Abstracts of Presentations

Type of Talk: Research Report; Author: Amanda Sawyer, James Madison University

Talk Title: "Exploring an Artificial Intelligence Chatbot as a Mathematics Curriculum Development Tool" 

Keywords: Artificial Intelligence, Mathematics Curriculum Development

Abstract: "Teachers believe in using new technologies in the classroom; thus, many researchers have investigated teachers' current resources based on technology trends. Researchers have also studied the quality of the materials found in virtual resource pools using Stein and Smith’s (1998) level of cognitive demand. However, MTEs do not know the level of cognitive demand for resources constructed by the newest technology trend, Artificial Intelligence (AI) chatbots like ChatGPT.

We investigated 191 text responses constructed by ChatGPT, each corresponding to a task created for one of the elementary Common Core mathematical standards from kindergarten to fifth grade, to answer the following research questions: What are the associated levels of cognitive demand of the ChatGPT’s text responses? Is there a statistically significant difference between lower and higher-level mathematical tasks? And what are the common characteristics of AI tasks? Between October 30, 2023 - November 6, 2023, we asked ChatGPT to “Create a mathematical task for [insert Common Core Standard].” As seen in Table 1, after analyzing the data, we determined that 70.1%, or 134 of the 191 constructed tasks, could be considered higher levels of cognitive demand. Of those, 67% were Procedures with Connections tasks. We also identified that the level of cognitive demand in kindergarten through second grade was not statistically significant. Yet, higher grade level tasks (3rd through 5th grade) had a statistically significant difference between cognitive demands. The data also indicated a statistical difference in the level of cognitive demand of the Common Core domains except for Counting and Cardinality and Number and Operations in Base Ten. As for common characteristics, ChatGPT’s responses were repetitive in nature, created inappropriate content for elementary students, and lacked mathematical content and pedagogical knowledge. We expanded on the Mathematical Critical Curation Questions to address these findings. Finally, since we only used a zero-shot prompt for this investigation or a single prompt without providing any examples of how we wanted the AI tool to answer the question, we explored how MTEs could support teachers using this tool in their classroom by using other forms of mathematics education prompt engineering.

Specific recommendations need to be conveyed to MTEs. First, MTEs must caution their mathematics preservice teachers about using AI as a curriculum development tool in their mathematics content or methods courses. Since teachers use these tools, MTEs need to know the limitations of this investigation to caution their teachers. The Mathematical Critical Curation Questions can be further extended to include questions that address AI specifically. Second, MTEs need to teach prompt engineering techniques to get responses that are not easily created. Third, MTEs need to provide critical insight to their teachers to help them understand the biased nature of this tool. ".

Type of Talk: Research Report; Author: James Drimalla, University of Virginia

Talk Title: An Inferentialist Investigation of a Prospective Secondary Mathematics Teacher’s Conceptual Understanding of Function

Keywords: Conceptual Understanding, Function, Inferentialism

Abstract: In this report, I draw on inferentialism—a theory of meaning that addresses linguistic and epistemic issues—to investigate a prospective secondary mathematics teacher’s (PST) conceptual understanding of function. Inferentialism has primarily been developed by the philosopher Robert Brandom (2000) and, in the last ten years, has been adopted by several mathematics education researchers to study a variety of topics. Inferentialism has not, however, been employed in the context of undergraduate mathematics education and thus the reported investigation is a unique contribution to research in undergraduate mathematics education. In short, inferentialism describes how the semantic content of a concept (e.g., a mathematical concept like function or inverse) exists in the social game of giving-and-asking-for-reasons with an emphasis on the claims, inferences, and social norms surrounding the concept. The unique capability of inferentialism to address both individual and social aspects of learning allowed me to analyze a PST’s understanding of the concept of function across two related sources of data. The first source was video data from the first of three mathematics content courses for PSTs prior to student teaching. Amidst the content course, clinical interviews were performed with nine of the PSTs as part of an overarching teaching experiment (Steffe & Thompson, 2000). The PSTs were interviewed twice—near the beginning of the semester and near the end of the semester. These clinical interviews were my second source of data. Analyses of students’ mathematical conceptual understanding often draw on theories based in individualistic theories like cognitive constructivism, but inferentialism allowed me to foreground the PST’s individual learning within classroom discussions and simultaneously attend to how ideas received normative status in the classroom. Ultimately, my analysis illustrates how the PST’s understanding of function grew from initially only applying a simple “vertical line test” to a more nuanced application of the concept of function that included the concept’s formal definition as well as its inferential relationship with the concepts of input and output. Implications for teaching mathematics for conceptual understanding are then considered.. 

