Publications | Blake A. Farman

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2026

ASEE

WIP: Using Career Panels to Spark Curiosity and Career Awareness Among Mathematics Majors

Ann W. Clifton and Blake Farman

In ASEE Annual Conference & Exposition, Jun 2026

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This Work-in-Progress study investigates how career panels shape students’ perceptions of mathematics and the opportunities available to them. The 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences specifically recommends that “mathematical sciences major programs should offer their students an orientation to careers in mathematics,” citing benefits such as recruitment and retention of majors. Similarly, the BIG (Business, Industry, Government) Math Network urges departments to engage students in pathways beyond academia. Responding to these national calls, the Mathematics & Statistics program at ___ began hosting career panels in Fall 2025, supported by (grant). To examine student responses to these panels, we adapted a survey instrument previously validated in the field of information systems. Items were modified to reflect mathematics-specific constructs, including perceptions of the field, job availability, perceptions of professionals, usefulness of studies, confidence in learning, effectance motivation, and intention. In addition, we introduce a curiosity construct to capture new questions and interests stimulated by the panels. These constructs align with expectancy-value and career development frameworks, which highlight the roles of perceived utility, confidence, and curiosity in shaping student motivation and career exploration. The survey was administered to students attending the career panels, including both mathematics majors and students from other STEM disciplines. This paper reports preliminary data from the initial implementation of the panels and instrument. We examine patterns in students’ perceptions of mathematics-related careers and analyze open-ended responses to identify common themes in students’ expressed curiosities and questions. Quantitative Likert-scale responses are summarized descriptively to provide context for the qualitative findings. Looking forward, we will extend the instrument and undertake formal validation in the context of mathematics majors. Ultimately, this work will contribute a validated instrument for use across the mathematics community. In the immediate term, we offer a formative tool for departments seeking feedback on career-oriented programming.

2025

ASEE

Fostering Growth Mindsets: Implementing Standards-Based Grading in College Algebra

Blake Farman, Ann W. Clifton, William C. Long, and 1 more author

In ASEE Annual Conference & Exposition, Jun 2025

Abs DOI PDF

In mathematics courses, grading practices often focus on performance through a cumulative point system, which can mask students’ true understanding of the material. For students at EPSCoR institutions, many of whom are first-generation college students from rural communities, this traditional grading approach does not always foster the depth of learning or resilience required for long-term success. With the goal of encouraging a growth mindset, faculty members at these institutions sought a grading system that would better reflect students’ comprehension and support their academic growth. Standards-based grading (SBG) was implemented to meet these needs by shifting the focus away from partial credit accumulation and toward a more meaningful assessment of learning. SBG was adopted in College Algebra courses to encourage students to master specific learning objectives through an iterative process of reassessment. Unlike traditional weighted average grading (WAG), SBG emphasizes mastery of content, giving students multiple opportunities to demonstrate their understanding. This paper will present a statistical analysis between student outcomes in SBG and WAG college algebra courses. Supported by recent literature, SBG has been shown to foster student engagement, reduce math anxiety, and cultivate a growth mindset. This paper additionally explores the logistical aspects of implementing SBG, the process of gaining administrative and student buy-in, and provides resources for instructors interested in adopting SBG in STEM fields to improve student outcomes.
ASEE

Work in Progress: First-Year Engineering Students’ Confidence in Communicating Mathematical Content

Ann W. Clifton, Mary Fendley, Blake Farman, and 1 more author

In ASEE Annual Conference & Exposition, Jun 2025

Abs DOI PDF

This Work in Progress study explores the impact of weekly journaling assignments on engineering students’ ability to communicate mathematical concepts effectively in design projects. At _ University, first-year engineering students participate in the "Living with the Lab" course sequence, culminating in the First-Year Projects Showcase. While students excel at explaining their product’s purpose and hardware, they often struggle to articulate the underlying STEM principles, especially in mathematics. To address this gap, a targeted journaling assignment was integrated into the calculus sequence to enhance reflection on mathematical concepts and their connection to engineering applications. Using surveys, written reflections, and project presentations, this mixed-methods study evaluates the effectiveness of journaling in improving communication skills and confidence. Initial findings from the fall quarter suggest promising improvements in students’ mathematical communication skills, with ongoing data collection in winter and spring quarters. Aligned with Kern Entrepreneurial Engineering Network (KEEN) principles of Curiosity, Connections, and Collaboration, this intervention encourages students to integrate math into their hands-on work and articulate these connections. Findings aim to offer strategies to develop mathematical communication skills, supporting deeper learning and better preparation for future challenges in engineering.

2021

JNCG

Kernels for noncommutative projective schemes

Matthew Ballard and Blake Farman

J. Noncommut. Geom., Nov 2021

Abs DOI PDF

We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of Artin-Zhang. As an immediate application, we give a noncommutative projective derived Morita statement along the lines of Rickard and Orlov.

