Volume 8, Number 19        July 20, 2005   ISSN 1099-839X 
 

 

Elaborating a Change Process Model for  

Elementary Mathematics Teachers’ Beliefs and Practices 

 
Stephanie Z. Smith 

Georgia State University  

 

Marvin E. Smith 

Georgia Southern University 

 

Steven R. Williams  

Brigham Young University 

 
This report focuses on the processes of change in beliefs and practices 

experienced by practicing elementary school teachers during a sixteen-session 

course using two of the modules from the Developing Mathematical Ideas 

(DMI) materials. We identify a collection of six metaphors for knowing, 

learning, and teaching mathematics to succinctly describe and categorize 

teachers’ beliefs. We present three case studies representing a continuum of 

change in beliefs observed among the participants of the DMI course. We relate 

this continuum both to the beliefs teachers brought to the course and to their 

degree of engagement in various components of a change process model. Using 

this model of change, we analyze the ways in which teachers expressed interest 

in change, problematized their beliefs, experimented with possible solutions, and 

reflected on experimental results leading to changes in beliefs and practices. The 

results of our analysis indicate that variations in change among participants can 

be explained by variations in their levels of engagement in particular elements of 

the change model by the learning activities of the DMI course.  

Paradigmatic and systemic changes 

recommended in the standards of the National 

Council of Teachers of Mathematics (NCTM, 2000) 

and supported by teacher enhancement grants from 

the National Science Foundation involve complex 

issues associated with teacher beliefs, teacher 

knowledge, and changes in teaching practices. Friel 

and Bright (1997) integrated and summarized a 

number of issues of particular importance for these 

changes, including: (1) working to change teacher 

beliefs is the starting point for most professional 

development, and (2) changing beliefs and 

experimenting with teaching practices are intertwined 

in an iterative process.  

In discussing programs that are successful in 

changing beliefs, Richardson (1996) singled out 

“programs that approach learning to teach in a 

constructivist manner” as being particularly 

successful in “engaging their participants in 

examining and changing their beliefs and practices” 

(p. 113). Among the common features of successful 

constructivist professional development programs 

 

 

 

 

 

 

 

 



Current Issues in Education Vol. 8 No. 19 
 
which she listed, two are of particular importance for 

this study: (1) participant’s beliefs and 

understandings are a major concern in the program, 

and (2) the goal of the program is to facilitate 

participants’ understanding of their beliefs as well as 

their considering and experimenting with new beliefs 

and practices. Richardson (1996) also noted, 

“attitudes and beliefs are important concepts in 

understanding teachers’ thought processes, classroom 

practices, change, and learning to teach” (p. 102).  

In the context of an innovative elementary 

mathematics professional development program 

using two modules from Developing Mathematical 

Ideas (DMI) (Schifter, Bastable, & Russell, 1999a 

and 1999b), this study (1) provides a framework for 

organizing, describing, and analyzing teachers’ 

beliefs about knowing, learning, and teaching 

mathematics; (2) explores a sequence model for the 

processes of teacher change; and (3) reports effects of 

a specific teacher development program on 

participating teachers’ beliefs and practices.    

Conceptual Framework 

As we frame the discussion that follows, we 

draw from research on teachers’ beliefs, propose 

metaphors that summarize teachers’ beliefs and 

mental models for teaching mathematics, and 

summarize research on the processes of changing 

beliefs through professional development 

experiences. This literature frames our decision to 

use case-study methodology to study the effects of 

the course on teachers’ beliefs and practices as well 

as our presentation and interpretation of the results. 

Teachers’ Beliefs and Practices  

Although it seems obvious that teachers’ 

personal beliefs and experiences effect how they 

teach, Thompson (1992) pointed out that there are 

numerous ways of interpreting the idea of a belief. 

We focus here on the beliefs of teachers in our study 

in three areas: their views of what it means to know 

mathematics, to learn mathematics, and to teach 

mathematics.  

A useful way of talking about such beliefs is 

the approach taken by Bullough (1992) and his 

associates (Bullough & Stokes, 1994). Bullough 

discussed metaphors as mental images or models that 

briefly summarize the elaborate and complex 

theories, assumptions, and understandings upon 

which people act. He used the notion of personal 

teaching metaphors as one image that helps teachers 

discuss and reflect upon their assumptions about 

teaching and their roles as teachers. Bullough & 

Stokes (1994) discussed at some length one particular 

metaphor of a teacher as “‘one who knows,’ or 

master” and students as “disciples, imitators of higher 

authority.” They went on to suggest that “for the 

master, teaching is telling,” knowledge is “fixed and 

stable,” and the teaching and learning process is “a 

matter of compelling others to dance to the beat of 

the master’s drum” (p. 201).  

This master teaching metaphor is coherent 

with traditional views of knowing and learning 

mathematics. Kuhs and Ball (1986) summarized this 

view as follows: “rules are the basic building blocks 

of all mathematical knowledge and all mathematical 

behavior is rule-governed; knowledge of mathematics 

is being able to get answers and do problems using 

the rules that have been learned; computational 

procedures should be automatized; it is not necessary 

to understand the source or reason for student errors; 

further instruction on the correct way to do things 

will result in appropriate learning; in school, knowing 

mathematics means being able to demonstrate 

mastery of the skills described by instructional 

objectives” (p. 2, bullets removed). 

This description has much in common with 

Lampert’s (1990) description of the cultural 

assumptions about mathematics in schools: “doing 

mathematics means following the rules laid down by 

the teacher; knowing mathematics means 

remembering and applying the correct rule when the 

teacher asks a question; and mathematical truth is 

determined when the answer is ratified by the 

teacher” (p. 32). Discussions by Stigler and Hiebert 

(1999) and Ma (1999) suggest that this traditional 

view still dominates the culture of mathematics 

teaching. 

In contrast to the master metaphor, research 

from a social constructivist perspective (Vygotsky, 

1978; Bakhtin, 1981; Rogoff & Wertsch, 1984; 

Wertsch, 1985, 1991; Tharp & Gallimore, 1989; 

Rogoff, 1990) provides an alternative model for 

teaching centered on joint productive activity and 

instructional conversation (Tharp, 1997; Dalton, 

1998). This view of teaching, which could be 

summarized by a facilitator metaphor, is founded on 

alternative beliefs about what knowledge is, how it is 

learned, and the role of teachers in guiding and 

facilitating learning. Learning from this perspective 

involves social interactions among students and more 

capable others (teachers and peers) working together 

in authentic, goal-directed activities (joint 

productivity) and conversing about what they are 

doing, thinking, and learning (instructional 

conversations). The role of the teacher during these 

instructional conversations is to listen carefully to 

what students are saying and to use questioning 

techniques to guide students’ thinking as they 

develop understanding and connect it to individual 

and community knowledge (Tharp 1997). This view 

of teaching is clearly evident in the NCTM 

Professional Standards for Teaching Mathematics 

(1991). 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

3 

 

In Table 1 [see Appendix D], we provide a 

set of six metaphors that summarize the traditional 

and standards-based views of school mathematics. 

The traditional view includes (1) a toolbox metaphor 

for knowing mathematics, (2) a behaviorist metaphor 

for learning, and (3) the master teaching metaphor. 

The standards-based view of school mathematics 

includes (1) a flexible problem solving with 

understanding metaphor for knowing mathematics, 

(2) a social constructivist metaphor for learning 

mathematics, and (3) a facilitator metaphor for 

teaching mathematics. For us, these two sets of 

metaphors summarize the “from what” and “to what” 

of the current mathematics education reforms.  

Processes of Changing Beliefs and Practices  

In the case of school mathematics, changing 

from the traditional beliefs and practices summarized 

by one set of metaphors to fundamentally different 

standards-based beliefs and practices summarized by 

another set of metaphors constitutes a revolutionary 

paradigm shift rather than incremental change. 

Beginning to share new exemplars for what it means 

to know mathematics, to learn mathematics, to teach 

mathematics, and to practice in this community is a 

part of the basic approach of both the NCTM 

Standards and the DMI materials. As we examine the 

processes for encouraging and supporting these 

changes, we can benefit from research on learning 

and professional development of teachers.  

The same research that justifies the shift to 

social constructivist views of learning for students in 

school mathematics supports a shift in designing 

professional development for teachers. Social 

constructivist views of learning support the 

perspective that changes in teachers’ beliefs will be 

experiential, developmental, and gradual. This is 

consistent with Guskey’s (1986) model of the process 

of teacher change (see Figure 1 [see Appendix D]), 

which claimed that changes in teachers’ beliefs (Step 

4) followed improvements in student outcomes (Step 

3) produced by changes in classroom practices (Step 

2) teachers made in response to professional 

development experiences (Step 1). 

Recognizing the complexity of teacher 

change, Clarke and Hollingsworth (2002) recently 

reviewed Guskey (1986) and other models for teacher 

change (Johnson & Owen, 1986; Lappan et al., 1988; 

Cobb, Wood, & Yackel, 1990; Clarke & Peter, 1993) 

and proposed what they called an interconnected 

model of teacher professional growth. Describing 

four domains affecting a teacher’s world (external, 

personal, practice, and consequence), they focused on 

two processes of change: reflection and enactment. 

Their model particularly emphasizes how these two 

mechanisms of change produce particular change 

sequences affecting these four domains, allowing for 

development of what they call growth networks. 

Although they thoroughly elaborated their model 

with examples from their empirical studies, they did 

not provide examples for how their model could be 

used to analyze and explain variations in change 

across individuals. 

In searching for a model that is sufficiently 

detailed to explain variations across participants, we 

noticed that the Guskey (1986) model could be used 

to apply a change sequence to Dewey’s (1933) model 

of reflective thought, which included five phases or 

non-hierarchical elements, including: problematizing 

the situation and recognizing the conditions of the 

problem; recognizing possible solutions to a problem; 

generating hypotheses for possible solutions to the 

problem; reasoning about these hypotheses to 

determine their potential for success; and testing one 

or more of these hypotheses in the context of the 

problem. 

To problematize is to recognize a situation 

as a problem and to acknowledge the conditions in 

the situation that affect the problem. Problematizing 

current practices is required to translate general 

interest or curiosity into an interest in changing 

practices. In applying Dewey’s model to preservice 

teacher education, Mewborn (1999) emphasized the 

importance of problematizing teaching situations. If 

problematizing is not sufficient to develop interest in 

complex fundamental change then interest narrows to 

simple incremental change within current paradigms 

and practices. Obviously, unless a situation is 

problematized, it is difficult to formulate hypotheses 

and test possible solutions. 

Examining potential solutions is relatively 

simple for incremental changes and relatively 

complex for fundamental changes. Incremental 

change poses only minor changes in strategies within 

current metaphors as possible solutions. Fundamental 

change requires development of understanding of 

new principles, metaphors, and curriculum needed to 

support major changes in practices needed to solve 

more fundamental and more complex problems. 

Similarly, interest in incremental improvement leads 

to simple experiments with minor changes, while 

interest in fundamental changes leads to complex 

experiments with major changes. 