Type of Talk: Research Report; Author: Allison (Allie) Olshefke-Clark, University of Delaware

Talk Title: Static and dynamic: How two textbooks introduce derivative

Keywords: textbook analysis, derivatives, covariational reasoning,

Abstract: Research has shown that students often successfully learn to compute derivatives but have difficulties grasping the underlying concepts. Thompson and Harel (2021) argue improving students’ covariational reasoning (i.e., reasoning about how two or more quantities change in relation to one another) may be one way to help them engage deeply with derivative concepts. Covariational reasoning is implied in the MAA’s recommendation for teaching the derivative dynamically as the “measure of sensitivity of one variable to change in another,” as opposed to the traditional, “very static interpretation” (Bressoud et al., 2015, p. 18). Recent textbook analyses (e.g., Mkhatshwa, 2022) suggest students’ opportunities to reason about the derivative in a dynamic way via covariation are limited.

This study expands research in this area by analyzing how the derivative is introduced as static and dynamic in two commonly used Calculus textbooks: Larson and Edwards (2018) and Hughes Hallett et al. (2013). I use Tasova et al.’s (2018) framework which combines Moore and Thompson’s (2015) shape thinking framework with Thompson and Carlson’s (2017) covariational reasoning framework. Using this framework, I coded instances in which the textbooks promoted static or dynamic reasoning about quantities related to the derivative.

Results from this analysis indicate though each textbook provides opportunities for both static and dynamic conceptions of the derivative, they vary in their emphasis. For instance, 76% of the instances coded in Larson and Edwards (2018) were categorized as static. Hughes Hallet et al. (2013) provides a more balanced approach, with 54% of the instances coded as dynamic. Further, the only instance of Continuous Covariation (the highest level) was present in Hughes Hallett et al. (2013). In my presentation, I will present a more detailed breakdown of these results, including examples of each code from the textbooks.

The results of this study indicate the specific calculus textbook students engage with may have implications on how students learn to conceptualize the derivative. It’s well-established that calculus textbooks have an influence on teachers’ planning and practice (e.g., Gerami et al., 2023), so selecting a book that promotes a more dynamic conception of the derivative may be beneficial in fostering students’ productive meanings for the derivative. Teachers, as well as school and district leaders should take this into account when selecting a textbook or other curriculum material. Future research should consider including other textbooks in similar analyses to provide more comprehensive information for decision makers. Similarly, curriculum developers should consider incorporating more opportunities for students to conceive of the derivative dynamically. I posit supplementing textbooks with more dynamic, digital tools may be necessary to counter the inherent limitations of textbooks’ static medium.

Type of Talk: Research Report; Author: Meiqin Li, University of Virginia

Talk Title: Students' Views on Programming  in Engineering Linear Algebra Classes

Keywords: engineering linear algebra class, numerical computation, perception, programming

Abstract: Researchers have suggested that incorporating programming languages such as MATLAB, Python, and Mathematica into linear algebra courses could be beneficial(Carlson et al., 1993) (Shankar, 2017). Considering this, we redesigned APMA 3080 - Linear Algebra by modifying research and evidence-based curricula development (Baxter et al., 2019, 2020; Sheppard et al., 2018), to include four numerical computational components, with MATLAB serving as the primary tool. This curriculum change is detailed in a published paper (Li, 2023). The four components are: (a) solve in-class worksheet with coding problems, (ii)code core linear algebra concepts, (iii) solve applications by algorithms, and (iv) visualization of abstract concepts.

While literature exists on integrating technology into linear algebra classes(Kalsi, 2010; Silva et al., 2022; Tang, 2021), most studies have focused on isolated aspects, used technology partially, or lacked a specific focus on engineering students. Additionally, thorough investigations into students' perceptions have been lacking, even with its importance on course design (Doppel & Schenn, 2008). In this study, we thoroughly examined students' perceptions of the effectiveness of integrating MATLAB-based numerical computational components, shifting the emphasis from expert opinions to student feedback on how these changes impact their success.

The key research questions we have investigated are:

1. How do students' perceptions of MATLAB as a program change, throughout the semester? Do these perceptions differ based on factors like academic major, gender, or race/ethnicity?

