Algebra

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
Lead Organization(s): 
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
Lead Organization(s): 
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
Lead Organization(s): 
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
Lead Organization(s): 
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
Lead Organization(s): 
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
Lead Organization(s): 
Year: 
2017
Short Description: 

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. Implications for teaching are explored in this article.

Tiering Instruction for Middle School Students

Differentiating instruction (DI) is a pedagogical approach to managing classroom diversity in which teachers proactively adapt curricula, teaching methods, and products of learning to address individual students' needs in an effort to maximize learning for all (Tomlinson, 2005). DI is rooted in formative assessment, positions teachers and students together as learners, and involves providing choices and different pathways for students. Although teachers can differentiate for many characteristics of students, we differentiate for students' diverse ways of thinking.

Author/Presenter: 
Amy J. Hackenberg
Robin Jones
Rebecca Borowski
Lead Organization(s): 
Year: 
2020
Short Description: 

In this article, we describe an example of differentiating instruction (DI) involving middle school students from a five-year project funded by the National Science Foundation.

Backward Transfer Effects when Learning about Quadratic Functions

Presentation slides from the 42nd Conference of the International Group for the Psychology of Mathematics Education.

Hohensee, C., Willoughby, L., & Gartland, S. (2018, July). Backward transfer effects when learning about quadratic functions. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, p. 65). Umeå, Sweden: PME.

Author/Presenter: 
Charles Hohensee
Laura Willoughby
Sara Gartland
Lead Organization(s): 
Year: 
2018
Short Description: 

Presentation slides from the 42nd Conference of the International Group for the Psychology of Mathematics Education.

Backward Transfer Effects on Action and Process Views of Functions

Presentation slides from the 41st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

Hohensee, C., Gartland, S., & Willoughby, L. (2019, November). Backward transfer effects on action and process views of functions. In S. Otten, A. G. Candela, Z. de Araujo, C. Haines, & C. Munter (Eds.), Proceedings of the 41st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. St Louis, MO: University of Missouri.

Author/Presenter: 
Charles Hohensee
Sara Gartland
Laura Willoughby
Lead Organization(s): 
Year: 
2019
Short Description: 

Presentation slides from the 41st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

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