Highly Effective Teachers’ Conceptions of Mathematics Education


Journal of Urban Mathematics Education 

 July 2009, Vol. 2, No. 1, pp. 52–80 

©JUME. http://education.gsu.edu/JUME 

 

 

RICHARD S. KITCHEN is an associate professor of mathematics education at the University of New 
Mexico; Departments of Educational Specialties and Mathematics & Statistics, MSC 05 3040, Albuquerque, 
NM 87131; email:   kitchen@unm.edu.  His research interests include mathematics education at highly effec-
tive schools that serve poor communities and assessment of English Language Learners. 

  
FRANCINE CABRAL ROY is a mathematics education consultant and assistant adjunct professor at the 

University of Rhode Island, Kingston, RI 02881; e-mail: franroymathed@gmail.com. Her research interests 
include urban education, equity in mathematics education, and teacher learning. 

  
OKHEE LEE is a professor in the School of Education at the University of Miami, 1551 Brescia Ave-

nue, Coral Gables, FL 33146; e-mail: olee@miami.edu.  Her research areas include science education, lan-
guage and culture, and teacher education. 

  
WALTER G. SECADA is senior associate dean of the School of Education and chair of the Department 

of Teaching and Learning at the University of Miami, 5202 University Drive, Coral Gables, FL 33146; email: 
wsecada@miami.edu. His research interests include mathematics education, teachers' professional develop-
ment, school restructuring, equity in education, bilingual education, student engagement, and curricular 
reform.  

Comparing Teachers’ Conceptions of Mathematics 

Education and Student Diversity at Highly  

Effective and Typical Elementary Schools 
 

Richard S. Kitchen 

University of New Mexico 

 

Francine Cabral Roy 

University of Rhode Island 

 

Okhee Lee 

University of Miami 
Walter G. Secada 

University of Miami 

 
In this study, the authors examined what distinguished highly effective from typi-

cal elementary schools in mathematics by examining the conceptions of fourth-

grade teachers with regards to mathematics education (curriculum, instruction, 

and assessment) and student diversity (ability, culture, language, and socioeco-

nomic status). The study was conducted in two large urban areas with high pro-

portions of racially/ethnically and linguistically diverse student groups. Inter-

views were conducted with 32 fourth-grade teachers from 16 elementary schools, 

including 10 highly effective and 6 typical schools in the two areas. Compelling 

evidence was found that teachers at highly effective schools had better developed 

and better articulated conceptions of mathematics education and student diversi-

ty. While similar findings were found across the two areas (e.g., teachers’ beliefs 

about the influence of high-stakes assessment and the academic ability of low-

achieving students), differences were also found across the two areas (e.g., teach-

ers’ beliefs about the importance of implementing multiple instructional strategies 

and expectations for college attendance). 

 
KEYWORDS: conceptions of mathematics, highly effective schools, student diver-

sity, urban schools 
 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       53 

The purpose of this study was to examine what distinguishes highly effec-
tive from typical elementary schools in mathematics achievement by examining 

the conceptions (knowledge and beliefs) of fourth-grade teachers. Specifically, we 

compared teachers at highly effective schools with teachers at typical schools 

with regards to their conceptions in mathematics education (curriculum, instruc-

tion, and assessment) and student diversity (ability, culture, language, and socioe-

conomic status).  

It is clear from research that teachers‘ conceptions of mathematics content, 

instruction, and assessment are strongly linked to classroom practice and student 

learning in interesting and complex ways. What remains under researched in the 

literature on teachers‘ conceptions and practices is the role of race/ethnicity, cul-

ture, and language in mathematics education (Rodriguez & Kitchen, 2005; Rous-

seau & Tate, 2003). In this study, we pursued an integrated research project that 

examined the influence of teachers‘ conceptions with regards to both mathematics 

education and student diversity on student learning and achievement. Two specif-

ic research questions guided the study:  

 
1. How do teachers at highly effective schools articulate their conceptions of mathe-

matics education compared to their colleagues at typical schools?  

 
2. How do teachers at highly effective schools articulate their conceptions regarding 

student diversity compared to their colleagues at typical schools?  

 

Another distinguishing feature of this study is that we searched for characte-

ristics that differentiated schools rather than classrooms. This perspective necessi-

tates that we consult the effective schools literature. Much of this research identi-

fies factors other than instruction as distinguishing more from less effective 

schools (Edmonds, 1979; Good & Brophy, 1986; Martin, Mullis, Gregory, Hoyle, 

& Shen, 2000; Purkey & Smith, 1983). These factors include school-level charac-

teristics such as principal leadership, orderliness and safety, or opportunities at 

home such as access to reading materials. Interestingly, the Third International 

Mathematics and Science Study (TIMSS) results indicate that although the nature 

of mathematics and science instruction was important, it was not a defining cha-

racteristic distinguishing high from low performing schools (Martin, et al., 2000).  

In light of increasing pressure being placed on schools to enhance student 

learning and achievement in mathematics, there is a danger that policymakers 

view the effective schools literature as suggesting that instructional quality fails to 

distinguish more from less effective schools. We sought to contest this potential 

pitfall by providing empirical evidence to address the gap in the research literature 

on the link between school-level effectiveness and student learning and achieve-

ment. To accomplish this goal, we investigated teachers‘ conceptions of and ex-

planations for their practices in mathematics with diverse student groups.  



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       54 

Conceptual Framework 
 

We examined the conceptions of teachers from multiple school districts 

across two of the USA‘s major metropolitan areas. Teachers‘ responses were ex-

amined through a complex lens that includes constructs related to school organi-

zation, mathematics education, and multicultural education. The corresponding 

areas of literature that provided the conceptual framework for this study are: (a) 

school restructuring, (b) teachers‘ conceptions in mathematics education, and (c) 

teachers‘ conceptions in addressing student diversity. 

 
School Restructuring 
 

In research on school restructuring and its relationship to student perfor-

mance on high-level tasks, Newmann and Associates (1996; Newmann, Secada, 

& Wehlage, 1995; Newmann & Wehlage, 1995) found that students enrolled in 

schools where the curriculum content and instruction focused on depth over cov-

erage, analytic reasoning over memorization, and the construction of value over 

doing tasks as ends in themselves out performed students whose classrooms 

lacked these instructional features. Lee and Smith‘s (2001) study of secondary 

schools supports Newmann et al.‘s findings. At secondary schools where mathe-

matics and science course offerings were predominantly academic and where 

teachers as a whole tended to report instruction that focused on depth and higher 

order thinking, students evidenced higher achievement in mathematics and 

science and began to close the achievement gap based on social class, compared 

to schools where these instructional characteristics were lacking. 

In this study, we sought to examine whether teachers in highly effective 

schools were more likely than their counterparts at typical schools to articulate 

their goals to implement mathematics curricula and instruction as reported in 

Newmann and Associates (1996) and Lee and Smith (2001). 

 
Teachers’ Conceptions in Mathematics Education 
 

In the past several decades, notions about teachers‘ conceptions (knowledge 

and beliefs) of mathematics content (Ma, 1999), instruction (Ball & Cohen, 

1999), and assessment (Kulm, 1994) have dramatically changed. Current reforms 

in mathematics education stress the need for problem-solving approaches to pro-

mote students‘ reasoning and communication skills (National Council of Teachers 

of Mathematics [NCTM], 1989, 2000; National Science Foundation [NSF], 

1996). Furthermore, the mathematical processes of making connections, commu-

nicating, representing, and problem solving are no longer viewed as ends in them-

selves but as means to learning mathematics with understanding (Hiebert & Car-



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       55 

penter, 1992; Hiebert, et al.1996). The incongruity between this vision and the 

conceptions of many practicing mathematics teachers (e.g., mathematics as a se-

quential, rule-based discipline in which memorizing facts and following proce-

dures are highly valued) has been well documented (e.g., Romberg, 1992). Three 

perspectives on teachers‘ conceptions of mathematics education are described 

next. 

