Drew Lewis

Drew Lewis

Mathematician and education researcher

Publications

You can also find my articles on my Google Scholar profile.

Humanizing Defining in Mathematics with Sandwiches

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Abstract

This paper discusses multiple versions of an activity in which participants create, discuss, and analyze possible definitions of the term “sandwich.” This activity has helped us humanize mathematical practices in a low-stakes context, build community in mathematical spaces, model and reflect on effective collaboration for active learning, and peer under the hood of mathematics disciplinary practices around constructing definitions and examples. We have used this activity virtually, both synchronously and asynchronously, and face-to-face, with a wide range of participants, including for secondary, major and non-major undergraduate, and graduate mathematics students and for teaching professional development. This paper will discuss motivation, various implementations, and some outcomes of the activity.

Recommended citation:

Katz, B., Lewis, D., Bagley, S., Bockting-Conrad, S., Ho, A., Libertini, J., Pilgrim, M., Salomone, M., & Shilito, B. (2025). Humanizing Defining in Mathematics with Sandwiches. PRIMUS


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Community-driven data science practices

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Abstract

Mathematics researchers are becoming more involved with research questions at the interface of data science and social justice. This type of research needs to be grounded in the needs of the community in order to have significant impact. In this paper, we examine two examples of community-research partnerships in data science for social justice co-authored by both community members and mathematical researchers. The first, VECINA, is a place-based community-research partnership focused on environmental justice. VECINA introduces a framework for developing fruitful local collaborations. The second example, SToPA, originates in citizens’ request for an analysis of their town’s policing data, but focuses on how to scale this work beyond that place-based setting. SToPA’s research helps us imagine how we can continue to actively collaborate with community members even when working to scale projects beyond a single community. In both of these case studies, we examine the harmonies between established principles of power, process, and perspective with our framework for research-community partnerships. We use a duoethnography approach, directly illustrating the experiences of researchers. We also offer a set of reflections on the impact of these research-community partnerships.

Recommended citation:

Barreda II, A., Diaz Eaton, C., Hansen, S., Hibdon Jr., J. Gordon, L., Greenwald, R., Gutiérrez Paz, M.J., Ince, K., Kelling, C., Lewis, D., Mendible, A., Mercado, J., Piercy, V., Thompson, B. (2026). Community-driven data science practices. Preprint


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Promoting students’ sense-making in row reducing matrices

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Abstract

In this lesson analysis, the topic of the lesson is using operations to calculate the Reduced Row Echelon Form of a matrix. The course context is an introductory linear algebra course taught using Team-Based Inquiry Learning. The instructional challenges we seek to address are: (1) Students often do not see the reasons for why the RREF is a desirable form for a matrix, or why performing row operations is a reasonable thing to do to a matrix; and (2) Students tend to think about row reduction as a computation/algorithm, and we want to shift them into a reasoning/sense making stance about the problem. To address these challenges, we use an approach grounded in Inquiry-Based Learning. Students begin by identifying a set of operations that preserves the solution set of a linear system, and then build intuition around the idea that one linear system may look easier to solve than another even though they have the same solution set, motivating the definition of reduced row echelon form. Students then practice determining if a matrix is in RREF and, if not, identifying row operations that bring it closer to its final RREF.

Recommended citation:

Kostiuk, J., Lewis, D., Borges, T., Brandt, M., Chang-Lee, M., Creech, C., Freedman, S., Griffith, S., & Hashimoto, S. (2025). Promoting students’ sense-making in row reducing matrices. PRIMUS


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Quantifying Inequities and Documenting Elitism in PhD-granting Mathematical Sciences Departments in the United States

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Abstract

Using data about PhD-granting institutions in the United States and publicly available funding data from the National Science Foundation, we highlight inequalities based on gender in mathematical sciences departments at US institutions of higher education. We provide an example of the application of quantitative techniques, tools, and topics from mathematics and data science to analyze the mathematics community itself in order to quantify and document inequalities in our discipline. Specifically, we determine that a small fraction of mathematical sciences departments receive a large majority of federal funding awarded to support mathematics in the United States. Additionally, we quantify the extent to which women faculty members are unequally represented in mathematical sciences departments at PhD-granting institutions in the United States. We also show that this underrepresentation of women faculty is even more pronounced in departments that received more federal grant funding.

