma City, Oklahoma
Evaluating the Quality of Instruction in Post-secondary Mathematics (EQIPM) is a video coding instrument that provides indicators of the quality of instruction in community college algebra lessons. It grew out of two instruments that assess the quality of instruction in K-12 settings—the Mathematical Quality of Instruction (MQI) instrument (Hill, 2014) and the Quality of Instructional Practices in Algebra (QIPA) instrument (Litke, 2015). We present preliminary results of an exploratory factor analysis that suggests that the instrument captures three distinct dimensions of quality of instruction in community college algebra classes.
2019
Oklahoma City, Oklahoma
Using the Mathematical Association of America’s Characteristics of Successful Programs in College Calculus dataset (CSPCC) of 13,965 students from a variety of institutions nationwide, student characteristics and experiences were analyzed via pre- and post-course survey responses. This research evaluated the effect of student background, student-reported teaching behaviors, and institutional environments on academic achievement and student confidence. The findings of this research could lead to a better understanding of the impact of calculus teaching practices and the implications of retaking calculus for students of all experience levels.
2019
Oklahoma City, Oklahoma
This preliminary report explores data from a larger study investigating the nature of preservice elementary teachers’ content knowledge and pedagogical content knowledge (PCK) in the area of number theory. A prominent theme emergent from the data – a contributing factor in participants’ PCK – was the theme of tutoring experiences. Participants explicitly and regularly referenced their tutoring experiences when responding to hypothetical students in PCK tasks. The influential nature of preservice elementary teachers’ tutoring experiences on their PCK holds implications for teacher-training, but further investigation is necessary. Questions concerning the design of a future study are proposed for discussion.
2019
Oklahoma City, Oklahoma
An increase in general quantitative literacy and discipline-specific Physics Quantitative Literacy (PQL) is a major course goal of most introductory-level physics sequences—yet there exist no instruments to assess how PQL changes with instruction in these types of courses. To address this need, we are developing the Physics Inventory of Quantitative Literacy (PIQL), a multiple-choice inventory to assess students’ sense-making about arithmetic and algebra concepts that underpin reasoning in introductory physics courses—proportional reasoning, covariational reasoning and reasoning about sign and signed quantities. The PIQL will be used to not only to assess students’ PQL at specific points in time, but also to track changes in and development of PQL that can be attributed to instruction. Data from early versions of the PIQL suggest that students experience difficulty reasoning about sign and signed quantities.
2019
Oklahoma City, Oklahoma
In this work we examined modeling routes, mathematical thinking styles and teaching foci of two prospective secondary teachers as they considered two modeling tasks so to consider connections between the candidates’ own modeling processes and their approaches to teaching modeling. Two questions guided the study: How might pre-service teachers’ preferred mathematical thinking styles impact their modeling routes? How might pre-service teachers’ preferred mathematical thinking styles impact their focus while contemplating how they would teach mathematical modeling? Close links were found between teacher candidates’ validation methods within the modeling process and their decisions regarding what would be important for school learners to consider. Recognizing teacher candidates’ natural approaches to modeling tasks might help teacher educators to be better positioned in developing tasks that motivate reliance on a larger repertoire of representations.
2019
Oklahoma City, Oklahoma
Although students face many challenges in learning to construct mathematical proofs in general, proof by contradiction is believed to be particularly difficult for them. We investigate whether this is true, and what factors might explain it, using data from an “Introduction to Proof” course. We examined proofs constructed by students in homework and examinations, and conducted stimulated-recall interviews with some students about their thought processes while solving proof problems. Preliminary analysis of our data suggests that students’ background knowledge about the typical content domains that appear in indirect proof plays a larger role than the logical structure of the proof technique itself.
2019
Oklahoma City, Oklahoma
Interpreting graphs and drawing conclusions from data are important skills for students across science, technology, engineering, and mathematics fields. Here we describe a study that seeks to better understand how students reason about graphs in the context of enzyme kinetics, a topic that is underrepresented in the literature. Using semi-structured interviews and a think-aloud protocol, our qualitative study investigated the reasoning of 14 students enrolled in a second-year biochemistry course. During the interviews students were provided a typical enzyme kinetics graph and asked probing questions to make their reasoning more explicit. Findings focus on students’ mathematical reasoning, with analysis indicating students tended to focus on surface features when describing related equations and graphs, which limited their understanding of the chemical phenomena being modeled.
2019
Oklahoma City, Oklahoma
Colleges and universities are increasingly providing drop-in tutorial assistance through institutions’ learning or resource centers. In this study, we examine one-on-one mathematics tutoring interactions to discover how tutors naturally respond to student requests for assistance with knowing-to act, where a student may be familiar with a procedure, but not know-to use that procedure in the current situation. We contrast three 5-10 minutes episodes; in the first, the tutor appears not to recognize that the student knows-to. In the second, the tutor prevents the student from needing to know-to. In the final episode, a tutor incrementally narrows the vision of her student until the student knows-to.
2019
Oklahoma City, Oklahoma
This paper reports about using eliciting personification (Zazkis, 2015) as a means to study pre-service teachers’ (PSTs) beliefs. The method has the PSTs’ create a character named Math and describe their relationship with the character. The authors analyzed 68 personifications from college sophomore PSTs’ in an elementary math content course. At the end of the semester, the PSTs; revisited the assignment by describing a new character based on the math learned in class and writing a dialogue to themselves. At the beginning of the semester, the PSTs described math as having multiple personalities, out to hurt them, and having a relationship that fell apart throughout the years. The math described at the end was more compassionate, welcoming, and easier to understand.
