Journal of Urban Mathematics Education 

December 2016, Vol. 9, No. 2, pp. 153–184 

©JUME. http://education.gsu.edu/JUME 

 

 
 

EMILY JOY YANISKO is the Lead Clinical Faculty for Mathematics in the Baltimore site of Ur-

ban Teachers at the Johns Hopkins School of Education; email: emily.yanisko@urbanteachers.org. 

Her research interests include urban mathematics teacher education and development, discourse in 

the mathematics classroom, and issues of equity in urban schools.  

Negotiating Perceptions of Tracked Students: 

Novice Teachers Facilitating High-Quality 

Mathematics Instruction 
 

Emily Joy Yanisko 

Urban Teachers at Johns Hopkins University 

 
In this article, the author reports on a participant-observation case study that ex-

plored how alternatively certified, middle school teachers’ expectations of tracked 

students affect their ability to learn to teach in ways that promote students’ mathe-

matical struggle and participation in productive mathematical discussions. Two 

teachers—one teaching a “high-tracked” course and the other a “low-tracked” 

course—were participants. Both teachers initially held perceptions of their students 

that limited their efficacy and self-efficacy with respect to providing high-quality 

mathematics instruction. However, through program- and school-based mentoring, 

including participation in a modified reflective-teaching cycle, the teachers learned 

to learn from their teaching and modify their practice. Both teachers began to al-

low their students opportunities to struggle with rigorous mathematics and partici-

pate in student-centered discussion. 

 

KEYWORDS: academic tracking, African American/Black students, high-quality 

mathematics instruction, reflective teaching 

 

he National Council of Teachers of Mathematics (NCTM) states, “An excellent 

mathematics program requires that all students have access to a high-quality 

mathematics curriculum…[and] high expectations” (2014, p. 5). High-quality 

mathematics, according to the Common Core State Standards for Mathematical 

Practice (CCSSMP), provides students, among other things, opportunities to partic-

ipate in mathematics instruction that asks them to “make sense of problems and 

persevere in solving them” (CCSSMP, 2016, ¶2) and to “construct viable argu-

ments and critique the reasoning of others” (¶4). In the interest of justice, all stu-

dents should have access to rigorous, high-quality instruction. Both the NCTM and 

the CCSSMP claim equity as a driving purpose for the importance of the imple-

mentation of standards of high-quality mathematics instruction. The development 

of both the NCTM and the CCSSMP was guided by the desire to make high-quality 

mathematics available to all students, regardless of race, SES, language, school 

placement, or course placement, so that all students become college and career 

ready and are internationally competitive. The language of all espoused throughout 

both the NCTM and the CCSSMP documents, however, may not be sufficient to 

T 

http://education.gsu.edu/JUME
mailto:emily.yanisko@urbanteachers.org


 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       154 

ensure equity (Apple, 1992), but rather suggests what may be considered a neces-

sary step in the road to equity: equality of opportunity. Martin (2015) contends that 

the documents perpetuate the goals and ideals of middle- and upper-class White 

privilege and decenter the needs of those children and youth who he calls “the col-

lective Black” (p. 21). Therefore, we must first think about how the mathematics 

being taught in schools to students of non-dominant backgrounds mirrors (or not) 

what the NCTM and the CCSSMP consider high-quality mathematics instruction. 

Historically, the perceived student responsibility for learning mathematics fo-

cused on following rules presented by the teacher or the textbook, memorizing and 

applying those rules, and verifying correctness through an authority such as the 

teacher or the textbook (Cobb & Yackel, 1996; Lampert, 1990; Schoenfeld, 1992). 

This type of instructional practice can be formal and restricting and, for many stu-

dents, can limit opportunities to develop their mathematical reasoning (Brown, Col-

lins, & Duguid, 1989; Stein, Grover, & Henningsen, 1996). In addition to the strong 

language around problem solving and mathematical discussion of the CCSSMP, the 

Principles and Standards for School Mathematics (NCTM, 2000) also states that 

students should be able to “make and investigate mathematical conjectures” and to 

“develop and evaluate mathematical arguments and proofs” (pp. 57–58). Further-

more, this type of reasoning should be augmented through communication as stu-

dents—  

 
organize and consolidate their mathematical thinking through communication; com-

municate their mathematical thinking coherently and clearly to peers, teachers and oth-

ers; analyze and evaluate the mathematical thinking and strategies of others; [and] use 

the language of mathematics to express mathematical ideas precisely. (p. 60) 

 

If these understanding, communication, and reasoning goals and standards are to 

characterize norms of practice as evidenced in schools, there will need to be a fun-

damental adjustment in participation structures within the mathematics classroom 

(Foreman & Ansell, 2001; Herrenkohl & Guerra, 1998). Through mathematical 

communication, students and teachers may work and reason together as they “do 

mathematics” in a way that augments the mathematical knowledge that students are 

expected to know (Cobb & Yackel, 1996). 

Therefore, it is important to investigate how new teachers learn to teach in 

ways that encourage students’ perseverance in problem solving and participation in 

productive mathematical discussion with equitable practice and meaningful learn-

ing as the ultimate goal of these practices. In the study discussed here, I investigate 

two novice, alternatively certified, middle school teachers’ perceptions of their 

tracked students and how those perceptions affected their efficacy and self-efficacy 

with regard to infusing their classrooms with perseverance in problem solving, 

mathematical sense making, and productive discussions. Two research questions 

guided the inquiry:  



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       155 

1.    How might alternatively certified, middle school teachers’ expectations of 
tracked students affect their ability to learn to teach in ways that promote 

students’ mathematical struggle and participation in productive mathemat-

ical discussions? 

 

2.    How might these teachers negotiate their perceptions of students and learn 
to teach in ways that promote students’ mathematical struggle and partici-

pation in productive mathematical discussions? 

 
Review of Relevant Literature 

 

This study integrates and is informed by two primary bodies of literature, 

tracking and alternative certification. Taken together, this literature shaped the ra-

tionale and purpose for this study. 

  

Expectations in Schooling, Urban Schooling, and Tracking 
 

Novice teachers from alternative certification programs are often used to fill 

vacancies in “hard-to-staff” districts or schools that are more times than not popu-

lated with underserved students (Darling-Hammond, 2000). Too often, both novice 

and experienced teachers in these hard-to-staff schools have deficit perspectives of 

low-income students of color (Delpit, 2012; Habermann, 1991). These deficit per-

spectives may be more prevalent for teachers prepared in alternative certification 

programs than for those prepared in traditional teacher education programs 

(Brantlinger & Smith, 2013). 

Furthermore, academic tracking may further compound deficit perspectives 

that many teachers hold of low-income students, particularly, low-income students 

of color. Tracking is pervasive in education, and teachers of tracked students may 

have particular perceptions of these students’ ability, negatively altering their in-

struction based on these perceptions (Futrell & Gomez, 2008; Oakes, 2005). Not 

only are low-income students of color often placed in “low-track” courses, denying 

them equal access to high quality curricula, but also they are less likely to have cer-

tified, high-quality teachers (Furtrell & Gomez, 2008; Oakes, 2005). This unequal 

access to both high-quality curricula and teachers is doubly detrimental in mathe-

matics. Access to high-quality mathematics instruction can provide a gateway to 

future academic achievement, and can be considered a new civil right (Moses & 

Cobb, 2001; Oakes, 1990; Smith, 1996; Spielhagen, 2010). 

Oakes and Lipton (1996) contend that students placed in high-track courses 

have access to richer mathematics instruction as well as more content. Access to 

cognitively demanding mathematics instruction and content provides students en-

trée to college preparatory high school courses and higher probability of engaging 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       156 

in mathematics related fields post-secondary school (Oakes, 1990; Oakes & Lipton, 

1996; Smith, 1996; Spielhagen, 2010). When analyzing the Third International 

Mathematics and Science Study (TIMSS) data, Stigler and Hiebert (1997) found 

that education in the United States, in general, is often dominated by the teacher, 

following a pattern where teachers demonstrate a particular concept or skill, and 

then ask students to practice what they have “learned” by rote in an application 

phase. This prescription for instruction is reinforced over generations, regardless of 

reform efforts, because teachers often resort to providing their students the same 

instruction as they themselves experienced (Lortie, 1975). Furthermore, teachers in 

schools serving low-income students of color often are directed or feel the need to 

administer scripted curriculum that results in students participating in low-level 

tasks far more often than teachers placed in schools serving affluent, predominantly 

White populations (Delpit, 2003).  

Student expectations. Teachers of low-income students of color often convey 

low expectations of the educational potential of their students (Delpit, 2012; Ha-

bermann, 1991; Oakes, 1992), regardless of the way students are sorted into tracks, 

which can bear little relation to the students’ actual ability in mathematics (Oakes, 

2005). These low-expectations cause many teachers to engage their students in edu-

cational experiences that require little higher-order thinking, discussion, or sense 

making through problem solving. Teachers of tracked courses often modify their 

instructional methods based on their perceptions of the educational potential of the 

students in those classes (Braun, Nielsen, & Dykstra, 1975; Eder, 1981; Oakes, 

1992; Raudenbush, Rowan, & Cheong, 1993; Rosenbaum, 1980; Watanabe, 2008). 

Reciprocally, tracking, as well as race and SES, can be deterministic with regard to 

the teachers’ and other stakeholders’ expectations for their students’ educational 

potential (Oakes, 1992, 2005).  

Student testing used to evaluate both schools and teachers is prevalent in 

schools that serve low-income students of color. The emphasis on testing, in turn, 

too often results in teaching methods that are likened to “a thin gruel of test prepa-

ration … [t]he drill-and-kill practices that guarantee students will not be ready for 

college, skilled employment, lifelong learning, or effective citizenship” (Neill, 

2012, p. 24). The tradition of administering low-level curriculum that is specifically 

focused on test-preparation may be pervasive in any track, regardless of the “objec-

tive” level of student ability (Bol & Berry, 2005; Bol & Nunnery, 2004; Delpit, 

2003). 

Student placement in tracks due to expectations and resultant treatment due to 

both expectations and subsequent placement is a reciprocal relationship (Braun et 

al., 1975). Teachers believe that students are placed appropriately, whether due to 

race, SES, or perceived ability level (Oakes, 2005), and therefore treat their students 

differently. Teachers expect more from those whom they perceive as highly moti-

vated or high achieving, and therefore ask questions that are more rigorous and 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       157 

provide higher-level educational experiences (Braun et al., 1975; Brophy & Good, 

1970; Cooper, 1979; Dusek, 1975; Oakes, 1992; Page, 1990; Watanabe, 2008). 

