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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
D. I. Cross Francis
123
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
D. I. Cross Francis
123
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
123
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,
D. I. Cross Francis
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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
D. I. Cross Francis
123
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
123
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
123
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
123
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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