As described in Sections 1 and 2 of this framework, while robust standards and high-quality curriculum materials (HQCMs) are essential to providing all students the opportunities to learn what they need for success in college and a career of their choosing, high-quality instruction is also needed. Standards define what students should know and be able to do. HQCMs that are aligned to the standards provide educators with a roadmap and tools for how students can acquire that knowledge and skill. It is high-quality instruction that makes the curriculum come alive for students. High-quality instruction gives all students access and opportunity for acquiring the knowledge and skills defined by the standards with a culturally responsive and sustaining approach. “When teachers have great instructional materials, they can focus their time, energy, and creativity on meeting the diverse needs of students and helping them all learn and grow.” (Instruction Partners Curriculum Support Guide Executive Summary, p. 2) Executive-Summary-1.pdf (curriculumsupport.org)
The process of translating a high-quality curriculum into high-quality instruction involves much more than opening a box and diving in. This is because no single set of materials can be a perfect match for the needs of all the students that educators will be responsible for teaching. Therefore, educators must intentionally plan an implementation strategy in order to have the ability to translate HQCMs into high-quality instruction. Some key features to attend to include:
- Set systemic goals for curriculum implementation and establish a plan to monitor progress,
- Determine expectations for educator use of HQCMs,
- Craft meaningful opportunities for curriculum-based embedded professional learning,
- Factor in the need for collaborative planning and coaching (Instruction Partners Curriculum Support Guide Executive Summary, page 4) Executive-Summary-1.pdf (curriculumsupport.org), and
- Develop systems for collaboratively aligning HQCMs to the WIDA ELD Standards.
Thus, with a coherent system in place to support curriculum use, teachers will be well-positioned to attend to the nuances of their methods and make learning relevant and engaging for the diverse interests and needs of their students.
Given the above, what constitutes high-quality instruction? In short, high-quality instruction is defined by the practices that research and evidence have demonstrated over time as the most effective in supporting student learning. In other words, when teaching is high quality it embodies what the field of education has found to work the best. Therefore, this section provides a synthesis of research- and evidence-based practices that RIDE believes characterizes high-quality instruction in mathematics. This section begins by describing the high-quality instructional practices that apply across content areas and grades with details and examples that explain what these instructional practices look like in mathematics, and later explains other specific instructional practices that are at the core of high-quality instruction in mathematics. The instructional practices articulated in this section are aligned with and guided by best practices for multilingual learners and for differently-abled students, and specific information and resources are provided about how to support all students in their learning while drawing on their individual strengths. These instructional practices also contribute to a multi-tiered system of support (MTSS) in which all students have equitable access to strong, effective core instruction that supports their academic, behavioral, and social-emotional outcomes. This section on instruction ends with a set of resources and tools that can facilitate high-quality instruction and professional learning about high-quality instruction, including tools that are relevant across content areas and grade levels and those that are specific to mathematics.
In reviewing this section, use Part 2 to understand what high-quality instruction should look like for all students in mathematics. Use Part 3 to identify resources that can promote and build high-quality instruction, as well as resources for learning more about how to enact high-quality instruction.
In order to effectively implement high-quality curriculum materials, as well as ensure that all students have equitable opportunities to learn and prosper, it is essential that teachers are familiar with and routinely use instructional practices and methods that are research- and evidence-based. Below are instructional practices that are essential to effective teaching and learning in mathematics. The first set of instructional practices are those common across all disciplines and curriculum frameworks. These are followed by instructional practices specific to mathematics. For additional guidance, there are also descriptions and references to instructional practices that support specific student groups, such as multilingual learners and differently-abled students.
High-Quality Instruction in All Disciplines

Below are five high-quality instructional practices that RIDE has identified as essential to the effective implementation of standards and high-quality curriculum in all content areas (See figure to the right). These practices are emphasized across all the curriculum frameworks and are supported by the design of the HQCMs. They also strongly align with the instructional framework for multilingual learners, the high-leverage practices (HLPs) for students with disabilities, and RIDE’s teacher evaluation system. Below is a brief description of each practice and what it looks like in mathematics.
Assets-Based Stance
Teachers routinely leverage students’ strengths and assets by activating prior knowledge and connecting new learning to the culturally and linguistically diverse experiences of students while also respecting individual differences.
What This Looks Like in Mathematics
Taking an assets-based stance in mathematics instruction requires an educator to understand and acknowledge that all of their students can be competent learners and doers of mathematics. This begins with the recognition that each student brings with them a wealth of relevant knowledge that can serve as a bridge in refining and expanding their mathematical thinking and understanding. All students, regardless of their age, enter a classroom having experienced diverse quantitative situations, both in and outside of school, from which they have learned a collection of things about number (National Research Council, 2001). An assets-based stance challenges teachers to view their students’ unique cultural and linguistic experiences as intellectual resources (Celedón-Pattichis et al., 2018).
Effective instruction makes use of students’ unique quantitative understandings and cultural experiences by creating scaffolds between their prior knowledge and new knowledge. By providing students with the opportunity to engage with rigorous mathematical tasks, allowing students to share their mathematical ideas and thinking, acknowledging there are different ways to do mathematics, and connecting to prior knowledge, teachers empower students to be creative and powerful mathematical thinkers. All students should have routine and supported mathematical experiences that link to their identities and foster an understanding of both higher order concepts and foundational skills. If “teachers treat students as brilliant mathematical thinkers and expect students to demonstrate such mathematical brilliance in the classroom space (Jett, 2013),” they increase the likelihood of mathematical competence and empower students to continue to grow in their mathematical understanding (Gay, 2010; Ladson-Billings, 2009).
What this looks like in relation to Universal Design for Learning (UDL)
Differentiated core instruction based in UDL provides access and equity for each student providing multiple options for learning and expression without changing what is being taught. Differentiation is proactive with the goal of adjusting the how, based on understanding learner assets and needs, so students may achieve maximum academic growth. High-quality curriculum and instruction implemented through UDL and differentiation support access to grade-level curriculum as part of Tier 1 of a multi-tiered system of supports (MTSS).
What this looks like for Multilingual Learners (MLLs)
Educators with MLLs in their class will advance student learning by drawing on MLLs’ home languages, lived experiences, and world views. Although RIDE encourages student use of academic registers, it is important that educators and administrators maintain an asset-oriented stance in facilitating academic discourse and student understanding of standard English conventions, particularly when working with learners from minoritized groups. Educational agencies can play a role in sustaining the linguistic traditions of their students. Thus, classroom discourse, when done well, will reflect the discourse practices of local communities—capturing the rich ways families actually use language, rather than making prescriptive judgements about how students and their families ought to talk.
