To keep pace with a rapidly changing world, our nation’s students must be prepared for the challenges they face after high school—in college, career, and citizenship. A strong mathematics preparation is essential to their future success. However, our students score below average on international tests of mathematical knowledge and skills (Loveless, 2011), and nearly two-thirds of our nation’s eighth graders do not meet current mathematics standards (National Center for Education Statistics, 2015). Less than half of our college-bound students are prepared for post-secondary education and beyond (College Board, 2011). The nation’s current equation for math education is not producing students ready to thrive in a 21st-century economy. The country needs a new equation that works.
A Demand for Rigor—an Opportunity for Growth
Across the country, states have adopted new, more rigorous standards for mathematics. Developed around a coherent progression of mathematical ideas and skills, the grade-by-grade objectives focus on core concepts that build on one another and incorporate mathematical practices and habits of mind. These new standard’s have come with new assessments which seek to provide a more accurate measure of student readiness for college and career. The demands of increased expectations provide an opportunity for innovations in curriculum to enhance and deepen teaching and learning (Conley, Drummond, de Gonzalez, Rooseboom, & Stout, 2011). The new standards bring national attention to mathematics learning that is unprecedented, outlining what mathematics our students are to learn and ways in which they are to engage with content from kindergarten through high school.
College- and career-readiness standards begin as early as Grade 6, where students are expected to solve problems involving ratios, rational numbers, and plane figures—key ideas in the development of algebraic knowledge and skills. Algebra is often identified as the gatekeeper to higher mathematics, and thereby, a key to success in college and career (National Mathematics Advisory Panel [NMAP], 2008). Therefore, it is imperative that students have the mathematical foundation necessary for success to and through algebra. Indeed, this is an educational and social justice imperative.
In addition to providing a renewed perspective on the mathematics content students learn, these next generation standards require a focus on the act of doing mathematics and developing robust mathematical reasoning and thinking skills for all students. Mathematical knowledge and thinking skills have been identified as essential for college success (Conley, 2007).
Developing strong understandings of mathematics and keen abilities with the practices of mathematical thinking is particularly relevant for today’s students, as jobs utilizing mathematical knowledge and skills are growing dramatically. The number of jobs in the fields of science, technology, engineering, and mathematics (STEM) is projected to increase 17% by 2018, leading to 2.4 million available STEM positions between 2008 and 2018 (Carnevale, Smith, & Melton, 2011).
As we look to improve students’ mathematical preparedness for college and career, we must emphasize the role of the teacher in the implementation of new standards. The practices teachers employ have been learned and developed through participation in their own educational experiences as students (Ball, 1990; Cooney, 2001) and participation as teachers in communities of practice (Cobb & McClain, 2001). Additionally, teacher practices are filtered through what each teacher knows and believes. Therefore, teachers will implement the new standards upon negotiation of the objectives with their held knowledge and beliefs and current teaching practices (Cohen & Ball, 1990). Those designing effective support for teachers must consider this reality (Lappan, 1997) and the challenges teachers and their students face in adapting to these new expectations.
MATH 180—Revolutionary Math Intervention
With the goal of ensuring that struggling students become equipped with the knowledge, reasoning, and confidence to thrive in college and career, Houghton Mifflin Harcourt® has developed a revolutionary mathematics program that motivates students to learn the content critical for success in algebra. Since the development of emotional and social competencies essential for success in college and career works hand in hand with efforts to improve students’ content knowledge (Dweck, 2000), MATH 180 is rooted in relevant and realistic world connections, providing a rich landscape for learning in multiple domains. With a focus on the rigors of next generation math standards, MATH 180 leverages the research on effective mathematics teaching and learning and the need for educator support in implementing educational innovations to provide the support students need to develop key knowledge and skills essential for college and career success.
This report provides a detailed description of how MATH 180 utilizes the latest research to prepare students in Grades 5 and above with the mathematical understanding and skills they need to thrive in the 21st century. It summarizes the key research principles underlying the development of MATH 180 and delineates the specific program features that are designed to engage and empower learning experiences, support teachers in maximizing instructional effectiveness, and give administrators the tools to ensure high-quality implementation.
Three research-based principles have been engineered into MATH 180 to transform math instruction so that students believe in the possibility of success and their teachers have cutting-edge tools to accelerate them to meet the rigors of next generation math standards.
Focus on What Matters Most: Accelerating Student Learning
For middle-school students who are two or more years behind in math, reteaching every missed skill and concept simply isn’t possible. MATH 180 focuses on rebuilding the essential concepts and skills that underlie later math learning along a progression to algebra. Carefully curated by expert mathematicians and next-generation standards architects, including Dr. William McCallum and Dr. Sybilla Beckmann, the MATH 180 scope and sequence is built around a focused and coherent curriculum that enables struggling students to progress quickly and effectively toward grade-level curriculum.
Force Multiplier for Teaching: Build Teacher Effectiveness
A force multiplier is an approach that dramatically increases—or multiplies—effectiveness. Teachers are the key force behind effective math instruction, yet for most school districts, teacher preparedness has become a critical issue. MATH 180’s professional learning scaffolds less experienced teachers and provides a wealth of sophisticated supports to veteran math teachers. Guided by Dr. Deborah Ball, the country’s most respected voice in building teaching capacity, MATH 180 helps teachers become force multipliers by surrounding them with the resources they need to be greater at what they do best.
Growth Mindset: Attitudes Toward Intelligence Can Impact Performance
Too many students and their teachers have come to believe that math success is impossible. This “fixed mindset” undermines effort and becomes self-fulfilling. MATH 180 incorporates the work of Dr. Carol Dweck’s organization, Mindset Works®, in all aspects of the program to create a “growth mindset” culture in the classroom. Students learn that intelligence is malleable and reflect together on effective strategies for learning while building confidence and monitoring individual growth in an adaptive software environment.
MATH 180 Instructional Model for Blended Learning
The flexible instructional model in MATH 180 maximizes instructional time with a clear organization for whole-class, group, and individualized learning. This simple yet powerful design allows both the teacher and technology to deliver efficient, effective instruction based on their respective strengths.
Whole-Class Do NowThis classroom management routine develops mathematical thinking and makes connections to prior topics.
Group InstructionThe teacher facilitates instruction to build conceptual understanding, develop reasoning and communication skills, and interpret student thinking.
Personalized SoftwareThe MATH 180 software adapts to each student’s needs, providing added practice for those who need it and accelerating those ready to move on.
Brain ArcadeAvailable anytime, anywhere, the Brain Arcade provides each student with a personalized playlist of games that build strategic and procedural fluency.
MATH 180 picks up the progression to algebra at whole number multiplication and builds a coherent narrative of understanding through fractions and decimals, proportional reasoning, and functional thinking. Concepts, visual models, and procedural strategies build on one another, opening up and facilitating new learning. Students enter the progression at different points based on their existing knowledge and progress at their own pace. Instruction is organized into two courses, each with nine blocks of instruction, featuring high-interest career themes. The focused content helps students make connections while learning to think algebraically.
The Course I Learning Matrix
Designed for students who lack numerical understanding and reasoning skills, Course I focuses on key foundational concepts that enable students to make connections while learning to think algebraically.
The Course II Learning Matrix
The Course II curriculum transitions students to pre-algebra with an emphasis on building proportional reasoning with rates, ratios, linear relationships, and functions.
