Algebra

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
Lead Organization(s): 
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
Anne E. Adams
Annelise W. Nielsen
Erin C. Corwine
Lead Organization(s): 
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
Lead Organization(s): 
Year: 
2019
Short Description: 

Authors share 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.

Linear Algebra and Geometry

Linear Algebra and Geometry is organized around carefully sequenced problems that help students build both the tools and the habits that provide a solid basis for further study in mathematics. Requiring only high school algebra, it uses elementary geometry to build the beautiful edifice of results and methods that make linear algebra such an important field. 

Author/Presenter: 
Al Cuoco
Kevin Waterman
Bowen Kerins
Elena Kaczorowski
Michelle Manes
Year: 
2019
Short Description: 

Linear Algebra and Geometry is aimed at preservice and practicing high school mathematics teachers and advanced high school students looking for an addition to or replacement for calculus. The materials are organized around carefully sequenced problems that help students build both the tools and the habits that provide a solid basis for further study in mathematics. Requiring only high school algebra, it uses elementary geometry to build the beautiful edifice of results and methods that make linear algebra such an important field.

Thinking scientifically in a changing world

Shifting people’s judgments toward the scientific involves teaching them to purposefully evaluate connections between evidence and alternative explanations.

Lombardi, D. (2019). Thinking scientifically in a changing world. Science Brief: Psychological Science Agenda, 33(1). Retrieved from https://www.apa.org/science/about/psa/2019/01/changing-world.aspx

Author/Presenter: 
Doug Lombardi
Lead Organization(s): 
Year: 
2019
Short Description: 

Shifting people’s judgments toward the scientific involves teaching them to purposefully evaluate connections between evidence and alternative explanations.

A Student Asks About (-5)!

A first-year algebra student’s curiosity about factorials of negative numbers became a starting point for an extended discovery lesson into territory not usually explored in secondary school mathematics.
Author/Presenter: 
E. Paul Goldenberg
Cynthia J. Carter
Year: 
2017
Short Description: 

A first-year algebra student’s curiosity about factorials of negative numbers became a starting point for an extended discovery lesson into territory not usually explored in secondary school mathematics.

Resource(s): 

Problematizing and Assessing Secondary Mathematics Teachers’ Ways of Thinking

STEM Categorization: 
Day: 
Thu

Engage with presenters as they discuss assessment and rubrics designed to measure secondary teachers’ mathematical habits of mind.

Date/Time: 
9:30 am to 11:00 am
Session Materials: 

Work in secondary mathematics education takes many approaches to content, pedagogy, professional development and assessment. This session aims to illuminate the richness of hte content of secondary mathematics and the field of secondary mathematics education by sharing two such approaches and reflecting on the differences and commonalities between the two.   

Session Types: 

Constructing and Role-Playing Student Avatars in a Simulation of Teaching Algebra for Diverse Learners

From the perspectives of Graduate Research Assistants (GRAs), this study examines the design and implementation of a simulated teaching environment in Second Life (SL) for prospective teachers to teach algebra for diverse learners. Drawing upon the Learning-for-Use framework, the analyses provide evidence on the development of student avatars in construction and role-playing activities. The study reveals challenges, procedures, and suggestions for future simulations. This study also calls for research efforts toward preparing mathematics teachers for cultural diversity.

Author/Presenter: 
Tingting Ma
Irving A. Brown
Gerald Kulm
Trina J. Davis
Chance W. Lewis
G. Donald Allen
Lead Organization(s): 
Year: 
2014
Short Description: 

This study examines the design and implementation of a simulated teaching environment in Second Life for prospective teachers.

Illuminating Coordinate Geometry with Algebraic Symmetry

A symmetric polynomial is a polynomial in one or more variables in which swapping any pair of variables leaves the polynomial unchanged. For example, f(x, y, z) = xy +xz + yz is a symmetric polynomial.

Author/Presenter: 
Ryota Matsuura
Sarah Sword
Year: 
2015
Short Description: 

A symmetric polynomial is a polynomial in one or more variables in which swapping any pair of variables leaves the polynomial unchanged. For example, f(x, y, z) = xy +xz + yz is a symmetric polynomial. If we interchange the variables x and y, we obtain yx + yz + xz, which is the same as f(x, y, z); likewise, swapping x and z (or y and z) returns the original polynomial. These polynomials arise in many areas of mathematics, including Galois theory and combinatorics, but they are rarely taught in a high school curriculum. In this article, we describe an application of symmetric polynomials to a familiar problem in coordinate geometry, thus introducing this powerful tool in a context that is accessible to high school students.

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