Key Recommendation: Middle school mathematics should dismantle inequitable structures, including the tracking of teachers as well as the practice of ability grouping and tracking students into qualitatively different courses.

We have to recognize that even if people are just, even if they desire to be just, a society may not be just if its structures and practices are not also just.

(Su 2017, p. 489)

Are each and every student in all mathematics classrooms empowered as confident and capable learners and doers of mathematics? How do current state, provincial, district, and school policies and practices restrict or support equitable, just, and inclusive mathematics education? “Children of certain racial, ethnic, language, gender, ability, and socioeconomic backgrounds experience mathematics education in school differently, and many are disaffected by their mathematics education experience” (Aguirre et al. 2017, p. 125). Stakeholders should reflect on ways to individually and collectively examine the current practices in their middle schools to make systemic changes to dismantle inequitable structures.

Consider this scenario at one local middle school where Ms. Jacon and Ms. DeWitt are seventh-grade mathematics teachers. Ms. Jacon teaches the classes labeled as “general” and Ms. DeWitt teaches the “advanced” classes. Ms. Jacon has 28 students in her second period, of which 43 percent are Black, 28 percent White, 25 percent Latinx, and 4 percent Multiracial, with 11 percent identified with a learning disability, 8 percent categorized as emergent bilinguals (also called multilingual learners because English may be their third or fourth language), and 58 percent who receive free or reduced price for school lunch. Ms. DeWitt also has 28 students in her second period, of which 81 percent are White, 11 percent Asian, 4 percent Black, and 4 percent Latinx, with 7 percent who receive free or reduced price for school lunch. The seventh-grade mathematics teachers co-plan to ensure they are teaching the same or similar topics each day. In the lesson that follows, the topic is proportional relationships.

Ms. Jacon begins her lesson by telling her students that a proportion is two equivalent ratios and shows students that the ratio 4/5 is the same as the ratio 88/110. She further tells her students they can determine whether two ratios are equivalent by finding the cross products. Ms. Jacon states, “If the cross products are the same, then the ratios are equivalent.” Ms. Jacon shows students how to find the cross products, and then has her students work 10 similar problems as she walks around the room to make sure they are on task. After giving students the problems, the following dialogue occurs:

Ms. Jacon:

Now, I am really going to challenge you! I am going to tell you these [writes on the board] are equivalent.

78=x78

I want you to solve for x. Our first step is to multiply 7 and 78. Everyone write this down as I write it [writes on the board]. We are going to set 7 × 78 equal to x times 8. What is 7 times 78?

7×78=x×8

Arabella:546.
Ms. Jacon:Perfect! So, now we write 546 = 8x. Our next step is to divide both sides by 8. Joquain, why do we have to divide both sides by 8?
Joquain:Because you are multiplying the 8 with the x and you have to get x by itself.
Ms. Jacon:Excellent thinking! Yes, you have to get x by itself. What is 546 divided by 8?
Thomensina:68.25.
Ms. Jacon:You guys are on the ball today! So, x equals 68.25. Now I want you to continue to work on the problems on your own.

Students individually work on the problems.

Meanwhile, Ms. DeWitt has her students seated in small groups collaborating to solve the following problem:

A cookie jar contains 3 chocolate chip cookies and 7 oatmeal cookies. You want to bake cookies with the same ratio of chocolate chip cookies to oatmeal cookies to bring to your friend’s party because you know the guests attending tend to prefer oatmeal cookies. You know you want to bring 24 chocolate chip cookies. How many oatmeal cookies should you bring? Explain how you know.

As the students work on the problem, Ms. DeWitt circulates throughout the room listening to students’ conversations. As she goes from group to group, she asks her students questions such as, “How do you know?” “How could you prove that?” or “Do you agree or disagree with what [Philip] is saying? Why?”

At the end of the day, Ms. DeWitt went to Ms. Jacon’s room to debrief. Ms. DeWitt shared how the task she gave really helped her students reason and think about proportional relationships. She expressed that her students were coming up with a variety of strategies to make sense of the problem. Ms. Jacon commented, “My low students are not able to solve problems like these. If I gave them a problem like this, there would be mass chaos! We did have a successful lesson at the beginning of class today. They were following along and could do each step of the problem, but when I let them begin to work individually, they acted like they did not know what to do. I tell them and tell them, yet they still don’t get it. I’m not sure what to do.”

School and district leaders began to notice that even when the same content was taught, the mathematical learning experiences students received varied greatly. Further, district leaders found that during the past five years, Black and Latinx students, students with identified disabilities, students from low socioeconomic backgrounds, and emergent bilinguals were not proportionally represented in advanced mathematics classes; rather, they were overrepresented in lower-tracked classes. The district leaders invested time talking with teachers and discovered that although the teachers in each grade taught similar mathematical topics across the existing tracks, the pedagogical practices implemented and the types of tasks within the classrooms varied greatly, resulting in qualitatively different learning opportunities and noticeably different learning outcomes. The leaders knew that changing the course pathway in their middle school mathematics program would not be enough. They would have to invest both time and resources to fully support their teachers, who cared deeply about their students, in this effort.

Shifting from Deficit-Based to Strength-Based Beliefs about Students

Students in middle school are diverse in many ways, such as in their racial and ethnic background, socioeconomic status, language, exceptionalities, religion, sexual orientation, and physical and health differences, including vision, speech, and hearing as well as critical illness. Students in middle school are also developing their own identities of who they are and who they will become as individuals while they simultaneously develop their mathematical identities. One way that young adolescents’ mathematical identities are influenced is through biases of others manifested in mathematics education.

