PROCEEDINGS OF THE 17TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2014
Denver, Colorado
Number of pages: 1203
Number of papers: 144
- TIM FUKAWA-CONNOLLY
- GULDEN KARAKOK
- KAREN KEENE
- MICHELLE ZANDIEH
2014
Denver, Colorado
Number of pages: 1203
Number of papers: 144
2014
Denver, Colorado
Proof is a central concept in mathematics education, yet mathematics educators have failed to reach a consensus on how proof should be conceptualized. I advocate defining proof as a clustered concept, in the sense of Lakoff (1987). I contend that this offers a better account of mathematicians’ practice with respect to proof than previous accounts that attempted to define a proof as an argument possessing an essential property, such as being convincing or deductive. I also argue that it leads to useful consequences for research and teaching.
2014
Denver, Colorado
We analyzed multivariable calculus students’ meanings for domain and range and their generalization of that meaning as they reasoned about domain and range of multivariable functions. We found that students’ thinking about domain and range fell into three broad categories: input/output, independent/dependent variables, and/or as attached to specific variables. We used Ellis’ (2007) actor-oriented generalizations framework to characterize how students generalized their meanings for domain and range from single-variable to multivariable functions. This framework focuses on the process of generalization – what students see as similar between ideas in multiple contexts. We found that students generalized their meanings for domain and range by relating objects, extending their meanings, using general principles and rules, and using/modifying previous ideas. Our results about how students understand and generalize the concepts of domain and range imply that the domain and range of multivariable functions is a topic instructors should explicitly address.
2014
Denver, Colorado
Studies about students’ understanding of the formal definition of a limit, or the epsilon delta definition suggest that the temporal order of delta and epsilon is one of the most challenging aspects of the formal definition. While multiple studies have documented this difficulty for some students, patterns of students’ reasoning about the temporal order are largely unknown. This study investigates ways that students make sense of the temporal order by focusing on the justifications students provided for their claim about the temporal order. diSessa’s Knowledge in Pieces provides a suitable framework to explore the context specificity of students’ knowledge as well as the potential productivity of their prior knowledge in learning.
2014
Denver, Colorado
We studied students’ understanding of the Fundamental Theorem of Calculus (FTC) in graphical representations that are relevant in physics contexts. Two versions of written surveys, one in mathematics and one in physics, were administered in multivariable calculus and introductory calculus-based physics classes, respectively. Individual interviews were conducted with students from the survey population. A series of FTC-based physics questions were asked during the interviews. The written and interview data have yielded evidence of several student difficulties in interpreting or applying the FTC to the problems given, including attempting to evaluate the antiderivative at individual points and using the slope rather than the area to determine the integral. The interview results further suggest that students often fail to make meaningful connections between individual elements of the FTC.
2014
Denver, Colorado
We analyzed the interviews of three prospective secondary mathematics teachers to examine their conceptions of proof and how they validated arguments in the context of students’ answers. Our participants had differing views of the definition of proof and its role in mathematics, and they operationalized their conceptions of proof through differing emphases on generality, logical structure, and form or appearance of arguments. Their work when validating arguments in large part aligned with their professed views of proof, with some deviations on the part of one participant. Further research must examine whether this consistency is prevalent across prospective teachers and how this relates to teachers’ work with proof in classrooms.
2014
Denver, Colorado
We conducted an analysis of 17 modern, introductory linear algebra textbooks to investigate presentations of matrix multiplication. Using Harel’s (1987) textbook analysis framework, we examined the sequencing of matrix multiplication and its accompanying rationale. We found two principal sequences: one which first defines the operation as a linear combination of column vectors before introducing the dot product method (LC to DP), and another which invokes the dot product method before linear combinations (DP to LC). The rationale for these two trajectories varied in interesting ways. LC to DP demonstrates that solving a system of linear equations is equivalent to solving its corresponding matrix equation Ax=b. The rationale for DP to LC was less focused, opting in several cases to postpone the explanation until linear transformations are covered. We hope to initiate a discussion about the effectiveness of and pedagogical implications for these two contrasting approaches.
2014
Denver, Colorado
Prior formative assessment research has shown positive achievement gains when classes using formative assessment are compared to classes that do not. However, little is known about what, if any, benefits students that are not participating regularly in formative assessment gain from these assignments. The purpose of this study was to investigate the achievement of the students in two introductory calculus courses using formative assessment at the three different participation levels observed in class. Although there was no significant difference on any demographic variable other than gender and no significant difference in any achievement predictive variables between the groups of students at the different participation levels, there were significant differences in achievement on all but the first activity write-up and the final exam.
2014
Denver, Colorado
The purpose of this study is to determine the reliability and validity of the Mathematics Classroom Observation Protocol for Practices (MCOP2) in undergraduate mathematics classrooms, an observation instrument designed to measure the degree to which a mathematics classroom aligns with the standards put forth by national mathematics organizations. To examine the reliability and validity of the MCOP2 in the undergraduate setting, over thirty undergraduate mathematics classrooms at a large southeastern university were observed during the fall semester of 2013. The exploratory factor analysis conducted from the data collected indicates there are two main factors to consider in an undergraduate mathematics classroom: “lesson content” and “student engagement and classroom discourse”. The internal reliability of each of these factors was verified using classical test theory to measure well at the group level.
2014
Denver, Colorado
While a significant amount of research has been devoted to exploring why university students struggle applying logic, limited work can be found on how students actually make sense of the notational and structural components used in association with logic. This project borrows the theoretical framework of unitizing and reification, which have been effectively used to explain the types of integrated understanding required to make sense of symbols involved in numerical computation and algebraic manipulation, to investigate students’ conceptualizations of truth tables and implication statements. We use a continuum as a framework to analyze the degree to which students’ thinking of each is compartmentalized versus unified. Results indicate that students tend to treat the constituent pieces that make up these mechanisms independently without an understanding of each as a whole or an integrated view of the two together. Consequently, students manipulate symbols without an appreciation of the overarching meaning.
2014
Denver, Colorado
The primary goal of this work is to articulate a theoretical foundation based on Realistic Mathematics Education (RME) that can support the analysis of student learning. I first describe two RME design heuristics, guided reinvention and emergent models, and explicate each of these heuristics in terms of related theoretical constructs. I then consider how the RME design heuristics could inform how one conceptualizes of and documents student learning (where learning is viewed in terms of the creation of a new mathematical reality). To do so, I draw on two metaphors for learning and, by considering the design heuristics in light of these two perspectives, I propose two ways to conceive of “new mathematical reality” and discuss what could be considered as evidence for student learning.
2014
Denver, Colorado
Research on mathematicians’ pedagogical practice in advanced mathematics is sparse. The current paper contributes to this literature by reporting a case study on a mathematics professor’s presentation of a proof in a real analysis course. By interviewing the professor, we focus on his pedagogical goals when presenting this proof and link those with the actions that he took to achieve these goals. By interviewing six students, we investigate how they interpreted the proof and what they learned from it. Our analysis provides insight into why students did not learn what the professor intended to convey in his presentation.
2014
Denver, Colorado
Although counting problems are easy to state and provide rich, accessible problem solving situations, there is evidence that students struggle with solving counting problems correctly. With combinatorics (and the study of counting problems) becoming increasingly prevalent in K- 12 and undergraduate curricula, there is a need for researchers to identify potentially effective instructional interventions that might give students greater success as they solve counting problems. We tested one such intervention – having undergraduate students engage in systematic listing of what they were trying to count. We show that even creating partial lists of outcomes led to statistically significant improvements in students’ performance on problems, implying that systematic listing may be worthwhile for students to engage in as they learn to count. Our findings suggest that instructional interventions that facilitate listing warrant more attention.
2014
Denver, Colorado
Counting problems provide an accessible context for rich mathematical thinking, yet they can be surprisingly difficult for students. To foster conceptual understanding that is grounded in students’ thinking, we engaged a pair of undergraduate students in a ten-session teaching experiment. The students successfully reinvented four basic counting formulas, but their work revealed a number of unexpected issues concerning justification in counting. In this paper, we describe the students’ successful reinvention of the four counting formulas, we critically examine their combinatorial reasoning in terms of Lockwood's (2013) initial model of students' combinatorial thinking, and we offer several directions for further research.
2014
Denver, Colorado
The purpose of this paper is to investigate a theory about the nature of mathematical development, in which mathematics is characterized as the objectification of action. Informed by existing research on how students construct new mathematical objects, we consider as an example the psychological construction of cohomology and related objects of algebraic topology. This example extends neo-Piagetian theories of mathematical development from elementary school to graduate-level mathematics, while integrating existing research on students’ learning of abstract algebra. Results of the investigation affirm the objectification of action as a distinguishing feature of mathematics in general, while indicating the kinds of mental actions that undergird the objects of advanced mathematics.
2014
Denver, Colorado
Classroom teaching in multiple sections of Calculus I at a large comprehensive research university was observed and coded using the Teaching Dimensions Observation Protocol (TDOP). Multiple teaching styles were identified ranging from low engagement to moderate engagement to high engagement sometimes including student group work. Student performance on two course-wide uniform exams and on the Calculus Concept Inventory (CCI) was analyzed for any correlations with teaching methods. Significant correlations were found between high engagement teaching styles and performance on both the first exam and the final exam. However, section normalized gains on the CCI were found to be significantly correlated only with the presence of student group work or desk work and with no other measures of teaching practice or student performance.
2014
Denver, Colorado
In this study, a secondary school teacher’s goals for student learning were characterized using a framework that emerged from prior work. Observed lessons spanning the use of both conceptually rich and skill-based curricula were analyzed. The findings suggest that both challenges and opportunities exist for professional development endeavors that center around perturbing a teacher's goals.
2014
Denver, Colorado
This study examined how mathematical modeling activities within a collaborative group impact on students’ perceived ‘value’ of mathematics. With a unified framework of Makiguchi’s theory of ‘value’, mathematical disposition, and identity, the study identified the elements of the value-beauty, gains, and social good-with the observable evidences of mathematical disposition and identity. A total of 60 college students participated in ‘Lifestyle’ mathematical modeling project. Both qualitative and quantitative methods were used for data collection and analysis. The result from a paired-samples t-test showed the significant changes in students’ mathematical disposition. The results from the analysis of students’ written responses and interview data described how the context of the modeling tasks and the collaborative group interplayed with students’ perceived value.
2014
Denver, Colorado
We present the results of a study of the observed proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to- proof course. Students were interviewed individually towards the end of the course using the same protocol that we had used earlier at the beginning of a similar course (Selden and Selden, 2003). Results include a description of the students’ observed validation behaviors, a description of their proffered evaluative comments, and the, perhaps counterintuitive, suggestion that taking an inquiry- based transition-to-proof course does not seem to enhance validation abilities. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and the need for research on their interrelations.
