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Dispelling the notion of inconsistencies in teachers’ mathematics beliefs and practices: A 3-year case study Dionne I. Cross Francis Ó Springer Science+Business Media Dordrecht 2014 Abstract Researchers in the field of mathematics education have focused on beliefs as a significant area of study because of the influence of beliefs on what is taught and learned. Much of the research in this area speaks about inconsistency between teachers’ beliefs about mathematics teaching and learning and their classroom practices. In this case study, I look beyond two elementary teachers’ perceived inconsistencies to gain a better understanding of the nature of their beliefs and how they are organized. Data were gathered from individual and focus group interviews, classroom observations, email communications, and researcher memos over the course of 3 years. Results showed that non-mathematics beliefs and contextual factors took precedence in certain classroom situations and contextual factors had an intervening influence on the actualization of beliefs. Several theoretical, methodological and practical implications of the findings are discussed. Keywords Teacher beliefs Á Teacher practices Á Mathematics beliefs Á Beliefs system What a teacher does in the classroom is shaped by knowledge: content knowledge, pedagogical knowledge, and pedagogical content knowledge (Ball et al. 2001; Hill et al. 2007; Shulman 1986); school factors: curriculum, standards, and resources (Herbel-Eisenmann et al. 2006); psychological factors: goals and efficacy (Speer 2005); and socio-historic and contextual factors (Sztajn 2003). However, beliefs are considered to be one of, if not the most influential factor on teachers’ instructional practices (Pajares 1992; Philipp 2007; Richardson 1996). As such, researchers across Electronic supplementary material The online version of this article (doi:10.1007/s10857-014-9276-5) contains supplementary material, which is available to authorized users. D. I. Cross Francis (&) Mathematics Education, Indiana University, WW Wright Education Building, 201 N Rose Avenue, Bloomington, IN 47405, USA e-mail: [email protected] 123 J Math Teacher Educ DOI 10.1007/s10857-014-9276-5

Dispelling the notion of inconsistencies in teachers’ mathematics beliefs and practices: A 3-year case study

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Dispelling the notion of inconsistencies in teachers’mathematics beliefs and practices: A 3-year case study

Dionne I. Cross Francis

� Springer Science+Business Media Dordrecht 2014

Abstract Researchers in the field of mathematics education have focused on beliefs as

a significant area of study because of the influence of beliefs on what is taught and

learned. Much of the research in this area speaks about inconsistency between teachers’

beliefs about mathematics teaching and learning and their classroom practices. In this

case study, I look beyond two elementary teachers’ perceived inconsistencies to gain a

better understanding of the nature of their beliefs and how they are organized. Data were

gathered from individual and focus group interviews, classroom observations, email

communications, and researcher memos over the course of 3 years. Results showed that

non-mathematics beliefs and contextual factors took precedence in certain classroom

situations and contextual factors had an intervening influence on the actualization of

beliefs. Several theoretical, methodological and practical implications of the findings are

discussed.

Keywords Teacher beliefs � Teacher practices � Mathematics beliefs � Beliefs

system

What a teacher does in the classroom is shaped by knowledge: content knowledge,

pedagogical knowledge, and pedagogical content knowledge (Ball et al. 2001; Hill

et al. 2007; Shulman 1986); school factors: curriculum, standards, and resources

(Herbel-Eisenmann et al. 2006); psychological factors: goals and efficacy (Speer

2005); and socio-historic and contextual factors (Sztajn 2003). However, beliefs are

considered to be one of, if not the most influential factor on teachers’ instructional

practices (Pajares 1992; Philipp 2007; Richardson 1996). As such, researchers across

Electronic supplementary material The online version of this article (doi:10.1007/s10857-014-9276-5)contains supplementary material, which is available to authorized users.

D. I. Cross Francis (&)Mathematics Education, Indiana University, WW Wright Education Building, 201 N Rose Avenue,Bloomington, IN 47405, USAe-mail: [email protected]

123

J Math Teacher EducDOI 10.1007/s10857-014-9276-5

educational fields, including educational psychology and teacher education, have

focused significant research efforts on better understanding this construct (e.g., Clarke

and Peterson 1986; Cobb et al. 1991; Lumpe et al. 2000; Nespor 1987; Pajares 1992;

Philipp 2007; Philipp et al. 2007; Raymond 1997; Thompson 1992; Torff 2005;

Wilson and Cooney 2002).

Although several theorists purport that beliefs are determinants of actions, a sig-

nificant portion of the beliefs research in mathematics education documents mis-

alignment between teachers’ professed beliefs (beliefs stated by teachers) and practices

(e.g., Cooney 1985; Raymond 1997; Thompson 1984). It has been argued (e.g.,

Leatham 2006) that this conclusion is a result of the theoretical assumptions and the

methodological approach of researchers. However, few have sought to determine

whether these belief–action discrepancies are real or apparent, find alternative expla-

nations through the use of nonstandard (with respect to beliefs research) methodo-

logical approaches or to develop theory that would provide insight into these findings.

This paper aims to address some of these issues. Working from the perspective that

most teachers feel their beliefs and practices are consistent (Philipp 2007), I draw on

Leatham’s (2006) notion of ‘‘beliefs as sensible systems’’ (Leatham 2006, p. 92) as a

lens through which to examine and interpret teachers’ beliefs and actions. I contend

that observed inconsistencies should be considered sites for further inquiry and

exploration to obtain a deeper understanding of the teachers’ beliefs and how they are

organized.

In this study, I explore the belief systems of two elementary teachers, describe ways in

which their beliefs and practices appeared to be misaligned, and provide reinterpretations

of these perceived inconsistencies. I focus on answering the following questions:

1. Based on the elementary teachers’ descriptions of their mathematics-related beliefs

and experiences, what are their mathematics beliefs profiles?

2. What are the researcher-perceived inconsistencies between the teachers’ mathematics-

related beliefs and their practices?

a. What factors contribute to these perceived inconsistencies?

b. What is the nature of the beliefs with which the teachers’ practices cohere?

c. How can these perceived inconsistencies be reinterpreted as coherent?

I conclude with a discussion of the theoretical, methodological, and practical implica-

tions of the findings on the study of teachers’ beliefs.

Theoretical perspectives and related literature

As Pajares (1992) stated ‘‘Attention to the beliefs of teachers should be a focus of edu-

cational research and can inform educational practice in ways that prevailing research

agendas have not and cannot’’ (p. 307). The increased wave of research on this construct in

mathematics education in the early 80s was motivated by the push for instructional reform

and the need to understand what motivated teachers’ actions in the classroom. There was

the belief that developing deeper understandings of teachers’ beliefs would provide a

roadmap to strategies for transforming teachers’ practices to be more problem-solving,

reform-oriented (Skott 2009). In many ways, research in this area has not met these

expectations, as it is currently debatable the extent to which beliefs predict teachers’

D. I. Cross Francis

123

instructional practices. On the one hand, some research findings suggest misalignment

between beliefs and practices (e.g., Raymond 1997; Skott 2001), while other results show

that teachers’ beliefs greatly influence their pedagogical decisions and overall effectiveness

(e.g., Artz and Armour-Thomas 1999; Cross 2009), which ultimately shape the kinds of

learning experiences students have (Mewborn and Cross 2007). In addition, research in this

area still has several limitations. Two commonly noted limitations are (a) the lack of a

clear definition of beliefs that distinguishes it from orientations, attitudes, and dispositions

(see Pajares 1992; Philipp 2007 for discussion of this issue), and (b) the assumption that

teachers can clearly describe their beliefs and there is a perfect alignment between what

teachers say and researchers’ interpretations of these statements (Leatham 2006). The

purpose of this article was not to resolve these conceptual issues; however, for clarity, in

the next sections, I provide the definition of beliefs that frames this article and describe the

conceptual frameworks and research findings from which this study draws.

Teachers’ beliefs

Definition of beliefs

Drawing from the work of Pajares (1992), Thompson (1992), and Green (1971), I define

beliefs as embodied conscious and unconscious ideas and thoughts about oneself, the

world, and one’s position in it developed through membership in various social groups,

which are considered by the individual to be true. Beliefs are considered to be fairly stable

and very influential in determining an individual’s actions (Lumpe et al. 2000; Rimm-

Kaufman and Sawyer 2004; Thompson 1992; Torff and Warburton 2005). However, the

relationship between teachers’ beliefs and their instructional practices is not considered

linear, as contextual factors such as school climate, parental expectations, standardized

testing, and teacher-related factors including teacher efficacy and content knowledge tend

to influence teachers’ enacted practices (Herbel-Eisenmann et al. 2006; Richardson 1996;

Skott 2001).

Teachers’ mathematics beliefs

Many teacher educators and researchers within mathematics education consider beliefs that

align closely with a view of teaching and learning described in the National Council of

Teachers of Mathematics [NCTM]’s Principles and Standards for School Mathematics

(2000) to be most supportive of inquiry-based classroom practices (Cobb and McClain

2006; Lampert 1990). However, many teachers see mathematics as a system of facts and

procedures to be memorized and reproduced with the sole purpose of obtaining one correct

answer to routine, textbook-type problems (Stipek et al. 2001; Thompson 1992). These

traditional beliefs tend to align with a teacher-centered view of teaching and learning

where the learner is passive and the teacher acts as the knowledge-giver (traditional view).

Over the past few decades, various researchers (Ernest (1989): problem-solving, Platonist,

instrumentalist, and Kuhs and Ball (1986): learner-focused, content-focused, classroom-

focused) have labeled and classified these sets of beliefs differently (see Thompson (1992)

for a full description of these classifications). Nevertheless, beliefs that support problem-

solving and critical thinking are often foregrounded as they are considered most beneficial

for high-quality instruction.

