The Relationship Between Proof and Certainty in Mathematical Practice

Many mathematics educators believe a goal of instruction is for students to obtain conviction and certainty in mathematical statements using the same types of evidence that mathematicians do. However, few empirical studies have examined how mathematicians use proofs to obtain conviction and certainty. We report on a study in which 16 advanced mathematics doctoral students were given a task-based interview in which they were presented with various sources of evidence in support of a specific mathematical claim and were asked how convinced they were that the claim was true after reviewing this evidence. In particular, we explore why our participants retained doubts about our claim after reading its proof and how they used empirical evidence to reduce those doubts.

Responsiveness to Students’ Mathematical Thinking in Middle-Grades Classrooms

Responsiveness to students’ mathematical thinking is a characteristic of classroom discourse that reflects the extent to which students’ mathematical ideas are present, attended to, and taken up as the basis for instruction. Using the Mathematically Responsive Interaction (MRI) Framework and data from 11 middle-grades classrooms, we illustrate varied enactments of responsiveness and describe fluctuations in and relationships among different components of responsiveness. We found positive associations between different components of responsiveness, but they were not entirely predictive of one another. Individual classrooms appeared more or less responsive depending on which component was foregrounded. Our findings offer a more comprehensive characterization of responsiveness that documents the intertwined nature of teacher moves and student contributions during all whole-class instruction.

Fixing the Crooked Heart: How Aesthetic Practices Support Sense Making in Mathematical Play

We build on mathematicians’ descriptions of their work and conceptualize mathematics as an aesthetic endeavor. Invoking the anthropological meaning of practice, we claim that mathematical aesthetic practices shape meanings of and appreciation (or distaste) for particular manifestations of mathematics. To see learners’ spontaneous mathematical aesthetic practices, we situate our study in an informal context featuring design-centered play with mathematical objects. Drawing from video data that support inferences about children’s perspectives, we use interaction analysis to examine one child’s mathematical aesthetic practices, highlighting the emergence of aesthetic problems whose resolution required engagement in mathematics sense making. As mathematics educators seek to broaden access, our empirical findings challenge commonsense understandings about what and where mathematics is, opening possibilities for designs for learning.

Undergraduate Students’ Combinatorial Proof of Binomial Identities

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.

Using Moment-by-Moment Reading Protocols to Understand Students’ Processes of Reading Mathematical Proof

This article documents differences between novice and experienced undergraduate students’ processes of reading mathematical proofs as revealed by moment-by-moment, think-aloud protocols. We found three key reading behaviors that describe how novices’ reading differed from that of their experienced peers: alternative task models, accrual of premises, and warranting. Alternative task models refer to the types of goals that students set up for their reading of the text, which may differ from identifying and justifying inferences. Accrual of premises refers to the way novice readers did not distinguish propositions in the theorem statement as assumptions or conclusions and thus did not use them differently for interpreting the proof. Finally, we observed variation in the type and quality of warrants, which we categorized as illustrate with examples, construct a miniproof, or state the warrant in general form.

The Cultural Production of Racial Narratives About Asian Americans in Mathematics

Although considerable literature illustrates how students’ experiences and identities are racialized in mathematics education, little attention has been given to Asian American students. Employing ethnographic methods, this study followed 10 immigrant Chinese-heritage families to explore how the racial narrative of the model minority myth was locally produced in mathematics education. We draw on constructs of racial narratives and cultural production to identify the local production of the narrative Asians are smart and good at math during K–12 schooling. Specifically, the Asian American students (re)produced racial narratives related to three cultural resources: (a) Their immigrant parents’ narratives about the U.S. elementary school mathematics curriculum; (b) the school mathematics student tracking system; and (c) students’ locally generated racial narratives about what being Asian means.

Let’s Spend Time Together: Text Messaging Parents to Help Math-Anxious Students

Even when parents have the time required to support their children’s education, they can increase their children’s anxiety about school when they try to help, especially if they are not confident in their own abilities. This study measures the effects of having parents complete nonacademic schoolwork with their teenage children. Half of the 422 participating parents were randomly assigned to receive weekly assignments for nonacademic activities to complete with their children, whereas the other half received information about upcoming mathematics tests. Mathematics-anxious students benefited from working on the nonacademic assignments, performing significantly better on their mathematics tests and decreasing their mathematics anxiety after treatment. These findings highlight the importance of involving parents in ways that feel nonthreatening to their children.