Recommended citation:

Buckmire, R., Diaz Eaton, C., Hibdon Jr., J.E., Kauba, J., Lewis, D., Ortega, O., Pabon, J., Roca, R., & Vindas-Melendez, A.R. (2025). Quantifying and Documenting Gender-Based Inequalities in the Mathematical Sciences in the United States. In Garcia-Cardona, C., Lee, H. (eds) Advances in Data Science. Association for Women in Mathematics Series, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-031-87804-6_16


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Key network disruptors: A structural analysis of the #DisruptJMM network influencers and actions

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Abstract

Twitter (now known as X) has been used by an active group of aca-demic mathematics communities to connect and share ideas and informa-tion. Like many other Academic Twitter communities, conference hash-tags have been widely used to share information between conference atten-dees and communicate new ideas presented at conferences to the broaderacademic community. However, in 2020, Dr. Piper Harris proposed a dif-ferent kind of academic hashtag — #DisruptJMM — as a counternetworkhashtag to an in-person conference. This study looks at the use of thishashtag through and in between the 2020 and 2021 Joint MathematicsMeetings, which are advertised as the largest mathematics gatherings inthe wold. Through qualitative coding, we investigate the many ways wecan conceptualize and define “influencers” in this context. We find thatsome definitions consistently identify the same individuals as “communityorganizers”, whereas some definitions of influencers allow us to identify al-ternative ways that individuals can influence the growth and propagationof hashtags. This study contributes to our understanding of the myriadways that individuals can contribute to build hashtag movements thatmove between in-person and online spaces.

Recommended citation:

Lewis, D., Roca, R., Marshall, S., Hibdon Jr., J., & Diaz Eaton, C. (2025). Key network disruptors: A structural analysis of the #DisruptJMM network influencers and actions. Preprint


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Checkit OER assessment generator

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Abstract

The CheckIt framework is an open educational technology for authoring randomized exercises, banks, and assessments for mathematics. Technological means facilitate the easy sharing and modification of open-source templates and generators authored in the framework to be used both in classrooms as well as to support educational research; case studies in this paper illustrate its use in both contexts, and describe its use in both small and large classrooms.

Recommended citation:

Clontz, S., Lewis, D., & Krinsky, S.. (2025). Checkit OER assessment generator. In E. Elkhoury, J. Olivier, and T. N. Thurston, editors, Open Educational Resources for and as Assessment. Utah State University, Logan, UT.


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Community-Based, Open Source, Continuous OER Authoring (COCOA): A Novel Model for Open Educational Resources

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Abstract

As demand for open education resources (OER) grows among instructors, methods of supporting OER authors that leverage new technologies and toolkits are needed to support this growing community. In this article, we describe our Community-based, Open source, Continuous OER Authoring (COCOA) model of OER development. In addition to describing the model in the abstract, we also detail the software and services we are currently using to support authoring teams using this model and provide reflections on the relative strengths and challenges this model presents.

Recommended citation:

Clontz, S. & Lewis, D. (2025). Community-Based, Open Source, Continuous OER Authoring (COCOA): A Novel Model for Open Educational Resources, Journal of Open Educational Resources in Higher Education 3(3), 273-290. https://doi.org/10.31274/joerhe.20094


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Supporting instructors in implementing team-based inquiry learning

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Abstract

Team-Based Inquiry Learning (TBIL) is a novel active learning pedagogy designed to facilitate the use of inquiry-based learning in lower division courses. This preliminary report examines supports provided by the TBIL project to instructors, as well as the fidelity of implementation of TBIL by participants of the project. Initial findings suggest that classroom-ready materials and ongoing support, both synchronous and asynchronous, were most helpful to faculty in their TBIL implementations.

Recommended citation:

Lewis, D., Clontz, S., Parrish, C., Estis, J., & Chaudhury, S.R. (2024). Supporting instructors in implementing team-based inquiry learning. Proceedings of the 26th Anuual Conference on Research in Undergraduate Mathematics Education, 1152–1157.


