Standard P.3. Implementing Ambitious and Equitable Mathematics Instruction 

• Establish mathematics goals to focus on learning

• Implement tasks that promote reasoning and problem solving

• Use and connect mathematical representations

• Facilitate meaningful mathematical discourse

• Pose purposeful questions

• Build procedural fluency from conceptual understanding

• Support productive struggle in learning mathematics

• Elicit and use evidence of student thinking

As EMSs intentionally and explicitly attend to equity and access during instruction, they also draw from the following five equity-based mathematics teaching practices: 

• Going deep with mathematics

• Leveraging multiple mathematical competencies

• Affirming mathematics learners identities

• Challenging spaces of marginality

• Drawing on multiple resources of knowledge.

During lessons, EMSs focus on developing a deep understanding of mathematics, drawing from their knowing students (Standard P.1), planning (Standard P.2), and assessing (Standard P.4). EMSs support teachers in understanding and enacting this instruction. Support for teachers developing ambitious and equitable instruction could be provided via professional learning that includes participating in lesson study, analyzing classroom video and student work, and engaging in mathematical tasks, along with modeling instruction, co-planning, co-teaching, and observing instruction then providing feedback and engaging in thoughtful reflection.

P.3.a. Fostering a mathematics learning community

In EMSs’ work with students, they foster an inclusive mathematics learning community that centers on and elevates all students’ knowledge and experiences (79). They challenge and disrupt spaces of marginality, as they understand and strive to ensure that all students’ voices and ideas are a necessary condition for learning mathematics. EMSs establish classroom expectations for participation (which could be co-created with students), position all students as capable, monitor how students position one another and manage status issues as they arise, and press for the academic success of all learners. When students’ social and emotional needs are met, including their feelings of competence, safety, and belonging within the mathematics classroom, they more readily exhibit curiosity, take risks, and ask questions. A safe environment supports students in understanding that productive struggle is part of the process of learning mathematics, allowing them to willingly grapple with emergent or partial ideas. Further, within a context of valuing students’ voices and choices, EMSs provide options during lessons related to materials, knowledge expression, questions for exploration, and learning tasks. 

In cultivating a mathematics learning community, EMSs understand the importance of sharing mathematical authority with students by positioning them as holders of expertise. EMSs encourage all students to reason and make sense of their mathematical thinking and value their multiple, varied strategies, grounded in a firm belief that all children are highly capable mathematical learners and doers (80). As EMSs notice students’ mathematical strengths, they assign competence through naming or publicly drawing attention to students’ intellectual contributions when solving problems (81). This elevated positioning of student contributions as competent raises the status of each and every learner. Over time, students learn to recognize, value, and rely on their own mathematical strengths as well as those of others when engaging in mathematical tasks.  

EMSs foster the social nature of learning mathematics by ensuring active student participation and collaboration in a discourse-rich environment. They prompt for students’ mathematical thinking by using purposeful questioning techniques that facilitate productive classroom discussions focused on and responsive to student reasoning and sense making. Questions should assess and advance students’ understandings, with EMSs using students’ statements and work to build shared mathematical understandings for the class. This discourse-rich environment provides opportunities to make competence explicit by highlighting and amplifying students’ contributions and recognizing emergent ideas. Throughout classroom interactions, EMSs elicit students’ mathematical ideas, and attend, interpret, and respond to students’ thinking as it unfolds, which serve as a guide for instructional decisions and enacted teaching moves (82). Further, student groupings during learning experiences are intentionally mixed to leverage a variety of strengths, where expectations are high for all students and they collectively learn from one another.

P.3.b. Focusing on a deep understanding of mathematics

In their support of students’ learning, EMSs develop clear, rigorous learning goals, situate these goals within research on how children learn mathematics, and facilitate cognitively-demanding instruction focused on meaningful, important, and relevant mathematics (83). Mathematics learning goals should guide instructional decisions, including the selection and implementation of cognitively-demanding tasks. EMSs use tasks that: engage students in problem solving, reasoning, and sense making; allow for multiple solution strategies and the use and connection of multiple representations (e.g., visual, physical, symbolic, contextual, verbal); support connections across mathematical concepts; and develop conceptual understanding as a foundation for procedural fluency. EMSs support students in analyzing, comparing, justifying, and proving their solution strategies, which often occurs through collaborative student discussion and debate. Generally, EMSs support students’ learning of content through engagement in mathematical processes and practices (e.g., representing and connecting, explaining and justifying, contextualizing and decontextualizing, noticing and using mathematical structures), rather than memorization of tips, tricks (e.g., mnemonics without connections), and rules and practice of step-by-step procedures and skills. In addition, in developing students’ understanding of mathematics, EMSs use the subject as a lens for understanding, critiquing, and creating change in their world.  