Track placements are often permanent, and those students in low tracks experience 

less rigorous instruction and lower-quality materials (Rosenbaum, 1980). 

Tracking in mathematics. Tracking is pervasive in mathematics programs, and 

is used to sort students based on many different characteristics, including perceived 

ability, academic performance, test scores, or non-academic reasons (Bol & Berry, 

2005; Oakes & Lipton, 1996). Although tracking in mathematics has been defended 

as a mechanism to ameliorate the “achievement gap” (Bol & Berry, 2005), there is 

“no empirical evidence to justify unequal access to valued … mathematics curricu-

lum, instruction, and teachers” (Oakes, 1990, p. xi; emphasis in original). African 

American and Latin@ students are over-represented in low-track courses and often 

have less qualified teachers, and are subjected to low expectations. However, low 

expectations of low-tracked students often begin before placement in tracks. There 

are many instances where African American and Latin@ students are placed in re-

medial mathematics programs even when the measures of their academic ability are 

equal to or better than their White and Asian peers (Education Trust, as referenced 

in Love, 2002). These low expectations of students are reflected in instructional and 

assessment methods used in mathematics courses, which result in less emphasis on 

high-quality mathematics instruction (Flores, 2007; Irvine & York, 1993). 

Tracking in mathematics has long-term effects on the trajectory of low-

income students of color. For example, access to algebra in eighth grade influences 

students’ mathematical knowledge, productive disposition, course taking patterns, 

and future mathematics achievement (Smith, 1996). Taking an Algebra I course in 

middle school offers students experience with demanding mathematics programs 

and allows them access to the mathematics pipeline, as Algebra I is often consid-

ered a “gatekeeping course” (Moses & Cobb, 2001; Oakes, 1990; Spielhagen, 

2010). Enrollment in algebra in middle school also has a positive effect on students’ 

access to a 4-year college (Spielhagen, 2010). Students of color, however, have less 

chance of being admitted to middle school Algebra I courses (Spielhagen, 2010). 

 

Novice Alternatively Certified Teachers and Teacher Education 
 

Novice mathematics teachers seem to be fighting an uphill battle. They are 

faced with providing instruction that is consistent with the new Common Core State 

Standards for Mathematical Practices (see CSSSMP, 2016). They are affected by 

the “apprenticeship of observation,” which dictates that when they encounter a situ-

ation in which they are unsure, they revert all too often to the instructional methods 

of their own K–12 experience (Lortie, 1975). Furthermore, if they are placed in 

schools that serve low-income students of color which are historically low-

performing, they may have expectations of their students that results in planning 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       158 

and enacting instructional routines which are low level in the intent to be test-

preparatory (Delpit, 2003; Habermann, 1991). 

Hiebert, Morris, and Glass (2003) suggest that it is unreasonable to expect 

that a teacher will learn everything they need to know to be an effective teacher 

during their teacher preparation program. This gap in knowledge is even more sig-

nificant in the context of alternative certification, where the teacher education pro-

gram is shorter in duration (Darling-Hammond, 2000). Such gaps may affect their 

efficacy in providing high-quality mathematics education as well. Hiebert and col-

leagues argue that in order for teachers to “learn to learn to teach” (p. 201), they 

must treat their teaching experience as an experiment, and be taught to learn from 

their own practice as a scientist would learn from an iteration of an experiment. 

With help, novice, alternatively certified teachers may be able to learn to learn to 

teach in ways that engage students in productive, high-level mathematics discussion 

regardless of their race, SES, or track placement.  

The literature reviewed highlights the importance of investigating how new 

teachers’ perceptions of tracked, low-income students of color affect their instruc-

tion, and how they may be supported to negotiate the challenges that the teachers’ 

notions of student ability pose in order to provide high-quality mathematics instruc-

tion.  

 
Conceptual Framework of Mathematics Instruction 

 

An in-depth understanding of mathematics includes not only the knowledge 

of rules and procedures but also the ability to engage in mathematical sense mak-

ing, participate in productive mathematical discussions, and have productive dispo-

sitions toward mathematics (National Research Council [NRC], 2001). For the pur-

poses of this article, mathematical sense making is defined as valuing and applying 

abstraction and using those tools to understand mathematical structures (Schoen-

feld, 1994). Productive mathematical discussions are those student-centered discus-

sions where ideas are communicated mathematically so the shared ideas can be 

publicly understood, critiqued, and guided toward a learning goal (Smith & Stein, 

2011). And perseverance in problem solving is describe as when students make 

conjectures about different solution paths and monitor and evaluate their progress, 

making changes whenever and wherever necessary (CCSSMP, 2016).  

Mathematics classrooms that have students make sense of and persevere in 

mathematics problem solving through productive discussions and collaborations 

can be considered a more indirect form of instruction. Some researchers have ar-

gued that when instructional schemes in mathematics classrooms focus on problem 

solving, communication in groups, and more indirect pedagogy (i.e., student-

centered and focused on student investigation and sense making), students from the 

dominant culture may be privileged (Apple, 1992; Delpit, 2006; Lubienski, 2000). 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       159 

Other researchers have found that classrooms organized around these principles can 

be equitable and beneficial to low-income students of color if support systems are 

in place, and careful attention is given to assigning competence and developing 

norms for problem solving (see, e.g., Boaler & Staples, 2008; Gutiérrez, 2000; 

Kitchen, DePree, Celedón-Pattichis, & Brinkerhoff, 2007). This conditional clarifi-

cation is consistent with the recommendations of researchers who suggest that stu-

dents from outside the dominant culture must be given explicit access to the “cul-

ture of power” that will allow them to be successful (Bourdieu, Passeron, & de 

Saint Martin, 1994; Delpit, 2006). 

In theorizing about how to dismantle the culture of power in mathematics ed-

ucation as a shift toward equity, Gutiérrez (2007) offers four dimensions to consid-

er: access, achievement, identity, and power. Access relates to the tangible re-

sources that students have available to them to participate in mathematics, including 

teachers, technology, curriculum, and classroom environments that foster participa-

tion. Achievement includes enrollment and participation in mathematics courses, 

standardized test scores, and persisting in the mathematics pipeline. Identity ad-

dresses students seeing themselves in the mathematics curriculum, as well as seeing 

themselves and mathematics in the world. Power refers to the presence or absence 

of student voice and decision-making abilities, ability to apply mathematics as a 

critical lens through which to see society, alternative forms of knowledge, and the 

representation of mathematics as a field that needs people.  

Providing students with access to teachers who can provide a rigorous and af-

firming mathematical experience is beneficial to equity. Incorporating tasks of 

high-cognitive demand and opportunities to participate through student-centered, 

productive discussion allows students access to achieving a type of mathematical 

proficiency that allows them the ability to apply their knowledge outside of the 

mathematics classroom (see NRC, 2001; Smith & Stein, 2011; Stein, Smith, Hen-

ningsen, & Silver, 2000). Having student voices and decisions around mathematical 

problems with multiple possible solutions or solution strategies as a central feature 

of the mathematics classroom facilitates students’ power in the classroom.  

To facilitate student sense making and productive discussion, teachers may 

structure lessons around problematic or investigative tasks, set up these tasks to aid 

student access, and require students to complete the tasks collaboratively in small 

groups (Boaler, 2002a, 2002b, 2006). Teachers may assert and maintain expecta-

tions, and develop norms for students’ production of explanations of mathematical 

solutions strategies in small- and whole-group discussions, as well as require stu-

dents’ active engagement with making sense of, responding to, and attempting to 

understand student explanations and justifications (Wood, 1999; Yackel, 2001; 

Yackel & Cobb, 1996). Such instructional strategies mean that students must be 

able to answer questions such as “how” and “why,” and that teachers may ask prob-

ing, leading, or advancing questions to encourage students to develop these answers 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       160 

(Bowers, Cobb, & McClain, 1999; McClain & Cobb, 2001; Yackel, 2001; Yackel 

& Cobb, 1996). Having students pose questions to others is another important com-

ponent of sense making through discussion (Borasi, 1992; Ciardello, 1998; Zack & 

Graves, 2001). 

To promote this exchange between students, teachers may direct student ques-

tions to other students, asking students to revoice a students’ explanation or to pose 

a question if they cannot revoice. To structure discussions that promote sense mak-

ing, teachers may also carefully choose solutions to be presented so that a discus-

sion of reasonableness or correctness of those solutions may result (Smith & Stein, 

2011). Teachers may use instructional strategies of their own design, or use strate-

gies that are presented to them during teacher preparation courses or through teach-

er-support systems, such as teacher mentoring. 

Integrating these facets of equitable instruction is challenging work for any 

mathematics teacher; it is especially challenging for new teachers (Achinstein & 

Athanases, 2005). In particular, it is important to attend to the expectations that new 

teachers have of students. Teachers’ expectations of students may affect teachers’ 

ability to provide students access to high-quality mathematics instruction (Oakes, 

2005). Teacher’s expectations of students may be influenced by race, SES, and per-

ceived ability level (Delpit, 2012; Habermann, 1991; Oakes, 2005). In particularly, 

there is a dangerous and harmful essentialism present in the discourse around low-

income students of color and education: that most members of this collective sub-

population are inherently low performing, specifically so in mathematics (Faulkner, 

Stiff, Marshall, Nietfeld, & Crossland, 2014; Martin, 2012; Stinson, 2006). 

 
Methods 

 

Design 
 

This study used participant observation within a case study (Yin, 2009). Yin 

defines case study as “an empirical inquiry that investigates a contemporary phe-

nomenon in depth and within its real-life context, especially when the boundaries 

between the phenomenon and the context are not clearly evident” (p. 18) The meth-

odology employed allowed me to describe in depth the instructional strategies con-

sidered and co-constructed by the novice teachers and me within the context of 

mentoring sessions and a teacher seminar (subsequently described). Due to the na-

ture of the study, and my relationship with the teachers, I learned much about them, 

both professionally and personally. The collected data provided firsthand notations 

and records describing their experiences and professional change evidenced through 

the talk and action of the teachers within the context of their mentoring and seminar 

support as well as their actions within their classrooms and schools. 