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The mathematics of mathematics: Using mathematics and data science to analyze the mathematical sciences community and enhance social justice

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Abstract

We present and discuss a curated selection of recent literature related to the application of quantitative techniques, tools, and topics from mathematics and data science that have been used to analyze the mathematical sciences community. We engage in this project with a focus on including research that highlights, documents, or quantifies (in)equities that exist in the mathematical sciences, specifically, and STEM (science, technology, engineering, and mathematics) more broadly. We seek to enhance social justice in the mathematics and data science communities by providing numerous examples of the ways in which the mathematical sciences fails to meet standards of equity, equal opportunity and inclusion. We introduce the term “mathematics of Mathematics” for this project, explicitly building upon the growing, interdisciplinary field known as “Science of Science” to interrogate, investigate, and identify the nature of the mathematical sciences itself. We aim to promote, provide, and posit sources of productive collaborations and we invite interested researchers to contribute to this developing body of work.

Recommended citation:

Buckmire, R., Hibdon Jr., J.E., Lewis, D., Ortega, O., Pabon, J., Roca, R., & Vindas-Melendez, A.R.(2024). The mathematics of mathematics: Using mathematics and data science to analyze the mathematical sciences community and enhance social justice. La Matematica 4, 110–125 (2025). https://doi.org/10.1007/s44007-024-00146-6


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#DisruptJMM: Online social justice advocacy and community building in STEM.

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Abstract

In 2019, #DisruptJMM, a Twitter hashtag, began circulating after an Inclusion/Exclusion blog by Dr. Piper H pointing to the need to make commonplace conversations about human suffering in the Joint Mathematics Meetings (JMM). While the #DisruptJMM hashtag has been used since 2019, the vast majority of use was in the JMM 2020 meetings. Twitter hashtags are used by activists to push forward conversations, join communities around a single idea, and create change. In this article, we draw on frameworks from community building seen in other equity and inclusion advocacy hashtags such as #GirlsLikeUs (Jackson et al., 2018) to qualitatively code and analyze tweets which used the #DisruptJMM hashtag. This analysis gives us a glimpse into the evolving conversations on social justice, equity, diversity and inclusion in the mathematical community. We see an emergence of community around recentering humanity in mathematics. This community supported each other’s efforts, reflected on who is represented, and amplified discussions of power and privilege with a particular emphasis on bringing visibility to colonialism and sexual harrassment.

Recommended citation:

Roca, R., Eaton, C. D., Lewis, D., Hibdon, J. E., Jr., & Marshall, S. (2023). #DisruptJMM: Online Social Justice Advocacy and Community Building in Mathematics. Journal of Humanistic Mathematics, 13(2), 39–63. https://doi.org/10.5642/jhummath.IQAC8746


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On definitions of mathematician

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Abstract

The definition of who is or what makes a “mathematician” is an important and urgent issue to be addressed in the mathematics community. Too often, a narrower definition of who is considered a mathematician (and what is considered mathematics) is used to exclude people from the discipline – both explicitly and implicitly. However, using a narrow definition of a mathematician allows us to examine and challenge systemic barriers that exist in certain spaces of the community. This paper explores and illuminates tensions between narrow and broad definitions and how they can be used to promote both inclusion and exclusion simultaneously. In this article, we present a framework of definitions based on identity, function, and qualification and exploring several different meanings of “mathematician”. By interrogating various definitions, we highlight their risks and opportunities, with an emphasis on implications for broadening and/or narrowing participation of underrepresented groups.

Recommended citation:

Ron Buckmire, Carrie Diaz Eaton, Joseph E. Hibdon Jr., Katherine M. Kinnaird, Drew Lewis, Jessica Libertini, Omayra Ortega, Rachel Roca, and Andrés R. Vindas-Meléndez. On Definitions of “Mathematician” (2023). Journal of Humanistic Mathematics, 13(2), 8–38. https://doi.org/10.5642/jhummath.ZRUZ1463


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Impacts of Standards-Based Grading on Students' Mindset and Test Anxiety

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Abstract

Standards-based grading is an alternative grading method with many claimed benefits. This paper reports on quantitative studies investigating several of these oft-made anecdotal claims, such as reducing students’ test anxiety and fostering a growth mindset. We found that standards-based grading did reduce students’ test anxiety; moreover, the typically found difference in test anxiety between male and female students was eliminated in standards-based grading courses. We found no change in students’ growth mindset, but that students’ mastery avoidance goals were reduced.