Dewey’s (1904/1965, 1933) arguments for 

reflective thinking support the notion that teachers 

need to develop habits of reflection to be able to do 

the level of problematizing needed to improve their 

teaching and students’ learning. He argued that 

teachers who avoid reflective thinking develop an 

intellectual dependency on and interest in explicit 

directions from others on what and how to teach. 

This dependence can reinforce the master metaphor 

and focus teachers’ interest on acquiring additional 



Current Issues in Education Vol. 8 No. 19 
 
activities and teaching strategies to be accommodated 

within existing practices. This could support minor, 

incremental changes in teaching, but would not lead 

to fundamental changes in practices that produce big 

benefits in student learning, such as understanding 

mathematics. 

Both Dewey and Guskey also discussed 

elements of interest in change. Dewey (1933) 

mentioned curiosity as one of three personal 

resources essential for reflective thinking. Guskey 

(1986) was more specific, claiming that most 

teachers engage in professional development because 

of their interest in students and their desire to become 

better teachers to the benefit of their students. This 

desire to improve student outcomes provides both the 

basis for problematizing current practices and for 

changing beliefs when evidence of improved student 

outcomes can be tied to specific changes in teaching 

practices. 

Again considering both Dewey (1933) and 

Guskey (1986), reflecting on teacher experiences and 

student outcomes leads to decisions about whether 

experiments were successful or not for improving 

students’ learning and were practical or not for long-

term use. Success then leads to changes in beliefs; 

failure leads to ending the experiment, reversion to 

previous practices, and no changes in beliefs. The 

complexity of the needed reflection depends on the 

complexity of both the problem posed and the 

solutions attempted in the teaching experiments. 

Summary 

The literature summarized above suggests to 

us that two important characteristics would mediate 

the effects of the DMI course: (1) beliefs about 

knowing, learning, and teaching mathematics; and (2) 

sequenced processes for changing beliefs and 

practices. These two themes framed our exploration 

of variations in the impact of the DMI course on 

teachers’ beliefs and practices. For the purposes of 

this study, we have chosen to formulate and 

empirically support a time-sequenced (and therefore 

linear) process model that supports the detailed 

analysis of variations in teacher change yet fits within 

Clarke and Hollingsworth’s more global (and 

therefore interconnected and iterative) model of 

professional growth as a specific change sequence 

between two domains (the personal domain and the 

domain of professional practice). 

Integrating the frameworks for change from 

Dewey and Guskey provides the components of a 

model for analysis of the essential elements and 

processes of changing beliefs, including: 

 Teachers’ interests in change; 

 The extents to which they problematize 
current practices and pose possible 

solutions; 

 Their activities in exploring and testing 
these alternative practices; and 

 Their reflective analyses of the benefits of 
these changes for students, leading to lasting 

changes in beliefs and practices. 

Consequently, this study focuses on (1) 

describing the beginning and ending beliefs of 

participants in the DMI course (including 

summarizing those beliefs with metaphors for 

knowing, learning, and teaching mathematics) and 

(2) elaborating and testing a sequenced process 

model that is useful for explaining the variability in 

the documented changes in these teachers’ beliefs. 

Methodology 

The DMI Course and Participants  

Thirteen inservice elementary teachers 

participated in a one-semester university course using 

the Developing Mathematical Ideas (DMI) materials 

developed at the Educational Development Center. 

Participating teachers were employed in two local 

school districts. Their teaching experience ranged 

from 1-32 years and averaged about 10 years. Their 

teaching assignments were distributed across grades 

1-5, with one teacher in a K-6 mathematics lab. 

The course used two DMI modules: 

Building a System of Tens (Schifter, Bastable, & 

Russell, 1999a) and Making Meaning for Operations 

(Schifter, Bastable, & Russell, 1999b). The big ideas 

in Building a System of Tens focused on 

understanding place-value and Making Meaning for 

Operations focused on understanding addition, 

subtraction, multiplication, and division operations. 

These two DMI modules supported a series of sixteen 

three-hour class sessions designed to help teachers 

think about these particular big ideas of mathematics 

and to examine how children learn to understand 

those ideas. The essence of the course experiences 

included “a deep exploration of mathematics 

content—including the base-10 structure of our 

number system, the meaning of operations, and 

methods for calculating with multi-digit numbers and 

fractions—as well as analyzing children’s thinking 

about that content” (Davenport, 2001, p. 6).  

The DMI materials were designed to help 

teachers learn additional mathematics content and 

make mathematical connections, appreciate students’ 

thinking and learn how to foster such thinking, and 

analyze mathematics lessons and activities to uncover 

the mathematics students will learn. At the heart of 

the DMI materials are: (1) sets of classroom episodes 

(cases) illustrating student thinking (as described by 

the students’ teachers), (2) various learning activities 

providing opportunities for participants to reflect on 

their own and their students’ understandings of 

mathematics, and (3) assignments providing 

opportunities for participants to reflect on teaching 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

5 

 

mathematics. Participants collected their written 

work and responses from the facilitator in a portfolio, 

which provided another tool for reflection. 

The DMI materials provided a variety of 

models for course organization. The course in this 

study met one evening each week for a full semester 

and was led by the first author as facilitator. One 

major role of the facilitator was to organize and 

moderate small-group and whole-class discussions 

about: (1) focus questions about the written cases of 

classroom episodes; (2) videotapes of mathematics 

classrooms and clinical interviews; (3) samples of 

student work from teachers’ classrooms; and (4) 

research reports related to students’ thinking. The 

teachers also (5) explored the mathematical ideas in 

the videotapes; (6) planned, conducted, and analyzed 

clinical interviews of their own students; (7) wrote 

additional cases about their students; (8) 

experimented with the ideas discussed in class while 

teaching in their own classrooms, and (9) wrote 

personal reflections.  

Data Collection and Analysis  

Data for this study were taken from (1) 

audio taped whole-class and small-group discussions 

during the course; (2) field notes taken by the 

authors; (3) written materials collected from the 

teachers as part of the course; (4) a group interview 

with participants conducted by an independent 

evaluator following the course; (5) a post-course 

observation of participants’ teaching practices in their 

regular classrooms; and (6) a post-observation 

interview on changes in practices resulting from the 

DMI course.  

We audio taped each class session. During 

small group discussions, the third author usually sat 

with a target group and recorded their conversation. 

These audiotapes were transcribed for analysis. We 

used our field notes to clarify and provide context for 

the recorded conversations. 

Analysis of the data proceeded in a manner 

consistent with a naturalistic inquiry approach 

(Lincoln & Guba, 1985). First, two researchers read 

the transcripts independently and identified emergent 

themes. They paid particular attention to teachers’ 

beliefs (especially images and metaphors for 

teaching) and participation in processes of change. 

As these researchers discussed their reading of the 

transcripts, common themes began to emerge and 

they developed codes for these themes. Second, these 

same researchers re-read the transcripts and coded the 

conversations according to the emergent themes. The 

next step in the analysis of the data involved the 

isolation and validation of the major themes wherever 

they appeared in the data by triangulation across the 

various data sources and across time. Lastly, these 

themes were used to analyze all of the other data 

from the study. 

At the end of the semester following the 

conclusion of the DMI course, the second author 

observed several of the DMI course participants’ 

classroom teaching and then interviewed them about 

changes in teaching practices they attributed to the 

DMI course. These interviews were audio taped and 

transcribed. Some of the observed teaching episodes 

were also audio taped using a wireless microphone 

worn by the teacher. Field notes were taken for all 

observations, providing the primary data source for 

those cases in which teachers declined to wear the 

wireless microphone. This data provided an 

additional source for teachers’ reflections and for 

triangulation and confirmation of the conclusions 

from the primary data collected during the DMI 

course.  

Results 

As we analyzed the data we noticed that the 

impact of the course varied across the participants. 

To illuminate this variation, we present three cases of 

individual teachers (indicated by the pseudonyms 

Christine, Linda, and Paula). In many ways, the cases 

of Christine and Paula represent the two extremes of 

the course’s impact, and the effects of the course on 

Linda are similar to those of the majority of the 

participants. These three cases also articulate the 

variety of images and metaphors held by the 

participants and illuminate the coherence in their 

mental models. These three individual stories are 

largely self-narratives by the teachers that we have 

pieced together from the various data sources 

indicated. We selected these statements for their 

consistency across time and data sources. 

The descriptions of these three cases 

(located in the Appendices) are organized around the 

framework for changing beliefs and practices that 

emerged from the analysis of the data and the review 

of the literature: interest in change, problematizing 

and posing solutions, exploring/testing alternatives, 

and reflective analysis of benefits and changing 

beliefs and practices. This section includes brief 

descriptions of these teachers’ beginning beliefs as 

well as summaries of our claims about these teachers’ 

changes in beliefs and practices. 

Christine 

At the time of the DMI course, Christine 

was teaching third grade for the second consecutive 

year. She had previously taught for ten years in the 

resource room with students she characterized as two 

or three years behind grade-level in mathematics.  

Beginning beliefs. Christine spoke about 

her mathematical background and beliefs on several 

occasions, which clearly indicated a toolbox 

perspective on knowing mathematics, a behaviorist 



Current Issues in Education Vol. 8 No. 19 
 
view of learning mathematics, and a strong 

commitment to the master metaphor for teaching 

mathematics, primarily because she had been taught 

that way. She believed mathematics should focus on 

skills and facts, expressed the goal of getting correct 

answers using procedures memorized to the point of 

automaticity, and believed that standard algorithms 

were most efficient both for her and for her students. 

Christine was comfortable with direct instruction of 

algorithms and believed that it was “the best way” to 

teach mathematics. This approach had been 

successful for her as a student, and it provided a 

foundation for her confidence as a successful 

mathematics teacher. For example, she said: [I have 

thought about] why I like algorithms so much. [It’s 

because] I was taught that way. I am comfortable 

using an algorithm, and I am good at it (Portfolio 

Assignment 1.7). 

See the Case of Christine (Appendix A) for 

evidence of her engagement in the change processes.  

Changes in beliefs and practices. On the 

whole, the evidence in the Case of Christine indicates 

that her view of mathematics changed very little as a 

result of the DMI course. She had enlarged her view 

of what it means to do mathematics somewhat by 

allowing invented strategies into her toolbox, but she 

made little progress as a result of the course toward 

revising her preferences for behaviorist learning and 

the master metaphor for teaching. She learned how to 

ask students to explain their solutions and allowed 

some temporary informal inventing, but only if it led 

students to the traditional algorithms which she 

continued to emphasize through her direct 

instruction. (See Table 2 [see Appendix D] for a 

comparison of Christine’s beginning and ending 

beliefs and practices.).  

Linda 

 Linda had taught kindergarten for 16 years 

and was teaching first grade for the first time during 

the DMI course. 

Beginning beliefs. At the beginning of the 

course, Linda embraced the traditional algorithmic 

emphasis in elementary mathematics and felt that it 

was fast and accurate. She noted that she was able to 

perform mathematical tasks and arrive at correct 

answers, but had not developed meaning for the 

mathematics she had learned to do. Because Linda’s 

beliefs about mathematics were firmly based in 

algorithmic processes, she was surprised at first by 

the variation in students’ thinking exhibited in the 

DMI materials. Initially, Linda emphasized direct 

instruction; she believed that the traditional 

algorithms needed to be taught directly. 