2. How do students' perceptions of MATLAB as a learning tool for linear algebra change, if at all, over the course of the semester? Do these perceptions differ based on factors above?

3. How do students perceive the various numerical computational components and the associated support resources in this course? Is there a correlation between their perceptions of different components?

~140 students participated in this study. They were asked to complete a pre-survey at the start of the semester and a post-survey at the end of Fall 2023, serving as the primary data sources. These surveys were designed to investigate the above three research questions and reviewed by education experts. The data gathered from participants was analyzed using statistical methods, such as boxplots, linear regressions, and hypothesis tests.

In this study, it was found that students felt significantly more comfortable to use MATLAB and improved their abilities to solve problems in MATLAB after the incorporation of four numerical components. Also, students reflected that MATLAB was an efficient tool to solve linear algebra problems and they used MATLAB for various tasks at different comfortable levels. The findings are going to be presented in aggregate form (for quantitative and quantized qualitative data) and individually (for qualitative data).

Type of Talk: Ideas for Research Studies; Author: Hui Ma, University of Virginia

Talk Title: The impact of mastery grading on subsequent course performance

Keywords: Calculus, mastery grading, long-term effects, student success 

Abstract: "This preliminary research study aims to investigate the long-term effects of mastery grading on student performance by comparing outcomes in a subsequent course with those of peers who experienced traditional grading.

Mastery grading has been increasingly adopted in STEM education due to its focus on student growth and well-being. Mastery grading was introduced in our Calculus I course in the fall 2022. Students in the mastery-graded class reported reduced test anxiety, earned higher letter grades through penalty-free reattempts, and felt more confident in their math abilities. In fall 2023, we expanded mastery grading to more sections and made several modifications to the system. The common final exam results showed notable benefits from mastery grading for students with the lowest diagnostic scores entering Calculus I.

Currently in the design phase, we want to see how well students who took mastery-graded sections in Calculus I do in their Calculus II class compared to students who took traditionally graded sections. This will help us understand how the grading method affects student success in the long run. We have collected students' performance data in Calculus II from Spring 2024 and plan to analyze their trends in performance throughout the semester, their final grades, and their DFW rates. We invite feedback and suggestions to refine our study design and ensure our results are meaningful.

Type of Talk: Ideas for Research Studies; Authors: Brian Tyrrell-Nic Dhonncha,

 California Polytechnic University, SLO; Vladislav Kokushkin, James Madison University

Talk Title: The Role of Signers’ Gestures in Offloading Cognitive Demands on Working Memory During High-Level Mathematics Tasks

Keywords: Proofs, ASL, Working Memory, gestures, cognition

Abstract: "Applications of cognitive psychology to mathematics education have revealed a gamut of cognitive challenges experienced by undergraduate students during high-level mathematics tasks (such as proof comprehension, validation, and explanation). These challenges are related to students’ attention, emotions, consciousness and reflection, persistence, memory, and many other aspects of everyday cognition. One of them, working memory (WM), is a psychological model for humans’ ability to simultaneously store and process a limited amount of information over a short period of time (e.g., Pascual-Leone, 1970). It provides a platform for higher cognition and is actively involved in various mathematical activities that range from performing basic arithmetic calculations (e.g., Alibali & DiRusso, 1999) to working with mathematical proofs (Kokushkin, 2022).

WM capacity is limited and varies markedly across individuals and groups. Nevertheless, it is natural for a learner to alleviate the experienced cognitive burden via the mechanism known as cognitive offloading. Traditionally, cognitive offloading refers to using embodied resources to navigate a task's cognitive challenges. Examples of cognitive offloading include drawings, the use of technology, and hand gesturing. Recent studies (Kokushkin, 2022, 2024) suggest that gesturing may be a powerful offloading mechanism in high-level mathematics.

The propensity to gesture depends on age, gender, ethnicity, culture, and many other factors (e.g., Hostetter & Hopkins, 2002). Given that gesture production may help learners to reduce the experienced cognitive load, how (if at all) does this type of offloading change for ASL-trained individuals who are “hardwired” to communicate in the physical modality?