Although teachers bring many strengths to their classrooms, deficit views are deeply rooted in the broader culture of mathematics education and are grounded in long-standing structures and practices. In many cases, students, particularly those historically marginalized (such as students from low socioeconomic backgrounds, students of color, emergent bilinguals, and students with disabilities), may receive fewer opportunities to engage in high-quality academic experiences (TNTP 2018, n.p.). In a study of five different school systems (e.g., urban and rural, charter and school district) across the United States, TNTP researchers (2018) observed approximately 1,000 lessons, reviewed about 5,000 assignments, analyzed more than 20,000 samples of student work, and collected about 30,000 student surveys. They found that in four out of ten classrooms where the majority of students were students of color, students were not given the opportunity to work on assignments that were at grade level (TNTP 2018). It is critical that beliefs that lead to such practices are acknowledged and confronted so that efforts are implemented to challenge and change beliefs and practices across schools and districts.

Systemic change is critical. The brilliance and capability of each and every student must be acknowledged and purposefully planned for in mathematics instruction so that all students are positioned as active participants in the mathematics classroom. To do so, teachers must intentionally position students as competent mathematics learners (Berry 2018c) and be supported in these efforts by school and district leaders. Affirming students’ strengths daily positions students in ways that cultivate positive mathematical identities.

Conscious and unconscious beliefs exist about what each and every student can learn and do in the mathematics classroom. These beliefs translate into equitable or inequitable practices by either positioning each and every student to be or not to be learners and doers of mathematics. “Positioning students as capable is not just a mindset but also an explicit practice that requires teachers to purposefully notice and highlight students’ strengths rather than to attend to what they are lacking” (Wager, Pietz, and Klehr 2017, p. 103). Blacks, Latinx, students with disabilities, emergent bilinguals, and other marginalized groups have historically been positioned in the mathematics classroom as less capable (Borgioli 2008; Gutiérrez and Irving 2012; Lambert 2018; Langer-Osuna et al. 2016; Martin 2013; Mosqueda 2010). A growing number of studies show that variation in gender-based trends of mathematics achievement and participation are shaped by students’ gendered mathematics experiences in which teachers’ beliefs, differential treatment, and opportunities for participation impact students’ mathematics success (Levya 2017). Specific to race, Battey and Leyva (2018) suggest implicit racial attitudes that are “unconscious feelings and beliefs … [that] can reside in individuals or teachers even though they profess overt beliefs in being equitable” (p. 23) exist in today’s society. Implicit racial attitudes inadvertently lead to lower teacher expectations for Black students, which often results in mathematics instruction that focuses on memorized facts and procedures and the use of a single strategy to solve a mathematics problem (Lubienski 2002). Principles to Actions (NCTM 2014b) asserts, “The question is not whether all students can succeed in mathematics but whether the adults organizing mathematical learning opportunities can alter traditional beliefs and practices to promote success for all” (p. 61). While inequities have many intricate influences and considerations, the key is that each and every student has access to a high-quality mathematics program.

Disrupting deficit practices requires educators to intentionally unpack their deficit beliefs and take steps to focus on strengths. One step in this process is to identify, explore, and recognize their own strengths to then leverage them to address areas of growth (Kobett and Karp 2020). Once educators build on one area, they can use that success as a catalyst to build more success. This kind of purposeful self-examination positions educators to think differently about what is (and is not) working in classrooms. Another step toward developing a strengths-based perspective about students requires educators to identify and acknowledge deficit beliefs they may espouse regarding the diverse students in their mathematics classroom (Kobett and Karp 2020). For example, an educator might espouse a deficit-based view about a student such as “Jaden comes from a broken home. His family does not care. They never attend parent-teacher conferences. They do not care about Jaden’s education. Jaden does not have any family support, and he has a difficult time understanding mathematical concepts. Unfortunately, he will turn out to be just like his older brother.” This educator would need to question what evidence there is for such beliefs about Jaden and his family, and then work to change the deficit-based narratives and to reframe their perspective to provide opportunities for Jaden to be viewed as a learner and doer of mathematics. Part of this process requires collectively holding each other accountable for shifting such beliefs. For example, in the case of Jaden, an appropriate response from one colleague to another could be, “What supports does Jaden have? What mathematical concepts does Jaden understand? When is Jaden most successful?” It is important to remember that families and student caretakers show that they care about their student’s education in a myriad of ways, such as helping with homework or finding someone else to provide support, making sure their student has good attendance, or believing that if something is not going well, a teacher will reach out so as to best support the student (Bartell and Drake 2019).

Shifting deficit beliefs is often challenging as people generally do not recognize the deficit perspectives they hold; instead they view these perspectives as normal rather than developed over time through stereotypical, and then institutionalized, cultural narratives (Gorski 2011; Sleeter 2008). Thus, it may be difficult to identify and implement needed changes in beliefs in isolation; rather, support from colleagues or through professional development opportunities can initiate the confrontation, identification, and examination of deficit-based beliefs about students (or their families and communities). Critical conversations and self-reflective discussions among instructional leaders, middle school mathematics teachers, special education teachers, English as a second language teachers, and instructional support staff should become commonplace to ensure that all students are taught mathematics from a strengths-based perspective. The commitment to openly discuss and realign current beliefs and to seek out counterevidence to stereotypical assumptions will shift the narrative to one in which all students are viewed as learners, contributors, and doers in the mathematics classroom.

