Chapter 4: Implementing Equitable Mathematics Instruction

Key Recommendation:Mathematics instruction should be consistent with research-informed and equitable teaching practices that foster students’ positive mathematical identities and strong sense of agency.

Building upon a strong mathematics foundation developed in the elementary grades, middle school is a critical time to empower students to use that strong foundation as they become confident learners and doers of mathematics. Equitable instruction, the focus of this chapter, occurs as teachers implement the eight effective Mathematics Teaching Practices (NCTM 2014b) while ensuring each and every student develops a deep understanding of mathematics, a positive mathematical identity, and a strong sense of mathematical agency. As such, implementing instruction equitably requires that teachers take direct action stemming from careful and intentional planning and reflection informed by data from their students. Importantly, many of the ideas described in this chapter aim to capture the essence of a key conclusion reached by the National Academy of Sciences, Engineering, and Medicine (2018) regarding instruction:

Effective instruction depends on understanding of the complex interplay among learners’ prior knowledge, experiences, motivations, interests, and language and cognitive skills; educators own experiences and cultural influences; and the cultural, social, cognitive, and emotional characteristics of the learning environment. (pp. 6–7)

A teacher can begin or further develop in the area of implementing equitable mathematics instruction to immediately improve students’ mathematical learning experiences. Dedication to implementing equitable instruction should be adopted as a career-long professional commitment. This chapter synthesizes research-informed guidance on equitable instruction and is organized as follows: (1) identity, agency, and shared classroom authority; (2) equitable mathematics teaching practices; and (3) supporting the implementation of equitable mathematics instruction.

The learning of mathematics is closely tied to students’ mathematical *identity* (defined in chapter 2, p. 9) and how they and others see them as learners and doers of mathematics (Aguirre, Mayfield-Ingram, and Martin 2013; Jackson and Delaney 2017; Wood et al. 2019). It is important for teachers (and other stakeholders) to “… recognize the diverse cultural wealth that students use to bridge their in- and out-of-school mathematical practices and experiences” (Nasir and McKinney de Royston 2013, p. 283). The ways in which a student expresses their mathematical identity and participates in mathematics represents their level of mathematical *agency* (NCTM 2018) and should be meaningful both personally and socially (Berry 2016). Agency can be defined as the expression of one’s identity (Murrell 2007). For example, emergent bilinguals can productively position themselves in the mathematics classroom and use their linguistic resources in creative, competent ways to navigate mathematical interactions (Langer-Osuna et al. 2016). The ways in which students participate in the middle school mathematics classroom—that is, the mathematical discourse in which they engage, the way they approach problems, and their willingness to take risks and articulate their mathematical thinking—are all evidence of a student’s level of agency. “As students author ideas, decide and justify whether particular mathematical ideas are reasonable or correct, and press one another for explanations, they take on forms of intellectual authority that fuel the collaborative mathematics classroom” (Langer-Osuna 2017, p. 238).

Mathematics classrooms that empower students to develop a strong sense of agency and that foster positive mathematical identities do so through a *shared classroom authority*. That is, students are positioned as having valuable knowledge to contribute to the classroom conversation, are given time to develop their ideas, and engage in meaningful discourse (Berry 2019). Ensuring students have shared authority aligns with the beliefs that each and every student should have access to high-quality mathematics instruction, is capable of doing mathematics, and is mathematically competent. Through these foundational beliefs and the establishment of shared classroom authority, equitable teaching practices can be successfully realized. Teachers play a vital role in making this aspiration a reality (Aguirre, Mayfield-Ingram, and Martin 2013), especially when making instructional decisions on which mathematics tasks to implement with their students. Mathematics tasks that have multiple entry points and require a range of reasoning strategies position students as mathematically competent as they solve the task (Langer-Osuna et al. 2016), and as a result foster students’ positive mathematical identities.

The unique needs of middle school students demand mathematics programs that intentionally meet their needs. The Association for Middle Level Education (AMLE) in their position paper *This We Believe: Keys to Educating Young Adolescents* (2010, n.p.) identified the following characteristics of successful middle schools related to the topics of curriculum, instruction, and assessment:

- Educators value young adolescents and are prepared to teach them.
- Students and teachers are engaged in active, purposeful learning.
- Curriculum is challenging, exploratory, integrative, and relevant.
- Educators use multiple learning and teaching approaches.
- Varied and ongoing assessments advance learning as well as measure it.