MATH 180 is the result of Houghton Mifflin Harcourt’s collaboration with the leading math education researchers, thinkers, and practitioners.
Dr. Deborah BallDr. Deborah Ball, the Dean of the School of Education at the University of Michigan, is one of the nation’s foremost voices on building teacher effectiveness and the founder of TeachingWorks—an organization that aims to transform how teachers are prepared and supported. Serving as the lead advisor on teacher-facilitated instruction, Dr. Ball has helped to embed TeachingWorks’ research-based High Leverage Teaching Practices into every MATH 180 lesson.
Dr. Ted HasselbringDr. Ted Hasselbring is a Professor of Special Education at Peabody College of Vanderbilt University and author of READ 180®, System 44®, and FASTT Math®. Dr. Hasselbring researched the use of technology for enhancing learning in students with mild disabilities and those who are at risk of school failure. With MATH 180, Dr. Hasselbring shares his expertise in adaptive technology to build a personalized learning experience for struggling math students.
Dr. William McCallumDr. William McCallum is the award-winning Distinguished Professor of Mathematics at the University of Arizona and founder of Illustrative Mathematics, a nonprofit that curates high quality mathematical lessons to support next generation math standards. Dr. McCallum helped construct the scope and sequence for the MATH 180 curriculum.
Dr. Sybilla BeckmannDr. Sybilla Beckmann is a Professor of Mathematics at the University of Georgia. She has worked on the development of several state mathematics standards and was a member of the mathematics writing team for the Common Core State Standards Initiative. Dr. Beckmann helped design the scope and sequence for the MATH 180 Course I curriculum.
Dr. David DocktermanDr. David Dockterman is a leading developer of educational technology. Dockterman serves as Chief Architect, Learning Sciences at Houghton Mifflin Harcourt, identifying and guiding the effective application of research to practice. David advised on Brain Arcade and the integration of growth mindset.
Harold AsturiasHarold Asturias is the Director of the Center for Mathematics Excellence and Equity (CeMEE) at the University of California, Berkeley, where he designed and implemented professional development for K–12 math teachers who teach English language learners. Harold has integrated explicit language goals and vocabulary routines as well as support for English language learners into MATH 180.
Dr. Lee Peng YeeDr. Lee Peng Yee is an adjunct senior fellow in mathematics education at the National Institute of Education (NIE), Nanyang Technological University, Singapore. His experience in curriculum development and textbook writing spans the globe. From proven visual models to problem-solving routines, Dr. Lee applied his extensive research in mathematical analysis to enhance MATH 180’s rigorous curriculum.
Dr. Wong Khoon YoongDr. Wong Khoon Yoong is the now retired associate professor in mathematics education at the National Institute of Education (NIE), Nanyang Technological University, Singapore. Dr. Wong has participated in the design of the national mathematics curriculum in Malaysia, Brunei, and Singapore. During the development of MATH 180, Dr. Wong advised on applying best practices from the highly effective Singapore math curriculum and stressed the importance of visualization, communication, and making connections.
Mindset Works®Mindset Works is a social venture that helps human beings realize their full potential. Founded by leading motivational researcher Dr. Carol Dweck and her colleague Dr. Lisa S. Blackwell, the organization focuses on translating lessons into programs that schools can use to increase student motivation. Mindset Works has been influential in driving MATH 180’s transformative growth mindset principle.
Math Solutions®Math Solutions is the nation’s leading provider of math professional development. Math Solutions was instrumental in the program-embedded Professional Learning, which gives teachers content and pedagogic background for every topic.
Center for Applied Special Technology (CAST)Founded in 1984 as the Center for Applied Special Technology, CAST is a nonprofit research and development organization that works to expand learning opportunities for all individuals, especially those with disabilities, through Universal Design for Learning.
SRI EducationSRI Education is a division of SRI International, one of the leading research and development organizations in the world. The SRI team helped to identify key research and advisors to guide MATH 180 development and in the implementation of MATH 180’s dynamic visual models.
Carefully curated by leading thinkers and practitioners, including Dr. Bill McCallum and Dr. Sybilla Beckmann, the MATH 180 scope and sequence is built around a focused and coherent curriculum that enables struggling students to make connections while learning to think algebraically. MATH 180 teaching practices focus on building conceptual understanding with a capacity for disciplined reasoning, analysis, argument, and critique.
MATH 180 focuses on the specific set of concepts and mathematical practices along the progression to algebra. Research shows that to accelerate learning, students and teachers must be both effective and efficient, developing deep conceptual understanding and then persistently activating and building on prior knowledge to foster new neural connections. Struggling math students often lack the explicit mathematical vocabulary they need to understand the problems they are expected to solve or to communicate their lack of understanding. MATH 180 considers an understanding of the language of mathematics across all subject areas to be a vital part of the progression to algebra. Similarly, the eight Standards for Mathematical Practice provide an affective-behavioral profile of proficient mathematical learners.
Our approach relies on four foundational principles:
1. Focus: Concentration on the concepts along the progression to algebra
2. Coherence: Emphasis on the interdependence and cumulative nature of mathematics
3. Rigor: Robust opportunities for reasoning and high-order thinking and reasoning in routine and nonroutine problems
4. Communication and Standards for Mathematical Practice: Explicit mathematical vocabulary instruction, English language support, and active development of the behaviors employed by proficient mathematics learners
MATH 180 ensures that older students who have not been successful now have an explicit, accelerated path to algebra. This learning progression is divided into two courses. Students who are significantly below grade level have not developed the numerical understanding and fluency needed to make sense of more complex math. Building these critical foundations is the goal of Course I. Other students whose difficulties lie with abstract pre-algebra concepts will benefit from Course II’s instructional focus on developing proportional reasoning.
To accelerate learning for students below grade level, MATH 180 maintains a tight focus on the concepts, strategies, and content knowledge that matter: those that constitute the progression to algebra. Guided by experts involved in the development of the new standards and Singapore’s National Institute of Education, MATH 180 delivers a concise and highly focused mathematics framework designed to maximize instructional time.
Research and Expert Opinion
How MATH 180 Delivers
MATH 180 accelerates students to algebra by providing a concise, logical curriculum tightly focused on building deep conceptual understanding and fluency. Each lesson provides a new concept or strategy that will be used again and again in future lessons, as students continually expand their toolbox of mathematical vocabulary terms, procedural strategies, and interactive virtual manipulatives. Division strategies are extended when students study fractions, and fraction strategies are extended when students begin to understand the relationship between decimals and fractions. Fraction strategies from Course I are further extended when modeling ratios and graphing linear relationships in Course II.
MATH 180 focuses on the progression to algebra as well as the Singapore national curriculum, which emphasizes the use of visual models and a strong, transparent conceptual progression. Within each lesson of MATH 180, interactive mTools, such as the bar model, fraction strips, and motion model, are used in conjunction with novel problem-solving contexts to demonstrate the applications of mathematical strategies. In the Learn Zone, students extend their new strategies and models with the proper scaffolds, gradually moving from guided to independent practice with the aid of a metacognitive coach.
New standards state that the application and extension of foundational mathematics concepts form the core of algebra instruction: rational expressions are an extension of rational numbers, while polynomials are a natural extension of integers. Students in MATH 180 are explicitly shown how to make these connections, continually applying and extending their understanding of arithmetic operations, rational numbers, and proportional reasoning into new contexts. The student software ensures that students make such connections actively, and the classroom routines, guided practice, and activities in the student mSpace ensure that students are extending their previous understandings consciously and logically.
