# Algebra

## 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.

## “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
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
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.

Resource(s):

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
Year:
2018
Short Description:

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

Resource(s):

## 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
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.

Resource(s):

## LEAP Early Algebra Curriculum

The LEAP program is the first early algebra curriculum for students in grades 3-5. The program includes 18-20 one-hour lessons at each grade level and teacher support and assessment. Professional development  is also available.

Author/Presenter:
Maria Blanton
Year:
2020
Short Description:

The LEAP program is the first early algebra curriculum for students in grades 3-5. The program includes 18-20 one-hour lessons at each grade level and teacher support and assessment. Professional development  is also available.

## Teaching Practices for Differentiating Mathematics Instruction for Middle School Students

Three iterative, 18-episode design experiments were conducted after school with groups of 6–9 middle school students to understand how to differentiate mathematics instruction. Prior research on differentiating instruction (DI) and hypothetical learning trajectories guided the instruction. As the experiments proceeded, this definition of DI emerged: proactively tailoring instruction to students’ mathematical thinking while developing a cohesive classroom community.

Author/Presenter:
Amy J. Hackenberg
Mark Creager
Ayfer Eker
Year:
2020
Short Description:

This study is a case of using second-order models of students’ mathematical thinking to differentiate instruction, and it reveals that inquiring into research-based knowledge and inquiring responsively into students’ thinking are at the heart of differentiating mathematics instruction.

## 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:
David A. Yopp
Rob Ely
Annelise W. Nielsen
Erin C. Corwine
Year:
2019
Short Description:

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 eliminating counterexamples (ECE) framework.

## Growth in children’s understanding of generalizing and representing mathematical structure and relationships

We share here results from a quasi-experimental study that examines growth in students’ algebraic thinking practices of generalizing and representing generalizations, particularly with variable notation, as a result of an early algebra instructional sequence implemented across grades 3–5.

Author/Presenter:
Maria Blanton
Isil Isler-Baykal
Rena Stroud
Ana Stephens
Eric Knuth
Angela Murphy Gardiner