Recommended citation:

D. Lewis, Impacts of Standards-Based Grading on Students' Mindset and Test Anxiety. Journal of Scholarship of Teaching and Learning 22, no. 2 (2022), 67-77.


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A unified approach to embeddings of a line in 3-space

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Abstract

While the general question of whether every closed embedding of an affine line in affine $3$-space can be rectified remains open, there have been several partial results proved by several different means. We provide a new approach, namely constructing (strongly) residual coordinates, that allows us to give new proofs of all known partial results, and in particular generalize the results of Bhatwadekar-Roy and Kuroda on embeddings of the form $(t^n,t^m,t^l+t)$.

Recommended citation:

D. Lewis, A unified approach to embeddings of a line in 3-space. Communications in Algebra 52(1), 189–200 (2024). https://doi.org/10.1080/00927872.2023.2239344


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Alternative Grading

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Two page handout summarizing commonalities and distinctions among the most popular varieties of alternative grading. Licenced under CC-BY-SA-4.0

Recommended citation:

D. Lewis, Alternative Grading.

Team learning in a technology-driven era

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Abstract

Today’s students are team-oriented, confident, and dependent on technology. These attributes are coupled with a desire for immediate feedback to promote improvement. However, they are dampened by a lack of socialization, collaboration, critical thinking, problem-solving, and communication skills (Shatto and Erwin, Creative Nurs 23:24-28, 2017). Educators must adapt to address these needs and promote attainment of these skills for both collaboration and competitiveness in the workplace. To accomplish this goal, an evolution from traditional learning to team learning using technology is imperative. An overview of active learning strategies is discussed with a focus on team-based learning (TBL), including the additional benefits of TBL and the use of complimentary technology.

Recommended citation:

J. K. Takemoto, D. Lewis, C. W. Parrish, L. Coyne, & C. M. Burns. Team learning in a technology-driven era. In L. O. Campbell, R. Hartshorne, and R. F. DeMara, editors, Perspectives on Digitally-Mediated Team Learning, (2021), 33-51. Springer International Publishing, Cham.


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Team-Based Inquiry Learning

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Abstract

Team-Based Learning (TBL) is a cooperative learning strategy blending elements of flipped learning, inquiry-based learning, and problem-based learning. Although used quite frequently in other disciplines, use of this strategy in mathematics has been limited. In this article, we describe how TBL can be implemented in math courses with adherence to essential elements of TBL and introduce modifications specific to mathematics instruction. In particular, we introduce a particular style of TBL, which we term Team-Based Inquiry Learning, that satisfies the defining pillars of inquiry-based learning.

Recommended citation:

D. Lewis, S. Clontz, & J. Estis, Team-Based Inquiry Learning. PRIMUS, no. 2 (2021), 223-238.


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Fostering Cognitive Presence, Social Presence and Teaching Presence with Integrated Online Team-Based Learning

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Abstract

This paper describes the development of the Integrated Online Team-Based Learning (IO-TBL) model and details students’ perceptions of IO-TBL using the Community of Inquiry framework. IO-TBL is an online team-based learning course design that combines the flexibility of asynchronous engagement with the connectedness offered through synchronous meetings. Student comments from small group instructional feedback sessions and end-of-course teaching evaluations were grouped into clusters of similar statements about what was going well and suggestions for improvement, which were then assigned to one of the three presences of the Community of Inquiry framework. While students most commonly identified increased learning, synchronous meetings, teamwork, and the instructor as going well in the course, students found IO-TBL to impose a heavy workload and require a significant amount of time. Clusters were most often related to teaching presence, followed by social presence, and then cognitive presence.

Recommended citation:

C. Parrish, S. Guffey, D. Williams, J. Estis, D. Lewis, Fostering Cognitive Presence, Social Presence and Teaching Presence with Integrated Online Team-Based Learning. TechTrends (2021).