Answering this broad question, we seek to explore two (not necessarily mutually exclusive) hypotheses. On the one hand, there is a significant overlap between signs and gestures (e.g. Goldin-Meadow & Brentari, 2017). For example, it has been shown that signers’ gestures are compatible with speakers’ gestures on certain mathematical tasks (Goldin-Meadow et al., 2012). As such, it might be the case that ASL-trained students employ hand gestures similar to hearing students or even adapt sign language to a mathematical task. Second, the relationship between signs and gestures is more complex than it first appears (cf. Müller, 2018): the necessity to juxtapose multiple embodied modalities may place additional cognitive burden on a student’s WM and create even greater cognitive overload.

During our presentation, we will share the theoretical foundation, broad research questions, and preliminary design of our study. We will ask the audience for feedback. Undergraduate mathematics education for deaf students is a pressing national problem. Studying the resources students can employ during high-level mathematical cognition should promote better access to learning mathematics at universities and colleges.  

Type of Talk: Research Report; Author: Ted Townsend, West Virginia University

Talk Title: Institutional Differences Between Undergraduate Mathematics Students’ Knowledge About Graduate School

Keywords: Graduate School Applications, Graduate School, Liberal Arts, R1

Abstract: The Carnegie Classification sorts institutions by their different styles of education (e.g. high research activity universities (R1) and liberal arts focused colleges (LA)). Previous studies identified different cultural characteristics that affect graduate school aspirations. Studies showed that attending an R1 institution negatively affects graduate school aspirations but attending an elite or high quality college positively affects it (Astin, 1997; Eide et al., 1998; Zhang, 2005). Researchers argued that good teaching practices, integration within the social and academic systems of the institution, supportive departmental climate, and personal relationships with professors have positive effects on persistence to graduate school (Blaney & Wofford, 2021; Ethington & Smart, 1986; Hanson et al., 2016; Hearn, 1987; Ostrove et al., 2011). Advising is a crucial characteristic since it affects student learning outcomes (Mu & Fosnacht, 2019). R1 schools are more likely to have centralized advising which has been shown to have positive effects compared to faculty advising or no advising at all (Faurot et al., 2013; Kennedy-Dudley, 2007; Kot, 2014). We address two research questions: (a) Are there differences between R1 and LA mathematics students’ interest in attending graduate school? Are there differences between their knowledge about graduate school and applying to graduate school in mathematics? (b) Are there differences between R1 and LA students with respect to the quality of their relationships with their mathematics departments? Do these relationships have an impact on the answers to the previous question? We used the theories of social and cultural capital to interpret our results. Coleman (1988) defines social capital as a variety of different entities that consist of some aspects of social structures that facilitate actions of actors within the social structure. Bills (2000) defines cultural capital to be “the degree and ease of familiarity that one has with the dominant culture of society” (p 90). We analyzed responses from a national undergraduate mathematics major sample using chi-squared and Mann-Whitney U tests to identify differences between students’ knowledge about graduate school and its application process by the R1 and LA institution types. We also used chi-squared tests to explore the differences between departmental (professors, advisors, and mentors) support by institution type. We found that our LA and R1 participants did not differ on what they know about graduate school and how to apply to graduate school. However, in comparison to their LA peers, our R1 participants perceived a lack of connections and support from their departments. Thus, we conclude that the LA participants possess social capital within their departmental relationships to acquire their information. We hypothesize that our R1 participants utilize their cultural capital with a social capital existing within their peer and math club relationships.

Type of Talk: RUME Proposals for Feedback; Author: Dan Velleman, Amherst College

Talk Title: Using Software to Teach Proof

Keywords: Proof, software, Lean

Abstract: "Interactive theorem-proving software has the potential to help students learn to write proofs.  The software gives students immediate feedback, identifying mistakes in an incorrect proof or certifying the correctness of a correct proof.  The software also provides guidance during the construction of a proof, and it forces students to use precise language and proof techniques based on rules of logic.  A potential disadvantage of using such software is that students must learn the formal syntax required by the software.

I will describe teaching materials I have developed for using the proof assistant Lean to teach proofs, and I will demonstrate Lean.  I have not done any research on the use of these materials, but I believe there is interesting research to be done in this area.

There have been a few studies of the use of Lean to teach proof.  Thoma and Iannone (2021) compared students in a transition to proof course who participated in voluntary workshops on Lean to students who did not participate.  They analyzed proofs written by students in both groups using pencil and paper (not Lean) and found:

""The analysis shows two characteristics of proofs written by students who engaged with the programming language. The first concerns proof writing and includes the accurate and correct use of mathematics language and symbols, together with the use of complete sentences and punctuations in proofs. The second concerns proof structure and includes the overt break down of proofs in goals and sub-goals.""

