Alison Mirin

• Doctor of Philosophy
• Postdoctoral Research Associate at University of Arizona

While there has been research on students' understanding of the meaning of the equals sign, there has yet to be a thorough discussion in math education on a strong meaning of the equals sign. This paper discusses the philosophical and logical literature on the identity relation and reviews the math education research community's attempt to characte...

This article explores how, within formal mathematics, there are two conflicting definitions of "function".

In studies of children's reasoning about equations, a major finding is that many children understand equality to be asymmetric. In this study, we investigate how experts interpret equations in order to determine whether and why they interpret equations asymmetrically. We do so by using a breaching experiment in which we present nine mathematicians...

This study investigates when and how university students in first-semester introductory calculus interpret multiple representations of the same function. Specifically, it focuses on three tasks. The first task has students give their definitions of ‘function sameness’, the results of which suggests that many students understand a function as being...

In mathematics education, knowledge is often divided into conceptual knowledge and procedural knowledge. These two knowledge types are sometimes seen as competing for teachers' attention and curricular focus. Similarly, there exists a perceived dichotomy between proof-based mathematics and procedure-based mathematics. In this context, learning proc...

We designed the Relative Risk Tool (RRT) to help people assess the relative risks associated with COVID-19 vaccination and infection. In May 2022 (N = 400) and November 2022 (N = 615), U.S. residents participated in a survey that included questions about the risks of vaccination and infection. In both cohorts, we found an association between relati...

The relationship between mathematical induction (MI) and recursion compels us to ask how we could leverage recursive functions to bolster students' understanding of MI. We describe task-based interviews that utilized concurrent interactions with MI tasks and recursive functions that mirrored those induction tasks via a character-based user-interfac...

In order to learn more about student understanding of the structure of proofs, we generated a novel genre of tasks called "Proof Without Claim" (PWC). Our work can be viewed as an extension of Selden and Selden's (1995) construct of "proof framework"; while Selden and Selden discuss how the structure of a proof can be discerned by the claim it prov...

We designed the Relative Risk Tool (RRT) to give people access to the same quantitative peer-reviewed information about the risks and benefits of vaccination that professionals use to make determinations about approving vaccines. Based on our initial qualitative research and the known associations between concern about vaccination risk and vaccine...

Academic disability accommodations are essential for providing equitable access to education. This study examines the barriers that college students face in obtaining their disability accommodations in mathematics classrooms. It shows that even when a student has school-approved disability accommodations, implementing them is often not a straightfo...

Isomorphism and equality are important aspects of mathematics and are both types of sameness. However, these are not identical concepts; objects can be isomorphic without being equal. We discuss the difference between equality and isomorphism as types of sameness and the way that popular introductory abstract algebra textbooks distinguish between a...

Reasoning with inequalities and their solutions is important in mathematics. We provide a two-layered APOS genetic decomposition of the meaning of number line graphs as representing solution sets.

Much of the education research on implicit differentiation and related rates treats the topic of differentiating equations as an unproblematic application of the chain rule. This paper instead problematizes the legitimacy of this procedure. It develops a conceptual analysis aimed at exploring how a student might come to understand when and why one...

In the mathematical community, two notions of "function" are used: the set-theoretic definition as a univalent set of ordered pairs, and the Bourbaki triple. These definitions entail different interpretations and answers to mathematical questions that even a secondary student might be prompted to answer. However, mathematicians and mathematics educ...

By analyzing the responses of 100 introductory calculus students to two questions, this study addresses how students understand the fundamental theorem of calculus as it relates to function identity. One question involves students’ understandings of the fundamental theorem of calculus, and the other involves their concept definitions of function sa...

This paper addresses how students understand number line graphs. Utilizing a Think Aloud interview followed by a reflection-eliciting interview, we investigate how two successful College Algebra students understand what it means to graph a statement with one free variable on a number line. These particular students show a mathematically non-normati...

This paper examines the various characterizations of the operational (non-normative) meanings of the equals sign discussed in math education literature. It provides both an exposition and a critique of the various classifications of students' misunderstandings of the equal sign and of equations. This meta-analysis provides valuable starting points...

Calculus is about change and thus dictates a need for instruction involving dynamic imagery. DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus) utilizes animations to support students' dynamic imagery. This paper investigates how students use and understand animations in the DIRACC textbook in connection with associate...

This paper examines the various characterizations of the operational (non-normative) meanings of the equals sign discussed in math education literature. It provides both an exposition and a critique of the various classifications of students’ misunderstandings of the equal sign and of equations. This meta-analysis provides valuable starting points...

The study of calculus focuses on change and thus dictates a need for instruction involving dynamic imagery. DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus) utilizes animations to support students' dynamic imagery. This poster investigates how students use and understand animations in the DIRACC textbook in connectio...

Investigating student interpretations of animations in a conceptual calculus course

This paper discusses the conceptual basis for differentiating an equation, an essential aspect of implicit differentiation. We explain that implicit differentiation is more than merely the procedure of differentiating an equation and carefully provide a conceptual analysis of what is entailed in understanding the legitimacy of this procedure. This...

This study focuses on students' understanding of multiple representations of functions. It examines student responses to a task in which calculus students are asked to evaluate the derivative at a point of the cubing function when represented piecewise. Results suggest that attending to the graph of the piecewise function does not improve students'...

This study reports calculus students’ failure to differentiate the cubing function when represented piecewise as f(x)=x^3 if x/=2, f(x)=8 if x=2. The data reported here suggest that students did not fail simply due to inattention to the function definition; when reminded that 2 cubed is 8 and prompted to compare the graph of f to that of the cubing...

Exploring students' conceptions of sameness is an avenue for exploring their understandings of the objects being compared. More specifically, finding what students think it means for functions to be identical can help us figure out what students think it means for something to be a function, since identity within a category (in this case the catego...

250 calculus students were asked to evaluate f'(2) when f(x)=x^3 if x/=2 and f(x)=8 if x=2. Responses were coded, and eight students were interviewed about their answers. The data provide insight into students’ understandings of function, derivative, and graph. See "Piecewise Functions as Instructions" for follow-up study.

This work is a mathematical and historical exposition of relation algebras. I explore Chin and Tarski's Distributive and modular laws in the arithmetic of relation algebras, as well as Jónsson and Tarski's Boolean Algebras With Operators. The proofs given in the papers themselves are very brief and often incomplete, so I supply elaboration, clarifi...

In De Interpretatione 9, Aristotle famously argues against the principle of bivalence. Lesser known is an argument that Lukasiewicz makes, which is essentially the same argument. While on the surface their arguments are about bivalence, a careful examination of their arguments shows us that their line of reasoning - in particular, their corresponde...