See the Case of Linda (Appendix B) for 

evidence of her engagement in the change processes.  

Changes in beliefs and practices. The 

evidence shows that during the DMI course, a major 

focus for Linda became a new view of number—

what she called number sense. She came to believe 

that differences among students’ strategies for 

solving number problems in sensible ways indicated 

real understanding. She came to desire her own 

students being able to visualize number in the same 

ways as the students on the DMI videotapes. As part 

of this, she wanted her students to be able to explain 

their thinking and be flexible in working with 

numbers. Although she came to value invented 

algorithms as useful for mental computation, she still 

placed some importance on speed and efficiency 

using conventional algorithms. 

Linda pushed herself to solve problems in 

non-traditional ways and began to enjoy talking to 

others about her own mathematical ideas. She began 

to value these new ways of thinking and started to 

think she had been cheated by the narrow focus of 

her early mathematical experiences. By the end of the 

course, she had expanded her view of learning to 

include children being able to think, explore, take 

risks, and struggle with a problem in coming to 

understand mathematics. 

During the course, it became apparent to 

Linda that children who have developed number 

sense can approach problems in a variety of ways. 

Thus she began to value letting children explore their 

own ways of approaching problems as a way to help 

them “grasp a concept.” At the same time, she felt 

that this exploration needed “some structure and 

explanation” and that direct instruction of algorithms 

still had its place. By the end of the course, changes 

in her goals for learning allowed changes in her 

teaching practices. Her expectations had changed 

from recall of number facts to explaining or modeling 

answers to story problems that allowed her to assess 

errors in students’ understanding. She began to focus 

more on questioning individual students’ problem 

solutions and to keep track of their strategies. In 

order to do this, she began to keep detailed notes on 

her students daily, rather than just writing down a 

score. Linda recognized the need to become a better 

questioner, to help her students articulate their 

strategies and understandings and explain their 

thinking processes. Finally, she said that she had 

begun to question and evaluate how she taught. She 

noted that she was having fun approaching teaching 

in this way; she found it both exhilarating and 

frightening. (See Table 2 [see Appendix D] for a 

comparison of Linda’s beginning and ending beliefs 

and practices.) 

Paula 

Paula had been teaching fifth grade for six 

years and had previously taught grades two and four.  



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

7 

 

 Beginning beliefs. Paula enjoyed 

mathematics and had taken other courses to help her 

become a better mathematics teacher. She entered the 

course with a view of mathematics as a mixture of 

the traditional toolbox of algorithms and sense 

making. She said she valued critical thinking and 

problem solving skills and wanted her students to 

“understand what they are doing,” nevertheless she 

expressed a concern that her fifth grade students were 

not proficient in the traditional algorithms. She 

questioned the traditional emphasis on algorithms, 

even though she was then teaching from a textbook 

that emphasized direct instruction of standard 

algorithms. She expressed the concern that if the 

goals of mathematics instruction included critical 

thinking and problem solving, more direct instruction 

in procedural skills would not help meet those goals. 

This led her to question the efficacy of the traditional 

master teaching methods she was still using, but there 

was no evidence that she understood that achieving 

these additional learning goals required social 

constructivist learning processes and alternative 

teaching methods.  

See the Case of Paula (Appendix C) for 

evidence of her engagement in the change processes.  

Changes in beliefs and practices. The 

evidence shows that she developed a deeper and 

more interconnected view of mathematics and 

adopted this as a goal of her instruction. She also 

valued an increased focus on conceptual 

understanding. This development was supported by 

her personal mathematical problem solving in the 

course. She noted that she was becoming more 

flexible in her problem-solving approaches, and this 

in turn allowed her to value flexibility in her students. 

She articulated this as a change in her philosophy 

about knowing mathematics.  

Paula’s beliefs about learning grew deeper 

and more interconnected. She began to focus more on 

students’ thinking processes and variations in 

thinking among students evidenced through dialogue 

and written representations. She viewed this as 

children taking back the responsibility for their 

learning.  

The changes in Paula’s beliefs about 

teaching focused on movements away from the 

standard practice of following a page-by-page 

sequence in the textbook and a concern about 

“covering the material.” She felt that she had moved 

toward a focus on individual children’s thinking, 

cautious planning of problems, and questioning 

students to get at their understanding. She began to 

focus more on individual students’ mistakes and 

problematic thinking in order to help them build a 

“bridge” to understanding. Because she wanted to 

press for deeper understanding, she began to be 

concerned with building deeper, more coherent 

curriculum materials. She found this approach to 

teaching very enjoyable, but more importantly, she 

felt it was a better way of educating students. (See 

Table 2 [see Appendix D] for a comparison of 

Paula’s beginning and ending beliefs and practices.) 

Discussion 

These three cases represent a continuum of 

engagement in the process of change. On the one 

extreme, Christine started with very traditional 

beliefs about knowing, learning, and teaching 

mathematics, and made the least progress toward 

alternative beliefs and practices. More typical of the 

results of course participants, Linda started with very 

traditional beliefs and practices and made significant 

shifts in both, although only beginning to understand 

the coherent alternative provided by the standards-

based perspective. At the other extreme, Paula started 

the course already wondering about children’s 

understanding of mathematics and made the most real 

progress by connecting additional complexity to her 

previous understanding and developing a robust and 

coherent philosophy of standards-based mathematics 

education. 

The purpose of this section is to examine 

how varying levels of engagement in the process of 

change during the DMI course, linked to various 

beginning beliefs, resulted in these variations in 

change. In the process of this analysis, the 

applicability of the analytical framework proposed 

from the review of literature and used to organize the 

descriptions of the cases will also be demonstrated. 

This analysis is organized in terms of the following 

four variables in the change process: (1) interest in 

change, (2) problematizing/posing solutions; (3) 

exploring/testing alternative practices, and (4) 

reflective analysis/changing beliefs and practices.  

Interest in Change 

By enrolling in the course, all of the 

participants expressed an interest in learning 

something that could possibly improve or add greater 

variety to their teaching, constituting at least a low 

level of interest in change. Most of these interests 

were expressed in the context of the participants’ 

initial expectations for the course. These expectations 

focused largely on acquiring immediately applicable 

“teaching-as-telling” strategies, “exciting” activities, 

or “make-it-take-it” materials consistent with 

incremental change in traditional beliefs and 

practices. Although a few participants came with a 

desire to explore ways of enhancing their own 

mathematical understanding and that of their 

students, most came looking for better ways to be a 

master teacher rather than for fundamentally different 

ways to think about mathematics knowing, learning, 

and teaching. 



Current Issues in Education Vol. 8 No. 19 
 

At first, these expectations created a 

mismatch between what the course actually offered 

and what students expected. The critical point here is 

that almost all of the participant’s expectations for 

the course were closely aligned with the traditional 

beliefs of knowing, learning, and teaching 

mathematics. The fact that some teachers made 

progress in changing their beliefs given these initial 

expectations is encouraging. 

From examining the data, we recognized 

that these teachers’ interest in change, as it originated 

at the beginning of the course or developed during 

the course, involved three factors: (1) curiosity, (2) 

recognition of differences in beliefs and practices, 

and (3) dissatisfaction with current beliefs and 

practices.  

Curiosity. Some teachers enrolled in the 
DMI course with high initial levels of curiosity, while 

others increased in curiosity during the early course 

sessions. In both cases, this curiosity encouraged 

their becoming more dissatisfied with the status quo, 

more aware of alternatives, and, consequently, more 

interested in change. For example, Linda expressed 

curiosity about the ways in which some of her 

students thought about numbers in ways that she 

could not. At first, she could not understand the 

thinking of those students. This led her to become 

curious about their thinking and to question, “What is 

it that they know? What are they trying to do? What 

is in this problem that is causing them to think this 

way?” She became anxious to explore children’s 

thinking and her own understanding of mathematics 

in order to answer some of these questions. This 

contributed to her increased interest in change. 

Recognition of differences in beliefs and 

practices. This factor in developing greater interest 

in change became evident in the discourse during 

class sessions that verbalized, compared, and 

contrasted the variety of beliefs and opinions about 

teaching practices held by course participants (e.g., 

the value of using manipulatives) or evident in the 

DMI cases (e.g., facilitating invention of strategies 

from conceptual understandings). Teachers in the 

course had many opportunities to see and discuss 

these differences of opinion about various aspects of 

teaching. Recognition of these differences also 

contributed to increased interest in change. 

For example, regarding the value of using 

manipulatives, Christine expressed the opinion that 

“when you do use manipulatives you’ve got to drill 

and drill so that they understand exactly what those 

manipulatives are representing.” Paula expressed a 

different view. “I read a research article saying 

manipulatives are not the answer—be careful and 

sure that they understand. The emphasis here is on 

the child’s understanding and the meaning the child 

makes of the materials they use” rather than on the 

use of the manipulatives, which can be 

proceduralized without understanding. This 

discussion situated one of the differences between 

Christine’s and Paula’s beliefs about knowing 

mathematics: Christine used manipulatives in 

particular ways to further her goals of learning 

procedures and Paula recognized that manipulatives 

had to be used in different ways to meet her goal of 

developing conceptual understanding. 

Regarding the example of facilitating 

children’s invention of strategies from 

understandings as evidenced in the DMI cases, 

Christine analyzed the teaching practices evident in 

those cases from the perspective of standard 

algorithms as the best pathway to correct answers, 

with her interest in students’ invented strategies only 

as temporary alternative pathways to the same 

destination, pathways with which particular students 

may be more comfortable. However, she paid little 

attention to the specific thinking and understanding 

students used to invent those strategies. Linda, in 

comparison, was fascinated by the student thinking in 

the cases and responded that she wanted to become 

better at helping students communicate their 

understanding by improving her questioning 

techniques focused on flexible problem solving with 

understanding. Paula demonstrated in her response to 

the cases that she recognized the connection between 

classroom discussions of students’ thinking and their 

understanding, and she could even see the logic in 

students’ incorrect strategies. 

Dissatisfaction with current beliefs and 

practices. As with curiosity, teachers can come with 

this factor of interest in change or it can be created in 

the initial course sessions. More important than 

curiosity or awareness of differences, this factor is 

the essential element of an interest in change 

sufficient to lead to problematizing current beliefs 

and practices. This factor originates as dissatisfaction 

with the outcomes of teaching, not yet sufficiently 

detailed as to have identified the particular problem 

or cause of the dissatisfaction. For example, engaging 

in mathematical problem solving and examining 

children’s thinking and understanding of mathematics 

during the DMI course led Linda and Paula to 

express dissatisfaction with their own early 

mathematical experiences. Linda recognized that she 

had not developed an understanding of the 

mathematical procedures she had been taught. Paula 

had found mathematics enjoyable, but recognized 

that not all of her friends “got it” and was thus 

dissatisfied with some aspects of her experience. 