These characteristics rightfully attend to the unique needs of young adolescents. More specific to mathematics, the eight research-informed Mathematics Teaching Practices in NCTM’s *Principles to Actions: Ensuring Success for All* (2014b) speak to the actionable ways in which teachers of mathematics can provide students with access to high-quality mathematics instruction.

NCTM’s eight Mathematics Teaching Practices provide a foundational framework that, at its core, embodies high-quality mathematics instruction, and simply put, good teaching. Good teaching is necessary but not sufficient in achieving equitable instruction. Additional ideas and strategies beyond NCTM’s Mathematics Teaching Practices are necessary to address equitable instruction in classrooms, schools, and districts. Equitable instruction embraces access points for developing and sustaining students’ development of a positive mathematical identity, strong sense of agency, and mathematical competence. Deficit views about some students’ intellectual potential, which are often based on students’ race, class, gender, perceived ability, status in the classroom, and other factors, should not be tolerated to any degree. These deficit views can operate at a structural level (as described in chapter 3) or at the classroom level, in terms of who has access to participate, do, write, and speak mathematics.

Privilege and oppression (e.g., racism, sexism, classism, heterosexism, ableism) are factors operating in many classroom spaces even today, and they constrain or deny some students opportunities to participate in mathematics in ways that are needed, deserved, and of high quality. The racial ideology of whiteness privileges those who are White while oppressing those who are not, all under the guise of being color-blind where low academic achievement for Black and Latinx students is blamed on the students, uncaring parents, or devaluation of education (Battey and Leyva 2016).

Privilege and oppression is not a figment of “other peoples’ ” imagination, but hold a great deal of explanatory power related to achievement and success differentials in mathematics in the United States (and throughout the globe). For the tide to change in regard to mathematics opportunities, we, as mathematics educators, must be vigilant in examining and re-examining our work, our commitments, and ourselves. Then, we must do the hard work of making things right. (Stinson and Spencer 2013, p. 5)

Instructional practices play a key role in addressing inequitable systemic structures that lead to privilege and oppression. Privilege and oppression in mathematics education is evidenced in what content gets taught, to which students, and by which teachers.

In the next section, NCTM’s eight Mathematics Teaching Practices are defined and described in the context of a sample task, and the intersectionality of the eight teaching practices is discussed. A collection of ideas and strategies is also shared so that all stakeholders can critically examine current teaching practices in classrooms, schools, and districts and the extent to which they are equitable, just, and inclusive. This section concludes with a table that aligns the eight Mathematics Teaching Practices with examples of equitable teaching to bring coherence to the ideas in this section. Consider the following Equivalent Expression task and its accompanying classroom dialogue adapted from Roy et al. (2017, pp. 101–104), which will be used to ground the discussion of the eight Mathematics Teaching Practices.

Equivalent Expressions TaskSofia and Andre are working on a mathematics task. Sofia writes 2(12

x+ 24); Andre writes 6(4x+ 8). Sofia thought the expressions were equivalent but Andre did not.

- Who is right? How do you know?
- What are other equivalent ways of writing 2(12
x+ 24)?

After presenting the task, the teacher gave students individual time to think about and record their initial thoughts. This independent work was intentionally planned to position students as capable mathematical contributors. Students then shared their thinking with a shoulder partner, which provided the teacher with an opportunity to monitor and carefully consider which students’ reasoning to strategically highlight in the subsequent whole-class discussion (Smith and Stein 2018). The intentional move to be inclusive by strategically focusing on students’ strengths and contributions (rather than on what they did not yet know) helped the teacher address barriers to participation and equalize students’ status during a whole-group conversation, making each students’ contributions equally valued among their peers and the teacher (as in Wood et al. 2019). The following vignette contains portions of the whole-class discussion:

Teacher: | Let’s talk about it … I am going to have you make sense of what Lynn is doing. [Student writes 24x + 48 on the board.] … Nico, tell us what do you think she did? |

Nico: | I think she multiplied 2 by 12 and 2 by 24. She distributed the 2 to the 12 and 24. |

Pat: | You do 2 times 12x. |

Shawn: | I have a question … Don’t we need to do what’s in the parentheses first before multiplying by 2? |

Teacher: | Let’s talk about that. What about Shawn’s question? You know the rule that parentheses come first. |

Ayala: | I don’t think so. You can’t do what is in parentheses first because you can’t add 12x and 24. |

Jamie: | They are not like terms. |

Students continued to offer additional equivalent expressions with the teacher, bringing a more diverse array of expressions forward and pressing students to make sense of them. As the conversation continued, students began to get more creative and took more risks regarding the equivalent expressions they proposed to the class. In the next segment, students realized that they could use negative integers.