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Ability Grouping and Student Tracking

Ability grouping and tracking sort middle school students in two different ways. Ability grouping is placing students in small groups based on their perceived ability level within a classroom and is not included on students’ records such as report cards and transcripts (Houghton n.d.; Loveless 1998, 2009). Tracking places students into vastly different mathematics classes, in terms of content and pedagogy, on the basis of their perceived ability (Keiser 2002; Oakes 1985), test scores, or teacher evaluations, which are documented on students’ records and transcripts (Kelly 2007; Loveless 1998, 2009). The practices of student ability grouping and tracking have been prominent in schools since the beginning of the 20th century. The systemic structure of homogeneous grouping that reified and validated the unequal power dynamics in education more than 100 years ago still continues even today (Ellis 2008).

ABILITY GROUPING

Ability grouping, which occurs within individual mathematics classrooms, has been argued by some to be an effective practice as it allows teachers to homogeneously group students for the purpose of targeted instruction. Within this structure, students are not forced to wait for other students who are currently struggling nor are they rushed through the content if they need more time. Students who understand a mathematical concept have the opportunity to progress at a faster pace or work on a more challenging problem. Ability grouping in mathematics education is an inequitable structure that perpetuates privilege for a few and marginality for others. It is important to note that there is a distinction between ability grouping, which is an inequitable structure, and appropriate differentiation. Ability grouping places small groups of students with similar perceived achievement together within a classroom. Conversely, differentiation appropriately targets instruction on the basis of students’ current needs and plays a needed role in flexible instructional supports described at the end of this chapter.

Consider the following classroom episode. Mr. Mabson, a sixth-grade mathematics teacher at a local middle school, has taught mathematics for 15 years. Mr. Mabson has 25 students in his second period class of which 52 percent are White, 28 percent Black, 12 percent Latinx, and 8 percent Asian, with 8 percent of the group identified with a learning disability (George and Adanya) and 8 percent of the class categorized as emergent bilinguals (Ciara and Tony). Upon entering Mr. Mabson’s classroom, one would notice his students’ desks are in groups of four, with the exception of four desks that are spread throughout the classroom; there are two desks in the back on opposite sides of the room and a desk on either side of his desk, which is at the front of the classroom on the far right. Mr. Mabson purposefully arranges his students to sit in homogeneous groups according to his perception of their mathematical ability. Mr. Mabson explains that he believes his practice of class ability grouping improves his students’ learning. He contends that his classroom arrangement makes it easier for him to help students who are currently struggling and to provide extension tasks to his “advanced” group of students. He intentionally groups his students with identified learning disabilities together, provides them with direct instruction, and tells them that they will learn more when they go and see Ms. Whitewater, the special education teacher. He also groups the emergent bilingual students together, spending considerable time teaching them English by emphasizing mathematics vocabulary and telling this group of students to speak only in English.

After taking attendance, Mr. Mabson begins the lesson by providing the definition of a ratio and asks students to solve the following problem:

Jackson is buying invitations for his party. The party store sells a box of 24 invitations for $10 or a box of 60 invitations for $25. Which box is the better buy and why?

As the students are working in their small groups, Mr. Mabson circulates to each group and listens to the discussions. The following are some of the conversations Mr. Mabson overheard. First, Mr. Mabson went to the group he identified as “advanced”, which was having the following conversation:

Tara:Well, we know that 24 invitations cost $10. So, to find the price of one invitation, we just divide. Okay, 10 divided by 24, what is that?
Jason:It is 0.416666…. So, one invitation costs about $0.42.
Jasmine:That is a cheap invitation!
Tara:No kidding!
Timothy:Now, let’s figure out how much one invitation is in the box of 60.
Jasmine:We do the same thing … divide the total cost by the number of invitations. Let’s see … 25 divided by 60.
Tara:It’s the same amount! I got 0.41666.
Timothy:Mr. Mabson! Are you trying to trick us? The invitations in both boxes cost the same. They are both good deals!
Mr. Mabson:Wow! This group is so advanced and so fast! Now, I want to stretch your brains a bit. I want you to find a different size box of invitations where the invitations each cost the same as those in the boxes of 24 and 60.

Next, Mr. Mabson went to his students that he identified as “currently struggling,” who were grouped with the students with identified learning disabilities. He approaches the group and starts the following conversation:

Mr. Mabson:What if you take the numbers you were given in the problem and divide them? Okay, to solve this problem you will divide the numbers to see which one is the better price.
George:Okay, 24 divided by 10 is 2.4.
Adanya:I am not sure what 25 divided by 60 is … I need to use my calculator.
Billy:[Entering the numbers into his calculator] It is 0.416666667.
Mr. Mabson:No George, you have to divide the numbers the other way. You need to take 10 divided by 24.
Micah:Why?
Mr. Mabson:Ms. Whitewater will explain it to you more when you go see her this afternoon.
Adanya:So, is neither the better buy?
Mr. Mabson:That is correct.

Finally, Mr. Mabson goes to his group of emergent bilinguals where he sees them drawing on their papers. The conversation unfolds like this:

Mr. Mabson:[Pointing to the problem] Ciara, what are you supposed to be working on?

Ciara looks down at her paper and shrugs her shoulders.