Ernest (1988, 1991) identified three possible views of the nature of an aca-

demic discipline. The first, called instrumentalist, views a discipline as an isolated 

body of discrete skills. The second, called Platonist, regards a discipline as a body 

of connected and unified knowledge. The third, called problem-solving, deems a 

discipline as a process of inquiry that is continuously expanded by human crea-

tion. These three distinctions have also been cast as a duality between absolute 

(e.g., instrumentalist) and fallible (e.g., problem-solving) views that take shape in 

beliefs about mathematical knowledge and legitimate mathematical activities 

(Romberg, 1992; Thompson, 1992). 

Teachers who embrace an instrumentalist view often look at mathematics as 

a sequence of fixed skills or concepts. Mastery of pre-requisite skills is deemed 

necessary for subsequent learning. This view assumes that ―rules are the basic 

building blocks of all mathematical knowledge and all mathematical behavior is 

rule governed‖ (Thompson, 1992, p. 136). Thompson (1992) argued that there is a 

consistent relationship between teachers‘ beliefs and instructional practices in ma-

thematics. Because mathematics is characterized as static and predetermined in 

the instrumentalist philosophy, those who adhere to this view emphasize mathe-

matical facts and pursue the drill-and-practice approach to teaching (Schifter & 

Fosnot, 1993). 

In contrast, teachers who adhere to a Platonist view of mathematics emphas-

ize the logic that connects concepts. These relationships are assumed to be fixed 

and often require explanations by the teacher. This view models a top-down ap-

proach in which instruction begins with the knowledge of the expert, rather than 

that of the learner (Hiebert & Carpenter, 1992). An example of teachers who em-

brace the Platonist view of mathematics is provided in Liping Ma‘s book, Know-

ing and Teaching Elementary Mathematics (1999). A theme that runs throughout 

Ma‘s book is that teachers must possess a deep and broad knowledge of mathe-

matics to make conceptual connections between mathematical ideas. 

On the other hand, teachers who adopt a problem-solving or inquiry view of 

mathematics see their role as posing questions and challenging students to think 

and reason. Teachers whose primary objective is to advance mathematical prob-

lem solving advocate that students are the ones ―doing mathematics‖ (Davis & 

Hersh, 1980; Ernest, 1991; Lakatos, 1976; Tymoczko, 1986). From this view, 

―learning is primarily a process of concept construction and active interpreta-

tion—as opposed to the absorption and accumulation of received items of infor-



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       56 

mation‖ (Schifter & Fosnot, p. 8, 1993). Pedagogy inspired by this view engages 

students in posing and solving problems, making and proving conjectures, explor-

ing puzzles, sharing and debating ideas, and contemplating the beauty of ideas in 

an academic discipline. 

Alternative or multiple assessment formats that do not simply assess facts 

and skills in isolation, but that also require students to apply their knowledge in 

real-life contexts align more with the Platonist or problem-solving view. Accord-

ing to Kulm (1994): 

 
Alternative assessment approaches that include open-ended questions, presentation 

of solutions in both written and oral form, and other performances send very differ-

ent messages to students about what is important in mathematics learning….The 

shift from an emphasis on producing correct answers to the expectation that students 

think and communicate is a major one for many students and teachers. (p. 6) 

 

This shift of emphasis corresponds to the philosophical change necessary for 

an instrumentalist to become a Platonist or to promote problem-solving. Such a 

shift is supported by researchers who advocate revising assessment practices to 

bring about changes in instruction based on how children learn (O‘Day & Smith, 

1993). In addition, alternative assessment approaches and the use of multiple as-

sessment formats require students to communicate their thinking and elicit a range 

of student responses (Wiggins & McTighe, 1999). 

It is important to note that teachers‘ conceptions may not be easily classified 

into solely instrumentalist, Platonist, or mathematical problem-solving perspec-

tives and that their classroom practices may not align well with their stated be-

liefs. Kitchen, DePree, DePree, Celedón-Pattichis, and Brinkerhoff (2007) found 

that teachers at highly effective schools in low-income communities often ex-

pressed multiple perspectives on the teaching and learning of mathematics and 

their classroom practices reflected these multiple perspectives. According to 

Thompson (1992), teachers may hold multiple perspectives that helps explain 

why they express one philosophical view while practicing another.  

 
Teachers’ Conceptions of Student Diversity 
 

The existing knowledge base for promoting academic achievement with a 

culturally and linguistically diverse student population is limited and fragmented, 

in part because disciplinary knowledge and student diversity have traditionally 

constituted separate research agendas (Lee, 1999). In recent years, studies consi-

dering the interaction between academic disciplines and students‘ linguistic and 

cultural experiences have begun to emerge in mathematics education (e.g., Civil, 

& Andrade, 2002; Gutiérrez, 2002). In this review of the research literature, two 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       57 

areas related to student diversity in mathematics education are considered: (a) 

teacher expectations and (b) culturally relevant instruction. 

 

Teacher expectations. Teachers hold varied conceptions pertaining to stu-

dents, including students‘ potential for success, cultural backgrounds, strengths 

and weaknesses, and future placement in society. High expectations are lauded as 

a key to successful teaching of students often underserved by schooling (Lipman, 

1998; Zeichner, 1996). Zeichner (1996) argued that high expectations are a neces-

sary attribute of classrooms that have the potential to narrow the achievement gap 

in urban schools. The research literature contains a growing consensus about what 

teachers need to be like, to know, and to be able to do in order to teach all stu-

dents to high academic standards. This vision rests on teachers believing that all 

students can learn and taking the responsibility for this task regardless of students‘ 

economic circumstance or skin color. The belief that ―all children can learn‖ and 

advocacy for high expectations have become part of mainstream educational rhe-

toric. 

Although high expectations are necessary for effective teaching, they are far 

from the norm—especially with racial/ethnic minorities and low socio-economic 

status (SES) students. Teachers hold lower expectations for students of color and 

the poor than they do for White middle-class students (Ferguson, 1998; Grant, 

1989; Knapp & Woolverton, 1995; Zeichner, 1996). Low expectations are be-

lieved to be at the root of ineffective pedagogy with students of color and the 

poor. Low expectations appear to result in transmission, authoritative, and con-

trolled forms of pedagogy emphasizing basic skills (Haberman, 1991; Knapp & 

Woolverton, 1995). Ladson-Billings (1994) described how some teachers expect 

African American students to be more difficult to control, and therefore work 

harder at restraining these students in the classroom. In their review of social class 

and schooling, Knapp and Woolverton (1995) claimed that controlled forms of 

instruction teach low-SES students that little is expected from them except com-

pliance to a rigid classroom environment. This ―pedagogy of poverty‖ (Haber-

man, 1991) is prevalent in urban and/or low-SES schools. 

 

Culturally relevant instruction. Teachers who believe that all children can 

learn may have high expectations but may not necessarily provide culturally rele-

vant instruction if they teach only dominant content but ignore issues of culture. 