Christine, in comparison, expressed satisfaction with 

what she had been taught and how she had learned, 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

9 

 

and was content with teaching her students the same 

mathematics in the same manner.  

In summary, Christine expressed a low level 

of interest in change, focused primarily as adding 

some “spice” to her teaching. Linda expressed a 

moderate level of interest in change, primarily 

curiosity about children’s thinking that was different 

from her own and improving her questioning 

techniques to facilitate students’ communication. 

Paula expressed a high level of interest in change, 

centered on sufficient dissatisfaction with traditional 

practice for her to be interested in exploring the 

rationale behind particular alternative teaching 

practices intended to building conceptual 

understanding. (See Table 3 [see Appendix D] for a 

summary of our appraisal of these three teachers’ 

levels of interest in change as low, moderate, and 

high, respectively.) 

Problematizing Current Beliefs and Practices and 

Posing Possible Solutions  

 One element of problematizing current 

beliefs and practices is becoming aware of possible 

alternatives. The task of incremental change is 

primarily experimenting with minor improvements to 

practices within existing beliefs and reflecting upon 

the results of these incremental changes in practices. 

However, without problematizing current beliefs and 

practices, the need for revolutionary change is 

unrealized and the focus remains on incremental 

improvement. Two examples are provided here of 

how revolutionary alternatives problematize 

traditional beliefs and practices: (1) comparing the 

learning of algorithms with developing conceptual 

understanding, and (2) comparing “fast and accurate” 

computational skills with flexible problem solving. 

Comparing algorithms to understanding. 
Paula questioned the traditional emphasis on 

algorithms. She expressed a concern that her fifth 

grade students were not proficient in the traditional 

algorithms. This led her to question the efficacy of 

traditional teaching methods. She also expressed the 

concern that if the goals of mathematics instruction 

included critical thinking and problem solving, more 

instruction in procedural skills would not help meet 

those goals. She even felt constricted by the need to 

continue to spend time teaching algorithms to satisfy 

parents and prepare students for standardized tests. In 

comparison, Linda translated her sensitivity about her 

own supposed lack of understanding into a greater 

interest in her students coming to thoroughly 

understand mathematics and be able to clearly 

communicate that understanding.  

 Comparing fast and accurate to flexible. 

Recognition that the traditional elementary school 

emphasis on “fast and accurate” computation 

(automaticity) and direct instruction in “one way” to 

solve particular problems was challenged by a more 

interconnected and conceptual view of mathematics, 

including understanding of concepts, invented 

strategies, and flexibility in solving nonroutine 

problems. Paula came to understand the connection 

between depth of conceptual understanding and 

degree of flexibility in problem solving strategies. 

Linda also recognized that children who understand 

concepts of number show greater flexibility in 

problem solving and develop very efficient invented 

strategies for mental computations. She reinforced 

this perspective as her own understanding grew 

during the course and she began to develop greater 

flexibility in her own problem solving. She admitted 

that her invented mental strategies were often faster 

than her traditional computational methods. Linda 

also recognized that this shift in learning goals 

required a corresponding shift in teaching practices. 

Recognizing problems as occurring within 

one’s own experience requires that these problems be 

stated and that the conditions that affect them be 

examined carefully and in detail. Thorough 

statements of problems of practice include the 

contexts where they typically arise and the 

relationships between the stated problems and other 

aspects of practice. Mewborn (1999) indicated that 

preservice elementary teachers rarely specifically 

stated problems and even less frequently questioned 

what was problematic about particular situations for 

others. However, the inservice teachers participating 

in the DMI course frequently stated problems 

specifically and often questioned others about 

particular teaching situations in which those 

problems occurred. Their attempts to personally solve 

these problems were assisted by the readily available 

alternatives in the DMI materials. Absent these 

workable solutions, attempting to solve such 

Significant problems of practice would likely have 

been very difficult.  

In summary, Christine expressed a low level 

of problematizing and solution posing as she 

remained confident in her current success with direct 

instruction and worried about how she could explain 

a variety of invented strategies without confusing her 

students further. However, Linda focused on how to 

create an appropriate environment for student 

exploration and sharing of strategies, reflecting a 

high level of problematizing knowing, learning, and 

teaching and posing solutions in each of those areas. 

Paula began to consider changes in her curriculum to 

provide problem situations that could develop critical 

thinking, problem solving, and understanding, also 

reflecting a high level of problematizing and solution 

posing in all three areas of beliefs and practices. See 

Table 3 [see Appendix D] for a summary of our 

appraisal of these three teachers’ levels of 



Current Issues in Education Vol. 8 No. 19 
 
problematizing/posing solutions as low, high, and 

high, respectively. 

Exploring/Testing Alternative Beliefs and 

Practices  

To engage in exploring/testing alternatives, 

the process of problematizing and posing possible 

solutions in the DMI course needed to create 

sufficient hope that the revolutionary alternatives 

would result in better outcomes and more satisfying 

results to generate the needed experimentation. Yet at 

this stage of the process, many of the teachers’ 

understandings of the alternative practices were 

minimal. Actually experimenting with these 

alternatives helped them become more 

knowledgeable about the specifics of the alternatives.  

One of the strengths of the DMI course was 

that it provided two sites for the 

experimenting/testing of solutions to problematic 

practice. One was the DMI classroom, where peers 

had conversations about elementary mathematics and 

children’s solutions. The second site was the 

participants’ own classrooms, where they interviewed 

their own students and tried out different teaching 

ideas. Participants’ experiences suggested that these 

two laboratories were portable in the sense that they 

could remember the first one as they tried out ideas in 

the second, and vice versa. This provided a very rich 

context for reflection on practice. 

Many of the participants indicated that the 

things they read in the cases and saw on the 

videotapes were actually happening in their 

classrooms and interviews with their students. By 

design, the teachers’ classrooms provided 

laboratories for validating what was taking place in 

the DMI course. Having both sites available for 

exploration of ideas—the DMI course and individual 

classrooms—allowed participants to reflect on their 

mathematical thinking and their teaching in both 

venues. The extent to which participants enjoyed 

talking to each other about their mathematical ideas 

and the results of their experiments were evidenced 

through the participants’ dialogue during class 

sessions and their written responses in portfolios. 

Change processes in the DMI classroom 

involved what Clarke and Hollingsworth (2002) 

referred to as the teacher’s personal domain, while 

change processes occurring in the school classrooms 

involved the teacher’s domain of practice. Being able 

to problematize, experiment, and reflect in both the 

DMI class and one’s own classroom provided an 

iterative element with a short turn-around time and 

increased the impact of the DMI course on teachers’ 

beliefs and practices. 

As expected, attempts to experiment with 

alternative practices were constrained by a variety of 

factors even after current practices had been 

problematized. In particular, participants’ 

conceptions (formed during several years of school 

mathematics instruction) and traditional expectations 

of administrators or the teacher next door both served 

to discourage some attempts to experiment with 

alternative beliefs and practices. 

In summary, Christine tried asking some of 

her students to share their thinking, but her focus in 

teaching mathematics continued to be on the correct 

answer and providing students with hints if they 

didn’t get to the answer she wanted. Linda departed 

from her emphasis on direct instruction and tried 

letting children explore their own ways of 

approaching problems as a way to help them “grasp a 

concept.” In doing so, she began to focus more on 

questioning individual students’ problem solutions 

and keeping daily detailed notes on her students’ 

strategies rather than just recording their scores for 

number of correct answers. Paula began to focus 

specifically on developing understanding; cautiously 

planning problems to provide a deeper, more 

coherent curriculum; and questioning students to get 

at their understanding. (See Table 3 [see Appendix 

D] for a summary of our appraisal of these three 

teachers’ levels of experimenting/testing alternatives 

as moderate, high, and high, respectively.) 

Reflective Analysis of Benefits and Changing 

Beliefs and Practices  

 We have learned that reflecting on practices 

and changing beliefs is a complex activity. It involves 

comparative analysis of the evidence of results from 

current and alternative beliefs and practices and 

making connections among these experiences. 

Reflecting also involves making judgments about the 

relative efficacy and relative satisfaction from these 

alternatives and making generalizations about their 

potential efficacy in somewhat different 

circumstances or contexts. In addition, reflecting 

involves examining the level of implementation of 

traditional or alternative practices for opportunities 

for incremental change within existing or alternative 

beliefs. Interest in change and problematizing are 

highly interactive, with new awareness of possible 

alternatives strengthening dissatisfactions and leading 

to additional problematizing of ineffective practices. 

All of these elements of the change process 

occurred in both laboratories: the DMI course and 

teachers’ classrooms. For some, well-developed 

theories emerged from their analysis of personal 

experiments with alternative practices. For others, the 

effects of the DMI course were less dramatic. 

In summary, although Christine engaged in 

some exploring of alternative practices, her low 

interest in change and low problematizing of current 

beliefs and practices left her without a significant 

problem to be solved by alternative practices. 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

11 

 

Consequently, she saw only limited usefulness for 

students’ invented strategies and made only 

insignificant changes in her beliefs, curriculum, and 

teaching practices. She thought students’ strategies 

were interesting and worth sharing in class 

occasionally, but failed to recognize them as the 

foundation for building students’ understanding. 

In comparison, Linda and Paula both 

recognized the results of their experiments as 

beneficial for students, and made significant progress 

in changing beliefs and practices involving knowing, 

learning, and teaching mathematics. Linda began to 

expand her students’ opportunities to explore 

mathematics in a supportive environment and to build 

their number sense, particularly for use in mental 

computations. She increased her ability to ask good 

questions, rephrase what students said, and assess 

errors in their thinking. However, she also held on to 

some traditional tools and emphases in the 

curriculum. This represents significant progress given 

her moderate interest in change and her current 

understanding of the concepts of the standards-based 

perspective. 

Paula’s reflections on the benefits of her 

experiments resulted in high levels of change in her 

beliefs about knowing, learning, and teaching 

mathematics. Her teaching metaphor became that of a 

facilitator who poses interesting problems, uses 

questioning to find out what students think, and leads 

them to build deep, well-connected understandings. 

She enlarged her curriculum to include more 

emphasis on understanding big ideas, problem 

solving, and critical thinking. She also recognized 

that understanding occurs through problem solving, 

dialogue about solution processes, and creating 

written representations for those solutions. All of 

these changes were responsive to her initial interest 

in the whys of the standards-based perspective and 

her ability to develop a new personal theory of 

mathematics education. (See Table 3 [see Appendix 

D] for a summary of our appraisal of these three 

teachers’ levels of reflecting/ changing beliefs and 

practices as low, moderate, and high, respectively.  

Conclusions  

We learned from this study that metaphors 

can provide concise yet powerfully descriptive 

summaries of widely held beliefs about knowing, 

learning, and teaching mathematics. Although the 

titles of these metaphors immediately conjure up 

particular images in our minds, the ways we used 

these metaphors still allowed us to describe 

additional details about specific teachers’ beliefs so 

that our overall characterization of any particular 

teacher’s beliefs remained true to the individual 

subject. Yet, for purposes of comparing beginning 

and ending beliefs, analyzing change, and 

summarizing the effects of this particular 

professional development experience, the level of 

simplification afforded by these metaphors proved to 

be very helpful. 