What this looks like for Differently-Abled Students (DAS)
Implementation of HLP 3: Collaborate with Families to Support Student Learning and Secure Needed Services promotes an assets-based stance for students with IEPs. Effective collaboration between educators and families is built on positive interactions in which families and students are treated with dignity. Educators affirm student strengths and honor cultural diversity maintaining open lines of communication with phone calls or other media to build on students’ assets and discuss supports or resources. Trust is established with communication for a variety of purposes and not just for formal reasons such as report cards, discipline reports, or parent conferences.
To Learn More
Below is a variety of links to resources to learn more about this practice.
Clear Learning Goals
Teachers routinely use a variety of strategies to ensure that students understand the following:
- What they are learning (and what proficient work looks like),
- Why they are learning it (how it connects to what their own learning goals, what they have already learned and what they will learn), and
- How they will know when they have learned it.
What This Looks Like in Mathematics
Establishing clear goals and expectations linked to the standards, high-quality curriculum, and learning trajectories or progressions serve to guide teachers’ decision-making and focus students’ attention during a lesson. Moreover, they assist the student in monitoring their own progress in learning the mathematics (NCTM, 2014). Studies indicate students perform at higher levels when expectations for learning are clear (Haystead & Marzano, 2009; Hattie, 2009). Key questions for both teachers and students to consider with respect to goals are:
- What mathematics content is being taught and what mathematical practices will support the learning?
- Why is the mathematics important — both for school and for life?
- How does the mathematics relate to what has already been learned — both in school and from lived experiences?
- Where are these mathematical ideas going — how will they help with future understandings? (NCTM, 2014)
Part of clarifying learning goals for students is to communicate criteria for proficiency and to model for students a process to self-assess their progress in meeting the criteria. With guidance and practice, students can compare their own understanding and work to the established criteria in order to identify their successes, be mindful of areas of improvement, and engage in strategies to grow their understanding to a level of proficiency (New South Wales Government – Education, 2021).
What this looks like for Multilingual Learners (MLLs)
For educators with one or more active MLLs on their roster, clear learning goals for MLLs will consist of explicit language goals to guide instruction in mathematics. Educators will model effective use of disciplinary academic vocabulary and syntax, creating opportunities every day for explicit disciplinary language development, aligned to the WIDA ELD Standards.
What this looks like for Differently-Abled Students (DAS)
HLP 14, Teach Cognitive and Metacognitive Strategies to Support Learning and Independence, supports the high quality instruction practice of Clear Learning Goals. Through task analysis, educators can support DAS by determining the steps they need to take to accomplish goals, then create and teach a procedure to help the student meet the goals. The educator uses explicit instruction (HLP 16) to teach the student self-regulation strategies such as self-monitoring, self-talk, goal-setting, etc. Clear, step-by-step modeling with ample opportunities for practice and prompt feedback coupled with positive reinforcement (HLP 22) in different contexts over time ensure that DAS become fluent users of metacognitive strategies toward understanding and achieving learning goals.
To Learn More
Below is a variety of links to resources to learn more about this practice.
Student-Centered Engagement
Teachers routinely use techniques that are student-centered and foster high levels of engagement through individual and collaborative sense-making activities that promote practice, application in increasingly sophisticated settings and contexts, and metacognitive reflection.
What This Looks Like in Mathematics
Today’s mathematics classroom should be a vibrant space in which students take charge of their own learning, think deeply, communicate and collaborate with their peers, and persist to solve meaningful problems. The Standards for Mathematical Practice (MPs) clearly articulate these behaviors and others that students should engage in while learning mathematics from kindergarten through high school. Quality mathematics instruction grounded in the practices facilitates dynamic learning experiences for students, discouraging passivity and encouraging student agency.
Learning is optimized when students engage in the mathematical practices while solving rigorous tasks requiring reasoning (MP2) and higher order thinking skills. Teachers should select high-quality tasks that offer multiple points of entry, allow for a wide array of solution strategies, and connect to students’ current level of understanding, prior knowledge, and lived experiences. When students are able to gain access to these tasks, the mathematics at hand becomes meaningful. The potential to expand or solidify a student’s current level of understanding is increased. Additionally, intentional and repeated exposure to higher level tasks provides students with insight into how long and hard (MP1) they may need to work at a task and an implicit understanding of how mathematics works (NCTM, 2014).
Engagement with rigorous mathematical tasks requires deep thinking and reasoning on the part of the student. As educators, it is necessary to overcome the desire to “rescue” students when they become frustrated with a task. Avoid the inclination to stop at the first sign of frustration and walk students through tasks step by step (Reinhart, 2000). Instead, during the planning process, consider where frustration and missteps may occur and intentionally design strategies to support students’ sense-making and engagement while maintaining the rigor of a task (NCTM, 2014). Encourage students to take responsibility for their learning by having them ask questions of both you and their peers (MP3). Help them discover entry points into a problem by probing their thinking, thus giving them license to make mistakes and connections to content and relationships they already understand. Foster the mindset that making sense of mathematics is not a product of innate ability, but rather one of hard work and perseverance (Dweck, 2008). By allowing students to engage in productive struggle, they become empowered mathematical learners and deeper mathmatical thinkers.
What this looks like for Multilingual Learners (MLLs)
Educators with MLLs in their class can promote student-centered engagement by providing scaffolded opportunities for students to build conceptual understanding and fluency with core disciplinary skills, appropriate to their English language proficiency levels.
What this looks like for Differently-Abled Students (DAS)
Student-centered engagement is maximized when educators implement HLP 7, Establish a Consistent, Organized, and Respectful Learning Environment. DAS benefit from educators who explicitly teach consistent classroom procedures and expected behaviors while considering student input. Viewing behavior as communication, reteaching expectations and procedures across different school environments, and helping students understand the rationale for the rules and procedures as part of HLP 7 implementation will enhance student-centered engagement for DAS. In any content area, this may mean providing additional opportunities to for DAS to learn and practice routines that some peers might already have mastered. Some IEPs may call for self-monitoring checklists and visual schedules to support students in active participation in learning activities. Individual DAS will need specific supports unique to their learning profiles. Educators can implement HLP 7 in conjunction with HLP 18, Use Strategies to Promote Active Student Engagement, and HLP 8, Provide Positive and Constructive Feedback to Guide Students’ Learning and Behavior, for individualized student supports.
To Learn More
Below is a variety of links to resources to learn more about this practice.