2019
Oklahoma City, Oklahoma
Deaf and hard of hearing (D/HH) students face many challenges in the study of undergraduate mathematics. Unfortunately, minimal literature exists in this area, evidencing the need for further research. Through five qualitative survey responses from D/HH students, we identified common themes of concern in addition to a number of specific struggles (and a few successes) encountered by each of the respondents in their own undergraduate mathematics courses. From these students’ experience, we can identify further areas of research with the goal of developing new educational tools for mathematics instructors with deaf or hard of hearing students. In doing so, we can help give equal opportunity to mathematics students regardless of their level of hearing.
2019
Oklahoma City, Oklahoma
When considering undergraduate mathematics education, gender equity is an ongoing issue and it has been suggested that inquiry-based instruction could make classes more equitable for men and women. In this study, we analyze data from 42 undergraduate instructors and courses and 681 students in the context of inquiry-oriented instruction in either abstract algebra, differential equations, or linear algebra. Specific instructional units were video recorded, watched, and coded to see how teachers distributed opportunities to participate in whole class discussion, how these opportunities were taken up by students, and what teachers did with student ideas. Mathematically substantial opportunities were not distributed equitably between men and women, which was consistent with inequitable student participation observed. Further, instructors tended to leverage women’s ideas less than men’s ideas when building on formalizing students’ mathematical contributions.
2019
Oklahoma City, Oklahoma
This hermeneutical phenomenological study explored the experiences of students in the College of Humanities and Social Sciences (CHASS) in an undergraduate mathematics course and their perceptions of its utility. Field observations and semi-structured interviews were conducted. Six themes emerged from the collected data. The phenomenon of being a CHASS student in Topics in Contemporary Mathematics is perceived as enjoyable, but impractical and useless. Moreover, what moves students to be successful are mostly (or only, in some cases) external regulators that do not promote autonomy. A set of implications is provided.
2019
Oklahoma City, Oklahoma
In this report, we present preliminary findings from clinical interviews examining inservice teachers’ understandings of quadratic growth and exponential growth. The purpose of this pilot study is to investigate how teachers may naturally leverage covariational reasoning to distinguish between the two types of growth. In this report, we first present relevant constructs pertaining to teachers’ covariational reasoning and then describe one task we used in clinical interviews. We then present preliminary findings regarding how teachers’ leveraged (or did not leverage) covariational reasoning as they addressed this task to differentiate between quadratic and exponential growth. We conclude with preliminary implications and questions regarding how these preliminary findings may have implications for a larger study with pre-service secondary mathematics teachers.
2019
Oklahoma City, Oklahoma
Research in undergraduate mathematics education has identified various research-based instructional practices to support students’ learning. However, little is known about how students experience those practices or how helpful they perceive those practices to be for their learning. As part of a larger national project of first-year mathematics, this study focused on classroom experiences in the Precalculus to Calculus 2 (P2C2) sequence. Using survey data from 4,969 students, we considered how helpful students find various teaching practices and then compared student and instructor reports of how characteristic these practices are of their P2C2 class. Here we report students’ ratings of twelve different teaching practices in terms of helpfulness for their learning in and descriptiveness of their P2C2 experience.
2019
Oklahoma City, Oklahoma
The purpose of this quasi-experimental study is to measure the impact of a contextualized-instruction intervention on student performance expectations, utility-value, and interest in college calculus courses. Six calculus sections were selected for this study, and three were randomly assigned to take the intervention. Students in the three intervention sections completed calculus tasks that were contextualized to various STEM disciplines, whereas students in the comparison sections did not. Three tasks, one from each in computer science, physics, and engineering were contextualized. The results indicated that the impact of the intervention on student motivation was not statistically significant. However, student motivation significantly changed over time.
2019
Oklahoma City, Oklahoma
This work focuses on the preparation of graduate Teaching Assistant in the mathematics department of a large midwestern R1 university. We explore the author’s experience with three different versions of the course in three varying capacities: as a student, a mathematics education researcher and a co-facilitator. As we delve into the author’s reflections on the evolution of the teaching preparation within this timeframe, we highlight some issues with the course structure and execution. This leads to the development of another, more realistic version of the preparation course.
2019
Oklahoma City, Oklahoma
Mathematics achievement, both in high school and early in college, is one of the strongest predictors of college completion. Research conducted within the framework of expectancy-value theory has shown that math interest, utility, engagement, self-efficacy, and identity are related to mathematics achievement. Hence, this study uses structural equation modeling to evaluate Ford’s (2017) empirical model linking mathematics beliefs and achievement with a sample of students enrolled in multiple sections of two algebra-focused remedial math courses at a community college near a midsize metropolitan southern city in the United States.
2019
Oklahoma City, Oklahoma
This study explores the relationship between preservice teachers (PTs) representational fluency and functional thinking. We conducted task-based interviews with five preservice teachers within a growing rectangle context. Although PT create, interpret, and connect representations with covariational and corresponding approach, they identified relationship between area and height an exponential rather than a quadratic. These findings illustrate a challenge of building rich conceptions of functions from emerging representational fluency and functional reasoning.
2019
Oklahoma City, Oklahoma
When implementing lessons connecting advanced undergraduate mathematics to school mathematics, challenges arise for faculty and for the undergraduate students. The Mathematical Education of Teachers as an Application of Undergraduate Mathematics (META Math) project has created, piloted, and field-tested lessons for undergraduate mathematics and statistics courses typically part of a mathematics major that leads to secondary mathematics teacher certification. Lessons in calculus, discrete mathematics, algebra, and statistics explicitly link topics in college mathematics with high school mathematics topics prospective teachers will eventually teach. The goal of this poster presentation is to discuss our preliminary observations of the challenges faced by faculty and undergraduate students when implementing or using these lessons. We also wish to gather feedback and suggestions on the study design and potential directions for further research.