Overall, it has been encouraging to see that 

some teachers made real progress in changing their 

beliefs during the DMI course, even though they 

were initially looking for additional strategies to use 

while teaching within their existing metaphors. 

Examining the processes through which these 

teachers changed their beliefs has also been a useful 

tool for understanding the nature of the changes and 

the variations across individuals.  

Figure 2 [see Appendix D] shows our 

completed process model for describing and 

analyzing the variations in changes in beliefs and 

practices observed in the participants of the DMI 

course and evidenced by the three cases presented. It 

was clear to us that participants’ beliefs about 

knowing, learning, and teaching mathematics were 

complex, varied, and interrelated. Creating and using 

this model has allowed us to elaborate on key 

elements and processes of teacher change which 

integrate the literature on professional development 

with our empirical experiences researching the 

effects of the DMI course. 

Acceptance of redefined learning goals 

emphasizing flexible problem solving with 

understanding, such as those found in NCTM’s 

Principles and Standards (2000), requires 

corresponding shifts toward paradigmatic theories of 

learning and teaching practices that are coherent with 

and capable of achieving those new learning goals. 

The results of this study suggest that one’s teaching 

practices are unlikely to be problematized unless 

beliefs about knowing and learning mathematics are 

problematized first. Without problematizing beliefs 

about knowing and learning mathematics, interest 

tended to follow the typical pattern of remaining 

focused on incremental change within current 

teaching metaphors rather than participation in the 

paradigmatic changes offered by the DMI course. 

Important elements of interest in change, which is 

essential to this problematizing process, included 

curiosity, recognition of differences, and 

dissatisfaction with current beliefs and practices. 

In analyzing variations in the amount of 

change achieved by participants in the DMI course, 

we observed that participants with low interest in 

change and/or low levels of engagement in these 

change processes, particularly in problematizing and 

experimenting, made only limited changes in their 

beliefs and practices. Conversely, those participants 

who had higher levels of initial interest and more 

fully engaged in each of the change processes, 

experienced greater changes in beliefs and practices. 



Current Issues in Education Vol. 8 No. 19 
 

 

 

Implications  

Although situated in the context of a specific 

professional development course in elementary 

mathematics education, the use of metaphors as a tool 

for describing and categorizing complex, 

interconnected beliefs is generally applicable. With 

some modification to the metaphors for knowing, this 

collection of metaphors may be useful in examining 

teachers’ beliefs about knowing, learning, and 

teaching in other content areas that also focus on 

developing conceptual understanding and thinking 

processes. Use of metaphors like these can facilitate 

analyses of the types of changes that can be achieved 

by various professional development activities. 

This change process model merges practical 

elements of professional development targeted at 

changing beliefs and practices with a potentially 

generalizable framework from the literature on 

teacher education. To the extent that this change 

model has elaborated a change sequence that has 

been useful for us in explaining variations in the 

outcomes of this DMI course, this process model may 

be helpful to others interested in instigating or 

studying paradigmatic changes in teachers’ beliefs. 

Additional work clearly remains. This study 

represents our effort at developing theory grounded 

in the specifics of three individual cases that are 

representative of the participants in one university 

classroom engaged together in one series of 

teaching/learning experiences. Additional utility can 

be added to this theory by further development and 

testing of suitable collections of metaphors for 

knowing, learning, and teaching. Utility can also be 

improved by development and testing suitable 

methods for gathering comparable data relevant to 

teachers’ participation in the various components of 

this change process, which would allow comparison 

of the results of this study to data from larger 

numbers of participants in similar systematic 

professional development activities.  

References 

Bakhtin, M. M. (1981). The dialogic imagination (M. 

 Holquist, Ed.; M. Holquist & C. Emerson, 

 Trans.). Austin, TX: University of Texas 

 Press.  

Bullough, R. V. (1992). Beginning teacher 

 curriculum decision making, personal 

 teaching metaphors, and teacher education. 

 Teaching and Teacher Education, 8(3), 239-

 252.  

Bullough, R. V., & Stokes, D. K. (1994). Analyzing 

 personal teaching metaphors in preservice 

 teacher education as a means for 

 encouraging professional development. 

 American Educational Research Journal, 

 31(1), 197-224.  

Clarke, D., & Hollingsworth, H. (2002). Elaborating 

 a model of teacher professional growth. 

 Teaching and Teacher Education 18, 947-

 967. 

Clarke, D. J., & Peter, A. (1993). Modelling teacher 

 change. In B. Atweh, C. Kanes, M. Carss, & 

 G. Booker (Eds.), Contexts in mathematics 

 education. Proceedings of thee 16th annual 

 conference of the Mathematics Education 

 Research Group of Australasia (MERGA). 

 Queensland: Mathematics Education 

 Research Group of Australasia. 

Cobb, P. Wood, T., & Yackel, E. (1990). Classrooms 

 as learning environments for teachers and 

 researchers. In R. B. Davis, C. A. Mayer, & 

 N. Noddings (Eds.), Constructivist views on 

 the teaching and learning of mathematics 

 (pp. 125-146). Reston, VA: National 

 Council of Teachers of Mathematics. 

Dalton, S. S. (1998). Pedagogy matters: Standards 

 for effective teaching practice. Santa Cruz, 

 CA: Center for Research on Education, 

 Diversity & Excellence, University of 

 California. 

Davenport, L. (2001). Elementary mathematics 

 curricula as a tool for mathematics 

 education reform: Challenges of 

 implementation and implications for 

 professional development. Newton, MA: 

 Center for the Development of Teaching at 

 Education Development Center.  

Dewey, J. (1933). How we think: A restatement of the 

 relation of reflective thinking to the 

 educative process. Boston: Heath.  

Dewey, J. (1965). The relation of theory to practice 

 in education. In M. L. Borrowman (Ed.), 

 Teacher education in America: A 

 documentary history (pp. 140-171). New 

 York: Teachers College Press. (Originally 

 published in 1904).  

Friel, S. N., & Bright, G. W. (eds.). (1997). 

 Reflecting on our work: NSF teacher 

 enhancement in K-6 mathematics. Lanham, 

 MD: University Press of America.  

Guskey, T. R. (1986). Staff development and the 

 process of teacher change. Educational 

 Researcher15(5), 5-12.  

Kuhs, T. M., & Ball, D. L. (1986). Approaches to 

 teaching mathematics: mapping the domains 

 of knowledge, skills, and dispositions. East 

 Lansing, Michigan: Michigan State 

 University, Center for Teacher Education.  

Lampert, M. (1985). How do teachers manage to 

 teach? Perspectives on problems in practice. 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

13 

 

 Harvard Educational Review, 55(2), 178-

 194.  

Lappan, G., Fitzgerald, W., Phillips, E., Winter, M. 

 J., Lanier, P., Madsen-Nason, A., Even, R., 

 Lee, B., Smith, J., & Weinberg, D. (1988). 

 The middle grades mathematics project. The 

 challenge: Good mathematics—taught well 

 (Final report of the National Science 

 Foundation for Grant #MDR8318218). East 

 Lansing, MI: Michigan State University.  

Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic 

 inquiry. Newbury Park, CA: Sage.  

Ma, L. (1999). Knowing and teaching elementary 

 mathematics: Teachers' understanding of 

 fundamental mathematics in China and the 

 United States. Mahwah, NJ: Lawrence 

 Erlbaum.  

Mewborn, D. S. (1999). Reflective thinking among 

 preservice elementary mathematics teachers. 

 Journal for Research in Mathematics 

 Education, 30(3), 316-341.  

National Council of Teachers of Mathematics. 

 (1991). Professional standards for teaching 

 mathematics. Reston, VA: Author.   

National Council of Teachers of Mathematics. 

 (2000). Principles and standards for school 

 mathematics. Reston, VA: Author.  

Richardson, V. (1996). The role of attitudes and 

 beliefs in learning to teach. In J. Sikula & T. 

 J. Buttery & R. Guyton (Eds.), Handbook of 

 research on teacher education (pp. 102-

 119). New York: Macmillan.  

Rogoff, B. (1990). Apprenticeship in thinking: 

 Cognitive development in social context. 

 New York: Oxford University Press.  

Rogoff, B., & Wertsch, J. V. (1984). Children's 

 learning in the zone of proximal 

 development. San Francisco: Jossey-Bass. 

Schifter, D., Bastable, V., & Russell, S. J. (with 

 Lester, J. B., Davenport, L, R., Yaffee, L., & 

 Cohen, S). (1999a). Building a system of 

 tens: Facilitator’s guide (Developing 

 mathematical ideas: Number and operations, 

 part 1). Parsippany, NJ: Dale Seymour.  

Schifter, D., Bastable, V., & Russell, S. J. (with 

 Lester, J. B., Davenport, L, R., Yaffee, L., & 

 Cohen, S). (1999b). Making meaning for 

 operations: Facilitator’s guide (Developing 

 mathematical ideas: Number and operations, 

 part 2). Parsippany, NJ: Dale Seymour  

Stigler, J. W., & Hiebert, J. (1999). The teaching 

 gap: Best ideas from the world's teachers 

 for improving education in the classroom. 

 New York: Free Press.  

Tharp, R. G. (1997). From at-risk to excellence: 

 Research, theory, and principles for practice 

 (Research Report 1). Santa Cruz, Ca: Center 

 for Research on Education, Diversity and 

 Excellence. Retrieved May 16, 2002 from 

 http://www.crede.ucsc.edu/products/print/re

 ports/rr1.html.   

Tharp, R. G., & Gallimore, R. (1989). Rousing minds 

 to life: Teaching, learning, and schooling in 

 social context. New York: Cambridge 

 University Press. 

Thompson, A. G. (1992). Teachers' beliefs and 

 conceptions: A synthesis of the research. In 

 D. A. Grouws (Ed.), Handbook of research 

 on mathematics teaching and learning (pp. 

 127-146). New York: Macmillan. 

Vygotsky, L. S. (1978). Mind in society: The 

 development of higher psychological 

 processes (M. Cole, V. John-Steiner, S. 

 Scribner, & E. Souberman, Eds. and Trans.). 

 Cambridge, MA: Harvard University Press.  

Wertsch, J. V. (1985). Culture, communication, and 

 cognition: Vygotskian perspectives. New 

 York: Cambridge University Press.  

Wertsch, J. V. (1991). A sociocultural approach to 

 socially shared cognition. In L. B. Resnick, 

 J. M. Levine, & S. D. Teasley (Eds.), 

 Perspectives on socially shared cognition 

 (pp. 85-100). Washington, DC: American 

 Psychological Association. 

. 

. 