Resource |
Description |
High-Leverage Practice (HLP) Leadership Guides from the Council for Exceptional Children |
Leadership Guides for the following HLPs:
#7: Establish a Consistent, Organized, and Respectful Learning Environment
#8: Provide Positive and Constructive Feedback to Guide Students’ Learning and Behavior
#17: Use Flexible Groupings
#18: Use Strategies to Promote Active Student Engagement
#21: Teach Students to Maintain and Generalize New Learning Across Time and Settings
|
Fundamental Skill Sheets Videos |
Video playlist from the Iris Center: Choice Making, Proximity Control, Wait Time, Behavior Specific Praise
Note: Video 6 focuses on proximity control in a high school mathematics class. The mnemonic used in the lesson is employed as a support after initial instruction that focused on developing conceptual understanding of the trigonometric functions and their relationship to right triangles.
|
High-Leverage Practices Video: Use Strategies to Promote Active Student Engagement |
Video highlighting HLP #18 which focuses on strategies to promote active student engagement |
HLP #21: Teach Students to Maintain and Generalize New Learning Across Time and Settings |
Leadership Guide for HLP #21: Teach Students to Maintain and Generalize New Learning Across Time and Settings |
Including Voice in Education: Addressing Equity Through Student and Family Voice in Classroom Learning |
Infographic on incorporating student voice and/or family voice into student learning, a promising strategy for teachers striving to foster culturally responsive and sustaining classrooms to enhance education access, opportunity, and success for students who are historically marginalized within the pre-kindergarten to grade 12 education systems |
SEL for Self-Management |
RIDE resources on Social Emotional Learning Indicators for Self-Management |
SEL for Social Awareness |
RIDE resources on Social Emotional Learning Indicators for Social Awareness |
WWC | Organizing Instruction and Study to Improve Student Learning (ed.gov) |
Guide including a set of concrete actions relating to the use of instructional and study time that are applicable to subjects that demand a great deal of content learning, including social studies, science, and mathematics. The guide was developed with some of the most important principles to emerge from research on learning and memory in mind.
- Space learning over time.
- Interleave worked example solutions with problem-solving exercises.
- Combine graphics with verbal descriptions.
- Connect and integrate abstract and concrete representations of concepts.
- Use quizzing to promote learning. Use quizzes to re-expose students to key content.
- Ask deep explanatory questions.
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Academic Discourse
Teachers routinely facilitate and encourage student use of academic discourse through effective and purposeful questioning and discussion techniques that foster rich peer-to-peer interactions and the integration of discipline-specific language into all aspects of learning.
What This Looks Like in Mathematics
The Rhode Island Professional Teaching Standards (RIDE, 2007, October) establish the expectation that teachers create a learning environment that is safe, secure, nurturing, and supportive of active engagement by all students. In this environment, mutual respect and intellectual risk-taking are modeled by the teacher and expected of all. Students are given opportunities to work both independently and collaboratively, and to take responsibility for their own learning. When these conditions are present in the mathematics classroom, the situation is ideal for the teacher to facilitate meaningful mathematical discourse.
The third Standard for Mathematical Practice calls for students to “construct viable arguments and critique the reasoning of others” (NGA Center & CCSSO, 2010). Classrooms that strategically promote student-to-student discourse in addition to student-to-teacher discourse provide the environment in which students can develop and nurture this skill. When planned effectively, mathematical discussions in the classroom promote an exchange of ideas that can clarify understanding, develop the use of academic language, enable students to see the mathematics from a variety of vantage points, and advance the conceptual understanding of all students. When students feel safe to share their ideas, their understandings, and their misunderstandings, they are better positioned to take charge of their own learning and to support the understanding of their peers.
It is important for the teacher, as facilitator, to set the stage for meaningful discourse and to have a variety of routines to stimulate participation and support communication for all. One routine suggested by Smith and Stein (2011) outlines these considerations:
- Anticipate student responses prior to a lesson.
- Monitor students’ work and task engagement.
- Select specific students to present their mathematical work.
- Sequence students’ responses in a specific order during discussion.
- Connect the different student responses to one another and to the key mathematical ideas of the lesson.
By following these key practices, discourse will remain focused on the goal of the lesson, build upon different approaches to addressing the problem of the lesson, and maximize access to understanding the content of the lesson for all students.
What this looks like in relation to Social Emotional Learning
The five core competencies of Rhode Island’s Social Emotional Learning standards and indicators support academic discourse across the content areas. Learners must engage effectively in a range of collaborative discussions with diverse partners, building on each other’s’ ideas and expressing their own clearly.
- Self-Awareness: Identifying one’s strengths and weaknesses while working within a group, staying motivated and engaged throughout the work.
- Self-Management: Controlling one’s emotions, responding calmly to comments, questions, and nonverbal communication.
- Social-Awareness: Understanding others’ perspectives and cultures, compromising with peers when the situation calls for it, accepting feedback from peers and teachers, listening to the opinions of others and taking them into consideration.
- Relationship Skills: Expressing one’s perspective clearly, following agreed upon rules of the group and carrying out assigned role(s), gaining peers’ attention in an appropriate manner, asking questions of group members, limiting the amount of information shared with others, and actively listening to peers when they speak.
- Responsible Decision Making: Coming to the group prepared, demonstrating independence with work tasks, dividing labor to achieve the overall group goal efficiently.
Social and emotional skills are implicitly embedded in the content standards, and students must learn many social and emotional competencies to successfully progress academically. Social Emotional Learning skills are instrumental for each student and are crucial for differently-abled students.
What this looks like for Multilingual Learners (MLLs)
Though beneficial for all students, academic discourse is especially important for MLLs because engaging in authentic interaction with discipline-specific oral language facilitates MLLs’ overall development of English language proficiency. In RIDE’s High-Quality Instructional Framework for MLLs to Thrive, academic discourse is defined as a sustained spoken interaction between two or more students in which knowledge is shared using the conventions of particular genres and disciplines.
What this looks like for Differently-Abled Students (DAS)
Educators plan mixed-ability small groups to increase DAS student engagement in academic discourse through a variety of cooperative learning structures consistent with HLP 17, Use Flexible Groupings. Effective groupings are monitored for learning and student interactions to meet various academic, behavioral, and interpersonal instructional objectives. DAS may require varied group sizes and types based upon specific IEP goals and accommodations. A student engaging in intensive instruction of a particular math or reading skill may do so in a supplemental homogenous group of only 2-3 peers while also having regular opportunities to engage in heterogeneous collaborative groups during core instruction with scaffolded supports.
To Learn More
Below is a variety of links to resources to learn more about this practice.
Formative Assessment
Teachers routinely use qualitative and quantitative assessment data (including student self-assessments) to analyze their teaching and student learning in order to provide timely and useful feedback to students and make necessary adjustments (e.g., adding or removing scaffolding and/or assistive technologies, identifying the need to provide intensive instruction) that improve student outcomes.