An essential element of MATH 180 is the delivery of a coherent curriculum in which concepts build on one another cumulatively and in a logical progression. Rather than being an unlinked series of tricks to solutions, math should make sense. Knowledge should build as progressions between grades and between topics (Core Standards, 2012). Students in MATH 180 progress from concrete to pictorial to abstract representations of each concept. In each successive unit of MATH 180, students are encouraged to activate prior learning and access the models and strategies common to multiple topics. This way, students build mental connections between topics and transfer knowledge with a reduced strain on memory-retrieval processes.
Research and Expert Opinion
How MATH 180 Delivers
Struggling students often fail to see the interrelatedness of subsequent math topics, and cannot, on their own, bring prior knowledge to bear when learning new concepts. MATH 180 celebrates the coherence, interdependence, and cumulative nature of mathematics concepts through the mechanics of interactive learning and game mechanics: mastering a concept “unlocks” the successive concepts that build on it. MATH 180 also supports coherence in blended learning by introducing instructional models in a consistent manner across all learning environments. A student will use common strategies on the daily Do Now, in Group Instruction lessons and in lessons on the student software.
Each block of instruction is designed to be internally cohesive and to unfold clearly and logically. Each concept builds on preceding concepts; during Group Instruction, students explicitly “connect” the new concept to prior learning, and on the student software, each student is reminded to apply and extend his or her interactive mTools—interactive virtual manipulatives—into new conceptual contexts. In accordance with the Singapore model, students gain understanding of abstract math by explicitly connecting each math concept across multiple modalities, including in context, pictures, symbols, and words.
Between Blocks, MATH 180 uses consistent mathematical language to activate prior learning as new concepts are introduced. In the scope and sequence of the Blocks, vital transitions—such as the transition from division to fractions or from proportional relationships to linear equations—are made explicit both by the virtual coaches and through the instructional model. Strands of instruction apply to multiple Blocks of instruction. When students learn how to work with place value in whole number operations and again in decimal operations, the mathematical language and models are consistent to reduce cognitive overload and support transfer.
Rigor refers to the degree that sets of standards address key content that prepares students for success beyond high school (Achieve, 2010). To meet the expectations of new, more rigorous standards, MATH 180 delivers on three aspects of rigor: conceptual understanding, procedural skills and fluency, and applications. Rather than trying to teach students a little bit of everything, learning and understanding must be deeper, with high learning expectations for each student.
Research and Expert Opinion
How MATH 180 Delivers
The lessons in MATH 180 intertwine conceptual understanding with procedural skill so that students learn the mathematical steps in conjunction with the reasons behind them. Targeted and deliberate practice builds fluency with those procedures. After watching a video that introduces the new concepts in a problem-solving context, the concept is then broken down from concrete to representational to abstract, through the use of interactive mTools and the introduction of mathematical language and symbols. Every lesson in MATH 180 models reasoning and provides rigorous scaffolded practice; as students advance, they build conceptual fluency and model reasoning in order to prepare for high-order performance tasks.
MATH 180 offers opportunities for procedural fluency in the Learn Zone and student mSpace, where students receive rigorous, scaffolded, and adaptive practice problems. In the Practice (guided practice) section of the software, students receive corrective feedback on each step of every problem, while in the Master (independent practice) section, students receive customized problem sets of gradually increasing difficulty. Students in MATH 180 develop procedural fluency in the Brain Arcade, which provides practice with the procedures, computational skills, models, and strategies in an engaging game environment.
MATH 180 provides multiple opportunities to apply the concepts and procedures to real-world situations. Every Block contains a career-based, problem-solving mathematics simulation which allows students to apply the concepts they are learning. In the Success Zone, students expand their understanding to nonroutine problems, including word problems, inspired by the next generation assessments. mSpace provides rigorous scaffolded practice; students apply their conceptual and procedural understanding on multiple performance tasks of gradually increasing difficulty, all situated in relevant and engaging college and career contexts.
Any successful mathematics learning environment includes engaged classroom discussions in which every student can clearly communicate his or her own mathematical reasoning. To foster a communicative classroom culture, MATH 180 provides explicit instruction in mathematical vocabulary and language support, including a Spanish translation of all mathematical terms and audio recordings of all mathematical instruction on the student software. The Standards for Mathematical Practice are varieties of expertise employed by mathematically proficient students. MATH 180 emphasizes these standards in all aspects of the program.
Research and Expert Opinion
How MATH 180 Delivers
MATH 180 provides extensive language support and vocabulary instruction designed to allow students to reason abstractly and quantitatively; mathematics vocabulary is both embedded into the instruction at point of use, supporting students’ language development, and instantly available to students in a searchable mathematics glossary provided in English and Spanish. Math Talk questions in the Success Zone assess a student’s mastery of math vocabulary. Each simulation in the Explore Zone is followed by a reflection exercise, in which students evaluate their thought processes in writing, to be reviewed by classroom teachers in the students’ digital portfolio. The student mSpace contains an Exit Ticket, which assesses learning through writing.
MATH 180 delivers on all the Standards for Mathematical Practice. At the beginning of every lesson, Do Now activities target and strengthen student skills with specific mathematical practices. Students construct viable arguments by sharing their reasoning, thought processes, and procedures with teachers and with one another in person and in writing.
The Teach section of Group Instruction supports teachers and students as they model problems with mathematics using math games and visual models. Students learn to use appropriate tools strategically both in groups and on the software, where they have full access to the suite of mTools, teach videos, Anchor Videos, metacognitive coaching, and the bilingual math glossary. Finally, students are encouraged to focus on expressing regularity in repeated reasoning in the Learn Zone and the mSpace student book, as well as through the use of mTools and in successive levels of games in the Brain Arcade.
Guided by Dr. Deborah Ball, the country’s most respected voice in effective teaching practices, MATH 180 helps teachers become force multipliers by surrounding them with the resources they need to be greater at what they do best. Smart, adaptive technology by Dr. Ted Hasselbring gathers data through every interaction and uses it to personalize learning and provide actionable intelligence to the teacher to make instruction more effective, efficient, and engaging.
Research shows that effective teaching is the driving force behind powerful math instruction and deep understanding. MATH 180 empowers teachers to be force multipliers, dramatically increasing student achievement by providing teachers with the tools, resources, and professional learning they need to improve learning outcomes and create an engaging classroom culture.
Every MATH 180 teacher receives a full suite of professional learning resources: implementation and leadership training and online professional development provided by Math Solutions and High Leverage Practices developed by Dr. Deborah Ball and TeachingWorks. The first two weeks of MATH 180 are designed in conjunction with Mindset Works to develop a mastery-oriented classroom culture where teachers foster a growth mindset, equating effort with achievement.
Four elements of MATH 180’s teaching support include:
1. High-Leverage Teaching Practices: Dr. Deborah Ball outlines best practices to elicit student reasoning, lead discussions, and surface misconceptions.
2. Data-Powered Differentiation: Teachers can access actionable data from the student software to create learning groups and differentiate instruction.
3. English Learners: Teachers are provided with language goals and supports to build fluency with academic and mathematical vocabulary.