Notions of culturally relevant instruction embrace the significance of high expec-

tations but move beyond it to highlight the significance of culture and power as 

both a means to an end and goals in and of themselves. Culturally relevant in-

struction views student culture and cultural experiences as a strength and use this 

knowledge as a bridge to learning (Ladson-Billings, 1994, 1995). 



 

 

 

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Journal of Urban Mathematics Education Vol. 2, No. 1                                       58 

Because mathematics tends to be presented as a set of objective and univer-

sal facts and rules, it is often viewed as ―culture free‖ and not considered as a so-

cially and culturally constructed discipline (Banks, 1993; Peterson & Barnes, 

1996; Secada, 1989). Instructional practices have traditionally relied on examples, 

analogies, and artifacts that are often unfamiliar to non-mainstream students 

(Ninnes, 2000). Recent efforts to provide culturally relevant instruction indicate 

that when their cultural and linguistic experiences are used as intellectual re-

sources, non-mainstream students are able to engage in academic learning and 

show significant achievement gains. For example, the Algebra Project (Silva & 

Moses, 1990) uses student knowledge of the subway system as a basis for under-

standing operations with integers. The focus on student strengths contrasts the 

remediation model of teaching urban students, in which curriculum and instruc-

tion are predicated on what students do not know and often emphasize rote skills 

(Haberman, 1991; Oakes, 1990). 

 
Methods 

 

Research Setting and Teacher Participants 
 

District selection. The study was conducted in two areas that had previously 

received Urban Systemic Initiative
1
 funding from the National Science Founda-

tion, ―President City Area
2
― and ―Arbor City.‖ Because President City consisted 

of three adjacent school districts with similar characteristics, we refer to it as the 

―President City Area.‖  

The Arbor City school district and three contiguous President City Area 

school districts were chosen to participate in this study because they served large 

low-income and minority student populations; had on-going mathematics educa-

tion reform efforts underway; adhered to standards in mathematics, science, and 

language arts; and gave standardized tests to 4
th
 graders in mathematics and lan-

guage arts. All four districts used standardized criterion-referenced tests
3
 (hence-

forth referred as the ―State Assessment‖) to evaluate 4
th

 graders in mathematics 

and language arts.  

                                                
1
 The National Science Foundation launched its Urban Systemic Initiative (USI) program in 1994, 

applying lessons learned from its earlier Statewide Systemic Initiative program to the problems of 

inner-city school systems. The USI program was offered to 25 major cities with the largest num-

ber of K–12 students living in poverty. 

 
2
 Names of participating cities are pseudonyms. 

 
3
 In Arbor City, 4

th
 graders were tested with the Stanford Achievement Test-9

th
 (SAT9) Edition: 

Intermediate I. Fourth graders in the President City Area were tested with the State Assessment of 

Academic Skills.  



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       59 

Arbor City, one of the nation‘s oldest cities, contained over 170 elementary 

schools. At the time the study was undertaken in 2000, the majority of the public 

school district‘s more than 200,000 students were African American (65%), with 

approximately equal numbers of White (16%) and Hispanic/Latino (13%) stu-

dents. Students in the public schools were predominantly low-income; 80% were 

eligible for free or reduced-price lunch. In Arbor City, 62% of the teachers were 

White, 36% African American, and 1% Hispanic/Latino. 

The President City Area schools, comprised of three contiguous districts, 

contained 116 elementary schools. Of over 103,000 public school students, 85% 

were Hispanic; 11% White; and 3% African American. Over 26% of these dis-

tricts‘ students were limited English proficient/English for speakers of other lan-

guages (LEP/ESOL); nearly 70% qualified for free or reduced-price lunch. At all 

but one of the President City Area schools, the majority of teachers were Hispan-

ic/Latino/a (60% or more). At all but this one school, 49 to 74% of the teachers 

reported the ability to speak fluent Spanish. 

 

School selection. To be selected for inclusion in this study, elementary 

schools had to meet two sets of criteria. First, in terms of demographic criteria, an 

elementary school had to serve predominately minority, low-income students. At 

least 50% of the students of a school‘s population had to be low-income as de-

fined by the district. Second, in terms of academic criteria, at schools that were 

designated as highly effective across the four participating districts, 90% or more 

of students were tested with the State Assessment. Additionally, student test 

scores in mathematics and science at the fourth-grade level from 1996 through 

1999 were either increasing or were consistently high. This requirement reduced 

the likelihood that a school considered highly effective one year would fall below 

the threshold the next year—a common finding in the effective schools literature 

(Good & Brophy, 1986). Finally, highly effective schools had to be among those 

with the highest percentage of 4
th

 grade students scoring at Basic or above in 

Reading in their district in 1999.  

 

In Arbor City, criteria to select highly effective schools also included:  

 
 66% or more of the 4

th
 graders tested scored at Basic or above in mathematics, and 

 30% or more of the 4
th

 graders tested scored at Proficient or above in mathematics. 

  

In the President City Area, criteria to select highly effective schools also in-

cluded: 

 
 90% or more of the 4

th
 graders tested scored at Passing or above in mathematics, 

 50% or more of the 4
th

 graders tested scored at Proficient or above in mathematics, 

and 



 

 

 

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Journal of Urban Mathematics Education Vol. 2, No. 1                                       60 

 30% or more of a school‘s students were Limited English proficient (LEP). 

 

Based on the two sets of criteria described above, three schools were se-

lected as typical schools matched to the five highly effective schools in both Ar-

bor City and in the President City Area, respectively. Typical schools were in the 

same district-administrative unit, called a cluster, as a highly effective school to 

ensure that the schools were in the same geographic region of the city and had 

similar access to professional development activities. They enrolled approximate-

ly the same percentage of low-income students and had approximately the same 

racial and ethnic breakdown. To the degree possible, typical schools and highly 

effective schools enrolled a similar number of students. One school that was in-

itially selected as a highly effective school in Arbor City was re-designated as a 

typical school because in 2000 their SAT9 scores plummeted and remained very 

low in 2001. All selected schools were given a $600 incentive for participation. 

 

Teacher selection. We selected fourth grade as the target because restricted 

resources limited us to studying a single grade. Schools had been selected as high-

ly effective or typical based on the achievement of their fourth graders, and fourth 

is also the grade when National Assessment of Education Progress (NAEP) test-

ing is done. Principals at participating schools were asked to make a list of the 

fourth grade teachers in their schools who taught both mathematics and science. 

Then, they ranked the teachers from ―most effective‖ to ―least effective‖ accord-

ing to some explicit criterion, such as a recent job evaluation. In both the Presi-

dent City Area and Arbor City, two teachers per school were selected to partici-

pate in this study based on their being at neither extreme of the principal‘s rank-

ing. Thus, a total of 32 teachers took part in the study (16 from each city), includ-

ing 20 from highly effective schools and 12 from typical schools across the two 

cities. 

 
Data Collection 
 

The data collection team consisted of three mathematics educators, includ-

ing one who was involved in the development of the classroom observation in-

strument and one who had extensive experience in using this instrument. No one 

associated with this study was aware of which schools were highly effective and 

which were typical until the results were aggregated at the level of school effec-

tiveness. Data collectors visited each school twice during a single year—once in 

the fall and once in the spring. During each visit, each participating teacher was 

observed teaching twice—two distinct lessons on consecutive days, for a total of 

four observations of each of the 32 participating teachers in the sample. 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       61 

We conducted semi-structured interviews with each teacher immediately af-

ter the two classroom observations. Each of the 32 teachers in the sample was to 

have been interviewed twice; however, due to teacher absenteeism, we finished 

56 of 64 possible interviews for the study. We obtained copies of academic tasks 

(e.g., handouts) during classroom observations, and then used these tasks and 

classroom activities as the context to examine teachers‘ conceptions during inter-

views. In addition, the concrete and immediate nature of the tasks and activities 

facilitated interviews with the teachers. 