Ali: | Let’s see, –2(–12x + –24). |

Teacher: | We are going to have to stop for a second because Cam got really excited. |

Cam: | I did. |

Teacher: | Why did you get excited? |

Cam: | ̕Cause I didn’t think of a negative number. |

Teacher: | You didn’t think of a negative number. Do you think it works? |

Students: | Yes. |

Cam: | I do! [Students laugh.] |

Next, a student realizes that non-integer values, whether written as fractions or decimals, can be used to create expressions equivalent to the original expression in the task. The teacher makes sure that this student’s expression is shared with the class and her thinking that led to it is discussed.

(Adapted from Roy et al. 2017, pp. 101–104)

Kerry:Once I considered a decimal number [as a possibility to create an equivalent expression], I figured, yes, that could be right because 32 is bigger than 12 but if you multiply it by a decimal, it is going to get smaller; that would allow it to become 24. Fran:Thirty-two is bigger than 12 just like Kerry said. We need to multiply it by a number less than 1 to make it smaller … three-fourths of 32 is 24. [ Pause; some students reach for calculators to confirm Fran’s calculation.]Teacher:Justice, help them out. You said you do not need a calculator to figure out what to multiply 32 by. Justice:You really don’t ‘cause kind of what Fran said, 3 times 32 is 96 and then divide it [96] by 4, which equals 24. Kerry:I have another one; 64 times (.375 x+ .75).Teacher:[ Adds Kerry’s new expression to the list on the board.] Which one of the other equivalent expressions is this like? Bailey?Bailey:[ Points to32(0.75x+ 1.5)] They multiplied the 32 by 2 to get 64 and then divided the numbers inside the parentheses by 2.Mackenzie:There are an infinite number of ways [to write equivalent expressions]. Teacher:Hope? You have a thought? Hope:Yes, I kind of have a formula [

writing a generalization for the distributive property on the board].

$ay\left(\frac{b}{y}+\frac{c}{y}\right)$ Every time you multiply the number outside the parentheses you divide the numbers inside the parentheses because if you multiply both of them, the number will go over what you need, and you need to keep it equivalent.

Teacher:[ Pauses for a moment.] So, you [students] agree with her then? Cameron?Cameron:If you divide or multiply the number outside the parentheses, you use the inverse operation for the numbers inside the parentheses. Teacher:What did Cameron just say? Jamie:OK, if you do multiplication on the outside, you’ll have to do division on the inside, but if you divide on the outside, you’ll have to multiply on the inside.

The teacher wrapped up the whole-class discussion by asking students to summarize Hope’s conjecture in their own words, to explain why they think it works or does not work, and to provide an example to support their claim.

The eight Mathematics Teaching Practices are described and connected to the Equivalent Expression task. The descriptions of the eight Mathematics Teaching Practices are based on *Principles to Actions* (NCTM 2014b) and *Taking Action: Implementing Effective Mathematics Teaching Practices* (Smith, Steele, and Raith 2017).