2019
Oklahoma City, Oklahoma
Two populations of undergraduate mathematics learners can benefit from extra support: preservice elementary (PsE) teachers and developmental mathematics (DM) students. In this project, PsE students tutored DM students; the former gained experience working with students, the latter received extra, structured support. Results of a survey given to DM students show that although perception of the program was tepid, they were remarkably successful in the course.
2019
Oklahoma City, Oklahoma
The IMAGEMath project combines inquiry based learning with an application-inspired approach. Students first learn about an application, and then, in an inquiry framework they develop the mathematics necessary to investigate the application. A novel feature of this approach is that the applied problem inspires the mathematics, rather than the applied problem being presented after the relevant mathematics has been learned. In this poster, we give an overview of the IMAGEMath modules that use image and data applications (radiography, tomography and heat diffusion) to inspire linear algebra topics. We present results from implementing the modules on a small scale at a few institutions, including student and faculty feedback. We also provide information for faculty interested in using IMAGEMath materials.
2019
Oklahoma City, Oklahoma
Observation protocols allow researchers to document moments of teaching and learning, as well as reveal inequities and opportunities for improvement. In two undergraduate mathematics courses, I used the OPAL protocol to understand if and how active learning strategies created equitable learning environments. In this poster, I share findings from observations and discuss possibilities for adapting observation protocols to align with equitable teaching practices.
2019
Oklahoma City, Oklahoma
It is well-known that too many students abandon a STEM career because of their calculus requirement. Therefore, being able to identify early on which students may be at risk of failing is important. Using indicators of mathematical readiness (SAT/ACT and PCA) and attitudes toward mathematics (MAPS), we build models predicting final grades. Our analyses show that all three indicators are significant predictors of success in calculus.
2019
Oklahoma City, Oklahoma
The Mathematical Knowledge for Teaching (MKT) theoretical framework describes effective mathematics teaching in a way that relies on an instructor’s subject matter knowledge (SMK) and on their pedagogical content knowledge (PCK). This proposal reports our initial effort to understand MKT within chemistry instruction, namely what MKT could support chemistry instructors’ efforts to help students develop a deeper understanding of the mathematics used in general chemistry. Coding of several types of general chemistry problems involving ratios and proportions and covariation are provided as examples.
2019
Oklahoma City, Oklahoma
Volume problems are a typical first type of integral application problem that students encounter in second-semester calculus. We will present students’ responses to the question, “Why does integration give a volume?” Participants were Calculus 2 and Calculus 3 students enrolled in summer classes at a large, public university. Task-based interviews consisted of students working on and discussing second-semester volume problems. Students had varied and interesting responses that included formula-, units-, and derivative/antiderivative-based reasoning. These results are part of a larger study on how students understand the underlying structure of the definite integral, and how they use pictures and visualizations when solving volume problems.
2019
Oklahoma City, Oklahoma
This proposal describes a pilot study of replacing a remediation mathematics class for undergraduates with a credit-bearing mathematics course designed for elementary education majors. The replacement course focused on number and operations. Results of pass rates, placement test scores (pre and post the course), as well as course feedback will be shared.
2019
Oklahoma City, Oklahoma
We observed recordings of instances from a graduate course in topology where students engaged in proving theorems on the whiteboard in a collaborative environment. We considered the written component on the whiteboard as “the proof”, which was aided, in 17 out of 20 instances by some form of verbal explanation. The peculiarity of the class structure allowed each lesson to be followed by an open discussion regarding “the proof”. As a result of the discussions, the written component of each proof would undergo improvements. When analyzing the developments of the proofs in this course, we employed the thematic of proof introduced by Mariotti. Stemmed by these proof-presentations, we introduce the idea of proving as a “work-in-progress” activity.
2019
Oklahoma City, Oklahoma
Introductory Statistics is a course commonly taken by students from a variety of wide-ranging majors, sometimes across departments; however, there is little known about the extent topics are covered generally across courses. Textbooks include more material than can reasonably be covered in a single course, but the non-linear nature of many topics means that from course to course the covered content can diverge greatly. We provide results of a nationwide survey of 148 introductory statistics instructors and assess how often concepts are covered in introductory courses across instructor experience, course audience and course pedagogy.
2019
Oklahoma City, Oklahoma
Our initial project focused on assessing conceptual understanding of key topics in Calculus I, specifically measuring changes in the achievement gap between underprepared and prepared students in Active and Traditional classrooms. However, a main hurdle is the lack of instrument for assessing Calculus readiness. In this poster, we present results of student understanding of continuity in Active vs. Traditional settings from 16 sections of Calculus I. We present ideas for refining this study to be able to better assess student growth by creating and validating questions regarding students’ initial understanding of Calculus topics: continuity, differentiability, limits, and area. We present our study design and initial findings; we look forward to feedback as we enter the latter half of our project.
2019
Oklahoma City, Oklahoma
We investigate links between units coordination structures and covariation schemes.
2019
Oklahoma City, Oklahoma
We present the methodology and preliminary findings from a pilot study undertaken at three institutions during Spring 2018. Our purpose is to uncover student reasoning around volumes of solids of revolution. Initial findings suggest issues arise in the Product layer of the Riemann Integral Framework (Sealey, 2014).
2019
Oklahoma City, Oklahoma
Students taking courses below calculus are an understudied population in undergraduate mathematics education, as are students with mathematics difficulty or disability. Students seeking additional help are likely to seek YouTube and other outside resources, which may not mesh with in-class instruction. Since secondary education research suggests that targeted tutoring is beneficial to students with mathematics difficulty or disability, this single case study investigated if there was a functional relationship between a Smartpen to create videos for a student with a mathematics disability to listen to during tutoring sessions and her achievement on synthetic division problems over a four week intervention.