To examine teachers‘ conceptions (knowledge and beliefs) of mathematics 

education with student diversity, our semi-structured interview protocols were 

adapted from existing instruments (Newmann, Secada, & Wehlage, 1995). During 

fall 2000, interviews were designed to examine teachers‘ (a) general perceptions 

of the lessons, (b) conceptions of the nature of mathematics, (c) conceptions of 

student diversity, and (d) conceptions of integrating mathematics with student di-

versity (see Appendix I for Fall 2000 teacher interview protocol). During spring 

2001, interviews were designed to examine teachers‘ (a) general perceptions of 

the lessons, (b) influences of high-stakes assessments and accountability on class-

room practices, (c) understanding of individual students‘ strengths and weak-

nesses, and (d) use of instructional strategies to meet students‘ learning needs in 

mathematics (see Appendix II for Spring 2001 teacher interview protocol). Each 

question was followed with probes. Each interview lasted about 45 minutes to 1 

hour. All the interviews were audio-taped and transcribed. 

 
Data Analysis 
 

The teacher interview transcripts were initially read as a complete dataset 

and coded for structural categories according to classroom data, interview ques-

tions, individual questions, students, and schools. With the use of NUD*IST, this 

structure provided the initial database, allowing for flexible grouping and coding 

of teachers‘ conceptions. Data were sorted by interview questions to examine (a) 

teachers‘ conceptions of curriculum, instruction, and assessment in mathematics 

and (b) teachers‘ conceptions of student diversity in mathematics. Then, for each 

of (a) and (b), the data were sorted by district (President City Area and Arbor 

City). 

The data subsets were analyzed using interpretive methods (Erickson, 1986). 

Each data subset was read as a whole, followed by a period of open coding to al-

low for the emergence of themes. An iterative process of coding, memo writing, 

focused coding, and integrative memo writing followed (Emerson, Fretz, & Shaw, 

1995). Creation of the codes went through multiple revisions, as the data were 

repeatedly read to check the consistency of themes. This process continued until 

either no new categories were developed or consistency was achieved. After we 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       62 

obtained the set of themes from the dataset, we searched for commonalities and 

differences in teachers‘ responses between highly effective and typical schools 

with respect to each of the themes. We sought both confirming and disconfirming 

evidence by searching for supportive and non-supportive evidence (Erickson, 

1986). We considered teachers‘ conceptions in each city to be replication of the 

other. 

 
Results 

 

The results about teachers‘ conceptions are grouped into (a) mathematics in 

terms of curriculum, instruction, and assessment and (b) student diversity in terms 

of teachers‘ expectations and understanding of culture. Frequencies (%) of teach-

ers‘ responses for each key theme are presented; on occasion, an illustrative ex-

ample is presented to clarify a theme.  

 
Teachers’ Conceptions of Mathematics Curriculum, Instruction, and Assessment 
 

We investigated elementary teachers‘ conceptions of mathematics curricu-

lum, instruction, and assessment between highly effective and typical schools. 

Findings are presented in Table 1. 

 

Teachers’ conceptions of mathematics curriculum. We identified two 

themes that differentiated teachers at highly effective and typical schools across 

the two cities: (a) student knowledge of addition and subtraction facts and (b) 

teaching more content than simple problem solving and basic facts. 

Teachers at highly effective schools were less concerned about their stu-

dents knowing the basic addition and subtraction facts (i.e., could recall sums and 

differences of two single-digit numbers). In contrast, the majority of teachers at 

typical schools included basic addition and subtraction as among the most impor-

tant mathematical ideas that they believed they should teach. In the President City 

Area, a higher percentage of teachers from typical schools (67%) prioritized 

teaching students basic addition and subtraction skills compared to only 30% of 

teachers from highly effective schools. In Arbor City, all but one of the teachers 

from typical schools (93%), but only 25% of teachers from highly effective 

schools identified the importance of students knowing their basic addition and 

subtraction facts. Because we did not study students‘ prior educational expe-

riences in mathematics, we do not know if this difference can be attributed to stu-

dents at highly effective schools having access to a more challenging mathematics 

curriculum. 

 

 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       63 

Table 1 

Teachers’ Conceptions of Mathematics Curriculum, Instruction, and Assessment 
 

 
Theme School Type 

President 

City Area 

Arbor 

City 

 

Curriculum 

 

Basic skills 

 

 

Highly Effective 

Typical 

 

30% 

67% 

 

 

25% 

93% 

 

 Teaching  more than problem solving 

and basic facts 

 

Highly Effective 

Typical 

 

90% 

67% 

50% 

67% 

 

 

Instruction 

 

Make meaning of mathematics 

 

 

Highly Effective 

Typical 

 

 

40% 

17% 

 

7% 

0% 

 
 Multiple instructional strategies Highly Effective 

Typical 

 

30% 

33% 

 

50% 

0% 

 

 Meeting students‘ affective needs Highly Effective 

Typical 

 

40% 

17% 

 

50% 

17% 

 

 

Assessment 

 

Influenced by test 

 

Highly Effective 

Typical 

 

 

100% 

100% 

 

63% 

100% 

 Teach beyond test Highly Effective 

Typical 

63% 

  0% 

11% 

  0% 
 

 

Many teachers cited problem solving and basic facts (i.e., being able to op-

erate on two single-digit numbers with the four basic operations) as among the 

three most important mathematical ideas to teach during fourth grade. Yet, there 

were noticeable differences between highly effective and typical schools in the 

President City Area, although such differences were not as evident in Arbor City. 

In the President City Area, teachers at typical schools were generally not particu-

larly articulate about what their students needed to learn in mathematics. In con-

trast, 90% of the teachers from highly effective schools talked at great length 

about going beyond teaching only problem solving and the basic facts and identi-

fied additional mathematical ideas that they wanted their fourth graders to learn. 

For example, Mr. Bruno
4
 from a highly effective school in the President City Area 

discussed the importance of his students analyzing the reasonableness of their so-

lutions and learning algebra, geometry, and measurement: 

 

                                                
4
 All names of participating teachers are pseudonyms. 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       64 

Reasonableness is the first thing. I want them to be able to relate to reality. The 

second one would be being able to work as a whole, different kinds of math prob-

lems. Multiplication, division, subtraction, even algebra, if possible, so that when 

they go up to higher grades or even on to college, they‘ll be able to perform mathe-

matical situations in an easier way for them. I‘ve seen many students struggle with 

that, so I want to make it easier for them. So, the second one is to work problems. 
The third one is for them to get to know measurement in every aspect—weights, 

length, depth, volume. Lastly, they need to get to know geometry. All those things 

together will make a mathematician. (Bruno, Highly Effective, President City Area) 

 

Teachers’ conceptions of mathematics instruction. Our analysis identified 

two themes that differentiated teachers at highly effective and typical schools 

across the two cities: (a) making meaning of mathematics and (b) using multiple 

instructional strategies. 

