2021 Research in Undergraduate Mathematics Education Reports
2021
N/A
Number of pages: 417
Number of papers: 49
- Shiv Smith Karunakaran
- Abigail Higgins
2021
N/A
Number of pages: 417
Number of papers: 49
Inquiry Based Learning (IBL) professional development workshops are designed to increase participants’ capacity to teach using IBL methods. This study used a sample of 312 participants from workshops held in 2010-2018 to examine the relationship between professional development participation, IBL capacity, and use of IBL teaching practices. We found that instructors’ IBL capacity, meaning the beliefs, knowledge and skills that prepare them to use IBL, and use of IBL teaching practices increased after participating in professional development. Using the Theory of Planned Behavior as a conceptual framework, we used a structural equation model to explain the effects of workshop participation and other factors on the use of IBL teaching practices. Findings indicated that workshop participation, collegial support, prior IBL experience, class size, and course coordination influenced workshop participants’ use of IBL teaching practices. These findings support the use of well-designed, intensive professional development as a means to change teaching practices.
As VITAL faculty continue to teach more university mathematics courses, departmental efforts to improve instruction and shift toward active learning should align with instructors’ teaching approaches and decisions. In this study, three instructors described their classroom norms and justified teaching moves via professional obligations. External factors, such as the physical learning spaces, interactions with students, and institutional constraints, influenced their pedagogy. This report emphasizes the support systems required within mathematics departments to sustain instruction improvement efforts among VITAL faculty.
I present the results from a set of clinical interviews conducted during the 2019-2020 academic year to produce generative models of student thinking about rates of change towards the end of a university-level calculus sequence at a large State University in the Southwestern United States. The data presented here presents a case for supporting students’ symbolization as emergent from their previous conceptualization and mental activity. I provide a conceptual analysis to illustrate ways of thinking that may be propitious for understanding rates of change in a calculus context and symbolizing with the intent to convey meaning.
This study examined the nature of Calculus II student’s engagement with MATLAB modules and the ways in which the computational modules mediated student’s understanding of Taylor series. Through analysis of observations and interviews, using an instrumental genesis framework, a pattern of student’s views on the relationship between mathematics and computation developed in relation to student ability to conjecture and engage in mediated epistemic interactions with the MATLAB modules. This study highlights how the conceptualization of the two areas as distinct poses barriers and challenges for students, thereby questioning how to develop computational environments that support mediated epistemic interactions.
In advanced mathematics courses, proofs are instrumental in conveying mathematical knowledge. As a result, in many upper-division undergraduate math courses, proof comprehension is a crucial aspect of learning mathematics. This study examines proof reading strategies that eleven mathematics doctoral students profess to use to facilitate proof comprehension. This paper identifies 12 strategies that participants in this study employed to improve proof comprehension. I argue that those 12 proof reading strategies occur in three phases. In particular, doctoral students in this study engaged in three types of reading: preliminary reading, engaged reading, and reflective reading. The paper concludes with a brief remark on this research's implication for teaching proofs in undergraduate mathematics.
This paper explores how undergraduate engineering students engage with puzzle problems related to first-order differential equations. One hundred and thirty- five undergraduate engineering students engaged with four puzzle problems related to first-order differential equations in self-selected groups of two or three students while their communications were audio recorded. The findings related to one of these problems are reported here. The results show that many students struggled with identifying how DEs could be used for modeling real-world situations, and as a consequence of that, they solved the puzzle using their physics knowledge. The findings suggest that more focus should be paid to including modeling (or puzzle) tasks in DEs courses to help engineering students engage in higher-order thinking, develop thinking skills and problem-solving strategies needed for their future careers and advanced courses.