Mr. Mabson:Tony, read the problem out loud.
Tony:[Translating the problem to Spanish and reading it to Ciara] Jackson está comprando invitaciones para su fiesta. La tienda de fiestas vende una caja de 24 invitaciones por $10 o una caja de 60 invitaciones por $25. ¿Cuál es la mejor oferta y por qué?
Ciara:

Tenemos que encontrar el precio de uno …

(Translation: We need to find the price of one …)

Mr. Mabson:[Interrupting]. No, what does the problem say in English? You need to speak English.
Tony:[Reading the problem in a mixture of Spanish and English] Jackson is buying invitaciones para su fiesta. The party store sells una caja de 24 invitaciones por $10 o una caja de 60 invitaciones por $25. Which is the better buy and why?
Mr. Mabson:No, the problem says, “Jackson is buying invitations for his party. The party store sells a box of 24 invitations for $10 or a box of 60 invitations for $25. Which is the better buy and why?” Now I want you to say it after me.

Mr. Mabson reads the problem again and the students repeat what he says.

While Mr. Mabson fosters agency and extends his “advanced” students mathematical thinking, he believes he is helping his students who are currently struggling and the ones who have an identified learning disability by telling them exactly what to do to successfully solve the problem. Even though this practice, on the surface, appears to be productive, it is not. In fact, it minimizes the opportunity for this group of students to reason mathematically. It does not position the students to be learners and doers of mathematics. Moreover, instead of allowing Tony to use his language as an asset to engage Ciara in participating in reasoning with mathematics, Mr. Mabson negatively regards Tony’s bilingualism as a deficit to his learning. Although Mr. Mabson has the best intentions for all his students, his perceptions of what his students can and cannot do are manifested in his instruction, which results in vastly different learning outcomes for his students.

Now, consider Ms. Roderick, who teaches across the hall from Mr. Mabson. Ms. Roderick’s sixth-grade mathematics classroom has similar student demographics as Mr. Mabson’s, but she heterogeneously groups her students and maintains high expectations for collaboration within each group. Ms. Roderick is teaching a similar lesson on ratios, and as she circulates the classroom, she overhears the following conversation:

Laura:Let’s use a ratio table.
John:Okay. Me and Jorge will work on the first one, and you and Jessica can do the ratio table for the second box of invitations.
All:OK.
John:We are finished. Our table looks like this. What does yours look like?
Number of Cards24487296
Total Cost10203040
Jessica:Ours look like this.
Number of Cards60120180240
Total Cost255075100
Laura:OK. I think Jessica’s and mine is the better deal because we can always get more cards with more money.
John:I disagree. I think mine and Jorge’s is the better deal because we spend less money.
Ms. Roderick:Jorge, do you agree with John?
Jorge:Si.
Ms. Roderick:Why do you think yours and John’s is the better deal?
Jorge:This number [pointing to the 40] lesser than cein (Translation: 100).
Ms. Roderick:I see how you are thinking about that, but how many cards do you have for each?
Jessica:We have 240 and they have 96. Wait a minute! We can’t compare the cost yet.
Ms. Roderick:Talk in your group about what you need to do next.

Ms. Roderick walks away and the students begin talking about their next steps.

Ms. Roderick positions all of the students as learners and doers of mathematics. She did not allow any students’ perceived ability to dictate who could or could not participate, learn, or make sense of the mathematics. Instead, she fostered students’ ability to reason and make sense of the mathematics by asking them questions to push their thinking forward. It is important that each and every student is given agency and is positioned as a mathematical learner and doer. In sum, this emphasis on agency rarely occurs for all students in a classroom where they are ability grouped; it more often happens in heterogeneously grouped classes.

STUDENT TRACKING

Tracking is the practice of placing students into qualitatively different course pathways or qualitatively different mathematical learning experiences. It reinforces the misguided notion that only some people are capable of achievement in mathematics (Boaler 2011). These fixed sequences of mathematics courses are problematic because they often place students into vastly different tracks, or levels of the same course, with frequently assigned “deterministic” labels for each track, such as “gifted,” “honors,” “college prep,” “regular,” “intensive,” or “remedial.” A major concern is that a student’s placement into these tracks is often based on nonacademic factors, including race, gender, language, socioeconomic status, perceived but not potential academic ability, classroom behavior, or other variables (Stiff and Johnson 2011). Even though research documents that tracking is detrimental for many students because it can lead to qualitatively different learning experiences, the inequitable structure of tracking is still prevalent and unquestioned in many school districts (e.g., Boaler 2007; Chmielewski, Dumont, and Trautwein 2013; George 1988; Oakes 1985). Among the systemic questions that must be raised are “Why does tracking continue to exist in many middle school mathematics programs?” and “Who primarily benefits?” Oakes and Lipton (1999) argue in their essay, “Access to Knowledge: Challenging the Techniques, Norms, and Politics of Schooling” that—

Those who promote ability grouping, special education, gifted programs, and the myriad other homogeneous instructional groups in schools claim that these classifications are objective and color blind, rather than, as Goodlad suggests, reflecting myths and prejudices. Advocates of grouping sometimes explain the disproportionate classification of white students as gifted or advanced and students of color as slow or basic as the unfortunate consequence of different backgrounds and abilities. These claims are sometimes based on objectivity and century-old (and older) explanations of differences that are neither scientific nor bias-free. (p. 133)

As a result of the practice of student tracking, Blacks, Latinx, and students from low socioeconomic status are often overrepresented in remedial intervention courses (Boaler 2011; Larnell 2016; Steele and Huhn 2018). The racial and socioeconomic stratification caused by the practice of tracking reveals that students of poverty, Blacks, and Latinx are disproportionately placed in low-tracked courses, which have been characterized as having an “exclusive focus on basic skills, low expectations, and the least qualified teachers” (Heubert and Hauser 1999, p. 282). Emergent bilinguals are generally placed in lower tracks based on their developing English proficiency rather than their academic ability (Callahan 2005; Kanno and Kangas 2014). The majority of students enrolled in advanced mathematics courses are disproportionately White and Asian (National Science Board 2018).