Teachers from highly effective schools were more likely to express the im-

portance of helping their students make meaning of mathematical ideas by engag-

ing them in mathematical investigations. As shown in Table 1, teachers from 

highly effective schools across both participating cities were more likely than 

their counterparts at typical schools to describe helping their students make mean-

ing of mathematics as one of their instructional strategies. Ms. James, a teacher 

from a highly effective school in Arbor City described a multiplication algorithm 

that she taught and believed helped her students develop an understanding of mul-

tiplication: 

 
But multiplication is where they get a new algorithm; they don‘t do it the traditional 

way. This one, they do it according to an array that they draw and according to how 
many sections there are in the array. That is the way the algorithm is broken down 

and they have to explain each section. So this algorithm is much longer and it‘s de-

tailed but it shows you exactly what steps you went through…. And it helps you un-

derstand the value of the number as opposed to ‗why am I doing this‘ or just going 

through a process and not understanding why you‘re doing it. (James, Highly Effec-

tive, Arbor City) 

 

In Arbor City, a significantly higher percentage of teachers from highly effective 

schools (50%) than teachers from typical schools (0%) discussed the importance 

of implementing multiple instructional strategies. This finding was not replicated 

in the President City area.  

Ms. Birge, another teacher from a highly effective school in Arbor City, ex-

plained how her instruction changed as the academic year progressed to emphas-

ize students justifying their ideas while problem solving in groups: 

 
I just keep plugging at it. Every lesson that I teach, tell me how you came up with 

that. Well, the answer is wrong. It doesn‘t matter; what made you say that? They also 

have math journals, which you didn‘t get a chance to see. In their math journals, 

they‘ll start to write more about how to solve problems and what they took from their 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       65 

problem and, more importantly, how it relates to everyday life…. They don‘t even 

do as much group work now as they will in the coming months. And that‘s when I 

really get to see how you think. So right now, all of them are doing relatively well 

but this is skill stuff. This is not really digging in—the meat and gravy of it, just the 

top layer. (Birge, Highly Effective, Arbor City) 

 

Teachers’ conceptions of mathematics assessment. The teachers in this 

study were forthright about the influence of standardized testing on their mathe-

matics curriculum and instruction. Our analysis identified two themes that diffe-

rentiated teachers at highly effective and typical schools across the two cities: (a) 

influenced by test, and (b) teaching beyond test. 

In the President City Area, all of the teachers described how their mathemat-

ics instruction was strongly influenced by the mathematical content of the State 

Assessment. They were candid that their assessment goal was to learn what ma-

thematical topics students needed to study to prepare for the State Assessment. 

They were expected to get specific training to prepare their students to be success-

ful on the State Assessment. This expectation appeared to be particularly true for 

teachers from typical schools, most of whom described attending the Evaluation 

Program workshops focused exclusively on preparing students for success on the 

State Assessment. Interestingly, 30% of teachers from highly effective schools, 

but no teacher from a typical school, challenged using only the Evaluation Pro-

gram curriculum. 

A significant finding was that none of the teachers from the typical schools 

in the President City Area discussed how their mathematics instruction went 

beyond simply preparing their students to be successful on the test. In contrast, 

63% of teachers from highly effective schools stated that they were either more 

concerned about teaching to the state curriculum framework than preparing their 

students to be successful on the State Assessment or that they tried to do more 

than simply teach to the test. Ms. Padron described how she taught to the state 

curriculum framework: 

 
I always use my bible, which is my state curriculum framework, and that‘s aligned to 

the State Assessment….It‘s our state curriculum and we have to go by that. And 

that‘s what the State Assessment also tests, but sometimes....I tell them that we‘re 

not learning just because of the State Assessment….We go beyond that. We do more 

than that. That‘s just the minimum that you should know. (Padron, Highly Effective, 

President City Area) 

 

The Arbor City teachers devoted considerable time to developing and implement-

ing lessons to prepare students to be successful on the district‘s test. Few teachers 

alluded to how they worked to teach beyond the learning objectives of the test.  

 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       66 

Summary. The main themes that emerged in our analysis of the participating 

teachers‘ conceptions of mathematics curriculum, instruction, and assessment in-

volved: (a) the mathematics content that should be taught, (b) the instructional 

strategies that should be used for lessons, and (c) the influence of standardized 

testing. Teachers from highly effective schools across the two cities expressed 

more elaborate conceptions (i.e., conceptions aligned with recommendations 

found in the mathematics education reform literature) with respect to these themes 

and others (see Table 1). Though teachers across both cities were under pressure 

to prepare students to be successful on a standardized test in mathematics, teach-

ers at highly effective school expressed more elaborate beliefs and knowledge 

about mathematics, placed less emphasis on preparing their students for the test 

(in the President City Area), and implemented more diverse instructional strate-

gies (in Arbor City) than their colleagues from typical schools. 

 
Teachers’ Conceptions of Student Diversity 
 

In this study, we also examined and compared teachers‘ conceptions with 

regards to student diversity across highly effective schools and typical schools. 

Findings are summarized in Table 2. 

 

Teacher expectations. Four themes differentiated teachers at highly effective 

and typical schools across the two cities with regards to teacher expectations: (a) 

high-level thinking, (b) academic excellence, (c) academic potential of a low 

achieving student, and (d) expectation of attending college. 

First, high-level thinking (HLT) in mathematics includes reasoning, problem 

solving, evaluating, predicting, synthesizing, making connections, building mod-

els or simulations, and communicating about the content (NCTM, 2000; NSF, 

1996). The emphasis is on students engaging in these high-level cognitive 

processes as they make sense of mathematics by connecting new to existing 

knowledge and learn content with understanding (see, e.g., Hiebert & Carpenter, 

1992; Lampert, 1990). Teachers‘ comments about high-level thinking fell into 

three sub-themes, including student generated solutions, mathematical inquiry, 

and logical or mathematical reasoning. The more common of these responses was 

student-generated solutions; some teachers described students deriving alternative 

solution strategies that had not been taught by the teacher: 

 
Sometimes it takes awhile and then they have a whole different way of doing it. I 

will show them the math way and they would show another way and I would be like 

wow! I never thought of it that way then again I don‘t know how a 9-year-old mind 

works so yeah, they surprised me. It is nice when it happens you know. (Haney, Typ-
ical, President City Area) 

 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       67 

Table 2 

Teachers’ Conceptions of Student Diversity 
 

 
Theme School Type 

President 

City Area 

Arbor 

City 

 

Teacher  

expectations 

 

High level thinking 

 

 

Highly Effective 

Typical 

 

89% 

20% 

 

 

30% 

33% 

 

 Academic excellence 

 

Highly Effective 

Typical 

 

78% 

60% 

60% 

83% 

 

 Academic ability of low-achieving 

student (low) 

Highly Effective 

Typical 

22% 

40% 

  0% 

33% 

  

College attendance 

 

Highly Effective 

Typical 

 

 

100% 

100% 

 

57% 

17% 

 

Culture 

 

Cultural connection 

 

 

Highly Effective 

Typical 

 

 

60% 

50% 

 

50% 

17% 
 

 Multicultural curricula Highly Effective 

Typical 

 

60% 

33% 

 

30% 

33% 

 

 Not culture Highly Effective 

Typical 

 

50% 

83% 

 

70% 

83% 

 

     

HLT was a strong distinguishing factor between teachers from highly effec-

tive and typical schools in the President City Area. Most of the teachers from 

highly effective schools (89%) saw HLT as strengths compared to 20% of teach-

ers from typical schools. HLT was less prominent with the Arbor City teachers, 

with about one-third of the teachers from both groups citing HLT. 