 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       161 

Program Context 
 

The context of this study was first-year teachers who were enrolled in an al-

ternative certification program. Alternatively certified teachers were chosen for two 

reasons. First, alternatively certified teachers are often less prepared to teach in 

ways that result in high levels of student mathematics achievement due to the trun-

cated nature of their teacher preparation (Brantlinger, Cooley, & Brantlinger, 2010). 

Second, as previously noted, alternatively certified teachers may hold lower expec-

tations for low-income students of color than traditionally certified teachers in the 

same context (Brantlinger & Smith, 2013; Delpit, 2012; Habermann, 1991). It is 

imperative, in the interests of equity, to investigate these teachers’ expectations of 

their tracked students, as well as how they may be taught to negotiate these expecta-

tions. 

The program for which I served as a mentor and in which the participants 

were enrolled was a collaborative effort affiliated with a large mid-Atlantic univer-

sity and a large local school district. Locally, it was not unusual for schools in this 

district to be described as hard to staff; in that, many of the schools had a history of 

low student performance on the state’s high-stakes standardized assessments, had a 

high rate of teacher turnover, and had a predominant population of students who 

were low-income children of color. By design, this alternative certification program 

recruited prospective teachers who were committed to the community and/or the 

population of the student body. Those who enrolled in and successfully completed 

the requirements of the program were certified as teachers of middle school math-

ematics or science and one other core middle school subject. While it encompassed 

some of the features of alternative resident-teacher certification programs, the pro-

gram was designed so that prospective teachers were slowly introduced to teaching. 

During the summer, they were enrolled in a short field experience. When they en-

tered teaching in the fall, they were partnered with a cooperating teacher for a 

month-long internship, and then subsequently assumed a half-time (rather than a 

full-time) teaching load while partnered with another novice teacher from the co-

hort. In this way, a pair of novice teachers filled a full-time teaching vacancy. 

Through pairing, teachers were able to gain experience with the work of teaching 

and to observe other teachers during their non-teaching time. 

 

Participants 
 

The participants in this study were two novice teachers enrolled in the alterna-

tive certification program described. During the year in which I collected data, there 

were 13 teachers enrolled in the program. Eleven of those teachers’ primary certifi-

cation was in mathematics. Of those 11 teachers, nine agreed to participate in the 

study. After conducting preliminary observations of these nine teachers in their 

summer methods course, I selected two based on my perception of their initial pre-



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       162 

dispositions for and conceptions of teaching and placement to solicit data from two 

contrasting cases (Yin, 2009). The first, Jack Davis (all proper names are pseudo-

nyms), was selected due to his unabashed commitments to student-centered instruc-

tion. The second, Michelle Miller, was selected because of my perception of her 

reticence to teaching in a conceptual, student-centered manner. These teachers were 

my mentees in the alternative certification program and participated in a teacher 

seminar while simultaneously participating in the requirements of the teacher prep-

aration program. Although both teachers are African American, it was not inten-

tional; the alternative certification program attracted a high number of teachers of 

color as the program was focused on recruiting candidates that had an investment 

and commitment to the urban community in which they were going to teach. 

Michelle and Jack shared a vacancy in a K–8 public school academy. The 

students in the middle school program were tracked in all of their classes based on 

their reading scores. That is, the students’ scores on a reading assessment deter-

mined whether they were placed in the honors track for all of their classes or the 

comprehensive track (i.e., the low track) for all of their classes. Michelle was as-

signed to teach the low-track courses; Jack was assigned to teach the high-track 

courses. 

Michelle Miller. Michelle Miller is an African American woman. She had 

earned both a bachelor’s degree in physics and a master’s degree in mechanical en-

gineering and previously worked as an automotive engineer before deciding to be-

come a teacher. Her experiences in school and in her subsequent profession were 

characterized by sexism. During her undergraduate education, all of her mathemat-

ics and science professors were men who rarely provided her with the substantive 

feedback necessary to achieve. As an engineer, she felt isolated and excluded from 

any employee bonding activities. She had no prior teaching experience before en-

tering the program. The licensures that she subsequently earned through enrollment 

in the program were in middle grades mathematics and science. Michelle’s initial 

one-month internship was in the same school where she was subsequently assigned 

as a paired, half-time teacher. This school was a public K–8 academy. Michelle’s 

permanent placement assigned her to teach the low-tracked seventh-grade mathe-

matics and science courses; she taught one section of each subject. 

Jack Davis. Jack Davis is an African American man. He entered the program 

immediately after receiving his undergraduate degree in economics. His K–12 ex-

periences with mathematics were racialized; even though he was high-achieving he 

and his family still had to fight for his placement in the high-track mathematics 

courses. He also received his education in the same school district in which he was 

teaching. His previous instructional experience encompassed mentoring and provid-

ing tutoring for middle school students. The licensures that he earned through en-

rollment in the program were in middle grades mathematics and social studies. Jack 

interned and was permanently placed in the same K–8 school as Michelle. He 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       163 

taught one section of seventh-grade honors mathematics and one section of sev-

enth-grade social studies. 

 

Setting 
 

The school in which Jack and Michelle were placed enrolled nearly 700 ele-

mentary and middle school students in 2011. Ninety-five percent of the students 

enrolled at the school were African American and 2% were Hispanic or Latin@. 

Sixty-nine percent of the middle school students were considered low income, ac-

cording to the data on students receiving free or reduced-price meals. Fourteen per-

cent of the middle school students were receiving special education services. Both 

Jack and Michelle taught tracked mathematics courses, where the students were 

placed into all their courses based on a standardized reading inventory. Although 

mathematics and reading scores may be correlated, this placement is significant be-

cause Oakes (2005) suggests that often teachers tacitly accept the placement proce-

dures as objective and appropriate regardless of the methodology and appropriate-

ness of placement. This setting is also of particular importance because studies of 

the differential expectations and educational experiences of students in tracked 

courses are often studied in more diverse settings (e.g., Oakes, 2005). There is a 

need to investigate the implications of tracking on the expectations of teachers in 

tracked settings where the majority of the population is low-income students of col-

or, where expectations are already low, and the cognitive demand of the curriculum 

is often lowered across all tracks (Delpit, 2012, 2013). 

 

Researcher Positionality 
 

As a researcher, I worked closely with the participants in this study. I was 

their program provided mentor and supported them during their first year of teach-

ing. Furthermore, I was an active participant in a seminar during their field place-

ment. That is to say, during their initial year as a teacher, I influenced what instruc-

tional strategies were considered and potentially discussed as mechanisms for sup-

porting and maintaining high cognitive demand during instruction (Stein & Smith, 

1998). 

I am a white, middle-class woman. I had a middle-class upbringing, and at-

tended majority-minority schools in my K–12 education. I benefitted from a type of 

tracking discussed here, specifically, a gifted program that was originally created as 

a way to desegregate schools by incentivizing White student attendance. I recall 

being offended as a young person by the comments my parents’ friends would 

make about their decision to send me to school with “those kids.” After undergrad-

uate education in computer science and mathematics, I came into teaching through 

a grant-funded alternative certification program. This program was a partnership 

between the same university and school district that partnered together for Jack and 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       164 

Michelle’s teacher education. I taught high school for 8 years in that same district 

before attending the university as a full-time doctoral student to conduct this study. 

While I was teaching high school, I was also continuously enrolled in gradu-

ate coursework. The school in which I taught served a population of about 75% Af-

rican and African American and 25% Latin@ students. I once again noticed, and 

was offended by, the reactions and negative comments about my students that I re-

ceived when I would tell people where I worked. I increasingly began to recognize 

how important it was for me to educate myself on critical perspectives in education, 

culturally relevant pedagogy, issues of inequity and racism in education, and the 

cultures and lived experiences of my students if I was going to effectively conduct 

my duties as an educator. I enrolled in as many courses at the university that ad-

dressed issues of equity, power, race, and class in education as I could, and I made a 

concerted effort to engage in conversations about equity and diversity with my col-

leagues of color, both in the school and at the university. When I enrolled as a full-

time student to complete my dissertation, my fellowship provided me the oppor-

tunity to work with an alternative certification program and mentor new teachers. I 

felt connected to this work because I had also been alternatively certified, I had 

taught in the same district, and the alternative certification program was attempting 

to bring in teachers who had a dedication or a connection to the district community. 

My experiences drove my dedication to mentor teachers to provide high-quality 

mathematics education for the low-income students of color whom my district 

served. 

 

Mentoring 
  

Features. The focus of the mentoring was on two interconnected features of 

instruction. First, I encouraged the teachers to find or develop tasks that provided 

students opportunities to perseverance with and make sense of mathematics to un-

cover relevant mathematics on their own through application of their prior 

knowledge to an accessible problem. Second, I encouraged the teachers to provide 

students opportunity to have productive small-group and whole-class discussions 

(Smith & Stein, 2011) about the relevant mathematics and to develop strategies of 

effectively facilitating that discussion without taking ownership of the discussion 

from the students. Because these teachers were novice teachers, many of the in-

structional strategies that I proposed arose either in discussion with them during 

individual mentoring sessions or group seminars. I served as a researcher collecting 

data and as mentor of these individuals operating with the intent of supporting each 

of them throughout their first year of teaching. 

This mentoring is important because several researchers argue that mentoring 

should be integrated into transformation of the teaching profession as well as an 

inquiry-based approach to incorporating all the features of education reform, includ-

ing standards-based practices (Feiman-Nemser & Parker, 1993; Hargreaves & Fullan, 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       165 

2000; Stanulis & Floden, 2009). Research suggests that focusing on “balanced” 

approaches to instruction would improve the actual practice of teaching, rather than 

simply focusing on emotional support of new teachers (Stanulis & Floden, 2009). 

Furthermore, there is a call for more research on how to support novices in their 

development of reform-based teaching by incorporating the standards into the base 

conception of induction mentoring practice (Wang & Odell, 2002). This mentoring 

treatment foregrounded two of the Standards of Mathematical Practice (CCSSM, 

2016), “make sense of problems and persevere in solving them” (¶ 2), and “construct 

viable arguments and critique the reasoning of others” (¶ 4), while simultaneously 

incorporating content standards, mathematics teaching practices (NCTM, 2014), oth-

er standards of mathematical practice (CCSSM, 2016), and the NCTM Principals 

and Standards for School Mathematics (2000).  