When students are placed in lower-track courses, they continue on this same limited pathway throughout high school (Domina et al. 2019; Ellis 2008; Oakes 2005). Tracking provides few opportunities for upward student mobility (Burris and Garrity 2008; Domina et al. 2019; Kelly 2007; Lucas and Good 2001; Mulkey et al. 2005). For example, Burris and Garrity (2008) examined student transcripts from their school district and patterns emerged based on the track in which students were placed. Although students were often told they would have the opportunity to move up from different tracks, none of the transcripts showed evidence of this occurring. In fact, student transcripts revealed the opposite, where students who took courses on the middle track were sometimes moved down to the lowest track. When students who had the mathematical ability to be successful at higher levels were inadvertently placed on a lower level track, their mathematics achievement decreased (Stiff, Johnson, and Akos 2011), and students’ performance became a self-fulfilling prophecy (Flores 2007).

Pushing middle school students to race to algebra I (or first course in high school sequence if other than algebra I) in middle school is not appropriate, and this recommendation aligns with Catalyzing Change in High School Mathematics (2018), which states that students should not rush to calculus. However, both of these practices are often promoted through inequitable policies at the state or provincial, district, and school levels. Students of marginalized groups are disproportionally underrepresented in algebra I in middle school. Historically, algebra I in middle school was only offered to a perceived elite group of students (i.e., white middle- to upper-middle class males [Chazan 2008; Clements et al. 2012]). Moses and Cobb (2001) argued that algebra was the new civil right initiative for students of color from low socioeconomic backgrounds, and as a result Moses started the Algebra Project in a middle school. The Algebra Project and other such efforts are a catalyst for the field in embracing and acting on the belief that each and every student is mathematically capable and deserves access to courses that increase opportunities for their professional and personal futures.

During the last three decades, the movement to have more students take algebra I in middle school was well intentioned, but in the end, it contributed to inequities. For example, Faulkner and colleagues (2014) conducted a secondary analysis of the Early Childhood Longitudinal Study that followed students from kindergarten to eighth grade and found that “… the odds of placement in algebra by eighth grade for Black students were reduced by two-thirds to two-fifths compared to their White peers” (p. 304). In another study, Spielhagen (2006) found that students who received free or reduced-price lunch were less likely to be enrolled in algebra in eighth grade. The practice of student tracking reinforces the social inequities experienced by Blacks, Latinx, and students living in poverty (Larnell 2016).

In summary, tracking is inequitable and leads to qualitatively different mathematical learning experiences for students. The systemic structure of tracking continues to stand as a barrier to middle school students. When students are treated as capable and competent learners and doers of mathematics, they are more likely to be successful (Ladson-Billings 1995).

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Teacher Tracking

Catalyzing Change in Middle School Mathematics recommends that middle school mathematics teachers have teaching responsibilities that are balanced across grade levels and do not promote tracking. Middle school mathematics teachers are often tracked, with the most qualified or those perceived to be the most effective or more experienced teaching either the upper grades in middle school or the “advanced” mathematics course at a particular grade level, and novice, less experienced teachers teaching lower grades in middle school or the “remedial” or “regular” courses (Joseph 2015). In a study of upper elementary and middle schools in 29 diverse school districts, historically marginalized students had less access to high-quality teachers’ instruction when compared to their peers (Isenberg et al. 2013). Often, teachers who teach the lower-level mathematics courses have low expectations of their students, are not provided adequate resources, and focus on basic skills; teachers who are tracked to teach the “advanced” mathematics courses, however, have high expectations of their students and engage their students in rich, problem-solving tasks (Callahan 2005; Miller 2018). Given the significant impact instruction has on student learning (Borman and Kimball 2005; Darling-Hammond 2000; Schmoker 2006), educators should acknowledge and consider how tracking teachers into course assignments hinders equitable practices (Lubienski 2007). As such, the structural inequity where the more effective middle school mathematics teachers teach the classes labeled as advanced and the less effective teachers teach the classes labeled as remedial or regular must be eliminated.

Tracking teachers in middle school does not provide each and every student access to the highest quality mathematics instruction. Gamoran and Berends (1987) claim teachers who consistently teach lower-tracked classes get progressively worse over time, and both the teachers and students lose confidence and adhere to low expectations. The unjust practice of teacher tracking leads to unjust differential student learning outcomes and contributes to fallacious beliefs about which middle school students are and are not capable of learning and doing mathematics. To obtain the goal of supporting each and every student in reaching their potential as learners and doers of mathematics, the insidious practice of teacher tracking must be eliminated.