Dispelling inconsistencies between beliefs and practice

123

Philosophical organization of beliefs

Researchers often assume that individuals can clearly articulate their beliefs. This is not

necessarily so as beliefs tend to reside below an individual’s immediate level of consciousness

(Torff and Warburton 2005). Consequently, beliefs cannot be directly observed, rather they

must be inferred from what people say and do (Pajares 1992). This covert and resolute nature

can be attributed to the organization of beliefs as systems (Green 1971). Green provided a

three-dimensional framework for the organization of belief systems. This first dimension deals

with how the way beliefs are held. The connection between two or more beliefs is not based on

their content but on the individual’s idea that one belief implies the other. For example:

Belief A: Mathematics is about problem-solving and thinking.

Belief B: Quality teaching involves encouraging students to explain their thinking and

justify their responses.

Belief A functions as the premise from which belief B is implied. Whether or not belief

B is true or logically follows from belief A is somewhat irrelevant. The key issue is that for

the teacher they are seen as related.

The second dimension, the psychological strength of a belief, describes the importance

of the individual places on that belief. Beliefs that are held with great psychological

strength are called core beliefs and the others are referred to as peripheral beliefs (Green

1971). The strength of a belief is related to its connectedness to other beliefs, such that

strongly held beliefs tend to cohere with other beliefs in the system. The third dimension to

this system is related to how the beliefs are clustered. This clustering feature provides

protection and support for different sets of beliefs, allowing seemingly incompatible and

inconsistent beliefs to coexist. The clusters tend to reside in isolation from each other; thus,

it can appear to an external observer that the individual holds conflicting beliefs, although

these beliefs may not seem contradictory to the belief holder.

To deeply understand how the belief–action relationship works and to explain perceived

discrepancies within this relationship, researchers must keep in mind that beliefs’ systems

are dynamic in nature (Thompson 1992), they are organized in clusters, and individuals

tend to organize their beliefs within the system so they cohere (Pehkonen 2008). Below I

describe a framework grounded in these assumptions, thereby providing a mechanism

through which to rethink perceived inconsistencies.

Sensible systems framework

Philipp (2007) stated that as researchers we must assume that contradictions between

teachers’ beliefs and practices do not exist. Rather, when we observe apparent con-

tradictions, we must assume they exist in our minds and not within the teacher’s. In

accordance with this view, Leatham’s (2006) sensible systems framework holds as a basic

assumption that ‘‘teachers are inherently sensible rather than inconsistent beings’’ (p. 92).

In other words, individuals’ beliefs are organized in systems that make sense to them. As

Op’t Eynde et al. (2002) stated ‘‘People always strive for a coherent belief system; only

then are they able to function in an intelligible way’’ (p. 25). As such, we find that certain

beliefs may have stronger influence over certain actions in certain situations as the belief to

which the action is aligned becomes prioritized in the context. Consistent with Green’s

model, beliefs are not held based on logical relationships or based on their indisputable

nature. Rather, beliefs are justified and upheld because they cohere with other beliefs in the

system, which provide support and protection. Therefore, if the individual seeks to justify a

D. I. Cross Francis

123

belief, she will not necessarily find indisputable evidence for support; instead, she will

make adjustments within the system until coherence is obtained. Thus, ‘‘whenever beliefs

that might be seen as contradictory come together, the person holding the beliefs finds a

way to resolve the conflict within the system’’ (Leatham 2006, pg. 95), thereby making the

system sensible, to the person. As such, it is indeed possible that the observer may not

readily identify these clusters or see how this clustering is justified. However, this does not

make them less coherent to the individual.

Within the mathematics education literature, there are several studies that document

inconsistencies between teachers’ beliefs and their practices (cf. Raymond 1997; Cooney

1985; Skott 2001). In these studies, and others, researchers often enter the field investi-

gating a particular set of beliefs. For example, mathematics education researchers tend to

focus on beliefs specific to the discipline of mathematics and its teaching and learning. As

such, they tend to make assumptions about the belief–action relationship that aligns with

the beliefs under study (in this case, mathematics beliefs), thereby making it possible and,

even likely, that other beliefs (such as beliefs about particular students or efficacy beliefs)

with which the observed action more closely cohere go overlooked. Based on these often

premature connections, the conclusion that there exists inconsistencies between teachers’

beliefs and actions is drawn. Therefore, in situations where there are observed inconsis-

tencies, there are several other likely possibilities. One, it is possible that either the belief–

action relationship inferred by the observer was erroneous; two, the observer may have

failed to notice another belief that was foregrounded in that situation; or three, there were

intervening contextual factors (discussed below) that influenced the observed action. It

follows then that observations that show possible inconsistencies are sites for further

investigation and exploration and present opportunities to learn.

Teacher-related and contextual factors Prior research has shown that the relationship

between teachers’ beliefs and practices is mediated by contexts (Raymond 1997; Sztajn

2003). In Raymond’s (1997) case study of Joanna, an elementary teacher who held tradi-

tional beliefs about mathematics but non-traditional beliefs about mathematics teaching and

learning, Raymond concluded that time constraints, scarcity of resources, concerns about

standardized tests, and students’ behavior were factors that impacted her teaching. Similarly,

in the case of Christopher, the teacher in Skott’s (2001) study, school mathematics images

(SMI)1 were overshadowed by the broader educational issues present in his classroom.

Sztajn (2003) also concluded that context played an important role in the teaching behavior

of the two teachers in her study, although context was framed somewhat differently. The

participants in Sztajn’s study taught students with vastly different socioeconomic situations.

Concerns about how to best prepare these students for the future, given their backgrounds,

impacted their teaching. Unlike Joanna, the major factor that accounted for these teachers’

practices was not school-based factors, but their beliefs about children, society, and edu-

cation. Herbel-Eisenmann et al. (2006) have also cautioned that in examining teachers’

actions, we must attend to (a) the political and social contexts in which teachers work,

(b) teachers’ access to curricular materials and resources, and (c) the expectations of both

parents and students, as they greatly impact teachers’ enacted instruction.

In addition to external, contextual factors, there are also teacher attributes that impact

how they instruct. Teachers’ self-efficacy also influences their willingness to adopt and

1 Skott (2001) defines school mathematics images as ‘‘teachers’’ idiosyncratic priorities in relation tomathematics, mathematics as a school subject, and the teaching and learning of mathematics in schools’(p. 6).

Dispelling inconsistencies between beliefs and practice

123

enact particular instructional practices. Teacher efficacy refers to the teacher’s beliefs

about her capacity to affect how students learn and their overall performance (Stein and

Wang 1988; Tschannen-Moran et al. 1998). Research suggests that teachers with a lowered

sense of teaching efficacy tend to be less open to trying innovative pedagogical approaches

(Czerniak and Schriver 1994; Guskey 1988). Elementary teachers, in comparison with

secondary-level teachers, tend to have lower mathematics self-concept (Ball 1990) and

teacher efficacy (Swars 2005). Given the many factors that can influence teachers’

instructional decisions and enacted practices, as mathematics education researchers, we

must seek to explore not only mathematics beliefs, but also beliefs beyond mathematics

(Sztajn 2003), and examine the circumstances surrounding teachers’ practices (Philipp

2007) to fully understand the belief–action relationship. In this study, I chose to hold the

view that teachers’ beliefs and practices are consistent and to seek explanations for per-

ceived inconsistencies.

Research methods

The research that is the focus of this article was a part of a 3-year (Summer 2008–Spring

2011) professional development project. The goals were to improve the teachers’ mathe-

matics content knowledge and expose them to reform-based teaching strategies. Although

the project involved 15 elementary teachers, this study specifically focused on the thinking

and beliefs of two early childhood teachers from the group.

Participants and setting

This study included a first-grade teacher, Ms. Wicker, and a kindergarten teacher, Ms.

Edmonds, who taught at Walton Elementary School located in a Midwestern state in the

United States. The population was 99 % African American with approximately 81 % of

the students eligible for free and reduced-priced lunch. Walton Elementary School was

located in a district that was ranked in the lower tier compared to all districts in the state.

Over the 3 years of the project, Walton received a D (on an A–F scale) rating in the first

2 years of the project and an A at the end of the third year. Within the state, there are

mandated and optional assessments that are administered during the year. Due to the low

ranking of the district and school, Walton Elementary students took all the mandatory

assessments and most of the optional. Specific to the early grades, students in grades K-2

took mClass/DIBELS (three times per year—some districts administered this assessment

more frequently), district comprehensive assessment (DCA—at the end of the year) and

district quarterly assessment (DQA—quarterly).

Ms. Wicker2 is African American and prior to teaching at Walton elementary, she

taught for 6 years at another predominantly African American elementary school in the

same state. Given her passion for teaching and empowering black students, she enjoyed

teaching at Walton and considered it a good fit. She was from a family of teachers and

although she considered pursuing other careers during college, she ‘‘felt drawn to teach-

ing.’’ Desiring the best for her students, she continually sought out opportunities for

professional development, so she could learn about new teaching methods and ways to

2 Names include in this study are pseudonyms.

D. I. Cross Francis

123

work with struggling students. It was this desire to improve their teaching that motivated

both teachers to participate in the project.

Although Ms. Edmonds had over 20 years of teaching experience, she believed that the

best teachers were those who were continuously learning so she was eager to get involved

in the PD project. Her dedication to her students was exemplified in her continued efforts

to educate herself about her students and about innovative ways to teach. Prior to coming

to Walton, Ms. Edmonds had previously taught at four other elementary schools, all high

needs with large minority populations. As a Latina with lots of experience in urban, low

socioeconomic schools that serve African American students, she understood the plight of

low-income, minority students so she focused on ensuring that their schooling experiences

were enjoyable and meaningful.

Prior to the start of the project, both Ms. Wicker and Ms. Edmonds had been engaged

in efforts to improve their teaching of science through involvement in a year-long PD

program focused on project-based science at Walton Elementary. At the end of the

science program, approximately 95 % of the teachers volunteered to participate in the

mathematics PD project that began the following year. However, Ms. Wicker and Ms.

Edmonds were the only two teachers to volunteer for the extended PD (video discus-

sions—described below) and data collection (weekly videotaping). Although the teachers

were similar in this regard and in other aspects related to personality and teaching, there

were also differences. These similarities and differences will be highlighted in the

findings section.