During this mentoring process, I provided time and guidance during collabora-

tive sessions for the teachers to reflect on videos of instruction as well as their own 

teaching by focusing on teacher moves that promoted or did not promote students’ 

sense making and productive discussion (Smith & Stein, 2011). During the planning 

sessions, Michelle, Jack, and I met as a group to either plan individual lessons or 

whole units. I shared information about tasks that, I believed, would promote student 

discovery of mathematics; the ordering of topics in a way that made mathematical 

sense; the different topics that would need to be incorporated; how to promote discus-

sion; and what manipulatives would be appropriate for hands-on, concrete learning 

opportunities. During observations, I usually simply took notes, but there was one 

occasion that I co-taught with Michelle while Jack observed as a response to a re-

quest from Michelle. During reflection sessions, I would focus on giving feedback on 

specific teacher moves or plans that, from my perspective, supported or inhibited stu-

dent sense making and productive discussion. We would also collaboratively review 

student data and make decisions on next steps for student learning. The content of 

each mentoring cycle is summarized in Table 1. The first cycle is disaggregated be-

tween Jack (J) and Michelle (M), as that planning session was conducted individual-

ly, while all the others were conducted in a group of three (Jack, Michelle, and me). 

Differences in mentoring of Michelle and Jack. When specifically focusing 

mentoring resources on Michelle, the work of mentoring became focused on lesson 

and unit planning. Together, we worked on planning lessons that included tasks that 

were of a high level of cognitive demand, and included opportunities for student talk.  

When specifically focusing mentoring resources on Jack, I attuned Jack to the on-task 

discussions that the students had without his direct intervention, as well as mentored 

him to reduce the amount of scaffolding (e.g., leading questions) that he provided to 

small groups during their time working on problems that had high cognitive demand. 

Also, because Michelle and Jack observed each other due to their half-time teaching 

arrangement, they were each able to participate in the other’s reflection conversation 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       166 

with me and support the positive strides that both students and they were making with 

respect to having students persevere with, make sense of, and discuss mathematics. 

 

Table 1 

Mentoring Cycle Content 
 

Cycle   Content  

1 (J) 

 

 Task with high cognitive demand (Stein & Smith, 1998) 

 Individual work, small-group work, and whole-class discussion (Lampert, 2003) 

 Calling on students randomly during whole-class discussions 

 Moving toward questions that elicit mathematical thinking rather than leading questions (Dris-
coll, 1999) 

 Using talk moves to engage more students in whole-class discussion (Chapin, O’Connor, & 
Anderson, 2009) 

 

1 (M) 

 Task with high cognitive demand (Stein & Smith, 1998) 

 Individual work, small-group work, and whole-class discussion (Lampert, 2003) 

 (teacher suggested) Student roles in groups (Johnson & Johnson, 1999) 

 Feedback on student investigation implemented as direct instruction 

 Feedback on Initiation-Response-Evaluation model of student talk 

 Using talk moves to give students agency in their learning (Chapin, O’Connor, & Anderson, 
2009) 

 Allotting enough time for student sense making 

2 

 Co-planning open-ended task with high cognitive demand (Stein & Smith, 1998) 

 Co-teaching and facilitating student whole-class discussion with Michelle as Jack observed 

 Feedback on pacing of lesson; do not have to wait until all students have the right answer to 
have a summary discussion 

 Feedback on focusing on mathematical goals 

 Analyzing student work 

 (Jack) Feedback on effective use of talk moves (Chapin, O’Connor, & Anderson, 2009) 

3 

 Teacher reported that administration wanted more use of hands-on activities and manipulatives 

 Task with high cognitive demand (Stein & Smith, 1998) 

 Unit planning – Expressions and Equations 

 Guess-Test-Generalize (CME Project, Algebra I) and Polya’s Problem-Solving Techniques 
(Polya, 1945) 

 Guidance to avoid using keywords to decode word problems (NRC, 2001) 

 Concrete-Representational-Abstract sequence of instruction (Witzel, Mercer, & Miller, 2003) 
and the National Library of Virtual Manipulatives 

 Activating prior knowledge (Hollingsworth & Ybarra, 2009) 

 Facilitating teachers doing the math 

4 

 Having students construct their own mathematics definitions based on examples 

 Using multiple different representations of figures 

 Unit planning – Geometry 

 Task with high cognitive demand (Stein & Smith, 1998) 

 Heterogeneous grouping 

 Small-group stations 

 Using rubrics to hold students accountable for participation in small-group instruction 

 Asking a random student to explain to the teacher to encourage students collaborating to devel-
op an explanation (Boaler & Staples, 2008) 

 (Michelle) Feedback on slow pacing and lowering of cognitive demand 
 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       167 

5 

 Unit planning – Data Analysis 

 Tasks with high cognitive demand (Smith & Stein, 1998) 

 Developing a project for Project-Based Learning (Krajcik & Blumenfeld, 2006) 

 Working through why constructions “work” based on properties of circles with the teachers 

6 

 Review of project for Project-Based Learning (Krajcik & Blumenfeld, 2006) 

 (Jack) Review of the mathematics behind misleading graphs 

 (Michelle) Reflection on successful whole-class discussion on which data displays are appropri-
ate for which situations 

7 

 Reflection on the project 

 Discussion of the upcoming unit 

 Discussion of upcoming testing and field trips 

 Benefits of including tasks with high cognitive demand and small-group discussion 

 Closing out coaching 

  

Data Collection and Analysis 
 

I collected data from seven teacher-support reflection cycles. Each support 

cycle took place over approximately two weeks. These sources allowed me access 

to teacher expectations of students as well as their instruction trajectory (see Table 

2).  

 

Table 2 

Description of Data Sources 
 

Data Collected Description 

Collaboration: Teacher Seminar 
All mentees (four mentees and I) met to review teaching videos 

and their own teaching with an eye to student sense making and 

discussion. Sessions were videotaped. 

Planning: Mentoring Session Jack, Michelle, and I met bi-weekly to plan lessons and units. 

Observations 
Lessons were observed bi-weekly. There was one instance of co-

teaching with Michelle while Jack observed. 

Reflection: Mentoring Session 
After the observations, the mentor and the participants met one-

on-one to debrief the lesson. 

Baseline 
Baseline data was collected through observations of participants’ 

summer coursework, reading of their submissions in summer 

coursework, and interviews about how to teach with a task. 

Follow-up Interview 
Participants were interviewed after the original treatment in 

order to reflect on questions on what transformed their teaching. 

 

Within the teacher-support reflection cycle, all of the planning and reflection 

mentoring sessions were audio recorded and transcribed. Each teacher seminar, as 

conducted between November 2011 and March 2012, was also video recorded and 

transcribed. Although seven classroom observations of each participant were con-

ducted, only three of these were sources for data collection. The teaching observa-

tions that served as sources for data collection were conducted in November 2011, 

during Cycle 1 of the teacher-support reflection cycle; in January 2012, during Cy-

cle 4 of the teacher-support reflection cycle; and in March 2012, during Cycle 7 of 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       168 

the teacher-support reflection cycle. I identified these particular classroom observa-

tions as sources for data collection to establish whether instructional change had 

transpired as documented by the beginning, middle, and end of the study. I took 

field notes during each of these three observations and made notations that indicat-

ed particular features of the lesson that were relevant to analysis pursuant to an-

swering the research questions. These notations consisted of codes that I hypothe-

sized before conducting the study. Each teacher also carried an audio-recorder to 

document their verbal teacher moves as well as responses from students with whom 

they were interacting. These audio-recordings were transcribed. My observational 

data consisted of field notes and audio transcripts; however, my audio transcripts 

are limited to statements of teachers and the responses of students in interaction 

with their teachers. 

The codes for this corpus of data were centered on strategies and challenges. I 

looked for instances of strategies that the teachers were using to incorporate math-

ematical sense making and productive discussion, both those that the teacher used 

independently and those that we developed together. I also looked for the teacher 

using language that suggested they were struggling to implement high-cognitive-

demand tasks that allowed students room for discussion, and what they expressed 

as their challenge. Working with other colleagues in the doctoral program, we cod-

ed pieces of data independently to ensure the coding method was reliable. A sub-

code that arose during analysis in the “challenge” codes was student ability. Teach-

er talk that referenced their notions of their students’ ability to participate in the 

mathematics arose regularly.  

 
Findings 

 

During their interactions with me in the mentoring sessions, both Michelle 

and Jack expressed their expectations of their students’ ability due to their track and 

their experiences interacting with their students. They also, through mentoring and 

reflecting on their experiences with their students, began to develop the ability to 

provide high-quality mathematical experiences that allowed their students to engage 

in mathematical sense making through participation in productive mathematical 

discussion. 

 

Michelle: Low-Tracked Course 
 

Initially, Michelle had low expectations for her students in the low-tracked 

course. Through mentoring and experience with her students, however, she began 

to teach in a way that provided opportunities for students to engage with high-

quality mathematics tasks and participate in mathematical sense making through 

productive discussion. 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       169 

Michelle’s perception. As we discussed mathematical content in a manner 

that focused more on overarching mathematical ideas and topics rather than presen-

tation of single, isolated skills, Michelle’s perception of her students’ ability and the 

successes they would be able to experience when approaching mathematics through 

both concepts and skills often troubled her. In one mentoring session, I suggested 

that a teacher could teach equation writing and solving from a conceptual perspec-

tive based on generalization from arithmetic and working backwards to undo arith-

metic procedures. I posited that then it would be unnecessary to teach solving one-

step equations using addition, subtraction, multiplication, and division, and then 

two-step equations with combinations of those as individual elements. I suggested 

that initially thinking about an equation as a procedure with a result would allow 

students to think about what was done to the variable and then what would have to 

be undone to return back to the value of that variable. Michelle responded by stat-

ing that she would separate one-step and two-step equations and then further break 

down one-step equations to focus on those that involved addition and subtraction as 

a single skill, and then subsequently address those one-step equations that involved 

multiplication and division: 

 
Michelle: Well, I don’t know, I probably would for [my class]. Maybe I wouldn’t 

take 2 days for each [operation]…. I can combine addition, subtraction and then mul-

tiplication, division. 

Jack: I can see that. 

Michelle: I would put those together but I would do step-by-step. I wouldn’t com-

bine them all at once and do several different operations with [my students]. So, 

okay, well I’m happy that it gives us a time to teach conceptually in step-by-step 

with the operations. Okay. 

