A Collaborative Classroom-Based Approach To Professional Development A Bilingual Teacher Explores Issues Of Language And Mathematics Problem

Classroom-based Professional Development- 1
DRAFT – Please, do not cite or quote without authors’ authorization
RUNNING HEAD: Classroom-based Professional Development
A Collaborative Classroom-based Approach to Professional Development: A
Bilingual Teacher Explores Issues of Language and Mathematics Problem Solving1
Sandra I. Musanti, Sylvia Celedón-Pattichis & Mary E. Marshall
The University of New Mexico
Paper presented at the Annual Meeting of the Association of Teacher Educators
New Orleans, Louisiana.
February 24-27, 2008
1
This research was supported by the National Science Foundation, under grant ESI-0424983, awarded to
CEMELA (The Center for the Mathematics Education of Latino/as). The views expressed here are those of
the author and do not necessarily reflect the views of the funding agency.

Introduction
Currently, reform initiatives and research concur on the need to afford
mathematics educators with learning opportunities centered on understanding students’
mathematical thinking (Franke, Carpenter, Levi & Fennema, 2001; NCTM, 2000).
Literature on teacher growth has explained that teachers develop understanding of their
practice as they deepen their comprehension of student learning (Franke et al., 2001;
Sykes, 1999). Understanding how students solve problems, how their thinking develops
and how language impacts learning can foster teacher understanding of how instruction
can promote mathematical learning. Therefore, research efforts have been made to
understand how to better prepare teachers to make meaning of student work in relation to
the students’ problem solving and verbal explanations (Kazemi & Franke, 2004; Little,
2004). This paper presents a case study of a professional development initiative in which
a first grade bilingual teacher engages in learning about Cognitively Guided Instruction
(CGI) (Carpenter, Fennema, Franke, Levi, & Empson, 1999), a framework for
understanding student thinking through context-rich word problem lessons. This case
study also explores the importance of problem solving in mathematics learning in the
context of a prolonged collaboration with a group of researchers.
Recently, the field of bilingual education has been deeply affected by policies
aimed at dismantling it. A clear example of the tremendous consequences of these events
is the English-Only movement, where 17 states currently have monolingual policies in
effect, among them California, Arizona, and Massachussets (Varghese, 2004). In this
political context, the education of bilingual teachers and their role in the education of

Latino students is under close scrutiny. Research has also indicated the need for more in
depth exploration of the formation and ongoing development of teachers of Latino
students who are English language learners (ELL) (Téllez, 2004). This study contributes
to the emergent research in the field of the professional development of Latino students’
teachers and bilingual teachers (Hart & Lee, 2003) by exploring the impact of a
classroom-based professional development initiative aimed at expanding bilingual
teachers’ knowledge of CGI while integrating it in the context of mathematics reform
curriculum.
Throughout a year-long collaboration, a teacher and researchers explored the
nuances of students’ thinking and the role of language and culture in mathematics
instruction. As a team, we engaged in planning and delivering mathematics lessons
focused on developing students’ problem solving strategies as well as students’
capabilities to communicate mathematical reasoning in their first language, Spanish. This
case study contributes to comprehending the bilingual teacher’s development over time
as she participated in ongoing conversations about Latinos/as’ student thinking, problem
solving, issues of language, and adaptation of instruction to meet students’ needs.
Specifically, the study explores: (a) the impact of classroom-based professional
development on teacher's understanding of teaching mathematics to Latino students; and
(b) issues of language and culture with which teachers grapple while engaged in
reflecting on students’ thinking about mathematics.