2019
Oklahoma City, Oklahoma
In an attempt to bring more realistic situations into college mathematics classroom environments, lessons were created that utilized culturally relevant pedagogy for a college algebra course at a large historically black college/university (HBCU) in the south. These lessons were aimed at the growing population of diverse students in an effort to gauge their effectiveness with students, in regards to achievement and self efficacy. This poster will illuminate the conceptual developmental process of four “experimental” lessons and provide some preliminary findings of the course that utilized these lessons in comparison with a control class that did not.
2019
Oklahoma City, Oklahoma
Minimal research has been conducted surrounding how best students learn trigonometric functions in a precalculus course. Using motivation from a study conducted by Weber (2005), results from this study indicate that students who participated in a college level precalculus course where the unit circle was taught before right triangle trigonometry were better able to utilize a unit circle but struggled to conceptualize some of its properties. This report has implications for mathematics programs looking to determine best practices for the instruction order of precalculus courses.
2019
Oklahoma City, Oklahoma
The trigonometry is first framed on a right triangle, next on a unit circle with a parametrized pair of coordinates (r cos t, r sin t), and then on a complex frame, r (cos t + i sin t) unifying the Cartesian pair (Ekici, 2010). Students often struggle in understanding the connections and the transitions among triangle, circle, and complex trigonometry which serve as a critical mathematical foundation in many STEM fields. It is a challenge for students and teachers to coordinate the multiplicity of these trigonometric frames to develop coherent meanings. To support this transition, dynamic manipulatives using GeoGebra are here developed for student experimentation in modeling with modified circular and complex trigonometric functions. The results show that inquiry-based modeling using these multiple yet interconnected frames facilitate the emergence of coherence observed while validating these trigonometric models.
2019
Oklahoma City, Oklahoma
As part of an ongoing project to redesign a calculus sequence centered around core calculus concepts through an active learning approach, we aim at understanding the knowledge students need in order to be successful in this setting. In particular, we are interested in exploring what conceptual understandings of precalculus concepts support students in an active learning intensive calculus sequence. We present preliminary results of an analysis carried out to answer the question: What is the relationship between students’ precalculus understandings and performance in this newly redesigned calculus sequence?
2019
Oklahoma City, Oklahoma
As part of an ongoing effort to understand how mathematics departments in the U.S. can better support graduate students teaching in precalculus and calculus courses, we are interested in investigating plans (or potential plans) departments are making toward improving their graduate teaching assistant (GTA) professional development (PD) programs. Contributing to a larger national project of first-year mathematics, this study looked at mathematics departments’ survey responses to three items regarding changes to GTA PD programs. Out of the 223 departments that responded to the survey 66 of them indicated some level of plans of change to their program. For those schools, we analyzed the open-ended responses elaborating on the current status of the GTA PD program and found several noticeable themes regarding changes or plans to change their programs.
2019
Oklahoma City, Oklahoma
I report on the first stage of my dissertation project which sought to understand engagement in a Precalculus course at a four-year public university. Breaching instructional activities, student interviews, and classroom recordings were used to study the development of several sociological and psychological constructs to help characterize students’ engagement. Despite the instructor’s attempts to negotiate productive norms, data analysis shows that some students’ detrimental practices and beliefs remained unchanged or were even supported by the course. I examine the roots and consequences of this phenomenon.
2019
Oklahoma City, Oklahoma
In this poster we introduce a new learning modality called MathChavrusa. Inspired by the ancient rabbinic approach to Talmudic study, the chavrusa model pairs students in a partnership of deep text-based analysis, discussion, and debate. Over centuries the model has proved its ability to generate thorough understanding, build skills, develop the courage to question, and demonstrate to students the value of both independent thinking and collaboration. MathChavrusa is a complementary model to other accepted modalities for generating student understanding in mathematics. It is particularly effective when employed after a lecture class. In teaching about the model, we will discuss its origins, how it facilitates deep learning and understanding in mathematics, and techniques for implementation. We have begun to utilize the model in our classes, and are gathering data about its real-world effectiveness. Preliminary data implications will be discussed.
2019
Oklahoma City, Oklahoma
Upgrading Learning for Teachers in Real Analysis is a project in which we designed and implemented an innovative real analysis course for pre-service and in-service mathematics teachers (PISTs). More generally, this project provides an alternative model to teaching advanced mathematics to PISTs, a model that more meaningfully connects the teaching of secondary mathematics to the advanced mathematics content. This poster describes the theoretical model, the means of developing connections between real analysis and secondary mathematics content, the 12 modules we designed and how they fit in a standard real analysis curriculum, and presents evidence for their efficacy. The instruction is built from and returns to authentic secondary mathematics classroom situations.
2019
Oklahoma City, Oklahoma
Through their coding of survey responses, researchers can create spaces to humanize students’ attitudes toward math. To account for complexity in students’ attitudes beyond positive or negative, we developed three additional codes: mixed, ambiguous, and detached. In our coding methods, we account for a diversity, rather than a binary, of student attitudes.
2019
Oklahoma City, Oklahoma
Integrating innovative pedagogical initiatives within the learning environment at the Lebanese American University in Beirut, Lebanon, has been set as a strategic goal. Active learning, as one medium of instruction, has seen widespread implementation in mathematics classrooms. This study reports on an inquiry oriented differential equations class offered in spring 2018. The focus is on the role of the curriculum in guiding students reinvent successfully key mathematical notions covered in any introductory differential equations class.
