# Algebra

## Conceptions and Consequences of Mathematical Argumentation, Justification, and Proof

This book aims to advance ongoing debates in the field of mathematics and mathematics education regarding conceptions of argumentation, justification, and proof and the consequences for research and practice when applying particular conceptions of each construct. Through analyses of classroom practice across grade levels using different lenses - particular conceptions of argumentation, justification, and proof - researchers consider the implications of how each conception shapes empirical outcomes.

Author/Presenter:
Kristen N. Bieda, AnnaMarie Conner, Karl W. Kosko, Megan Staples
AnnaMarie Conner
Karl W. Kosko
Megan Staples
Year:
2020
Short Description:

This book aims to advance ongoing debates in the field of mathematics and mathematics education regarding conceptions of argumentation, justification, and proof and the consequences for research and practice when applying particular conceptions of each construct. Through analyses of classroom practice across grade levels using different lenses - particular conceptions of argumentation, justification, and proof - researchers consider the implications of how each conception shapes empirical outcomes. In each section, organized by grade band, authors adopt particular conceptions of argumentation, justification, and proof, and they analyse one data set from each perspective. In addition, each section includes a synthesis chapter from an expert in the field to bring to the fore potential implications, as well as new questions, raised by the analyses. Finally, a culminating section considers the use of each conception across grade bands and data sets.

## Domain appropriateness and skepticism in viable argumentation

Year:
2020
Short Description:

Several recent studies have focused on helping students understand the limitations of empirical arguments (e.g., Stylianides, G. J. & Stylianides, A. J., 2009, Brown, 2014). One view is that students use empirical argumentation because they hold empirical proof schemes—they are convinced a general claim is true by checking a few cases (Harel & Sowder, 1998). Some researchers have sought to unseat students’ empirical proof schemes by developing students’ skepticism, their uncertainty about the truth of a general claim in the face of confirming (but not exhaustive) evidence (e.g., Brown, 2014; Stylianides, G. J. & Stylianides, A. J., 2009). With sufficient skepticism, students would seek more secure, non-empirical arguments to convince themselves that a general claim is true. We take a different perspective, seeking to develop students’ awareness of domain appropriateness (DA), whether the argument type is appropriate to the domain of the claim. In particular, DA entails understanding that an empirical check of a proper subset of cases in a claim’s domain does not (i) guarantee the claim is true and does not (ii) provide an argument that is acceptable in the mathematical or classroom community, although checking all cases does both (i) and (ii). DA is distinct from skepticism; it is not concerned with students’ confidence about the truth of a general claim. We studied how ten 8th graders developed DA through classroom experiences that were part of a broader project focused on developing viable argumentation.

## Eliminating counterexamples: A case study intervention for improving adolescents’ ability to critique direct arguments

Students’ difficulties with argumentation, proving, and the role of counterexamples in proving are well documented. Students in this study experienced an intervention for improving their argumentation and proving practices. The intervention included the eliminating counterexamples (ECE) framework as a means of constructing and critiquing viable arguments for a general claim. This framework involves constructing descriptions of all possible counterexamples to a conditional claim and determining whether or not a direct argument eliminates the possibility of counterexamples.

Author/Presenter:
Carolyn Maher
Year:
2020
Short Description:

Students’ difficulties with argumentation, proving, and the role of counterexamples in proving are well documented. Students in this study experienced an intervention for improving their argumentation and proving practices. The intervention included the eliminating counterexamples (ECE) framework as a means of constructing and critiquing viable arguments for a general claim. This framework involves constructing descriptions of all possible counterexamples to a conditional claim and determining whether or not a direct argument eliminates the possibility of counterexamples. This case study investigates U.S. eighth-grade (age 13) mathematics students’ conceptions about the validity of a direct argument after the students received instruction on the ECE framework. We describe student activities in response to the intervention, and we identify students’ conceptions that are inconsistent with canonical notions of mathematical proving and appear to be barriers to using the ECE framework.

## Eliminating counterexamples: An intervention for improving adolescents’ contrapositive reasoning

Students’ difficulties with contrapositive reasoning are well documented. Lack of intuition about contrapositive reasoning and lack of a meta-argument for the logical equivalence between a conditional claim and its contrapositive may contribute to students’ struggles. This case study investigated the effectiveness of the eliminating counterexamples intervention in improving students’ ability to construct, critique, and validate contrapositive arguments in a U.S. eighth-grade mathematics classroom.

