We explore student performance on True-False assessments with statements in the conditional form “If P then Q” in order to better understand how students process conditional logic and to see whether logical misconceptions impede students’ ability to demonstrate mathematical knowledge. We administered an online assessment to a population of Calculus II students. We find that students make logical errors on True-False items of a certain logical form, and that these errors are unrelated to their calculus knowledge.

The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their f-vectors and some additional constraints.

We explore student performance on True-False assessments with statements in the conditional form “If P then Q” in order to see whether logical misconceptions impede students’ ability to demonstrate mathematical knowledge in the context of this kind of assessment. We administered an online assessment to a population of Calculus II students. Half of these students were given a standard True-False assessment, and half were given an experimental assessment with three choices. We find that students do make logical errors on True-False items of a certain logical form, and that these errors are unrelated to their calculus knowledge. However, this effect is eliminated by the experimental three-choice assessment. We believe that this provides an easy and practical alternative to True-False items in the problematic logical form for early undergraduate mathematics students.

In this paper, we will study the Möbius polynomial, an invariant of ranked posets that arises in the study of splitting algebras. We will present a formula for the Möbius polynomial of the direct product of posets in terms of the Möbius polynomials of the factors. We will then use this formula to calculate Hilbert series and graded trace generating functions associated to the splitting algebras of the Boolean algebra and the poset of factors of a natural number n.

In this work we will study the universal labeling algebra A(Gamma), a related algebra B(Gamma), and their behavior as invariants of layered graphs. We will introduce the notion of an upper vertex-like basis, which allows us to recover structural information about the graph Gamma from the algebra B(Gamma). We will use these bases to show that several classes of layered graphs are uniquely identified by their corresponding algebras B(Gamma).