## Technical Appendix

### Module Overview

**Enhancing Core Mathematics Instruction for Students at Risk for Mathematics Disabilities.**(Doabler et al., 2012). This practitioner-focused article provides 8 guidelines that support teachers in making their mathematics instruction more explicit and systematic.

**Challenging the Ideology of Normal in Schools.**(Annamma et al., 2013). This theoretical article uses critical frameworks to examine the idea of “normal” and explores ways in which ideas about normality can be more inclusive and expansive.

**Sines of Disability: Disrupting Ableism in Mathematics and Beyond.**(Reinholz & Torres-Gerald, 2022). This blog post is about the launch of Sines of Disability, an anti-ableist and anti-oppression mathematics community. The post explains the ways in which ableism is often integrated into mathematics instruction and offers ideas (and a community) for dismantling systems of oppression in mathematics.

**Culture, Race, Power, and Mathematics Education.**(NCTM, 2007). This book chapter explores the ways in which students’ culture, racial background, and personal experiences interact in the context of teaching and learning mathematics.

**“Good” Mathematics Teaching for Students of Color and Those in Poverty: The Importance of Relational Interactions within Instruction.**(Battey, 2012). This research article (case study) focuses on four ways in which “relational interactions” serve to mediate students’ access to mathematics learning.

**A Systematic Review of the Teacher Expectation Literature Over the Past 30 Years.**(Wang et al., 2018). This literature review provides a scoping look at literature on teacher expectations from 1989-2018, looking specifically at four major themes related to expectations: factors, mediators, moderators, and effects of teacher expectations.

### Section 1

#### Section Overview and Objectives

**Dis/ability Critical Race Studies (DisCrit): Theorizing at the Intersections of Race and Dis/ability.**(Annamma et al., 2013). This article introduces Dis/ability Critical Race Studies (DisCrit), including the tenants of DisCrit and the motivation behind developing the theoretical framework.

*Featured Video:***LeDerick Horne on Growing Up With Learning and Thinking Differences.**

#### Establishing Common Language

**Disability****.**(American Psychological Association, 2021) This guide to bias-free language provides an overview of language use — including person-first and identify first language — when writing (or speaking) about disability.

**Disability Rights and Wrongs Revisited.**(Shakespeare, 2013). This book, written by a disability activist and academic, takes a “critical realist approach” to examine the history, present state, and future directions of the disability studies research field.

**An Insider’s View of a Mathematics Learning Disability: Compensating to Gain Access to Fractions.**(Lewis & Lynn, 2018). This research article leverages the lived experience of the second author, an individual with a math learning disability (MLD). In their study, the authors explore the ways in which mathematics learning disabilities can be reframed as related to issues of access, rather than individual deficit.

**Rethinking Disability: The Social Model of Disability and Chronic Disease.**(Goering, 2015). This article defines the social model of disability and discusses how it might be applied in the medical field, specifically in regard to chronic disease.

**Brain Organization Underlying Superior Mathematical Abilities in Children with Autism.**(Iuculano et al., 2014). This article uses functional brain imaging and cognitive assessments to examine the mathematical problem-solving strengths of individuals with autism spectrum disorder (ASD).

**Prevalence of Developmental Dyscalculia.**(Shalev, 2007). This article explores the definition, prevalence, and environmental influences of learning disabilities and, specifically, mathematics learning disabilities (i.e., dyscalculia).

**Reaching the Mountaintop: Addressing the Common Core Standards in Mathematics for Students with Mathematics Difficulties.**(Powell et al., 2013). This article explores how Common Core Mathematics Standards can be achieved for students with mathematics disabilities and how teachers might support their learning, including through teaching foundational skills.

**Students with Disabilities.**(National Center for Education Statistics, 2022). This annual report includes a variety of data points related to students with disabilities (e.g., distribution of services by disability classification and racial/ethnic identity).

*Featured Video:***My Body Doesn’t Oppress Me, Society Does.**(Sins Invalid & Barnard Center for Research on Women, 2017). In this video (5 minutes), Patty Berne and Stacey Milburn talk about the social model of disability and the ways in which societal barriers (e.g., legal, physical, economic) are disabling and oppressive, especially for individuals with certain impairments.