Hanna, Larvor, and Yan (2024) give an extensive discussion of conceptual and cognitive reasons for using Lean to teach proof.  They also did an exploratory study of three students using Lean to write a proof.  In describing this study, they wrote:

""The findings of our exploratory study, admittedly tentative, suggest that the rigorous nature of Lean is not an obstacle for students and does not seem to lead to stifling students’ creativity in proving. On the contrary, proving with Lean offers a great deal of flexibility. …  We claim that what is gained in proving with Lean is the adventurous aspect of proving and the autonomy to go about a proof in one’s own way. … As Lean provides instant feedback with no shame attached, it allows students to play with proof ideas. This playfulness would likely enhance student engagement with proof, and further change how they perceive the creation and presentation of proof.""".

Type of Talk: Ideas for Research Studies; Author: Marcus Wolfe,  James Madison University

Talk Title: Future  Proposed Studies Around Gesture in Games for Fraction Reasoning

Keywords: Games, Gesture, Reasoning

Abstract: "In my 2024 study, it was explored how students gesture while playing video games designed to evoke fraction knowledge. The gestures were made when players were trying to recognize and make connections between the game and fraction content. Three categories of gesturing showed up in results of that study; players gestured using the in-game avatar, they gestured with one hand on the controller  or they gestured without the controller. The study mentioned that the ways gestures were used between those using and not using the controller were similar. Adopting a Gestures as Simulated Action framework suggests that two different phenomena occurred. One was that the player who gestured in the game did not experience the game controller as an inhibiting factor because they gestured similarly to their peers, retaining similar reasoning strategies. However those who were unable to use the in game avatar to gesture and remove the hand off the controller were inhibited. These players lost access to the digital resources of the game .

There are several potential reasons for the controller being an inhibiting factor but the ones that I want to further study are:  players isolating math reasoning to physical gestures  and the role of math knowledge. Students not having a history of explaining gameplay could mean that they would try to separate connection making from the gameplay experience, seeing connection making as a math-oriented activity. This separation might mean they want to leave the game behind to do the math. In addition, part of the reason why players may not gesture in game is that it might require more mathematical knowledge to use the abstract game world as a cognitive resource.

I am considering two different studies to make sense of the phenomena. I want to do a study where players of the game are asked only to use the characters and environment in the game to make connections between fraction content and game. I want to see if being asked only to use in game resources impacts the amount of math reasoning being made. I suggested that part of the reason why players gesture outside the game is because they see gameplay and connection ( a math activity) as separate tasks. If constricted to use the digital resources will prevent the controller from being an inhibiting factor.  I am also considering a study where two groups are created. I want to have one group asked to generate connections without physical gestures and then the other only with physical gestures. I want to see if the types of gestures used between these groups are different and as a result what types of fraction knowledge do they get access to.

Type of Talk: Ideas for Research Studies; Author: Praveen Chhikara, University of Illinois, Urbana-Champaign

Talk Title: Knowledge of History of Mathematics:  A Document Analysis

Keywords: History of Mathematics, Teacher Knowledge, Pedagogical Content Knowledge, Mathematics Instruction

Abstract: "Research documents the importance of integrating the history of mathematics (HoM) into instruction to enrich students' learning of mathematical concepts (Jankvist, 2009;  Kjeldsen and Blomhøj, 2012). Concurrently, there is a body of literature on teacher’s knowledge as the knowledge required to perform teaching tasks (Hill et al., 2008; Shulman, 1995; Speer et al., 2015). The analysis seeks to bridge the gap by exploring how a teacher's knowledge of the history of mathematics aligns with the kinds of knowledge required for effective teaching.

Background and Rationale: Jankvist (2009) conceptualized the integration of HoM in mathematics teaching and learning. He identified arguments that provide rationales for the role of HoM and approaches that integrate HoM in mathematics teaching, which he called why arguments and how approaches, respectively. He used the concepts called in-issues and meta-issues of mathematics to conceptualize whys and hows more clearly. The in-issues of mathematics mean the “issues related to mathematical concepts, theories, discipline, methods, etc. - the internal mathematics” (Jankvist, 2009, p. 240).  The meta-issues mean evolutionary, human/cultural, application, and epistemological/ontological aspects in the development of the discipline of new concepts, procedures, or theories in mathematics.