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Student anxiety in standards-based grading in mathematics courses

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Abstract

This paper describes a study examining how mathematics anxiety, test anxiety, and communication apprehension are related to student behavior in courses using standards-based grading. An observational study of mathematics courses with 221 participants found that test anxiety increased over the semester although many students reported lower stress or anxiety in an open-ended survey question. Mathematics anxiety and test anxiety were positively correlated with the number of voluntary reassessments students attempted, while communication apprehension was negatively correlated. These findings indicate that standards-based grading is an assessment framework that can provide alternate methods for some students to demonstrate content mastery. While this study was conducted in mathematics courses, the findings on test anxiety are likely to extend to other disciplines.

Recommended citation:

D. Lewis, Student anxiety in standards-based grading in mathematics courses. Innovative Higher Education, 45 (2020). 153-164.


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Normal subgroups generated by a single polynomial automorphism

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Abstract

We study criteria for deciding when the normal subgroup generated by a single special polynomial automorphism of $\mathbb{A}^n$ is as large as possible, namely, equal to the normal closure of the special linear group in the special automorphism group. In particular, we investigate m-triangular automorphisms, i.e., those that can be expressed as a product of affine automorphisms and m triangular automorphisms. Over a field of characteristic zero, we show that every nontrivial 4-triangular special automorphism generates the entire normal closure of the special linear group in the special tame subgroup, for any dimension n = 2. This generalizes a result of Furter and Lamy in dimension 2.

Recommended citation:

D. Lewis, Normal subgroups generated by a single polynomial automorphism. Transformation Groups, 25 (2020), 177-189.


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Improving mathematics content mastery and enhancing flexible problem solving through team-based inquiry learning

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Abstract

This article examines how student learning is affected by the use of team-based inquiry learning, a novel pedagogy in mathematics that uses team-based learning to implement inquiry-based learning. We conducted quasi-experimental and observational studies in intermediate level mathematics courses, finding that team-based inquiry learning led to increased content mastery and that students took a more flexible approach to solving problems. We also found that in the courses using this pedagogy, women (but not men) had a reduction in communication apprehension over the course of a semester. We conclude that team-based inquiry learning effectively enhances student learning and problem solving, preparing students for future academic success and fostering career readiness.

Recommended citation:

D. Lewis & Julie Estis, Improving mathematics content mastery and enhancing flexible problem solving through team-based inquiry learning. Teaching & Learning Inquiry, 7, no. 2 (2020), 165-183.


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Gender effects on re-assessment attempts in a standards-based grading implementation

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Abstract

This article describes a Standards-Based Grading (SBG) implementation used in a variety of lower-level mathematics courses, notably including the use of oral re-assessments. A retrospective study of this implementation was conducted that showed that female students take advantage of optional oral re-assessments at greater rates than male students (p < 0.0001). This paper also discusses the potential ways the assessment structure of SBG can foster a more equitable learning environment, such as through reducing anxiety and stereotype threat in students.

Recommended citation:

D. Lewis, Gender effects on re-assessment attempts in a standards-based grading implementation. PRIMUS, 30, no. 5. (2020), 539-551.


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Comparing demographics of signatories to public letters on diversity in the mathematical sciences

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Abstract

In its December 2019 edition, the Notices of the American Mathematical Society published an essay critical of the use of diversity statements in academic hiring. The publication of this essay prompted many responses, including three public letters circulated within the mathematical sciences community. Each letter was signed by hundreds of people and was published online, also by the American Mathematical Society. We report on a study of the signatories’ demographics, which we infer using a crowdsourcing approach. Letter A highlights diversity and social justice. The pool of signatories contains relatively more individuals inferred to be women and/or members of underrepresented ethnic groups. Moreover, this pool is diverse with respect to the levels of professional security and types of academic institutions represented. Letter B does not comment on diversity, but rather, asks for discussion and debate. This letter was signed by a strong majority of individuals inferred to be white men in professionally secure positions at highly research intensive universities. Letter C speaks out specifically against diversity statements, calling them “a mistake,” and claiming that their usage during early stages of faculty hiring “diminishes mathematical achievement.” Individuals who signed both Letters B and C, that is, signatories who both privilege debate and oppose diversity statements, are overwhelmingly inferred to be tenured white men at highly research intensive universities. Our empirical results are consistent with theories of power drawn from the social sciences.