The COVID-19 pandemic has had unprecedented ramifications on higher education. In this paper we describe how the pandemic impacted a community of practice of instructors involved in the instruction of first year mathematics courses. In particular, we use qualitative data from interviews and open-ended survey responses to describe how members of this community responded to the changes in instruction brought about by the pandemic, and in what ways the community adapted to support instructors in engaging in this new instructional format in order to sustain itself through this transition
Peer mentoring programs are one approach to improving the pedagogical development of mathematical sciences graduate students. This paper describes the peer mentoring experiences at three institutions that have implemented a multi-faceted GTA professional development program. Data was collected from surveys and focus groups conducted with graduate teaching assistants at each institution regarding mentees’ ratings of their mentors, mentors’ ratings of their impact on mentees, mentors’ impressions of the benefits and challenges of peer mentoring, and mentees and mentors’ ratings of program components related to support from mentors, their TA coach, program staff, and other graduate students. Most GTAs found value in participating in the peer mentoring program. While the mentees found their mentors to be significant to their own success and effectiveness, the mentors did not rate themselves as high as the mentees rated them with respect to their own significance in impacting the effectiveness of their mentee.
Retention in Science, Technology, Engineering, and Mathematics (STEM) continues to be a problem in the U.S. Prior research indicates that students leave STEM due to poor instruction and low sense of belonging in introductory STEM courses. Incorporating active learning into these courses, especially Calculus, has potential to support students in developing a greater sense of belonging and staying in STEM. This study investigates students’ sense of belonging in two versions of introductory Calculus – a standard course and a non-standard course infused with active learning. Results indicate that students in the two courses recognized differential opportunities to engage in active learning. While there was no significant change in students’ sense of belonging over the course of the semester in either course, students in the active learning course reported significantly higher sense of belonging than students in the standard course, both early in the semester and at the end of the semester.
Making sense of proofs and statements is a fundamental part of advanced mathematics classes; however, researchers have established that students have limited approaches to reading proofs and may struggle to comprehend them. Converting between representation systems can play an essential role in comprehending formal mathematics including proofs and statements. While navigating representation systems, students are likely to evoke an array of personal meanings that can lead to semiotic conflicts in communication. In this study, we examine what conflicts arose as a group of students collectively worked to comprehend the Fundamental Homomorphism Theorem. Our results show that the students had conflicts related to functions and quotient groups that arose when converting between the formal and other representation systems. Although these conflicts can be problematic, we believe that with a productive discussion and instructor intervention (when necessary) these conflicts can be resolved.
This study investigated how mathematics instructors’ instructional goals for teaching constant rate of change (CROC) influences their perception and use of applets in mathematics instruction. This report presents results from a clinical interview with graduate teaching instructors (GTIs) to illustrate the degree to which their mathematical meanings for teaching (MMT) impact their image for how an applet should be used and how the GTIs’ use of the applet influences their MMT. I use instrumentation theory as a lens for explaining the relationship that develops between an instructor and particular features of the applet. I conclude with a discussion of how GTIs MMT for constate rate of change impacts their intended use of an applet’s feature and how they imagine they can use an applet to advance students’ mathematical understanding of CROC.
Creativity is central to mathematics and mathematics education. One hindrance to research in mathematical creativity is the complex nature of defining and measuring creativity. Some efforts have been made to study mathematical creativity at the K-12 level, but just recently researchers have begun exploring mathematical creativity at the undergraduate level. Proof is essential to an undergraduate mathematics education, and as university mathematics classrooms evolve to incorporate more active and collaborative learning, it is imperative to understand the relationship between collaboration and creativity in proving. This study seeks to apply the Creativity-In-Progress Rubric (CPR) on Proving (Savic et al., 2017) to two collaborative small-group proving episodes and to evaluate its ability to present a holistic image of a group’s creative proving process. Findings of this evaluation led to the three suggestions for future use of the CPR on Proving in collaborative settings.
Recent large-scale research points to evidence of inequitable outcomes between women and men in inquiry-based mathematics education (IBME) courses. One explanation for differing outcomes may be that women are having different experiences in these courses than men. Specifically, the ways in which students garner mathematical authority and leverage their authority in both whole class and small group contexts may differ between students. Framing authority as a relation between people determined by their mathematical activity, we present an exploratory analysis of the authority relations between students as they engage with tasks developed for an IBME abstract algebra course. Findings suggest there are indeed discrepancies in the amount of time students have mathematical authority. We present examples of situations in which discrepancies are visible to begin examining the underlying nature of these discrepancies.