Author/Presenter:
David Yopp
Year:
2020
Short Description:

Students’ difficulties with contrapositive reasoning are well documented. Lack of intuition about contrapositive reasoning and lack of a meta-argument for the logical equivalence between a conditional claim and its contrapositive may contribute to students’ struggles. This case study investigated the effectiveness of the eliminating counterexamples intervention in improving students’ ability to construct, critique, and validate contrapositive arguments in a U.S. eighth-grade mathematics classroom. The intervention involved constructing descriptions of all possible counterexamples to a conditional claim and its contrapositive, comparing the two descriptions, noting that the descriptions are the same barring the order of phrases, and finding a counterexample to show the claim is false or viably arguing that no counterexample exists.

## NCTM Presentation Line of "Good" Fit in Grade 8 Classrooms

Year:
2018
Short Description:

This presntation addreses 4 research cquestions

What extant criteria do Grade 8 students use to choose the better line
of fit between two lines “fit” to a set of data, when both lines express
the trend of the data?

Is a residual criterion accessible and useful to Grade 8 students when

How does introducing a residual criterion impact student
understanding of line of fit and their understanding mathematical
modeling process?

What stages of learning do students express as they engage in our
lesson?
Resource(s):

## Advancing Reasoning Covariationally (ARC) Curriculum

The Advancing Reasoning Covariationally (ARC) curriculum is a curriculum for working pre-service and in-service teachers. ARC targets and develops quantitative and covariational reasoning as connecting threads to major secondary mathematics ideas, particularly in the area of algebra, precalculus, and calculus.
Author/Presenter:
The ARC Team
Year:
2018
Short Description:
The Advancing Reasoning Covariationally (ARC) curriculum is a curriculum for working pre-service and in-service teachers. ARC targets and develops quantitative and covariational reasoning as connecting threads to major secondary mathematics ideas, particularly in the area of algebra, precalculus, and calculus.

## Math Pathways & Pitfalls Algebra Readiness: Lessons and Teaching Guide, Grades 7–8

The Math Pathways & Pitfalls Algebra Readiness mathematics intervention is intended to help students tackle stubborn pitfalls head-on and transform those pitfalls into pathways for learning key standards. It offers an entire year’s worth of lessons that focus on the critical areas of algebra readi­ness, using the same research-backed principles that informed the original series.

Author/Presenter:
The Math Pathways & Pitfalls Team
Year:
2019
Short Description:

The Math Pathways & Pitfalls K-8 curriculum was designed with built-in support for teachers, alignment to the Common Core State Standards and Mathematical Practices. The curriculum can be flexibly used as an intervention, as part of the core curriculum, or in after-school or small group settings.

## Backward Transfer Influences from Quadratic Functions Instruction on Students’ Prior Ways of Covariational Reasoning about Linear Functions

The study reported in this article examined the ways in which new mathematics learning influences students’ prior ways of reasoning. We conceptualize this kind of influence as a form of transfer of learning called backward transfer. The focus of our study was on students’ covariational reasoning about linear functions before and after they participated in a multi-lesson instructional unit on quadratic functions. The subjects were 57 students from two authentic algebra classrooms at two local high schools.

Author/Presenter:
Charles Hohensee
Sara Gartland
Laura Willoughby
Matthew Melville
Year:
2021
Short Description:

The study reported in this article examined the ways in which new mathematics learning influences students’ prior ways of reasoning. Authors conceptualize this kind of influence as a form of transfer of learning called backward transfer. The focus of the study was on students’ covariational reasoning about linear functions before and after they participated in a multi-lesson instructional unit on quadratic functions.

## Cognitive Instructional Principles in Elementary Mathematics Classrooms: A Case of Teaching Inverse Relations

Instructional principles gleaned from cognitive science play a critical role in improving classroom teaching. This study examines how three cognitive instructional principles including worked examples, representations, and deep questions are used in eight experienced elementary teachers’ early algebra lessons in the U.S. Based on the analysis of 32 videotaped lessons of inverse relations, we found that most teachers spent sufficient class time on worked examples; however, some lessons included repetitive examples that also included irrelevant practice problems.

Author/Presenter:
Meixia Ding
Ryan Hassler
Xiaobao Li
Year:
2020
Short Description:

This study examines how three cognitive instructional principles including worked examples, representations, and deep questions are used in eight experienced elementary teachers’ early algebra lessons in the U.S.

## Understanding of the Properties of Operations: A Cross-Cultural Analysis

This study examines how sampled Chinese and U.S. third and fourth grade students (NChina=167,NUS=97) understand the commutative, associative, and distributive properties. These students took both pre- and post-tests conducted at the beginning and end of a school year. Comparisons between students’ pre- and post-tests within and across countries indicate different learning patterns. Overall, Chinese students demonstrate a much better understanding than their U.S. counterparts.
Author/Presenter:
Meixia Ding
Xiaobao Li
Ryan Hassler
Eli Barnett