As EMSs focus on developing a deep understanding of mathematics, they know the meaning of conceptual understanding and procedural fluency and recognize that student learning must begin with and build from a solid foundation of deep conceptual knowledge (84). EMSs recognize that conceptual understanding (i.e., the comprehension and connection of mathematical concepts, operations, and relationships) precedes and is necessary for developing procedural fluency, and they structure student learning experiences accordingly. Further, EMSs know that procedural fluency includes much more than remembering facts or applying standard algorithms; this fluency is complex and involves applying procedures efficiently, flexibly, and accurately. EMSs’ instruction focuses on students becoming skillful over time in procedural fluency as they engage in contextualized problem solving, reasoning, and sense making. When procedures are connected with the underlying concepts, students benefit through improved retention, the ability to apply procedures in different and unfamiliar problems and contexts, and more productive disposition.   

EMSs use mathematics-specific tools, such as physical models and technological tools, to support students' problem solving and reasoning. EMSs recognize that students can use physical models to represent and readily adjust their thinking. They know physical tools and manipulatives also provide access to problems and support communication with others. Technological tools also have the potential to enhance mathematical understandings by amplifying the mathematics beyond what can be accomplished with physical materials (85). Technology has the potential to support mathematical visualization, modeling, and sense making. EMSs use technology to pose inquiry-based problems and facilitate creation of conjectures and justification of generalizations about mathematical topics. Additionally, EMSs encourage the thoughtful use of various physical and technological tools and support both teachers and students in understanding the affordances of each.

P.3.c. Leveraging multiple mathematical competencies

In their work with students, EMSs support the mathematical thinking and contributions of students with varying mathematical understandings and levels of confidence. This support is provided through implementing cognitively-demanding tasks and structuring student collaborations and discussions that allow them to use their differing mathematical knowledge and experiences to solve problems together. Students think, reason, and solve problems in different ways, and EMSs intentionally highlight and affirm this variability. For example, EMSs use cognitively-demanding tasks that support student access through multiple entry points to a problem, and they provide scaffolds and prompts or extensions based upon students’ needs. In this, students with varying understandings and confidence can engage and make contributions. The tasks also allow for varied and invented solution pathways and knowledge expression (86). This knowledge expression could include use of drawings, models, language, gestures, or other methods. The leveraging of multiple mathematical competencies also includes EMSs accessing, connecting to, and building upon students’ prior mathematical knowledge. Throughout these learning opportunities, EMSs emphasize students’ mathematics strengths so their confidence in their mathematics abilities is fostered.

P.3.d. Affirming students’ diversities and mathematics identities 

EMSs facilitate student learning experiences that account for and leverage students’ diversity and develop their positive mathematics identity, as described in P.1.a. and P.2.a (87). In responsively teaching students, EMSs differentiate the content, process, or products without lowering the cognitive demand for students. Instructional tasks concurrently have high levels of cognitive demand and to the extent possible connect to children’s questions, interests, and lives by building on family, community, and cultural funds of knowledge (88). Further, EMSs use story contexts and participants that mirror their own students’ identities, experiences, and values as well as those of others (89). Learning experiences for multilingual students emphasize multiple modes of thinking and communication (e.g., direct modeling, drawing, writing, speaking, gesturing), aiming to concurrently develop students’ mathematical understanding and academic language (90). EMSs understand, value, and leverage varying student conceptions, centering on student thinking and validating knowledge and experiences, both those in-school and out-of-school. 

Too often students with disabilities have limited opportunities to take on the role of doers of mathematics and to see themselves or have others see them as mathematically competent (91). EMSs recognize the critical importance of students with disabilities, as well as neurodiverse students, coming to a robust understanding of mathematics through opportunities to: engage with the subject using their own solution strategies and reasoning, communicate their ideas, respond to others’ ideas, and see connections between ideas. As elaborated in the indicators above, in working with students with disabilities, EMSs intentionally cultivate inclusive learning spaces and account for, leverage, and highlight multiple forms of mathematical competencies and knowledge. EMSs collaborate with others to develop meaningful mathematical goals in Individualized Educational Plans (IEPs) and to design and implement ambitious and equitable instruction. EMSs serving in the role of interventionist are highly knowledgeable of mathematics education research that focuses on students with disabilities. They draw from this knowledge for multiple purposes, including understanding and using effective instructional interventions and the multiple layers of support for students in schools. They also use these research-based understandings to advocate for students’ capabilities and rebut deficit notions, such as the widely circulating belief that students who struggle automatically need explicit instruction and cannot benefit from instruction focusing on problem solving, reasoning, and sense making (92).