2019
Oklahoma City, Oklahoma
Research has shown that faculty benefit from support and collaboration when introducing student centered instruction into their teaching (Henderson, Beach, & Finkelstein, 2011; Speer & Wagner, 2009). The RUME community has some knowledge about how these supports take shape and grow (e.g., Hayward, Kogan, & Laursen, 2015), but work is still needed. A crucial component is researching the facilitation of these supports. In this study, we focus on how the facilitation of online working groups occurs. Our preliminary results indicate that the actions facilitators take play crucial roles in how to use discussions of mathematics to proactively engage in student thinking.
2019
Oklahoma City, Oklahoma
There now exist resources (e.g., text- and video-based case activities) for use in the professional development of novice college mathematics instructors. We do not yet know much about the characteristics of effective use of those resources. Even less is known about what facilitators need to know to use the resources successfully. A newly funded project is building meta-materials to help providers use resources. These Provider Packages are a virtual facilitation partner, taking on some of the cognitive load of facilitation (e.g., audio tracks that can be turned on and off, notes from a more experienced peer that are virtual whispers in the ear). The project will examine use of the facilitation support tools in the Provider Packages to identify characteristics of effective facilitation of activities. At the poster, we will seek conversations about research designs that can leverage the opportunities of the Provider Packages.
2019
Oklahoma City, Oklahoma
Survey data for community college algebra students reveals relationships between a student’s attitudes towards mathematics and the student’s STEM career interests. Results show that while students may not always have a clear understanding of the tasks related to a chosen STEM career area, the student’s math interest predicts interest in some STEM careers and not others.
2019
Oklahoma City, Oklahoma
Students are engaged in various reasoning and proving tasks corresponding to the increased emphasis on reasoning and proving in mathematics education. Students routinely encounter differing language present in prompts for these reasoning and proving tasks. The semantic meaning of the language used in these prompts is not usually explicitly discussed and thus may cause inconsistencies in students’ responses to these tasks and in the assessment of their work. The preliminary results imply Calculus I students have various conceptions for prompts such as “prove”, “explain”, “show”, and “convince”. This poster will focus on students’ various conceptions on the two prompts “prove” and “show.”
2019
Oklahoma City, Oklahoma
This study attempts to answer the question: why do prospective elementary teachers (PTs) have high levels of anxiety learning mathematics and low levels of confidence to teach mathematics? Using a survey of 300 undergraduate PTs and a mixed methods approach, I report the results from the analysis. Identifying and addressing the causes of negative beliefs about mathematics is crucial to ending the negative cycle of beliefs in their future students. A survey of 300 undergraduate PTs was taken and a mixed methods analysis was done.
2019
Oklahoma City, Oklahoma
In order to determine preservice secondary mathematics teachers’ (PSMTs) conceptual understanding following an inquiry-based lesson on the constructed meanings of the equals sign and the distinctions between the concepts of function and equation, we utilized a scripting task in which the PSMTs individually continued a dialogue between two hypothetical students with opposing viewpoints with respect to an equation arising from a function context. This study is part of the Enhancing Explorations in Functions for Preservice Secondary Mathematics Teachers Project which is developing research-based tasks and explorations together with instructor materials to be used in mathematics courses for PSMTs. The goal of this poster presentation is to discuss our implementation of the scripting task to gauge PSMTs’ understanding of the nuances between function and equation. We also wish to gather feedback and suggestions on the study design and potential implications of our research
2019
Oklahoma City, Oklahoma
In this poster we share a qualitative study aimed at investigating creativity in problem solving for non-mathematics tracked students enrolled in a calculus course. Three task-based semi-structured interviews with volunteered participants were analyzed using a modified whole-to-part inductive approach (Erickson, 2006). Our findings suggest that even though students may perceive creativity as a process, this understanding may not necessarily be reflected in their written work.
2019
Oklahoma City, Oklahoma
Research-Based Instructional Strategies have been show to increase learning and retention of students in undergraduate STEM classes but have not been widely implemented in classrooms across this country. While there is research indicating the level of usage of RBIS across the country in gateway chemistry, mathematics, and physics courses, less is known about why instructors choose to use RBIS or not. We report on the design of an ongoing research study to assess the relative impact of individual, departmental, institutional, and disciplinary factors on instructional decisions in key courses for postsecondary STEM-intending students.
2019
Oklahoma City, Oklahoma
In an online environment that promotes self-learning and online interaction between teacher and students, this poster proposal presents the nature of students’ meanings of introductory trigonometry while they interact with each other. Students’ communication in the online interactive forum is crucial as their reasoning influence others to make their own meaning of the problem. The poster proposal also presents the difficulties students encounter in developing trigonometric understandings when they work online independently.
2019
Oklahoma City, Oklahoma
This poster presents conceptual analysis and hypothetical learning trajectory for learning proportionality which was previously limited only to figure out missing value using cross-multiplication. Based on a series of clinical interview that investigated students’ meaning of proportionality in an online format, I found that students tend to use only cross-multiplication strategy to reason proportionally which did not help them to reason proportionally. That emphasizes how learning proportionality along with the constant rate of change among quantities given in any specific word problem helps to reason proportionality conceptually.
2019
Oklahoma City, Oklahoma
Recommendations for the teaching and learning of introductory statistics at the tertiary level have been set forth by the research community, including recommendations outlining desirable pedagogical strategies, such as the use of student-centered instruction and the integration of technology and resampling methods to support the development of students’ conceptual understanding. Yet, surprisingly little is known about how introductory statistics is being taught at colleges and universities across the United States. The research presented here aims to shed light on these aspects of the introductory statistics course by reporting preliminary findings from an instructor survey that was recently completed by 148 instructors nationwide.