4. Students with Special Needs: In accordance with the Universal Design for Learning principles, teachers receive support for their students with special needs or learning disabilities, including support for IEP alignment and active development of the behaviors employed by proficient mathematics learners.
Deborah Ball’s High-Leverage Practices for mathematics teaching are designed around the principle that great teachers are not born, but taught. Research-based and proven in thousands of classrooms, the High-Leverage Practices provide a road map for clear, implementable point-of-use professional learning. In MATH 180, teachers learn how, why, and when to differentiate at point of use, foster meaningful class discussions, and elicit and respond to student reasoning.
Research and Expert Opinion
How MATH 180 Delivers
We recognize teachers as the most influential factor in student success. MATH 180 empowers teachers to focus on what they do best—elicit student thinking, develop reasoning skills, and facilitate communication to create an enriching, supportive classroom culture. Dr. Deborah Ball’s High-Leverage Practices support teachers as they establish a mathematical mindset, model reasoning, assess student learning across lessons, surface misconceptions, and foster engaging and effective class discussions.
MATH 180 lessons highlight exactly which High-Leverage Practices will be the most powerful and relevant at every step of a lesson. One moment may be an ideal time to engage the entire class in a discussion, while another may be a prime opportunity to elicit and interpret an individual student’s thinking. Teachers will differentiate at point of use by modifying tasks to match their students’ abilities and learn how to establish effective routines for classroom discourse and work. Teachers learn how to respond to common patterns of student thinking with the appropriate strategy or instructional response, correcting misconceptions as they occur.
The High-Leverage Practices are embedded into every MATH 180 lesson, available in the printed Teaching Guide and through SAM Central, the teachers’ digital dashboard for lesson planning. High-Leverage Practices serve as embedded professional learning, transforming the work of teaching by providing a common vocabulary of effective techniques and strategies across an entire teaching faculty, even those working with multiple curricula and with different grade levels.
MATH 180 is dedicated to providing the resources to educate students with a wide variety of abilities, interests, and learning needs. Technology-based learning provides assessment and instruction data for every student through ongoing formative assessments and progress monitoring, and MATH 180 translates this data seamlessly into meaningful data snapshots and interactive analytics to target instruction and group students according to their needs.
Research and Expert Opinion
How MATH 180 Delivers
In the age of big data, technology is an essential part of the education process. Every moment that a student spends on the MATH 180 software provides continuous, embedded formative assessments and a wealth of actionable data—performance data, formative and summative assessment data, granular data about student timing and engagement, and data about a student’s changing mindset and affect toward mathematics.
Teachers receive anytime, anywhere access to student data. From the minute teachers log onto SAM Central, their dashboards are populated with Data Snapshots of class performance. Student and classroom analytics allow teachers to see growth and progress toward mastery, and assessment readiness among multiple learning objectives at a glance, all in a digestible fashion designed specifically to meet the time constraints of working professionals.
In MATH 180, analytics are always actionable; teachers are empowered by interactive reports, which recommend the resources and lesson plans that are most effective for each class, group, or individual student. The Groupinator® organizes students around common needs and readiness for extension, allowing teachers to create customized, data-driven groups for intensive differentiation throughout the school year. Recommended lessons are provided both in SAM Central and in the Resources for Differentiated Instruction & Problem Solving Guide provided in MATH 180: targeted Boost lesson plans are provided for students who need additional support, while Stretch lessons provide stimulating performance tasks to students who have proven that they are ready for additional challenges.
Students who are learning English may need additional scaffolds to understand mathematical terms. In MATH 180, instruction begins with progressive language development goals; all mathematical terms are provided in Spanish and English, and on the student software many mathematical instructions are translated and all written instructions are read aloud.
Research and Expert Opinion
How MATH 180 Delivers
Classrooms with large numbers of English learners or students with low-level language development or literacy face many additional challenges in mathematics instruction. Working with language expert Harold Asturias, the MATH 180 advisors have developed instructional and classroom routines that foster student language learning through mathematical discourse that focuses on consistent academic language.
MATH 180 provides progressive “language development goals” within each section of every lesson, explicitly defined and outlined in all teacher materials. New mathematics vocabulary is introduced during guided practice with a consistent routine of “hear it, see it, say it, and define it.” Students discuss, dissect, and understand their mathematical thinking through language-rich classroom routines and discussions. Students write about mathematics and practice expressing their reasoning in the mSpace, supported by sentence frames. Language goals culminate in rich performance tasks and multistep word problems.
The student software provides a personal, confidential, risk-free environment in which students can respond, reflect, and access language resources, such as the interactive bilingual mathematics glossary, at their own pace. Every written mathematical instruction in the student software, has an accompanying audio recording that can be played aloud multiple times by students with low English-literacy levels. All mathematical concepts are supported by multiple forms of representation: mathematics tools and vocabulary have visual and oral representations, and engagement is stimulated through images and videos.
Students with special needs receive a wide variety of supports in MATH 180 that may be unavailable in traditional classrooms. Instructional tools and methods that have been proven effective for students with learning disabilities, such as immediate corrective feedback at each step of a problem, visual models to promote deep understanding, systematic and explicit instruction of concepts, cooperative and peer-mediated learning techniques, and differentiated lesson design, all contribute to a classroom culture that supports all learners.
Research and Expert Opinion
How MATH 180 Delivers
Students who have been identified as in need of special education services, or older struggling students, are fully supported in MATH 180. Individualized data reports allow teachers and parents to continually measure progress against IEP annual goals and benchmarks. The adaptive student software allows students to receive targeted instruction and to move at their own pace, with gradually released scaffolds and a support ecosystem. A study conducted by independent researchers found that, after using MATH 180, students who were designated as having a special education status made significant gains on Math Inventory over their peers who used different mathematics intervention programs.
Working with the Center for Applied Science Technologies (CAST), UDL principles are integrated into MATH 180 at all levels to facilitate a mathematical growth mindset for all learners. Students consistently encounter math through multiple means of representation, including videos, print, audio recordings, animations, games, and interactive tools on the student software and whiteboard. Students have multiple means of expression and communication with which they can express their mathematical knowledge.
Perhaps most of all, MATH 180 provides multiple means of engagement for learners. Learners understand the value and purpose of mathematics through high-quality videos and animations devoted to real-world career contexts, make choices to express their autonomy in authentic math simulations and Brain Arcade games, and are exposed to a mastery environment through the application of badges, stars, points, and other game mechanics.
MATH 180 builds student competence and confidence with the help of Dr. Carol Dweck’s important research on mindset. In MATH 180, teachers are guided to praise students for their effective effort, reinforcing the values of hard work and perseverance. Students learn that making a mistake is a natural part of learning and an opportunity to identify and correct errors.
Students with a fixed mindset about math—a belief that one’s ability to understand or excel at math is innate and immutable—do not see the value in putting forth the effort to learn. Struggling students, in particular, are more likely to have a negative, fixed mindset toward their own mathematical abilities (Pashler et al., 2007).
Students with a growth mindset, on the other hand, believe that their intelligence and abilities can be developed and improved over time through effort and dedication. Research states that not only are sustained effort and deliberate practice over time crucial to achieving mastery and expertise, but that students who understand this show more resilience and, ultimately, effectiveness in their pursuit of mastery (Dweck, 2000; Ericsson, 2006).