Most advanced mathematics courses are taught via lecture, in which it is hard to foster meaningful student engagement, agency, and community. In this paper, we discuss an academic support resource called The Students as Partners Program that was implemented in an Introduction to Analysis course with 26 students at a private institution in the Northeastern United States. This model of support positioned three currently-enrolled students in this course as “Student Partners” who were tasked with communicating information to the instructor about student experiences that the instructor could then use to respond dynamically to student needs. In this exploratory study, we examined how student-partnerships can promote student agency, which allowed instructors to implement instructional design structures that supported and facilitated student engagement in- and out-of-class and classroom community.
Research has shown that college students struggle to understand the operation of composition and the compositive structure of functions. A study of the treatment of composition in written US curricula identified the opportunities that were provided to teachers and students. This report presents the types of functions and representations used to communicate composition in mathematics curricula across the transition from secondary mathematics to college calculus. The presentation of composition was overwhelmingly algebraic and less than four percent involved multiple representations. The results also revealed that composition was mainly presented with linear binomials, monomials, and the inverse cancellation of transcendental functions. The opportunities in written curricula were similar to the types of tasks on which researchers have reported students being successful. Diversifying textbook problems and examples has potential to increase student and teacher knowledge on composition and the compositive structure of functions.
The abrupt switch from in-person instruction and tutoring to remote or online instruction and tutoring as a result of the COVID-19 pandemic in March 2020 was difficult for even the most experienced instructor. In this paper, we explore how graduate teaching assistants (GTAs) at three different institutions responded to and experienced this change. Data was collected from surveys and focus groups conducted with graduate teaching assistants at each institution, as part of our ongoing collaborative NSF-funded project focusing on equipping mathematical sciences GTAs to become better teachers. In their responses, the graduate teaching assistants discussed topics ranging from what they did in their remote classrooms to the challenges they faced and supports they received from their department, university, and fellow classmates and faculty.
Proof is a medium of mathematical communication and has distinct characteristics related to it. Among various characteristics, we focus on the words in proving tasks prompts (PTP), such as “prove,” “show,” “justify,” or “explain” because students’ arguments can be changed depending on the keyword, which can impact assessing their learning when the students and the instructor do not share the same interpretation for the same keyword in PTP. Thus, in this study, we observed an Introduction to Proofs course to see which and how the keywords in PTP are presented to students and interviewed the instructor and a focus group interview of her three students. The finding indicates that the instructor’s interpretation and students’ interpretations were different. Although students are adjusting their pre-existed meaning for those words to the context, this study suggests that instructors cannot assume that students share the same meaning of the keywords in PTP with them.
There is a need to address student understanding of the role of definitions in undergraduate mathematics, and research is needed to determine pedagogical strategies that facilitate development of this understanding. In geometry, research shows students can better develop their understanding of concepts and their definitions by observing properties and making conjectures in non-Euclidean geometry. In this study, students learned concepts in Taxicab geometry in a College Geometry course in which theory in Euclidean geometry is the primary focus. The triad of stages of schema development and a model of schema interaction were used as frameworks in the analysis of student responses to a questionnaire and follow-up interview about the definition of circle in Taxicab and Euclidean geometry. As a representative illustration of a level of schema interaction, insight into the thinking of one student is presented in this report. Pedagogical suggestions are provided as a result of this data analysis.
Mathematical problem solving research that focuses on the development of problem solving practices can provide mathematics educators the tools with which to foster the use of such practices in their students. This study hopes to identify influences on the development of emerging mathematicians’ problem solving technique. Emerging mathematicians from an undergraduate real analysis course and first-semester graduate course in linear algebra were invited to a sequence of two interviews, during which they completed problem solving tasks and reflected on their growth as mathematical problem solvers. In particular, they were asked to expound on formative experiences that affected the development of their problem solving strategies. Participants reported ways that both teaching mathematics and learning mathematics from skilled instructors influenced the way they solved novel problems.
This study explored the grasp of square roots among 11 students in a bridging course, with a special focus on instances where the same student generated seemingly conflicting responses. Building on the commognitive framework, the analysis indicated that individual students square-rooted differently in a range of situations, such as cases where roots were extracted from square numbers and from squared radicands, where roots “stood alone” and where they were incorporated in an exercise. Differences were found in the procedures that students employed and the tasks that they pursued. A theoretical account was offered, suggesting that what may appear as a conflict within a student’s discourse could be a sensible difference of actions taken in task situations that this student construed as incompatible.