Professional development activities

The project involved a range of activities; however, I will only describe those from

which data were drawn. These activities are particularly relevant because as Speer

(2005) recommended ‘‘…data on beliefs should come from sources that are tied to the

particular practices that one seeks to understand … one should begin with practices

and gather data related to beliefs in connection with those practices and contexts’’

(p. 373).

Year two (August 2009–June 2010)

During year two, we3 focused on extending teachers’ understandings of number and

geometry concepts. Teachers worked on inquiry-based tasks such as those included in

Beckman’s (2008) Mathematics for Elementary Teachers and similar sources. To support

the development of a student-centered pedagogy, teachers watched, critiqued, and reflected

on practices observed in video exemplars (e.g., classroom videos from Cognitively Guided

Instruction (CGI) Video Series4).

3 The author and four graduates conducted the professional development sessions. Some of these graduatestudents, along with a colleague, collaborated on portions for the analyses for this article. The use of ‘‘we’’throughout the methods section refers to these groups of individuals.4 The videos in this series show student mathematics interviews and classroom episodes from researchconducted at University of Wisconsin described in Carpenter, Fennema, Franke, Levi and Empson’s (1999)Children’s Mathematics: Cognitively Guided Instruction.

Dispelling inconsistencies between beliefs and practice

123

Video discussions During year two, the two teachers who are the focus of this study

volunteered to participate in additional PD activities. Ms. Wicker and Ms. Edmonds were

the only teachers who volunteered, although the offer was made to all the teachers. This

involved having their classes videotaped three to four times per week, then meeting

bimonthly with the PD facilitators to discuss the video clips. As we were also interested in

the classroom activities and teacher moves the teachers found noteworthy, we asked them

to select 2–3 classroom episodes they wanted to discuss. We also selected video clips that

were representative of the teachers’ practices—some aligned with and others seemed

inconsistent with teachers’ prior descriptions of their beliefs. In the discussion below, I

focus on describing instances where their practices did not appear to align with what they

stated. Given the time constraints, the discussions were usually focused around one clip

from each teacher and one from the facilitators.5 The discussions were positioned more as

group reflections so we would all watch the clip, then the teacher of focus would talk about

the instructional decision-making process around the lesson. Specifically, she would state

what the instructional goals were for the lesson, how she tried to achieve those goals

focusing her comments on what was visible in the clip, the reasoning behind teacher moves

in the clip, and how close she came to achieving the instructional goals. The general tone

of the sessions was reflective, so the teachers would consider their teacher moves in light of

the instructional goal and determine how well they thought their actions supported progress

toward those goals.

For example, from the first 2 weeks of video (equivalent to seven class sessions), we

observed that Ms. Wicker tended to provide answers to her own questions, instead of

allowing students time to think about the question and respond independently. Based on

what we observed, it was difficult for us to determine what the students were thinking or

how they were progressing mathematically. Instead of assuming that Ms. Wicker had

similar struggles and to better understand her approach to classroom discourse, we selected

two 10-min video clips that showed these teacher actions (along with others) for the first

video discussion. Then, we met with the teachers to discuss the videos. We would begin the

discussion with the members of the PD team (the author and four graduate students), along

with the teachers watching the video. Either Ms. Wicker or Ms. Edmonds (depending on

the video clip) was then asked to provide a brief context for the video, namely the

instructional goals for the day and how the teacher would determine whether the goals

were met. We would then narrow the focus to the specific clip and ask the respective

teacher to describe what she was doing in the video and what the students were doing in the

video. Based on the responses, the PD team would try to elicit the reasons behind the

actions to understand her decision-making process during the classroom episodes. These

conversations were recorded.

Research design and data sources

In order to infer a person’s beliefs with any degree of credibility, one needs numerous and

varied resources from which to draw those inferences (Leatham 2006; Pajares 1992). Thus,

I employed a qualitative case study approach (Stake 2000) in which I gathered multiple

data repeatedly over an extended period of time. The multiple and varied data sources

5 The professional developers were mindful of the possibility of the teachers responding in sociallydesirable ways. Although we understood that truthfulness is an issue that researchers have to contend with inself-report studies, we also made deliberate efforts to build trust and rapport with the participants so it was asafe environment to share their thoughts and ideas.

D. I. Cross Francis

123

gathered in this study allowed for data triangulation (Stake 2000; Yin 1984) and afforded a

multifaceted understanding of the cases.

Interviews

Each participant was interviewed at the beginning and end of each year during the 3 years

of the project, for a total of 6 individual interviews. The interviews ranged between 45 and

90 min and consisted of a range of questions. In order to comprehensively capture the

teachers’ authentic views about mathematics and mathematics teaching and to avoid the

participant stating what she thought the interviewer wanted to hear, beliefs were indirectly

targeted (see Appendix B/Online Resource 2). Questions also targeted reflections on PD

activities, how useful they thought they were, and if and how they incorporated the

strategies explored during PD. Speer (2005) stated that ‘‘…perceived discrepancies

[between teachers’ beliefs and actions] are sometimes the result of incomplete or inac-

curate understanding of terms and descriptions used by teachers and researchers’’ (p. 371).

To avoid any misunderstandings, I tried to exclude education jargon from the questions as

much as possible. Also, in situations where the teachers used these terms in their responses

(e.g., hands-on, inquiry, and problem-solving), I followed by asking the teachers to explain

what they meant by those terms (see Appendix A/Online Resource 1 for examples).

Additionally, three joint interviews were conducted with both teachers, one at the end of

each year, to elicit teachers’ thoughts about the process of changing their practices,

emotions elicited, and the impact of environmental factors on their teaching. Notes were

taken during both the individual and joint interviews (nine in total) that captured my

interpretations of the participants’ responses. These interviews were audio-recorded and

transcribed.

Email communication

Because the research site was a sizeable distance from my university, emails were used as a

main source of communication. Emails contained discussions about logistics pertaining to

the PD sessions, clarification of any vague meaning in the interview transcripts, reflections

on classroom episodes, and the participants’ evolving ideas about teaching and students’

learning.

Classroom observations

In the first and third years of the project, classroom observations were done bimonthly (see

Appendix C/Online Resource 3). The observation protocol was designed to capture various

aspects of the teachers’ practices as well as the classroom context. It provided details about

the major instructional approaches used (e.g., whole-class discussion and small-group work

time) and the time allotted to each approach; the types of tools/manipulatives used during

instruction, who used them, how they were used, and the time allotted for use; and the

summaries of the classroom context at 10-min intervals. The summaries also included

transcriptions of teachers’ and students’ statements. During year two, the participants’

classrooms were observed (using the protocol)6 twice per month and video-recorded three

times per week for 45 min to an hour. The observations helped me to better understand the

6 Although we had video-recordings, the camera was focused on the teacher. The observation notes pro-vided details about the overall classroom context that was often not well captured on the video.

Dispelling inconsistencies between beliefs and practice

123

teachers’ methods, the nature of their interactions with students, and the overall classroom

context. The data captured on the video-recordings during the second year were the focus

of the video discussions. The analyses focused on the observation data from years one and

two specifically.

Video discussions

In contrast to interviews that are somewhat decontextualized (i.e., conducted external to

the teaching context), video discussions allowed for the examination of beliefs and prac-

tices together in the context. Eight video discussions were conducted, which were audio-

recorded and then transcribed.

Data analysis

I employed generic thematic analysis by reducing the extensive text of interviews into core

themes (LeCompte and Preissle 1993) that reflected the teachers’ beliefs about mathe-

matics, mathematics teaching, and student learning. I, along with a colleague, indepen-

dently coded each transcript and then compared the codes. The refined codes were

compared, contrasted, and aggregated. Similar patterns were categorized and developed

into themes that represented the teacher’s mathematics-related beliefs (see Appendix

A/Online Resource 1 for examples). We also employed data source triangulation tech-

niques by adding, modifying, and merging the interview data with the observation data and

memos (Patton, 2002). The themes and findings were compiled and summarized in detailed

narratives of the teachers’ mathematics-related beliefs, referred to as belief profiles.

To be able to characterize teachers’ practices, we first examined the observation pro-

tocols of the teachers from years one and two (32 per teacher). We were most interested in

the teachers’ behavior that supported students’ mathematical engagement, both related to

physical actions and to discourse.7 We used the observation protocol to first identify

classroom episodes/clips where there was teacher–student interaction. These were mainly

during whole-class and small-group conversations. As year one data of teachers’ practices

only included the observation notes, we focused on the summaries and segments of the

notes (in the observation summaries) that included transcriptions of teachers’ and students’

statements. For year two data, we focused on the videos for which we also had observation

notes (32 between teachers). We identified chunks of video (using the observation notes as

a guide) that captured teacher–student interaction. Only those sections of the videos were

transcribed. To analyze the classroom talk, we drew on prior research (e.g., Stein), applied

a priori codes (such as questioning, telling, and revoicing) to one transcript, and then met to

discuss the coding. Discrepancies were discussed until we reached agreement. This dis-

cussion also led to the emergence of additional codes (e.g., praise and push for rigor) and

more detailed descriptions of the codes. We also examined the observation notes and

videos for teacher actions highlighting those that were specifically mentioned by the

teachers in the interviews such as manipulative use, collaborative work, small-group, and

whole-class discussion. Table 1 includes a summary of these data relevant to the

discussion.

7 Physical actions referred mainly to how the teachers organized the classroom for work, such as usingmanipulatives and arranging desks for collaborative work. Discourse referred to the nature of the talk duringteacher–student interaction.

D. I. Cross Francis

123

Based on the belief profiles, I examined the classroom observation data for the evidence

of classroom practices that seemed both consistent with and contradictory to the teachers’

beliefs. For example, when asked about ways to support the development of problem-

solving skills, Ms. Wicker stated, ‘‘You know not just giving directions, more like asking

questions … I think it’s the difficult part but it [is] the most important part rather than

giving them the answers….’’ However, observations of her classroom practices (described

above) showed her asking questions but for the most part, she also gave the answers. In this

instance, her statement along with this excerpt of video data (and others that showed

similar practices) was linked and coded as inconsistent. I highlight this clip as it provides

an example of a clearly stated belief by Ms. Wicker regarding practices she believed best

supported student learning; in particular, her belief that asking questions was more

important than providing explicit directions. Given our coding scheme, I was able to

determine the proportion of her talk that was questioning and the extent to which this

practice aligned with her stated belief.