2014
Denver, Colorado
The purpose of this study was to gain knowledge about undergraduate transition-to- proof course students’ proving difficulties. We analyzed the final examination papers of students in one such course. Our perspective included drawing inferences about students’ sometimes automated links between situations and mental, as well as physical, actions. We have identified process, rather than mathematical content, difficulties such as not constructing a proof framework, not unpacking the conclusion, and not using definitions correctly. The ultimate goal is to contribute to an understanding of some of these kinds of difficulties as pedagogical content knowledge with which to teach or redesign transition-to-proof courses.
2014
Denver, Colorado
Most recently, the Conference Board of the Mathematical Sciences has advocated for incorporating technology in secondary mathematics classrooms. Colleges and universities across the United States are incorporating technology to varying degrees into their mathematics teacher preparation programs. This study examines preservice secondary mathematics teachers’ opportunities to expand their knowledge of algebra through using technology and to learn how to incorporate technology when teaching algebra in mathematics classrooms. We explore the research question: What opportunities do secondary mathematics teacher preparation programs provide for PSTs to encounter technologies in learning algebra and learning to teach algebra? We examine data from a pilot study of three Midwestern teacher preparation programs conducted by the Preparing to Teach Algebra (PTA) project. Our data suggest that not all secondary mathematics teacher preparation programs integrate experiences with technology across mathematics courses, and that mathematics courses may provide few experiences with technology to PSTs beyond strictly computational.
2014
Denver, Colorado
Abstract algebra is recognized as a highly problematic course for most undergraduate students. Despite these difficulties, most mathematicians and mathematics educators affirm its importance to undergraduate mathematical learning. The purpose of this research was to formulate a list of the important concepts in abstract algebra as perceived by graduate students in mathematics, understand how they define these concepts, and recognize any relationships or connections between these concepts perceived by the students. The theoretical perspective of concept images and concept definitions as described by Tall and Vinner (1981) and Vinner (1983) was used to investigate participants’ understanding of abstract algebra concepts. Through an interview study, the students’ perceptions were analyzed through the creation of concept maps. The results revealed the participants had great difficulty articulating their concept images and concept definitions. In addition, they had differing views of major concepts and relationships within the course.
2014
Denver, Colorado
This is a preliminary report of a study where the purpose was to examine how participation in a mathematics master’s program for in-service teachers affected student growth on a state mathematics assessment taking into account student demographic variables. We collected data from school districts for 5 academic years spanning from pre- program through program completion. We conducted a four-step hierarchical multiple linear regression analysis. We can conclude that the combination of teachers’ years of participation in the Math TLC, teachers’ total years of experience, student ethnicity, and student free and reduced lunch eligibility provided a joint effect on the student growth percentiles on the state mathematics assessment. We explain future plans for continued research on this project.
2014
Denver, Colorado
Studies have shown that students have difficulties with the concepts of slope and derivative, especially in the case of real-life contexts. I used a written survey to collect data from 74 differential calculus students. Students answered questions about linear and nonlinear relationships and interpretations of slope and derivative. My analysis focused on students’ understanding of slope as a constant rate of change and derivative as an instantaneous rate of change, and what these meant in the context of the problems. Preliminary results indicate that students have more success with slope questions than derivative questions (McNemar’s test, p<0.05), and that while students correctly use the slope of a linear relationship to make predictions, they do not demonstrate an understanding of the derivative as an instantaneous rate of change and an estimate of the marginal change.
2014
Denver, Colorado
To build a supplementary theory from which we can derive a practical way of fostering inquiring minds in mathematics, this paper proposes a theoretical perspective that is compatible with existing ideas in mathematics education (radical constructivism, social constructivism, APOS theory, David Tall’s framework, the framework of embodied cognition, new materialist ontologies). We focus on the fact that descriptive and prescriptive statements can be treated simultaneously, and consider both descriptive and instantiated models in our minds. This indicates that descriptive statements in mathematics come from our descriptions of models, and prescriptive statements come from the instantiatedness of the instantiated models and non-existence of counterexample. As a practical suggestion from the proposed perspective, we point out that careful communication is needed so that students do not recognize the refutation of their arguments as a denial of their way of mathematical thinking.
2014
Denver, Colorado
Investigating teacher practice at all educational levels has become an important research arena. We consider the teaching of inquiry-oriented differential equations in undergraduate classrooms by comparing two enactments of the same fragment of a student-centered curriculum by two mathematics professors. We highlight differences in the professors’ practices and the consequent classroom results by analyzing the professors’ participation in whole-class discussions and the decisions they made during class. By considering how the same written curriculum can be enacted in very different ways in undergraduate level mathematics classrooms, we call for greater attention to research on the relationships between a written curriculum, an enacted curriculum, and student outcomes.
2014
Denver, Colorado
Although the research literature concerning topics in trigonometry is growing, explorations of students’ conceptions of trigonometric identities is scant. This study aimed to contribute to this area by considering the extent that students developed a structural understanding of trigonometric identities and the implications their understanding had on the solving of problems involving identities. Through task-based interviews involving verifying trigonometric identities, students appeared to view certain identities differently than other identities, holding a deeper understanding of them. Their understanding and trust in these identities allowed for the students to use them in the construction of new identities. In order to verifying identities involving complicated function arguments, some students either ignored the argument or replaced it with a single letter, typically an x. Although doing so allowed the students to focus on the structure of the identity, students’ conceptions of variable remained at a conceptually-weak level.
2014
Denver, Colorado
The purpose of this paper is to argue that attention to students’ ways of thinking should complement a focus on students’ understanding of specific mathematical content, and that attention to these issues can be leveraged to model the development of mathematical knowledge over time using learning trajectories. To illustrate the importance of ways of thinking, we draw on Harel’s (2008a, 2008b) description of mathematical knowledge as comprised of ways of thinking and ways of understanding. We use data to illustrate the explanatory and descriptive power that attention to the duality of ways of understanding and ways of thinking provides, and we propose suggestions for constructing learning trajectories in mathematics education research.
2014
Denver, Colorado
Studies about students’ understanding of the formal definition of a limit, or the epsilon delta definition suggest that the temporal order of delta and epsilon is an obstacle in learning the formal definition. While such difficulty has been widely documented, patterns of students’ reasoning are largely unknown. This study investigates the degree of difficulty students have with the temporal order, along with justifications that students provide to support their claim. diSessa’s Knowledge in Pieces provides a suitable framework to explore the context specificity of students’ knowledge as well as the potential productivity of their prior knowledge in learning.
2014
Denver, Colorado
In this paper we report a study designed to investigate the impact of logical reasoning ability on proof comprehension. Undergraduates beginning their study of proof-based mathematics were asked to complete a conditional reasoning task that involved deciding whether a stated conclusion follows necessarily from a statement of the form “if p then q”; they were then asked to read a previously unseen proof and to complete an associated comprehension test. To investigate the broader impact of their conditional reasoning skills, we also constructed a composite measure of the participants’ performance in their mathematics courses. Analyses revealed that the ability to reject invalid denial-of-the-antecedent and affirmation-of-the- consequent inferences predicted both proof comprehension and course performance, but the ability to endorse valid modus tollens inferences did not. This result adds to a growing body of research indicating that success in advanced mathematics does not require a normatively correct material interpretation of conditional statements.
2014
Denver, Colorado
The quality of education in introductory calculus classes is an issue of particular educational and economic importance. In work related to a national study of college calculus programs conducted by the MAA, I report on a study of four different pedagogical approaches to Calculus I at a single institution in the Fall 2012 semester. Using statistical methods, I analyze the effects of these four approaches on students’ persistence in STEM major tracks, attitudes and beliefs about mathematics, and procedural and conceptual achievement in calculus. Using qualitative methods, I draw links from the statistical results to differences and commonalities in the four classroom strategies.
2014
Denver, Colorado
We studied students’ understanding of the Fundamental Theorem of Calculus (FTC) in graphical representations that are relevant in physics contexts. Two versions of written surveys, one in mathematics and one in physics, were administered in multivariable calculus and introductory calculus-based physics classes, respectively. Individual interviews were conducted with students from the survey population. A series of FTC-based physics questions were asked during the interviews. The written and interview data have yielded evidence of several student difficulties in interpreting or applying the FTC to the problems given, including attempting to evaluate the antiderivative at individual points and using the slope rather than the area to determine the integral. The interview results further suggest that students often fail to make meaningful connections between individual elements of the FTC.
2014
Denver, Colorado
In the United States, many students enter their first year of college unprepared to take college level mathematics: at public 4-year institutions, 16% of entering freshmen took remedial (developmental) mathematics in 2000 (U.S. DOE, 2003, p. 18). These students are less likely to enter or persist in STEM majors, and even less likely to graduate from college than students who are prepared for, and succeed in, college mathematics in their first year (Adelman, 2006). According to the recent AAU (2012) draft discussion document announcing their STEM initiative, about 25% of freshmen across the country intend to go into a STEM field, but only 15 -17% of graduates complete a STEM major (AAU, pp. 2-3). Most students who drop STEM majors do so during the first two years of college, often because of trouble in their first year mathematics courses (AAU, 2012, p. 4).
2014
Denver, Colorado
Learners of mathematics, including preservice teachers, often explore online resources when investigating mathematical problems. When asked to search online for resources that would help them be able to better explain a proof of the Pythagorean theorem and its converse, preservice teachers used a variety of different searching strategies to locate information. Further, the ways in which this information was incorporated into their understanding of mathematics became evident through concept maps. This proposal describes the study conducted and initial results from the data and asks the reader to consider possible ways this research might be extended and refined.
2014
Denver, Colorado
This paper shares findings from a three-phase study exploring students’ conceptions of non- constructive existence proofs. Data are used to illustrate students’ tendency to apply a naïve Brouwerian lens to non-constructive proofs; that is, a perspective in which learner’s proof conceptions are governed by a potentially subconscious anticipation of construction, which enables the learner to construe proofs of existence (be they constructive or non-constructive) as providing actual instances of (or algorithms for producing) mathematical phenomena. Questions concerning researchers proof scheme inferences are raised.
2014
Denver, Colorado
Although deciding on the truth value of mathematical statements is an important part of the proving process, students are rarely engaged in making such decisions. Thus, little is known about the ways in which students use intuitive, informal, and formal reasoning to evaluate conjectures. In this study, task-based interviews were conducted with undergraduate students in which they were asked to determine the truth value of five mathematical statements on functions and relations. Students’ reasoning on these tasks will be classified as intuitive, informal, or formal, and then further categorized according to the findings of current research, with new categories added as needed. This study should contribute to our understanding of the ways in which students reason when dealing with uncertainty in the proving process. Additionally, this study may suggest ways in which educators can assist students in navigating the often difficult process of proving and refuting mathematical statements.