Theoretical Underpinnings and Related Literature
This case study is part of a longitudinal study of a professional development
initiative in which elementary bilingual teachers engage in learning about CGI (Carpenter
et al., 1999) and the importance of problem solving in mathematics learning. CGI claims
that teachers need to understand why children, even in the early grades, should be
afforded repeated opportunities to solve a variety of word problems and communicate
their thinking about their solutions. As Franke et al. (2001) explain:
CGI focuses on helping teachers understand children’s mathematical thinking by
helping them construct models of the development of children’s mathematical
thinking in well-defined content domains. No instructional materials or
specifications for practice are provided; rather teachers develop their own
instructional materials and practices from watching and listening to their students
and struggling to understand what they see and hear. (p. 657)
Creating professional development opportunities in which teachers can “learn
with understanding” (Franke et al., 2004) about students’ mathematical thinking requires
‘situating’ teacher learning in relation to their practice, as an integral part of their
teaching lives (Wenger, 1998). Current conceptualizations of professional development
emphasize generating communities of practices in which teachers learn in collaboration
with others, have opportunities to reflect on their practices, and collegially design
teaching approaches that respond to students’ needs. From a sociocultural perspective,
communities of practice can afford teachers with opportunities to actively participate in
their own development and transform their understanding of students’ learning,
ultimately changing their practice and themselves (Rogoff, 1995). Relevant research has
documented the impact over time of professional development that sees teachers as
ongoing learners and reflective practitioners (Schön, 1983). For instance, Franke et al.

(2004) developed a follow-up study of a group of 22 teachers who had participated in a
prolonged professional development experience on learning about CGI. The study
showed that all teachers continued to consider students’ thinking in their practice even
though in different manners, and that 10 teachers continued to grow over time. Their
growth was characterized by in depth understanding of children’s thinking, their
continued emphasis in constructing their own comprehension of students’ mathematical
understanding, and their tendency to seek collegial learning opportunities.
In the field of mathematics teachers’ development, of particular relevance is the
research that looks at how teachers learn in the context of professional development
experiences that promote the sustained analysis of student work (e.g. Franke, Fennema,
Carpenter, Ansell, & Behrend, 1998). Using student work as a catalyst for teacher
learning allows the shifting of “teachers’ focus from one of general pedagogy to one that
is particularly connected to their own students” (Kazemi & Franke, 2004, p.204). An
important tenant underlying this professional development practice is the importance of
teachers “learning in and from practice” (Ball & Cohen, 1999, p. 10) and from students’
ideas and understandings. Little (2004) concluded that research exploring organized
opportunities for teachers to learn in and from practice is still scarce. However, her
review of related studies indicated that professional development “designed to focus
teachers’ attention closely on children’s learning, may have a positive effect on outcomes
of interest: teacher knowledge, teaching practice and (in some cases) student learning”
(p.105). Zhao & Cobb (2007) thoroughly discuss the theoretical and practical
implications of a framework aimed at explaining the interplay between teacher learning
and classroom practices in the context of professional development. They assert that

efforts such as analyzing student work requires a deep understanding of teachers’
instructional realities that involve not only what is observable in terms of their practices
but “the underlying rationales that teachers hold towards teaching and learning” (p. 21).
Scholarship on bilingual teachers’ development and current explorations of what
is relevant in the formation of teachers of English language learners (ELLs), especially in
the field of mathematics education is growing. However, more needs to be done. As
Téllez (2004/2005) points out, there are few studies that investigate issues regarding the
education and professional development available for teachers of Latino students. Flores
(2001) surveyed 176 bilingual teachers and concluded that teachers’ positive personal
experiences in relation to language and culture affected teachers’ beliefs regarding the
benefits of bilingual education. Most importantly, the study showed that teachers’ beliefs
can be revised and transformed as the result of professional development, in particular
through experiences that foster the reflective examination of practices and beliefs. In
another extensive survey study of teachers who taught immigrant and refugee ELLs in a
midwestern suburban district, Karabenick & Clemens Noda (2004) found that a high
percentage of teachers held positive attitudes towards ELLs and bilingualism. However,
most of these teachers felt unprepared to serve this population, especially in what relates
to cultural awareness and second language theory.
Studies investigating issues of professional development for bilingual and ESL
teachers from a subject matter perspective are also sparse. An important study conducted
by Hart & Lee (2003) explored the impact of a professional development program
designed to improve the achievement of ELLs in science and literacy on teachers’ beliefs
and practices. The study shows that to further ELL students’ achievement, elementary

