# Algebra

## 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

Year
2021
Short Description

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.

## Teaching Early Algebra through Example-based Problem Solving: Insights from Chinese and U.S. Elementary Classrooms

Drawing on rich classroom observations of educators teaching in China and the U.S., this book details an innovative and effective approach to teaching algebra at the elementary level, namely, "teaching through example-based problem solving" (TEPS).

Author/Presenter

Meixia Ding

Year
2021
Short Description

Drawing on rich classroom observations of educators teaching in China and the U.S., this book details an innovative and effective approach to teaching algebra at the elementary level, namely, "teaching through example-based problem solving" (TEPS).

## The Role of Balance Scales in Supporting Productive Thinking about Equations Among Diverse Learners

This research focuses on ways in which balance scales mediate students’ relational understandings of the equal sign. Participants included 21 Kindergarten–Grade 2 students who took part in an early algebra classroom intervention focused in part on developing a relational understanding of the equal sign through the use of balance scales. Students participated in pre-, mid- and post-intervention interviews in which they were asked to evaluate true-false equations and solve open number sentences. Students often worked with balance scales while solving these tasks.

Author/Presenter

Ana Stephens

Yewon Sung

Susanne Strachota

Ranza Veltri Torres

Karisma Morton

Angela Murphy Gardiner

Maria Blanton

Eric Knuth

Rena Stroud

Year
2020
Short Description

This research focuses on ways in which balance scales mediate students’ relational understandings of the equal sign.

## Does Early Algebra Matter? The Effectiveness of an Early Algebra Intervention in Grades 3 to 5

A cluster randomized trial design was used to examine the effectiveness of a Grades 3 to 5 early algebra intervention with a diverse student population. Forty-six schools in three school districts participated. Students in treatment schools were taught the intervention by classroom teachers during regular mathematics instruction. Students in control schools received only regular mathematics instruction.

Author/Presenter

Maria Blanton

Rena Stroud

Ana Stephens

Angela Murphy Gardiner

Despina A. Stylianou

Eric Knuth

Isil Isler-Baykal

Susanne Strachota

Year
2019
Short Description

A cluster randomized trial design was used to examine the effectiveness of a Grades 3 to 5 early algebra intervention with a diverse student population.

## Does Early Algebra Matter? The Effectiveness of an Early Algebra Intervention in Grades 3 to 5

A cluster randomized trial design was used to examine the effectiveness of a Grades 3 to 5 early algebra intervention with a diverse student population. Forty-six schools in three school districts participated. Students in treatment schools were taught the intervention by classroom teachers during regular mathematics instruction. Students in control schools received only regular mathematics instruction.

Author/Presenter

Maria Blanton

Rena Stroud

Ana Stephens

Angela Murphy Gardiner

Despina A. Stylianou

Eric Knuth

Isil Isler-Baykal

Susanne Strachota

Year
2019
Short Description

A cluster randomized trial design was used to examine the effectiveness of a Grades 3 to 5 early algebra intervention with a diverse student population.

## “Approximate” Multiplicative Relationships between Quantitative Unknowns

Three 18-session design experiments were conducted, each with 6–9 7th and 8th grade students, to investigate relationships between students’ rational number knowledge and algebraic reasoning. Students were to represent in drawings and equations two multiplicatively related unknown heights (e.g., one was 5 times another). Twelve of the 22 participating students operated with the second multiplicative concept, which meant they viewed known quantities as units of units, or two-levels-of-units structures, but not as three-levels-of-units structures.

Author/Presenter

Amy J. Hackenberg

Robin Jones

Ayfer Eker

Mark Creager