2014
Denver, Colorado
Engaging students in the construction of proofs often does not include conversations about what does and does not count as proof, to the detriment of the students. The critiques of student-generated arguments should be communicated in a language common to instructor and student; such a language can be developed via an assessment tool that is accessible to both parties. This paper describes the development of an argument assessment tool that will be useful for instructors and researchers both to assess students’ and participants’ ability to construct proofs and to communicate those assessments. The tool is introduced and two assessed student arguments are shared to illustrate the tool’s application. Future work with the argument assessment tool will include its use in a classroom as an instructional tool for establishing a common language for instructor and students and providing the foundation for discussions about proof production.
2014
Denver, Colorado
This study was designed to investigate the types of questions college students generate as they watch video lectures in a business calculus class. Thirty-six students taking an undergraduate calculus course participated in the study. In this paper we share the preliminary results of our qualitative analysis. We have found nine mutually exclusive categories that uncover the thoughts, struggles, and successes our students go through as they experience this new teaching modality of video-viewing. We also include three questions for the audience to help further our analysis and open up new research opportunities for the improvement of collegiate teaching through the study of students’ questions.
2014
Denver, Colorado
I report initial findings of a study that seeks to investigate the change in developmental (remedial) mathematics students’ mathematical problem solving skills. I report on the analysis of one-on-one interviews with six students before a four-week Intermediate Algebra course. The ultimate goal is to see the extent to which their skills changed after the course. Using a framework of reasoning developed by Lithner (2000), I describe events in which one particular student shows plausible reasoning and also reasoning based on established experience. I seek input with regard to alternative frameworks or analysis of the data that may help me interpret the findings.
2014
Denver, Colorado
This study will demonstrate the ways in which students’ ideas about convergence of infinite series are deeply connected to the particular representation of the mathematical content, in ways that are often conflicting and self-contradictory. Specifically, this study explores the different limiting processes that students attend to when presented with five different phrasings of a particular mathematical task - ∑(1/2)n - and the ways in which each phrasing of the task brings to light different ideas that were not evident or salient in the other phrasings of the same task. This research suggests that when attempting to gain a more robust understanding of the ways that students extend the ideas of calculus – in this case, limit – one must take care to attend to not only students’ reasoning and explanation, but also the implications of the representations chosen to probe students’ conceptions, as these representations may mask or alter student responses.
2014
Denver, Colorado
This poster aims to present a modified version of SPOT diagrams (Structure Perceived Over Time) (Yoon, 2012) – an aspect of analysis and data presentation used to present interactive student video data, during which perceptual shifts may occur. The larger study in which this tool was employed (Champney, in preparation) explored undergraduate calculus students’ self-generated representations (SGR) used during interviews in which they were asked to explain to an absent peer the events of the day(s) during which infinite series were introduced and discussed. While typical studies ask students to address tasks and issues framed by a researcher, this study instead asked students to explain the content, thereby providing a broader window into “what counts” from the student perspective.
2014
Denver, Colorado
In this study the prospective teachers’ understanding of extreme probabilities is studied via their examples. Watson and Mason’s Learner Generated Examples (LGE) theory is employed to justify the type of data used in this study and to emphasize the importance of examples in learning about different levels of the learners’ probabilistic thinking.
2014
Denver, Colorado
This research focuses on the cognitive challenges that students face and how they resolve these challenges while transitioning from intuitive reasoning to constructing a more formal mathematical structure of Riemann sum while modeling “real life” contexts. A pair of Calculus I students who had just received instruction on definite integral defined using Riemann sums and illustrated as area participated in ten interviews. They were given three contextual problems related to Riemann sums but were not informed of this relationship. The intent was to observes students' transitioning from "model-of" to “model-for” reasoning based on Gravemeijer and Stephan (2010). Findings indicate that it was not the end results but records of their ways of acting and reasoning about their contextual problem through multiple representations along with real life intervention that served as tools for supporting their transition from “model-of” informal activities to “model-for” more formal mathematical reasoning.
2014
Denver, Colorado
A student who has completed both Linear Algebra and Quantum Mechanics should have a wealth of conceptual and procedural knowledge that has been obtained from mathematics and physics classes. However in practice, students seem to struggle with this task. This investigation casts light on students’ thinking about matrix multiplication and how their thinking appears to be influenced by their framing of the problem as either a mathematics or physics question. Using Framing and Resources as a theoretical lens can provide insight into the ideas and concepts that a student accesses from domains of mathematics and physics. Using lexicon analysis, it appears the student shifts from a “mathematical frame” to a “physics frame” and back again, but struggles to successfully transfer concepts between these two frames. I will highlight the markers for these frame shifts and demonstrate why framing and resources is the appropriate lens for this investigation.
2014
Denver, Colorado
Stochastic conceptions undergird development of conceptual connections between probability and statistics and support development of a principled understanding (Greeno, 1978) of probability distribution. This study employed mixed research methods to investigate the impact of an instructional course intervention designed to support development of stochastic understanding of probability distribution. Instructional supports consisted of supplemental lab assignments comprised of anticipatory tasks designed to engage students in coordinating thinking about complementary probabilistic and statistical notions along a hypothetical learning trajectory aimed at development of stochastic understanding of probability distribution. Participants were 184 undergraduate students enrolled in a lecture/recitation, calculus-based, introductory probability and statistics course. Results of quantitative analyses showed completion of stochastic lab assignments had a statistically significant impact on students’ stochastic understanding of probability distribution. Student interviews revealed those who held stochastic conceptions also indicted integrated reasoning related to probability, variability, and distribution and presented images supporting principled understanding of probability distribution.
2014
Denver, Colorado
We analyzed the interviews of three prospective secondary mathematics teachers to examine their conceptions of proof and how they validated arguments in the context of students’ answers. Our participants had differing views of the definition of proof and its role in mathematics. Their work when validating arguments in large part aligned with their professed views of proof, with some deviations on the part of one participant. Further research must examine whether this consistency is prevalent across prospective teachers and how this relates to teachers’ work with proof in classrooms.
2014
Denver, Colorado
This paper reports on an exploratory study of 10 algebraists designed to investigate the reasoning behind their selection of examples for their own teaching and research. Variation theory provided a lens with which to analyze the algebraists’ goals for their collections of examples and to speculate about the resulting pedagogical implications. Though findings from this exploratory study should be regarded only as preliminary and in need of further justification, our results provide some initial evidence that mathematicians use a relatively small number of very well- chosen classes of examples in both their teaching and their research (suggesting that this might be a useful pedagogical strategy for students as well). We also report on the examples of groups and rings that the algebraists deemed to be the most important for students of introductory abstract algebra.
2014
Denver, Colorado
Abstract algebra is a fascinating field of study among mathematics topics. Despite its importance, very little research has focused on the teaching of abstract algebra. In response to this deficiency, in this study we present an abstract algebra professor’s daily activities and thought processes as shared through his teaching diaries with a team of two mathematics educators and another abstract algebraist over the period of two semesters. We examined how he was able to live in the formal world of mathematical thinking while also dealing with the many pedagogical challenges that were set before him during the lectures.
2014
Denver, Colorado
We conducted an analysis of 17 modern, introductory linear algebra textbooks to investigate presentations of matrix multiplication. Using Harel’s (1987) textbook analysis framework, we examined the sequencing of matrix multiplication and its accompanying rationale. We found two principal sequences: one which first defines the operation as a linear combination of column vectors before introducing the dot product method (LC to DP), and another which invokes the dot product method before linear combinations (DP to LC). The rationale for these two trajectories varied in interesting ways. LC to DP demonstrates that solving a system of linear equations is equivalent to solving its corresponding matrix equation Ax=b. The rationale for DP to LC was less focused, opting in several cases to postpone the explanation until linear transformations are covered. We hope to initiate a discussion about the effectiveness of and pedagogical implications for these two contrasting approaches.
2014
Denver, Colorado
This study investigates the understandings of mean, median, distribution and standard deviation that undergraduate students have at the end of an introductory statistics course. The goal was to explore their understandings as a follow-up to previous studies documenting incoming student difficulties with the concepts and determine whether a course would help them achieve a more statistically appropriate understanding. They overwhelmingly think about the mean as the “average” and via the calculating formula, meaning they understand it as a process. Similarly, they understand the median in terms of the process for determining it, or via the location-based term, “middle.” As a result, students do not generally understand the two measures to be describing a similar concept. Students do, reliably connect the shape of a distribution to standard deviation, but that connection varies by type of display and is not based on a reliable rule.
2014
Denver, Colorado
Mathematical modeling tasks are used to help students learn mathematics and also to improve their modeling skills. Validating has been identified as the process by which students check and revise their models, but little is known about when or how students choose to do so. This study examined engineering students’ validating activity and characterized it into a typology of different kinds of validating activity satisfying different roles in ensuring accuracy of the model.
2014
Denver, Colorado
Recent work has shown that there is a lack of coherence from calculus to differential equa- tions (Czocher, Tague, & Baker, 2013). We define lack of coherence as the gap between the knowledge students are expected to gain by the end of the calculus sequence versus how cal- culus knowledge is expected to be used in differential equations (Czocher, Tague, & Baker, 2013). In this report, we describe how we have exploited the flipped classroom model to begin to address some of these issues with coherence. We share our theoretical perspective, how it was enacted, and also a preliminary evaluation of students’ perceptions of the coher- ence of the course and its content.
2014
Denver, Colorado
Many consider logic a hallmark of mathematical practice and an integral part of proof- oriented mathematical instruction. This is true of the term logic whether it refers to a domain of mathematical study or to aspects of reasoning, but I claim that these formalized and psychological senses of the term must be carefully distinguished in mathematics education research. In the course of identifying how the abstraction criterion has been misapplied across various types of logic in psychological and mathematics education research, I outline a framework for the disambiguation of the range of research constructs referred to as logic. By distinguishing the types of logic pertinent to mathematics education instruction, I hope to provide a language by which future research can better specify the constructs they investigate. Clearer research constructs should help the community to understand the role various logics play in students’ apprenticeship into the practices of advanced mathematics.
2014
Denver, Colorado
Functions are a crucial topic in the study of mathematics. Research has found that a lack of deep understanding of functions is one of the main reasons why students struggle in calculus (Eisenberg, 1991; Ferrini-Mundy & Graham, 1991; Lauten, Graham, Ferrini-Mundy, 1994; McDonald, Mathews, & Strobel, 2000; Monk, 1994). In light of these studies, we investigate – using traditional paper- and-pencil assessments, concept maps, and an interview – what pre-calculus students’ understanding of functions is, to what extent students have a repertoire of functions at their disposal, how students’ understanding evolves over a semester, and what non-traditional assessments can tell us about this understanding. We found that (1) As Williams (1998) suggested, concept map assessments do reveal something that traditional assessments do not; (2) participants have trouble giving non-examples of functions, and (3) there does not seem to be a major change in participants’ understanding of functions over time.