#### Executive Functions and Math Difficulties

**Against the Odds: Insights from a Statistician with Dyscalculia.**(Lewis & Lynn, 2018). This research article uses an asset-based framework to examine the ways in which the second author, an individual with a mathematics learning disability (dyscalculia), worked to become a statistician at the University of California, Berkley. The article includes an examination of societal barriers to success in mathematics for students with disabilities and challenges existing deficit-oriented frameworks in research and practice.

**Understanding the Cognition Related to Mathematics Difficulties: A Meta-Analysis on the Cognitive Deficit Profiles and the Bottleneck Theory.**(Peng et al., 2018). The meta-analysis examines 75 independent studies related to the cognitive profile of students with mathematics difficulties, examining characteristics identified across studies, as well as potential moderators and mechanisms.

**Learning Disabilities: From Identification to Intervention (Second Edition).**

**Executive Function & Self-Regulation.**

**Executive Functions.**(Diamond, 2013). This article explores what executive functions are, their relationships with other cognitive skills, and ways to improve them through practice.

**Handbook of Learning Disabilities (Second Edition).**(Swanson et al., 2014). This textbook presents a general reference for research on learning disabilities (LD), including ways to use the response to intervention (RTI) framework to support students identified with LD. This link includes an option to view a sample chapter that overviews the basic framework and history of some research on LD.

**A Meta-Analysis of Mathematics and Working Memory: Moderating Effects of Working Memory Domain, Type of Mathematics Skill, and Sample Characteristics.**(Peng et al., 2016). This meta-analysis includes 110 studies on working memory (WM) and mathematics, looking at the relationship between WM and mathematics skills, as well as potential moderators.

**Cognitive flexibility: exploring students’ problem-solving in elementary school mathematics learning.**

*Featured Video:***EduTalks: Katherine Lewis & Difference Not Deficit**(2016). In this video (5 minutes), Professor Katherine Lewis describes compensatory strategies students with disabilities develop to support their learning in mathematics.

#### Teacher Expectations

**ADHD Alien: Comics About the Daily Struggle with ADHD.**(ADHD Alien). This is a series of comics about the lived experience of having ADHD, created by an artist with ADHD.

**A Systematic Review of the Teacher Expectation Literature Over the Past 30 Years.**(Wang et al., 2018). This literature review provides a scoping look at literature on teacher expectations from 1989-2018, looking specifically at four major themes related to expectations: factors, mediators, moderators, and effects of teacher expectations.

**Adjusting Expectations or Maintaining First Impressions? The Stability of Teachers’ Expectations of Students’ Mathematics Achievement.**(Timmermans et al., 2021). This research article is about the stability of teacher expectations of students, based on the results of a study that spanned the course of a school year and found relatively stable expectations of students’ mathematics performance.

**Integrating Professional Development on Mathematics and Equity: Countering Deficit Views of Students of Color.**

**January 2022 Working Definition of Ableism.**(TL Lewis, 2022). TL Lewis, an activist and advocate in the disability justice community, defines ableism. The link includes additional context around the definition, past definitions, and an opportunity to explore more of TL’s work and publications.

**Demarginalizing the Intersection of Race and Sex: A Black Feminist Critique of Antidiscrimination Doctrine, Feminist Theory and Antiracist Politics.**

**Dis/ability Critical Race Studies (DisCrit): Theorizing at the Intersections of Race and Dis/ability.**(Annamma et al., 2012). This theoretical article presents Dis/ability Critical Race Studies (DisCrit), a combination of Critical Race Theory and Disability Studies, as a framework with which to examine the ways in which students of color with disabilities are historically multiply marginalized on the basis of race and disability. The article includes the tenets of DisCrit as well as a discussion of implications, applications, and limitations.

**School Segregation and Racial Gaps in Special Education Identification.**

**Significant Disproportionality in Special Education: Current Trends and Actions for Impact.**(National Center for Learning Disabilities, 2020). This research brief summarizes current literature and research regarding disproportionality in special education, especially as it relates to overidentification of students of color for special education enrollment, their placement in more restrictive educational settings, and increased discipline rates for students of color identified with disabilities.

**Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades.**(Institute of Education Sciences, 2021). This practice guide, developed by the What Works Clearinghouse (established by the Institute of Education Sciences) provides a breakdown of evidence-based practices for elementary mathematics instruction for students struggling with mathematics.

**Mathematics Instruction for Students with Learning Disabilities: A Meta-Analysis of Instructional Components.**(Gersten et al., 2009). This meta-analysis looks across 42 interventions to examine the effectiveness of approaches to mathematics instruction for students with learning disabilities.

#### Supporting Students’ Strengths, Interests, and Needs

#### Closing

**Funds of Knowledge for Teaching: Using a Qualitative Approach to Connect Homes and Classrooms.**(Moll et al., 2009). This article presents findings from a collaborative, asset-based qualitative research partnership between the classrooms and households of students living in Mexican communities in Tuscon, Arizona. The article not only presents the findings, but also an explanation of the research design and a discussion of the implications of the design in other contexts.

**The Impact of a Comprehensive Tier 1 Kindergarten Curriculum on the Achievement of Students At-Risk in Mathematics.**

**Enhancing Core Mathematics Instruction for Students At Risk for Math Disabilities.**(Doabler et al., 2012). This practitioner-focused article provides 8 guidelines that support teachers in making their mathematics instruction more explicit and systematic.

**Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades.**(Institution of Education Sciences, 2021). This practice guide, developed by the What Works Clearinghouse (established by the Institute of Education Sciences) provides a breakdown of evidence-based practices for elementary mathematics instruction for students struggling with mathematics.

### Section 2

#### Section Overview and Objectives

**Common Core Mathematics Standards.**

**Word Problem-Solving by Students with and without Mild Disabilities.**

**Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics.**(Hiebert et al., 1996). This conceptual article reframes problem solving in mathematics education, emphasizing the role of “reflective inquiry” about the subject of mathematics in promoting students’ understanding.

**Models of Problem Solving: A Study of Kindergarten Children’s Problem-Solving Processes.**(Carpenter et al., 1993). This article presents the results of a study of 70 kindergarten students’ mathematics problem solving methods, exploring modeling as a framework for understanding students’ problem solving.

**Effective Word-Problem Instruction: Using Schemas to Facilitate Mathematical Reasoning.**(Powell & Fuchs, 2018). This practitioner-oriented article presents ways of using attack strategies and schema-based instruction to promote students’ reasoning in mathematics.

#### Benefits and Challenges of Word Problems

**Making Sense: Teaching and Learning Mathematics with Understanding.**(Carpenter et al., 1997). This practitioner-oriented textbook describes several components of mathematics classrooms designed to support students’ understanding of mathematical concepts and practices.

**Adding It Up: Helping Children Learn Mathematics.**(National Research Council et al., 2001). This report explores mathematics learning for students in grades K-8 and includes recommendations for adjusting preservice teacher preparation to enhance mathematics instruction for K-8 students.

**Common Core Mathematics Standards.**

**The Role of Rewording and Context Personalization in the Solving of Mathematical Word Problems.**(Davis-Dorsey et al., 1991). This article presents the results of a quantitative study about the effects of personalization and rewording of word problems for 2^{nd}and 5^{th}grade students.

**Revising Tasks to Be Culturally Relevant.**

**Elementary and Middle School Mathematics: Teaching Developmentally.**(Van de Walle et al., 2019). This textbook (linked in full) presents a framework and related practices for a student-centered approach to teaching mathematics.

**Thinking with Numbers.**(Rathmell et al., 2017). This website includes numerous practice problems and activities for teaching mathematics.

**Principles to Actions.**(National Council of Teachers of Mathematics, 2014). This guidebook presents teaching practices that promote the development of competence, confidence, and mathematical understanding for all students.

**Twitter Profile: Meg Arieno.**This links to the Twitter profile of Meg Arieno, an elementary school educator who has posted photographs of classroom anchor charts and word problems she wrote for her kindergarten students.