What This Looks Like in Mathematics
With goals and expectations established and clearly communicated to students, teachers should continually monitor and assess their students’ understanding. Using a variety of assessment strategies, both formal and informal, teachers can use the associated data in a formative way to adjust instruction with the goal of improving student understanding, and not just to assign a grade. While well-designed tests and quizzes are two traditional sources for assessing student understanding, eliciting evidence of student thinking in a formative manner, integrated into instruction, has proven to be most efficacious in nimbly responding to student needs and improving student outcomes (NCTM, 2014).
Gathering evidence of student thinking requires significant planning and forethought. It is essential to align the goals of instruction with the evidence needed to measure attainment of those goals. Educators need to first identify indicators of what is important to take note of in students’ mathematical thinking and then plan ways to make that thinking transparent. Questions, tasks, and activities should be constructed to lay bare students’ patterns of reasoning, including missteps and misconceptions in that reasoning (NCTM, 2014). There should be “a constant stream of information about how student learning is evolving toward the desired goal” (Heritage, 2008).
Formulating and posing well-crafted questions is a powerful tool for eliciting evidence of student thinking, and there is abundant research on effective questioning techniques in mathematics. Generally speaking, research indicates the type of questions and the patterns in which those questions are posed should encourage students to both explain and reflect upon their thinking. There are times when teachers should pose questions to gather information, probe thinking, make the mathematics visible, and encourage reflection and justification. Each type of question elicits different types of information that are essential to enabling a teacher to assess and advance a student’s learning (NCTM, 2014).
The pattern of questioning also contributes to students making sense of the mathematics. Questions should be focused to allow for students to clearly communicate their thoughts and rationale, and to foster reflection on those thoughts and those of their peers. The objective is not to get to a single predetermined answer, but rather to open a window to the world of student thinking, thus allowing the teacher to employ their content knowledge to respond to student needs (NCTM, 2014).
The process of eliciting data and adjusting instruction based on that data are just two parts of an overall strategy to improve student learning and assist them in becoming responsible and committed to that learning. Providing descriptive, targeted, and actionable feedback is the glue that cements these efforts into a meaningful and effective whole. The research of Clarke (2003) and Hattie and Timperley (2007) drives home the importance of providing feedback linked to effort and perseverance, identifying successes, and lending support and advice on how to improve (OER4 Schools, 2013).
Hattie and Timperley posit feedback should be targeted to the task, process, or self-regulatory, and not the personal level. “It is most effective when it aids in building cues and information regarding erroneous hypotheses and ideas and then leads to the development of more effective and efficient strategies for processing and understanding the material.” Teachers and students must be adept at providing and reacting to feedback, respectively, thus necessitating educators and families coaching students in how to proactively respond to feedback (Hattie & Timperley, 2007).
What this looks like for Multilingual Learners (MLLs)
For educators with one or more active MLLs on their roster, formative assessment practices should include the collection of discipline-specific language samples and progress-monitoring of MLLs’ language development in mathematics. These language samples and assessment practices will give educators the data needed to provide students with language-focused feedback aligned to their language goals for mathematics. When designing or amplifying formative assessments for disciplinary language development, educators should draw on the English language proficiency level descriptors for their grade level(s) in the WIDA ELD Standards Framework. For additional information about how these descriptors can assist educators in offering targeted feedback based on the word, sentence, or discourse level dimension of students’ language samples, please see Section 4.
What this looks like for Differently-Abled Students (DAS)
HLP 4, Use Multiple Sources of Information to Develop a Comprehensive Understanding of a Student’s Strengths and Needs, describes assessment as a collaborative process that includes informal assessments to plan instruction that is responsive to individual needs. DAS participation in formative assessments may require specific accommodations specified in IEPs. Implemented in conjunction with HLP 22, Provide Positive and Constructive Feedback to Guide Students’ Learning and Behavior, DAS will receive immediate and specific feedback on their performance that is goal-directed and thoughtful in considering the specific learner profile. Feedback on formative assessment is positive and constructive when it avoids words like “should, but, however” and includes statements that highlight what they did appropriately followed by a question (what is another way?) or a suggestion (try adding or continuing with). A diagram or image can support DAS to understand feedback and their progress on formative assessments.
To Learn More
Below is a variety of links to resources to learn more about this practice.
Develop Procedural Skill and Fluency through Conceptual Understanding
Teachers routinely attend to building student fluency with procedures by grounding instruction in the conceptual underpinnings of those procedures. They provide varied types of practice through which students become flexible and efficient users of procedures when solving mathematical and real-world problems (NCTM, 2014).
The standards call for a balance of instructional attention between developing conceptual understanding, procedural skill and fluency, and the ability to apply mathematics when solving rigorous mathematical and real-world problems — the three elements of rigor identified by the standards. The goal of instructing students to gain fluency has an interesting history in the United States. Traditionally, American instruction has approached this goal only from a procedural vantage point. Classroom materials and instruction were characterized by step-by-step instructions leading to a specific outcome. If students were individually successful in gaining some insight into why a procedure or algorithm worked, all the better. Research has come to demonstrate that procedural skill and fluency is best developed with a sound grounding in conceptual understanding, strategic reasoning, and problem-solving (NCTM, 1989, 2000; NGA Center & CCSSO, 2010). When facts and procedures are connected to understanding, they are easier to use and to transfer to varied situations. A student who is fluent in some aspect of mathematics is one who is able to flexibly select a method or strategy to solve mathematical or real-world problems. There is an element of efficiency in the selection as well as the ability to explain or justify the selection and the process (NCTM, 2014).
With conceptual understanding in place, students are able to build and strengthen their fluency through practice. As with any facet of instruction, designing opportunities for student practice must be done intentionally and with a specific goal in mind. Giving students too many practice problems too soon is not an effective approach to building fluency. Small sets of problems, connected to conceptual understanding, distributed over a period of time, and accompanied by timely and meaningful feedback are best for building strong fluency with procedural skill (NCTM, 2014). Interestingly, another tactic for fluency practice, although frequently disregarded, is to use rich problem-solving tasks in which students need to employ specific computational skills while designing strategies to navigate the tasks. This so-called embedded practice can really pack a bang for the buck (National Research Council, 2001).
Use and Connect Multiple Representations
Teachers routinely incorporate the use of multiple representations into their mathematics instruction. They challenge their students to do the same knowing that when connections are made between different representations, there is a deepening of conceptual understanding and the ability to apply procedures when solving problems (NCTM, 2014).