2014
Denver, Colorado
Prior formative assessment research has shown positive achievement gains when classes using formative assessment are compared to classes that do not. However, little is known about what, if any, benefits students that are not participating regularly in formative assessment gain from these assignments. The purpose of this study was to investigate the achievement of the students in two introductory calculus courses using formative assessment at the three different participation levels observed in class. Although there was no significant difference on any demographic variable other than gender and no significant difference in any achievement predictive variables between the groups of students at the different participation levels, there were significant differences in achievement on all but the first activity write-up and the final exam.
2014
Denver, Colorado
Mathematicians use examples strategically while working on mathematical conjectures, and this strategic usage helps them gain a lot of insight about mathematical phenomenon. However, students do not always have the same strategic example usage; instead, they tend to over rely on examples without understanding of example based reasoning. This study examines college algebra students’ responses on a written assessment in the function domain and discusses students’ example spaces. The results reveal that students have very limited example space in the function domain that affects their strategic example usage. Student example usage was very limited to conventional example spaces that they learned during instruction or from their textbook. This study suggests that having conventional example spaces does not guarantee that students can use examples strategically which can help them better understand the mathematical conjectures.
2014
Denver, Colorado
Examining the understanding of formal logic by early undergraduates leads to the examination of a cycle within the broader scope of math education. Some early undergraduate students will eventually become pre-service teachers. The pre-service teachers will in turn become primary and secondary educators. Finally over time, some of their students will become undergraduates leading to a cycle. This poster proposes the idea that any change of curricula in undergraduate mathematics should have a ripple effect on the overall understanding of formal logic by students of all levels of education over a period of time as the cycle flows.
2014
Denver, Colorado
Units of measure are critical in many scientific fields. While instructors often note that students struggle with units, little research has been conducted about the nature and extent of these difficulties or why they exist. This study investigated calculus students’ unit use in area and volume computations. Seventy-three percent of students gave incorrect units for at least one task. The most common error was the misappropriation of length units in area and volume computations. Analyses of interview data indicate that some students think that the unit of the area or volume computation should be the same as the unit specified in the task statement. Findings also suggest that some students have difficulties correctly indicating the units for computations that involve the quantity π. In addition, findings suggest that calculus students’ difficulties with units are linked to their difficulties with understanding area and volume as arrays.
2014
Denver, Colorado
We analyzed multivariable calculus students’ meanings for domain and range and their generalization of that meaning as they reasoned about domain and range of multivariable functions. We found that students’ thinking about domain and range fell into three broad categories: input/output, independent/dependent variables, and/or as attached to specific variables. We used Ellis’ (2007) actor-oriented generalizations framework to characterize how students generalized their meanings for domain and range from single-variable to multivariable functions. This framework focuses on the process of generalization – what students see as similar between ideas in multiple contexts. We found that students generalized their meanings for domain and range by relating objects, extending their meanings, using general principles and rules, and using/modifying previous ideas. Our results about how students understand and generalize the concepts of domain and range imply that the domain and range of multivariable functions is a topic instructors should explicitly address.
2014
Denver, Colorado
In this report I present findings from a large, national study focused on Calculus I instruction. Graduate student Teaching Assistants (GTAs) contribute to Calculus I instruction in two ways: : as the primary teacher and as recitation leaders. As teachers, GTAs are completely in charge of the course just as a lecturer or tenured track/ tenured faculty would be, although they lack the experience, education, or time commitment of their faculty counterparts. In this study, I investigate how GTAs compare to tenure track/tenured faculty, and other full/part time faculty on their (a) beliefs about mathematics; (b) instructional practices; and (c) students’ success in Calculus I. Findings from this report point clearly to a need to prepare GTAs adequately for the teaching of calculus but also for further examination of the nature and implications of the differences between GTA and other instructor types’ beliefs about teaching and teaching practices.
2014
Denver, Colorado
Previous studies have indicated that effective mathematics teaching relies on teachers’ knowledge of both student thinking and mathematical content. Very little is known about the integration (combination) of teacher’s mathematical knowledge and science knowledge for teaching important topics like derivative and applied derivative problems. The goal of this study is to examine the knowledge of mathematics and science that teachers draw on when teaching the concept of derivative and applied derivative problems. We conducted task-based interviews with nine graduate assistants (GTAs). Findings revealed that GTAs made use of their knowledge of science as well as of mathematics when discussing how to teach applied derivative problem. In this proposal, we only look at the results of two interviews and try to shed light into the nature of science and mathematics knowledge the teachers use for Teaching and how that can lead into opportunities in professional development for the novice teachers.
2014
Denver, Colorado
Despite the large amount of time university students are expected to spend studying material and learning on their own outside of the classroom, little is known about what specific student study habits look like. This study sought to start developing a description of what activities students engage in when studying together in self-formed groups outside of the classroom. By identifying a set of macrotasks, verbally-cued transactions that identify what activity the group is currently engaged in doing, this study provides a way to compare how different study groups allocate their time and distinguish between the enactment of social and sociomathematical norms outside of the classroom.
2014
Denver, Colorado
The purpose of this study was to document mathematics teachers’ models of quantitative reasoning as they participated in a Model Eliciting Activity (MEA) grounded in their classroom practice. This MEA was designed and implemented in a master's course of 21 in- service mathematics teachers. The documents produced by the teachers were analyzed using a models and modeling perspective to determine how teachers’ models of quantitative reasoning developed through the MEA. Findings from this study included a framework describing the two ways teachers developed their model of quantitative reasoning. First, teachers’ models of quantitative reasoning became more coherent by being better articulated and connected between abstract and practical settings. Second, the middle school teachers’ models became more aligned with mathematics education literature by referring to quantities and quantitative relationships as aspects of quantitative reasoning, while most high school teachers’ models did not become more aligned with literature.
2014
Denver, Colorado
We describe the development of a new observation protocol instrument for classroom instruction that is mathematics-specific, spans K-16 mathematics, improves validity and reliability compared to existing instruments, and encompasses the Standards for Mathematical Practice. The instrument may be helpful for educators/researchers engaged in classroom evaluations of K-16 mathematics teaching.
2014
Denver, Colorado
Statistics educators consider statistical literacy a vital skill because it supports students in thinking critically about the way data is used in everyday social, political and medical contexts. Statistical literacy goes beyond simply reading graphs to include interpreting their meaning and evaluating, with a questioning attitude, the information being presented (Shaughnessy, 2007; Gal, 2002; Watson & Moritz, 1997). Watson (1997) describes statistical literacy in terms of a three-tiered hierarchy delineating the skills necessary for interpreting stochastic information.
2014
Denver, Colorado
The CSPCC (Characteristics of Successful Programs in College Calculus) project is a large empirical study, investigating mainstream Calculus 1, that aims to identify the factors that contribute to successful programs. The CSPCC project consists of two phases. Phase 1 entailed large-scale surveys of a stratified random sample of college Calculus 1 classes across the United States. Phase 2 involves explanatory case study research into programs that were identified as successful based in part on the results of the Phase 1 survey. This second phase will lead to the development of a theoretical framework for understanding how to build a successful program in calculus and in illustrative case studies for widespread dissemination. Technology was one of the topics we explored with students, instructors, administrators, and other individuals that we interviewed during our case study site visits. In this preliminary report, we will focus on calculus instructors’ views on instructional technology.
2014
Denver, Colorado
Although teacher quality is positively correlated with student achievement, easily quantified measures of teacher quality are not accurate measures of quality; teacher pedagogical content knowledge and skills are better predictors, but difficult to measure. Professional development may be a cost-effective vehicle for developing new skills in in-service teachers, but there is conflicting research on whether professional development measurably raises student achievement on high stakes standardized tests. The purpose of this causal- comparative study was to examine Andrew, an in-service, high school teacher participant in the master’s program. State mathematics assessment and student demographic data were collected from school districts for 4 academic years spanning from pre-program through program completion. One-way ANOVA analysis on student scale scores factoring by year showed a significant decrease in student mathematics scale scores potentially attributable to differences in population. Independent-samples t tests on the final two years showed a statistically insignificant increase in student growth percentiles.
2014
Denver, Colorado
Previous gesture studies conjecture that as individuals develop expertise in a field of mathematics their gestures tend to become more metaphoric, iconic, and dynamic. In this mixed-methods study, we compared the gestures of six experts and four pairs of novices as they geometrically described the complex number arithmetic operations z+w, zw, and 1/z. An ANOVA revealed that the factors Task and Gesture were statistically significant, but there was no statistically significant difference between the two groups’ gesture use. A Hierarchical Cluster Analysis directed the qualitative analysis where we found that novices exposed to technology appeared to produce gestures that were innovative or similar to the experts’ gestures. These findings suggest that facilitating students’ awareness of their own and the instructors’ gestures as well as exposing students to technology may help them develop more dynamic gestures and in turn possibly facilitate a more geometric perspective of the arithmetic of complex numbers.
2014
Denver, Colorado
While a significant amount of research has been devoted to exploring why university students struggle applying logic, limited work can be found on how students actually make sense of formal logic itself and the logical mechanisms used to communicate logical equivalence. This project borrows the theoretical framework of unitizing and reification, which have been effectively used to explain the types of integrated understanding required to make sense of symbols involved in numerical computation and algebraic manipulation, to investigate students’ conceptualization of truth tables and implication statements. By using a continuum as a framework to analyze the degree to which students’ thinking of each is compartmentalized versus unified, results indicate that students tend to favor one logical mechanism over another, without establishing a holistic view of both or an integrated view of the two together.
2014
Denver, Colorado
Many funding agencies require evaluation of the impact of professional development projects they support. However, improved student outcomes, the ultimate goal, may take longer to be realized than the project time frame allows. Instructors need time to implement and refine new skills before positive student outcomes are realized, a delay that may be exacerbated in classes that are not taught frequently. We report on one example of an efficient and cost- effective self-report measure designed to detect the initial changes in teaching practices that lead to improved student outcomes over time. We discuss the ability for timely and accurate measures through this instrument. Results support the interpretation that instructors’ reported teaching practices show changes consistent with methods taught at professional development workshops on Inquiry-Based Learning in mathematics. Additionally, correlations with self-reported level of implementation suggest that instructors are reporting honestly, and not just socially desirable changes consistent with their concept of “real Inquiry-Based Learning.”