**Handbook of Learning Disabilities (Second Edition).**(Swanson et al., 2014). This textbook presents a general reference for research on learning disabilities (LD), including ways to use the response to intervention (RTI) framework to support students identified with LD. This link includes an option to view a sample chapter that overviews the basic framework and history of some research on LD.

**Cognitive Load Theory: An Applied Reintroduction for Special and General Educators.**(Kennedy & Romig, 2021). This practitioner-oriented article defines and explains types of cognitive load and their relevance in teaching and learning.

**On Metaphor, Memory, and John King.**(Willingham, 2016). In this blog post, Professor Daniel Willingham describes models of cognition and metaphors for thinking. This is the original source of his simplified cognitive model displaying the relationship between working and long-term memory.

**Impact of Authenticity on Sense Making in Word Problem Solving.**(Palm, 2008). This article presents the results of a qualitative research study identifying problem solving strategies used by elementary students and the impact of authentic problem contexts on the use of problem solving strategies.

#### Components of Word Problem Solving

**Making Sense: Teaching and Learning Mathematics with Understanding.**(Carpenter et al., 1997). This practitioner-oriented textbook describes several components of mathematics classrooms designed to support students’ understanding of mathematical concepts and practices.

**What Evidence-Based Mathematics Practices Can Teachers Employ?**(Vanderbilt University IRIS Center, 2022). This learning module, designed for preservice teachers, presents the practice of schema instruction as an evidence-based practice for mathematics instruction for students with disabilities.

#### Problem Types

**Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades.**(Institute of Education Sciences, 2021). This practice guide, developed by the What Works Clearinghouse (established by the Institute of Education Sciences) provides a breakdown of evidence-based practices for elementary mathematics instruction for students struggling with mathematics.

**Making Sense: Teaching and Learning Mathematics with Understanding.**(Carpenter et al., 1997). This practitioner-oriented textbook describes several components of mathematics classrooms designed to support students’ understanding of mathematical concepts and practices.

**Common Core Mathematics Glossary: Common Addition and Subtraction.**(Common Core State Standards Initiative, 2022). This table provides examples of addition and subtraction problem types defined by the Common Core Mathematics Standards.

**Effective Word-Problem Instruction: Using Schemas to Facilitate Mathematical Reasoning.**(Powell & Fuchs, 2018). This practitioner-oriented article presents ways of using attack strategies and schema-based instruction to promote students’ reasoning in mathematics.

*Featured Video:***Addition and Subtraction Problem Types.***Children’s Mathematics: Cognitively Guided Instruction*.

#### Factors that Influence Challenge

**Improving Student Learning of Ratio, Proportion, and Percent: A Replication Study of Schema-Based Instruction.**(Jitendra et al., 2019). This research article presents the results of a quantitative study on the impact of a schema-based strategy instructional program on 7^{th}grade students’ mathematical problem solving performance.

**Analyzing the Word-Problem Performance and Strategies of Students Experiencing Mathematics Difficulty.**(Powell et al., 2020). This research article presents the results of a study examining the impact of a mathematics instructional intervention for 3^{rd}grade students identified with mathematics difficulty, the results of which indicate the impact of explicit schema instruction and metacognitive instruction.

**Making Sense: Teaching and Learning Mathematics with Understanding.**(Carpenter et al., 1997). This practitioner-oriented textbook describes several components of mathematics classrooms designed to support students’ understanding of mathematical concepts and practices.

**Solving Arithmetic Word Problems: An Analysis of Classification as a Function of Difficulty in Children With and Without Arithmetic LD.**

**Models of Problem Solving: A Study of Kindergarten Children’s Problem-Solving Processes.**(Carpenter et al., 1993). This article presents the results of a study of 70 kindergarten students’ mathematics problem solving methods, exploring modeling as a framework for understanding students’ problem solving.

**Supporting Counting to Problem Solving.**

**Effective Word-Problem Instruction: Using Schemas to Facilitate Mathematical Reasoning.**(Powell & Fuchs, 2018). This practitioner-oriented article presents ways of using attack strategies and schema-based instruction to promote students’ reasoning in mathematics.