teachers need extensive professional development to increase their knowledge of content-
specific teaching strategies and “to help ELLs and other students with limited literacy to
become English proficient” (p.493). In addition, the study provided important insights on
the impact of long-term professional development on teachers’ practices and beliefs, as
well as demonstrated the need for ongoing support if the goal is to implement reform
curriculum.
Our study explores how a bilingual teacher’s comprehension of student learning
impacts the way she understands and frames her practice. In the area of mathematics
education, research has shown the connections between teacher knowledge and the
decisions teachers make in relation to their mathematics instruction (Aguirre & Speer,
2000) and the role of language and teacher talk in Latino student mathematics learning
(Khisty & Chval, 2002). Understanding how students’ mathematical thinking develops
and how language impacts learning can foster teacher understanding of how instruction
can promote mathematical learning.
Methodology
Participant
Ms. López2
, the participant in this study, is originally from Peru and Spanish is
her first language. She finished high school in her native country and then emigrated to
the United States where she obtained her college degree and teaching license. From her
college years she remembers her struggles as an ELL: “It was the hardest thing. I
couldn't express what I wanted to say, the way I wanted to say it - it was very hard” (Ms.
López, Fall 2006). Ms. López has been teaching for 12 years in a large city of a
2
All names are pseudonyms.

southwestern state in the United States. She has taught 3 years in a 4th
and 5th
combined
class, and 9 years in first grade. She is a bilingually certified teacher who teaches 90% in
Spanish. She has been teaching in the same school for nine years.
This urban elementary school consists of culturally and linguistically diverse
students: Hispanic 86.3%, Native American 6.4%, White 4.3%, African American 1.5%,
Asian 0.8%, and other 0.9%. Almost all her students are from Mexican immigrant
families. At the time of the study, 99.5% of her students spoke Spanish as their first
language. This school has adopted a mathematics reform curriculum and Ms. López has
been teaching with it for the past 7 years. The curriculum provides materials for teacher
and students in Spanish, but was not designed specifically for this population. The
Spanish materials are translations from the English originals, reflecting classroom
learning dynamics and communication more closely aligned with mainstream U. S.
culture. The lessons follow the NCTM standards (NCTM, 2000) and are based on a
spiraling approach where concepts are introduced but not convered in depth at the time of
introduction. Concepts are reintroduced frequently as the curriculum spirals through the
various areas of mathematics. The curriculum emphasizes the mathematical processes of
problem solving, communication, reasoning, connections and representations.
Portrait of Ms. López’s Beliefs
As Aguirre & Speer (2000) contend, “beliefs play a central role in a teacher’s
selection and prioritazation of goals and actions” (p. 327). Beliefs affect practice and they
play an important role in how teachers intepret and implement curriculum. Because of the
importance we give to teacher beliefs in our work, we provide a succinct portrait of Ms.

López beliefs in order to ground our analysis of her reflections, insights and questions
raised during the professional development process.
Especially relevant for our research is Ms. López’s belief on the importance for
her students to learn about how mathematics is embedded in their lives. During our first
interview, she explained that:
If a child has…a solid number sense concept, then the child may be able to not
only relate a problem to the problem itself, but to life. I think the big idea is that
they build, not only for first grade but for later in the other grades and for life _ is
that whatever they are learning, that they can see that connection: how related it
is to life, to everyday life. (November 2006)
This belief is at the core of Ms. López’s practice and her interest in learning more about
CGI.
Ms. López started her teaching career as a bilingual teacher. In her first year
teaching she was placed in a dual language classroom, teaming with the teacher in charge
of the Spanish portion of instruction. During this year she taught math in English to two
different groups of students. That initial experience and the struggles her students
confronted while trying to make sense of the mathematics concepts raised many
questions in her mind around the effectiveness of ESL instruction and framed her actual
conception of bilingual education.
I used to ask, you know, “How is it working? How can they explain a problem?
Do they know enough English that they can explain their reasoning, their
thinking? (. . .). And in some cases they could, and in others they would lack the
vocabulary, and then the concepts. . . . And then again the first year that I taught.
. . . I was doing the math in English (. . . ); but the kids were not understanding. . .
. And then I realized, you know: what's the point of teaching in English? I think
it's important that they build their own native language, in this case Spanish, they
[need to] build academic language to be successful later. (Interview, November
2006)