Using social learning theory with the central concept of a community of practice, we situate this work within a secondary mathematics methods course to unpack preservice secondary mathematics teachers (PSMTs) development through the use of video case studies. We analyzed six sessions of the course in which PSMTs engaged in discussions about video segments of mathematics teaching rooted in the Teaching for Robust Understanding (TRU) framework for high-quality instruction. Analysis of this data showed opportunities for PSMTs to develop critical skills for teaching (Hiebert et al., 2007). Our contribution includes the addition of a new skill to Hiebert’s framework, Understanding the Mathematics, as an important component of PSMT learning that may precede the original four skills. Future research should focus on connections between PSMTs’ evolving mathematical understandings and analysis of video to better understand the impact of their content knowledge on developing the critical skills for teaching.
The question of what high-quality undergraduate-mathematics instruction entails has been examined in prior studies mainly by directly analyzing lecturers’ teaching practices. The current exploratory study aims to provide a complementary angle by inquiring into a student perspective on this topic. Based on student-centered data consisting of interviews and questionnaires, the findings point towards 8 characteristics of “excellent lecturers” as perceived by undergraduate students. These provide student-driven support for some of the prior lecture-centered research claims, though also highlight the importance students attribute to the affective dimension in teaching, which has not yet received sufficient research attention.
Combinatorics is a growing topic in mathematics with widespread applications in a variety of fields. It has become increasingly prominent in both K-12 and undergraduate curricula. There is a clear need in mathematics education for studies that address cognitive and pedagogical issues related to students’ conceptions of combinatorial ideas. I investigate students’ perceptions of the option of not choosing while solving counting problems. In this report, as part of a larger study, I focus on experiences of one undergraduate student who was interviewed in the larger study. The interview was conducted as they solved combinatorial tasks that included the possibility of not choosing a particular attribute when enumerating choices. The data analysis highlighted detecting choices as one of the factors contributing to students’ success in solving counting problems. I suggest an extension of a model of students’ combinatorial thinking that has been introduced in mathematics education literature.
This study qualitatively explored the effects of math anxiety (MA) on undergraduate college students. Four undergraduate students were selected using the Abbreviated Mathematics Anxiety Rating Scale (A-MARS) and then interviewed about their beliefs about MA and the symptoms they experience from the anxiety—including the origins of the anxiety and methods with which they cope with (or attempt to cope with) the symptoms of MA. Students reported the onset of their anxiety occurred around elementary school and reported some common experiences such as strong feelings and a physical response. However, participants also showed the individualized nature of MA with different effects and coping mechanisms. This has implications for undergraduate instructors who want to help math anxious students.
As part of a summer research project, an undergraduate named Anabella (pseudonym) and I (author) collaborated in efforts to prove an unsolved conjecture from the field of graph theory. Anabella is a Hispanic mother who one day desires to be an elementary or middle school teacher. The researcher collected and analyzed extensive data in order to understand what Anabella learned about the nature of mathematics through the research experience as well as how she perceived this new knowledge is relevant to her future as a teacher of mathematics. Anabella learned that her own mathematical ideas are valuable and reflected that it will be important to entertain and consider (rather than dismiss) the ideas of her future mathematics students. She also gained a new appreciation of the importance of mathematical communication.
Numerous calculus concepts rely on the ability to conceive of and represent distances in the Cartesian plane. Yet, research has found that undergraduate students, even those who have studied calculus, may not readily do so (Parr, 2020). This study reports on findings from the development of a hypothetical learning trajectory aimed at supporting students in representing distance in the Cartesian plane. We hypothesized that this skill would require (a) the ability to represent a distance on a number line as a difference and (b) drawing on the Cartesian connection to represent a horizontal or vertical distance in terms of x and in terms of y. We analyze the types of reasoning used by a Calculus I student while working on these tasks, as well as obstacles she faced relative to these two target understandings.
In this paper, we introduce an RME-based (Freudenthal, 1991) task sequence intended to support the guided reinvention of the linear algebra topic of vector spaces. We also share the results of a paired teaching experiment (Steffe & Thompson, 2000) with two students. The results show how students can leverage their work in the problem context to develop more general notions of Null Space. This work informs further revisions to the task statements for using these materials in a whole-class setting.