Experts in mathematics education stress the power of multiple representations — physical, visual, contextual, verbal, and symbolic. When students are asked to use different representations in mathematics and talk about the similarities in those representations, they not only gain improved access to the mathematics at hand, but also develop a deeper understanding of that mathematics. Perhaps Tripathi (2008) expressed it best when saying, “Different representations is like examining the concept through a variety of lenses, with each lens providing a different perspective that makes the picture (concept) richer and deeper.” Flexibility in the use of representations is linked to increased success in problem-solving and the ability to see and grapple with the essential structures in mathematics (NCTM, 2014).
Young students are no strangers to representing their mathematical thinking with diverse representations. They commonly use drawings or act out situations using manipulatives to gain entry into a problem and to formulate and justify a solution (National Research Council, 2001). These simple representations can serve as the connections to creating deeper mathematical understanding, evolving into efficient symbolic representations that can be justified and explained.
Visual and physical representations should not be the territory of solely the very young. It is interesting to note that visual representations are particularly helpful in increasing access to the mathematics for many learners. This is expressly true for differently-abled students and MLLs. Students’ visual representations tend to leave more overt clues to their thinking used when solving a problem and can serve as a platform for classroom discourse, allowing other students to gain access to and understanding of the mathematics (NCTM, 2014).
To Learn More
Below is a variety of links to resources to learn more about this practice.
Resource |
Description |
Instructional Approaches for Math Rigor |
This chart created by Achievement Network identifies instructional strategies that are especially effective for addressing each aspect of rigor in the standards (conceptual understanding, procedural skill and fluency, and application). |
NCTM Position: Procedural Fluency in Mathematics |
This position paper by NCTM answers the question, “What is procedural fluency, and how do we help students develop it?” |
Middle School Math Fluency Guide |
This UnboundEd guide offers strategies and rationale for busy teachers who want to develop procedural fluency in middle school mathematics. Connect and build on students’ conceptual understanding through structured practice that will help students make sense and expertise with procedures. Learn how to effectively support student development of their productive disposition toward mathematics. |
RIDE is committed to increasing access to high-quality STEM educational opportunities for all students. An understanding of STEM concepts and development of STEM-related skills is needed to prepare future generations to make informed choices and increase the number of qualified candidates for careers in Rhode Island’s growing STEM industries.
The acronym, STEM, was coined by the National Science Foundation (NSF) in 2001 to describe occupations that required knowledge and skills from the disciplines of Science, Technology, Engineering, and Mathematics. Beyond conceptual understanding, STEM occupations require the application of concepts across disciplines. In 2010, the Rhode Island School of Design (RISD) campaigned to add Art and Design to the acronym by revising it to STEAM. This addition shifted the term to highlight the more innovative aspects of creativity and problem-solving. The U.S. Bureau of Labor Statistics anticipates the number of STEM occupations to grow an additional 8 percent by 2029, compared with 3.7 percent for non-STEM occupations in the same period. To ensure that students have the knowledge and skillset to be successful in STEM occupations, all students need to engage in STEM experiences that focus on application and problem-solving throughout their education. Engaging in well-designed, grade-level appropriate STEM activities from an early age will give all students experiences where they can apply concepts and skills acquired in core classes to develop innovative solutions to local and global problems.
The individual subject areas of science, technology, engineering, and mathematics are the disciplines of STEM, where a solid foundation is built. Building from this, students need to engage in Integrated STEM, where experiences apply the knowledge and skills from several (or all) of the STEM disciplines. Time for Integrated STEM should be provided beyond the time allotted for mathematics and science instruction since these subjects have tightly packed curricula that need to be followed with fidelity. An increasing number of schools have supplemental, in-school STEM/STEAM programs for elementary students and STEM/STEAM courses for middle and high school students. These in-school opportunities need to be part of every student’s experience, not just offered as electives or as enrichment for high achieving students.
Planning for STEM (or STEAM) Integration
- Integrated STEM experiences should support the development of disciplinary knowledge while making cross-discipline connections explicit to students. Educators must thoughtfully design Integrated STEM experiences that provide intentional support for students to build knowledge and skill both within the disciplines and across disciplines.
- Educators need to ensure that STEM experiences reinforce the student learning in science and mathematics, but do not undermine or duplicate the core subject curriculum.
- When designing Integrated STEM experiences, it is important to use the grade-level science and mathematics standards and learning progressions. Additionally, the Standards for Technology and Engineering Literacy (STEL) (ITEEA, 2020) should be used to guide to ensure that the experiences are appropriate for the developmental level of the students and develop students’ technology and engineering proficiencies.
- Instructional models such as project-based/problem-based learning provide authentic opportunities for students to engage in Integrated STEM. Educators should design experiences that are grade-level appropriate and draw on student and/or community interest. The learning experiences should be iterative, annually reviewing them to incorporate new ideas or technology and to include novel student interests or community concerns.
- Integrated STEM education should not take the place of high-quality education focused on the individual STEM subjects, but it should require students to apply the knowledge and skills of the STEM subjects. While teachers should integrate STEM into math and science courses where it naturally fits, students need more opportunities to engage in Integrated STEM in school. Since the Next Generation Science Standards (NGSS) include engineering performance expectations as well as the practice of analyzing and interpreting data, there is some expected integration of the other STEM disciplines. Additional opportunities to engage in Integrated STEM will give students motivation to apply what they are learning in STEM discipline areas and advance their learning.
Real-World & Career Connections
- All students should view a career in STEM as accessible; engaging all students in STEM throughout K–12 is an important part of creating this perspective. Providing access to STEM experiences where students are challenged but can find success can lead to an interest in STEM careers. Schools and educators need create a climate that provides all students, especially those from underrepresented groups in STEM career fields, access and the opportunity to be successful in STEM learning. Partnering with local STEM organizations and industries will allow students to better understand the opportunities that exist through interaction with STEM professionals, exploration of potential careers, and understanding the variety of STEM-related workplaces. Industry partnerships can start at an early age as part of career awareness, later progressing to career exploration, and potentially including high school internships or pre-graduation training programs.
- Even if students do not follow a STEM career path, they will still need to acquire STEM literacy. STEM literacy includes the ability to be a critical consumer of information, be a creative problem solver, and develop critical thinking skills. Thoughtfully designed Integrated STEM experiences also build the Cross-Curricular Proficiencies of collaboration, communication, problem-solving and critical thinking, reflections and evaluation, and research. These skills will support all students to be lifelong learners and have success in college and career.