Research identifies many models of teacher knowledge for mathematics teaching (Ball et al., 2008; Krauss et al., 2008; Turner & Rowland, 2008). I will focus on Ball et al.’s (2008) Mathematical Knowledge for Teaching (MKT) framework, which expanded Shulman’s work in mathematics education. They defined “mathematical knowledge for teaching” as “mathematical knowledge needed to carry out the work of teaching mathematics” (Hill et al., 2008, p. 395). For MKT framework, Ball et al. (2008) define six domains of mathematical knowledge for teaching and divide them into two groups, each including three domains of knowledge: (1) Subject Matter Knowledge includes three domains viz. Common Content Knowledge (CCK), Horizon Content Knowledge (HCK), and Special Content Knowledge (SCK). (2) Pedagogical Content Knowledge (PCK) includes three domains viz. Knowledge of Content and Students (KCS), Knowledge of Content and Teaching (KCT), and Knowledge of Content and Curriculum (KCC).

Research Objectives: The primary objectives of this document analysis is to identify how the kinds of teacher knowledge relates to the integration of the history of mathematics into undergraduate and general mathematics education

Methodology: The study will include a systematic review of existing research literature on two research areas: (1) the integration of HoM into teaching and (2) teacher’s knowledge. Using a qualitative content analysis approach, the study aims to identify patterns and relationships between these two bodies of research.

The findings have potential implications for curriculum development and instructional practices in mathematics education.

Type of Talk: Ideas for Research Studies; Author: David Wizer, Towson University

Talk Title: PrimeTime element of supportive community 

Keywords: Advanced math, student led support 

Abstract: This presentation will feature PrimeTime as a support element for mathematics students at Towson University. PrimeTime is a student led help session for students enrolled in advanced mathematics courses.

Prior research: There are several research threads that guided this research, two are highlighted in this summary. We explored prior research on mathematics identity. What follows is a summary highlight on identity (Darragh, L. (2016) that relates to this research. “Identity is generally agreed to be multiple or referred to in the plural. Furthermore, …influential theorists treat identity in terms of an action rather than an acquisition. Wenger (1998) sees identity as Not an object, but a constant becoming.”

In addition, mathematics and STEM learning community research is central to this research work.

Theoretical foundation for this research draws on learning community research. Carrino, and Gerace, (2016), identify four psychosocial (affect) learning factors that students said improved because of their participation in the learning community: academic self- regulation, STEM professional/science identity, metacognition, and self-efficacy.

Methodological justification and Data analyzed: A mixed methods approach was employed in this research. We believe this mixed methods choice provides for more detailed understanding of the mathematics students. First, quantitative survey analysis led to preliminary results. The surveys included senior students, participants in research projects, and other department activities including PrimeTime. This research relied heavily on qualitative analysis of graduating student interviews. A theme base recursive review process was employed, based on coding of themes and observing trends are some of review techniques used by multiple researchers.

Major findings: This presentation will highlight the intended elements and purpose of PrimeTime as well as early data collection and analysis outcomes based on several data sources. (observations of PrimeTime, interviews of graduating seniors and surveys of mathematics students).

Preliminary results converge to the following themes:

1. PrimeTime apparently helps some students develop advanced knowledge and skills.

2. Psychologically, PrimeTime has been a supportive environment that builds confidence.

3. PrimeTime can be a setting in which mathematics majors develop enhanced math identity.

Contribution to the field: This work contributes information on evolving mathematics identity and becoming a mathematician. This research is a building block to better understand mathematics learning community. The intention is to enhance understanding of psychosocial learning factors that may improve because of their participation in a learning community.

Type of Talk: Poster; Author: Heze Chen, University of Virginia

Talk Title: Interactive Learning Modules with Visualization in Multivariable Calculus

Keywords: Graphical User Interfaces; Interactive Learning; Undergraduate Students

Abstract: This study addresses the educational challenges faced by engineering undergraduates when learning from diverse sources, including projects and worksheets. We developed a comprehensive web-based platform that seamlessly integrates web applications with high-quality graphical user interfaces, enhancing interactive learning and accessibility. Additionally, we condensed multivariable calculus lecture notes and implemented them into Jupyter Notebook, incorporating Python programming to facilitate an interactive and engaging learning environment. This integrated approach aims to improve comprehension and retention of complex engineering concepts through the use of advanced computational tools and tailored educational resources.