**Establish mathematics goals to focus learning.**Mathematics learning goals should include both the concepts and procedural fluency students will develop, be clearly connected to learning progressions, and be used to guide decisions made during instruction. “A strong mathematics goal should be more focused than a standard. Also, if the ultimate goal is for students to learn the conceptual underpinnings behind a particular procedure or set of procedures, the goal should not be so narrow that it specifies a single algorithmic approach” (Smith, Steele, and Raith 2017, p. 18). Importantly, each and every student should have access to challenging mathematics learning goals and be given adequate time and instructional supports as needed to meet such goals. At the same time, productive classroom norms should be established regarding classroom participation. Doing so positions middle school students with a high level of mathematical agency because students’ status is equalized and each and every student’s contributions are valued (as described in Wood et al. 2019). In the Equivalent Expressions task, the mathematics learning goals were (1) for students to use numbers and properties flexibly to develop understanding of equivalence and (2) for students to discover a generalization through exploring equivalent forms of the algebraic expression. Both learning goals focused on conceptual underpinnings related to equivalence.**Implement tasks that promote reasoning and problem solving.**Effective mathematics instruction includes implementing tasks that engage students in ways that promote mathematical reasoning and develop mathematical understanding through problem solving. These types of tasks allow for multiple entry points and value students’ varied solution paths and strategies. Such tasks have a high cognitive demand, as outlined in two of the four Levels of Demands (see figure 4.1, Smith and Stein 1998). The levels of demand provide a tool for analyzing the cognitive demand of a given task and for considering how to increase the cognitive demand of a task. Teachers, schools, and districts should move toward implementing a greater proportion of tasks that are classified as having a higher level of cognitive demand. Measures should be taken to maintain the high cognitive demand of a task rather than actions that might lead to a decline in cognitive demand as the task is implemented. As the continued implementation of higher-level demand tasks become a classroom reality for students, they become empowered by doing the mathematics and by deeply connecting the procedures they use with their conceptual underpinnings, building student agency as they develop an understanding for why things in mathematics work the way they do. During the Equivalent Expressions task, there were multiple entry points for exploring equivalence, multiple mathematical arguments that could be made to justify the conclusion, and procedures that were used in the service of conceptual understanding. These cognitive demands placed on students are categorized as higher-level demands as identified in figure 4.1.

**Use and connect mathematical representations.**Effective teaching of mathematics includes students engaging in using different mathematical representations to make connections as they deepen their understanding of mathematics concepts and how those concepts connect to procedures. Students should connect within and across visual, physical, symbolic, contextual, and verbal representations for the mathematics they are learning (Lesh, Post, and Behr 1987), and doing so takes time (see figure 4.2). The straight arrows show the importance of connecting across representations and the curved arrows show the importance of connecting within different representations of the same type. Importantly, students should have the opportunity to select which representation(s) they will use to explain their thinking. Selecting tasks that are relevant to students and allowing them to make connections across representations in ways that authentically connect to their lives can positively support their developing mathematical identities and help them see purpose in the mathematics they are learning. While the Equivalent Expressions task did not connect authentically to students’ lives, it did draw on their prior knowledge of numbers and properties to dive deeply into considering multiple equivalent expressions represented symbolically to build flexibility with numbers. Because the teacher highlighted specific expectations at the beginning of the class, students knew they would be expected to explain and justify their thinking as well as make sense of another student’s symbolic representations verbally. These expectations aided students in developing a deeper understanding of equivalence. Working flexibly within and across two types of representations—symbolic and verbal—provides a snapshot of what is possible when students work across representations. While this short episode did not involve other representations, many meaningful mathematics and real-world tasks provide opportunities for students to use most or all of the representations in a single task.