Balancing teaching responsibilities will strengthen and support both beginning and experienced teachers. One way to approach this is by establishing collaborative teams within each grade level (Larson 2017b). This process can support teachers in attending to the ways in which the mathematical ideas in middle school build, allowing them continued opportunities to learn and provide equitable instruction to all students. A sense of collective responsibility will begin to develop among each teacher at each grade level because teachers will have the pedagogical stance of doing what is most effective for all students and will differentiate their instructional practices accordingly (NCTM 2014b; Williams 2003). In other words, when tracking and ability groups are eliminated, teachers will have a shared sense of responsibility for each and every student, including students with disabilities and emergent bilinguals, and will begin to view the students collectively as their own students (NCTM 2014a). Additionally, this structure creates opportunities for shared responsibilities and collaboration among middle school teachers of mathematics, special education teachers, English as a second language teachers, and other instructional support teachers and paraprofessionals.

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Dismantling Tracking

Principles to Actions (NCTM 2014b) states, “An excellent mathematics program requires that all students have access to a high-quality mathematics curriculum, effective teaching and learning, high expectations, and the support and resources needed to maximize their learning potential” (p. 59). Catalyzing Change in Middle School Mathematics recommends a common shared pathway for middle school mathematics where students are not tracked into qualitatively different courses, and students are provided with and have access to the highest quality mathematics program that adequately prepares them for their bright futures. The common shared pathway in middle school should be focused on teaching mathematics deeply and coherently by not rushing through the content. Middle school mathematics should be about quality rather than quantity, and it should be about students developing their skills as thinkers and doers of mathematics.

School districts are working to dismantle tracking and are seeing positive results. District leaders, policymakers, and other stakeholders should invest both time and resources to fully support the process of de-tracking in their schools and districts. Consider three examples of districts engaged in de-tracking. San Francisco Unified School District (San Francisco Unified School District [SFUSD] Case Study, NCTM 2019a), for example, stopped separating students in mathematics courses by ability. The goal of their mathematics curriculum is to interweave algebraic concepts into mathematics instruction as early as kindergarten, culminating in an eighth-grade mathematics course focused on linear equations and functions and their relationships (Daro 2014; Tintocalis 2015). Since de-tracking in 2014, SFUSD has decreased disparities in mathematics achievement and increased the number and diversity of students taking upper-level mathematics courses in high school. Moreover, there was a decrease in the number of students repeating algebra 1 from 40 percent to only 8 percent (Knudson 2019). Even students who were previously enrolled in higher-tracked mathematics courses are benefitting. While the district experienced some pushback from parents, the district is committed to dismantling inequitable structures and remains de-tracked.

In a second example, Ithaca City School District in New York (Ithaca City School District NY n.d.) is currently de-tracking its middle school mathematics program. Prior to 2019, sixth-and seventh-grade students were tracked into non-accelerated and accelerated mathematics courses. Beginning in the fall of 2019, the school district has a single sixth-grade mathematics course (Math 6), and in the fall of 2020, a single seventh-grade course (Math 7) will be offered. Successful de-tracking is about more than just grouping students heterogeneously (Berry 2018b). The Ithaca district has plans to fully support the teachers in this process. Teachers will co-teach the new courses and receive professional development on flexible grouping, co-teaching with differentiation, and advanced mathematical content. This altered grouping practice requires commitment, time, and support to engage each and every student in learning mathematics that “cultivates [students’] mathematics identities, conceptual understanding, and critical problem-solving and thinking skills” (NCTM 2018, p. 17).

The Escondido Union High School District (Escondido Union High School District Case Study, NCTM 2019b) provides a third example. This district made gradual structural and systemic changes in high schools that positively benefited the mathematics learning of each and every student, particularly students of color. Prior to making the systemic changes, the Escondido Union High School District’s mathematics program tracked both teachers and students. In 2013, the district began making systemic changes to their mathematics curriculum, which now focuses on engaging teachers and students in rigorous, challenging mathematics and on improving teachers’ instructional practice, which is resulting in improved student experiences and outcomes. The district provided teachers common preparation time and summer workshops, implemented an integrated problem-based mathematics curriculum, and eliminated tracking in ninth and tenth grades. As a result of these changes, instruction in the mathematics classroom is more discussion-based, more students are enrolled in upper-level mathematics classes, nearly all of the students are on grade level in mathematics, and the district is moving away from viewing students from a deficit paradigm to one that is strengths-based.

Eliminating tracking supports each and every student’s positive mathematical identity and strong sense of agency (Boaler 2011), including students who have been identified as “high achievers” (Boaler, Wiliam, and Brown 2000). Although proponents of tracking contend high achievers would not be challenged in de-tracked classrooms, research shows no difference in performance when compared to students with similar “ability” in a tracked classroom (Rui 2009). Thus, all students have the opportunity to be doers of mathematics. In essence, when middle school students who would have been placed in a lower-track mathematics course are given the supports they need to be successful in a de-tracked mathematics course, their performance increases, and they are more likely to continue to enroll in upper-level mathematics courses (Boaler and Staples 2008). How does a school or district begin and continue the process of de-tracking their middle school mathematics program, particularly when some districts and schools track students based on teacher recommendations, test scores, grades, parental requests, and perceived student motivation (Burris and Garrity 2008), all of which are influenced by implicit biases?