 

In this explanation, Michelle immediately followed her statement of prefer-

ence for isolating addition and subtraction one-step equations from one-step equa-

tions involving multiplication and division with the statement that she was happy to 

have the time to teach conceptually. This response suggests that although she want-

ed to use manipulatives to have the students visualize the individual operations, she 

still felt the need to isolate a single skill into a lesson. When I suggested that she 

could address solving equations as one concept, she resisted. She stated, “Yeah, you 

probably could do it in your class” (Michelle, Mentoring Session, December 9, 

2012), referring to Jack’s class. This highlights her concern about her students’ abil-

ity level, because the school labeled Jack’s class as an honors class and labeled 

Michelle’s class as comprehensive, ostensibly populated by students with a lower 

level of ability. 

Michelle’s change in perception through experience. It may be that Michelle 

was experiencing some dissonance between her perception of what she felt her stu-

dents were able to accomplish successfully and what they actually were able to do. 

She had been seeing her students have successes in the classroom on problems that 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       170 

she felt might have been too difficult for her students to approach. She had stated 

previously: “Sometimes you’re kind of leery in giving them stuff and then run off 

on their own. But sometimes they surprise you” (Michelle, Seminar, November, 21, 

2012). The successes that she was seeing her students experience in the classroom 

from more discourse-based, conceptually focused instruction challenged her previ-

ous ideas regarding the limitations associated with perceptions of her students’ lim-

ited ability and allowed her to try more difficult mathematics problems with her 

students. 

Not only did Michelle begin to incorporate richer mathematics problems into 

her teaching but also she began to include more small-group work. She had students 

work together on real-world problems and allowed students to reason through the 

problems concretely before she asked questions that would help them think about 

the mathematical ideas behind them. She found that her students were successful in 

translating their thinking into mathematical terminology: 

 
So, for example, a lot of them, they started off with 30 and then $10 was for food, so 

they took out $10 out of the total 30 and I said, “What are you doing? What are you ac-

tually doing when you’re doing that?” And they get it! They say, “Oh, I’m taking 

away. Oh, what’s the mathematical term [for that]? Oh, subtracting.” (Michelle, Men-

toring Session, December 30, 2011) 

 

Michelle was happy to see that her students were “getting it” when they were 

approaching word problems, as Michelle had initially considered word problems to 

be too difficult for her students. She saw her students have success working on a 

problem in their small groups without her intervention and without her leading the 

students through the whole process of problem solving. These instances of students 

working together to solve difficult mathematics problems allowed her to feel that 

she could take more risks in her mathematics classroom. She began to change her 

perception of scaffolding. Originally, she thought of scaffolding as the practice of 

offering leading questions that directed students through the intended individual 

procedures and skills in the classroom. Now she was scaffolding by asking guiding 

questions that facilitated students’ mathematical thinking while they investigated 

problems using concepts and ideas. 

Michelle seemed to use what she knew or learned about student difficulties in 

a lesson as an insight for influencing her future instructional decisions, instead of 

abandoning her plans entirely. Michelle taught a mathematics course and a science 

course to the same students. To engage her students in a conceptual discussion 

about different systems within the body, she had given her science class a home-

work assignment that required them to read about a certain system in the human 

body and then come prepared to discuss their knowledge with the class in small 

groups. However, she was disappointed that her students did not come to class pre-

pared. She assigned blame to the students for not doing the homework assignment 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       171 

and therefore for sabotaging a lesson that would have focused on discussion. She 

noted, because of their lack of preparation, the students simply did not have any-

thing to add to the discussion: 

 
I don’t know. I had this nice thing planned, and it’s one thing too when you’re trying to 

have group work and facilitate discussion, you have to have something to add. I had a 

homework assignment, they were supposed to investigate things that they identified that 

they were going to do. If half the class doesn’t do that, it’s like I wanted to rotate them 

so that they would [teach] each other and provide information for one another. It’s like, 

when they don’t do the task; it just shuts the discussion down. Okay, you’ve got nothing 

to add. You know, they’ll just be sitting there. (Michelle, Seminar, February 1, 2012) 

 

Michelle expressed frustration with her students for their inability to partici-

pate in the discussion. In this case, it seemed that the students were not able to par-

ticipate in the discussion because of the individual, at-home nature of the preparato-

ry assignment. Michelle seemed to recognize this limitation, as she did not give up 

on having conceptual-themed discussions with this group of students. Instead, she 

modified her instructional strategies to allow time in class for students to formulate 

their responses in their small groups before engaging in the discussion. Subsequent-

ly, she tried again to organize a class based on the assumption that students would 

prepare for class in advance. But this time, Michelle did so with her mathematics 

class. In a unit on data analysis, she provided her students with a list of scenarios 

and asked the students to decide which data display would be the most appropriate 

in the given scenario. She also provided them with a reference sheet of key terms, 

definitions of those terms, and their exemplar applications, to allow students access 

to a discussion where there could be multiple correct responses to a single question. 

Instead of Michelle’s initial scaffolding design, which was to assume control of the 

conversation and carefully guide the trajectory of student talk in the classroom, she 

positioned the students centrally in the discussion. She provided the students with a 

means of facilitating their discussion through their reference sheet and then allowed 

them to express their thinking to each other, only providing direction and comments 

when necessary to continue the flow of discussion. She began to place herself in a 

position of facilitator rather than director of classroom talk. After Michelle made 

these different instructional adjustments, she saw that her students could successful-

ly participate in a conceptual discussion: 

 
And we’re at the point [where] we’re talking about which data display to choose and 

why, depending on the circumstance. So I had two scenarios, and I asked each group to 

talk amongst themselves to determine which data display they’ll choose for [each of] 

the scenario[s]. And so, once they did that I opened it up to the floor. … and I called a 

group, a table, and they shared their response, and I said, “Okay, who agreed or disa-

greed and want[s] to add to it?” So, that spawned a lot of discussion. “Oh, look, I disa-

gree because this, this, and this.” And they were able to use the terminology of why… 

and recommend something else. And so, that kind of went on back and forth for a good 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       172 

10 minutes, and it was a really good discussion. So I got to hear what they were think-

ing. So it was really good. Positive. (Michelle, Seminar, March 7, 2012) 

 

Although Michelle had experienced earlier failures when expecting the stu-

dents to prepare for and participate in whole-class discussions that focused on 

mathematical concepts and had multiple solutions, she had tried again and found 

success. She negotiated this challenge by providing not only class time to prepare in 

their small groups but also a reference sheet to remind students of the mathematical 

tools that they could use to respond to the tasks and to explain and justify their rea-

soning and solutions. This experience provided her with another example of student 

success in her classroom. 

Summary of Michelle. Michelle taught 70-minute class periods throughout the 

year. In November, Michelle incorporated 5 minutes of small-group work, some 

guided presentation, and individual work. In the January observation, she attempted 

to allow students to collaborate in small groups for 5 minutes, then co-opted their 

discussion when she felt the student were struggling. By March, she spent 34 

minutes of her 70-minute class period facilitating students’ discussion in small 

groups, followed by student presentations and discussions of findings in the whole 

group. This transition can be attributed, in part, to the consistent mentoring that re-

sulted in her having successful experiences with her students working on rigorous 

mathematics and having productive discussions. 

Throughout the year, Michelle saw more successes in terms of her students’ 

ability to access different mathematical problems in her classroom as she changed 

the types of scaffolds she provided. Instead of leading students through systematic 

procedures that she considered difficult for her students, she began to facilitate their 

interaction with mathematical concepts and their constituent skills. Where earlier in 

the year she would stop small-group time to demonstrate solutions to the exercises 

that the students were working on, later in the year she provided more indirect scaf-

folding and allowed the students to work together to solve problems. She discov-

ered that her students were capable of working with other students on difficult 

mathematics with less teacher direction. She reflected on her surprise and happiness 

about her students’ successes: “They do it and then you’re shocked…. It went very, 

very well” (Michelle, Seminar, March 14, 2012). She seemed to continue allowing 

students more autonomy in their small groups as a result of her negotiation of this 

challenge, in addition to allowing her students more opportunities for working to-

gether. Furthermore, instead of removing instances in which she thought her stu-

dents would struggle, she made different instructional decisions to support her stu-

dents’ efforts to working through not only mathematical skill but also the meaning 

of concepts. 

 

 

 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       173 

Jack: High-Tracked Class 
 

Although Jack initially believed that his honors tracked students would easily 

participate in sense making through productive discussion, his initial experience in 

the classroom gave him a different perception. Due to that experience, he became 

unsure and reticent about implementing instruction that promoted problem solving 

and discussion. 

Jack’s perception. Jack perceived that there were differences in his students’ 

and Michelle’s students’ willingness to participate in student-centered instruction 

that included discussion. He stated: 

 
When I sit at my desk and watch [Michelle] teach and I see her students doing…it’s 

easier for them to work off script…. I think my kids were technically supposed to be in 

honors classes, they like much more direct instruction and they don’t like to be asked 

to do something first. They’re a lot more resistant to it…. I think, like maybe, if you 

were in an honors class, you probably would be good at “doing school” and you’ll be 

good at “doing school” if you were good at just listening and taking notes. (Jack, Sem-

inar, November 21, 2011). 

 

Jack felt that his students were acclimated to instructional techniques that di-

rected them as to what to do and what to recall. Furthermore, he felt that his “hon-

ors” students were familiar with “doing school” in a particular way, a way that re-

quired them simply to sit quietly, listen, and take notes. He felt that Michelle’s stu-

dents might not have been as completely acculturated to direct instruction in the 

way that the honors students were, given that they were not considered to be “good 

at doing school.” Therefore, he felt “being good at school” caused his students to be 

more resistant to participation in student-centered lessons and mathematical discus-

sion, as compared to Michelle’s students. 

Jack also believed that his students’ familiarity with particular norms of 

schooling prevented them from productively collaborating; in that, instead of ex-

plaining and justifying solution strategies, asking clarifying questions, making 

sense of different solution strategies, and sharing the work, they were used to simp-

ly providing and sharing answers. He felt that his students were more comfortable 

with working individually rather than in small groups: 

 
I hope they drag [the students who do not understand] with them and not just leave 

them alone…. It’s hard to make them cooperate you know…. Sometimes they just like 

to do it on their own. (Jack, Mentoring Session, January 18, 2012) 

 

Jack wanted his students to work together so that a student who had a greater 

understanding of a particular topic could assist the other students who were not as 

secure. However, the students’ comfort with working individually, rather than col-



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       174 

laboratively, made it difficult for Jack to get them to assist each other when ad-

dressing a mathematics problem in small groups. 