**Facilitate meaningful mathematical discourse.**Facilitating meaningful discourse in the mathematics classroom engages students in contributing to the shared understanding of mathematical concepts and advances the learning of mathematics for the entire class. Preparing to implement meaningful mathematical discourse requires intentional planning and anticipation by the teacher as well as the careful selection of a task that lends itself to discussion.*The Five Practices for Orchestrating Productive Discussions*(Smith and Stein 2018) provides guidance to teachers as they work to plan and implement instruction rich with mathematical discourse. Creating a classroom environment that is rich with mathematical discourse, where each and every student is expected and positioned to participate, supports students’ positive mathematical identities, establishes shared mathematical authority in the classroom, and aids in equalizing student status (Wood et al. 2019). Facilitating mathematical discourse equitably means recognizing that students bring multiple forms of discourse and language to the classroom and that they are positive resources (Bartell et al. 2017). Acknowledging a student’s home language and that their home language is valid for doing mathematics (e.g., Setati 2005) and fostering positive classroom discourse in ways that are familiar to students (e.g., Howard 2001) are examples of being inclusive and equitable. The teacher that implemented the Equivalent Expressions task was careful to be inclusive when facilitating the whole-class discussion by engaging a wide range of students, allowing them to explain in words that made sense to them, promoting opportunities for students to comment on one another’s work, and keeping ownership of new ideas with students. As described by Gresalfi and Cobb (2006), students were “entitled, expected, and obligated to interact with one another as they work on content together” (Gresalfi and Cobb 2006, p. 51). Throughout the vignette there was evidence that a classroom environment was cultivated where students felt comfortable speaking up during the whole-class discussion, contributing ideas that were unique, and extending the ideas of their classmates (see Steele 2019 for extended analysis).**Pose purposeful questions.**Using purposeful questions can support teachers in assessing what students know about mathematical ideas and advancing students’ understanding by building on their existing knowledge. Teachers who consistently ask purposeful questions that gather information, probe thinking, make mathematics visible, and encourage reflection and justification position students as mathematically competent. This type of intentional questioning helps teachers to assess and advance students’ thinking, and it should be done in ways that focus learning rather than funnel learning (as described in Herbel-Eisenmann and Breyfogle 2005). Posing purposeful questions in ways that are equitable means that the teacher intentionally uses questioning to ensure that each and every student progresses in their thinking, is learning important and challenging mathematical ideas, and is developing a positive mathematical identity (Aguirre, Mayfield-Ingram, and Martin 2013). In the dialogue shared from the Equivalent Expression task, the teacher uses purposeful questions to make connections across student responses to reiterate and make sense of key ideas by expecting students to build upon the thinking of their peers. For example, the teacher used Shawn’s question, “Don’t we need to do what’s in the parentheses first before multiplying by 2?” to bring the class back to the idea of like terms.**Build procedural fluency from conceptual understanding.**“Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems” (NCTM 2014b, p. 42). Students often struggle when learning mathematics in ways that primarily focus on procedures, but when procedures are well connected and grounded in concepts, students can retain information better and apply what they know to new situations in more informed ways. Grounding teaching in conceptual understanding is essential to providing access to mathematics for each and every student because students have “a wider range of options for entering a task and building mathematical meaning” (Smith, Steele, and Raith 2017, p. 74). Further, building procedural fluency from conceptual understanding empowers students to make sense of the mathematics they are doing, helps them see the purpose of mathematics, and increases their level of agency. In the case of the Equivalent Expressions task, a focus was placed on strengthening students’ ability to use numbers flexibly and building their conceptual understanding of equivalence. During the task, students were able to use their newly developed ideas about equivalence to make connections to their prior knowledge of the distributive property, building the foundation for generalization and procedural fluency. This task and lesson stand in contrast to a set of exercises in which students might be asked to simplify expressions independently and check the accuracy of their answers.**Support productive struggle in learning mathematics.**Students should be given consistent opportunities, with support, to individually and collectively engage in productive struggle as they consider relationships among mathematical ideas and grapple with new learning. Students need time to wrestle with new ideas and develop conceptual understanding through productive struggle (Hiebert and Grouws 2007). Students are supported in their productive struggle by a teacher encouraging reflection, questioning, acknowledging effort, and providing adequate time, which requires intentional planning and anticipation by the teacher. Students’ perseverance through situations that involve productive struggle is empowering and supports positive mathematical identities and a strong sense of agency. The teacher in the Equivalent Expression task anticipated that students would need time to productively struggle past the use of only whole numbers to write equivalent expressions. The teacher also anticipated that discovering a generalization would be challenging. During the partner discussions that took place prior to the whole-class discussion, the teacher actively listened to each group and posed questions to push their thinking while mentally keeping track of different ideas that students could contribute to the whole-class discussion. As a result, the teacher carefully considered how to facilitate the whole-class discussion in ways where students’ individual contributions (which in part derived from the thinking with their shoulder partner) pushed forward the thinking of the group collectively and built systematically toward a generalization, moving through the use of whole numbers, then integers, and then rational numbers.**Elicit and use evidence of student thinking.**Eliciting evidence of student thinking is essential to assessing student progress and continually informing instruction as students progress in their understanding of mathematics. Using evidence of student thinking and understanding is how teachers can determine what students know at a given point and support students’ learning to move them forward in their mathematical understanding. When the implementation of high-level tasks is paired with students explaining and justifying their thinking, teachers are positioned to use questioning that builds on what students know to extend their thinking. Listening to students’ ideas to highlight important mathematics in the classroom can have a positive influence on students’ mathematical identities (Crespo 2000). Through this practice, students share in the authority of mathematical knowledge and are positioned as capable doers of mathematics and as contributors to the collective mathematics classroom conversation. The classroom discussion from the Equivalent Expression task built from the unique reasoning elicited from individual students. It is with this reasoning in mind that the teacher shared mathematical authority with the students in the class, and as a result, empowered them as important contributors to their mathematical knowledge as well as the collective knowledge of the whole class.