2019
Oklahoma City, Oklahoma
This study examines 22 future middle school teachers’ problem solutions on proportional relationships. Using the appropriateness attribute (Izsák, Jacobson & Lobato, 2011) as a framework, we explored to what extent future middle school teachers were able to appropriately identify multiplication situations (MS), as well as identify and classify as partitive division situation (PDS) or quotitive division situation (QDS). Findings reveal that participants succeeded in identifying multiplicative situations, but more interestingly, with problems requiring division, PDS was recognized four times as often as QDS.
2019
Oklahoma City, Oklahoma
Recent research shows the promise of using tasks that situate mathematics in a pedagogical context for secondary teachers, including tasks where teachers are asked to explain a solution to a mathematical task. We use a theory of positionality (Aaron, 2011; Herbst & Chazan, 2003, 2011) to make sense of why explanations might differ when the solver is positioned as a secondary teacher as compared to positioned as a university mathematics student.
2019
Oklahoma City, Oklahoma
Preparing graduate teaching assistants (GTAs) well for their teaching roles is a high-leverage opportunity to improve undergraduate mathematics education. The College Mathematics Instructor Development Source (CoMInDS) seeks to assist people who build and lead teaching-focused GTA professional development (TAPD) at their own institutions. CoMInDS offers direct support to these TAPD providers and seeks to enhance the development and use of research-based TAPD practices. We draw upon project evaluation data and team members’ reflections to identify progress, opportunities and challenges in this work.
2019
Oklahoma City, Oklahoma
The purpose of this study is to look at the covariational reasoning necessary to define surfaces parametrically in 3-space and the mathematics involved in writing statements in Graphing Calculator (GC). This study offers evidence on how developed ideas of covariational reasoning and parametric explanation have an impact on visualizing animations, especially in the case of solids of revolution. Additionally, the analysis of seven calculus textbooks focusing on parametrically defined relationships reveals that the use of parametric explanation is confined to representing familiar graphs parametrically and the trends of describing parametric relationships in the textbooks will be discussed as well. Through the structured way of thinking embedded in the statements in GC, students will be able to understand how the surfaces are formed as the parameters vary.
2019
Oklahoma City, Oklahoma
This study explored a pedagogical way to contribute to students’ consolidation of their newly formed mathematical knowledge. We applied problem posing activities in complex analysis course. Students posed a problem in a group and then solved this. In the process of these activities, we could observe three modes of consolidation in the students’ mathematical knowledge and the epistemic actions emerging in aspects of problem posing. A problem posing activity can act as a practical way to stimulate students’ consolidation.
2019
Oklahoma City, Oklahoma
Many undergraduate students experience significant difficulty in learning to prove mathematical propositions nationwide. A previous study by David & Zazkis (2017) used document analysis of publicly available syllabi to create a national portrait of approaches to supporting students’ transition to proof across a large sample of R1 and R2 universities. Liberal arts colleges (LACs) operate under different sets of institutional constraints and thus offer the possibility of different approaches to this issue. We report results of a preliminary survey study, the goal of which was to enhance previous work on approaches to the transition to proof by specifically focusing on the case of LACs. Analysis of the survey data show that LACs’ approaches have distinctive features as compared to R1 and R2 universities. Notably, discrete mathematics courses served as a transition to proof course in almost half of the surveyed institutions.
2019
Oklahoma City, Oklahoma
This study (a) validates a measure “bold problem solving” for postsecondary students and (b) examines patterns in bold problem solving tendencies within and across various math classes. A confirmatory factor analysis demonstrates the general construct holds for the postsecondary population. Course and gendered differences in bold problem solving tendencies exist.
2019
Oklahoma City, Oklahoma
The author is part of a multi-institution grant that is attempting to implement the recommendations of the MAA Curriculum Foundations Project CRAFTY report on mathematics for partner disciplines. The author’s institution is focusing mathematics for business students. Although the CRAFTY report is nearly 15 years old there seems to be little in the literature looking at the effectiveness of implementation of the report’s recommendations. This report looks at how implementation changes student’s attitudes toward mathematics.
2019
Oklahoma City, Oklahoma
This poster presentation will highlight the results of a qualitative, multicase study which explored the use of alternative scoring practices in collegiate mathematics classes. Specifically, the researchers explored the use of two different scoring practices: one in an entry-level College Algebra course and one in an upper-level Modern Geometry course. In addition to classroom observations, data collection for each case consisted of two interviews for each course instructor, one interview with each course designer, and interviews with students in each course. This poster presentation will detail themes from cross case analysis which suggest important details for successful implementation of alternative scoring practices in collegiate mathematics courses.
2019
Oklahoma City, Oklahoma
Undergraduate students in a History of Mathematics course engaged with various historical mathematical problems. Reflective journals and interviews were used to analyze their perspectives on meta-issues of mathematics. The results indicate some revision of their metaperspectives and new cultural awareness.
2019
Oklahoma City, Oklahoma
Student success in introductory calculus is imperative to obtaining a degree in STEM. Calculus I is a main gatekeeper course for STEM majors, and many students leave the class with a diminished motivation to pursue further courses related to mathematics. This poster reports a qualitative case study from a larger mixed-methods project aimed at exploring the relationship between course structures (hybrid, traditional, and large active learning) and student motivation in calculus. Using the theoretical framework of self-determination theory (SDT), six students were interviewed to investigate how each course structure was related to students’ perceptions of their competence, autonomy, and relatedness. Emerging themes showing differences in student motivation between the three course types will be presented.