Jack attributed his students’ resistance to student-centered mathematics and 

participation in discussion to the type of instruction with which his honors students 

were familiar. This resistance was not what he had expected as he entered into 

teaching believing that students who were labeled honors would find it easier, as 

compared to the “comprehensive” students, to participate in discussion and student-

centered instruction. His students’ resistance to collaborative problem solving and 

discussion, however, led him to believe that his students had repeatedly experienced 

instruction in their years of honors course placement which were organized around 

lecture, note taking, and independent seatwork. He reflected during a follow-up in-

terview during the following school year: 

 
If anyone, I would expect them to be able to handle it more…like I said…it was harder 

for them to make the jump to doing something extra. I guess what they saw as some-

thing extra. As long as they could write it down, I think they thought that was suffi-

cient. Um, but I guess I had to spend time explaining like, “You learn more, you learn 

more by teaching.” (Jack, Follow-Up Interview, May 7, 2013) 

 

Because Jack believed that his students previously had solely been required to 

complete problems individually and to record their answers in written form, they 

were resistant to doing something they felt was unnecessary, or “extra.” Therefore, 

Jack had to reinforce his expectations consistently and to provide a rationale as to 

why students would benefit from discussing mathematics to entice his students into 

participating fully. 

Jack’s change in perception through experience. Initially, Jack attempted to 

implement a procedure whereby those students who finished a particular problem 

first would serve as experts and assist other students in completing that problem. 

During an early seminar, Jack explained the problem with this particular strategy: 
 

So, one thing I tried…in terms of getting students to appreciate each other, [or] what 

each other has to…say…when they work on something,…. Whoever finishes first and 

gets it correct, they get to go around and explain it to the rest of the class and like check 

off the papers and stuff. But what I noticed was that…they weren’t explaining. (Jack, 

Seminar, November 7, 2011) 

 

Jack wanted his students to discuss and explain their solution strategies to 

others in order for each of the students to develop an appreciation of other students’ 

thinking. The students who were serving as experts, however, were simply telling 

the others the answer and having them change their approaches to earn a check on 

their work. This telling was not the desired outcome; Jack wanted students to learn 

to collaborate and explain their thinking to others. Jack realized that the strategy of 

using student experts was not yielding the result of collaboration. 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       175 

In November, Jack taught a lesson where students investigated patterns to dis-

cover how to calculate numbers raised to the zero power and negative exponents. 

To investigate negative exponents, Jack had the students complete a table of posi-

tive exponents and extend the pattern backwards to a zero exponent. Jack circulated 

around the classroom and encouraged students to explain their thinking about why 

the result of raising a number to a zero exponent would be 1. 

 
Jack: Can you explain why you think it may be 1? Do think it may be 1? Can you ex-

plain, [student name]?  

Student 1: No. 

Jack: So you just think it’ll be 1 but don’t know why. 

Student 1: Oh, I used the calculator (laughing). 

Jack: (To another student) OK, you think you can [explain it]? 

Student 2: Yeah. 

Jack: Go ahead and explain it.  

Student 2: I think it would be 1 because yeah I would say 3 [to the] 1. Three [to the] 1s 

gonna equal 3 but as you have 9 over 9 that’s gonna be 1 whole. (Jack, Classroom Ob-

servation, November 17, 2011) 

 

When one student said that she used the calculator to find her answer and 

could not explain her thinking further, another student in that group suggested that 

she could explain it and then proceeded to explain her thinking. Jack persisted in 

pressing students to explain their thinking rather than just finding a solution. He 

asked for explanations from individual students as well as encouraged them to 

speak to each other, saying, “I need you to make sure that everybody at this table 

[understands]” (Jack, Classroom Observation, November 17, 2011). Jack was not 

only eliciting explanations from students so that they could relay their ideas to him, 

but also encouraging them to explain to each other when he was not present. 

Later in January, Jack began to incorporate center rotations with his students. 

In this particular lesson, half of the class was completing a review “scavenger 

hunt,” where folded papers were arranged around the classroom with a multiple-

choice problem on the inside, and the answer to a different problem on the outside. 

Students would move about the classroom searching for the answers to their prob-

lems, and, in this way, complete each of the problems. The other half of the class 

was split into two groups, one that was directed by Jack to take notes on the defini-

tions related to congruence, and the other was attempting to discover the surface 

area of a rectangular prism through the use of nets. Most of Jack’s focus was on the 

group to which he was providing direct instruction, so the students in the other 

group had to work collaboratively and mostly autonomously, as Jack would check 

back with them only intermittently. Jack previously challenged a small group of 

students to refine their formula by double-checking if the area of each face of the 

rectangular prism was the same. While Jack worked with the other group, the fol-

lowing conversation transpired: 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       176 

Student: Mr. Davis, we got it. 

Jack: Do [all the sides] go together? 

Student: No, because [different side lengths of the different pieces]. 

Jack: Check each other, [student name] and [different student name] check each oth-

er’s. (Jack, Classroom Observation, January 27, 2012) 

 

Two things are salient about this conversation. It shows that Jack was still, 

consistently, not only requiring students to work together, but also asking them to 

collaborate through validating their answers with other students. Also, there is a 

shift in the language that the students were using. In the initial observation, the stu-

dents used the work “I” when explaining their thinking. Here, the student used the 

word “we.” Jack’s consistent, high-level expectations were changing the norms of 

student discussion in the classroom. 

In March, Jack reflected in seminar about a lesson in which he engaged stu-

dents in error correction of a past assessment. By then, Jack was aware of students 

engaging autonomously in their small groups in discussions that included both pro-

cedural and conceptual talk. I also affirmed, through observation, that his students 

were making sense of problems, procedurally and conceptually, in small groups, 

through discussion. Jack was also aware of occasional incidents when students 

would give answers to each other and then would quickly redirect their conversa-

tion to explanation. He reflected: 

 
I think there was a mix [of procedural and conceptual conversation among students]. 

Of course, there’s going to be some people just saying, “Oh, the answer was ‘a’,” and 

so I did hear that. And so for those people I had to say, “But why is it ‘a’?” (Jack, Sem-

inar, March 7, 2012) 

 

Summary of Jack. Jack negotiated the challenge of his students’ receptiveness 

and ability to persevere, make sense of, and have productive discussions about 

mathematics. In November, Jack’s students spent 50 minutes of his 70-minute class 

period working individually, while the rest of the time was spent participating in 

teacher-led talk. By the time I observed in January, Jack had his whole class work-

ing in centers for the duration of the period, where they worked in small groups ei-

ther autonomously, in student-centered investigations, or teacher-led notes. During 

the observation in March, Jack spent 45 of 70 minutes in whole-class discussion 

where students presented their findings, challenged each other, and justified their 

arguments. Although there had been a marked shift in both the occurrence and the 

manner through which students were discussing in the classroom, there still was an 

occasional deviation from that pattern. While Jack was cognizant of these devia-

tions, he persisted in enforcing his high expectations for these students by remind-

ing them quickly to explain their thinking to each other. Jack’s commitment to ex-

cellence in teaching and education motivated him to work diligently to change the 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       177 

culture of discussion in the class through teacher moves that reminded students of 

his persistent expectation of explanation and discussion. 

 
Discussion 

 

For Michelle and Jack, the fact that their students were tracked had at least an 

initial effect on their instructional decision-making. Research indicates that tracking 

has an influence on teachers’ perceptions of students, especially their assumptions 

about their lower-tracked students (Oakes, 2005). In this case, tracking affected the 

perceptions that these teachers had with respect to their students both in the courses 

labeled comprehensive and in those labeled honors. 

Michelle taught a course that was labeled as comprehensive, although this 

tracking was based on her students’ prior reading scores and not on any prior as-

sessments of their mathematics proficiency. Nevertheless, Michelle often used her 

preconceived notions of her students’ low ability as an excuse for not engaging 

them in challenging, student-centered mathematics lessons that would require prob-

lem solving and provide the students with opportunities for sense-making discus-

sions. In addition, Michelle did not recognize the differing strengths or needs of her 

students. She spoke of her students as an aggregate, as a whole class only composed 

of students with low levels of ability. Although she made attempts at incorporating 

opportunities for discussion in her class, her perception of her students’ ability 

caused her to fear student struggle, to provide direct instruction of procedures, and 

to focus much of her attention on rote memorization of vocabulary terms. Nonethe-

less, although Jack and Michelle’s students took the exact same district-prescribed 

unit assessments, she never indicated that she had any evidence that her students 

were performing any differently that Jack’s honors students were. Her instructional 

decision making, particularly her decision to slow down her pacing and to include 

more direct instruction, seemed to be based solely on her preconceived notion of 

what the “limited” ability level of students placed in a lower-tracked course meant 

to her. However, she was able to negotiate this challenge and engage her students in 

productive mathematics learning in whole- and small-groups discussions that al-

lowed students opportunities to make sense of mathematics. 

Michelle used strategies to mitigate her students’ perceived struggle to en-

courage their participation in class discussions. She gave students both individual 

and small-group thinking time before requiring them to share their thinking to the 

class. She also provided reference sheets with mathematical terminology so that 

students could use the “correct” mathematical vocabulary in their discussions. She 

required them to verify individually calculated solutions with other small-group 

members. Also, Michelle began to appreciate using small-group work because it 

provided her with more contact time with individual students. During this time, she 

could correct any misconceptions held by individuals. She may not have changed 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       178 

her mind about her students’ initial ability, but she was able to see her students’ 

successes in mathematical sense making as the year progressed. 

Jack, on the other hand, taught the class that was labeled honors; students 

were placed in this course by the same prior reading assessment that determined the 

placement of Michelle’s students. Although Jack stated that he had automatically 

assumed that his honors students would be more capable of engaging in student-

centered mathematics investigations and discussions, he found his students resistant 

to this type of instruction. He attributed their resistance to the students’ assignment 

to high-tracked courses. He felt that because his students had been tracked into 

honors courses, a feature of scheduling that had been in place for more than one 

year, his students had become acclimated to learning in a particular manner: 

through listening, note taking during directed instruction, and completion of written 

solutions to mathematics problems reflecting their individual and independent ef-

forts. He felt that his students’ prior experiences with tracked courses, and their pri-

or teachers’ mode of instruction in these tracked courses, made his students appre-

ciate and expect specific types of instruction over others. They were not only famil-

iar with these instructional routines; they had learned how to be “good at school” in 

environments marked by these practices. Jack felt that he had to struggle harder 

than Michelle to teach students how to participate in student-centered learning envi-

ronments where the expectation was for students to explain their thinking to others, 

and he attributed that struggle to the fact that his students were tracked into the 

honors section. 