Second, teachers‘ efforts to promote learning or raise achievement beyond 

what was expected in the standard curriculum were coded under academic excel-

lence. This term is consistent with Ladson-Billings‘ (1994) notion of academic 

excellence as a core component of culturally relevant pedagogy. Strategies that 

teachers stated they used to promote academic excellence included (a) differen-

tiating instruction, (b) pushing students to strive academically, and (c) referring 

students to gifted and talented programs. Similar to the HLT analysis, there were 

differences with regards to academic excellence across districts. In the President 

City Area, teachers from highly effective schools (78%) were more likely to pro-

mote academic excellence than teachers from typical schools (60%). The opposite 

was true for Arbor City, with 60% from highly effective schools and 83% from 

typical schools respectively. 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       68 

The third signifier of teachers‘ expectations involved their beliefs about stu-

dent ability. Teachers who exhibit high expectations believe that students have the 

potential to succeed (Ladson-Billings, 1994). The results show some variation in 

teachers‘ expectations of a low-achieving student with regard to student behavior 

including good or poor academic performance, potential for success, and descrip-

tors of the student (e.g., smart). Of these three, the most frequent comments in-

volved academic performance. Across the participating districts, teachers at typi-

cal schools were more likely than their counterparts at highly effective schools to 

state that students they had designated as ―low‖ had less potential for academic 

success than their other students. 

Finally, an analysis of the data shows differences in expectations for college 

attendance between the two cities. All teachers from the President City Area 

(highly effective and typical) stated that they believed a majority, if not all, of 

their students would attend college. Such was not the case with Arbor City. In-

stead, teachers often cited the negative influences of students‘ neighborhoods and 

lack of college-educated role models as barriers to college attendance. Neverthe-

less, teachers from highly effective schools (57%) were more likely to see college 

as an attainable goal for many of their students compared to teachers from typical 

schools (17%). 

 

Culture. The culture analysis is an examination of teachers‘ conceptions of 

students‘ cultures (including race, ethnicity, language proficiency, and social 

class), the influence of culture on teachers‘ practices, and the relationship of cul-

ture to achievement. Three themes emerged from our analysis: (a) cultural con-

nection, (b) multicultural curricula, and (c) not culture. With the exception of 

multicultural curricula, these themes were not specific to mathematics but encom-

pass education issues generally. 

First, some teachers commented on sharing cultural characteristics with stu-

dents. The teachers described cultural connections in terms of race/ethnicity, 

shared language (Spanish), and common life experiences:  

 
One of the ways that I win the parents over is by having a conference with the fami-

ly. Once they see that I am interested in their culture—it‘s my culture too. It‘s my 

background too. I let them know how I learned Spanish because I had a blind 

grandmother and she just corrected me as I went along....I think that I‘ve been able to 

help the students not only be successful, but I‘ve been able to reach them….I‘m very 

much a part of them. (Mercado, Highly Effective, President City Area) 

 

Across the two cities, teachers from highly effective schools were more likely 

than teachers from typical schools to mention connecting culturally with students. 

The difference among teacher groups was greater in Arbor City (50% versus 

17%) than the President City Area (60% versus 50%). 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       69 

Second, some teachers talked of incorporating culture into the teaching of 

mathematics. The most common strategy was situating mathematics within con-

texts that were familiar to students from either a personal or cultural perspective. 

The purpose of this strategy was to promote meaning and understanding of the 

content: 

 
My example that I give them, when we got to Mexico. Because the kids have been to 

Mexico. Is there only one way to get to Mexico? Well, no. We go by the freeway, we 

go through the downtown, we take this bridge, so I say how many ways is that? Like 

five. That‘s the way you do the math. There are different ways to get to the answer. 

When we get to Mexico, we just don‘t go the same way. There are different ways 

and you can still get to Mexico. So in math, there are different ways to get to the an-
swer. So they laugh about it. But they remember it. (Rincon, Highly Effective, Presi-

dent City Area) 

 

Another strategy was to infuse into the curriculum current professional or histori-

cal figures in mathematics who come from diverse backgrounds. In President 

City, teachers from highly effective schools were almost twice as likely (60%) to 

embed multicultural curricula aspects into their instruction of mathematics than 

teachers from typical schools (33%). This difference was not found in Arbor City 

where the rates were relatively equal at 33%. 

Finally, a major theme involved teachers‘ statements that culture does not 

play a significant role in learning in general or the teaching and learning of ma-

thematics (i.e., ―not culture‖). As alternatives, teachers offered ―kids are kids,‖ or 

the universality of mathematics as significantly affecting achievement. Across 

both cities, teachers from typical schools were more likely than teachers from 

highly effective schools to state that culture was not a significant influence on 

learning (83% versus 50% in President City Area, and 83% versus 70% in Arbor 

City). However, when we examined subthemes, the ―kids are just kids‖ subtheme 

emerged as the most significant: 

 
I really try hard not to make it an issue of English and Spanish or Hispanic or non-

Hispanic. That is not the issue. The issue is that you are a child in this world that has 

value. (James, Highly Effective, Arbor City) 

 

Summary. The principal themes that emerged in our analysis of the partici-

pating teachers‘ conceptions of student diversity involved: (a) teachers‘ academic 

and future expectations for their students, and (b) teachers‘ conceptions of stu-

dents‘ cultures. Teachers from highly effective schools across the two cities ex-

pressed more elaborate conceptions (i.e., conceptions aligned with recommenda-

tions made by researchers in multicultural education) with respect to these themes 

and others (see Table 2). Teachers at highly effective schools across the two dis-

tricts were more likely to state that all their students had potential for academic 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       70 

success. In addition, teachers from highly effective schools were more likely than 

teachers from typical schools to report sharing a cultural connection with students 

and this difference was greater in Arbor City than in the President City Area. 

Lastly, while multicultural curricula did not distinguish the teacher groups in Ar-

bor City, teachers at highly effective schools in the President City Area were al-

most twice as likely as teachers at typical schools to state that they used cultural 

contexts or cultural icons to teach mathematics. 

 
Discussion and Implications 

 

A guiding principle for this study was that teachers at highly effective 

schools possess desirable characteristics or engage in desirable instructional prac-

tices more so than teachers at typical schools. While other research has focused on 

non-instructional, school-level characteristics (Edmonds, 1979; Martin et al., 

2000; Purkey & Smith, 1983), in this study we focused on teacher conceptions of 

mathematics and student diversity. Following the literature on school restructur-

ing (Lee & Smith, 2001; Newmann and Associates, 1996), we set out to examine 

how teachers in highly effective schools articulated their learning goals in ma-

thematics for diverse student groups in comparison to their colleagues at typical 

schools. 

 
Major Findings and Relevant Findings 
 

The research findings help to explain the differentiation between highly ef-

fective and typical schools.  

 

Finding #1: We found consistent evidence that teachers at highly effective 

schools expressed more elaborated conceptions of mathematics education than 

teachers at typical schools. Across the two cities, teachers at effective schools 

stated that mathematics is more than just addition and subtraction, that instruction 

should help students make sense of mathematics, and that they employ multiple 

instructional strategies.  

While all teachers acknowledged that their curricular decisions were heavily 

influenced by high-stakes mathematics tests, the effective schools‘ teachers were 

more likely to state (very eloquently, at times) that they tried to teach beyond the 

test. Furthermore, teachers at effective schools in the President City Area tried to 

incorporate more advanced mathematics content in their fourth-grade curriculum. 

These teachers‘ conceptions were more aligned with the Platonist view (Hiebert 

& Carpenter, 1992) or the problem-solving view (Ball & Cohen, 1999) of mathe-

matics than the instrumentalist view. 