Recommended citation:

C. Topaz, J. Cart, C. Diaz Eaton, A. Hanson Shrout, J. Higdon, K. Ince, B. Katz, D. Lewis, J. Libertini, & C. Smith. Comparing demographics of signatories to public letters on diversity in the mathematical sciences. PloS one, 15, no. 4 (2020), e0232075.


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Applying a standards-based grading framework across lower level mathematics courses

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Abstract

This article describes a framework of Standards-Based Grading suitable for use in a variety of mathematics courses. We detail our experience adapting this framework to various courses, ranging from Pre-calculus to Differential Equations, at four different institutions, including both small liberal arts colleges and large universities, and in class sizes ranging from small to large. Of particular note, we describe our structure of reassessments, including the use of oral reassessments. We also give examples of how this framework provides data allowing instructors to make evidence-based changes to their teaching practice.

Recommended citation:

J. Elsinger & D. Lewis, Applying a standards-based grading framework across lower level mathematics courses. PRIMUS, 30, no. 8 (2020), 885-907.


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Co-tame polynomial automorphisms

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Abstract

A polynomial automorphism of $\mathbb{A}^n$ over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms of $\mathbb{A}^n$, including nonaffine $3$-triangular automorphisms, are co-tame. Of particular interest, if $n=3$, we show that the statement “Every m-triangular automorphism is either affine or co-tame” is true if and only if $m=3$; this improves upon positive results of Bodnarchuk (for $m=2$, in any dimension $n$) and negative results of the authors (for $m=6$, $n=3$). The main technical tool we introduce is a class of maps we term translation degenerate automorphisms; we show that all of these are either affine or co-tame, a result that may be of independent interest in the further study of co-tame automorphisms.

Recommended citation:

E. Edo & D. Lewis. Co-tame polynomial automorphisms, International Journal of Algebra and Computation, 29(5), (2019), 803-825.


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An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms

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Abstract

We study the interaction between two structures on the group of polynomial automorphisms of the affine plane: its structure as an amalgamated free product and as an infinite-dimensional algebraic variety. We introduce a new conjecture, and show how it implies the Polydegree Conjecture. As the new conjecture is an ideal membership question, this shows that the Polydegree Conjecture is algorithmically decidable. We further describe how this approach provides a unified and shorter method of recovering existing results of Edo and Furter.

Recommended citation:

D. Lewis, K. Perry, & A. Straub. An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms, Journal of Pure and Applied Algebra, 223, no. 12 (2019), 5346-5359.


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The affine automorphism group of A^3 is not a maximal subgroup of the tame automorphism group

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Abstract

We construct explicitly a family of proper subgroups of the tame automorphism group of affine three-space (in any characteristic) that are generated by the affine subgroup and a nonaffine tame automorphism. One important corollary is the titular result that settles negatively the open question (in characteristic zero) of whether the affine subgroup is a maximal subgroup of the tame automorphism group. We also prove that all groups of this family have the structure of an amalgamated free product of the affine group and a finite group over their intersection.

Recommended citation:

E. Edo & D. Lewis, The affine automorphism group of $\mathbb{A}^3$ is not a maximal subgroup of the tame automorphism group, Michigan Mathematics Journal, 64(3) (2015), 555-568.


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Some families of polynomial automorphisms III

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Abstract

We prove that the closure (for the Zariski topology) of the set of polynomial automorphisms of the complex affine plane whose polydegree is $(cd-1,b,a)$ contains all automorphisms of polydegree $(cd+a)$ where $a,b \geq 2$ and $c\geq 1$ are integers and $d=ab-1$. When $b=2$, this result gives a family of counterexamples to a conjecture of Furter.

Recommended citation:

E. Edo & D. Lewis, Some families of polynomial automorphisms III, Journal of Pure and Applied Algebra, 219, no. 4 (2015), 864-874.