To make sense of apparent inconsistencies, I conducted a second round of analyses of

the interviews and video discussions using a constant comparative method. I examined the

teachers’ statements from the interviews about their interpretations of their changing

practices and how environmental factors influenced their instruction. Also, from the video

discussions, I identified statements that described their specific intentions and the reasons

underlying their actions observed in the videos, comparing these statements to the math-

ematics-related belief statements. As Goetz and LeCompte (1981) stated ‘‘As events are

constantly compared with previous events, new topological dimension, as well as new

relationships, may be discovered’’ (p. 58). Data that aligned (Dye et al. 2000), including

classroom episodes and teacher statements, were grouped and then compared. Compari-

sons made within each data group (consisting of observation, interview, and discussion

data) showed that teacher actions that were previously associated with mathematics-related

Table 1 Physical and discourse codes by teacher

Physical CodesTeacher Action Ms. WickerManipulative Use % use in class % of classes not used

56 44Used for meaning-

makingUsed for demonstration

etc.33 67

Complex Problems % use in class % of classes not used81 19

Sustained Work around problem

Work not sustained

19 81Worksheet Use

(more than half the class time% use in class % of classes not used

74 26

Discourse CodesMs. Edmonds

Questioning, Telling and other codes

% of overall codesQuestioning followed by Telling Independent

TellingOther Codes (restating,

revoicing, etc.)52 26 9

Telling: declarative statements solely for the purpose of providing information to studentsQuestioning: questions posed at different levels asked with the goal of eliciting, engaging or challenging students’ thinking

Dispelling inconsistencies between beliefs and practice

123

beliefs were more aligned with beliefs beyond mathematics, contextual environmental

factors. For example, after the second round of analyses, the apparent inconsistency

described above was coded as beliefs beyond mathematics (described in more detail below

and represented in Table 2). In the next section, I present the results of these analyses in

narrative form.

Results

In this section, I provide a description of the teachers’ belief profiles, outline ways in which

these profiles appeared inconsistent with their teaching practices, and discuss how apparent

inconsistencies can be reconciled.

Teachers’ mathematical-belief profiles

In response to questions about her views on mathematics, Ms. Wicker responded

…you know it’s about problem solving, it’s more than formulas, because if you

memorize the formulas and you still can’t use it outside the classroom, then you can’t

help yourself. You need to understand why the formula makes sense and where it

comes from.

Ms. Wicker considered mathematics to be problem-solving, where students should think

their way through problem situations rather than memorizing formulas. These thoughts

about mathematics seemed to inform her views on teaching. In describing what she thought

was the best approach to teaching mathematics, she stated

You know not just giving directions, more like asking questions you know – I think

it’s the difficult part but it [is] the most important part rather than giving them the

answers. Making sure that they [the students] are thinking…so maybe model it, give

them some manipulatives.

Ms. Wicker believed that to expose students to mathematics of this nature, the teacher

should serve as more of a facilitator, asking good questions, encouraging active

participation, and thinking. She also thought that if students were physically engaged in

what they were doing, then it was highly likely that they were learning.

I wanted everyone to have that hands-on experience, because I think they understand

by going through the process … I want them to be able to illustrate what they know

using the manipulatives and pictures so I can see that higher order thinking. I also

like the enthusiasm and willingness to interact because if they are engaged, they are

learning.

Ms. Edmonds held similar conceptions of mathematics. She thought that mathematics

involved numbers but the emphasis should be on manipulating the numbers to apply it to

your everyday life. She stated

[When I think of mathematics] Numbers come to mind and how to use numbers in

your everyday life, thinking about how numbers are important to say a strategy, and

understanding how numbers apply to a concept … it’s about trying to figure things

out. It’s interesting, fun, interactive, problem solving, something that you would use

everyday in your life.

D. I. Cross Francis

123

Ta

ble

2A

nal

ysi

sg

uid

e

Bel

ief-

Pra

ctic

e In

cons

iste

ncy

Tab

leB

elie

fSt

ated

Bel

ief

Dat

a So

urce

s: I

nter

view

s &

Vid

eo D

iscu

ssio

ns

Des

crip

tion

of

Obs

erve

d P

ract

ices

Dat

a So

urce

s: C

lass

room

O

bser

vati

ons

Nat

ure

of th

e B

elie

f-P

ract

ice

Inco

nsis

tenc

yE

xpla

nati

on o

fIn

cons

iste

ncy

Dat

a So

urce

s: I

nter

view

s an

d V

ideo

Dis

cuss

ions

Ms.

Wic

ker

Bel

ief

abou

t mat

h…

you

know

it’s

abo

ut

prob

lem

sol

ving

, it’

s m

ore

than

for

mul

as, b

ecau

se if

yo

u m

emor

ize

the

form

ulas

an

d yo

u st

ill c

an’t

use

it

outs

ide

the

clas

sroo

m, t

hen

you

can’

t hel

p yo

urse

lf.

You

nee

d to

und

erst

and

why

the

form

ula

mak

es

sens

e an

d w

here

it c

omes

fr

om.

•Fe

w s

essi

ons

[app

roxi

mat

ely

18%

] w

here

ther

e w

as

sust

aine

dw

ork

arou

nd

com

plex

pro

blem

s

Tas

k Se

lect

ion:

Low

co

gniti

vely

-dem

andi

ng

task

s vs

Hig

h co

gniti

vely

de

man

ding

task

s

Tes

ting

Cul

ture

But

they

[th

e st

uden

ts]

have

a h

ard

tim

e co

nnec

ting

wha

t the

y do

on

thei

r fi

nger

s an

d w

ith th

e m

anip

ulat

ives

to w

hat t

hey

have

to

do

on th

e te

st s

o I

have

to s

tick

wit

h m

ore

of th

e as

sess

men

t-ty

pe q

uest

ions

Bel

ief

abou

t T

each

ing

You

kno

w n

ot ju

st g

ivin

g di

rect

ions

, mor

e lik

e as

king

qu

estio

ns y

ou k

now

–I

thin

k it

’s th

e di

ffic

ult p

art

but i

t [is

] th

e m

ost

impo

rtan

t par

t rat

her

than

gi

ving

them

the

answ

ers.

M

akin

g su

re th

at th

ey [

the

stud

ents

] ar

e th

inki

ng…

so

may

be m

odel

it, g

ive

them

so

me

man

ipul

ativ

es.

•U

se o

f w

orks

heet

s w

as c

omm

on

prac

tice

[app

roxi

mat

ely

74%

of th

e se

ssio

ns

incl

uded

the

use

of

wor

kshe

ets

for

over

ha

lf o

f th

e cl

ass]

•M

anip

ulat

ives

wer

e us

ed b

ut r

arel

y to

co

nstr

uct i

deas

or

to

exte

nd m

athe

mat

ical

th

inki

ng

Stud

ent E

ngag

emen

t: B

uild

ing

idea

s th

roug

h th

e us

e of

man

ipul

ativ

es v

s.

freq

uent

use

of

wor

kshe

ets

Par

enta

l Exp

ecta

tion

s[T

he p

aren

ts]

they

look

for

wor

kshe

ets

all t

he ti

me.

I w

ant t

o be

fe

arle

ss a

nd I

wan

t to

be a

ble

to h

ave

high

exp

ecta

tion

s an

d to

be

mor

e kn

owle

dgea

ble

abou

t thi

ngs.

..I w

ant t

o te

ll a

par

ent

“I'm

not

sit

ting

here

doi

ng w

orks

heet

s al

l day

!”

Lac

k of

Res

ourc

es…

the

stud

ents

like

mak

ing

conj

ectu

res

and

wri

ting

them

on

the

boar

d an

d w

orki

ng w

ith

man

ipul

ativ

es…

We

did

have

sol

ids

[at

one

tim

e], t

hey

[pas

t stu

dent

s] w

ould

hav

e th

eir

own

soli

ds.

The

n w

e st

arte

d lo

sing

them

so

I w

as a

lway

s bo

rrow

ing,

ev

eryd

ay. S

o th

ey [

curr

ent s

tude

nts]

act

uall

y m

ade

thei

rs f

rom

pa

per,

then

like

aft

er tw

o da

ys, i

t’s

gone

!

Tim

e C

onst

rain

ts a

nd T

esti

ng C

ultu

re“…

they

hav

e a

hard

tim

e co

nnec

ting

wha

t the

y do

on

thei

r fi

nger

s an

d w

ith

the

man

ipul

ativ

es to

wha

t the

y ha

ve to

do

on

the

test

so

I ha

ve to

sti

ck w

ith

mor

e of

the

asse

ssm

ent-

type

qu

esti

ons

Bel

iefs

ab

out

lear

ning

I w

ante

d ev

eryo

ne to

hav

e th

at h

ands

-on

expe

rien

ce,

beca

use

I th

ink

they

un

ders

tand

by

goin

g th

roug

h th

e pr

oces

s…I

•M

inim

al o

ppor

tuni

ties

crea

ted

for

stud

ent

expl

anat

ion

or

just

ific

atio

n of

re

ason

ing

(see

Tab

le 1

)

Cla

ssro

om T

alk:

Tea

cher

-dom

inat

ed ta

lk v

s.

Tal

k su

ppor

tive

of s

tude

nts’

th

inki

ng a

nd r

easo

ning

Low

tea

cher

eff

icac

y“…

Lik

e I

said

, I h

ave

a ha

rd ti

me

wit

h qu

esti

onin

g, a

ltho

ugh

I th

ink

it’s

impo

rtan

t, so

I w

ould

like

hel

p w

ith

that

.”