District leaders should listen to how teachers, parents, and others respond as underlying beliefs are often revealed through the language used. The deeply rooted and overly used labels such as “low” class, “advanced” class, “top” kids, “honors” students, or “remedial” group are detrimental to both the teachers’ instruction and the students’ overall mathematical learning experience because labels positively or negatively influence students’ mathematical identity and agency while maintaining deficit perspectives about groups of students. The language that teachers, district leaders, parents and families, policymakers, and other stakeholders use shapes their beliefs about students and what they can and cannot do (Burris and Garrity 2008; Hodge 2019). While it is challenging and in many cases an eye-opening experience to do an intentional self-evaluation of one’s beliefs, it is even more challenging to encourage others to self-reflect and self-evaluate their beliefs to bring about change. For this reason, the initial efforts of discussion on the systemic change of de-tracking should focus on the data as well as a deep understanding of privilege and oppression, in particular institutionalized oppression that plays out in policies and practices.

Perceptions are then made real as far as how African Americans and Latin@ are treated in mathematics classrooms, the forms of instruction available to them, and what courses schools provide; which in turn lead to different testing outcomes (gaps). Institutions make these ideologies concrete when they provided African Americans and Latin@ impoverished forms of instruction through tracking and reduced funding in the form of property taxes. This then serves to legitimize the ideology that African Americans and Latin@s are innately worse at mathematics rather than deconstructing the role of institutions or noting the efforts of educators and communities to combat these racist structures daily. (Battey and Leyva 2016, p. 59)

Because there is no reason to justify why a particular race does better or worse in mathematics, other than institutionalized racist structures that are prevalent in many U.S. schools (Battey and Leyva 2016), district leaders, teachers, and parents and families must acknowledge and begin to disrupt those structures. This may be the impetus of what is needed to transform beliefs and help others see the deleterious effects of tracking.

It is important to keep in mind that dismantling the systemic structure of tracking is a challenging process that will be met with some resistance from some families, teachers, and students. Families and student caregivers should be assured that de-tracking will not slow their student down and that their student will not be subject to engaging in a less rigorous curriculum. It should also be emphasized that students learn from diverse viewpoints (Ascher 1992). The systemic change of de-tracking should happen gradually to be effective and sustaining. Burris and Garrity (2008) suggest district leaders, policymakers, and teachers should examine how students and which students are tracked in each mathematics course. They further recommend three strategies school districts and schools can follow to begin the de-tracking process: (1) implement change at the point the tracking starts—simply changing the curriculum in each grade level is not enough; (2) begin with teachers who are interested and who will be vested in the process; and (3) gradually de-track by eliminating the lowest track first. It is important to begin the process by developing a shared vision for mathematics instruction among middle school mathematics teachers and by spending time helping administrators and other teachers in the building understand that dismantling tracking will help meet that shared vision for each and every student. With colleagues on board, time and care must be taken to develop a plan and a communication strategy for that plan and to engage the school board, parent and family organizations, and the community in discussions about that vision before moving to implementation (Steele and Huhn 2018). Aligned with the recommendation made here to de-track middle schools, Berry (2018b) recommends the following action steps to move toward de-tracking middle schools:

  • Identify, analyze, and evaluate policies, practices, and procedures to assess the impact of tracking in restricting student access to and success in mathematics.
  • Provide each and every student access to a grade-appropriate, academically rigorous, and intellectually challenging curriculum.
  • Provide students with additional targeted instructional time and other instructional supports to support their learning and success.
  • Analyze teacher assignments to develop balanced and supportive assignments to provide high-quality engaging learning experiences for all students.
  • Analyze where research-informed equitable instructional practices are implemented and where not, and facilitate changes. This includes the use of culturally relevant pedagogy, building on students’ interests and knowledge, incorporating real-life experiences into the curriculum, and using practices that showcase students’ strengths.
  • Provide teachers with access to mathematics coaches/specialists for ongoing real-time professional development and support. Ongoing professional development includes but is not limited to coaching, co-teaching, co-planning, and frequent interactions on teaching and learning.
  • Provide teachers and mathematics coaches/specialists with time and space to collaborate with one another on instructional issues and to continue their own professional learning of both mathematics and mathematics-specific pedagogy. Teachers need opportunities to share strategies, learn new teaching techniques, meet as a department or grade level, and collaborate for improved student learning.
(Berry 2018b, para. 5)

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Distinguishing Tracking from Appropriate Acceleration

Tracking should be distinguished from appropriate acceleration. Acceleration may be appropriate for students who have demonstrated a deep and coherent understanding of their current grade-level or course-based mathematics standards (Larson 2017a; NCTM 2018). Appropriate acceleration must ensure that no critical concepts are rushed or skipped, and such decisions should not be based solely on traditional assessment instrument results (NCTM 2016). Simply skipping a grade does not constitute appropriate acceleration. The practice of skipping a grade level of mathematics is a disservice to students because they miss foundational concepts and do not have an opportunity to develop conceptual understanding of the mathematics topics skipped. In cases where acceleration is warranted, it should reflect the demographics of the school population. As described in the NCTM Position Statement “Providing Opportunities for Students with Exceptional Mathematical Promise” (2016), “exceptional mathematical promise is evenly distributed across geographic, demographic, and economic boundaries” (para. 4). In other words, exceptional mathematical promise is not limited to specific racial groups, sex, or socioeconomic backgrounds (e.g., only white males from middle class to affluent backgrounds). Students with disabilities, including learning disabilities, are students who also demonstrate exceptional mathematical promise (Al-Hroub 2011; Bracamonte 2010; Brody and Mills 1997). Any student demonstrating exceptional promise warranting acceleration should progress through the common shared pathway with meaningful extensions that “deepen their mathematical curiosity, knowledge, and skills” (NCTM 2018, p. 89) so they are sufficiently challenged.