2014
Denver, Colorado
In the light of a rough description of the different contexts in which mathematics is learned and used in engineering studies, this report addresses epistemic relations between mathematics in higher mathematics lectures and mathematics in advanced engineering courses. In particular it elaborates on how different meanings of symbols, as subjectively relevant aspects of mathematical objects, are related to different institutional contexts and their dominant discourses. It is argued that modeling cycles are not an adequate tool in this context. Instead, we suggest using concepts from Anthropological Theory of Didactics (ATD). Inspired by (Castela & Romo Vázquez, 2011), exemplarily concepts from ATD are applied to topics and data from signal analysis. Finally, we claim this research could serve as a step towards investigating empirical questions relevant to students’ learning and competences and, in particular, optimizing curricula and teaching in undergraduate mathematics.
2014
Denver, Colorado
Science and engineering instructors often observe that students have difficulty using or applying prerequisite mathematics knowledge in their courses. Historically, transfer theory is used to investigate students’ issue applying their vector knowledge from a trigonometry course to a physics course, but this qualitative case-study is positioned differently epistemologically and theoretically from transfer theory to understand and describe the mathematical vector practices in the two courses. Saussure’s (1959) concept of signifier and signified provided a lens for examining the data during analysis. Multiple recursions of within-case comparisons and across- case comparison were analyzed for differences in what the instructors and textbooks explicitly stated and later performed as their practices. While the trigonometry and physics instruction differed slightly, the two main differences occurred in the nature and use of vectors in the physics course.
2014
Denver, Colorado
There has been “increasing awareness that reasoning is central to mathematics and mathematics learning” (Yackel & Hanna, 2003, p. 227) among education researchers, which raises questions about student conceptions of reasoning, argumentation and mathematical proof. Mathematicians and mathematics education researchers have consistently asserted the crucial roles deductive reasoning and proof play in discovering, communicating, verifying, understanding, and systematizing mathematics (Hanna, 2000; Ko, 2008; Thurston, 1998). In response to the many affirmations of the importance of proof for learning and understanding mathematics, there has been extensive study of students’ conceptions of mathematical proof, their abilities to construct and understand proofs, and the frequency with which inductive evidence is accepted as sufficient verification of mathematical conjectures (Bell, 1976; Healy & Hoyles, 2000; Ko, 2008; Stylianides, 2009; Varghese, 2009).
2014
Denver, Colorado
The primary goal of this work is to articulate a theoretical foundation based on Realistic Mathematics Education (RME) that can support the analysis of student learning. To do so, I will first frame the guided reinvention and emergent models design heuristics separately in terms of both increasingly general student activity and in terms of concept development. Then, I will consider how the RME design heuristics could inform how one conceptualizes student learning. To do so, I will draw on two metaphors for learning and, by drawing on these two perspectives, propose ways in which the RME design heuristics can inform the analysis of student learning.
2014
Denver, Colorado
This report draws on data collected by the Characteristics of Successful Programs in College Calculus project in order to investigate issues around coverage and pacing. This includes identifying what topics are being taught in Calculus I, determining the extent to which instructors and departments feel pressure to cover a set amount of material, and investigating possible relationships between concerns over coverage, instructional practices, and the nature of the material covered at five institutions selected for having successful Calculus programs.
2014
Denver, Colorado
Student understanding of the integral is a topic of recent interest in undergraduate education. We are just beginning to learn how different interpretations of the definite integral influence student thinking in both mathematics and science classrooms. This paper examines the relative “productivity” of three conceptualizations of the definite integral in mathematics and physics tasks. It appeared that the notion of the integral as an “addition over many pieces” was especially useful for understanding applied problems.
2014
Denver, Colorado
An undergraduate mathematics major should come to hold appropriate views about the conclusions reached by our disciplinary methods. This project explores the views about truth in axiomatic mathematics of a group of students who are (mostly) in their final proof-based course, Modern Geometry. Do these students hold expert-like views about truth in mathematics, and do those views change during a course that emphasizes epistemological themes? I find preliminarily that many of these experienced students do not distinguish the truth-value of theorems from that of definitions or axioms at the start of the term, but they develop more expert-like perspectives on truth during the course.
2014
Denver, Colorado
We present results from a classroom teaching data collection that involved practicing teachers as they participated in an inquiry-oriented differential equations (IO-DE) course. Data was collected to investigate how the teachers’ participation in this kind of course, different from any of their previous mathematics courses, may influence their conceptions of teaching, mathematics, and student learning. Preliminary results indicate that the perceptions of teachers were changed by their experience in the class, at least as expressed in interviews. The teachers were likely to attempt to use more student-centered methods in their classrooms and believe that student learning is better in the student-centered environment. Additionally, attitudes about non-lecture, although mixed, did indicate a positive tone towards the constructivist perspective on learning. Finally, the teachers’ participation in argumentation increased during the IO-DE course.
2014
Denver, Colorado
The purpose of this study is to examine the effectiveness of 5-minute preview video lectures for each lecture using podcasting in terms of mathematical achievement and mathematics self-efficacy in intermediate algebra and college algebra courses at a university. Data from 128 students in 6 sections collected for two semesters through first and final exams, questionnaires, a classroom observation checklist, and the Mathematics Self-Efficacy Scale. The preliminary findings indicate no significant difference on the mathematical achievement and mathematics self-efficacy between the control group who did not watch the preview lectures and the treatment group who watched the preview lectures while the treatment group slightly developed their mathematics self-efficacy and abilities for mobile technology. In addition, the treatment group was significantly satisfied with the preview lectures. When the treatment group was divided into intermediate low and high subgroups based on the first exam, the intermediate low subgroup significantly improved their mathematical achievement.
2014
Denver, Colorado
In this preliminary report, we share the design and results of the first phase of our on-going research study. Our three-phase study is designed to investigate individual student’s transfer of learning of linear algebra concepts along with social mathematical interactions in which such concepts developed in group-based courses. We first frame our study in relation to current literature, then discuss our initial analysis from the first phase. Finally, we give a description of upcoming phases along with questions we wish to discuss with the audience.
2014
Denver, Colorado
We present initial findings of ongoing research that investigates the nature of instructors’ concerns as they design and use assessments for their students using inquiry-based learning (IBL) approaches. Using data collected from biweekly online-teaching logs written by 39 instructors, we categorized concerns into three major themes: Item Design/Assessment, Course/Resources, and Student difficulty. We compare two areas of concerns (designing assessment and using quizzes, tests, and exams) according to the type of concern and the instructor’s experience with IBL, course level, and year by using the frequencies of each category cited for each log. Our work will contribute to IBL research by analyzing instructors’ challenges as a preliminary study to enhancing IBL teaching and learning in college mathematics education.
2014
Denver, Colorado
Mathematical knowledge for teaching (MKT) has been shown to be a measurable construct impacting instructional quality and student outcomes. The primary examples that educators have for MKT tasks are those that were designed and validated for assessment purposes. It is not known to what extent features of a task that support its use as assessment may support or hinder its use in instruction. We examine this tension by studying the use of two such MKT tasks in a course for prospective teachers. Key considerations for using MKT tasks in professional preparation tasks were how the MKT task represents teaching practice and the possible purposes of using that representation in teacher education.
2014
Denver, Colorado
The CSPCC (Characteristics of Successful Programs in College Calculus) project is a large empirical study, investigating mainstream Calculus 1, that aims to identify the factors that contribute to successful programs. The CSPCC project consists of two phases. Phase 1 entailed large-scale surveys of a stratified random sample of college Calculus 1 classes across the United States. Phase 2 involves explanatory case study research into programs that were identified as successful based in part on the results of the Phase 1 survey. During our case study site visits, we interviewed calculus instructors and asked a number of questions that prompted them to discuss their students. The purpose of the analyses we will present here is to characterize the ways that calculus instructors talk about their students. To do so, we will examine instructor survey responses and analyze instructor interviews conducted at the case-study institutions (PhD and Bachelors granting levels).
2014
Denver, Colorado
Over the past years, research in the RUME community has driven the development of inquiry- oriented instructional materials in a number of undergraduate mathematics content areas including abstract algebra, differential equations, and linear algebra. Literature at the K-12 level has documented challenges inherent to scaling up the implementation of this kind of instruction. In this study, we explore how instructors make sense of and implement inquiry- oriented instructional materials in undergraduate mathematics, and the nature of supports these instructors report using and wanting when planning for instruction. We consider instructors’ interpretations and desired supports as they relate to prior pedagogical experience and institutional setting. Data is taken from surveys, interviews, and video-taped instruction of three participating instructors at three different institutions as they work to implement two inquiry- oriented instructional units in undergraduate linear algebra.
2014
Denver, Colorado
In this study, we use Newton’s method as a means to examine first-semester calculus students’ understanding of tangent lines. Within that context, we found that many students had difficulty sketching and describing tangent lines. We examined the language students use to describe tangent lines as well as their graphical illustrations of tangent lines. Task-based interviews were conducted with twelve first-semester calculus students who were asked to verbally describe a tangent line, sketch tangent lines for multiple curves, and use tangent lines within the context of Newton’s method. Six prominent categories describing students’ concept images of tangent lines emerged, and we found that individual students often possessed multiple concept images. Furthermore, data shows that these concept images were often conflicting, and students were usually willing to modify their concept images in different contexts.
2014
Denver, Colorado
Over 50,000 students take trigonometry at two-year colleges in the U.S., yet little is knownabout their instruction. We report an analysis of activities in trigonometry classes taught at acommunity college attending to two dimensions, the type of knowledge used (Factual,Procedural, Conceptual, and Metacognitive) and the cognitive processes (Remember,Understand, Apply, Analyze, Evaluate, Create) intended in the activity as enacted by teachers intheir lessons. Most of the 163 activities were classified as applying procedural knowledge; overone-fifth of the activities were coded as remembering factual knowledge or understandingconceptual knowledge. We discuss these findings in light of the community college setting andoffer some questions for further research.
2014
Denver, Colorado
Teaching and learning calculus has been the subject of mathematics education research for many years. Although the body of research is mainly concerned with students’ difficulties with calculus, in this study we will be focusing our attention on the professors and instructors of calculus. In this research we used Schoenfeld’s framework to examine four instructors’ resources, orientations and goals in teaching calculus to low achieving students. So far, the preliminary results of the interviews show that although the professors thought differently about many aspects regarding teaching calculus, they all claimed that the first step to succeed in calculus courses is being prepared and having the right background.
2014
Denver, Colorado
We report a case study aimed at researching the rationale of a university mathematics professor for using diagrams in his analysis lectures, what he hoped his students would learn from these diagrams, the ways students understand these diagrams, and what they learn from them. Preliminary analysis suggest that by focusing on specific properties of the diagrams presented in mathematics lectures, or by attributing little importance to them, students fail to fully understand what professors hoped they would learn from these diagrams.
2014
Denver, Colorado
Research on mathematicians’ pedagogical practice in advanced mathematics is sparse. The current paper contributes to this literature by reporting a case study on a proof that a professor presented in a real analysis course. By interviewing the professor, we focus on his learning goals in this proof and the actions that he took to achieve these goals. By interviewing six students, we investigate how they perceived the proof and what they learned from it. Our analysis provides insight into why students did not learn what the professor desired from this lecture.