Jack, however, used his experience with building relationships with middle 

school students to establish a safe environment in his classroom. He built a rapport 

with his students that caused them to trust that he would not embarrass them public-

ly. Furthermore, he established norms in his classroom whereby students would be 

respectful of others during small- and whole-group discussions and would not de-

value the thinking of others. Through these techniques, he began to develop a cul-

ture of participation in small- and whole-group discussion in his classrooms. His 

students became familiar with the practices of struggling with, and making sense of, 

mathematics collaboratively. 

 

Implications 
 

Tracking has long been targeted for discussion in the mathematics education 

community. Many mathematics researchers have advocated for heterogeneous 

classrooms, claiming that they are better sites to support all students’ learning 

across and within all achievement levels, advancing both equity and mathematics 

achievement (e.g., Boaler & Staples, 2008; Burris, Hubert, & Levin, 2006). Track-

ing not only affects teachers’ preconceptions of students’ potential to achieve, but 

also it might affect students’ development of a productive disposition about mathe-

matics (e.g., Oakes, 2005). Nevertheless, there has been much resistance to group-



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       179 

ing students’ heterogeneously both in public discourse and in policy documents 

(e.g., Loveless, 1998). Education policy is not likely to eliminate tracking. Thus, it 

is important to prepare teachers for the possibility of teaching in a tracked course. 

The study reported here highlights two points. First, teacher education courses 

must include an opportunity for both prospective and practicing teachers to think 

about and discuss tracking critically. Teacher educators must provide counter-

narratives to the pre-conceptions that tracked former students bring to their class-

rooms as teachers. The teachers’ perceptions of tracked students influenced their 

desire and ability to engage their students in sense making and productive discus-

sion. Furthermore, it has been stated that teachers in alternative certification pro-

grams may be more likely to harbor low-expectations of tracked students than tradi-

tionally certified teachers (Brantlinger & Smith, 2013; Delpit, 2012; Habermann, 

1991). Therefore, it follows that critical discussion about tracking, the perceptions 

of students who are tracked, and how to actively work against tracking are arguably 

even more important in alternative certification programs than in traditional teacher 

education programs. It often is assumed that tracking is necessary to provide appro-

priate instruction for homogeneous groups of students. Research has shown that de-

tracking and heterogeneous instruction is powerful and equitable (Boaler & Staples, 

2008; Burris et al., 2006). 

Moreover, Oakes (2005) finds that the methodology of tracking is often tacit-

ly accepted as appropriate, and therefore teachers believe that instruction with dif-

ferent levels of rigor is appropriate for their tracked population. As teacher educa-

tors, we need to problematize tracking and its implications. In addition, teachers 

need to be pushed to continue to try to engage their students in sense making 

through participation in productive discussion so they can gain the experience 

through experimentation that can further challenge their pre-conceptions of their 

students’ ability. Through literature, hands-on experiences, and classroom discus-

sions, we may be able to disrupt the misconceptions that new teachers might have 

about students who have been tracked. This disruption will help to address situa-

tions similar to that evidenced by Michelle and Jack, as their assumptions about 

their students’ abilities contributed to the challenges that they faced when develop-

ing student-centered instruction, instruction that included opportunities for students 

to discuss and make sense of mathematics collaboratively.  

Furthermore, teacher educators need to prepare teachers to have discussions 

with their students that include counter-narratives to ability grouping. Teachers 

must be equipped with ways to talk to students about their ability so that students 

placed in low-track courses are able to build the same types of productive disposi-

tions that students in high-track mathematics courses might develop. Additionally, 

high-tracked teachers need to be equipped with ways to challenge the norms of 

schooling that may be prevalent in courses that are labeled honors. Often teacher 

educators focus solely on the low teacher perceptions of low-tracked students; how-



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       180 

ever, student resistance to a “new way” of participating in mathematics courses also 

needs to be a focus, considering that schools that serve low-income students of col-

or often heavily stress low-level direct instruction for test-preparation purposes 

even in those courses noted as honors (Delpit, 2003). In this way, teacher educators 

can help to reduce the challenges that novice teachers face. Teacher educators, 

however, cannot expect that teachers learn everything they need to know about 

teaching during their teacher preparation program (Hiebert et al., 2003). Therefore, 

it is important that teacher education programs prepare teachers to analyze their 

students’ abilities in ways that enable the teachers to recognize student strengths as 

well as weaknesses and to learn for themselves that students of any track are capa-

ble of being successful. 

Additionally, it is important for teacher education programs to help novice 

teachers develop ways to de-track mathematics within their classrooms and at their 

schools. In this instance, Jack and Michelle were encouraged to group students het-

erogeneously in their classrooms. Furthermore, because Jack and Michelle were 

planning together, the level and type of education that the students in the two dif-

ferent tracks were receiving were more similar than they were different. Therefore, 

as Jack and Michelle learned how to implement lessons that included tasks with 

high levels of cognitive demand (Stein & Smith, 1998) and productive discussion 

(Smith & Stein, 2011), they were effectively de-tracking the seventh grade at their 

school. It should be encouraged in teacher preparation programs that teachers, no 

matter the level of the track of their placement, consistently hold students to high 

expectations, allow them opportunities to make sense of the mathematics, and allow 

them to work in heterogeneous groups (Boaler & Staples, 2008). 

 
Concluding Thoughts 

 

The participants were intentionally placed in hard-to-staff schools where the 

population was high-minority and low-income. In addition, students in these 

schools are tracked, which is pervasive in education. Teachers who work in schools 

that serve a population that is high-minority and low-SES may have, or have to 

challenge, low expectations of the children in their classrooms (Delpit, 2006; Ha-

berman, 1991). Tracking may further the low expectations of this population of 

children. Research indicates that tracking has an influence on teachers’ perceptions 

of students, especially their assumptions about their lower-tracked students (Oakes, 

2005). Both the NCTM and the CCSSM, however, have goals for every student to 

experience an education that provides opportunity for mathematical success for 

each student at a high level. What this study elucidates is that tracking may nega-

tively affect teachers’ perceptions on their ability to engage both low-tracked and 

high-tracked students in persevering in, making sense of, and productively discuss-

ing mathematics.  



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       181 

References 
 

Achinstein, B., & Athanases, S. Z. (2005). Focusing new teachers on diversity and equity: Toward a 

knowledge base for mentors. Teaching and Teacher Education, 21(7), 843–862. 

Apple, M. W. (1992). Do the standards go far enough? Power, policy, and practices in mathematics 

education. Journal for Research in Mathematics Education, 23(5), 412–431. 

Boaler, J. (2002a). Experiencing school mathematics: Traditional and reform approaches to 

teaching and their impact on student learning (revised and expanded ed.). Mahwah, NJ: 

Erlbaum. 

Boaler, J. (2002b). Learning from teaching: Exploring the relationship between reform curriculum 

and equity. Journal for Research in Mathematics Education, 33(4), 239–258. 

Boaler, J. (2006). How a detracked mathematics approach promoted respect, responsibility and high 

achievement. Theory Into Practice, 45(1), 40–46. 

Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching 

approach: The case of railside school. Teachers College Record, 110(3), 608–645. 

Bol, L., & Berry, R. Q., III. (2005). Mathematics teachers’ perceptions of the achievement gap. The 

High School Journal, 88(4), 32–45. 

Bol, L., & Nunnery, J. A. (2004). The impact of high stakes testing on restructuring efforts in schools 

service at risk students. In G. Taylor (Ed.), The impact of high-stakes testing on the academic 

futures of non-mainstreamed students (pp. 101–117). Lewiston, New York: Edwin Mellon 

Press. 

Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth, NH: Heinemann. 
Bourdieu, P., Passeron, J. C., & de Saint Martin, M. (1994). Academic discourse: Linguistic misun-

derstanding and professorial power (R. Teese, Trans.). Stanford, CA: Stanford University 

Press. 

Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. 

Cognition and Instruction, 17(1), 25–64. 

Brantlinger, A., Cooley, L., & Brantlinger, E. (2010). Families, values, and class relations: The 

politics of alternative certification. In M. W. Apple, S. J. Ball, & L. A. Gandin (Eds.), The 

Routledge international handbook of the sociology of education (pp. 179–189). New York, 

NY: Routledge. 

Brantlinger, A., & Smith, B. (2013). Alternative teacher certification and the new professionalism: 

The pre-service preparation of mathematics teachers in the New York City Teaching Fellows 

program. Teachers College Record, 115(7), 1–44. 

Braun, C., Neilsen, A. R., & Dykstra, R. (1975). Teacher’s expectations: Prime mover or inhibitor? 

The Elementary School Journal, 76(3), 181–188. 

Brophy, J. E., & Good, T. L. (1970). Teachers’ communication of differential expectations for 

children's classroom. Journal of Educational Psychology, 61(5), 365–374. 

Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. 

Educational Researcher, 18(1), 32–42. 

Burris, C. C., Heubert, J. P., & Levin, H. M. (2006). Accelerating mathematics achievement. 

American Educational Research Journal, 43(1), 105–136. 

Chapin, S. H., O’Connor, C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to 

help students learn, grades K–6. Sausalito, CA: Math Solutions. 

Ciardello, A. V. (1998). Did you ask a good question today? Alternative cognitive and metacognitive 

strategies. Journal of Adolescent & Adult Literacy, 42(3), 210–219. 

Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the 

context of developmental research. Educational Psychologist, 31(3/4), 175–190. 

Common Core State Standards for Mathematical Practice (CCSSMP). (2016). Standards for Mathe-

matical Practice. Retrieved from http://www.corestandards.org/Math/Practice/ 

http://www.corestandards.org/Math/Practice/


 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       182 

Cooper, H. M. (1979). Pygmalion grows up: A model for teacher expectation communication and 

performance influnce. Review of Educational Research, 49(3), 389–410. 

Darling-Hammond, L. (2000). Solving the dilemma of teacher supply, demand, and standards: How 

we can ensure a competent, caring, and qualified teacher for every child. Kutztown, PA: 

National Commission on Teaching & American’s Future. 