**Word-Problem Performance Differences by Schema: A Comparison of Students with and without Mathematics Difficulty.**

**The Role of Rewording and Context Personalization in the Solving of Mathematical Word Problems.**(Davis-Dorsey et al., 1991). This article presents the results of a quantitative study about the effects of personalization and rewording of word problems for 2^{nd}and 5^{th}grade students.

**Content Knowledge for Teaching: What Makes It Special?**(Ball et al., 2008). This theoretical article explores different sub-types of content knowledge and the practice implications of each.

### Section 3

#### Section Overview and Objectives

**Promoting General Metacognitive Awareness.**(Schraw, 1998). This broad article presents a view of metacognition as multi-faceted, specifying two aspects of metacognition, along with four practices for promoting metacognition.

**High-Leverage Practices in Special Education.**(McLeskey et al., 2017). This textbook (linked in full) synthesizes four areas of research on high-leverage practices in special education: collaboration, assessment, social-emotional/behavioral, and instructional practices.

#### Metacognition to Support Executive Functions

**Adding It Up: Helping Children Learn Mathematics.**(National Research Council et al., 2001). This report explores mathematics learning for students in grades K-8 and includes recommendations for adjusting preservice teacher preparation to enhance mathematics instruction for K-8 students.

**Strategy Instruction for Students with Learning Disabilities (Second Edition).**(Reid et al., 2013). This textbook defines cognitive strategy instruction and explores how it can support learning, especially for students who are struggling with instructional content. The link includes a sample chapter about why strategy instruction is useful, particularly for students with learning disabilities.

#### Introduction to Metacognitive Modeling

**Explicit Instruction: Effective and Efficient Teaching.**(Archer & Hughes, 2010). This textbook defines and provides examples of explicit instruction and explores ways to implement explicit instruction in the classroom. The link includes multiple videos of explicit instruction being implemented at the elementary and secondary levels, as well as a link to a sample chapter on the foundations of the practice.

**Promoting General Metacognitive Awareness.**(Schraw, 1998). This broad article presents a view of metacognition as multi-faceted, specifying two aspects of metacognition, along with four practices for promoting metacognition.

**High-Leverage Practices in Special Education.**(McLeskey et al., 2017). This textbook (linked in full) synthesizes four areas of research on high-leverage practices in special education: collaboration, assessment, social-emotional/behavioral, and instructional practices.

**Teaching Students to Drive Their Brains: Metacognitive Strategies, Activities, and Lesson Ideas.**(Wilson & Conyers, 2016). This book explores the practice of teaching for and with metacognition. The linked chapter includes a definition of metacognition (and related terms) and a discussion of why metacognitive instruction can be useful instructionally.

**“Cognitive Apprenticeship” Revisited: Shining a Light on the Processes of Thinking to Understand Learning.**

**Practices That Cross Disciplines?: Revisiting Explicit Instruction in Elementary Mathematics and English Language Arts.**(Cohen, 2018). This article defines explicit instruction, focusing on the practices of strategy instruction and modeling, and explores ways in which the practices are applied in English Language Arts and mathematics instruction at the elementary school level.

**Intensive Intervention for Students with Mathematics Disabilities: Seven Principles of Effective Practice.**(Fuchs et al., 2008). This article focuses on cognitive challenges that students face in third-grade mathematics and includes 7 principles for effective intervention for students struggling with mathematics at this level.

#### Planning for Metacognitive Modeling

**Launching Complex Tasks.**(Gibbons & Cobb, 2012). This practitioner-oriented article presents four recommendations for launching complex mathematical tasks with elementary students.

**Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades.**(Institute of Education Sciences, 2021). This practice guide, developed by the What Works Clearinghouse (established by the Institute of Education Sciences) provides a breakdown of evidence-based practices for elementary mathematics instruction for students struggling with mathematics.

**Effective Word-Problem Instruction: Using Schemas to Facilitate Mathematical Reasoning.**(Powell & Fuchs, 2018). This practitioner-oriented article presents ways of using attack strategies and schema-based instruction to promote students’ reasoning in mathematics.