Equitable Access to STEM
- Assuring access to STEM experiences for learners traditionally underrepresented in STEM fields can provide opportunities for individual success as well as broader changes to the STEM workforce. Additionally, engaging learners in STEM-related problem-based learning provides motivation and engagement not found in decontextualized academic knowledge. (Parker et al, 2016)
- Access is only one aspect of equity, schools also need to look carefully at how their designs and strategies encourage broadened participation through alternative ways of thinking about motivation, engagement, and persistence. Equity needs to be addressed with targeted strategies that align with the local context and realities of the learners, whether geographic (e.g., experiences of rural learners), social (e.g., experiences of girls), or experiential (e.g., experiences of students with disabilities). At the same time, strategies that are explicitly aligned to broadening participation in STEM also improve STEM experiences for all students. (Parker et al, 2016)
To Learn More
Below is a variety of links to resources to learn more about this practice.
Resource |
Description |
My PBLWorks from Buck Institute |
To help schools and districts visualize high-quality PBL in the classroom, the Buck Institute for Education (BIE) has videos showcasing PBL projects from K–12 schools nationwide, including several STEM-themed projects. Teachers can view videos of successful PBL projects that feature teacher interviews and actual classroom footage and highlight projects from a range of grade levels, settings, and subject areas, including STEM. |
STEMWorks at WestEd |
STEMworks is a searchable online honor roll of high-quality science, technology, engineering, and mathematics (STEM) education programs. STEMworks helps companies, states, and individuals make smart investments in their communities by evaluating and cataloging programs that meet rigorous and results-driven design principles. |
National Science Foundation (NSF) Resources for STEM Education |
NSF research and development projects have produced a rich array of principles, materials, and practitioner insights that are helpful guides to improved preparation and professional development of STEM teachers. The following examples illustrate the range of ideas and products available from that work. |
The development of a second, third, or fourth language is a lifelong process — one that cannot take place in isolation or within a stand-alone hour of the school day. If we are to ensure all students have meaningful access to core instructional programs, all educators must share responsibility for the education of MLLs, including teachers of ELA/Literacy. For those not certified in English to Speakers of Other Languages or Bilingual/Dual Language, shared responsibility might beg the question: What is high-quality instruction for MLLs? What practices are evidence-based in promoting content and language learning with MLLs?
RIDE offers in-depth guidance about the key components of high-quality MLL instruction in its High-Quality Instructional Framework for MLLs to Thrive, but the research is clear: language development is most effective when integrated within content area instruction. Integrated language and content teaching gives MLLs rich, highly contextualized opportunities to use disciplinary language, which in turn reinforces content learning. Rather than teaching a discrete set of grammar rules or vocabulary lists, devoid of disciplinary context, educators must reflect on the language demands of content-based tasks from the core curriculum, offering explicit language instruction and ample scaffolds so MLLs can linguistically access and engage in core content area instruction.
To Learn More
Below is a variety of links to resources to learn more about this practice.
Equity requires participation and a sense of belonging. To ensure that all students participate in mathematics instruction — not just the hand raisers — teachers will need a continuum of proactive strategies that increase opportunities for student engagement. Students with IEPs or a 504 plan are general education students who access the grade-level curriculum through the support of high-quality instruction, as described in the preceding sections, which utilizes data on learner characteristics to differentiate and scaffold. Accommodations determined by the IEP team or a 504 plan complement the differentiation and scaffolds to ensure that accessibility needs specific to the individual learner are met. General education and content area teachers are responsible for providing instruction that is differentiated, scaffolded, and where appropriate for individual learners, includes accommodations. Some learners will also require instructional modifications as determined by the IEP team. When students receive quality supplementary curricula as part of their specially designed instruction (SDI), then inclusion can provide accommodations for generalizing skills they mastered in SDI. Collaborative planning with special educators and related service providers will support general educators in developing their repertoire of rigorous and accessible instructional practices.
The Leadership Implementation Guides from the High Leverage Practices for Students with Disabilities include tips for school leaders to support teachers; questions to prompt discussion, self-reflection and observer feedback; observable behaviors for teachers implementing the HLPs; and references and additional resources on each HLP. These guides, referenced throughout this section, were developed to help leaders integrate the HLPs into professional development and observation feedback.
Understanding learner characteristics will help clarify what types of support to provide to DAS in their planning, organizing, and writing to promote DAS access and progress in the mathematics curriculum. A combination of techniques such as guided inquiry, mathematics notebooks and Self-Regulated Strategy Development (SRSD) provides scaffolding to promote the success of DAS. Any accommodations outlined per the IEP or a 504 plan that provide reading, writing, and math access will be important for mathematics (Collins & Fulton, 2017).
To Learn More
Below is a variety of links to resources to learn more about this practice.
Resource |
Description |
High-Leverage Practice (HLP) Leadership Guides from the Council for Exceptional Children |
Leadership Guides for the following HLPs:
#1: Collaborate with Professionals to Increase Student Success
#5: Interpret and Communicate Assessment Information with Stakeholders to Collaboratively Design and Implement Educational Programs
#14: Teach Cognitive and Metacognitive Strategies to Support Learning and Independence
|
Unit Co-Planning for Academic and College and Career Readiness in Inclusive Secondary Classrooms |
Article describing the UCPG, a tool to support general and special education teacher collaboration and planning in inclusive general education classrooms |
Big Ideas in Special Education: Specially Designed Instruction, High-Leverage Practices, Explicit Instruction, and Intensive Instruction |
Article describing the differences between specially designed instruction, high-leverage practices, explicit instruction, and intensive instruction |
IEP Tip Sheet: What are Supplementary Aids & Services? |
Tip Sheet from Progress Center on accommodations for instruction and assessment, modifications, and other aids and services |
IEP Tip Sheet: What are Program Modifications & Supports? |
Tip Sheet from Progress Center on program modifications and supports that promote access to and progress in general education programming and shares tips for implementation |
Can you implement DBI to support students with intellectual and developmental disabilities? |
In this brief video, Dr. Chris Lemons shares considerations for implementing data-based individualization (DBI) to support students with intellectual and developmental disabilities |
Classroom Supports: Universal Design for Learning, Differentiated Instruction CTE Series 3 | NTACT:C (transitionta.org) |
Webinar from the National Assistance Center on Transition — UDL at secondary: “Fundamentals of differentiated instruction to support effective teaching, individualized learning and maximize student engagement are shared.” |
TIES Center: Inclusive Instruction: Resources on Inclusive Instruction |
Resources on Inclusive Instruction:
TIES Brief #4: Providing Meaningful General Education Curriculum Access to Students with Significant Cognitive Disabilities
TIES Brief #5: The General Education Curriculum- Not an Alternative Curriculum!