 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       71 

Finding #2: We found evidence that teachers at highly effective schools 

possess more elaborated conceptions involving student diversity than teachers at 

typical schools. For instance, teachers at highly effective schools across the two 

cities advocated for explicitly including multicultural referents in the curriculum, 

whereas teachers at typical schools expressed ideas to the effect that mathematics 

is a culturally neutral subject. 

Interestingly, but unexpectedly, teachers‘ conceptions of student diversity 

varied by city. In the President City Area, teachers at highly effective schools 

couched their conceptions in terms of teacher expectations of their students. The 

teachers emphasized students‘ higher order thinking in mathematics, tried to im-

prove academic excellence in mathematics, saw potential in low-achieving stu-

dents, and expected the majority of their students to go to college (Lipman, 1998, 

Zeichner, 1996). In Arbor City, teachers at highly effective schools couched their 

conceptions in terms of their own and their students‘ cultural backgrounds. The 

teachers stated that they shared similar cultural backgrounds (either through 

race/ethnicity or upbringing) with their students, and noted that their students 

brought knowledge acquired in their homes and communities that could enrich or 

make the teaching of mathematics more meaningful (Banks, 1993; Ladson-

Billings, 1994, 1995).  

 

Discussion of the two findings: We found compelling evidence that teachers 

at highly effective schools have more elaborated conceptions of both mathematics 

education and of student diversity. In general, teachers at highly effective schools 

articulated more often than their counterparts at typical schools conceptions that 

align with recommended reforms in mathematics education and conceptions with 

respect to student diversity that parallel scholars‘ recommendations related to the 

constructs of teacher expectations and culturally relevant instruction. Because we 

did not study students‘ prior educational experiences, we do not know if these dif-

ferences could be explained in terms of students at highly effective schools having 

already learned and achieved at higher levels prior to the study. Moreover, these 

differences did not exist for each category (e.g., teachers at typical schools in Ar-

bor City were more likely to promote academic excellence than their counterparts 

from highly effective schools). Nevertheless, given the care that was taken to dif-

ferentiate highly effective schools from their typical counterparts, our findings do 

demonstrate that differences existed between these two groups of schools with 

regards to teachers‘ conceptions of mathematics education and of student diversi-

ty.  

Teachers at highly effective schools believed in the importance of teaching 

more than problem solving and the basic facts and stated that they used a variety 

of instructional strategies to help students make sense of mathematics. Teachers at 

highly effective schools also emphasized how they taught their students more than 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       72 

what was covered on the State Assessment or standardized tests. While instruction 

as described in the reform literature (see, e.g., NCTM, 2000; NSF, 1996) is very 

important, it is also the case that teachers do more than simply teach. Beyond 

teaching, teachers can exhort students to do well, show care for students and their 

parents, and provide students with reasons for staying in school and trying hard 

for their future lives. All of these features would seem to rely on the kinds of ex-

pectations and cultural knowledge that the teachers in our highly effective schools 

expressed. Thus, it seems reasonable to conclude that teachers‘ conceptions of 

student diversity, as well as their conceptions of mathematics education, both in-

fluence student learning (Rodriguez & Kitchen, 2005; Rousseau & Tate, 2003). 

Our results are certainly consistent with such a view. 

Highly effective teachers who participated in this study not only expressed 

high expectations for their students and/or valued their students‘ cultural back-

grounds, they also talked about their desire to utilize mathematics curricula and 

implement instructional strategies aligned with reforms in mathematics education. 

It was beyond the scope of this study to determine whether causality existed be-

tween differing teachers‘ conceptions, such as how teachers‘ conceptions about 

student diversity may influence their conceptions about mathematics education. 

Nevertheless, this study motivates questions about the interaction between teach-

ers‘ conceptions about mathematics education and student diversity. 

The results also suggest the need for a more nuanced analysis of how varia-

tion in student backgrounds could result in variations in teachers‘ responses to 

those backgrounds. In this study, teachers at highly effective schools in the Presi-

dent City Area, many of whom shared the same cultural backgrounds as their stu-

dents, said more about their expectations for their students than their shared back-

grounds and shared experiences. In contrast, teachers at highly effective schools 

in Arbor City reported relying on their shared cultural backgrounds more than 

they discussed the kinds of expectations they had for their students. Both groups 

of teachers, however, were very similar in their more elaborated conceptions of 

the teaching of mathematics with diverse student groups than teachers at typical 

schools. 

 
Implications for Teaching and Teacher Development 
 

To promote reform oriented instruction in mathematics, professional devel-

opment efforts should involve helping teachers develop deep understanding of 

content knowledge, foster students‘ initiation and exploration in problem solving, 

identify students‘ mathematical conceptions and misconceptions, and implement 

content-specific teaching strategies (Garet, Porter, Desimone, Birman, & Yoon, 

2001; Richardson & Placier, 2001). This need is great with elementary teachers in 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       73 

mathematics, because they are generally not adequately prepared with content 

knowledge or effective teaching practices. 

By their comments, many teachers in the study displayed limited knowledge 

of how to incorporate students‘ linguistic and cultural diversity into mathematics 

instruction. Some teachers, especially in typical schools, considered mathematics 

as ―culture-free.‖ In contrast, teachers in highly effective schools made statements 

about student diversity that could serve as a basis for the provisio n of professional 

development. That is, professional development should involve enhancing teach-

ers‘ knowledge of student diversity and the intersections of culture and language 

with teaching practices and student achievement. Such an effort may involve di-

versifying and tailoring professional development efforts to the specific chal-

lenges that different groups of teachers face with their students. For example, 

―pedagogy of poverty‖ (Haberman, 1991) may need to be addressed in Arbor 

City, whereas issues of English language learners should be integrated in the Pres-

ident City Area (Moschkovich, 1999). 

 
Recommendations for Further Research 
 

This study provided an in-depth analysis of fourth-grade teachers‘ concep-

tions across schools in geographically distinct areas. The focus on differences in 

teachers‘ conceptions affords one way to examine why some schools are more 

effective than others. Our data, however, did not provide the means to understand 

how significant a role teachers‘ conceptions actually are in differentiating highly 

effective schools from their more typical counterpart. Research needs to be under-

taken to understand the significance of teachers‘ conceptions in distinguishing 

schools. Furthermore, research needs to be carried out to examine the relationship 

between teachers‘ conceptions about mathematics education and student diversity. 

If teachers have more elaborate conceptions about the teaching and learning of 

mathematics, are they more likely to develop more elaborate conceptions towards 

student diversity and vice versa? 

Effective instruction requires both content coverage and reform-oriented 

teaching practices. Teachers at highly effective schools repeatedly stated that they 

taught more than what was covered on the State Assessment or standardized tests, 

and their conceptions of the mathematics curriculum were more elaborate than 

those of typical schools‘ teachers. The results of this study suggest that future re-

search should examine teachers‘ content coverage and seek to tie the content cov-

erage to their specific teaching practices. 

High-stakes testing had a strong influence on teachers‘ conceptions of the 

teaching and learning of mathematics with diverse student groups. The results 

suggest that teachers believe that high-stakes tests press them to focus on teaching 

limited content to the exclusion of other content and constrain the instructional 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       74 

practices that teachers believe they can employ. Further research on how high-

stakes testing influences teachers‘ conceptions and curriculum content coverage 

and how it also influences student learning and achievement seems warranted. 