 

http://www.crede.ucsc.edu/products/print/re%09ports/rr1.html
http://www.crede.ucsc.edu/products/print/re%09ports/rr1.html


Current Issues in Education Vol. 8 No. 19 
 

Appendix A 

 

The Case of Christine 

 

Interest in Change. Christine expressed some curiosity about whether there were other ways to teach 

mathematics that could provide ideas she could incorporate into her current practices in an incremental rather than 

revolutionary way. She wrote: 

Most of my students did well in math. I could bring their math skills up to grade level. I thought I was a good 

math teacher. I took this class because I thought there was always room for improvement—a way of learning 

to teach math in a better and more exciting way. (Portfolio Assignment 1.5) 

Christine’s expectations for the course included learning more about how students learn to think 

mathematically, along with some skills to make her a better math teacher. She hoped to transfer those skills into her 

classroom so that her students would be better at mathematics. She was only mildly curious about fundamentally 

different approaches to teaching mathematics that place greater emphasis on understanding mathematics concepts 

than on memorizing facts and algorithms. 

Problematizing and Posing Solutions. Various tasks and activities during the course provided 

opportunities for Christine to question what she believed and knew about children’s thinking. As she began studying 

the DMI cases of teachers’ experiences with children’s thinking, she expressed surprise and amusement at the 

different ways the children solved problems, some of which seemed “weird and difficult” to her (Portfolio 

Assignment 1.2). She wondered if the tediousness and complexity of the student’s work at solving what she thought 

were simple math problems was not the reason why some students hated mathematics. She questioned the 

underlying causes of some students’ difficulties. Maybe “it is because they do not have a good understanding of 

basic place value systems…. With this new way of teaching math, will more students change their minds about 

math? Will they do better in math? Is there a better way, a most efficient way, to solve math problems? I wonder!” 

(Portfolio Assignment 1.3) 

Exploring/Testing Alternatives. As Christine engaged in discussions about various strategies for solving 

problems, she observed how interesting it was “to see how different people had their own math style in solving a 

problem.” But her primary interest was in how this should affect her teaching. “In third grade, if you present a 

problem like this, do you explain all this that we have been doing? Or do you just say you can’t do that? Or do we 

just go on and confuse them more by showing all these ways? I’m not sure they’re ready for this” (DMI Session 

Discourse). 

More questions surfaced for Christine as she shared her experiences from interviewing some of her students.  

From the interview I found out I can’t assume that students can do those simple, simple things. I just was 

shocked at what they didn’t know. I wish I could do this with every one of my kids, so I could see just where 

they mess up. (DMI Session Discourse) 

Christine reflected in her portfolio on one student’s work that really concerned her. She titled it “Did I do the 

right thing?” 

Jessica was getting very discouraged. I could tell that she did not want to continue any longer. My question: 

Did I do the right thing to have her come up in front of the class to solve the problem? It was obvious that she 

had no understanding of the question. I was surprised at what she came up with. I tried to find out more by 

probing, redirecting, and asking questions to clear up her confusions. How do I get her to ask her own 

questions? How do I get her to stick with a problem even when it’s difficult without getting discouraged? Am 

I embarrassing her in front of the class? Should I give her more hints to come up with the answer, or should I 

allow her time to figure it out herself? (Portfolio Assignment 2.2) 

Christine struggled to generate thoughtful potential solution paths to her problematic activity of attending 

to differences in the classroom. Although she perceived the need to expose her students to different solution 

strategies, her metaphors limited the kinds of solutions she could seek. Rather than question her metaphors, she tried 

solutions that left her teaching metaphor intact, essentially deciding just to tell these other strategies to students. Her 

choices of which strategies to show in class were also linked to her metaphor for mathematics; she chose to show 

only those strategies that had a clear and direct path to the traditional algorithm. This resulted in a relatively 

superficial change in her practice. 

Reflective Analysis of Benefits and Changing Beliefs and Practices. After one classroom episode, 

Christine decided to interview Jessica. Christine had just finished a multiplication unit where her students first 

memorized the multiplication tables, then they all passed the “daily times test.” She then taught a unit about the 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

15 

 

meaning of the multiplication process. Christine wrote that she thought Jessica appeared to understand the basic 

concept of multiplication. However, during the interview it became clear to Christine that Jessica hadn’t 

remembered anything they had discussed in class and was unable to explain what “times” meant. Christine wrote: 

Now I am getting a little discouraged. Maybe I did not teach her well. Maybe she did not pay attention in 

class. Maybe she needs more one-on-one help in math. I was surprised at her [lack of] understanding. 

(Portfolio Assignment 2.5) 

Christine began reflecting on her own mathematical thoughts. She recorded in her portfolio that she had 

never done this before. It was enlightening to her to see how she solved problems differently than her colleagues and 

how that made her more aware of her thinking. However, throughout the entire course Christine still chose a 

standard algorithm as her preferred approach to solving problems. She said she felt her ways of problem solving had 

not changed and she was still “most comfortable” with standard algorithms. At times she would look at a problem 

from what she called “a different way,” but she said she felt it took much longer. 

As the course progressed, Christine broadened her learning goals for students a little bit, but continued to 

frame her teaching with the master metaphor. For example, during the first of the two DMI modules she recorded in 

her portfolio “the ultimate goal of doing mathematics is to get correct answers. However, in order to get correct 

answers understanding plays an important role” (Portfolio Assignment 1.5). She had noticed that children and adults 

could use different solution strategies in solving problems and still come up with the same answer. She provided 

some room in her goals for invented strategies as alternatives to standard algorithms. She even wanted her students 

to be exposed to what she called “different ways” of solving problems. However consistent with her master 

metaphor of teaching, she beganshowing those strategies to her students and encouraging them to try them. The 

value of these invented strategies for her was as a “stop gap” measure until students learned the more efficient 

standard algorithms. She said: 

This class has helped me see a variety of ways children solve problems. Maybe children can figure out 

problems other ways until they have had enough practice with the math facts and have drilled standard 

algorithms long enough that they will become more efficient with them, because many of these creative ways 

are so time consuming. But it is a way they can solve the problems without getting them marked wrong. Who 

cares anyway how you get it. As long as the answer is correct it doesn’t really matter how you got it—just get 

it! (DMI Session Discourse) 

As Christine reflected back over the first half of the course, she wrote that several things had changed for 

her. She felt she had learned more about the variety of children’s invented strategies. She noticed she had begun to 

ask students to tell how they had come up with an answer as opposed to just asking them “What is the answer?” In 

doing so, she believed she had “added a lot to her instruction.” 

I tend to show students different ways of solving math problems, to expose them to various ways. And I have 

gained a lot of respect for the many different ways problems can be solved, though some methods I do not 

really agree with. It has added a little more spice to my math curriculum. (Portfolio Assignment 1.7) 

As Christine saw other participants in the course bring up new ways to do mathematics, she continually 

questioned whether they were better than her traditional methods. At the end of the course, she revisited her beliefs 

about standard algorithms and wrote, “Now, I don’t think it is the best way of doing math. It may be one of the 

many ways of doing math, but it surprised me to admit that it is only one of many ways of doing math” (Portfolio 

Assignment 2.8).  

Five months after the completion of the DMI course, Christine taught a mathematics lesson during which 

she posed eight division story problems and asked students to solve them using manipulatives, pictures, graph paper, 

or traditional algorithms. She asked students to explain their solution strategies, and after the initial solution, asked 

for students who did it another way. For each of the eight problems, Christine asked for additional strategies until 

she had someone share the traditional long-division algorithm. If no one shared the standard algorithm she would do 

so, emphasizing how the traditional algorithm was faster, particularly with larger numbers (Observation Field 

Notes). 

During the post-observation interview that focused on Christine’s perceptions of what had changed in her 

teaching as a result of the DMI course, she responded: 

Most of the time I just pick out story problems from what we call the curriculum…. And once in a while I add 

a little bit to it…but most of the time I just follow the curriculum... I just ask more often, “Explain to me how 

you got this answer. Why do you say this?” So I ask more for the understanding... I try to use a lot of the 

visual aids. I try to… meet the different learning styles of the kids by using different ways of teaching. Some 



Current Issues in Education Vol. 8 No. 19 
 

may be more comfortable with the traditional way, so I am still doing that. I feel like they still need to be 

exposed to that. I think they still need to know that because of standardize testing. They go to the computer 

room, and they still give you basic division. You still have to do that. At the same time, I think the 

understanding part is very important, too, and so I like to see them doing it different ways. I want them to say, 

“Math is not just one way of doing it.” (End-of-Year Interview Transcript)  

 

  



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

17 

 

Appendix B 

 

The Case of Linda 

 

Interest in Change. Linda expressed an interest in coming to understand the mathematics she already knew 

how to do, but her new assignment in first grade focused her interest in learning new things that were immediately 

applicable to her teaching. “I had hoped for activities and ideas that I could incorporate into my classroom.... More 

like a big workshop, where you take the ideas and run with them; like a make-it-take-it” (Portfolio Assignment 1.2). 

Linda said she felt her way of looking at skills would be challenged in the course, but she still considered “the old 

tried and true ways” faster and more accurate than the children’s invented strategies introduced during the first class 

session. She felt skills still needed to be taught directly. 

Linda expressed curiosity about some children’s mathematical thinking when she said that her students 

thought about numbers in ways that she could not. At first, she did not understand how they were thinking, and she 

was curious about how they were able to manipulate numbers in their heads. This led her to question, “What is it 

that they know? What are they trying to do? What is it in this problem that is causing them to think this way?” 

Problematizing and Posing Solutions. As the course progressed, Linda gained an ease and confidence that 

allowed her to share her previous experiences with mathematics as a student and her perceptions of how those 

experiences had affected her current feelings toward mathematics.  

When I first came to this class, I was terrified. I wanted to keep my mouth shut because I was afraid someone 

would discover how little I know about mathematics…. When I was in school a very long time ago, the 

system expected me to know how to get the answers, and if I got the answer and had it right, that was the 

important thing…. I really feel bad looking back. I wanted to understand the algorithms, but I never could…. I 

don’t think I ever really understood why you multiplied this number and then you moved over a space and 

multiplied the next one. But you obeyed and did it. (DMI Session Discourse) 

Linda began to raise questions and to reflect on her assumptions about learning mathematics as she read cases about 

other teachers’ classroom experiences, viewed video recordings of children explaining their mathematical solutions, 

and discussed children’s thinking with peers. 

This bothered me last week, because it just seems so much faster and easier to do it my way—using the 

standard algorithm. And why shouldn’t I teach the kids that way? But then I watch these kids. They’re 

thinking, and they really picture the numbers much better than I do…. What I am finding I dearly love about 

this class is I can sit here and I look at these kids and I’m forced to answer these focus questions. I’m forced to 

step inside their shoes and say, “Okay, what do they know? What are they trying to do? What is in this 

problem that is causing them to think this way?” (DMI Session Discourse) 

Linda continued to question her beliefs about how children learn mathematics, and she commented on these 

beliefs in her portfolio. She noted that children who understand numbers show great flexibility in their approaches to 

problems and become very efficient in mental math. She remained uncertain about the need for direct instruction in 

algorithms even as her own computational strategies became more flexible. 