2014
Denver, Colorado
Although counting problems are easy to state and provide rich, accessible problem solving situations, there is much evidence that students struggle with solving counting problems correctly. With combinatorics (and the study of counting problems) becoming increasingly prevalent in K-12 and undergraduate curricula, there is a need for researchers to identify potentially effective instructional interventions that might give students greater success as they solve counting problems. In this study, we tested one such intervention – having students engage in systematic listing of what they were trying to count. We found that even creating partial lists of the set of outcomes was a significant factor in students’ success on problems. Our findings suggest that more needs to be done to refine instructional interventions that will facilitate listing. We discuss these findings, suggest follow-up studies, and request feedback from the audience.
2014
Denver, Colorado
Counting problems provide an accessible context for rich mathematical thinking, yet they can be surprisingly difficult for students. While some researchers have addressed these difficulties, more work is needed to uncover ways to help students count effectively. In an effort to foster conceptual understanding that is grounded in students’ thinking, we had two undergraduate students engage in guided reinvention in a ten-session teaching experiment. In this experiment, the students successfully reinvented four basic counting formulas. In follow-up problems, combinations proved to be the most problematic for them, however, suggesting that the learning of combinations may require special attention. In this presentation, we describe the students’ successful reinvention, and we discuss potential reasons for the students’ issues with combinations. We additionally present potential implications and directions for further research.
2014
Denver, Colorado
This preliminary report examines pre-service secondary mathematics teachers’ engagement with problems which contextualized mathematics in issues of social justice. A framework for Teaching Mathematics for Social Justice was employed and participant responses were analysed with respect to what mathematics they noticed and attended to in and after the problem solving. Results suggest participants had difficulty “seeing” the math in non-math contexts, and that their ability to notice the embedded mathematics was influenced by the specific social context as well as their orientation towards mathematics (both in general and regarding specific content). Implications for research and teacher education are described.
2014
Denver, Colorado
This paper explores pre-service secondary school mathematics teachers’ preferences when advising a student on how to determine the area of an irregular hexagon. The research attends to participants’ personal mathematical knowledge, as interpreted through the lens of Knowledge at the Mathematical Horizon. Philosophical notions of inner and outer horizons of conceptual objects are adapted to provide a refined analysis of participants’ personal strategies and preferences as evoked by an unconventional problem. The interplay amongst participants’ understanding of mathematical structure, their focus of attention when interpreting a problem, and the advice they offer to a student are of interest. Implications for teacher education and further avenues of research are suggested.
2014
Denver, Colorado
In this study, a secondary school teacher’s goals for student learning were characterized using a framework that emerged from prior work. Observed lessons spanning the use of both conceptually rich and conceptually poor curricula were analyzed and lead to unexpected findings, suggesting that both challenges and opportunities for professional development endeavors exist that center around perturbing a teacher's goals.
2014
Denver, Colorado
The mathematical practice of strengthening or weakening a theorem requires careful attention to hypothesis and conclusion. Selden and Selden (1987) reported that students often unintentionally weaken theorems raising concerns of undergraduates’ attention to hypothesis. In this paper, I consider both the prevalence of this error and what the practice of strengthening/weakening a theorem may look like. A survey of prove/disprove prompts was piloted with five graduate students. A subset of these prompts was then given to undergraduates in an introductory group theory course. Preliminary results indicate that the error of weakening the theorem is prevalent amongst both populations. The graduate students participated in follow-up interviews where they were prompted to strengthen/weaken conjectures to further examine their attention to the hypotheses. In this preliminary report, I will present the survey results and one graduate case to illustrate what the practice of strengthening/weakening a theorem may look like.
2014
Denver, Colorado
In this report we will draw on the Characteristics of Successful Programs in College Calculus data set in order to investigate instructor beliefs about the role calculus plays. Specifically, in this preliminary report, we have analyzed instructor interview transcripts in order to address the question: How do instructors perceive the role of calculus at successful four-year universities? Our preliminary analysis has uncovered six emerging themes. Each will be presented and illustrated with an instructor’s quote.
2014
Denver, Colorado
The CSPCC (Characteristics of Successful Programs in College Calculus) project is a large empirical study investigating mainstream Calculus 1 to identify the factors that contribute to success, to understand how these factors are leveraged within highly successful programs. Phase 1 of CSPCC entailed large-scale surveys of a stratified random sample of college Calculus 1 classes across the United States. From these surveys, successful institutions were selected as case studies. At each case study institution, Calculus I instructors, students and related administration were interviewed. In this report, we will present preliminary analysis on the five bachelor’s granting institutions selected. We will discuss common themes and factors that have emerged from the five institutions.
2014
Denver, Colorado
We present a textbook analysis of conceptions of key ideas associated with inverse trigonometric functions using Balacheff’s model of conceptions (Balacheff & Gaudin, 2010). We found conflicting conceptions of angles, trigonometric functions, and inverse trigonometric functions that may help explain difficulties that community college trigonometry instructors and their students face when explaining tasks associated with this topic. We make suggestions for further research.
2014
Denver, Colorado
We present an analysis of features common across four Calculus I programs at two-year colleges identified as successful in the Characteristics of Successful Programs in College Calculus (CSPCC) study. In this paper we discuss how these features emerged in the analysis of the four cases and their connection to theories of student academic and social integration. Student academic and social integration have been identified as closely related to student persistence in college. We used a constant comparative analysis to identify themes within and across institutions, using transcripts of 22 interviews with faculty, staff, and administrators, and student focus groups. We discuss three of the seven major themes that arose, High quality instructors, Faculty autonomy and trust in the teaching of calculus, Supporting students academically and socially, and Attention to placement, which support a model of student academic and social integration. We present further research steps and some implications for practice.
2014
Denver, Colorado
Although the gender gap in participation in undergraduate mathematics has narrowed, disparities persist at the doctoral level. Only 30 percent of recent doctoral recipients in mathematics were women (Hill, Corbett, & St. Rose, 2010). To increase retention of women in mathematics doctoral programs, it is critical to study the factors that are associated with success. A survey was distributed to 142 female mathematics professors to assess the impact of factors that could have contributed to their success. Results point to changeable factors that can be implemented to narrow the gender gap. Salient factors include persistence and dedication, strong undergraduate preparation and quality doctoral courses, and support from the doctoral advisor. Results show that gender still has an impact on the experiences of the participants, but there may be reason for optimism. Respondents who received their doctorates more recently reported less gender discrimination.
2014
Denver, Colorado
Definitions are an important part of the study of mathematics, yet many students struggle with successfully understanding and using this construct. It has been suggested that students may improve their understanding of mathematical definitions by engaging in the act of writing definitions (de Villiers, Govender, & Patterson, 2009). Through a mixture of survey and teaching experiment methodology this study explores pre-service elementary teachers’ understanding of mathematical definitions before and after engaging in a teaching experiment which provided many opportunities for the participants to write their own mathematical definitions for familiar and novel classes of quadrilaterals. Definitions were assessed as having necessary, sufficient and minimal conditions. It was found that while many students initially struggled to write definitions that meet these qualifications, the process of trying to construct their own definitions did improve students’ understanding of these characteristics of mathematical definitions.
2014
Denver, Colorado
Both prior research and national standards emphasize the importance of critical ideas, such as the Cartesian Connection and equivalence, in algebra problem solving. The mathematics education community, however, has yet to determine whether the secondary teachers who teach such ideas fully grasp these ideas themselves. To investigate this, I interviewed a cohort of nine preservice secondary teachers in a teacher education program with two algebra problems that embed these ideas. The results showed that many of the teachers failed to understand equivalence as a relation between geometric objects, and thus could not solve an algebra problem by relating algebraic equations to their corresponding graphs. Many also misinterpreted the meaning of the term “solution,” and thus could not use the Cartesian Connection to find a solution of an equation. It is advisable that secondary teacher education programs focus more on these critical ideas so that secondary teachers can impart such ideas on their students.
2014
Denver, Colorado
The purpose of this study was to identify some of the characteristics mathematicians value in good proof writing. Four mathematicians were interviewed. First, they evaluated and scored six proofs of elementary theorems written by students in a discrete mathematics or geometry course, and second, they responded to questions about the characteristics they value in a well- written proof and how they communicate these characteristics to students. Preliminary results indicate that these mathematicians agreed that the most important characteristics of a well- written proof are (a) correct logic and (b) clarity. Although these mathematicians differed in the attention they give to layout, grammar, punctuation, and mathematical notation, they agreed in giving these characteristics relatively little weight in the overall score. The results also showed that, in addition to demonstrating good proof writing in class, writing comments on students’ papers is an important way they teach their students to write good proofs.
2014
Denver, Colorado
The ability to translate a text into a mathematical process is a key goal of mathematics education. Knowing when students have the prerequisite knowledge to understand such a process is a perennial concern for instructors. Here we use Newton’s method to evaluate reader oriented theory as a means to illuminate these issues. Through clinical interviews with twelve first semester calculus students, we determined that knowledge of both tangent lines and roots is required for students to understand and apply Newton’s method. Analysis was done from the perspective of the empirical, implied, and intended readers and was examined for the extent to which the empirical and implied readers aligned. It was found that although the alignment of the empirical and implied readers was helpful in determining the success of the students, it was not in itself a deciding factor.
2014
Denver, Colorado
This is an investigation of the teaching philosophies of beginning mathematics graduate teaching assistants. Three teaching philosophy statements from each of four participants were collected at different stages of a semester-long teaching assistant preparation program and analyzed. Principal elements found in these statements before they underwent training and how their philosophies changed over time during training will be discussed.
2014
Denver, Colorado
Many preservice elementary teachers have a limited understanding of the mathematics that they will teach, including many topics in number theory (e.g., Zazkis & Liljedahl, 2004), which suggests that they may not be prepared to teach mathematics for understanding. The research also suggests that pedagogical content knowledge (PCK) is important for teaching (e.g., Ball, Thames, & Phelps, 2008; Shulman, 1986), but little is known about preservice elementary teachers’ PCK in number theory.
2014
Denver, Colorado
The purpose of this paper is to investigate a theory about the nature of mathematical development, in which mathematics is characterized as the objectification of action. Informed by existing research on how students construct new mathematical objects, we consider as an example the psychological construction of cohomology and related objects of algebraic topology. This example extends neo-Piagetian theories of mathematical development from elementary school to graduate-level mathematics, while integrating existing research on students’ learning of abstract algebra. Results of the investigation affirm the objectification of action as a distinguishing feature of mathematics in general, while indicating the kinds of mental actions that undergird the objects of advanced mathematics.