In particular, students who understand that the brain is a muscle that needs to be exercised value practice because they know that such activity will increase both the number and the strength of their neural connections. These same students are more likely to develop a growth mindset and increase their efforts (Blackwell, Trzesniewski, & Dweck, 2007).
Four distinct criteria have been proven to foster a growth mindset:
1. Purpose and Value: Students feel that their work is interesting, meaningful, and important.
2. Agency and Choice: Students are empowered to make autonomous decisions.
3. Success and Competence: Students experience a growing sense of mastery and self-efficacy as a result of their practice.
4. Community and Family Engagement: Students receive support for and affirmation of their efforts from others in their immediate environment.
87% of ACT®-tested high school students aspire to graduate from a postsecondary institution (such as a two- or four-year college, trade school, or technical school), but only 46% of students meet ACT’s mathematics benchmark for college readiness (ACT, 2012). When students understand how mathematics is accessible and personally valuable, they become engaged in the work, ready to take responsibility for their own improvement, and increasingly resilient and persistent in their pursuit of mastery.
Research and Expert Opinion
How MATH 180 Delivers
MATH 180 explicitly situates mathematics in college and career contexts, exposing students to a variety of potential visions of future success. Each MATH 180 unit begins with an Anchor Video that presents the upcoming content through high-interest situations. In one Course I unit, for instance, students learn how the performances of the world’s top athletes are often separated by only tenths or even hundredths of a second. In another Anchor Video in Course II, students see how marketers use ratios and social media to highlight the importance of social issues. Whether promoting a social cause or a consumer product, workers in marketing and sales use ratios to convey information to influence thinking and actions.
Every Block, or unit, of MATH 180 contains engaging, media-rich, multistep simulations, in which students take on the authentic tasks of social media marketers, restaurant managers, medical and educational professionals, and many other engaging, high-interest careers. Students get to experience the mathematical concepts as concrete, representational, and intensely purposeful, before they are asked to perceive them as abstract and algorithmic.
MATH 180 offers more than 50 career-themed badges, earned by students for demonstrating persistence, sustained focus, and accuracy in their mathematical practice. As students put forth effort and progress through the software, they “unlock” a wide variety of potential futures, making an explicit connection between current academic performance and college and career success.
Students who feel a sense of ownership and independence in the learning process demonstrate greater effort, motivation, and engagement with mathematics. However, many struggling students have few chances to make meaningful, autonomous decisions in their daily educational environments. Providing opportunities for agency and personal choice allows students to develop a positive affect and a growth mindset toward mathematics.
Research and Expert Opinion
How MATH 180 Delivers
MATH 180 provides opportunities for students to act autonomously and with agency in every area of instruction. At the onset of each session, students choose which zone of instruction they’d like to work in first. Throughout the Learn Zone, students have the opportunity to attempt an accelerated Fast Track if and when they feel confident and competent to meet the challenge. In the Success Zone, the traditional summative assessment is imbued with agency; students choose which questions and question types to tackle in order to earn points and move forward.
In the Brain Arcade, students can choose among the 15 different math games that improve procedural fluency and strategic thinking. Games are proven to be risk-free environments that destigmatize failure, demarcate progress, and reward persistence, and Brain Arcade games encourage students to pursue their personal mathematical goals with a sense of independence and agency.
Additionally, MATH 180 arms students with multiple strategies for tackling routine, nonroutine, and contextualized problems, and then allows students to choose among multiple solution paths. While working through a given problem, students receive immediate feedback that asks them to identify their own mistakes and correct them in real time; this provides them with ownership of the learning process, allowing them to internalize their loci of control and understand that their mistakes are learning opportunities, rather than personal failures. Students are also granted the agency to choose from a variety of scaffolding resources: instructional and Anchor Videos, worked examples, the math glossary, and a library of mTools, virtual manipulatives designed to foster deep concrete and representational understanding.
Students who perceive themselves as competent and capable of success are motivated to work persistently toward their learning goals. Conversely, students with a personal history of failure in mathematics, and especially those with a fixed mindset, may perceive themselves as incompetent and unable to improve. MATH 180 provides a safe, supportive environment that rewards improvement; students experience success from the beginning, establishing a foundation of self-efficacy and confidence on which to build.
Research and Expert Opinion
How MATH 180 Delivers
In MATH 180, students experience success early and often, allowing them to establish a baseline of confidence and self-efficacy, even if they have never perceived themselves as competent before. The adaptive software in MATH 180 allows each student to move at his or her own pace and repeat topics as necessary without any sense of stigma; as the challenge is gradually increased, students are confident in their ability to tackle upcoming topics.
MATH 180 builds self-efficacy by tracking and reporting student progress every day and across many dimensions, skills, and attitudes. The ThinkTracker shows students that they are progressing with every step of a problem, and the daily Newsfeed, the first thing students see on the software each day, celebrates progress and achievements in every zone of instruction. In the Success Zone, students are assessed not by how many questions they miss, but by how many questions they are willing to take on and answer correctly; each question is rewarded with a variable number of points, tracked by a student-facing progress meter.
MATH 180 employs the best practice of game design to convey success, competence, and progress. Students are rewarded with stars, points, and badges for their accuracy, focus, perseverance, and effort, as well as for hitting performance goals and showing content expertise. Students receive badges more frequently early in the program; these early rewards are designed explicitly for students who may not have received recognition for their mathematical efforts in the past, and serve as concrete symbols of mastery that remind students of their competence and ability to overcome obstacles.
Mathematics instruction does not happen in a vacuum; every student is part of a network of interactions and relationships, and a student’s mindset is strongly affected by his or her peers, teachers, community, and family. MATH 180 provides all the tools to transform the collective mindset of a classroom through class routines, teacher professional learning, and strategies for family engagement.
Research and Expert Opinion
How MATH 180 Delivers
Developed in conjunction with Dr. Carol Dweck’s organization, the first two weeks of MATH 180 are dedicated directly to building a classroom culture in which teachers and students deeply understand the principles, language, and tools to foster a growth mindset. Students complete a Mindset Scan at the beginning of the program to establish a baseline for monitoring their own mindsets. MATH 180 then periodically prompts students to reflect on their own mindsets and the learning strategies they are accumulating throughout the program.
Students in MATH 180 work in a safe, supportive classroom environment which values cooperative progress and growth. For teachers, MATH 180 provides lesson plans and classroom routines designed to foster a classroom culture in which students focus on improving their own skills and mastering the material cooperatively. The Professional Learning Guide and Teaching Guides provide teachers with guidance on how to establish vocabulary and principles of growth mindset and transfer them to their students, as well as models of “smart praise” to use with each lesson, in which students are encouraged for their effort, perseverance, and dedication to improvement. For parents and families, the MATH 180 Family Portal brings the learning home, providing guidance for parents to both understand the power of the growth mindset and cultivate mathematical learning opportunities.
Achieve, Inc. (2010). Comparing the Common Core State Standards and Singapore’s Mathematics Syllabus. Retrieved from http://www.achieve.org/CCSSandSingapore
ACT. (2010). A first look at the Common Core and college and career readiness. Iowa City, IA: Author.
ACT. (2012). The condition of college and career readiness, 2012. Iowa City, IA: Author.
Allen, F. B. (1988). Language and the learning of mathematics. A Paper presented at the National Council of Teachers of Mathematics Annual Meeting, Chicago, IL.