…an

d th

ere

are

just

som

e th

ings

that

I a

m n

ot s

ure

abou

t…w

ith

Dispelling inconsistencies between beliefs and practice

123

Ta

ble

2co

nti

nued

wan

t the

m to

be

able

to

illu

stra

te w

hat t

hey

know

us

ing

the

man

ipul

ativ

es

and

pict

ures

so

I ca

n se

e th

at h

ighe

r or

der

thin

king

. I

also

like

the

enth

usia

sm

and

will

ingn

ess

to in

tera

ct

beca

use

if th

ey a

re

enga

ged,

they

are

lear

ning

.

•L

ots

of q

uest

ioni

ng(s

ee T

able

1);

si

gnif

ican

t por

tion

of

gear

ed to

war

ds f

inal

an

swer

s w

itho

ut

expe

ctat

ion

for

expl

anat

ion

soli

ds, l

ike,

ahm

, doe

s th

e co

ne h

ave

edge

s? T

he b

ook

says

one

th

ing

-no

edg

es, t

hen

I lo

ok o

n th

e in

tern

et a

nd s

ays

a di

ffer

ent

thin

g. L

ike

whi

ch is

it?

Bel

iefs

bey

ond

mat

hem

atic

sI

thin

k it

’s v

ery

hard

wit

h pr

imar

y st

uden

ts [

to e

ngag

e in

in

quir

y]…

just

in g

ener

al th

ere

com

es a

poi

nt w

hen

you

just

ha

ve to

giv

e th

em th

e in

form

atio

n or

sev

eral

kid

s ju

st g

et le

ft

behi

nd.

The

y [p

rim

ary

stud

ents

] do

n’t h

ave

the

voca

bula

ry to

ex

plai

n…li

ke it

mig

ht b

e in

you

r [t

heir

] he

ad b

ut if

you

[th

ey]

don'

t hav

e th

e w

ords

to c

omm

unic

ate

it th

en th

at's

a pr

oble

m…

like

I s

aid

for

70 p

erce

nt, t

he v

ocab

ular

y is

just

not

th

ere.

Ms.

Edm

onds

Bel

iefs

ab

out M

ath

[Whe

n I

thin

k of

m

athe

mat

ics]

Num

bers

co

me

to m

ind

and

how

to

use

num

bers

in y

our

ever

yday

life

, thi

nkin

g ab

out h

ow n

umbe

rs a

re

impo

rtan

t to

say

a st

rate

gy,

and

unde

rsta

ndin

g ho

w

num

bers

app

ly to

a

conc

ept…

it’s

abo

ut tr

ying

to

fig

ure

thin

gs o

ut. I

t’s

inte

rest

ing,

fun

, int

erac

tive

, pr

oble

m s

olvi

ng,

som

ethi

ng th

at y

ou w

ould

us

e ev

eryd

ay in

you

r li

fe.

CO

NSI

STE

NT

: Act

iviti

es

conn

ecte

d to

stu

dent

s’

expe

rien

ces

Man

y ac

tivi

ties

wer

e cr

oss-

curr

icul

ar d

evel

oped

to c

onne

ct w

ith

stud

ents

’ ex

peri

ence

s w

ith

thei

r en

viro

nmen

t. T

he a

ctiv

ity

desc

ribe

d w

as c

onne

cted

to e

nvir

onm

enta

l obs

erva

tion

s th

ey

had

mad

e in

sci

ence

cla

ss a

s w

ell a

s th

e st

ory

the

stud

ents

wer

e re

adin

g in

lang

uage

art

s.

Tim

e C

onst

rain

ts a

nd T

esti

ngI

have

to k

eep

in m

ind

the

test

ing

requ

irem

ents

in te

rms

of th

e cu

rric

ulum

and

ass

essm

ent,

som

etim

es th

at in

flue

nces

wha

t I d

o w

ith

my

kids

….w

hen

you

focu

s on

pro

blem

sol

ving

that

take

s a

long

er ti

me,

mor

e ef

fort

, mor

e ti

me

than

if y

ou ju

st g

o di

rect

ly,

this

is w

hat y

ou n

eed

to le

arn,

boo

m, b

oom

, boo

m!

Bel

iefs

ab

out

Tea

chin

g

My

mai

n fo

cus

is

unde

rsta

ndin

g…I

thin

k ha

nds-

on a

ctiv

itie

s ar

e im

port

ant,

appl

ying

it

[mat

h] to

som

ethi

ng in

th

eir

surr

ound

ings

so

they

ca

n m

ake

that

con

nect

ion.

T

hat m

eans

you

hav

e to

let

your

cre

ativ

ity

flow

…yo

u ca

n’t j

ust s

tay

rest

rict

ed to

bo

oks,

you

hav

e to

go

and

•R

egul

arly

dev

elop

ed

less

on a

roun

d st

uden

ts’

inte

rest

s an

d ex

peri

ence

s•

Tea

cher

fre

quen

tly

aske

d qu

esti

ons.

H

owev

er, m

any

wer

e to

elic

it fi

nal

solu

tions

fro

m th

e st

uden

ts.

Que

stio

ns

that

targ

eted

Cla

ssro

om T

alk:

Tea

cher

-dom

inat

ed ta

lk

vs. T

alk

supp

orti

ve o

f st

uden

ts’

thin

king

and

re

ason

ing

Low

Tea

cher

Eff

icac

yS

o qu

esti

onin

g to

me

[is

one

of m

y m

ajor

are

as f

or

impr

ovem

ent]

…it

's v

ery

diff

icul

t and

I'm

ser

ious

, lik

e I

stil

l ha

ven'

t mas

tere

d li

ke th

e fo

rm o

f qu

estio

ning

that

will

hel

p th

e st

uden

ts.

One

of

the

thin

gs I

not

ice

is th

at, I

mig

ht g

ive

inst

ruct

ions

, but

th

ey d

on’t

list

en…

Whe

n th

ey s

how

me

thei

r w

ork,

wha

t I n

otic

e it’

s th

at it

is n

ot w

hat I

am

ask

ing

for.

So

som

etim

es I

hav

e to

sh

ow th

em w

hat I

am

ask

ing

for,

I n

otic

ed th

at in

the

less

on.

D. I. Cross Francis

123

Ta

ble

2co

nti

nued

try

to f

igur

e ou

t how

is th

is

less

on im

port

ant t

o th

is

chil

d an

d ho

w a

re th

ey

goin

g to

con

nect

it o

nce

they

leav

e yo

ur

clas

sroo

m…

they

hav

e to

ta

ke o

wne

rshi

p of

thei

r le

arni

ng a

nd w

hat t

hey

lear

n.

stud

ents

’ th

inki

ng

and

reas

onin

g w

ere

ofte

n an

swer

ed b

y th

e te

ache

r.

Bel

iefs

ab

out

Lea

rnin

g

The

y m

ust b

e tr

ying

to

unde

rsta

nd w

hat t

hey

are

doin

g, f

igur

ing

thin

gs o

ut

by th

emse

lves

and

not

just

m

emor

izin

g or

bei

ng to

ld

but a

ctua

lly

tryi

ng to

fin

d a

solu

tion

or

answ

er to

the

prob

lem

that

is th

ere

in

fron

t of

them

….

•M

inim

al

oppo

rtun

ities

for

st

uden

ts to

in

depe

nden

tly

mak

e se

nse

of

mat

hem

atic

al id

eas

Cla

ssro

om T

alk:

Tea

cher

-dom

inat

ed ta

lk

vs. T

alk

supp

orti

ve o

f st

uden

ts’

thin

king

and

re

ason

ing

Lac

k of

pre

para

tion

I w

ill s

ay th

at p

repa

ratio

n, I

don

’t th

ink…

it’s

stil

l not

as

deep

as

it n

eeds

to b

e be

caus

e m

any

tim

es w

hen

I am

in th

e le

sson

, I s

till

en

coun

ter

thin

gs th

at I

hav

en’t

pre

pare

d fo

r an

d it

just

thro

ws

me

off.

It j

ust r

eall

y th

row

s m

e of

f. S

o, it

’s li

ke w

here

do

you

go

wit

h th

is n

ow?

Wha

t que

stio

ns d

o I

ask?

Do

I go

bac

k an

d te

ach

this

?

Dispelling inconsistencies between beliefs and practice

123

Like Ms. Wicker, Ms. Edmonds also strongly advocated teaching for understanding so

students could appreciate the utility value of mathematics. To do this effectively, she tried

to anchor the mathematics in the students’ experiences, ensuring that she helped them to

apply the concepts they were learning to their own experiences.

My main focus is understanding … I think hands-on activities are important,

applying it [mathematics] to something in their surroundings so they can make that

connection. That means you have to let your creativity flow … you can’t just stay

restricted to books, you have to go and try to figure out how is this lesson important

to this child and how are they going to connect it once they leave your class-

room…they have to take ownership of their learning and what they learn.

Ms. Edmonds’ goal was for her students to understand the mathematics deeply so they

could apply it to problems that arose in their daily lives. In the excerpt below, Ms.

Edmonds described how she identified whether her students were learning.

They must be trying to understand what they are doing, figuring things out by

themselves and not just memorizing or being told but actually trying to find a

solution or answer to the problem that is there in front of them….

Both Ms. Wicker and Ms. Edmond’s stated views about mathematics learning were

similar in several aspects. They believed that students learned best by doing and through

engaging in activities rooted in their personal experiences. Memorization was not the key

component of learning; rather, students should be making sense of the ideas and concepts

and have opportunities to show what they know in multiple ways.

Perceived inconsistencies between beliefs and practice

From these initial analyses, it seemed that teachers’ beliefs aligned well with the goals of

the professional development and the kinds of beliefs that are supportive of reform-based

practices. However, based on the observations throughout the first year, their practices did

not always reflect these views. Observations of Ms. Wicker’s class showed few classroom

sessions where the students were actually solving complex problems. She would often

engage students in mathematically rich tasks, but her teaching style was generally teacher-

centered and the level of reasoning required of students tended to be low. Students were

seldom asked to explain their answers or reasoning with manipulatives or diagrams.