2014
Denver, Colorado
Prior research reflects a positive relationship between homework and student academic achievement in undergraduate mathematics courses. Additionally, recent research has indicated no significant difference in student learning based upon the medium of the assignment (on-line based versus paper-based). These findings led us to ask the question: How does the nature of Calculus I homework assignments at doctoral institutions with successful calculus programs compare to assignments at institutions with less successful calculus programs? Descriptive analyses of student and instructor responses from a large national survey given to mainstream Calculus I programs were conducted. Analysis revealed significant differences in the nature of homework between successful and less successful institutions, including differences in the content and frequency of assignments. The holistic approach to homework taken by successful institutions adds to the existing literature on homework at the undergraduate level and indicates an interesting relationship between homework and student success in Calculus I courses.
2014
Denver, Colorado
This study probes the thinking of students at different stages: a) secondary students taking calculus, b) college students taking calculus, and c) college students pursuing teacher certification taking a mathematics course other than calculus. The study asks: 1) what is the nature of student thinking when solving a graph problem, and 2) do students with different levels of mathematical experience solve a graph problem differently? A pilot investigation reveals many students estimate answers, even if they had studied calculus. For the current study, data will be collected during Fall, 2013. Oral interviews will be conducted with a subset of the participants and coded via Grounded Theory (Strauss & Corbin, 1990; Dick, 2005). This work follows physics education (McDermott, Rosenquist & van Zee, 1987; Thornton & Sokoloff, 1990; Kim & Kim, 2005), and mathematics education (Chiu, Kessel, Moschkovich & Munch-Nunez, 2001; Moschkovich, 1996) that describe difficulties students have with graph interpretation.
2014
Denver, Colorado
This study examined how mathematical modeling activities within a collaborative group impact on students’ perceived ‘value’ of mathematics. With a unified framework of Makiguchi’s theory of ‘value’, mathematical disposition, and identity, the study identified the elements of the value-beauty, gains, and social good-with the observable evidences of mathematical disposition and identity. A total of 60 college students participated in ‘Lifestyle’ mathematical modeling project. Both qualitative and quantitative methods were used for data collection and analysis. The result from a paired-samples t-test showed the significant changes in students’ mathematical disposition. The results from the analysis of students’ written responses and interview data described how the context of the modeling tasks and the collaborative group interplayed with students’ perceived value. The poster will present the main findings and the examples of students’ written tasks and responses.
2014
Denver, Colorado
This paper analyzes how student discourse about Calculus is situated in a graphical representation of a physics problem. Students were asked to identify three unlabeled graphs as representing the position, velocity and acceleration of a car. Findings showed that the students reasoned in three distinct contexts - static-graphical, covariational, and physical. While the students were able to communicate effectively between the first two contexts, and leverage them to find a solution to the problem, the students' discourse in the physical context did not communicate well with their discourse in the other two contexts, nor was it very fruitful in finding a solution to the problem.
2014
Denver, Colorado
In this study, we present preliminary findings regarding student understanding of linear independence of vector-valued functions. Students were given a series of homework questionnaires and participated in individual and paired interviews. The researchers used grounded theory to categorize student approaches for determining linear (in)dependence of functions. In order to gain insight into students’ intuitive notions, data were collected before any formal instruction about the definition of linear independence of functions. The researchers describe initial analyses of student approaches, conjecturing their treatment of vector-valued functions at specific t-values or for varying t as a potentially beneficial lens of analysis. Students who evaluated specific t-values determined the linear independence of a set of vectors in 2 rather than the linear independence of the set of functions, themselves elements of a function space. The analytical construct of process/object pairs (Sfard, 1991) could be a useful lens to explore this distinction.
2014
Denver, Colorado
In this report, we present initial findings from our case study analyses at five exemplary calculus programs at institutions that offer a doctoral degree in mathematics. Understanding the features that characterize exemplary calculus programs at doctoral degree granting institutions is particularly important because the vast majority of STEM graduates come from such institutions. Analysis of over 95 hours of interviews with faculty, administrators and students reveals seven different programmatic and structural features that are common across the five institutions. A community of practice and a social-academic integrations perspective are used to illuminate why and how these seven features contribute to successful calculus programs.
2014
Denver, Colorado
Undergraduates planning to be teachers often encounter mathematics content textbooks written specifically for preservice teachers. Elementary mathematics textbooks of this kind provide in-depth definitions of elementary school mathematics to foster deeper understanding of these basic concepts. I looked at measurement definitions (length, area, and volume) across six preservice textbooks and identified overarching themes, using an open coding method. The following themes emerged across the set of definitions: discrete/continuous, unit, no overlaps/full cover, interior/exterior, function, measurement as an attribute, and space filling. This links to graduate level mathematics and has implications for preservice teachers and their future elementary students.
2014
Denver, Colorado
This study explores mathematicians’ views on 1) transition-to-proof courses, 2) knowledge and skills students need in order to succeed in subsequent mathematics courses, and 3) differences in the proving process across mathematical content areas. Seven mathematicians from three different universities (varying in department size), were interviewed. Precision, sense-making, flexibility, definition use, reading and validating proofs, and proof techniques are skills that the mathematicians stated were necessary to be successful in advanced mathematics courses. The participants agreed unanimously that a content course could be used as a transition-to-proof course under certain conditions. They also noted differences in the proving processes between abstract algebra and real analysis. Results from this study will be used to frame a larger study investigating students’ proof processes in their subsequent mathematics content courses and investigating how these skills can be incorporated into a transition-to-proof course.
2014
Denver, Colorado
Several approaches and models, partly distinct and partly overlapping, shape the theoretical landscape in mathematics education research on teacher knowledge.
2014
Denver, Colorado
This paper presents the results of a study of the proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to-proof course. Students were interviewed individually towards the end of the course using the same protocol that we had used earlier at the beginning of a similar course (Selden and Selden, 2003). Results include a description of the students’ observed validation behaviors, a description of their proffered evaluative comments, and the, perhaps counterintuitive, suggestion that taking an inquiry- based transition-to-proof course does not seem to enhance validation abilities. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and the need for research on their interrelation.
2014
Denver, Colorado
The purpose of the study was to gain knowledge about undergraduate transition-to- proof course students’ proving difficulties. We analyzed the final examination papers of students in one such course. Our perspective included the sometimes automated links between situations and (mental, as well as physical) actions. We have tentatively identified process, rather than content, categories of difficulties such as nonstandard language/notation, insufficient warrants, and extraneous statements. The ultimate goal is to use an understanding of some of these categories as pedagogical content knowledge with which to redesign an existing transition-to-proof course.
2014
Denver, Colorado
Exponential functions are one of the most critical mathematical topics used by students in financial mathematics. This presentation explores university finance students’ notion of exponential function from two sets of data. First, I use data collected through surveys to examine students’ understanding of exponential function in general and, more specifically, to identify the extent to which students conflate exponential functions with polynomials. I then draw on data collected in an inquiry-based instructional sequence aimed at improving financial mathematics students’ understanding of exponential functions. Results include delineation of what ways of understanding exponential functions are critical to studying financial mathematics and insights into how best to guide students in developing these understandings within the context of their field of study.
2014
Denver, Colorado
This report will present preliminary findings from a research study designed to investigate calculus instructors’ perceptions of approximation as a central concept and possible unifying theme of the first-year calculus. The study will also examine the role approximation plays in participants’ self-reported instructional practices. A survey was administered through Qualtrics to a stratified random sample of 3930 mathematicians at higher education institutions throughout the United States with a desired N = 300. Quantitative and qualitative methods were used to analyze the data gathered. Findings from this research will contribute to what is known about the perceptions and teaching practices of calculus instructors regarding the role of approximation in first-year calculus courses. Research-based findings related to the role of the approximation concept in the first-year calculus could have implications for first-year calculus curricula.
2014
Denver, Colorado
Using Bakker and Hoffman’s (2005) framework on diagrammatic reasoning, we analyzed a video-taped interview to explore two undergraduates’ ability to reason geometrically about tasks related to complex variables. Our findings indicate that in order to provide a geometric interpretation, our participants needed to first perform algebraic computations that reduced the current task to a task they knew how to interpret geometrically. These computations appeared to provide them with the pieces required to construct a diagram. Once these pieces were in place the participants used dynamic gesture to enact their geometric interpretations with the aid of their diagram. It appeared that their dynamic gestures assisted with embodying geometric interpretations and as such one particular task was influential throughout the interview. Furthermore, the participants integrated less dynamic gesture as they progressed with similar tasks.
2014
Denver, Colorado
Findings from research into “mathematical knowledge for teaching” have informed the design of preparation and professional development programs for K-12 teachers. At the college level there has been limited research into mathematical knowledge for teaching. We lack findings that demonstrate that expert teachers of college mathematics know and make use of knowledge beyond solely mathematical content.. The goal of this study is to examine the knowledge of student thinking possessed by mathematicians who teach calculus. Data come from interviews on student thinking about core calculus concepts.. Interviewees were research mathematicians who have been recognized for their teaching excellence and mathematics graduate students.. Findings demonstrate that the mathematicians were more able to identify known student difficulties as well as to describe common strategies students use to successfully solve the problems. Implications for research and professional development for novice college mathematics instructors are discussed.
2014
Denver, Colorado
Most recently, the Conference Board of the Mathematical Sciences has advocated for incorporating technology in secondary mathematics classrooms. Colleges and universities across the United States are incorporating technology to varying degrees into their mathematics teacher preparation programs. This study examines preservice secondary mathematics teachers’ opportunities to expand their knowledge of algebra through the use of technology and to learn how to incorporate technology when teaching algebra in mathematics classrooms. We explore the research question: What opportunities do secondary mathematics teacher preparation programs provide for PSTs to encounter technologies in learning algebra and learning to teach algebra? We examine data collected from a pilot study of three Midwestern teacher education programs conducted by the Preparing to Teach Algebra (PTA) project investigating algebra. Our data suggest that not all secondary mathematics teacher preparation programs integrate experiences with technology across mathematics courses, and that mathematics courses may provide few experiences with technology to PSTs beyond strictly computational.
2014
Denver, Colorado
Abstract algebra is recognized as a highly problematic course for most undergraduate students. Despite these difficulties, most mathematicians and mathematics educators affirm its importance to undergraduate mathematical learning. The goal of this research was to establish a list of the important concepts in abstract algebra as perceived by graduate students in mathematics and understand how they believe these concepts are related. Through an interview study, the students’ perceptions of abstract algebra were analyzed through the development of concept maps. Participants’ concept images and concept definitions are highlighted in this article to understand their concept perceptions. The results revealed graduate students had great difficulty articulating what they learned and their concept definitions. Consequently, they had differing views of major concepts and relationships within the course. Their concept images regarding perceived concept importance seemed to be equated to the amount of time their class spent discussing that concept.
