Math

Digital Tool Dependency vs. Conceptual Understanding in Secondary Mathematics Presenters 

Nayeli Cordova
Our Lady of the Lake University

This qualitative study explores the intersection of digital tool proficiency and conceptual mathematical understanding in secondary education. As digital graphing tools like Desmos become integral to state-tested courses, there is a growing concern regarding student dependency on these tools at the expense of foundational reasoning. The purpose of this research is to investigate how mathematics teachers perceive this shift and its impact on student mastery. The primary goal is to answer how educators perceive the impact of digital graphing tools on conceptual understanding, the extent of tool dependency observed during high-stakes testing preparation, and what instructional strategies are used to ensure digital proficiency does not replace mathematical reasoning. Data will be collected through semi-structured interviews with 5-8 secondary mathematics educators recruited via purposeful sampling within the San Antonio area. Interviews will be conducted either in person or virtually via Microsoft Teams to ensure secure data management. To ensure minimal risk, all data will be de-identified using pseudonyms. The findings aim to provide insight into balancing instructional technology with traditional mathematical rigor, helping educators design strategies that leverage technology without compromising foundational problem-solving skills. 

Investigating Basis Elements of Certain Exterior Algebras and Constructing Fi nite Free Complexes in Relation to Carlsson’s Conjecture

Austin Brown
University of Texas at Arlington

In the 80’s, Gunnar Carlsson introduced a conjecture pertaining to the singular homology of certain finite CW complexes. In 2018, Iyengar and Walker demonstrated that a related conjecture was false using tools from homological algebra and a result on maximal rank of multiplication maps between exterior algebras over vector spaces in a paper by Conca, Herbig, and Iyengar. This presentation will showcase an investigation of maximal rank of a generalized form of multiplication map not discussed in the prior papers. In particular, such a generalized form of multiplication map between exterior algebras is induced by a certain partition of a finite dimensional vector space’s basis. Although more counterexamples can be produced in this way, this research indicates why vector spaces of dimension 2n induce certain multiplication maps that are well-behaved, and why more research is needed for the case of vector spaces of dimension dn. In particular, most partitions of bases yield unwieldy basis elements that may not necessarily lie in the image of this partition’s associated multiplication map.