Although Ms. Wicker asked a lot of questions, the questions were not geared toward

eliciting or extending students’ thinking, so often the stated (and accepted) responses

tended to lack depth and justification. Students on occasion had access to manipulatives but

they were often given to the students without much guidance on how to use them to

construct ideas. Worksheets were also fairly commonplace, often used during the latter half

of the period, reinforcing content covered during the first half. The excerpt below, taken

from a classroom activity about solids, provides an example (Fig. 1).

Prior to the classroom conversation, students had completed a worksheet that required

them to identify and name the solid and plane figures shown. The worksheet had two

columns: a picture of the figure was in a box on the left side of the paper and the students

were to write the name of the figure in the empty box on the right side of the figure. The

students had discussed solids in a previous class in which they made paper solids from nets

and were introduced to the names of the solids then. The goal of the current activity was for

students to distinguish between solids and plane figures (based on whether or not the object

had volume) using vocabulary words they had learned previously. The current lesson was

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an adaptation (done by Ms. Wicker) of an investigatory activity from the kindergarten

materials of a reform-based textbook series. The adaptation as described above was less

mathematically rigorous than the original version of the activity that required students to

sort, analyze, and compare two-dimensional and three-dimensional shapes. Ms. Wicker

often adapted activities from the district-approved curriculum and other curricula to align

with her instructional goals. In some cases, the adaptations she made were creative and

allowed for better connections with the students’ experiences. In other situations, like the

case describe above, her modifications lowered the cognitive demand of the task.

In the excerpt, Ms. Wicker asked the students to determine sameness or difference

between a plane figure and a solid by using ‘‘synonyms’’ for ‘‘thick’’ and ‘‘thin.’’ The

students provided several other words as synonyms to describe each of the objects shown.

This excerpt was selected as it is noteworthy in two distinct ways. The first is that Ms.

Wicker is the only person in the class with manipulatives. The students did not use them to

complete the activity nor were they referring to any during the discussion. Second, the

questioning does not encourage students to think about what the descriptors mean with

respect to the figure or object. Students could simply state synonyms for fat and flat without

really understanding the fundamental difference between the sphere and the two-dimen-

sional representation of the sphere—the circle drawn on the sheet of paper. Using a

diagram of a circle on a sheet of paper is problematic in this case. Expecting children to

think of objects that have no volume by creating a physical representation on paper creates

potential misconceptions because the paper itself has volume—if the object can be held, it

has volume. Although the standards do include analyzing and comparing two- and three-

dimensional objects, careful thought must be given to how to appropriately convey to

Ms. Wicker: Why aren’t they the same thing [holding up a spherical object and a circle]? Student: Because one is fat and one is flatMs. Wicker: Ok, because one is flat and one is fat. Anything else? Ariel?Ariel: Because one is skinny and one is bigMs. Wicker: Ok, I wouldn’t say skinny and big, ok. What is a vocabulary word that we have been

using?

[Students raise their hands. Ms. Wicker points to Gabby]

Gabby: One is thick and one is thinMs. Wicker: Which one is this [Holding up the sphere]?Sts[together]: Thick!Ms. Wicker: Ok. What is this? [Holding up the circle]Sts: Flat! Flat! Skinny!Ms. Wicker: If this is thick [holding up the sphere], then what is this? [holding up the circle]Sts: ThinMs. Wicker: Ok. What is another word for thin?

[Students raise their hands]

Ms. Wicker: Natalie?Natalie: FlatMs. Wicker: Ok. Let’s go to our second shape. What is that?Sts: PrismMs. Wicker: Look in the second box. The name should be in the second box.

Fig. 1 Excerpt of transcript from Ms. Wicker’s class

Dispelling inconsistencies between beliefs and practice

123

young students the distinction between an object with volume and a figure that does not.

From these responses, it was difficult to determine whether or not the students understood

this difference or could connect the terms ‘‘thin,’’ ‘‘flat,’’ and ‘‘skinny’’ to the object having

no space inside it or whether they could categorize all ‘‘thick’’ objects as solids. Given the

conceptual complexity of the distinction, most likely not.

These observations were particularly striking given Ms. Wicker’s earlier statements:

I wanted everyone to have that hands-on experience with the material,

But I also want them to be able to illustrate what they know using the manipulatives

and pictures so I can see that higher order thinking.

[Quality teaching is] You know not just giving directions, more facilitating. More so

like asking questions you know ….Making sure that they [the students] are thinking,

that they understand the process and procedure of it so maybe model it, give them

some manipulatives.

Similar to the observations of Ms. Wicker’s classes, there were also inconsistencies

between Ms. Edmonds’ beliefs and her teaching. The excerpt below (Fig. 2) was taken

from a cross-curricular lesson Ms. Edmonds taught to her kindergarteners (see Cross et al.

2012) for a full description of the lesson).

Prior to this lesson, the students had read a story as a part of their language arts class

about Farfallina, a caterpillar who changes into a butterfly. The students were given pic-

tures of Farfallina, with each body segment the shape of a hexagon. Students were

expected to manipulate different pattern blocks (hexagons, rhombi, triangles, and squares)

to cover the caterpillar in an effort to develop their spatial reasoning skills. In the excerpt

above, one student, Racquel, used triangles to make her caterpillar. Ms. Edmonds guided

her in figuring out how many triangles would make the caterpillar. Her interaction with

Misha however was quite different. Instead of allowing her to figure out whether squares

would fit within the border of the hexagon, she told her they would not fit. She confirmed

this with the class, then proceeded to tell the students what other shapes would not fit.

Based on the incoherent mumbles, it would appear that most students had not yet figured

Ms. Edmonds: Ok Racquel, how many shapes did you use to cover your caterpillar?Racquel: I used trianglesMs. Edmonds: So how many did you use?Racquel: Ahh…Ms. Edmonds: Ok, let’s count them…countRacquel: One, two, three [both Racquel and Ms. Edmonds count together]……Twenty-fourMs. Edmonds: Very good. You have twenty-four triangles.

[Ms. Edmonds walks around the class looking at other students’ work]

Misha: Look at mineMs. Edmonds: Yes, Misha…now look at your shapes. You can’t use squares, squares won’t fit. Will

squares cover your caterpillar class?Sts: [Students mumble]Ms. Edmonds: No, they won’t. Squares won’t fit on your caterpillar. What other shape won’t fit?Sts: [Students mumble]Ms. Edmonds: The tan shape, the other diamond

Fig. 2 Excerpt from Ms. Edmond’s class

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this out. By telling Misha and the other students that the square would not fit within the

hexagonal space, Ms. Edmonds essentially eliminated the opportunity for them to figure

this out for themselves through the trial-and-error process of physically manipulating the

object. This physical trial-and-error process was essential for students to develop their

abilities to mentally manipulate shapes to fit in desired spaces, thereby advancing their

spatial reasoning skills (Battista 2007; Clements et al. 2004).

Trying to reconcile Ms. Edmonds’ actions was difficult given statements she had made

during earlier conversations.

…just don’t present things where the student has to memorize because they will

forget it and not use it. It has to have an impact on them and they have to take

ownership of their learning and what they learn…

They must be trying to understand what they are doing, figuring things out by

themselves and not just memorizing or being told but actually trying to find a

solution or answer to the problem that is there in front of them…

Based on these statements, I found some of her teaching actions during the class to be

misaligned with her stated goals for mathematics teaching and her descriptions of how she

thought students learned best. In our conversations, there had been such an emphasis on the

importance of students’ thinking and figuring things out, that it was particularly surprising

that she simply ‘‘told’’ Misha the squares would not fit, instead of allowing her to ‘‘figure it

out.’’ I highlight the act of telling in this particular classroom episode for two reasons. One,

it was a prevalent teaching action in Ms. Edmond’s classes (approximately 788 % of the

videos showed ‘‘telling’’—see Table 1). Two, this particular teacher action most directly

contradicted how she described her beliefs about quality mathematics teaching and how

students learn best.

Exploring inconsistencies

Based on these observations, the teachers’ statements and actions seemed to be incon-

sistent. However, I saw this as an opportunity for exploration, to better understand the

broader set of beliefs that could be influential in the teachers’ decision making and

behavior. Data from years two (video discussions and interviews) and three (interviews)

were particularly revealing. Based on these conversations, I was able to better understand

the nature and extent of the influence of non-mathematics beliefs and contextual factors on

the teachers’ classroom behavior (see Table 2).

In reference to Ms. Wicker, I focus here on her decision-making process as it related to

the use of manipulatives, teacher questioning, and facilitating classroom conversations. My

goal was to identify reasonable explanations for the prevalence of worksheets, acceptance

of responses lacking explanation, and the general absence of student–student talk. Across

our numerous conversations, Ms. Wicker frequently referred to factors that influenced how

she organized her classroom—one major factor being parents. In a conversation that began

with a discussion about the happenings of her day, she discussed her arduous relationship

with parents and how this related to her long-term goals with respect to her teaching.

8 The telling code was defined as declarative statements solely for the purpose of providing information tostudents. This code was applied differently from revoicing and extending, which were also used fordeclarative statements made by the teachers but for which the purpose of the statements was to drivestudents’ thinking.

Dispelling inconsistencies between beliefs and practice

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I would just like to be fearless, like I think you’re under so many constraints you

know even like parent expectations, like sometimes you have to buckle … they look

for worksheets all the time. I want to be fearless and I want to be able to have high

expectations and to be more knowledgeable about things … I want to tell a parent

‘‘I’m not sitting here doing worksheets all day!’’ … Also, like I said, I have a hard

time with questioning, although I think it’s important, so I would like help with

that…

It is clear that Ms. Wicker’s perceptions of parental expectations influenced how she

planned her lessons. However, although Ms. Wicker thought that parents were sometimes

overbearing, she also thought it was important to safeguard relationships with the few

parents who actively engaged in the education process. Responding to questions about the

parent–teacher relationship in a joint interview, she concurred with Ms. Edmonds’

statement about parent involvement, ‘‘It makes me feel good to know that a parent cares

enough to come to school and sit with her child everyday … I know every parent can’t do

that … you know, she just makes me grateful for the few that do come.’’