The Mathematics Teaching Framework (see figure 4.3) shows the intersectionality among the eight Mathematics Teaching Practices and discusses how the framework can guide daily mathematics instruction in ways that foster students’ positive mathematical identity, strong sense of agency, and a shared classroom authority. The framework should be seen as a way to strengthen the existing daily work of teaching, not as something additional.

The Mathematics Teaching Framework begins with establishing mathematics goals to focus learning. Implementing any mathematics goals should be situated within norms for participation. In the vignette shared, students were challenged and empowered to author their mathematical ideas, share their explanations, and justify their reasoning—as a result, the mathematical authority was shared among the students and the teacher. After mathematics goals are established, tasks should be selected that promote reasoning and problem solving and in which procedural fluency is built on the foundation of conceptual understanding. Over the course of a unit of instruction, tasks that prioritize conceptual understanding occur early in a unit, with the development of procedural fluency taking place over time. Focusing mathematics instruction in this way helps to level the playing field, potentially allowing each and every student to engage with the task and participate in the mathematical understanding through connections across mathematical ideas. Students are continually positioned to make sense of the mathematics they are doing and to see the purpose in the mathematics they are learning, which helps to establish positive mathematical identities. The practices within facilitating meaningful mathematical discourse all have important implications for mathematical identity, agency, and authority as they work seamlessly together to ensure students see the connections and are positioned as capable mathematics contributors. Instruction is intentionally planned so questioning is purposeful and student thinking is used to assess and inform future instruction.

As previously described, good teaching is necessary but not sufficient in achieving equitable instruction. Here, a sampling (not intended as an exhaustive list) of ideas and strategies beyond NCTM’s eight Mathematics Teaching Practices are shared to invite collegial yet critical conversations about the equitable instructional practices used in schools or districts to meet the needs of each and every student.

Culturally Relevant Pedagogy, outlined by Ladson-Billings (1995), and Culturally Responsive Teaching (CRT), outlined by Gay (2000, 2018), provide important guidance focused on the role students’ cultures have in ensuring that instruction is equitable. Ladson-Billings’ work primarily seeks to influence attitudes and dispositions, ultimately describing a position a teacher might adopt to be culturally relevant. Culturally relevant pedagogy goes beyond superficial changes such as using different names in word problems to represent the students in the classroom or displaying posters and creating bulletin boards that include diverse images. Rather, culturally relevant pedagogy requires attention to three components simultaneously: (1) academic success, (2) cultural competence, and (3) sociopolitical consciousness. Ladson-Billings characterizes these components and states that academic success is the

… intellectual growth that students experience as a result of classroom instruction and learning experiences. Cultural competence refers to the ability to help students appreciate and celebrate their cultures of origin while gaining knowledge of and fluency in at least one other culture. Sociopolitical consciousness is the ability to take learning beyond the confines of the classroom using school knowledge and skills to identify, analyze, and solve real-world problems. (Ladson-Billings 2014, p. 75)

Gay’s work focuses on specific actions teachers can take to be culturally responsive. Based on the assumption that academic learning is more meaningful and rich when situated in the lived experiences of students (Gay 2013), culturally relevant teaching is an expression of the values, beliefs, and knowledge that recognize the importance of racial and cultural diversity (see example excerpts in Thomas and Berry 2019).

Cohen and colleagues (2002) describe complex instruction as a collection of strategies used for creating equitable classrooms that enable teachers to “… teach to a high intellectual level in academically and linguistically heterogeneous classrooms” (p. 1047). Complex instruction seeks to address challenges to group work that inhibit learning (Featherstone et al. 2011). As described by Featherstone and colleagues (2011), with complex instruction, students are given formal roles (e.g., facilitator, resource monitor) that define their individual duties within the work of their group. Other complex instruction strategies include establishing classroom norms for group work that includes ideas such as showing respect, helping does not mean giving the answer, and we are smarter when we all work together. Further, a key goal of complex instruction is addressing the concept of unequal student status. In particular, teachers implementing complex instruction attend to the social ranking of students in the classroom, which is often determined by who is perceived as good at mathematics. Complex instruction provides teachers with strategies to counter status issues, including assigning some students as competent by introducing tasks in ways that foster different expectations for different students. Horn (2012) describes five practices to help change teaching toward more equitable instruction: listen carefully (to students’ ideas and thinking), watch your pace, connect to students, have acceptance and high expectations for students, and model a stance of humility and courage (pp. 88–93). These ideas overlap in part with NCTM’s eight Mathematics Teaching Practices; for example, “listen carefully” aligns with “elicit and use evidence of student thinking.”