Lessons for All: The 5-15-45 Tool
|
TIES Center: TIES TIPS: Foundation of Inclusion TIPS |
TIES Inclusive Practice Series TIPS
#15 Turn and Talk in the Inclusive Classroom
#16 Making Inferences in the Inclusive Classroom
#19 Creating Accessible Grade-level Texts for Students with Cognitive Disabilities in Inclusive Classrooms
|
Evidence-based practices for children, youth, and young adults with Autism |
Report on evidence-based practice including a fact sheet for each that provides a longer description, information about participant ages and positive outcomes, and a full reference list. |
Enacting the high-quality instructional practices described above is an essential yet complex task for teachers. Thus, ensuring high-quality instruction for all students in school often requires a team effort involving grade-level/content-area teachers, specialists and educators working with multilingual learners and differently-abled students in particular, and the administrators, leaders, and coaches who support all the educators. In addition, effective professional learning that helps teachers enhance their knowledge and application of high-quality instructional practices should strategically integrate multiple types of professional learning, as described in this section.
First, as mentioned in earlier sections of this framework, high-quality instruction begins with a deep understanding of the standards since they provide the foundation for instruction by defining what students need to know and be able to do. Professional learning suggestions and guidance for deepening the understanding of standards can be found in Section 2 of this framework.
Professional learning for high-quality instruction must also focus on developing a solid understanding of the high-quality instructional practices listed above. Readers are encouraged to review the many resources listed with each instructional practice and to establish ‘book study’ groups with colleagues to read, review and discuss any of the resources shared in Part 2 of this section of the framework.
In addition, supporting effective professional learning requires supporting teachers’ application of the practices described above. As with any complex skill, when supporting the application of high-quality instructional practices, the key ingredient is timely and targeted feedback. For feedback to be provided in a targeted and timely fashion, practices must be made visible so that the application of instructional practices can be observed. Once observed, feedback can then be generated. Most of the professional learning tools designed to provide feedback align with three key phases of the instructional cycle where it is very helpful for teachers to receive feedback about their instruction. The first phase is during lesson planning, before instruction actually takes place. The next phase is the actual instruction where teachers can be observed engaging with students. The final phase is after teaching has taken place and focused on the review of student work and evidence of learning. Below is a variety of tools and resources that are designed to provide teachers with feedback during these three phases. They are organized into the following three categories: Planning Tools, Observation Tools, and Evidence of Learning Tools. These tools come from a variety of sources, but all are intended to guide coaches, professional learning providers, and other leaders in offering support to teachers in this work.
Planning Tools
Resource |
Description |
30-Minute Tuning Protocol |
Protocol designed to be used within collaborative teacher teams. It can be used to provide teachers with feedback on any artifact of their teaching and is a great tool to solicit feedback about lessons. In the protocol, a presenting teacher shares the goal, need, and plan of their professional work. Participants share feedback in rounds. The presenter then reflects on what was said that was helpful and what feedback they will try to incorporate to improve their plan. |
UDL Tip for Designing Learning |
Tip sheet with teacher questions, examples, and further resources to help anticipate learner variability and make instruction flexible and useful for all learners |
CAST | Key Questions to Consider When Planning Lessons |
One-pager of question prompts for teacher to improve lesson accessibility |
Whole-Group Response Strategies to Promote Student Engagement in Inclusive Classrooms |
Article on whole-group response systems paired with formative assessment charts to provide instruction that actively engages students in the learning process
“These strategies can be implemented easily in classrooms with minimal additional resources and are applicable across grade levels and content areas with appropriate modifications.”
|
Approaching Explicit Instruction Within a Universal Design for Learning Framework (See references section) |
Article on implementation suggestions for using EL and UDL in tandem to better support students access and understanding of lesson content with improved student engagement and demonstration of what they know and can do |
Achieve the Core’s Lesson Planning and Reflection: Quick Reference Question Guide |
The questions in this resource support a thoughtful planning and reflection process for mathematics instruction. |
Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades |
This practice guide, developed by the What Works Clearinghouse™ (WWC) in conjunction with an expert panel, distills contemporary mathematics intervention research into easily comprehensible and practical recommendations for teachers to use when teaching elementary students in intervention settings. |
Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students |
This What Works Clearinghouse™ (WWC) practice guide presents evidenced-based suggestions for how to improve algebra skills and knowledge for students in grades 6–12. The guide offers three recommendations that provide teachers with specific, actionable guidance for implementing these practices in their classrooms. |
Project STAIR Algebra
Project STAIR - YouTube
October |2018 | Project STAIR (smu.edu)
Project STAIR: Supporting Teaching of Algebra: Individual Readiness | Simmons School of Education & Human Development | SMU |
A U.S. Department of Education Office of Special Education Programs model demonstration project on a system of instructional practices for supporting the algebra-readiness of middle school students with specific learning disabilities in mathematics resulting in videos for teachers to learn instructional strategies for algebra readiness for students with IEPs or in intensive intervention. |
Principles for the Design of Mathematics Curricula: Promoting Language and Content Development |
Written with the goal of providing guidance to mathematics teachers for recognizing and supporting students’ language development processes in the context of mathematical sense-making, this document describes a framework of Mathematical Language Routines (MLRs) and includes supporting examples. The MLRs are designed to help teachers address the specialized academic language demands in mathematics when planning and delivering lessons. |
UnboundEd Middle School Math Fluency Guide |
The UnboundEd Middle School Math Fluency Guide offers strategies and rationale for busy teachers who want to develop procedural fluency in Middle School mathematics. Connect and build on your students’ conceptual understanding through structured practice that will help students make sense and expertise with procedures. |
TNTP Math Assignment Review Protocol |
Use this content-specific protocol to assess the quality of assignments that students are asked to complete. Are they aligned to grade-level standards? Do they give students the opportunity to engage with the mathematical practices? Do they make meaningful connections to the real world? |
Observation Tools
Resource |
Description |
30-Minute Atlas Protocol |
Protocol describing a collaborative process for examining students’ performance data to inform next steps in teaching. |
RESET Explicit Instruction Rubrics |
Rubric, webinar, manual, and related resources focused on explicit instruction. The Recognizing Effective Special Education Teachers (RESET) project, funded by U.S. Department of Education Institute for Education Sciences (IES) and led by Evelyn Johnson at Boise State University, developed a series of rubrics based on evidence-based practices for students with high-incidence disabilities. One set of rubrics focuses on explicit instruction. Based on the main ideas of Explicit Instruction, the Explicit Instruction Rubric was designed for use by supervisors and administrators to reliably evaluate explicit instructional practice, to provide specific, accurate, and actionable feedback to special education teachers about the quality of their explicit instruction, and ultimately, improve the outcomes for students with disabilities. |
Achieve the Core’s Instructional Practice Guide |
The Instructional Practice Guide (IPG) is a K–12 classroom observation rubric that prioritizes what is observable in and expected of classroom instruction when instructional content is aligned to college- and career-ready (CCR) standards. It purposefully focuses on the limited number of classroom practices tied most closely to content of the lesson. |
“What to Look For” Observation Guides |
Created by the Massachusetts Department of Education, these guides briefly outline what content and practices should be observed during instruction for a specific grade or course. |
Evidence of Learning Tools
Resource |
Description |
Student Work Analysis Protocol |
Protocol describing a process that groups of educators can use to discuss and analyze student work. It is intended to be applicable across subjects and grades, including literacy, mathematics, science, the arts, and others. Analyzing student work gives educators information about students’ understanding of concepts and skills and can help them make instructional decisions for improving student learning. |
Instructional Rounds / Atlas Protocol |
Protocol describing a process for conducting 8-minute instructional rounds in groups. |
Calibration Protocol for Scoring Student Work |
Protocol describing a process that groups of educators can use to discuss student work in order to reach consensus about how to score it based on rubric/scoring criteria. It is intended to be applicable across subjects and grades, including literacy, mathematics, science, the arts, and others. Examples of student work that can be used as practice for calibration are included as appendices. |
Additional Tools and Resources
Download a list of additional tools and resources.