Given the particular school context where the desires of parents were valued, especially

those who were actively involved, she thought it best to align her practice with those

expectations. It is however clear that this was not her preferred way of organizing her

classroom as she hoped that one day she would be ‘‘fearless,’’ bold enough to instruct in the

way she thought best supported students’ learning. In our prior conversations and obser-

vations, the teachers had not described the value they placed on parent–teacher relation-

ships, nor had we observed parent–teacher interaction. Although the clustering of beliefs

allows for the contextualization of beliefs, the individual rarely states these limitations or

qualifications to beliefs when beliefs are being described. Individuals tend to describe

beliefs as absolutes, rarely including exceptions to the rules. In this regard, Ms. Wicker saw

it as sensible to act in ways that would please those with power and influence within her

school and over her students.

Ms. Wicker also talked about pedagogical skills she valued but still struggled with.

Specifically, she pinpointed questioning as a skill she highly valued but by which she was

still challenged and about which she wanted to be more knowledgeable. Research studies

(Frykholm 2004; Smith 1996) have reported on the impact of teacher efficacy on teachers’

abilities and willingness to enact reform-based practices. The teacher-directed style often

serves as a mask for teachers who do not have (or do not believe they have) deep

knowledge of the content, students, or teaching (Artz and Armour-Thomas 1999). In this

regard, it appeared that Ms. Wicker’s low sense of efficacy in her questioning was a factor

in the pervasive initiate–respond–evaluate, teacher–student interaction style in her class-

room. Her low efficacy was often explicit in the video discussions. In the excerpts below,

she was more explicit about the content she was unsure about,

…and there are just some things that I am not sure about…with solids, like, ahm,

does the cone have edges? The book says one thing - no edges, then I look on the

internet and says a different thing. Like which is it?

…I have always referred to that shape [rhombus] as diamond … I didn’t know that

wasn’t a geometry name until the PD…

Ms. Wicker also described how the prevalent testing culture in the school impacted how

she designed her lessons. She explained,

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My favorite activities are when they are away from paper and pencil, when they are

using their hands to create things that they already have some knowledge about.

[This way] I am able to see their growth. With the manipulatives I can kind of get a

glance at what they’re thinking about.

However, despite the fact that she found manipulatives valuable for learning and for

formative assessment purposes, she faced challenges in using them during class. First,

there was the issue of availability; second, she had difficulties getting students to connect

their hands-on work with the paper-and-pencil problems. Below, she described these

challenges.

…the students like making conjectures and writing them on the board and working

with manipulatives. But they have a hard time connecting what they do on their

fingers and with the manipulatives to what they have to do on the test so I have to

stick with more of the assessment-type questions … We did have solids [at one

time], they [past students] would have their own solids. Then we started losing them

so I was always borrowing, everyday. So they [current students] actually made theirs

from paper, then like after two days, it’s gone!

From Ms. Wicker’s statements above, we see that she highly valued inquiry-based

teaching, students’ conjecturing, and using manipulatives as meaningful aspects of

teaching. They show that due to the lack of resources (manipulatives in this case), she did

not consistently use them in her lessons. Also, similar to her statements above, in several of

our conversations, she talked about the constraints of testing on her teaching. In the

excerpt, she discussed how the pressures of preparing the students for achievement tests

had influenced how she taught and the types of tasks she assigned her students. She spoke

specifically about her struggles with getting students to show what they know and

understand on paper. To address this challenge, she gave more assessment-type questions

(usually in the form of worksheets) as a way of attending to time constraints and test

preparation. Here, we see a situation where factors within the environment deem it non-

conducive to the actualization of particular belief(s). In this case, the lack of content

knowledge and lack of resources made it difficult to enact her beliefs. Keeping in mind that

teachers are considered to be ‘‘sensible beings’’ and therefore would not deliberately act in

contradictory ways, I also consider the sessions (56 % of the observed sessions—see

Table 1) where she did utilize the available manipulatives as evidence that the belief was

held and instances where the context supported enactment of the belief.

In addition to the above-mentioned factors, non-mathematics beliefs also took prece-

dence in certain classroom situations. Although Ms. Wicker thought inquiry-based

teaching was a meaningful instructional approach, she questioned its utility for her first-

grade students.

I’ve attempted [to incorporate inquiry-based practices in my teaching]! ‘‘For me, I

like the [the students’] enthusiasm and the willingness to explore and inter-

act … [now they have] a more specific understanding of what it means to investigate

and explore … I enjoyed that! … but, I think it’s very hard with primary students [to

engage in inquiry] … just in general there comes a point when you just have to give

them the information or several kids just get left behind.

In this statement, Ms. Wicker was referring to her attempts to take an inquiry-based

approach to teaching both science and mathematics. Although she thought her students

were more engaged when she used this approach, she had reservations about whether first

Dispelling inconsistencies between beliefs and practice

123

graders could make complete sense of the ideas on their own and thought it often required

her to step in and give them the information. Her beliefs about the capabilities of her

students seemed to also influence how she engaged her students in discussions.

They [primary students] don’t have the vocabulary to explain … like it might be in

your [their] head but if you [they] don’t have the words to communicate it then that’s

a problem … like I said for 70 %, the vocabulary is just not there.

She held beliefs about the abilities of her first-grade students (young students cannot

handle inquiry and they have limited vocabulary), and mathematical justification

(justification can be solely verbal), which were foregrounded in her actions. According

to Leatham (2006), ‘‘certain beliefs have more influence over certain actions in certain

contexts’’ (p. 96). In this context, her beliefs about primary students seemingly took

precedence in the sessions we observed. These examples illustrate that Ms. Wicker’s

beliefs and observed practices were not inconsistent, instead the actions I observed were

not aligned with the beliefs I had initially inferred.

Similar to Ms. Wicker, initially Ms. Edmonds’ mathematics-related beliefs and teaching

actions appeared to be inconsistent. In general, Ms. Edmonds’ lessons were very creative,

they tended to be well-connected to the students’ prior knowledge and experiences and

almost always incorporated the use of manipulatives. For example, the lesson described

above, which was fairly typical, was connected to the students’ prior explorations of the

life cycle of a butterfly and of polygons, and incorporated the use of pattern blocks. In her

descriptions of how she thought students learned best and ways she tried to support

students’ learning, there was an emphasis on providing opportunities for students to ‘‘figure

things out.’’ However, in many of the lessons we observed, instead of allowing students to

grapple with concepts and ideas, she often told students the answers or what they should do

to figure out the problem. Ms. Edmonds was also asked to state teaching skills she thought

she needed to acquire or improve in the long term. In her response, she stated

So questioning to me [is one of my major areas for improvement] … it’s very

difficult and I’m serious, like I still haven’t mastered like the form of questioning that

will help the students.

Ms. Edmonds recognized that questioning was an important part of teaching but admitted

that she found it difficult and still struggled to formulate and ask questions that would be

beneficial to students as they were solving problems. Observations of Mrs. Edmonds

teaching confirmed this. In addition to her low sense of efficacy with respect to teaching,

there also appeared to be teacher- and student-related beliefs as well as contextual factors

that impacted the implementation of her lessons. Commenting on her own practice in one

of the video discussions, she stated

I think I am still a little hard on my students … they have great potential and it’s just

some of them are just not listening and not focused … One of the things I notice is

that, I might give instructions, but they don’t listen … When they show me their

work, what I notice it’s that it is not what I am asking for. So sometimes I have to

show them what I am asking for, I noticed that in the lesson.

There may be several reasons that students are seemingly not following instructions. For

example, the instructions may not be clear or they may not understand the task so their

work reflects lack of focus. However, irrespective of these possibilities, what is key here is

how Ms. Edmonds interpreted and responded to these student actions. From her

perspective, her instructions were not always followed so she resorted to telling students

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what to do so the task or activity could be completed. What would have been informative

in this regard would be to know Ms. Wicker’s belief about how to deal with indolent

students. If so, I could determine whether there was also another belief taking precedence

in this regard.

During one of the video discussions, Ms. Edmonds elaborated on ways she tried to

improve her teaching. She identified some teacher-related factors that influenced how she

interacted with her students. She stated,

I will say that preparation, I don’t think … it’s still not as deep as it needs to be

because many times when I am in the lesson, I still encounter things that I haven’t

prepared for and it just throws me off. It just really throws me off. So, it’s like where

do you go with this now? What questions do I ask? Do I go back and teach this? I

realize they don’t know this, so do I need to just give them what they need at this

moment to finish the task because sometimes when they don’t get it, like the hexagon

thing [referring to fitting the square into the hexagonal shape], I was trying to explain

and it lasted like, well over time. Just way over time and I was concerned that the

whole lesson may be lost.

In this excerpt, Ms. Edmonds identified two teacher-related factors that influenced her

questioning and how she interacted with her students in the classroom. Although she spent

time thinking deeply about the content, making efforts to connect the concepts to students’

prior knowledge, she still thought her preparation was lacking. Moreover, she thought that

because of this lack of adequate preparation, she was often at a loss about how to

appropriately respond to students’ comments and for good questions to ask. Similar to Ms.

Wicker’s situation, described above, there were factors restricting enactment of Ms.

Edmond’s belief that students should be ‘‘figuring things out … and not just be told.’’

In addition to feeling unprepared and perplexed in those situations, she was often

pressed for time so in order for the ‘‘… whole lesson [not] to be lost,’’ she ensured that all

the students had solved the problem or completed the activity. This often meant that she

defaulted to telling the students what to do. Based on her statements, Ms. Edmonds’ also

believed that a successful lesson was a completed lesson. This excerpt also showed that

time played a critical role in Ms. Edmonds’ pedagogical decisions. Similar to Ms. Wicker,

time constraints and testing were significant factors impacting her pedagogical decisions.

In one of the video discussions about a lesson on number she had taught, Ms. Edmonds

tried to reconcile her beliefs about teaching and her practices observed in the video. In the

excerpt below, she explained why she sometimes taught in ways that did not align with

what she thought was in the best interest of her students.