Allsopp, D. H., Kyger, M. M., & Lovin, L. H. (2007). Teaching mathematics meaningfully: Solutions for reaching struggling learners. Baltimore, MD: Paul H. Brookes Publishing Co.
Ames, C. (1992). Classrooms: Goals, structures, and student motivation. Journal of Educational Psychology, 84, 261–271.
Anderson-Inman, L., & Horney, M. (2007). Supported eText: Assistive technology through text transformations. Reading Research Quarterly, 42(1), 153–160.
Andrade, H. (2007, December, 2008, January). Self-assessment through rubrics. Educational Leadership, 65(4), 60–63.
Artzt, A. F., & Armour-Thomas, E. (1999). A cognitive model for examining teachers’ instructional practice in mathematics: A guide for facilitating teacher reflection. Educational Studies in Mathematics, 40(3), 211–235.
Ashcraft, M. H., & Kirk, E. P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology, 130(2), 224–237.
Avalos, M. A. (2006). No two learners are alike: Learners with linguistic and cultural differences. In J. S. Schumm (Ed.), Reading assessment and instruction for all learners (pp. 59–86). New York: Guilford Press.
Ball, D. L. (1988). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1), 40–48.
Ball, D. L. (1990). Breaking with experience in learning to teach mathematics: The role of a preservice methods course. For the Learning of Mathematics, 10(2), 10–16.
Ball, D. L., & Forzani, F. M. (2011, Summer). Building a common core for learning to teach, and connecting professional learning to practice. American Educator, 35(2), 17–21, 38–39.
Ball, D. L., Mundy, J. F., et al. (2005). Reaching for common ground in K–12 mathematics education. Notices of the AMS, 52(9), 1055–1058.
Bandura, A. (1993). Perceived self-efficacy in cognitive development and functioning, Educational Psychologist, 28, 117–148.
Barry, N. H. (2007). Motivating the reluctant student. American Music Teacher, 56(5), 23–27.
Blackburn, B. (2008). Rigor and the Common Core State Standards. Retrieved from http://www.nassp.org/Content/158/rigor_and_CCSS.pdf
Blackwell, L. S., Trzeniewski, K. H., & Dweck, C. S. (2007). Theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention. Child Development, 78(1), 246–263.
Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.
Burnett, P. C., & Mandel, V. (2010). Praise and feedback in the primary classroom: Teachers’ and students’ perspectives. Australian Journal of Educational & Developmental Psychology, 10, 145–154.
Burns, M. (2007). About teaching mathematics: A K–8 resource (3rd edition). Sausalito, CA: Math Solutions.
Carnegie Council on Advancing Adolescent Literacy. (2010). Time to act: An agenda for advancing adolescent literacy for college and career success. New York: Carnegie Corporation of New York.
Carnevale, A., Smith, N., & Melton, M. (2011). STEM: science, technology, engineering, and mathematics. Washington, DC: Georgetown University: Center on Education and the Workforce.
Chapin, S. H., O’Connor, C. O., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn. Sausalito, CA: Math Solutions.
Cobb, P., Jaworski, B., & Presmeg, N. (1996). Emergent and sociocultural views of mathematical activity. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 3–19). Mahwah, NJ: Lawrence Erlbaum.
Cobb, P., & McClain, K. (2001). Supporting teachers’ learning. In F. L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 207–231). Netherlands: Kluwer Academic Publishers.
Cohen, D. K., & Ball, D. L. (1990). Relations between policy and practice: A commentary. Educational Evaluation and Policy Analysis, 12(3), 331–338.
College Board. (2011, September 14). Forty-three percent of 2011 college-bound seniors met SAT college and career readiness benchmark [Press release]. Retrieved from http://press.collegeboard.org/releases/2011/43-percent-2011-college-bound-seniors-met-sat-college-and-careerreadiness-benchmark
Common Core State Standards Initiative. (2010). Common Core State Standards initiative for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.
Confrey, J. & Krupa, E. (Eds.). (2010). Summary report of a Center for the Study of Mathematics curriculum conference: Curriculum design, development, and implementation in an era of Common Core State Standards. Retrieved from http://mathcurriculumcenter.org/conferences.php
Conley, D. T. (2007). Redefining college readiness, Volume 3. Eugene, OR: Educational Policy Improvement Center.
Conley, D. T. (2011). Building on the Common Core. What Students Need to Learn, 68(6), 16–20.
Conley, D. T., Drummond, K. V., de Gonzalez, A., Rooseboom, J., & Stout, O. (2011). Reaching the goal: The applicability and importance of the Common Core State Standards to college and career readiness. Retrieved from https://www.epiconline.org/files/pdf/ReachingtheGoal-FullReport.pdf
Cooney, T. J. (2001). Considering the paradoxes, perils, and purposes of conceptualizing teacher development. In F. L. Lin & T. J. Cooney (Eds.). Making sense of mathematics teacher education (pp. 9–31). Netherlands: Kluwer Academic Publishers.
Council of Chief State School Officers. (2012). Common Core State Standards for mathematics: Shifts and implications for mathematics instruction [PowerPoint slides]. Retrieved from http://www.ccsso.org/Resources/Digital_Resources/CCSS_for_Math_Shifts_and_Implications_for_Instruction.html
Cunha, F., & Heckman, J. J. (2008). Formulating, identifying and estimating the technology of cognitive and noncognitive skill formation. The Journal of Human Resources, 43(4), 738–782.
Distel, R. F. (2001). Evaluation Series LS Class—netschools.net. PLATO Learning, Inc. South Bloomington. Retrieved from http://www.netschools.net/medi/E/Eastern%20High%20School.pdf
Dockterman, D. (1984). The role of educational computer games in informal learning environments (pp. 12–14). Qualifying Paper, Harvard Graduate School of Education.
Duke, N., & Pearson, D. (2002). Effective practices for developing reading comprehension. In A. Farstrup and S. Samuels (Eds.), What research has to say about reading instruction (pp. 205–242). Newark, DE: International Reading Association.
Dukes, C. (2005). Best practices for integrating technology into English language instruction. SEIRTEC News Wire, Southeast Initiatives Regional Technology in Education Consortium, 7(1). Retrieved from http://www.seirtec.org/publications/newswire.html
Dweck, C. S. (1986). Motivation processes affecting learning. American Psychologist, 41, 1040–1048.
Dweck, C. S. (2000). Self-theories: Their role in motivation, personality, and development. Philadelphia, PA: Taylor & Francis.
Dweck, C. S., & Leggett, E. L. (1988). A social-cognitive approach to motivation and personality. Psychological Review, 95, 256–273.
Ee, J., & Wong, K. (2002). Understanding and overcoming pupils’ learning difficulties in mathematics. Teaching and Learning, 23(1), 49–58.
Ells, E. S., & Worthington, L.A. (1994). Research synthesis on effective teaching principles and the design of quality tools for educators (Technical Report No. 5). University of Oregon National Center to Improve the Tools of Educators.
Epstein, G., Cook, J., & Dihoff, R.E. (2005). Efficacy of error for the correction of initially incorrect assumptions and of feedback for the affirmation of correct responding: Learning in the classroom. The Psychological Record, 55(3), 401–418.
Ericsson, K. A. (2006). The influence of experience and deliberate practice on the development of superior expert performance. In K. A. Ericsson, N. Charness, P. Feltovich, & R. R. Hoffman (Eds.), Cambridge handbook of expertise and expert performance (pp. 685–706). Cambridge, UK: Cambridge University Press.