As we will show later, her mathematics teaching is embedded in her belief that
Latino/a students should be granted access to education in their native language in order
to have the opportunity to develop academic language.
Professional Development Design
Building on our grounding in teachers’ beliefs, we follow a central premise of the
CGI framework (Carpenter et al., 1999), problem solving, to guide our approach to
professional development. It is our goal from this premise to foster teachers’
understanding of the relevance of problem solving in mathematics education. Underlying
this CGI problem solving premise and contained within our goal is the centrality of
promoting “learning for understanding” and the need to form teachers who
know how to help students (a) connect knowledge they are learning to
what they already know, (b) construct a coherent structure for the
knowledge they are acquiring rather than learning a collection of isolated
bits of information and disconnected skills, (c) engage students in inquiry
and problem solving, and (d) take responsibility for validating their ideas
and procedures. (Carpenter et al., 2004, p. 5)
Another premise of our approach is that “professional development opportunities
should engage teachers in what teachers do” (Crockett, 2002). Therefore, teachers are
offered various opportunities to reflect on their practice, discuss activities and their daily
work, design lessons appropriate for students’ needs and grade level, and reflect on
student work. Recently, researchers have promoted the use of student work as a tool to
engage teachers in reflection on students’ learning and thinking (Ball & Cohen, 1999;
Kazemi & Franke, 2004; Little, 2004).
Our involvement with the school and the a group of K-2 bilingual teachers started
three years ago as part of a research project that pursued to explore how Latino/a students
developed mathematical understanding in the context of native language instruction

(Turner, Celedón-Pattichis, Marshall, & Tennison, in press). Several teachers at this
school site participated in CGI summer training facilitated by the Center for the
Mathematics Education of Latino/a Students (CEMELA). Overtime, they manifested
their interest in implementing CGI in their classrooms. Teachers invited CEMELA
researchers into their classrooms beginning in the fall of 2005 to help them develop and
implement problem-solving lessons with their students. The interest of a master
kindergarten teacher in particular helped pave the way for this research collaboration
through her enthusiasm for enriching the reform curriculum with CGI problem solving.
Research in that kindergarten classroom during the 2005-2006 school year showed the
effectiveness of context-rich problem solving and an emphasis on mathematical
communication in developing mathematical thinking with this particular population of
Latino students (see Turner, Celedón-Pattichis, and Marshall, in press).
Our collaborative work with Ms. López started in Fall 2006. We discussed with
her different options to support her interest in learning and integrating CGI into her math
curriculum. At that time, visiting her classroom on a weekly basis seemed to better suit
her professional development interest and our research goals. Per her request, the first
semester consisted of modeling CGI inspired lessons. In addition, we decided that it was
very important to provide for an unstructured debriefing time after each lesson to discuss
the outcomes, challenges, insights, questions, and topics related to issues of language and
mathematics learning pertaining the day’s lesson. During our class visits, typically
involving two or three researchers, we observed Ms. López work in small groups or with
the whole class implementing CGI problems. We provided support to her teaching by
facilitating small groups’ work. During the first semester of our work together, Ms.

López embraced this strategy as a way to introduce herself to facilitating problem solving
mini-lessons with one small group at a time. Every week she decided on the group
configuration. Typically, three groups were sometimes organized based on different
levels of skills, and sometimes were more homogenous in terms of the complexity of
strategies they were using for problem solving.
The second semester, Ms. López decided to conduct the CGI lessons. She crafted
her own way to approach each lesson that most likely fit with her usual configuration of
classroom activities. Typically, she initiated the lesson by presenting problems for the
whole class to think and solve. Students gathered on the carpet in front of the white
board. She would create a story with a context very familiar to the students. She would
begin by bringing them into the conversation and creation of the story. Once they were
engaged, she would develop the numbers of the stories. She emphasized a retelling of the
story by several students and then asked for volunteers to solve the problem. It was
important during this whole group time that students know the context of the story and
relate that verbally to explain their solutions. After whole group examples, students
moved to their desks to work on a whole class problem individually, or to work in a small
group. The mathematics lesson ended with a whole class debriefing and some students
showing the rest of the class their solution strategies.
The debriefing sessions proved to be an essential aspect of the professional
development. As researchers, we approached these conversations from a collegial
perspective to co-construct knowledge about students’ mathematical learning in their
native language as well as to establish a dialogic relationship through which to reflect on
our joint practice in the classroom context as we engaged in planning and implementing