Type of Talk: Poster; Author: Tony Mixell, Dickinson College

Talk Title: Preliminary Analysis of Undergraduate Student Problem Posing in a Scaffolded Calculus Intervention

Keywords: problem posing, Calculus, mathematical representations,

Abstract: Problem posing, that is, the reformatting of existing problems and the generation of new ones, has been promoted as a valuable practice for deepening students’ mathematical thinking (NCTM 1989, 2000). Problem posing has also been associated with problem-solving ability, conceptual understanding, and mathematical habits of mind such as divergent thinking. Research has shown that mathematical representations, such as equations, graphs, tables, and visuals, are important tools for developing and reflecting understandings of mathematical ideas, and policy (NCTM, 2000) has called for the use of multiple representations for promoting mathematical thinking. As Single-Variable Calculus serves as a gateway course for many college majors, I explored what kinds of mathematical problem scenarios (analogous, somewhat analogous, or original) undergraduate Calculus students created while receiving a scaffolded Calculus and representation-based intervention.

During two semesters, I, the principal investigator, led a nine-week intervention in four undergraduate Calculus courses in a small, mid-Atlantic college. Over the nine weeks, all students received instruction on average rate of change, limits, continuity, the derivative, and applications. Students were paired randomly by an outside researcher. As a preliminary analysis, six dyads with similar Calculus and problem-posing backgrounds were initially analyzed. For Phase 1 (five weeks), students were told to create problem scenarios for approximately 20 minutes per week based on the mathematical content taught that week. For Phase 2 (four weeks), students were instructed on the concept of problem posing, that is, the difference between analogous problems, “what if?” problems, and original problems (inversions), along with the incorporation of one, then two mathematical representations. Then, they were again instructed to create problem scenarios for approximately 20 minutes per week.  

Upon analysis of the problem scenarios, during Phase 1, almost 90% of the created scenarios were analogous to those received during instruction, no matter the students’ prior experiences with Calculus or with problem posing. However, during Phase 2, this number dropped to less than 20%. Problem scenarios created during Phase 2 varied in degree of complexity, that is, in the number of steps taken to solve a problem. Differences also existed regarding the degree of mathematical thinking and reasoning incorporated into the scenario. Implications of the intervention for practitioners and researchers are discussed.

Type of Talk: Poster; Author: Meiqin Li, University of Virginia

Talk Title: The Reliability and Predictivity of Online Placement for Calculus Courses

Keywords: Online Placement Test, Reliability, Predictivity, Calculus, Precalculus

Abstract: Placement Tests (PT) are essential tools for assigning incoming first-year college students to  appropriate calculus courses. However, inappropriate placements can cause significant stress for  students. At our institution, we observed discrepancies between online PT scores and actual  understanding, with some students scoring high online but significantly lower in-person, leading to  struggles in their courses. Conversely, students with better in-person PT scores performed well in  their assigned courses. This observation motivates a critical question: How reliable are online PTs in  accurately placing students?

While PTs have been extensively studied, most research focuses on their impact on student  sustainability and graduation rates or on PT design [1]. The reliability and validity of online PTs  have been underemphasized. The limited existing literature primarily focuses on liberal arts subjects [2]. Supported by an HHMI IE3 grant, our project investigates the reliability of online PTs in  accurately assigning students to their appropriate courses and the predictive validity of in-person  PTs for Calculus performance in a different approach: comparing the overall and topic-specific  scores of highly similar placement tests, one conducted online and the other in-person.

Our research questions are:

RQ1: To what extent, is the online placement test reliable for students in Calculus I and Calculus II  overall? Do these reliabilities vary based on factors such as AP scores, calculus performance and  gender?

RQ2: To what extent, do the reliabilities for different topics in online PT differ? Do these  reliabilities of different topics vary based on gender?

RQ3: If any, how predictive is in-person PT for student performance in Calculus I and Calculus II? The main data resources are online PT scores, in-person PT scores and Cal1/Cal II grades for  students enrolled in Calculus I and Calculus II (~500 students each year). The in-person PT is  administered at the beginning of the semester. Statistical tools will be used for data analysis.

For RQ1, we will examine the percentage differences between online and in-person scores at  various reliability levels (±4, 6, 8). For RQ2, we will divide PT problems into topic blocks and  analyze score differences. For RQ3, we will study correlations between in-person assessments and  final exam performance in Calculus I and II.

Via this research, we can improve PT structure, identify challenging topics for students, and provide  more targeted support. The study has IRB approval, and all data management adheres to  institutional policies.