This research topic is urgent as high-stakes assessment and accountability policy 

impacts more students, especially those who have traditionally been underserved 

in mathematics. 

The research findings described here combine constructs of effective peda-

gogy found in mainstream mathematics education with multicultural education. 

Although research on academic content and student diversity rarely overlap, both 

emerged as significant in our data. These findings then beget the need for re-

searchers, policymakers, and teacher educators to draw from multiple knowledge 

bases. Yet, little is known about how the constructs inform and interact with one 

another at the student, teacher, or school level. If research is to improve educa-

tion, there is a need to develop a research agenda that combines the expertise of 

mainstream academic content and multicultural education researchers. 

 
Acknowledgments 

 

This study is part of a larger research project that compared mathematics education between high-

ly effective and typical elementary schools in two cities that had participated in an Urban Systemic 

Initiative (USI). Both, the larger and this study, were supported by a grant from the National 

Science Foundation (NSF) to the Urban Institute in Washington, DC. We would like to thank Bea-

triz (Toni) Clewell and Patricia Campbell for their assistance and feedback on this manuscript; our 
research methodology section draws heavily from their book on the larger study (see Clewell, 

Campbell, & Perlman, 2007). The bulk of this work was done while Walter Secada was on faculty 

at the University of Wisconsin–Madison. Secada received additional support during his sabbatical 

year from the UW–Madison Graduate School and School of Education, by the National Center for 

Improving Student Learning in Mathematics and Science (funded by the U.S. Department of Edu-

cation Office of Educational Research and Improvement) and by Diversity in Mathematics Educa-

tion/Center on Learning and Teaching (funded by NSF). Additional support was provided to Ok-

hee Lee by a grant from the National Science Foundation. All findings and opinions are the au-

thors‘. No endorsement by any of the above organizations or funding agencies should be inferred. 

 
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Appendix I 

Teacher Interview Protocol 

Fall 2000 

 
At the completion of two consecutive visitations with teachers at participating schools, the inter-

viewer/observer conducts an interview with the teacher. A primary purpose of interviews is to 

understand teachers‘ conceptions (beliefs and knowledge) about mathematics teaching and learn-

ing with culturally and linguistically diverse students. A secondary purpose is to extend the infor-

mation about mathematics to other subject areas. A third purpose is to use interview responses in 

verifying or triangulating classroom visitations in their classrooms as well as survey responses of 
the entire teacher group in the research.  

 

Interviews examine teachers‘: (a) general perceptions of the lessons, (b) conceptions of the nature 

of mathematics, (c) conceptions of students, and (d) conceptions of integrating mathematics with 

students. Each question is followed with probes. After a classroom visitation, the observer obtains 

academic tasks, such as handouts or overhead transparencies, that are used as the context for the 

interview. To contextualize the interviews in relation to the lessons, both the interviewer/observer 

and the teacher use examples of salient events during the lessons. 

 

Teachers’ General Perceptions of the Lessons 
 

1. Do these lessons give us as a good sense of the kind of work that you typically have this 

group of students do in mathematics? 

 

PROBE: If yes, how is it typical? If no, what is typical? 

 

PROBE: Use specific incidents to probe teacher‘s perceptions. 

 
2. What were the mathematical ideas that you were trying to cover during these lessons? 

 

PROBE: Ask the teacher to define terms used, such as ―topics‖ and ―skills.‖ 

 

Topics – e.g., measurement, geometry, fractions 

Skills (basic, low-level) – e.g., computational algorithms 

Processes (higher-level) – e.g., observe, describe, classify, infer, conclude 

 

Teachers’ Conceptions of the Nature of Mathematics 
 

3. What are three most important mathematics ideas for your students to learn in general? 

  

PROBE: Why? 

 

4. What determines the mathematical content that you cover during the year? 

 

PROBE: To what extent do you rely on: textbooks, curriculum guides, standardized tests, new or 

innovative curricular, the standards, personal expertise, and/or beliefs about mathemat-
ics 

 



 

 

 

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Journal of Urban Mathematics Education Vol. 2, No. 1                                       79 

Note: Ask the teacher to explain each of these items. 

 

PROBE: How do the lessons fit into what you just said is important? 

 

Teachers’ Conceptions of the Students 
 

5. Are there some students who aren‘t doing as well in mathematics as they should? 

 

PROBE: Anyone who surprised you? Anyone who disappointed you? 

 

6. Why do you think some students are doing better than you expected? Why do you think some 

are not doing as well as they should? 

 
PROBE: Probe for teachers‘ accounts of high achievement 

Probe for teachers‘ accounts of low achievement 

 

7. What are you doing to meet the learning needs of these students? 

 

PROBE: Does your instruction change during the academic year to meet your students‘ learning 

needs? If so, how?  

 

PROBE: Probe for teacher efficacy (i.e., how teachers work to actually meet needs of students). 

What is the impact of your efforts on these students? 

 

Teachers’ Conceptions of Integrating Mathematics with the Students 
 

8. Is math ―culture free,‖ or does culture have any influence on math? 

 

PROBE: Give an example of how students‘ backgrounds (such as their home culture and gender) 

have any influence on mathematics. 

 
PROBE: The education literature says that students from certain cultural backgrounds do not 

question adults at home and, therefore, do not ask questions to the teacher. What do you 

think about the influence of your students‘ cultural backgrounds on math learning? 

 

9. Do you consider your students‘ culture, language, and gender in your teaching in order to 
promote their math learning? 

 

PROBE: Give an example of how you used your students‘ cultural experiences during the lesson 

(if possible). (If not), give an example from your teaching at other times or during other 

lessons during this year of culture, gender, English language proficiency, socioeconomic 

status, disability. 
 

 

 

 

 

 



 

 

 

Kitchen et al.                                                      Comparing Teachers’ Conceptions 

Journal of Urban Mathematics Education Vol. 2, No. 1                                       80 

Appendix II 

Teacher Interview Protocol 

Spring 2001 

 

 
1. Teachers’ General Perceptions of the Lesson 
 

What were the mathematical (scientific) ideas (broad, generic sense) that you were trying to 
cover during these lessons? 

 

PROBE: 
Probe distinctions among topics, skills, or processes 

 

(a) topics – e.g., measurement, geometry, fractions, biology, ecology, physical science 

(b) skills (basic, low-level) – e.g., measurement, computational algorithms 

(c) processes (higher-level) – e.g., measure, observe, describe, classify, infer, conclude 

 

2. Testing 
 

During our last visit, many teachers spoke about the impact standardized testing was having 

on instruction. In what ways, if any, do you feel your math (science) lesson today was influ-

enced by state/district standardized tests? 

 

3. For the two students who are the focus of the math (science) interview 
 

Could we talk for a few minutes about Student A? 

(Note: It will be hard for this to happen without teacher using student name.) 

 

 Did students A do anything that surprised you during your lesson? If yes, what did the 
student do? Why did that surprise you? 

 How well do you usually expect Student A does in math (science) in general? How well 
does Student A perform on math (science) tests? 

 What strengths does Student A bring to math (science)? What weaknesses? 

 Which, if any, strategies work particularly well with student A in math (science)? Why 
do you think Student A responds well to these strategies? 

 Which, if any, strategies DON‘T work particularly well with student A in math (science)? 
Why do you think Student A doesn‘t respond well to these strategies? 

 What, if any, in and out-of-class resources do you have that can help you better teach 
Student A mathematics (science)? 

 

Could we talk for a few minutes about Student B? 

Go through the same procedure for Student B