different problem solving lessons, discussed how these lessons articulated with Ms.
López’s curriculum, and how students’ work show evidence of mathematical
understanding. However, we were aware of the power differential affecting our
relationship with Ms. López. We made a conscious and continuous effort to move from
the traditional expert-learner dichotomy that affects professional development endeavors
such as this one. The following excerpt from a debriefing session illustrates the type of
interaction between researchers and teacher (the transcription has been edited to facilitate
reading):
Researcher: We have to think about (. . .) what do we want the kids to do, do
we want to see them being able to find a question and understand a
problem, do we want to see them working with big numbers, do we
want to see if they can come up with an equation, do we want to
see if they can take base ten blocks and work with those and do
exchanges, there are so many things (. . .) so many things going
on..
Teacher: To me there are two things that I’d leave for later even for second
grade, I would cross out big numbers and for the time being I’d
cross out explaining a problem using base ten blocks. I think it’s
more important that they understand the problem and that they
represent it with pictures, numbers and then with words. I think
that asking the questions is important and [also] writing the answer
in a complete sentence (. . .) I know that in their brains they can
reason and they can solve it but I’m also thinking [about] what is
required [by the State benchmarks and standards] is that they
explain the problem or write the sentence and [give the] answer to
the question. . . [in tests] they’ll ask them to give the answer and
maybe explain their answer.
Researcher: Right, right, I think that’s really important too. [Researcher’s
name] had suggested that we use larger numbers
Teacher: I’m not [sure] because in this group you can see [that] (. . . ) some
of them took their little chart and started counting and they came
up [with an answer], and the little boy Juan (…) I know that in his
brain he can solve it but it is hard for him to represent it.

Researcher: I sort of agree with you, I think smaller numbers are OK because
it’s the concept that they are trying to learn right now, the concept
of comparing (…) they are trying to develop (. . .) how to explain
their thinking and if you have smaller numbers it’s the same
concept. . . .they get all confused with that, with the large numbers.
It’d be really interesting if we could change the problem to keep
the numbers simple.
Overtime our conversations changed and our interactions gathered many of the
characteristics of a true collaboration having moved from the teacher being in a ‘listener
position’ to an active dialogue between team partners embedded in a common project
with a shared purpose and clear goals (John-Steiner, 2000; Musanti, 2005). Providing
teachers with the opportunity to collaborate with researchers in the classroom is central to
our belief that teachers should be afforded opportunities to learn from and within the
teaching context (Ball & Cohen, 1999).
In the context of our work with Ms. López and other teachers at this school site,
analysis of student work is an important catalyst for reflection on mathematical problem
solving and takes place in different manners. For instance, teachers have opportunities
during in-class support to reflect on different pieces of student work produced in their
own classrooms.
In our debriefings with Ms. López we tended to focus on student work to begin
our discussions, especially when we had each conducted a small group. We would go
through the papers briefly, or discuss how the students modeled problems with cubes and
each would explain what she observed in the students. These conversations took place on
the same day as the mathematics lesson and were particularly timely because the actions
of the students were still fresh in our minds. We also discussed how we had supported
students’ thinking, what worked and what didn’t. In this way we developed consensus in