Formalization: merged into mathlib4

mathlib4

feat(CategoryTheory/Abelian/Preradical): introduce and characterize radicals

Blake Farman

Mar 2026

Following Stenström, a preradical ‘Φ‘ is called radical if it coincides with its self colon. We encode this as the existence of an isomorphism ‘Φ.colon Φ ≅ Φ‘. We then prove a basic characterization of radical preradicals in terms of the vanishing of ‘Φ.r‘ on ‘Φ.quotient‘.

GitHub
mathlib4

feat(CategoryTheory/Abelian/Preradical): add Stenström’s colon construction for preradicals

Blake Farman

Mar 2026

Given preradicals ‘Φ‘ and ‘Ψ‘ on an abelian category ‘C‘, this file defines their colon ‘Φ : Ψ‘ in the sense of Stenström. Categorically, ‘Φ : Ψ‘ is constructed objectwise as a pullback of the canonical projection ‘Φ.π X : X ⟶ Φ.quotient.obj X‘ along the inclusion ‘Functor.whiskerLeft Φ.quotient Ψ.ι‘.

GitHub
mathlib4

feat(CategoryTheory/Abelian/Preradical): introduce basic definition of preradicals

Blake Farman

Mar 2026

A preradical on an abelian category ‘C‘ is a monomorphism in the functor category ‘C ⥤ C‘ with codomain ‘𝟭 C‘, i.e. an element of ‘MonoOver (𝟭 C)‘.

GitHub
mathlib4

feat(RingTheory/IdealFilter): add ideal filters and Gabriel filters

Blake Farman

Jan 2026

Introduces ideal filters on rings and Gabriel filters in mathlib4, including the uniformity conditions (T1–T3), Gabriel composition, and a characterization of Gabriel filters (T4). Preparatory infrastructure for relating Gabriel filters to Giraud subcategories; merged into mathlib4 via PR #33021

GitHub
mathlib4

feat(RingTheory/IdealFilter): topologies associated to ideal filters

Blake Farman

Feb 2026

Introduces additive and ring topologies on rings arising from ideal filters in mathlib4, including constructions of additive and ring filter bases, characterizations of uniform ideal filters via the existence of a ring filter basis, neighborhood descriptions of the induced topologies, and a proof that the resulting ring topology is linear; provides the topological counterpart to the algebraic theory of ideal filters; merged into mathlib4 via PR #33852.

GitHub
mathlib4

feat(RingTheory/Ideal): Generalize Submodule.colon to sets

Blake Farman

Jan 2026

Generalize Submodule.colon to accept S : Set M and update the surrounding API; merged into mathlib4 via PR #33390.

GitHub
mathlib4

feat(Algebra/Homology/ShortComplex): pullbacks and pushouts of short exact sequences

Blake Farman

Sep 2026

Given a short complex S and a morphism f : Y ⟶ S.X₃, constructs the short complex S.pull f, namely S.X₁ ⟶ pullback f S.g ⟶ Y, and shows it is short exact whenever S is, in an abelian category. Dually, for f : S.X₁ ⟶ Y constructs S.push f, namely Y ⟶ pushout f S.f ⟶ S.X₃, with the same conclusion. Adds ShortComplex.pull, ShortComplex.push, ShortComplex.ShortExact.pull and ShortComplex.ShortExact.push. Merged into mathlib4 via PR #44229.

GitHub
mathlib4

feat(CategoryTheory/ObjectProperty): closure properties of orthogonals and S. E. Dickson’s theorem

Blake Farman

Sep 2026

Studies how the closure properties of a property of objects P interact with the orthogonal complements P.leftOrthogonal and P.rightOrthogonal and with passage to the opposite category. In a well-powered abelian category with coproducts, if P is closed under quotients, extensions and coproducts then P.rightOrthogonal.leftOrthogonal = P, the hard direction of S. E. Dickson’s characterization of torsion classes. Merged into mathlib4 via PR #44226.

GitHub
mathlib4

feat(CategoryTheory/Abelian/TorsionTheory): Introduce torsion theory for abelian categories

Blake Farman

Mar 2026

Introduces torsion theories on an abelian category: a pair of object properties that are each other’s left and right orthogonals. Adds Abelian.TorsionTheory, the typeclasses IsTorsionClass and IsTorsionFreeClass, the torsion theories generated and cogenerated by a property, transport along op and unop, and Dickson’s theorem isTorsionClass_iff: in a well-powered abelian category with coproducts, a class is a torsion class if and only if it is closed under quotients, extensions and coproducts. Merged into mathlib4 via PR #36744.

GitHub