In this paper we share findings from six interviews with instructors on whether, and how, they attend to generalization in their teaching. In particular, the interviews highlighted a distinction between generalizing in the classroom as being teacher or student generated. This study furthers our understanding of generalization beyond student activity and opens us to further questions such as how, if at all, students understand this distinction in the classroom.
Helping students see conceptual connections between content areas is important for the development of flexible understanding, yet research on ways in which notions of sameness throughout math can be connected is limited. This study examines survey responses from mathematicians on the relevance of sameness to topics in abstract algebra and connections between sameness in algebra and other courses. Common connections are highlighted as are themes in the types of connections provided.
Students have demonstrated difficulties in adopting the various techniques of proof in their upper-level mathematics coursework. One of these techniques of proof which students struggle with is mathematical induction. In this paper, we present an analysis of three groups of students in an Introduction to Proofs course as they made sense of and analyzed two sample induction arguments. To aid in our analysis, we utilized Stylianides' (2007) definition of proof with three components (accepted statements, modes of argumentation, modes of argument representation) to describe the components of proof students focus upon when they encounter induction arguments for the first time in an Introduction to Proof course.
Addressing student affect around assessment is vital, given it is tightly interwoven with cognition. This study seeks to describe the relations between exam-specific affect and stress mindset in a university mathematics course. Participants (N = 356) completed a survey assessing their exam-related self-efficacy, achievement emotions, and stress mindset. The study demonstrated significant correlations between a stress-is-enhancing mindset with positive affect and a stress-is-debilitating mindset with negative affect. When controlling for prior achievement and gender, stress mindset was significant, and student exam-related emotions were dominant in explaining exam-related self-efficacy. The results are discussed with opportunities to adapt learners’ stress mindset and the development of exam-related self-efficacy.
In response to calls for more investigations of teachers’ mathematical meanings for teaching (MMT) and the need to improve US mathematics teachers’ and students’ mathematical meanings, I investigated mechanisms for advancing post-secondary instructors’ mathematical meanings and teaching practices in the context of their teaching precalculus mathematics using a research-based curriculum. This report presents results from clinical interviews with graduate teaching instructors (GTIs) to illustrate the impact of a video-reflection intervention on teachers’ MMT and image of effective teaching practices. More specifically, I present two GTIs’ expressed meanings for angle measure and image of effective teaching before and after attending a professional development seminar where a trigonometry video-reflection intervention was used. I conclude by hypothesizing that video-reflection interventions may orient teachers toward reflecting on the degree to which they are impacting student thinking by providing examples of high-quality teaching (Musgrave & Carlson, 2017) interactions.
Sameness is a notion that pervades mathematics through concepts like equality and isomorphism, but limited research has examined how mathematicians perceive sameness in mathematics. This study examines survey responses from mathematicians on the nature of mathematical sameness. Themes highlighted from responses include philosophical, proof-based, and informal notions of mathematical sameness.
This study addresses what students’ conceptions of circles are and how students’ interaction with a digital environment influences their development of those conceptions. Two undergraduates from calculus 1 courses participated in exploratory teaching interviews for two one-hour sessions in fall 2019. In this study, we focus on moments in which the students’ mathematical reasoning in regard to circles was on display. The two students demonstrated similar conceptions of and construction of circles on a physical medium at the start of Day 1. However, they interacted differently with the digital environment in the rest of the teaching interviews. Consequently, their conceptions at the end of Day 2 were markedly different. We will discuss the progression of each student and how their interactions with the digital environment influenced their conceptions of circles, paying particular attention to the students’ reasoning throughout.
An underutilized strategy for addressing the underrepresentation of Black, Indigenous, and people of color (BIPOC) in the U.S. STEM workforce is directing more attention to Minority Serving Institutions (MSIs). Unlike some MSIs, Hispanic Serving Institutions (HSIs) are defined by the current demographics of their student population rather than the intention of the institution to serve a specific population. As instructors are the mediators between students that attend the HSIs and mathematics content, we use the framework of professional obligations (Herbst & Chazan, 2012; individual, interpersonal, disciplinary, and institutional) to investigate if or how they are being intentional about the ways their practice can impact BIPOC students. We found from preliminary analysis of nine interviews that instructors are recognizing the four professional obligations in some common ways. A tension exists between the disciplinary obligation to portray mathematics as culture-free and attending to students as individuals while fostering an inclusive classroom environment.