If a middle school student is accelerated and begins the high school common shared pathway early, the student must receive the same qualitative mathematics learning experience and be held to the same expectations as they would if they were starting it in grade 9. In order for students to be accelerated to the next course, they must demonstrate their conceptual understanding and procedural fluency of the practices, processes, and content of that course before permitting acceleration. Students are sometimes able to do this for a variety of reasons. “When considering opportunities for acceleration in mathematics, care must be taken to ensure that opportunities are available to each and every prepared student and that no critical concepts are rushed or skipped, that students have multiple opportunities to investigate topics of interest in depth, and that students continue to take mathematics courses while still in high school and beyond” (NCTM 2016, para. 2).

An NCTM and Mathematical Association of America (MAA) joint statement asserts

[The] ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by grade 12 but to have established the mathematical foundation (and disposition toward mathematical work) that will enable students to pursue whatever course of study interests them when they get to college. (NCTM and MAA 2012, para. 2)

In today’s rapidly changing technological world, the advent and importance of data science and computational mathematics cannot be ignored. Students who have a solid foundation in middle school mathematics are better prepared to take advanced mathematics courses in high school and improve their success in college level courses (ACT 2009, 2012). Instead of pushing for acceleration in middle school, more time should be spent on making sure students understand the current mathematics content of middle school (NCEE 2013).

The National Assessment of Education Progress (NAEP) showed growth in eighth-grade mathematics scores from 2000 to 2007 in every group except the group of students who were accelerated, and this trend continued from 2005 to 2011 (Loveless 2008, 2013). The National Center on Education and the Economy (NCEE) report What Does it Really Mean to be College and Work Ready? (2013) recommends that it is vital students spend substantial time understanding middle school mathematics content and not try to advance through it too quickly because the middle school mathematics content is foundational to students’ college and career success. A lack of understanding of middle school mathematics may lead students to not pursue upper-level mathematics courses in high school and beyond (Boaler 2016; Larson 2017a) because of limited mathematical knowledge and lack of desire or motivation to continue to study mathematics throughout their high school career (Liang, Heckman, and Abedi 2012). Students who are not ready but are placed in an advanced mathematics class in eighth grade reported lower self-efficacy, a decline in motivation toward mathematics, and a decrease in task value for mathematics, which correlated with students not taking upper-level mathematics courses in high school (Simzar, Domina, and Tran 2016).

The system structures of middle school have a direct impact on students’ continued success in high school and in their personal and professional lives as adults. Regardless of the structure of the common shared pathway used in a high school mathematics program, there are substantial implications for feeder middle schools. Middle school mathematics teachers, instructional leaders, and administrators should be fully informed and knowledgeable regarding the common shared pathway their students will experience when they go to high school. Critical conversations between middle and high school mathematics stakeholders should occur in order to best ensure a seamless transition in students’ mathematics program as they transition from middle school to high school.

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Equitable Structures for Student Success

Teachers, district leaders, policymakers, and other stakeholders need to simultaneously work to understand how systems of privilege and oppression operate in schools and classrooms while also disrupting them (Stinson and Spencer 2013) when making decisions about extra class-time support. If students begin middle school without the strong foundation needed from elementary school, targeted instructional supports should be put into place so that each and every student has access to a high-quality middle school mathematics program. One approach to provide instructional support is the use of stacked or corequisite mathematics classes. This process places students in the typical grade-level course with their peers while also having students enrolled in an additional mathematics period to provide further time and support for the student (NCTM 2018). This additional time could be structured as a second period of grade-level mathematics or flexible intervention time where students attend when they need further targeted support on specific grade-level concepts in order to minimize instructional time lost in other subjects (Larson and Andrews 2015). Determinations about which students might need additional time should be decided by ongoing classroom-based formative assessments so that interventions are connected to grade-level learning, are timely, and used only when needed (Kanold et al. 2018). The additional class time should not be a “repeat” of what students did in the “regular” class, but should focus on instruction that would deepen students’ mathematical content knowledge, using technology and other engaging resources (Howard et al. 2015).

Appropriate instructional support is not a pullout program, which removes students from the grade-level mathematics classroom to purportedly “fill in” gaps in students’ learning (Mosqueda 2010). Such programs cause students to miss important mathematics content, continuing to exacerbate the students’ need for instructional support. Other ineffective intervention practices include placing students in separate mathematics courses that do not focus on grade-level content (e.g., Math 1 for Emergent Bilinguals).

Providing appropriate instructional supports is a collaborative endeavor among the general education mathematics teacher, special education teacher, English as a second language teacher, and other instructional support staff. Effective differentiation (Gamoran 2009) will support each and every student in the mathematics classroom. Making mathematics accessible through differentiating instruction and other support structures can help reshape deficit views regarding who can and cannot do mathematics (Lambert and Stylianou 2013). Effective interventions recognize the need to combat deficit views about students, particularly students who are labeled “lower-level” (Boaler 2011). Specifically, instructional supports and purposeful professional development that focus on shifting the mindsets of educators to more productive beliefs are needed (García and Guerra 2004; Jackson and Delaney 2017; NCTM 2014a; Rubin 2006; Watanabe 2006). Further, instructional supports and purposeful professional development should focus on supporting teachers’ understanding of the ways privilege and oppression operate in mathematics education, including at the classroom level (Battey and Franke 2015; Healy and Powell 2013; Stinson and Spencer 2013).

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