Advance CTE, Association of State Supervisors of Mathematics, Council of State Science Supervisors, and International Technology and Engineering Educators Association. (2018). STEM4: The power of collaboration for change. http://cosss.org/resources/Documents/STEM%20Collaboration%20paper.%20FINAL%20VERSION.%207.26.2018.pdf
Celedón-Pattichis, S., Lunney Borden, L., Pape, S.J., Clements, D.H., Peters, S.A., Males, J.R., Chapman, O., & Leonard, J. (2018). Asset-based approaches to equitable mathematics education research and practice. Journal for Research in Mathematics Education, 49 (4), 373–389. Asset-Based Approaches to Equitable Mathematics Education Research and Practice (d1wqtxts1xzle7.cloudfront.net)
Clarke, S. (2003). Enriching feedback in the primary classroom. Hodder and Stoughton.
Council for Exceptional Children. (2021). High-leverage practices for students with disabilities: Collaboration. https://highleveragepractices.org/four-areas-practice-k-12/collaboration
Council for Exceptional Children. (2021). High-leverage practices for students with disabilities: Instruction. https://highleveragepractices.org/four-areas-practice-k-12/instruction
Dweck, C. (2008). Mindsets and math/science achievement. Carnegie Corporation of New York Institute for Advanced Studies.
Foxworth, L.L., Hashey, A.I., Dexter, C., Rasnitsyn, S., & Beck, R. (2021). Approaching explicit instruction within a universal design for learning framework. TEACHING Exceptional Children, 1-8. DOI: 10.1177/004005992110101
Gay, G. (2010). Culturally responsive teaching (2nd ed.). Teachers College Press.
Hattie, J.A.C. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. Routledge.
Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1), 81–112.
Haystead, M.W., & Marzano, R. (2009). Meta-analytic synthesis of studies conducted at Marzano Research Laboratory on Instructional Strategies. Marzano Research Laboratory.
Heritage, M. (2008). Learning progressions: Supporting instruction and formative assessment. Council of Chief State School Officers.
Instruction Partners. (2019). Curriculum support guide: Executive summary. https://curriculumsupport.org/wp-content/uploads/2019/02/Executive-Summary-1.pdf
International Technology and Engineering Association. (2020). Standards for technological and engineering literacy: The role of technology and engineering in STEM education. www.iteea.org/STEL.aspx
Ladson-Billings, G. (2009). The dreamkeepers: Successful teachers of African American children (2nd ed.). Jossey-Bass.
Jett, C.C. (2013). Culturally responsive collegiate mathematics education: Implications for African American students. Interdisciplinary Journal of Teaching and Learning, 3 (2), 102–116. EJ1063224.pdf (ed.gov)
National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. NCTM.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for mathematics. NCTM.
National Council of Teachers of Mathematics (NCTM). (2014). Principles to action: Ensuring mathematical success for all. NCTM.
National Research Council. (2014). STEM Integration in K-12 Education: Status, Prospects, and an Agenda for Research. Washington, DC: The National Academies Press. https://doi.org/10.17226/18612.
National Research Council. (2001). Adding it up: Helping children learn mathematics. (J. Kilpatrick, J. Swafford, & B. Findell, Eds.). Mathematics Learning Study Commission, Center for Education,
Division of Behavioral and Social Sciences and Education. National Academy Press.
New South Wales Government – Education. (n.d.). Introducing student self-assessment. Retrieved May 5, 2021, from https://education.nsw.gov.au/teaching-and-learning/professional-learning/teacher-quality-and-accreditation/strong-start-great-teachers/refining-practice/peer-and-self-assessment-for-students/introducing-student-self-assessment
OER4 Schools. (2013, November 11). Shirley Clark video on feedback [video file]. YouTube. https://www.youtube.com/watch?v=DGNp0AJte_c
Parker, C.E., Pillai, S., Roschelle, J. (2016). Next Generation STEM Learning for All: A report from the NSF supported forum. Waltham, MA: Education Development Center.
Reinhart, S.C. (2000). Never say anything a kid can say! Mathematics Teaching in the Middle School, 5(8), 478–83.
Rhode Island Department of Elementary and Secondary Education. (2021). Rhode Island core standards for mathematics: Standards for mathematical practice. https://www.ride.ri.gov/InstructionAssessment/ContentStandards.aspx#44071958-mathematics-standards
Rhode Island Department of Education. (2021). Social and emotional learning (SEL). https://www.ride.ri.gov/StudentsFamilies/HealthSafety/SocialEmotionalLearning.aspx#31941087-core-sel-competencies
Rhode Island Department of Elementary and Secondary Education (RIDE). (2007, October). Rhode Island professional teaching standards. https://www.ride.ri.gov/Portals/0/Uploads/Documents/Teachers-and-Administrators-Excellent-Educators/Educator-Certification/Cert-main-page/RIPTS-with-preamble.pdf
Riccomini, P. J. & Morano, S. (2019). Guided practice for complex, multistep procedures in algebra: Scaffolding through worked solutions. TEACHING Exceptional Children, 51(6), 445–
454. DOI: 10.1177/0040059919848737
Smith, M.S., & Stein, M.K. (2011). 5 principles for orchestrating productive mathematics discussions. National Council of Teachers of Mathematics.
Tripathi, P.N. (2008). Developing mathematical understanding through multiple representations. Mathematics Teaching in the Middle School, 13(8), 438–445.