Two American Sign Language interpreters and two Deaf instructors were asked to sign various undergraduate mathematical terms and definitions. Various types of iconicity were identified in the signs and each term had more than one sign demonstrated for it. The participant’s preference and choice for signs reflected their mathematical beliefs and prior experience. While the motivation for all participants was to accurately translate the meaning of the mathematical concepts, not all participants shared the same type of preference for sign type. One out of the four participants preferred to use signs that represent the notation used for the concept, while the other three participants preferred to use signs that showed some conceptual meaning.
Systems of linear equations (SLE) comprise a fundamental concept in linear algebra, but there is little research regarding the teaching and learning of SLE, especially students' conceptions of solutions. In this study, we examine students’ understanding of solutions to SLE in the context of an experientially real task sequence. We interviewed two undergraduate mathematics majors, who were also preservice teachers, to see how they thought about solutions to SLE in ℝ3, especially linear systems with multiple solutions. We found participants used their knowledge of SLE in ℝ2 to think about systems in higher dimensions, sometimes ran into algebraic complications, and initially did not find the third dimension intuitive to think about geometrically. Our findings highlight students’ ways of reasoning with infinite solution sets, such as moving toward the notion of parametrization.
Student understanding of linear algebra concepts is a growing research area. This study explores students’ conceptualization of linear transformations and the various ways in which they use such conceptualization to reason through linear transformation problems after they have completed a linear algebra course. Three students participated in a task-based interview. Through analyzing interview data using a grounded theory approach, emerging themes were found indicating that students’ exposure to linear transformations in other courses and the nature of these experiences impacts how they further conceptualize linear transformations. Notably, the way that the participants were engaged with linear transformations within their other courses for their major seemed to influence their use of geometry, algebra, and proofs in determining whether a given transformation is linear. One implication of this study is a need to engage students with more real-life applications of linear transformations in linear algebra courses.
In this study we present the results of a discourse analysis of the interactions between two partners, Uma and Sean, through a lens of positional theory. During nearly five hours of small group work in a teaching experiment, the way in which each partner used language to position each other’s thinking as mathematically significant and establish a collaborative environment varied dramatically. Specifically, Uma shouldered the burden of continuously working to maintain collaboration, oftentimes at the expense of having her thinking positioned as mathematically significant. On the other hand, Sean regularly offered little opportunity for Uma to engage openly with his thinking, while simultaneously positioning his own thinking as mathematically significant. Language enacts and constructs identification with social groups and positions of privilege; thus, we also describe the role of Uma and Sean’s identities, particularly gender-roles, in potentially explaining the nature of their interactions.
Student engagement is one of the most robust predictors of student achievement and behavior (Klem & Connell, 2004). Therefore, a larger study was conducted to investigate undergraduate students’ engagement while learning mathematics. A pre-service mathematics teacher, Uma, reported drastically lower levels of engagement in the larger study. Given that the satisfaction of the basic psychological needs for autonomy, competence, and relatedness facilitates the internalization of motivation and, consequently, students’ engagement (Ryan & Deci, 2009), a case study was conducted to examine Uma’s low engagement through the Basic Psychological Needs Theory (BPNT) in a 5-week teaching experiment focused on developing notions of logarithmic change. Nine themes emerged from the analyses of the videos and an interview, which explain the low levels of engagement through the BPNT. The case of Uma provides rich information about how the satisfaction/frustration of the BPNs is related to engagement levels.
