Publications

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

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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.

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

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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 $(t^n,t^m,t^l+t)$.

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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

Team learning in a technology-driven era

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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.

Team-Based Inquiry Learning

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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.

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

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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.

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Student anxiety in standards-based grading in mathematics courses

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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.

Normal subgroups generated by a single polynomial automorphism

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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.

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

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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.

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

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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.

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

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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.

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

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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.

Co-tame polynomial automorphisms

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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.

An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms

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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.

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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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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.

Some families of polynomial automorphisms III

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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.

Strongly residual coordinates over A[x]

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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.

Vénéreau-type polynomials as potential counterexamples

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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.

Further Geometric Restrictions on Jordan Structure in Matrix Factorization

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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.

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

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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.