I have found myself trying to stretch and think [like] the children in the case studies…. In regards to addition 

where regrouping is required, I must admit that I have even found it faster than the traditional method whe n 

doing mental math. If it is written, I am still more comfortable with the old way. I still feel a need to check my 

answers using the traditional borrow and carry method. (Portfolio Assignment 1.5)  

Linda, who problematized not only mathematics and the learning of mathematics but also what it meant to 

teach, took the greatest risk in generating possible solution paths. She abandoned many of her earlier conceptions for 

new ways of teaching. Her shift from telling things to students to attending more to children’s thinking allowed her 

to consider a wide variety of practices as part of her solution: (1) keeping more detailed notes on children; (2) 

learning how to question children in order to help them articulate their thinking; (3) de-emphasizing the traditional 

algorithms; and (4) allowing children to struggle with problems without stepping in to help. All of these paths led 

Linda out of her comfort zone, but she could see that these paths also led to a significant solution to the problems in 

her practice.  

Exploring/Testing Alternatives. As Linda made progress in her own mathematical thinking she began to 

show some willingness to consider alternative approaches to teaching, and she began to confront the dilemma of 

how to reconcile her preference for direct instruction and what she was learning about teaching for understanding. 

I still find myself questioning how much instruction is necessary from the teacher. “How do I create an 



Current Issues in Education Vol. 8 No. 19 
 

environment where my students are willing to explore and to take risks? Is this method effective for children 

of all ability levels, or am I losing some of my class that doesn’t understand what their peers are doing?” 

(Portfolio Assignment 1.5) 

Partway through the semester the course participants were asked to identify the one case that had the 

greatest impact on them and describe why they had selected that case. It was at this time that we began to see more 

changes in Linda’s personal metaphor. Linda commented during class about the case she had selected. 

The first case had the greatest impact on me because I’m trying to teach my first graders number sense—very 

basic. But it moved me out of my comfort zone enough that I decided that I wanted my children to be able to 

visualize number like the students in those cases…. This class is helping me realize, as a primary teacher, we 

need to help our children develop number sense and do a better job helping children see relationships between 

numbers at a much younger age…. I’ll probably never teach them the algorithms. (DMI Session Discourse) 

Reflective Analysis of Benefits and Changing Beliefs and Practices. By halfway through the course Linda 

had made considerable progress from reflecting on the mathematics she wanted her students to know, what 

experiences would provide her students the opportunity to learn those things, and what would be her role as a 

teacher. In her portfolio she wrote: 

I think that my objectives have changed somewhat from things like “the student will demonstrate their 

knowledge of addition by adding numbers to 12” to “the student will be able to explain or model answers to 

story problems using addition.” I think that being able to explain what they are doing is not only helpful for 

them but for me as a teacher, so that I can see where they might have errors in their understanding of a 

process…. I struggle with how long to let them explore with some of these concepts before I demonstrate a 

method. I still feel some things in first grade need to be taught. You need to give them a beginning point. For 

instance, I need to teach that addition means to combine and subtraction means to take away. (Portfolio 

Assignment 1.6) 

However, Linda expressed some uncertainty about how to proceed as she began to move away from her 

expert metaphor. She recognized that rebuilding her teaching practice could be a lonely and uncertain process, even 

though she had seen alternative practices modeled in the videotapes and cases. “I feel like I’m kind of hanging out 

there, like Columbus, hoping that I don’t fall off the edge and take my class with me. It’s exhilarating and also very 

frightening” (Portfolio Assignment 1.6). 

Linda started to notice that changes in her teaching were beginning to have an important impact on the environment 

in her classroom and the thinking that her students were beginning to do. This required her to begin supporting 

students thinking in ways that were still new to her. 

It’s amazing that some of the things that are happening in these cases have actually been happening in my 

classroom. My children are finally starting to be able to tell me how they solved their problems…. It takes real 

restraint to sit back and let a child struggle with a problem and not point out what they should do. (DMI 

Session Discourse) 

Near the end of the first unit, Linda reflected on her personal knowledge growth with respect to various 

cases and how her growth had enlightened her thinking about her students’ understanding of mathematics. 

Each reading assignment has helped me view mathematical processes in a new light. I think the case studies in 

Section 1 have had the greatest impact on my thinking and teaching…. I am trying to become better at 

questioning my students to help them articulate what they are doing. Perhaps the reason I felt the greatest 

impact from the first case study is because it moved me out of my comfort zone and made me question and 

evaluate how I teach math in my classroom. (DMI Session Discourse) 

During the second DMI module, Linda noted several ideas that had become important to her. Gaining 

confidence as a problem solver and addressing her past experiences with mathematics had empowered alternative 

images of herself as a learner and teacher of mathematics.  

I am finding it exhilarating to talk with other teachers about the teaching of mathematical ideas. I have become 

concerned about individual understanding, as I have done interviews and case studies of my students. And I 

have learned to recognize the value of each voice in my class (Portfolio Assignment 2.3). 

I have found the focus questions stimulating. I love to come home and share them with my family. I would not 

have felt this way at the beginning of this class. At that time I felt insecure, and I was resisting the fact that my 

way of doing things was being challenged. (Portfolio Assignment 2.6) 

I hope that I will be able to continue to grow in my ability to ask questions, rephrase what the children tell me, 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

19 

 

and help them to explain their thinking and process in arriving at answers to problems. (Portfolio Assignment 

2.8) 

I’ve now come to the point where I feel that my comments are somewhat of value, and I feel much more 

comfortable. I’m not afraid to make a few mistakes and say, “What do you mean?” This class turned out to 

address a bigger picture than I had envisioned, and it turned out to be very exciting. It was such a revelation 

for my own self awareness that it became very exciting to see where I was coming from, from my past 

experiences as a child growing up with mathematics, and my feelings toward mathematics now. I have almost 

come to feel like I had been cheated. I will never look at mathematics the same way as I did before I had this 

class. (DMI Session Discourse) 

Linda’s shifts in mathematics learning goals for students provided a foundation for this shift in her image of 

a teacher as one who ask questions, rephrases what children say, and helps them explain their thinking and problem-

solving processes. 

Five months after the completion of the DMI course, Linda taught a first-grade mathematics lesson in 

which she asked students to use the “rodeo math” manipulatives they had created the previous day (consisting of a 

cowboy and multiple pants, shirts, and hats with numerals written on each piece of clothing). Working in groups of 

2-3, students exchanged pieces of clothing on their cowboys to produce addition number sentences. One student 

recorded the number sentences and all students in the group computed the sums. Linda encouraged students to look 

for patterns among the combinations in order to get all of the possible combinations. She asked students whether the 

same numbers in different orders were the same sum (e.g., Could 5 + 2 + 3 = 10 be 5 + 3 + 2 = 10 instead?). She 

also asked, “Which way could you dress the cowboy to get the highest number” (Observation Field Notes). 

During the post-observation interview, Linda commented about valuing students’ voices as they shared 

their solution strategies, but also showed that she was focusing on what children were thinking and asking follow-up 

questions. She said, “I can see that they have a much deeper knowledge of what they are doing if they can explain it 

and not just go through a ritual.... I am trying very hard to... ask questions and not tell answers.... That is a very 

difficult skill. I still struggle with it sometimes—not just telling them ‘This is where you’ve made your mistake.’” 

She had added two more realistic problem-solving tasks to the curriculum. She said, “We have a standard math 

book, and I have tried to stretch them beyond that and let them do things that I wouldn’t have done before. I took 

problems that were in the DMI book and brought them back, and I let them try to do them... This morning... we 

talked about the Jazz game, and I said, ‘If at the end of the first quarter, the Jazz had 20 points and Carl Malone had 

made 5 of those points, how many points had the other people made?’ It was interesting to watch them figure that 

out, and they all got the same answer. They used several different methods, and each of them shared it” (Post-

Observation Interview Transcript). 

She also commented on the dilemma she still faced in reconciling basic skills versus invented strategies: 

“I’ve felt like they passed the [state] core test very well. I don’t know that they’re any better, really, than other 

students... But I hope they are freer to explore. One thing that I still struggle with is that in first grade I think they 

have to have some basic skills. I am still trying to balance between drill and exploration. I still feel that... some of 

this has to be mastered for them to be able to explore” (Post-Observation Interview Transcript). 

She expressed plans for continued experimenting: “I’ve been thinking that perhaps I will change a little bit more. 

Maybe I’ve tried to take too big of a leap too fast... I’m going to start right at the first day of school, start doing some 

addition, some subtraction, all with these few numbers and just let it grow as we go along... Now that I’m seeing... 

the full scope of the curriculum a little better, it’s a little clearer to me. I think that I’m going to try to do all of it in 

connection with the few things that we’ve learned and then move on. That’s my thought for next year” (Post-

Observation Interview Transcript).  

 

  



Current Issues in Education Vol. 8 No. 19 
 

Appendix C 

 

The Case of Paula 

 

Interest in Change. From the beginning it was clear Paula questioned the metaphor of 

teacher as master and allowed room in her view of learning for sense making. Paula recorded 

five of her expectations for this course in her portfolio. 

 Explore the whys of basic algorithms. 

 Explore the whys of student processing. 

 Collect new games, techniques, and thought provoking problems to incorporate into my 

instruction. 

 Explore the hows in the management of so many toys, so many ideas, so many children, 

and so little time. 

 Enhance my own mathematical processing skills. (Portfolio Assignment 1.2) 

Later, Paula said she hoped “Teaching to the Big Ideas” would help her “create, 

duplicate, borrow, beg, or steal some intervention strategies” that would answer her concerns 

(Portfolio Assignment 1.5). Paula also expressed her concern with helping older learners 

“rebuild” a system of tens (Portfolio Assignment 2.3). 

Problematizing and Posing Solutions. Paula’s experience as a student differed from 

Linda’s. As a child, Paula was able to perform algorithms efficiently and enjoyed doing so. But 

she remembered noticing not all of her classmates could. This raised questions in her mind about 

whether too much emphasis was being placed on computational algorithms. 

When most of us went to school, we learned one way to add, one way to subtract, one way 

to multiply, and one way to divide. Several of us noticed, however, that not all of our 

friends we went to school with “got it.” The question I have is why is the entire nation 

seemingly committed to the algorithm anyway? Is it a universal truth or just a way to fit 

into society? Is an algorithm a tool? Is an algorithm really a good thing? Is there only one 

algorithm? I think we teach algorithms because we want to hold our jobs, so that people 

won’t say we were lousy teachers. Really, it’s so children’s parents can help them with their 

homework, right? What it all comes down to is how we are held accountable. It feels like 

we are being held back, because we have to teach this way so the children can pass tests. 

It’s too bad. (DMI Session Discourse) 

Exploring/Testing Alternatives. At the beginning of the course Paula indicated she was 

open to alternatives for what to teach and how to teach it and was willing to pursue what 

teaching for understanding might mean. 

Most of my fifth grade students are not proficient in traditional algorithms. Can I reinforce 

the students’ algorithms best through story context, or symbol context, or pictorial context, 

or a variety of all three? Is there a time when bare mathematical symbol context is better? 