Fisher, D., & Ivey, G. (2006). Evaluating the interventions for struggling adolescent readers. Journal of Adolescent & Adult Literacy, 50(3), 180–189.
Forman, E. A. (1996). Learning mathematics as a participation in classroom practice: Implications of sociocultural theory for educational reform. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 115–130). Mahwah, NJ: Lawrence Erlbaum.
Forster, M. (2009). Informative assessment: understanding and guiding learning. ACER Research Conference.
Frederickson, N., & Cline, T. (2002). Special educational needs, inclusion, and diversity: A textbook. (Chapter 12) Buckingham, United Kingdom: Open University Press.
Fuchs, L., Fuchs, D., Hamlett, C., Phillips, N., Karns, K., & Dutka, S. (1997). Enhancing students’ behavior during peer-mediated instruction. Journal for Research in Mathematics Education, 27(4), 403–434.
Garrison, L., & Mora, J. L. (1999). Adapting mathematics instruction for English language learners: The language-concept connections. In L. Ortiz-Franco, N. G. Hernandez, & Y. De la Cruz (Eds.), Changing the faces of mathematics: perspectives on Latinos (pp. 35–48.)
Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009–4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/
Gersten, R., Chard, D., Jayanthi, M., Baker, S., Morphy, P., & Flojo, J. (2008). Mathematics instruction for students with learning disabilities or difficulty learning mathematics: A synthesis of the intervention research. Portsmouth, NH: RMC Research Corporation, Center on Instruction.
Ginsburg, A., Leinwand, S., Anstrom, T., & Pollock, E. (2005). What the United States can learn from Singapore’s world-class mathematics system (and what Singapore can learn from the United States). Washington, DC: American Institutes for Research.
Grant, T. J., & Kline, K. (2004). Embracing the complexity of practice as a site for inquiry. In R. Rubenstein & G. Bright (Eds.), Perspectives on the teaching of mathematics, 66th yearbook of the National Council of Teachers of Mathematics (pp. 195–206). Reston, VA: National Council of Teachers of Mathematics.
Hall, T. E., Hughes, C. A., & Filbert, M. (2000). Computer-assisted instruction in reading for students with learning disabilities: A research synthesis. Education and Treatment of Children, 23(2), 173–193.
Hasselbring, T. S., & Glaser, C. (2000). Use of computer technology to help students with special needs. Future of Children: Children and Computer Technology, 10(2), 102–122.
Hasselbring, T. S., & Goin, L. (2004). Literacy instruction for older struggling readers: What is the role of technology? Reading and Writing Quarterly, 20(2), 123–144.
Hasselbring, T. S., Lewis, P., & Bausch, M. E. (2005). Assessing students with disabilities: Moving assessment forward through universal design. InSight, 5, 1–15.
Hasselbring, T. S., Lott, A. C., and Zydney, J. M. (2006). Technology-supported math instruction for students with disabilities: Two decades of research and development. Washington, DC: CITEd, Center for Implementing Technology in Education.
Henderson, A. T., & Mapp., K. L. (2002). A new wave of evidence: The impact of school, family, and community connections on student achievement. Austin, TX: National Center for Family and Community Connections with Schools, Southwest Educational Development Laboratory.
Heo, Y. (2007). The impact of multimedia anchored instruction on the motivation to learn of students with and without learning disabilities placed in inclusive middle school language arts classes (Doctoral dissertation, University of Texas, 2007). Dissertations Abstracts International. 6812A, 5031. Retrieved from https://www.lib.utexas.edu/etd/d/2007/heoy96433/heoy96433.pdf
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
Hiebert, J., & Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Charlotte, NC: Information Age Publishing.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Erlbaum.
Hoffer, T., Venkataraman, L., Hedberg, E. C., & Shagle, S. (2007). Final report on the national survey of algebra teachers for the National Math Panel. Chicago, IL: National Opinion Research Center at the University of Chicago.
Howard, K. L. (2004). Universal design for learning: Meeting the needs of all students. Learning and Leading With Technology, 31, 26–29.
Hrabowski, F. A., Maton, K. I., & Greif, G. L. (1998). Beating the odds: Raising academically successful African American males. New York, NY: Oxford University Press.
Hulleman, C. S., & Harackiewicz, J. M. (2009). Promoting interest and performance in high school science classes, Science, 326, 1410–1412.
Hupert, N., and Heinze, J. (2006). Results in the palms of their hands: Using handheld computers for data-driven decision making in the classroom. In M. van‘t Hooft & K. Swan (Eds.) Ubiquitous computing in education: Invisible technology, visible impact (pp. 211–229). Mahwah, NJ: Lawrence Erlbaum Associates.
Jitendra, A. K., & Star, J. R. (2011). Meeting the needs of students with learning disabilities in inclusive mathematics classrooms: The role of schema-based instruction on mathematical problem-solving. Theory Into Practice, 50(1), 12–19.
Jones, M. H., Audley-Piotrowski, S. R., & Kiefer, S. M. (2012). Relationships among adolescents’ perceptions of friends’ behaviors, academic self-concept, and math performance. Journal of Educational Psychology, 104(1), 19–31.
Kamii, C. (2000). Young children reinvent arithmetic: Implications of Piaget’s theory (2nd ed.). New York, NY: Teachers College Press.
Khisty, L. L., & Chval, K. B. (2002). Pedagogic discourse and equity in mathematics education: When teachers’ talk matters. Mathematics Education Research Journal, 14(3), 4–18.
Khisty, L. L., & Morales, Jr., H. (2004). Discourse matters: Equity, access, and Latinos learning mathematics. Retrieved from http://www.icme-organisers.dk/khisty.doc
Kloosterman, P. (2010). Mathematics skills of 17-year-olds in the United States: 1978 to 2004. Journal for Research in Mathematics Education, 41(1), 20–51.
Kortering, L. J., McClannon, T. W., & Braziel, P. M. (2008). Universal design for learning: A look at what algebra and biology students with and without high incidence conditions are saying. Remedial and Special Education, 29(6), 352–363.
Lager, C. A. (2007). Types of mathematics-language reading interactions that unnecessarily hinder algebra learning and assessment. Reading Psychology, 27(2), 165–204.
Lan, W., & Repman, J. (1995). The effect of social learning context and modeling on persistence and dynamism in academic activities. Journal of Experimental Education, 64(1), 53–67.
Lappan, G. (1997). The challenges of implementation: Supporting teachers. American Journal of Education, 106(1), 207–239.
Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, United Kingdom: Cambridge University Press.
Loveless, T. (2011). The 2010 Brown Center report on American education: How well are American students learning? With sections on international tests, who’s winning the real race to the top, and NAEP and the Common Core State Standards. Volume II, Number 5. Brown Center on Education Policy at Brookings.
Math Solutions. (2011). Getting started: Mathematics learning with classroom discussions. Retrieved from http://www.mathsolutions.com/documents/9781935099123_chpt1_1_14.pdf
Mayer, R. E., & Moreno, R. (2003). Nine ways to reduce cognitive load in multimedia learning. Educational Psychologist, 38(1), 43–52.
McGonigal, J. (2011). Reality is broken: Why games make us better and how they can change the world. London, United Kingdom: Jonathan Cape.