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Strongly residual coordinates over A[x]

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Abstract

For a commutative ring $A$, a polynomial $f \in A[x]^{[n]}$ is called a strongly residual coordinate if $f$ becomes a coordinate (over $A$) upon going modulo $x$, and $f$ becomes a coordinate (over $A[x, x^{-1}]$) upon inverting $x$. We study the question of when a strongly residual coordinate in $A[x]^{[n]}$ is a coordinate, a question closely related to the Dolgachev-Weisfeiler conjecture. It is known that all strongly residual coordinates are coordinates for $n = 2$ over an integral domain of characteristic zero. We show that a large class of strongly residual coordinates that are generated by elementaries over $A[x, x^{-1}]$ are in fact coordinates for arbitrary n, with a stronger result in the n = 3 case. As an application, we show that all Vénéreau-type polynomials are 1-stable coordinates.

Recommended citation:

D. Lewis. Strongly Residual Coordinates over $A[x]$, Automorphisms in birational and affine geometry, Springer Proceedings in Mathematics and Statistics, 79 (2014), 407340.


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Vénéreau-type polynomials as potential counterexamples

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Abstract

We study some properties of the Vénéreau polynomials $b_m=y+x^m(xz+y(yu+z^2)) \in \mathbb{C}[x,y,z,u]$, a sequence of proposed counterexamples to the Abhyankar-Sathaye embedding conjecture and the Dolgachev-Weisfeiler conjecture. It is well known that these are hyperplanes and residual coordinates, and for $m \geq 3$, they are $\mathbb{C}[x]$-coordinates. For $m=1,2$, it is only known that they are 1-stable $\mathbb{C}[x]$-coordinates. We show that $b_2$ is in fact a $\mathbb{C}[x]$-coordinate. We introduce the notion of Vénéreau-type polynomials, and show that these are all hyperplanes and residual coordinates. We show that some of these Vénéreau-type polynomials are in fact $\mathbb{C}[x]$-coordinates; the rest remain potential counterexamples to the aforementioned conjectures. For those that we show to be coordinates, we also show that any automorphism with one of them as a component is stably tame. The remainder are stably tame, 1-stable $\mathbb{C}[x]$-coordinates.

Recommended citation:

D. Lewis, Vénéreau-type polynomials as potential counterexamples.Journal of Pure and Applied Algebra, 217, no. 5 (2013), 956-957.


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Further Geometric Restrictions on Jordan Structure in Matrix Factorization

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Abstract

It is known that a nonsingular, nonscalar, n-by-n complex matrix A may be factored as A = BC, in which the spectra of B and C are arbitrary, subject to det(A) = det(B)det(C). It has been shown that when two matrices have eigenvalues of high geometric multiplicity, this restricts the possible Jordan structure of the third. We demonstrate a previously unknown restriction on the Jordan structures of B and C. Furthermore, we show that this generalized geometric multiplicity restriction implies the already known geometric multiplicity restriction, show that the new more restrictive condition is not sufficient in general but is sufficient in a situation that we identify.

Recommended citation:

C. Johnson, D. Lewis, & Y. Zhang. Further Geometric Restrictions on Jordan Structure in Matrix Factorization. Asian-European Journal of Mathematics, 5, no. 3 (2012) 1250036.


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Arbitrariness of Jordan structure in factorization: the geometric multiplicity restriction and the 3×3 case

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Abstract

It is known that a nonsingular, nonscalar, n-by-n complex matrix A may be factored as A = BC, in which the spectra of B and C are arbitrary, subject to det(A) = det(B)det(C). It has been shown that when two matrices have eigenvalues of high geometric multiplicity, this restricts the possible Jordan structure of the third. We demonstrate a previously unknown restriction on the Jordan structures of B and C. Furthermore, we show that this generalized geometric multiplicity restriction implies the already known geometric multiplicity restriction, show that the new more restrictive condition is not sufficient in general but is sufficient in a situation that we identify.

Recommended citation:

C. Johnson, D. Lewis, & Y. Zhang. Arbitrariness of Jordan structure in factorization: the geometric multiplicity restriction and the $3\times 3$ case. Asian-European Journal of Mathematics, 5, no. 2 (2012), 1250018.


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