Delpit, L. (2003). Educators as “seed people” growing a new future. Educational Researcher, 32(7), 

14–21. 

Delpit, L. (2006). Other people’s children: Cultural conflict in the classroom. New York, NY: The 

New Press. 

Delpit, L. (2012). “Multiplication is for white people”: Raising expectations for other people’s 

children. New York, NY: The New Press. 

Dusek, J. B. (1975). Do teachers bias children’s learning. Review of Educational Research, 45(4), 

661–684. 

Eder, D. (1981). Ability grouping as a self-fulfulling prophecy: A micro-analysis of teacher–student 

interaction. Sociology of Education, 54(3), 151–162. 

Faulkner, V. N., Stiff, L. V., Marshall, P. L., Nietfeld, J., & Crossland, C. L. (2014). Race and 

teacher evaluations as predictors of algebra placement. Journal for Research in Mathematics 

Education, 45(3), 288–311. 

Feiman-Nemser, S., & Parker, M. (1993). Mentoring in context: A comparison of two US programs 

for beginning teachers. International Journal of Educational Reserach, 19(8), 699–718. 

Flores, A. (2007). Examining disparities in mathematics education: Achievement gap or opportunity 

gap? The High School Journal, 91(1), 29–42. 

Foreman, E. A., & Ansell, E. (2001). The multiple voices of a mathematics classroom community. 

Educational Studies in Mathematics, 46(1), 115–142. 

Futrell, M. H., & Gomez, J. (2008). How tracking creates a poverty of learning. Reshaping High 

Schools, 65(8), 74–78. 

Gutiérrez, R. (2000). Advancing African American urban youth in mathematics: Unpacking the 

success of one mathematics department. American Journal of Education, 109(1), 63–111. 

Gutiérrez, R. (2007). Context matters: Equity, success, and the future of mathematics education. In 

T. Lamberg & L. R. Wiest (Eds.), Proceedings of the 29th annual meeting of the North 

American Chapter of the International Group for the Psychology of Mathematics Education. 

Stateline (Lake Tahoe), NV: University of Nevada, Reno. 

Habermann, M. (1991). The pedagogy of poverty versus good teaching. The Phi Delta Kappan, 

73(4), 290–294. 

Hargreaves, A., & Fullan, M. (2000). Mentoring in the new millennium. Theory Into Practice, 39(1), 

50–56. 

Herrenkohl, L. R., & Guerra, M. R. (1998). Participant structures, scientific discourse, and student 

engagement in the fourth grade. Cognition and Instruction, 16(4), 431–473. 

Hiebert, J., Morris, A. K., & Glass, B. (2003). Learning to learn to teach: An “experiment” model for 

teaching and teacher preparation in mathematics. Journal of Mathematics Teacher Education, 

6(3), 201–222. 

Hollingsworth, J. R., & Ybarra, S. E. (2009). Explicit direct instruction: The power of the well-

crafted, well-taught lesson. Fowler, CA: DataWORKS Educational Research. 

Irvine, J., & York, D. E. (1993). Teacher perspectives: Why do African-American, Hispanic, and 

Vietnamese students fail. In S. W. Rothstein (Ed.), Handbook of schooling in urban America, 

(pp. 161–173). Westport, CT: Greenwood. 

Johnson, D. W., & Johnson, R. T. (1999). Making cooperative learning work. Theory Into Practice, 

38(2), 67–73. 

Kitchen, R., DePree, J., Celedón-Pattichis, S., & Brinkerhoff, J. (2007). Mathematics education at 

highly effective schools that serve the poor: Strategies for change. Mahwah, NJ: Erlbaum. 



 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       183 

Krajcik, J. S., & Blumenfeld, P. C. (2006). Project-based learning. In R. K. Sawyer (Ed.), The 

Cambridge handbook of the learning sciences. Cambridge, MA: Cambridge University Press. 

Lampert, M. (1990). When the problem is not the question and the solution is not the answer: 

Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–63. 

Lampert, M. (2003). Teaching problems and the problems of teaching. New Haven, CT: Yale Uni-

versity Press. 

Lortie, D. C. (1975). Schoolteacher: A sociological study. Chicago, IL: University of Chicago Press. 

Love, N. (2002). Using data/getting results: A practicul guide for school improvement in 

mathematics and science. Norwood, MA: Christopher-Gordon. 

Loveless, T. (1998). Making sense of the tracking and ability grouping debate. Fordham Report 

(Vol. 8, No. 2). Washington, DC: Fordham Foundation.  

Lubienski, S. T. (2000). A clash of social class cultures? Students’ experiences in a discussion-

intensive seventh-grade mathematics classroom. The Elementary School Journal, 100(4), 

377–403. 

Martin, D. B. (2012). Learning mathematics while Black. The Journal of Educational Foundations, 

26(1/2), 47–66. 

Martin, D. B. (2015). The collective Black and Principles to Actions. Journal of Urban Mathematics 

Education, 8(1), 17–23. Retrieved from  

http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/270/169 

McClain, K., & Cobb, P. (2001). An analysis of development of sociomathematical norms in one 

first grade classroom. Journal for Research in Mathematics Education, 93(1), 236–266. 

Moses, R. P., & Cobb, C. E. (2001). Radical equations: From Mississippi to the Algebra Project. 

Boston, MA: Beacon Press. 

National Council of Teachers of Mathematics. (2000). Principles and standards for school 

mathematics. Reston, VA: National Council of Teachers of Mathematics. 

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical 

success for all. Reston, VA: National Council of Teachers of Mathematics. 

National Reserach Council. (2001). Adding it up: Helping children learn mathematics. In J. 

Kilpatrick, J. Swafford, & B. Findell (Eds.). Mathematics Learning Study Committee, Center 

for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: 

National Academy Press. 

Neill, M. (2012). A child is not a test score. In W. Au & M. B. Tempel (Eds.), Pencils down:  

Rethinking high-stakes testing and accountability in public schools (pp. 21–30). Milwau-

kee, WI: Rethinking Schools.  

Oakes, J. (1990). Multiplying inequalities: The effects of race, social class, and tracking on oppor-

tunities to learn mathematics and science. Santa Monica, CA: The RAND Corporation. 

Oakes, J. (1992). Can tracking research inform practice? Technical, normative, and political 

considerations. Educational Researcher, 21(4), 12–21. 

Oakes, J. (2005). Keeping track: How schools structure inequality (2nd edition). New Haven, CT: 

Yale University Press. 

Oakes, J., & Lipton, M. (1996). Developing alternatives to tracking and grading. In L. I. Rendon & 

R. O. Hope (Eds.), Educating a new majority: Transforming American’s educational system 

for diversity (pp. 168–200). San Francisco, CA: Jossey-Bass. 

Page, R. N. (1990). Games of change: The lower-track curriculum in a college-preparatory high 

school. Curriculum Inquiry, 20(3), 249–281. 

Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press. 

Raudenbush, S. W., Rowan, B., & Cheong, Y. F. (1993). Higher order instructional goals in 

secondary schools: Class, teacher, and school influences. American Educational Research 

Journal, 30(3), 523–553. 

http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/270/169


 

 

 

Yanisko                                                                   Perceptions of Tracked Students 

Journal of Urban Mathematics Education Vol. 9, No. 2                       184 

Rosenbaum, J. E. (1980). Social implications of educational grouping. Review of Research in 

Education, 8, 361–401. 

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition and 

sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics 

teaching and learning (pp. 334–370). New York, NJ: Macmillian. 

Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. H. Schoenfeld 

(Ed.), Mathematical thinking and problem solving (pp. 53–70). Hillsdale, NJ: Erlbaurn. 

Smith, J. B. (1996). Does an extra year make any difference? The impact of early access to algebra 

on long-term gains in mathematics attainment. Educational Evaluation and Policy Analysis, 

18(2), 141–153. 

Smith, M. S., & Stein, M. K. (2011). 5 Practices for orchestrating productive mathematics discus-

sions. Reston, VA: National Council of Teachers of Mathematics. 

Spielhagen, F. R. (2010). Algebra for everyone? Student perceptions of tracking in mathematics. 

Middle Grades Research Journal, 5(4), 213–224. 

Stanulis, R. N., & Floden, R. E. (2009). Intensive mentoring as a way to help beginning teachers 

develop balanced instruction. Journal of Teacher Education, 60(2), 112–122. 

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical 

thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. 

American Educational Research Journal, 33(2), 455–488. 

Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From re-

search to practice. Mathematics Teaching in the Middle School, 3(4), 268–275. 

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-

based mathematics instruction: A casebook for professional development. New York, NY: 

Teachers College Press. 

Stigler, J., & Hiebert, J. (1997). Understanding and improving classroom mathematics instruction: 

An overview of the TIMSS video study. In Australian Council for Educational Research 

(Ed.), Raising Australian Standards in Mathematics and Science: Insights from TIMSS (pp. 

52–65). Melbourne, Australia: Australian Council for Educational Research. 

Stinson, D. W. (2006). African American male adolescents, schooling (and mathematics): 

Deficiency, rejection, and achievement. Review of Educational Research, 76(4), 477–506. 

Wang, J., & Odell, S. J. (2002). Mentored learning to teach according to standards-based reform: A 

critical review. Review of Educational Research, 72(3), 481–546. 

Watanabe, M. (2008). Tracking in the era of high-stakes state accountability reform: Case studies of 

classroom instruction in North Carolina. Teachers College Record, 110(3), 489–534. 

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning 

difficulties: An investigation of an explicit instruction model. Learning Disabilities Research 

& Practice, 18(2), 121–131. 

Wood, T. (1999). Creating a context for argument in mathematics. Journal for Research in Mathe-

matics Education, 30(2), 171–191. 

Yackel, E. (2001). Explanation, justification, and argumentation in mathematics classrooms. In M. 

van den Heuval-Panhuizen (Ed.), Proceedings of the 25th Conference of the International 

Group for the Psychology of Mathematics Education (pp. 9–23). Utrecht, The Netherlands: 

Pychology of Mathematics Education. 

Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in 

mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. 

Yin, R. K. (2009). Case study research: Design and methods (4th ed.). Thousand Oakes, CA: Sage. 

Zack, V., & Graves, B. (2001). Making mathematical meaning through dialogue: “Once you think of 

it, the z minus three seems pretty weird.” Educational Studies in Mathematics, 46(1/3), 229– 

271.