Ministry of Education Singapore. (2006). Secondary Mathematics Syllabuses. Retrieved from http://www.moe.gov.sg/education/syllabuses/sciences/files/maths-secondary.pdf
Moschkovich, J. (2012, January). Mathematics, the Common Core and language: Recommendations for mathematics instruction for ELs aligned with the Common Core. Paper presented at the Understanding Language Conference at Stanford University, Stanford, CA.
Murray, A. (2011). Montessori elementary philosophy reflects current motivation theories. Montessori Life, 23(1), 22–33.
National Center on Response to Intervention. (2010). Essential Components of RTI—A closer look at response to intervention. Washington, DC: U.S. Department of Education, Office of Special Education Programs, National Center on Response to Intervention.
National Council of Supervisors of Mathematics. (2010). Improving student achievement in mathematics by promoting positive self-beliefs. The National Council of Supervisors of Mathematics Improving Student Achievement Series. No. 7.
National Council of Supervisors of Mathematics. (2013a). Improving student achievement by infusing highly effective instructional strategies into RtI Tier 1 instruction. The National Council of Supervisors of Mathematics Improving Student Achievement Series No. 12.
National Council of Supervisors of Mathematics. (2013b). Improving student achievement in mathematics by using manipulatives with classroom instruction. The National Council of Supervisors of Mathematics Improving Student Achievement Series No. 11.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2007). Principles and standards of school mathematics. Reston, VA: Author.
National Joint Committee on Learning Disabilities. (2008). Adolescent literacy and older students with learning disabilities: A report from the National Joint Committee on Learning Disabilities. Retrieved from http://www.ldonline.org/njcld
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
National Research Council. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.
National Research Council. (2001). Adding it up: Helping children learn mathematics, J. Kilpatrick, J. Swafford, & B. Findell (Eds.) Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
National Research Council. (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academy Press.
Pashler, H., Bain, P., Bottge, B., Graesser, A., Koedinger, K., McDaniel, M., & Metcalfe, J. (2007). Organizing instruction and study to improve student learning (NCER 2007–2004). Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ncer.ed.gov
Pintrich, P., & De Groot, E. (1990). Motivational and self-regulated learning components of classroom academic performance. Journal of Educational Psychology, 82(1), 33–40.
Pressley, M., Gaskins, I. W., Solic, K., & Collins, S. (2006). A portrait of a benchmark school: How a school produces high achievement in students who previously failed. Journal of Educational Psychology, 98(2), 282–306.
Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35, 352–388.
Rose, D. H., & Meyer, A. (2000). Universal design for learning. Journal of Education Technology, 15(1), 67–70
Rose, D. H., Meyer, A., Strangman, N., & Rappolt, G. (2002). Teaching every student in the digital age: Universal design for learning. Alexandria, VA: Association for Supervision and Curriculum Development.
Rose, H., & Betts, J. R. (2001). Math matters: The links between high school curriculum, college graduation, and earnings. San Francisco, CA: Public Policy Institute of California.
Russell, S. J. (2000). Developing computational fluency with whole numbers. Teaching Children Mathematics, 7, 154–158.
Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78.
Sagor, R. (2003). Motivating students and teachers in an era of standards. Alexandria, VA: Association for Supervision and Curriculum Development.
Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent curriculum, American Educator, 26(2), pp. 1–18.
Schoenfeld, A. H. (2007). What is mathematical proficiency and how can it be assessed? In A. H. Schoenfeld (Ed.). Assessing mathematical proficiency (Mathematical Sciences Research Institute Publications Number 53) (pp. 59–73). Cambridge, United Kingdom: Cambridge University Press.
Schweinle, A., Meyer, D. K. & Turner, J. C. (2006). Striking the right balance: Students’ motivation and affect in elementary mathematics. Journal of Educational Research, 99(5), 271–293.
Seifert, T. L. (2004). Understanding student motivation. Educational Research, 46(2), 137–149.
Seifert, T., & O’Keefe, B. (2001). The relationship of work avoidance and learning goals to perceived competence, externality, and meaning. British Journal of Educational Psychology, 71, 81–92.
Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., ...Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE #2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/pdf/practice_guides/fractions_pg_093010.pdf
Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23, 691–697.
Smith, M. (2010). Best practices for next-generation assessments. Chapel Hill, NC: MetaMetrics, Inc.
Sowder, J. (2007). The mathematical education and development of teachers. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 157–223). Reston, VA: National Council of Teachers of Mathematics & Charlotte, NC: Information Age Publishing.
Sousa, D. A., & Tomlinson, C. A. (2011). Differentiation and the brain: How neuroscience supports the learner-friendly classroom. Bloomington, IN: Solution Tree Press.
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.
Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132–135.
Stecker, P., Fuchs, L., & Fuchs, D. (2005). Using curriculum-based measurement to improve student achievement: Review of research. Psychology in the Schools, 42(8), 795–819.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press.
Students at the Center. (2012). Motivation, engagement, and student voice. Retrieved from http://www.studentsatthecenter.org/sites/scl.dl-dev.com/files/Motivation%20Engagement%20Student%20Voice_0.pdf
TeachingWorks. (2013). Responsible teaching. In TeachingWorks. Retrieved from http://www.teachingworks.org/w...
Thaler, R., Sunstein, C., and Balz, J. (2010). Choice architecture. Retrieved from http://ssrn.com/abstract=1583509
Torgesen, J. K. (2002). Lessons learned from intervention research in reading: A way to go before we rest. In R. Stainthorpe (Ed.), Learning and teaching reading (pp. 89–103). British Journal of Educational Psychology Monograph. London, United Kingdom: British Psychological Society.
U.S. Department of Education Institute of Education Sciences; National Center for Education Statistics, National Department of Education Progress (NAEP), 2015 Mathematics Assessment.
Usher, A., & Kober, N. (2012). Student motivation: An overlooked piece of school reform. Summary, Center on Education Policy.
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). New York: Pearson Education, Inc.
Vaughn, S., & Denton, C. (2008). Tier 2: The role of intervention. In D. Fuchs, L. Fuchs, & S. Vaughn. (Eds.), WIS Prevention—response to intervention: A framework for reading educators, 2008 (pp. 51–69). Newark, DE: International Reading Association. Retrieved from http://books.google.com/
Vaughn, S., and Roberts, G. (2007). Secondary interventions in reading: Providing additional instruction for students at risk. Teaching Exceptional Children, 39(5), 40–46.
Vygotsky, L. S. (1978). Mind in society: The development of the higher psychological processes. Cambridge, MA: The Harvard University Press.
Weiner, B. (1984). Principles for a theory of student motivation and their application with an attributional framework, Research on Motivation in Education: Student Motivation, 6, 15–38.
Weiner, B. (1985). An attributional theory of achievement, motivation, and emotion. Psychological Review, 92, 548–573.
Wertsch, J. V. (1991). Voices of the mind: A sociocultural approach to mediated action. Cambridge, MA: Harvard University Press.
Wong, K. (2004, July). Using multi-modal think-board to teach mathematics. A presentation for TSG 14: Innovative approaches to the teacher of mathematics, ICME-10, DTU: Technical University of Denmark, Copenhagen, Denmark.
Zaslavsky, O. (2005). Seizing the opportunity to create uncertainty in learning mathematics. Educational Studies in Mathematics, 60, 297–321.