2014
Denver, Colorado
2014
Denver, Colorado
2014
Denver, Colorado
Instruments called concept inventories are being used to investigate students' conceptual knowledge of topics in STEM fields, including calculus. One interactive instructional style called Interactive-Engagement has been shown to improve students' gains on such instruments in physics. In this paper, we discuss the development of a video coding protocol which was used to analyze the level of Interactive-Engagement in calculus classes and investigate the correlation with gains on the Calculus Concept Inventory.
2014
Denver, Colorado
Classroom teaching in multiple sections of Calculus I at a large comprehensive research university was observed and coded using the Teaching Dimensions Observation Protocol (TDOP). Multiple teaching styles were identified ranging from low engagement to moderate engagement to high engagement sometimes including student group work. Student performance on two course-wide uniform exams and on the Calculus Concept Inventory was analyzed for any correlations with teaching methods. Significant correlations were found with high engagement teaching styles on both the first exam and the final exam. However, no significant correlations were found on the Calculus Concepts Inventory, indicating that students may not have exerted much effort on this assessment.
2014
Denver, Colorado
Studies have shown that students have difficulty with the concepts of slope and derivative, especially in the case of real-life contexts. I used a written survey to collect data from 75 differential calculus students. Students answered questions about linear and nonlinear relationships and interpretations of slope and derivative. My analysis focused on students’ understanding of slope as a constant rate of change and derivative as an instantaneous rate of change, and what these meant in the context of the problems. Preliminary results indicate that students have more success with slope questions than derivative questions (McNemar’s test, p<0.03), and that while students correctly use the slope of a linear relationship to make predictions, they do not demonstrate an understanding of the derivative as an instantaneous rate of change and an estimate of the marginal change. Plans for a modified survey and interviews are in place for fall 2013.
2014
Denver, Colorado
To build a supplementary theory from which we can derive a practical way of fostering inquiring minds in mathematics, this paper proposes a theoretical perspective that is compatible with existing ideas in mathematics education (radical constructivism, social constructivism, APOS theory, David Tall’s framework, the framework of embodied cognition, new materialist ontologies). We focus on the fact that descriptive and prescriptive statements can be treated simultaneously, and consider both descriptive and exemplary models in our minds. This indicates that descriptive statements in mathematics come from our descriptions of models, and prescriptive statements come from the exemplarity of exemplary models. As a practical suggestion from the proposed perspective, we point out that careful communication is needed so that inquiring minds do not recognize the refutation of their arguments as a denial of their way of mathematical thinking.
2014
Denver, Colorado
Investigating teacher practice at all educational levels has become an important research arena. We analyze teacher practice by comparing two implementations of the same fragment of a student-centered curriculum by two mathematics professors. We highlight differences in their practices and the consequent classroom results by analyzing their participation in class discussions, and we show how Schoenfeld’s (2011) resources, goals, and orientations framework may be used to explain these differences. Using classroom and interview data, we identify resources that each instructor believed he lacked, we highlight prominent mathematical and social goals that each instructor held, and we infer orientations toward teaching and learning mathematics that guided each instructor’s practices. All of these in combination suggest explanations for the observed differences in the implementations and class outcomes. We believe that this analysis provides an important technique to understand and improve teaching and learning at the undergraduate level in mathematics.
2014
Denver, Colorado
Current research on algebraic and quantitative reasoning shows that many students experience mathematics as the manipulation of meaningless symbols (Smith & Thompson, 2007). In order to develop meaning in symbolic contexts, students must first conceive of relationships between the underlying quantities present in a particular context. Our project focuses on a quantitative reasoning approach to multivariable calculus, in particular the concepts of function, rate, area and volume by using physical surfaces. In this poster, we provide examples of identifying, measuring, and recording of essential quantities on physical surfaces.
2014
Denver, Colorado
The goals of the recently funded DIOIMLA research project are to produce: (a) student materials composed of challenging and coherent task sequences that facilitate an inquiry- oriented approach to the teaching and learning of linear algebra; (b) instructional support materials for implementing the student materials; and (c) a prototype assessment instrument to measure student understanding of key linear algebra concepts. Our poster will provide more detailed information about the DIOIMLA research project. Each of the three aspects of the project will be described in more detail and examples of each will be shared. The poster will also include an overview of the current status of the research project and a summary of the timeline for planned future work.
2014
Denver, Colorado
The purpose of this paper is to characterize students’ ways of thinking about parameters and variables to reason about the behavior of multivariable functions. I focus on two single variable calculus students, Lisa and Carl, as they participated in a sequence of semi-structured exploratory teaching interviews intended to gain insight into 1) their approaches to reasoning about the behavior of single variable functions, and 2) what role those approaches played in their initial thinking about the behavior of functions of two, three and four variables. The interviews suggest that the students’ ability to move flexibly between thinking about a function’s variables as parameters allowed them to generalize their reasoning patterns about functions of n variables and extend that to functions of n+1 variables. I argue that their ability to parameterize functions allowed them to reason about functions for which they could not initially visualize representations.
2014
Denver, Colorado
A common feature across STEM disciplines is the study of change. Mathematically, we express the concept of changing one parameter while fixing others by using partial derivatives. However, how we use partial derivatives and how we talk about partial derivatives vary dramatically across STEM disciplines. The purpose of this poster is to share our preliminary results from student and expert problem-solving interviews about partial derivatives.
2014
Denver, Colorado
The purpose of this paper is to argue that attention to students’ ways of thinking should complement a focus on students’ understanding of specific mathematical content, and that attention to these issues can be leveraged to model the development of mathematical knowledge over time using learning trajectories. To illustrate the importance of ways of thinking, we draw on Harel’s (2008a, 2008b) description of mathematical knowledge as comprised of ways of thinking and ways of understanding. We use data to illustrate the explanatory and descriptive power that attention to the duality of ways of understanding and ways of thinking provides, and we propose suggestions for constructing learning trajectories in mathematics education research.
2014
Denver, Colorado
Proof is a central concept in mathematics education, yet mathematics educators have failed to reach a consensus on how proof should be conceptualized. I advocate defining proof as a clustered concept, in the sense of Lakoff (1987). I contend that this offers a better account of mathematicians’ practice with respect to proof than previous accounts that attempted to define a proof as an argument possessing an essential property, such as being convincing or deductive. I also argue that it leads to useful pedagogical consequences.
2014
Denver, Colorado
The concept of a sampling distribution plays a central role in the process of making statistical inferences. However, students typically struggle to understand and reason about sampling distributions. This study seeks to characterize the ways undergraduate students think about sampling distributions in scenarios involving repeated sampling and making statistical inferences. Eight students in an introductory statistics class worked on problems involving sampling distributions during a semi-structured interview. A framework was developed based on their responses to describe the ways they discussed and coordinated various aspects of the population and sampling distributions by focusing on the processes of sampling and repeated sampling; these descriptions suggest that explicitly coordinating particular aspects of these processes may correspond to the robustness of students’ conceptions of sampling distributions.
2014
Denver, Colorado
This paper describes the utilization of word clouds within a research methodology. To explore student notions of the concept of “simplify” in a trigonometry course, students responded to the prompt “In your own words, what does it mean to simplify?” The researcher created a word cloud derived from the student responses to explore and identify themes. These themes formed an initial framework for an in-depth analysis of the responses. During the textual analysis, the word cloud was consulted to confirm findings. Using the word cloud in preliminary and confirmatory roles adhered to the framework put forth by McNaught and Lam (2010). From the analysis, students appeared to view the act of simplifying as a process of taking an expression to its most basic state in order to reduce the perceived size (physical or cognitive) of the expression. Moreover, word clouds played a valuable role, providing visual representations of data.
2014
Denver, Colorado
This preliminary study attempts to describe an initial genetic decomposition of a trigonometric identity for college students. Scant research exists into the concepts found in trigonometry. Thus, little is known about how students actually understand a trigonometric identity. Following the guidelines of APOS theory, an initial genetic decomposition for a trigonometric identity was proposed. According to this decomposition, students with action conceptions can verify identities explicitly using step-by-step manipulations while students holding a process conception are able to visualize steps to demonstrate that the identity is true. Having an object conception means students recognize the truth of the equality without verification and are able to then use the identity to verify other identities. After observing students in task-based interviews, needed modifications to the genetic decomposition became apparent. For example, students’ conceptions of the function argument appeared to influence the verification process.
2014
Denver, Colorado
As part of a National Science Foundation Noyce Scholarship Grant, one university substantially revised its preservice secondary (grades 7-12) math teacher preparation program. As one component of this program, preservice teachers take three credit hours of middle level number and operation and geometry, with a focus on mathematical knowledge for teaching. As a research component, we investigated the impact of this course on preservice teachers capacity to identify, analyze, and respond to student errors. This paper provides additional background and results from the first two offerings of the course, as well as ideas for further study.
2014
Denver, Colorado
We present findings from a revised framework created to analyze tasks that calculus teachers assign their students. In the presentation we will highlight the features of the analytical framework and the steps taken to ensure high inter-coder reliability. The framework has been used to analyze all tasks (N=2,996) present in homework, quizzes, and exams from six faculty teaching Calculus I in two two-year colleges. We highlight some insights we have gained in creating this framework and possible uses by other researchers and other contexts.
2014
Denver, Colorado
We report initial findings of a study that seeks to investigate the methods instructors’ use to assess their students’ learning and how these assessments affect the instruction in their classrooms. Using data collected from 23 instructors using inquiry-based learning methods, we seek to discuss the instructors’ goals for the students, the ways they measure the students’ progress towards these goals, the feedback they give students, and how these assessments affected their instruction. Our analysis of the data uses open coding of the transcripts and of the documents (e.g., syllabi, exams, homework assignments) that the instructors gave to the students. Instructors cite using informal assessments and focusing on presentations when asked about “knowing” that students are learning. They cite formal assessments and examinations when asked about “measuring” that students are learning. We seek input on the analysis of the materials as current results may depend on the coding system used.
2014
Denver, Colorado
The distinction between analytic (notation-based) and visual (diagram-based) representations within students’ mathematical problem-solving has been part of the cognitive phycology and mathematics education literature for more than 40 years. However, in spite of this long history there are many unanswered questions regarding how and why particular students choose particular representations, and what influences their social surroundings have on their individual representation use. This study coordinates analyses of calculus students’ analytic and visual reasoning across both one-on-one interview and group-work settings. This analysis helps clarify differences between individual representation use and representation use in group settings.
2014
Denver, Colorado
We examine perspective secondary teachers’ conceptions of what constitutes comprehension of a given proof and their ideas of how students’ comprehension can be evaluated. These are explored using a relatively novel approach, scripted dialogues. The analysis utilizes and refines Mejia-Ramos, Fuller, Weber, Rhoads and Samkoff’s (2012) proof comprehension framework. We suggest that this refinement is applicable to other studies on proof comprehension.