I have to keep in mind the testing requirements in terms of the curriculum and

assessment, sometimes that influences what I do with my kids … when you focus on

problem solving that takes a longer time, more effort, more time than if you just go

directly, this is what you need to learn, boom, boom, boom! There are days where I

have to teach the curriculum based on them being able to master that assessment on

paper because the way they present it on paper is different than the problem based

learning, if that makes sense

Ms. Edmonds valued inquiry-based approaches to instruction and found it meaningful to

have students talk about their thinking and strategies to solving problems. However, she

admitted that although effective, she believed inquiry approaches to be time-consuming

and the types of tasks and thinking the students were engaged in were not represented in

their state assessments. As such, she often felt there was an ongoing tug-of-war between

Dispelling inconsistencies between beliefs and practice

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meeting the demands of the assessment and meeting her students’ academic needs. Within

the time restraints of school day, Ms. Edmonds opted for the least time-consuming

instruction approach—a ‘‘sensible’’ decision given the context.

Discussion

Drawing on the tenets of the sensible systems framework and utilizing both contextualized

and de-contexualized data sources, I observed that teachers’ practices did not consistently

align with the teachers’ stated beliefs, the belief to which the researcher attributed the

practices. Rather, the observed practices were aligned with other sets of beliefs and were

the results of teacher- and school-based factors and constraints that were overlooked by the

researcher—some of which seemed to align with other beliefs that strengthened the

influence of the factor in the situation In this section, I summarize the findings drawing

connections with other relevant research, make suggestions for future research in the area,

and discuss the theoretical, methodological, and practical implications of this study.

‘‘Belief-aligned’’ personal and external factors

Low sense of efficacy and the teachers’ perceived lack of preparation were two factors that

impacted how the teachers taught and interacted with their students. Research (Guskey

1988; Stein and Wang 1988) suggests that teachers who have low efficacy are less open to

trying new ideas and implementing innovative teaching strategies. Although the teachers

believed that having students figure things out helped them make sense of mathematical

ideas, the way in which they implemented the tasks and scaffolded the students tended to

decrease the cognitive demand of the tasks and the students’ agency in the problem-solving

process. Exploring these issues in the video discussions and through more directed ques-

tioning confirmed that the teachers did in fact value these practices but admittedly

struggled with enacting them in desired ways because they were ill-equipped or unpre-

pared. These actions (or inactions), resulting from the teachers’ low sense of efficacy, was

misinterpreted in the initial analyses and considered evidence of a contradictory belief–

action relationship. Two issues are of note here. The first is that the teachers self-identified

these issues as deterrents to teaching in desired ways—this self-awareness bode well for

professional development work focused on teacher change. Second, although the teachers

were cognizant of these factors, they did not express them until the relevant situation arose

within the context of the video discussions. The video discussions were invaluable as a tool

for building trust and for procuring contextualized data.

In addition to psychological factors, there were factors external to the teachers that

impacted how they engaged students. In particular, the teachers identified time, testing

concerns, and the influence of parent expectations as constraints and I observed that they

impacted the teachers’ lesson planning and implementation in ways that were not con-

sistent with their descriptions of practices they valued. Given the high priority placed on

test scores, the teachers’ goals of covering the content were foregrounded over their beliefs

about what was best for optimal student learning. Some of these factors have been iden-

tified by others (e.g., Herbel-Eisenmann et al. 2006); however, in this study, I also

observed that there were existing beliefs that served to increase the influence of those

factors. Specifically, both teachers believed that it was important to nurture parent–teacher

relationships so they were likely to acquiesce to parents if they had concerns or demands.

Additionally, Ms. Edmonds believed that ‘‘a good lesson was a lesson that was completed

D. I. Cross Francis

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in the allotted time’’ and that ‘‘a problem-solving approach takes more time than more

traditional methods’’ so if pressed for time, as she often was, it is likely that she would act

in accordance with these latter beliefs. In this regard, it would be ‘‘sensible’’ that if there

were strict time constraints, then you use the less time-consuming instructional approach

especially if you have doubts about your capabilities to effectively implement more stu-

dent-centered strategies. This would suggest that even in cases where it appears that

contextual or environmental factors are exerting the greatest influence, that there is a belief

with which this factor aligns.

Beliefs beyond mathematics

Both Skott (2001) and Sztajn (2003) draw attention to the ways in which beliefs not

directly related to the teaching of mathematics play a role in teachers’ classroom behavior.

Both teachers had beliefs about their students that informed their decisions of how to teach

and interact with them. They believed that their students lacked some essential knowledge

(vocabulary) and skills (listening) essential to being successful in school, so they adapted

their instruction in ways they deemed appropriate. Their beliefs about younger students

were exceptions to their beliefs that students should figure things out instead of being told,

and teachers should not just give answers but engage students with questions. It was

difficult to determine whether these exception beliefs were specific to their kindergarteners

and first graders or whether they were more generalized beliefs about young children.

Nevertheless, these beliefs were central in the teachers’ decisions about when and how to

give information to students and the expectations they had for students’ responses.

In this regard, both Green’s (1971) notion of belief clusters and Leatham’s (2006)

sensible systems framework are useful for exploring this phenomenon. Positing that

teachers are complex, intelligent beings who act in ways that make sense to them, and that

beliefs cluster and can exist in isolation of each other, they encourage researchers to move

beyond the specific espoused (researcher-attributed) beliefs under investigation (in this

case, mathematics beliefs) to explore teachers’ broader beliefs network (including other

self-beliefs and beliefs beyond mathematics), the microsystem (factors closely related

connected teachers’ immediate school environment) and macrosystem (factors closely

connected to the wider environment) factors that impact their teaching, and the beliefs that

strengthen the influence of these factors. Methodological, theoretical, and practical

implications of these findings are discussed below.

Implications

Commenting on the findings of past beliefs research, Philipp (2007) wrote ‘‘…as a research

stance in studying teachers and their beliefs, [I propose] we researchers assume that

contradictions [between beliefs and practice] do not exist’’ (p. 276). In alignment with

Leatham’s approach, this stance suggests that concluding that there are inconsistencies

between teachers’ beliefs and practices is untenable. Several studies to date have high-

lighted the influence of external factors on teachers’ actions, thereby making the assertion

‘‘beliefs are precursors to actions’’ debatable. Based on the findings of this research that

also identified several of these factors, it would suggest that with further investigation,

beliefs that align with many of these factors might be uncovered. Therefore, from a

theoretical standpoint, if we hold that ‘‘beliefs are precursors to actions,’’ then we must also

hold that all teachers’ actions are aligned with a belief or set of beliefs. Given this, it would

Dispelling inconsistencies between beliefs and practice

123

be contradictory to then say that teachers’ beliefs and actions are incongruous. It would be

more accurate to state that teachers’ practices do not align with teacher-stated (researcher-

attributed) beliefs, but they may be aligned with other sets of beliefs that the researcher

overlooked. Thus, the onus is on the researcher to conduct further investigation to deter-

mine the roots of observed practices.

A second implication for theory and methodology aligns with the work of both Green

(1971) and Hoyle (1992). Indeed, some beliefs are more central than others; however,

particular beliefs become foregrounded relative to the context. Additionally, I observed

that teachers are complex individuals with complicated belief systems, and that actions are

influenced by a multiplicity of factors that sometimes act simultaneously. As such, how an

individual acts cannot be determined by examining one aspect of his psychological world

in limited contexts. Therefore, it behooves beliefs researchers to expand the scope of their

investigations to include multiple contexts, examining the role of macro- and microfactors

on instruction. These investigations require time, depth, and patience to uncover what may

not be explicit to the participant or initially identifiable by the researcher. In this regard,

data on beliefs should be collected through sources where the beliefs are tied to the

practices of interest. Video discussions or video interviews (similar to the work of Sherin

and van Es (2005, 2009) have proven to be particularly useful in the study of beliefs. As

Shavelson et al. (1986) stated, ‘‘Teachers’ decisions are not one-and-for-all, rather, they are

made incrementally and adjusted [based on information provided in the current con-

text]…’’ (p. 77). It is therefore only through long-term, in-depth investigation of teachers’

thoughts and actions across classes, students and contexts that we will yield results that will

allow us to make more accurate inferences that will enhance our work with teachers.

Understanding of the beliefs and factors that influence individual teachers’ practices can

greatly inform how we design our work with teachers to specifically meet their needs. This

aligns with prior work (Cross 2009), which concluded that the ‘‘one-size-fits-all’’ profes-

sional development is only marginally effective, and proposed more individualized pro-

fessional development to foreground beliefs that are more conducive to reform-based

practices. Although quite costly, for professional development to be effective, it must be

designed to directly address teachers’ needs, both practically and psychologically. In this

regard, for teachers who hold beliefs supportive of student-centered instructional practices,

our goal should be to answer the question, ‘‘How can we help teachers foreground these

beliefs when there exists conflicting beliefs or environmental constraints?’’ (Fang 1996,

p. 59). The approach applied in this study allowed for the identification of the multiple

factors influencing the teachers’ actions, as well teacher-identified deterrents to become

visible. Both these insights bode well for professional development work on teacher

change.

A second practical implication relates to belief change. With the insights garnered from

these extended explorations, we can determine whether belief change is warranted. For

example, Ms. Wicker believed that inquiry-based approaches were meaningful ways to

engage students in mathematical thinking. Simultaneously, she also thought that given the

time constraints and her students’ prior knowledge and ages, these approaches were not

optimal for her students. Given that this latter belief was most influential in her teaching

decisions, we could carefully design professional development work to help foreground the

mathematics-related beliefs (the former) she held and provide strategies that help to

address the contextual constraints. Although in some instances, steps toward belief change

are warranted, in cases such as Ms. Wicker, professional development can be designed to

help teachers reorganize their beliefs so beliefs that support conceptual understanding

become central in instructional decision making. It is only through first understanding

D. I. Cross Francis

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teachers’ beliefs systems and the motives behind their actions, then using these data to

inform the design of PD that we will see more long-term, sustained implementation of

reform-based practices.

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