Aguirre, Mayfield-Ingram, and Martin (2013), in their book *The Impact of Identity in K–8 Mathematics Learning and Teaching: Rethinking Equity-Based Practices* describe five equity-based mathematics teaching practices to both strengthen students’ mathematical learning and cultivate their positive mathematical identities (pp. 43–44). The first practice, *going deep with mathematics*, focuses on engaging students in high cognitive demand tasks that develop their conceptual understanding, procedural fluency, reasoning, problem solving, and mathematical proficiency. The second, *leveraging multiple mathematical competencies*, highlights the importance of drawing from students’ mathematical strengths, which all students have, and recognizing and positioning their backgrounds and knowledge in positive ways. The third practice, *affirming mathematics learners’ identities*, describes that “Instruction that values multiple mathematical contributions, provides multiple entry points, and promotes student participation in various ways (teams, groups, and so on) can aid the development of a student’s mathematical learning identity” (p. 43). The fourth practice, *challenging spaces of marginality*, stresses the importance of diminishing student statuses in the classroom, valuing multiple mathematical contributions, and embracing competencies. The fifth practice, *drawing on multiple resources of knowledge,* focuses on being intentional about drawing from students’ prior knowledge and experiences—which can include those that are mathematical as well as linguistic, cultural, peer, family, and community—as important resources for learning mathematics.

Bartell and colleagues (2017, pp. 11–12) identified nine equitable mathematics teaching practices and the research supporting those practices. The nine practices include (1) draw on students’ funds of knowledge, (2) establish classroom norms for participation, (3) position students as capable, (4) monitor how students position each other, (5) attend explicitly to race and culture, (6) recognize multiple forms of discourse and language as a resource, (7) press for academic success, (8) attend to students’ mathematical thinking, and (9) support the development of a sociopolitical disposition. The nine practices represent a synthesis of the literature recognizing the important contributions of many pivotal equity-focused mathematics education researchers but is not exhaustive of all aspects that may inform equitable teaching. Further, these nine practices do have overlap and complement NCTM’s eight Mathematics Teaching Practices as well as other equity-based instruction ideas.

The Universal Design for Learning (UDL) framework was developed to “improve and optimize teaching and learning for all people based on scientific insights into how humans learn” (CAST 2018, para. 2). The UDL framework is organized by the why, what, and how of learning, and the guidelines focus on providing multiple means of engagement, representation, and action and expression. While UDL has a strong presence in the special education literature, it is relevant to all students as it is grounded in the idea that if a teacher can anticipate the needs of individual students and plan strategies to meet those needs, then all students in the class can benefit (Buchheister, Jackson, and Taylor 2014).

Figure 4.4, from *Catalyzing Change in High School Mathematics* (NCTM 2018), crosswalks equitable teaching practices and the eight Mathematics Teaching Practices from *Principles to Actions* (NCTM 2014b). This crosswalk is meant to spark conversations regarding intentional steps that can be taken in middle school to implement the eight Mathematics Teaching Practices in the most equitable ways and to represent some of the key ideas described in this chapter and by other researchers in mathematics education.

Implementing equitable mathematics instruction is an ongoing, collective commitment that must be made by teachers, schools, districts, and beyond to ensure that each and every student has access to high-quality mathematics instruction. Investment solely by individual teachers is not enough. Action must be supported by school and district leadership and approached as a collective vision for implementing the highest quality mathematics programs. This collective effort is described through NCTM’s Professionalism Principle, which states:

In an excellent mathematics program, educators hold themselves and their colleagues accountable for the mathematical success of every student and for personal and collective professional growth toward effective teaching and learning of mathematics. (NCTM 2014b, p. 99)

A key component of the Professionalism Principle is recognizing and embracing the notion that one’s learning as a teacher is never complete and that building a culture of positive collaboration is needed for both the teacher as an individual and for the collective whole (Berry and Berry 2017). Sometimes middle school mathematics teachers work closely with an interdisciplinary group of teachers that collectively teach the same group of students (often known as a team or core model), whereas others are in schools that function more as a junior high school and work most closely with other mathematics teachers. Further, some teachers come with expertise in elementary education, some received preparation specific to middle school and working with young adolescents, and yet others come from secondary preparation programs. Regardless of the pathway to teaching, middle school teachers of mathematics need time to collaborate, plan, and reflect together as a learning community (NCTM 2014b).