What I’m most concerned about is that they are able to develop critical thinking in their 

problem solving, and they won’t get there if we keep saying that the standard algorithm is 

the only way. In the long run they need to understand what they are doing. (DMI Session 

Discourse) 

As the course progressed, Paula noted in her portfolio that her understanding of 

mathematics was deepening and her rigidity had given way to greater flexibility. She commented 

that breaking down various problems had given her a much better understanding of how 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

21 

 

algorithms work, the value of base-ten knowledge, and the connected and integrated nature of 

mathematics. She expressed similar ideas during class: 

I’m noticing that mathematics is not just single ideas taught separately in isolation, but 

rather more of a connected and integrated whole. It’s making me re-think what mathematics 

is—its wholeness versus its parts. It seems like the underlying factor for everything is base 

ten. We revisit it over and over again, just in different ways. (DMI Session Discourse) 

Along with shifting her beliefs about mathematics, Paula also changed what she did in 

the classroom. As she reflected on what she had begun to do differently, she wrote about 

increased frequency of “flights away from the book, workbook, and workshop files” with more 

time spent on exploring and dissecting students’ mistakes on assignments and tests (Portfolio 

Assignment 1.6). Later she noted: 

I have found that I can lead a whole class discussion with only two or three problems and 

use the children’s ideas to develop a much better session than a page of math problems can 

accomplish. What many of them are doing is rebuilding an understanding of mathematics. 

However, it takes longer than I anticipated. (Portfolio Assignment 2.3)  

They are taking back the responsibility for learning, and I am becoming convinced that this 

is not only a “funner” way to teach math but a better way as well. (Portfolio Assignment 

2.7) 

Paula attributed some changes in her teaching practices to a greater emphasis on 

children’s thinking and understanding. 

I am thinking more about students’ thinking individually rather than covering the 

mathematics. Not just to say, “Okay, that’s checked off” and continue going on hoping 

everybody in the class has gotten what I planned them to get. But more, “What did that 

student individually get?” You have to stop and reflect about what you are seeing, and it 

takes time. I noticed in the last few days I have been more cautious in my planning as to 

what kind of problems can get at thinking, that will make their mathematical understanding 

deeper. What I would like is to help bridge their thinking and to diagnose how to help them 

straighten out some of the problems they are having in that thinking. (DMI Session 

Discourse) 

Reflective Analysis of Benefits and Changing Beliefs and Practices. At the end of the 

course, Paula reflected in her portfolio on what had changed for her during the semester. She 

wrote that she knew her students better than before. She noticed her students could now track 

their thinking processes and pinpoint the variations in their ideas. She also noticed that their 

understanding of the mathematics was deeper and their dialogues and written representations 

were more focused. Paula had gained confidence in her ability to question and pose interesting 

problems. She commented in class that she had started to teach mathematics more as connected 

big ideas rather than separate disconnected pieces, which was consistent with changes in her 

philosophy of mathematics education. 

I thought this class was going to be more about ideas on how to teach. I think I had an idea 

that there was going to be a little more theory behind it, and not just ideas, but I think the 

thing that was most helpful was it helped me develop my philosophy of mathematics 

education. And, I think until you understand what your philosophy of something is, you are 

not as powerful at it. I think I had a philosophy, but it changed by the process that we went 

through and discussions we had. I knew questioning was important, but I now see it spilling 



Current Issues in Education Vol. 8 No. 19 
 

over into other subjects. The type of questions that you ask children to get them to 

understand is permeating my whole life, and I will never be the same. (DMI Session 

Discourse) 

By the end of the course, Paula had built upon her views of how children learn to 

redesign her teaching metaphor as one who uses questions to find out what students are thinking 

and then poses additional questions to lead students to build new understandings of well-

connected mathematical ideas. 

Five months after the end of the DMI course, Paula taught a mathematics lesson that 

posed a problem that focused on vocabulary and language development associated with the 

mathematical concepts of two- and three-dimensional shapes connected to the science of soap 

bubbles and strongest shapes. She fostered development of a problem-solving community as she 

engaged her students in a social constructivist experience and used questioning to prompt and 

facilitate students’ thinking about deep understandings of the mathematics in the lesson. 

(Observation Field Notes) 

During the post-observation interview focused on changes in Paula’s teaching practices 

resulting from the DMI course, she commented on how she now attended to children’s 

understanding and helped them explore their mistakes. 

I spend a lot more time discussing their ideas. I don’t do as many problems per se, but we 

spend more time on specific problems. We just go broader because we take the time to 

listen and talk about all the different ways they would approach it. . . . I find myself asking 

more questions and trying to generate more questions from them. . . . I am trying to build on 

what each child says and to connect it to either someone else in the room, something they 

said earlier, or a particular path I am hoping they will go down. . . . I try to point out to them 

how valuable it is when they find and correct themselves and figure out why they did it 

wrong. I use a phrase this year I have never used before. I will say, ‘Do you realize the most 

important thing that has happened today just happened for you in our class, because you 

corrected yourself, and you just figured out what you have done wrong. And that is the most 

valuable thing.’ I found that the more that I use that phrase the more children start saying 

that. They would try to find the mistakes they had made and then bring it to my attention. 

‘Look, this is what I did wrong. That is the most important thing I could do today. I just 

found what I did wrong.’ And that cognition. . . [and] celebration—of ‘Oh, I got that wrong, 

and now if I figure out why I got it wrong she is going to be even happier than if I had 

gotten it right’— is very different, and that is attributed directly to the [DMI] class. (Post-

Observation Interview Transcript) 

Paula recognized that changes to her fifth-grade curriculum were essential to achieving 

her goals for learning. 

I haven’t changed all my problems. I’m locked into district core in math, and the majority of 

the core deals with computation out of context. I do have to continue to work with them on 

math out of context, but I have given more real life math problems at other times in the day, 

such as the warm-up activity in the morning. I used to spend quite a bit of time with 

grammar or daily oral language, and now probably fifty percent of it would be a math 

contextual question or game, because I have found that if you put math in the context of real 

life that builds their grammar skills. If the math is in the context of a story, then you are 

killing two birds with one stone. I want more of the kinds of problems [we saw in the DMI 

course]. I want more of them because I am too exhausted to think them up myself. I am 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

23 

 

stretched too thin to keep generating them, and I have generated quite a few. I will take 

situations in the literature book we’ll be reading and use that as a springboard for a math 

problem, or a field trip we’ve been on. I did a lot more of that this year. Almost every field 

trip or outside experience we did I tied back into a math problem after we had done it, and I 

watch for those opportunities to do that. . . . Still, I haven’t moved everything completely to 

a big idea. (Post-Observation Interview Transcript) 

Her comments during the interview also indicated she understood that shifts in teaching 

required to facilitate learning flexible problem solving with understanding were revolutionary in 

nature and as yet unsupported by continuing standardized testing. In addition, she expressed 

some concern over vocal groups expressing opposing viewpoints that indicated that her changes 

in beliefs and practices were very fragile due to the limited evidence of their effectiveness. She 

said, “I think math education is [students’] ticket to the future, and my consciousness is open and 

sensitive to the need of better math education. I know what we have done traditionally all these 

years isn’t working as well as it should, and it is exciting but also frustrating to be caught in the 

middle of the revolution and not know if what you are doing is really [helping]” (Post-

Observation Interview Transcript). 
 

 

  



Current Issues in Education Vol. 8 No. 19 
 

Appendix D 

 

List of Table and Figure Captions  

 

Table 1. Metaphors 

Table 2. Changes in Metaphors/Beliefs 

Table 3. Interest in Change, Problematizing, Exploring, and Analyzing 

Figure 1. Guskey’s (1986) Model of the Process of Teacher Change 

Figure 2. Change Process Model for Teachers’ Beliefs and Practices 

 

Table 1. Metaphors 

 
  

 

 

 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

25 

 

Table 2. Changes in Metaphors/Beliefs 

 
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Current Issues in Education Vol. 8 No. 19 
 

Table 3. Interest in Change, Problematizing, Exploring, and Analyzing 

 
  

Figure 1. Guskey’s (1986) Model of the Process of Teacher Change 

 
  

 

 

 

 

 

 

 

 

 

 



Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

27 

 

Figure 2. Change Process Model for Teachers’ Beliefs and Practices 

 

 

  



Current Issues in Education Vol. 8 No. 19 
 
2005 Article Citation  

Smith, S.Z., Smith, M.E. & Williams, S.R. (2005, July 20). Elaborating a Change Process Model for Elementary 

 Mathematics Teachers’ Beliefs and Practices. Current Issues in Education [On-line], 8(19). Available: 

 http://cie.ed.asu.edu/volume8/number19/ 

 

Author Notes  
Stephanie Z. Smith 

Georgia State University 

 

Dr. Stephanie Z. Smith is an Assistant Professor in the Early Childhood  

Education Department at Georgia State University in Atlanta, GA. She received her PhD in Curriculum and 

Instruction (Mathematics Education) from the University of Wisconsin–Madison in 1998. Her scholarly interests 

include learning and teaching mathematics with understanding, children’s and teacher’s conceptions of mathematics, 

and teacher education and professional development. 

 

Marvin E. Smith 

Georgia Southern University 

 

Dr. Marvin E. Smith is an Assistant Professor in the Department of Teaching and Learning at Georgia Southern 

University in Statesboro, GA. He received his PhD in Curriculum and Instruction (Mathematics Education) from the 

University of Wisconsin–Madison in 2000. His scholarly interests include learning with understanding, classroom 

assessment, and teacher education and professional development. 

 

Steven R. Williams  

Brigham Young University 

 

Dr. Steven R. Williams is a Professor in the Mathematics Education Department at Brigham Young University in 

Provo, UT. He received his PhD in Curriculum and Instruction (Mathematics Education) from the University of 

Wisconsin–Madison in 1989. His research has focused on the learning of calculus, discourse in mathematics 

classrooms, and various topics in teacher education. 

 

 

Note from the 2015 Executive Editor, Constantin Schreiber  
July 13, 2015. This article was first published at the original Current Issues in Education website, located at 

http://cie.asu.edu/articles/index.html. In 2009, CIE changed online platforms to deliver the journal at 

http://cie.asu.edu. The original CIE website was from then on only used as an archival repository for published 

articles prior to Volume 12. After the new CIE website moved to a different server in 2014, the original website and 

original article URLs could not be accessed anymore. Therefore, this article had to be repurposed into the published 

format you are viewing now.  

All content from the original publication has been preserved. No content edits occurred. Spelling, grammar, 

and mechanical errors that may be found were present in the original publication. The CIE logo and publisher 

information in use at the time of the article’s original publication is unaltered. Please direct questions about this 

article’s repurposing to cie@asu.edu.  

 

 

2015 Article Citation  
Smith, S. Z., Smith, M. E., & Williams, S. R. (2005). Elaborating a change process model for elementary 

 mathematics teachers’ beliefs and practices. Current Issues in Education, 8(19). Retrieved from 

 http://cie.asu.edu/ojs/index.php/cieatasu/article/view/1650  

 

 

file:///D:/CIE/Volumes%20&%20Issues/cie-archive/2005,%20Vol%208,%20%231-25/number15/index.html
http://cie.asu.edu/ojs/index.php/cieatasu/article/view/1650


Elaborating a Change Process Model for Elementary Mathematics Teachers’ Beliefs and Practices 

  

29