This study details the embodied, symbolic, and formal reasoning of an undergraduate student when determining proper bounds for a triple integral problem within a virtual reality program called Calcflow. While the participant initially had difficulty inputting valid bounds, once he learned the appropriate constraints, he gradually connected various aspects of his symbolic reasoning with his embodied reasoning through progressive experimentation in Calcflow. Results suggest a virtual reality environment can help ease students’ difficulties in visualizing three-dimensional shapes and thereby provide them the opportunity to develop deeper reasoning about these shapes and their corresponding equations
This report describes a pilot study conducted in Spring 2019. We set out to accomplish two tasks. First, through semi-structured interviews, we sought to understand characteristics of women and underrepresented minority (WURM) undergraduate mathematics majors’ experiences pertaining to the secondary-tertiary transition (STT) in mathematics. Second, we designed and implemented a supportive intervention to address aspects of the STT that emerged from the outcome of the interviews. We draw upon Di Martino and Zan’s (2010) three-dimensional model of attitude, which reifies the role of affect in the STT, and focuses on students’ vision of mathematics, perceived competence in mathematics, and emotional disposition towards mathematics. In this report, we present data from three WURMs’ perspectives to document three related dimensions in the STT and discuss whether and how our intervention supported these students’ transition
We report the results of a survey of calculus instructors from colleges and universities across the US related to their reported awareness and usage of Inquiry Based Learning, instructional practices, and beliefs about teaching and learning. Cluster analysis of the data revealed three distinct types of self-identified IBL-users based on the proportion of in-class time students spent in different activities, including a large number of instructors who lecture for the majority of class time. The teaching practices of that sub-group did not differ significantly from those of the group of instructors who reported no knowledge of IBL. Variation in level of agreement with positive statements about lecture shows some association with in-class instructional practice; all four groups indicate strong agreement with statements that inquiry practices support learning.
The purpose of this study is to examine students’ meanings for the derivative at a given input value. While students may verbally state that the derivative represents instantaneous rate of change, that does not imply that they have a coherent meaning for the difference quotient as an average rate of change. This study explores students’ responses to a typical calculus 1 problem that requires the use of derivatives to determine a linear approximation. This study analyzes student responses about instantaneous rate of change and the differences in meanings that students may hold about it.
Local instructional theories (LITs) are often described as a generalized sequence of steps for students’ guided reinvention of some mathematical concept. They consist of a series of tasks and the accompanying rationale for the tasks, where the rationale is described in student’s mathematical activity. Literature suggests that the rationale can be used to adapt LIT’s into different instructional sequences however it fails to provide specifics as to how these adaptations are achieved. This report details how the design heuristic of didactical phenomenology can be used to create multiple instructional sequences for students with different mathematical backgrounds, thus contributing to our understanding of the theory of realistic mathematics education.
Fostering conceptual understanding is a main goal of mathematics education. Yet, students do not often see connections between topics. The recognition of connections can be fostered by making them explicit in instruction and by encouraging students to struggle with the relevant mathematics. Instances in which knowledge from one domain is brought to bear in another domain are examples of transfer; in particular, backward transfer describes the ways in which learning new information influences prior knowledge. We solicited narratives from individuals with a strong mathematics background describing instances during which they recognized a connection between two mathematics topics. We present two of these narratives which contain instances of backward transfer connecting advanced mathematics to the binomial theorem. These narratives suggest that backward transfer is also facilitated by explicit instruction and productive struggle. We discuss implications for education of backward transfer that is induced by learning advanced mathematics, particularly for pre-service teachers.
In this paper, I use two frameworks—Schoenfeld’s theory of teaching in context (2010) and Herbst and Chazan’s practical rationality theory (2012)—to make sense of an undergraduate mathematics instructor’s decision-making when teaching with inquiry-based learning (IBL) methods. After providing a brief review of the relevant literature and the frameworks, I illustrate the analysis of one decision-making situation. I then discuss the affordances and challenges of the theories. The goal is to better understand what each framework offers in relation to the context and the nature of the analyzed data, and how findings from each framework can contribute to research on decision-making in the context of undergraduate teaching and IBL.
This report presents a conceptual analysis that describes the productive ways of thinking of a hypothetical student in learning the idea of constant rate of change. The paper characterizes a hypothetical learning trajectory with a hierarchy of ideas that we conjecture to be foundational for understanding constant rate of change. The study briefly presents an instructional sequence of tasks guided by the hypothetical learning trajectory to promote learning the foundational ideas for understanding the constant rate of change.
The goal of this paper is to describe a cross-cutting way of thinking which may be productive for learning precalculus ideas. I describe the construct, “relative size reasoning” which consists of thinking about the relative size of two quantities as they change together. This way of thinking entails quantitative and covariational reasoning. I describe the mental operations involved in relative size reasoning and how it can be used by providing three illustrations of relative size reasoning in three precalculus ideas and conclude with future research.