2019
Oklahoma City, Oklahoma
This proposal describes findings of an ongoing project designed to support adjunct instructors’ teaching of undergraduate Precalculus. We are studying the impact of supports on Precalculus instructors’ knowledge through interview and assessment data. Using Shulman’s (1987) components for a teaching knowledge base, we discuss shifts in instructors’ perspectives
2019
Oklahoma City, Oklahoma
This study investigates student engagement while learning through use of an app that collected student engagement reported by participants during a classroom teaching experiment. This paper discusses preliminary results on students’ engagement in the process of learning. Though not anticipated, we observed differences between male and female students’ engagement while working in mixed-pairs worthy of investigation.
2019
Oklahoma City, Oklahoma
We suggest a 4-step cycle for helping prospective teachers transform their mathematical understandings from procedurally-based to more conceptually-based understandings. We use Mezirow’s (1991) idea of Transformative Learning Theory (TLT), which is an application of androgogy, or the methods of teaching adults. In this poster, we share our model of a TLT cycle and illustrate it using an example of a proportional reasoning problem for prospective teachers.
2019
Oklahoma City, Oklahoma
Small groups teaching as part of first semester was studied through observations of the seminars, analysis of the tasks, and students’ responses in surveys and interviews. The research question posed is how the teaching activities enabled the students to develop their conceptual and procedural knowledge. The results show a desire for a development of conceptual knowledge but the demands on procedural knowledge placed the students with conflicting aims.
2019
Oklahoma City, Oklahoma
This poster introduces early results from IOLA-G, a project exploring the possibilities of supporting inquiry-oriented instruction with digital videogames. During the poster presentation, we will describe our game design process, including the theories we have drawn upon during its development. We will also show small samples of gameplay from a corpus of data we collected with undergraduate students at three different universities. Additionally, we will demonstrate gameplay on laptops, providing poster visitors with first-hand experience playing the game.
2019
Oklahoma City, Oklahoma
This study explored adjunct instructors’ opportunities for learning as they faced challenges while implementing a research-based mathematics curriculum. Three case studies explored adjunct instructors’ experiences as they implemented a research-based precalculus curriculum for the first time over the course of two semesters. The similarities and differences between the challenges faced by the instructors and the opportunities for their learning were analyzed.
2019
Oklahoma City, Oklahoma
This work involves a methodological presentation of an analytic framework for characterizing mathematical reasoning, introducing the construct of “graphical forms”. Graphical forms build on Sherin’s (2001) symbolic forms (i.e., intuitive ideas about equations) by focusing on ideas associated with a pattern in a graph. In addition to providing an overview of the symbolic forms identified in the literature, we describe how we expand the symbolic forms framework to encompass graphical reasoning. Analysis involving the graphical forms framework is illustrated by providing examples of interpretations of graphs across disciplines, using introductory biology, calculus, chemistry, and physics textbooks. Our work suggests the broad applicability of the framework for analyzing graphical reasoning across different contexts.
2019
Oklahoma City, Oklahoma
In this study, we explore how quantum mechanics students understand linear algebra concepts in the context of two spin-½ probability problems, the second of which required a change of basis. In particular, our research question is: what problem solving approaches do students use, what mathematical concepts are involved in that approach, and how do students reason about basis and change of basis as they engage with the problems? Data come from individual, semi-structured interviews with twelve quantum mechanics students from two different universities. Our poster will share preliminary results for all parts of the research question.
2019
Oklahoma City, Oklahoma
This preliminary report will focus on how college precalculus teachers, mostly graduate teaching assistants, interact with department-provided curriculum materials. We specifically address what collegiate teachers notice in curriculum resources while planning. Comparisons will be drawn between first-time instructors and those with more experience, ultimately informing what and how collegiate teacher educators might incorporate experiences for precalculus teachers to develop curriculum use practices.
2019
Oklahoma City, Oklahoma
This poster explores how Introduction to Statistics students think about and compare the mean and variability of four datasets. They explored the datasets through various representations (e.g., balance beam, leveling off) and ranked them from most to least variance. When exploring the mean, the students found value in both the balance beam and leveling off representation, but preferred the balance beam for reasoning about variability. However, when reasoning about the variance using the balance beam representation, the students focused on the wrong properties and made faulty inferences. When reasoning using the leveling off representation, they focused on the correct properties and made sound inferences about the data.
2019
Oklahoma City, Oklahoma
In an effort to improve the quality of citizen engagement in workplace, politics, and other domains in which quantitative reasoning plays an important role, Quantitative Literacy (QL) has become the focus of considerable research and development efforts in mathematics education. QL is characterized by sophisticated reasoning with elementary mathematics. In this project, we extend the notions of QL to include the physics domain and call it Physics Quantitative Literacy (PQL). We report on early stage development from a collaboration that focuses on reasoning inventory design and data analysis methodology for measuring the development of PQL across the introductory physics sequence. We have piloted a prototype assessment designed to measure students' PQL in introductory physics: Physics Inventory of Quantitative Literacy (PIQL). This prototype PIQL focuses on two components of PQL: proportional reasoning, and reasoning with signed quantities. We present preliminary results from approximately 1,000 undergraduate and 20 graduate students.
2019
Oklahoma City, Oklahoma
In this work we examined the effects of programming as a problem solving heuristic on students’ mathematical work on tasks involving probability and expected value. Analysis of student performance on a unit of instruction that focused on students’ competence in both programming and calculating probabilities and expected values revealed that students can see programming as a valid problem solving strategy and use it effectively
2019
Oklahoma City, Oklahoma
Research on Mathematical Knowledge for Teaching has helped the education community understand the complex, knowledge-related factors that shape instructors’ practices and the learning opportunities they create for students. Much of this work has occurred in the context of K-12 teaching. Although expanding, research on knowledge for teaching undergraduate mathematics is not extensive. A similar situation exists in science education. To help support these research efforts and theory development, we analyzed existing literature on knowledge for teaching undergraduate STEM content. Findings take the form of cross-disciplinary themes and differences that can help inform research efforts in this area. We seek feedback from the RUME community about our representations of knowledge for teaching, ideas about findings from research on Mathematical Knowledge for Teaching that have been especially useful, and/or ideas for research investigations that would be particularly useful to inform curriculum development, professional development for teaching or theory.