Schools should purposefully prioritize creating time and space during teachers’ daily work time for teacher collaboration to focus both on planning and addressing patterns of inequities. Professional collaboration among teachers aligns with Hattie’s (2018) meta-analysis research on collective teacher efficacy. Collective teacher efficacy is the collective belief that a group of teachers share in their ability to positively affect students, and this construct has one of the highest effect sizes with regard to being correlated with student achievement. This finding provides strong support for building structures in schools and districts that prioritize time for teacher collaboration. The focus of this teacher collaboration time should be both on equitable instruction and content (discussed in chapter 5), and then determining how teachers will know whether students have learned the mathematics identified (Kanold and Larson 2015). Consistent and appropriate mathematical language, models, and agreed-upon strategies should be used across the entire school, which can be described as a Mathematics Whole School Agreement (Karp, Bush, and Dougherty 2016). Such collaborations should include investigating student work, discussing instructional strategies of lessons, data analysis, and critical conversations centered on implementing instruction in ways that are equitable as described in this chapter. Teachers and instructional leaders can work together to assess and reflect on instructional practice, keeping goals for instructional improvement in mind and celebrating successes (Boston, Candela, and Dixon 2019). While time can certainly be an obstacle, lack of time to collaborate leads to professional isolation (NCTM 2014b) and both “good” and “bad” teaching can go unnoticed (NCTM 2018, p. 35).

Furthermore, NCTM’s *Principles to Actions* (2014b) points to the following productive beliefs (adapted from pp. 102–103) related to professionalism and collaboration:

- The development of expertise as a teacher of mathematics is a career-long process.
- Knowledge for teaching mathematics includes mathematical content knowledge for teaching, mathematical pedagogical knowledge, and knowledge of students as learners.
- When mathematics teachers collaborate with colleagues both inside and outside of their school, they are more effective.
- All teachers of mathematics, even the most experienced teachers, can benefit from mathematics-focused instructional coaching.
- Schools and districts should prioritize content-focused collaborative planning time.
- Teachers become master teachers because of their continual dedication to improvement over time.
- Purposeful and collaborative planning leads to effective mathematics teaching.

Supporting incoming and early career teachers is an important endeavor, which often includes supports such as induction programs, mentoring, and additional supports coordinated by school districts, individual schools, partnering universities and their teacher education programs, and more. This includes dedication to teacher recruitment, including intentional efforts to recruit those who are underrepresented in the field of education, such as teachers of color. Supporting the newest teachers in the field is critical to overall mathematics teacher retention (Amick et al. 2020). Further, strong partnerships between university teacher education programs and school districts can provide invaluable benefits to both and support the recommendations of *Catalyzing Change*. The “well-prepared beginning teacher” participates in collaborative communities of practice, engages in ongoing professional learning communities, and actively partners and collaborates with community members, community-based organizations, middle school mathematics teachers, and other stakeholders (Association of Mathematics Teacher Educators [AMTE] 2017).

Education preparation providers should be guided by the vision and goals of the AMTE document *Standards For Preparing Teachers of Mathematics* (2017) when preparing teacher candidates to teach mathematics in middle school, including teacher candidates who will receive mathematics certification specific to middle school, grades PK–8, grades 7–12, and other grade bands, as well as English as a second language K–12 and special education K–12 teachers. AMTE provides guidance to the roles of mathematics teacher educators (AMTE 2017) as well as four standards for the well-prepared beginning teacher of mathematics that include (1) mathematics concepts, practices, and curriculum; (2) pedagogical knowledge and practices for teaching mathematics; (3) students as learners of mathematics; and (4) social contexts of mathematics teaching and learning (AMTE 2017, p. 6). In 2018, an AMTE equity committee survey completed by 292 mathematics teacher educators found that “more than half of respondents disagreed or strongly disagreed that their program completers understand issues of power, privilege, race or racism, or other systems of oppression that their students experience” (AMTE 2018, para 7). This notable finding and others from the survey identify areas of programmatic need for mathematics teacher educators and the teacher candidates they prepare.