2019
Oklahoma City, Oklahoma
Most universities and community colleges are struggling with how to prepare incoming students for the rigor of college-level mathematics courses. At Governors State University, developmental courses are not offered, although a significant number of students do not have the required prerequisite mathematics knowledge for college-level courses. This poster has three themes: an analysis of institutional data on freshmen mathematics, a discussion about navigating conflicting goals and ideas from university leadership, and an examination of mathematics interventions, both those that were tried and recommendations for next steps.
2019
Oklahoma City, Oklahoma
Using a stakeholder-centered design, this interactive poster presents a research framework for attending to equity and supporting transformative change for faculty who teach courses for future K-8 teachers. The poster reports on a research and development project that is creating and examining the impact of professional learning modules for these faculty. One aim of the project is intentional awareness development among faculty about their own views of mathematics and opportunities to learn it, those of their undergraduate students, and those of the children their students will one day teach. The particular focus of the poster is our research attempt to identify and capture aspects of equity that factor into instructor decisions in each phase of their professional learning experience (motivation, construction, and organization).
2019
Oklahoma City, Oklahoma
The goal of this study was to illustrate how notions of identity could be used as an analytical tool to account for such diverse perspectives along with issues of power in the context of Latinx and Native American students. Interviews and classroom observations revealed an array of perspectives regarding what counts as mathematics within a classroom, yet is reflective of an ongoing assimilationist practices that have negatively impacted Indigenous peoples for centuries. I argue for the need for mathematics educators to identify dehumanizing practices in mathematics by seeking the perspective of Indigenous educators.
2019
Oklahoma City, Oklahoma
There is increased focus on exploring the role of computation in students’ learning of mathematical concepts, and the notion of computational thinking has gained prominence. In this poster, we demonstrate ways in which students make connections among different combinatorial representations, and we argue that computational thinking mediated such connections.
2019
Oklahoma City, Oklahoma
Providing students with explanation feedback on their homework has been shown to highly and effectively support their learning. However, there is limited research on how students might play an active role, in collaboration with teachers, to tailor feedback, to meet students’ needs. This study, carried out as a form of practitioner-inquiry in a teacher’s own classroom, explores how a teacher used their own and their students’ shared experiences to refine and develop explanation feedback that supports students’ mathematics learning in an undergraduate Probability and Statistics class. Data include records of co-generative dialogues between the teacher and her ten students, and the students’ homework worksheets. Emergent findings show that co-generative dialogues provide an effective opportunity for teachers to learn from their students how to improve their pedagogical practices, especially in providing effective explanation feedback.
2019
Oklahoma City, Oklahoma
David Tall’s Three Worlds of Mathematics provides a framework through which to study the development of mathematical thought, the three worlds being the embodied, the symbolic, and the formal. This poster shares initial investigations and findings into how the concept of numeral is situated in the symbolic world, beginning with developing a concept image for numeral that is held by pre-service elementary teachers. An analysis of eight mathematics textbooks for pre-service teachers provided an initial definition. Additionally, students in a college mathematics content course for pre-service elementary teachers completed a survey. The participants in the survey were given several symbols and prompted to respond whether or not they believed the symbol represented a numeral. Results of the textbook analysis and the survey were used to understand the students’ concept image of numeral.
2019
Oklahoma City, Oklahoma
This research explores preservice teachers (PSTs) views of students’ mathematical capabilities (VSMC) within mediated field experiences (MFEs) and the role of beliefs on instructional decisions. In MFEs, teacher educators serve as instructors, coaches and supervisors as PSTs plan, enact, and debrief instruction (McDonald et al., 2014). The research questions were: What is the nature of PSTs VSMC across an MFE cycle? How might PSTs beliefs impact instructional decisions? What role might MFEs play in developing productive VSMC? Findings showed that some teachers believed students were incapable of engaging in rigorous instruction, and consequently, would not always respond to student difficulty in ways that helps students participate in rigorous mathematical environments. Results suggest the need to study how teachers might develop more productive VSMC and better support students who struggle. The analysis also revealed how daily debriefs within MFEs supported PSTs to glean general instruction principles to inform their teaching.
2019
Oklahoma City, Oklahoma
The study of calculus focuses on change and thus dictates a need for instruction involving dynamic imagery. DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus) utilizes animations to support students’ dynamic imagery. This poster investigates how students use and understand animations in the DIRACC textbook in connection with associated calculus topics.
2019
Oklahoma City, Oklahoma
The purpose of this study is to examine the meanings and interpretations a student has about the derivative at a point. The responses given by the student is representative of many Calculus 1 students and their beliefs about derivative.
2019
Oklahoma City, Oklahoma
Roughly half of the nearly 44% community college students referred to developmental mathematics never make it into, let alone through, the college-level mathematics courses required for their academic major. The disproportionate number of these students who are from underrepresented groups combined with the low success rates has prompted many community colleges to undertake developmental mathematics reform. The purpose of this study is to provide a multi-perspective account of one community college’s program redesign using Gutiérrez’s equity framework, Tinto’s model of persistence, and activity theory to analyze and interpret the experiences of students, instructors, and administrators.