our understanding of what had occurred and what the next step should be to help students
move forward.
Data Collection
As part of the ongoing ethnographic longitudinal research, multiple data were
collected. Overall, we worked with 7 teachers on integrating CGI as part of their
instructional approach to teaching mathematics to Latino/a students. For this particular
case study we are drawing from data we collected during 3 intense semesters of work
with Ms. López and her students. Data collection included detailed field notes from
problem solving lessons, audio-recording of the debriefing sessions and three semi-
structured interviews.
Observations. We observed Ms. López on a regular basis while implementing
CGI problem solving lessons, and as part of the in-classroom support provided by
researchers. Field notes were collected of each of these lessons (N=29). During Fall 2007
three of these lessons were audiotaped; segments of these lessons were later transcribed.
Debriefing sessions. (N=21) After each lesson, researchers and teacher spent
some time discussing the outcomes of the lesson, reflecting on students’ strategies,
analyzing samples of their work, planning future lessons, and discussing issues regarding
mathematics learning and students’ language and culture. These conversations were
audiotaped, and selected segments were transcribed. Selection of relevant segments was
done based on researchers’ notes and memos taken during and after each encounter.
Interviews. (N=3) Three semi-structured interviews were held with the teacher,
each lasted approximately 45 minutes, and they took place between the Fall 2006 and
Fall 2007. The interviews explored teachers’ perceptions of mathematics curriculum and

teaching, the nature of mathematics, curriculum integration, the impact of culture and
language, and teachers’ knowledge of students. Interviews were audiotaped and later
transcribed.
Data Analysis
Using the principles of grounded theory (Strauss & Corbin, 1998), interviews
with teachers, debriefing session transcripts, and field notes of classroom observations
were coded. This process involved chunking the data into meaningful units, and then
coding selected statements or interactions using words or phrases that specifically
addressed the research questions (Erlandson, Harris, Skipper et al., 1993). A first stage of
our data analysis focused on coding teacher’s interviews. The interviews were coded with
particular attention to teacher comments in regards to integrating problem solving into the
curriculum, supporting students in problem solving and communicating their
mathematical thinking in Spanish, and adapting the curriculum to meet their students’
needs. Teacher perceptions about the relation between mathematics learning, language
and culture were also coded. We used TAMS
Analyzer, a computer-based qualitative
research tool, to code interview data. Interview transcripts were coded by at least two
members of the research team to establish reliability (Miles and Huberman, 1994).
Differences in interpretation were discussed until agreement was reached. The TAMS
Analyzer tool allowed us to search across transcripts to establish recurring patterns, or
themes. Each theme was triangulated across data of various forms (i.e., debriefing
sessions, field notes, teacher interviews) (Erlandson et al., 1993; Merriam, 1998).
The debriefing sessions were scrutinized in terms of themes, and questions raised
by the teacher. Patterns were identified in relation to teacher’s insights and issues she

raises in relation to the intersection of mathematics teaching and/or learning, and
language. Field notes and videotapes were analyzed to identify recurrent patterns in
terms of how the teacher approached problem solving instruction, the way she presented
the problems for kids to solve, and how she elicited students’ mathematical thinking.
In the section that follows we discuss four themes that contribute to explain
insights gained and issues teacher grappled with during this process. First, data provided
evidence on practices Ms. López developed as part of her instructional repertoire to
integrate a CGI approach into her reform curriculum based teaching. Second, Ms. López
reflections involved discussing the actual reform curriculum she is implementing and
how she perceives it in relation to Latino/a students’ learning needs. Third, Ms. López
grappled with issues of language and how it affects students’ mathematical understanding
even though instruction happens in their native language. Lastly, understanding student
thinking was an ongoing area of reflection and learning. These themes are not mutually
exclusive but intimately interconnected. We discuss the interconnectedness further in the
paper.
Findings
Contextualizing Problem Solving and Scaffolding Students’ Thinking
Teacher practice is a complex and situated configuration constituted not only by
what teachers do in the classroom but also the decision making process behind their
actions, the planning process, assessment of students, and teachers’ thinking process
involving beliefs, knowledge, understandings and emotions. Following Simon &Tzur
(1999), “we see teacher’s practice as a conglomerate that can not only be understood by

looking at parts split off from the whole (i.e. looking only at beliefs or methods of
questioning or mathematical knowledge)” (p. 254).
In what follows, we attempt to illuminate the complexity of a bilingual teacher’s
practice through the lenses of the insights and changes observed in the context of the
professional development described above. Ms. López’s ways of teaching articulate her
understandings about teaching and learning and how mathematics should be taught, her
expectations in regards to students’ learning, her knowledge of students’ culture and
language, the curriculum she is implementing, and the local context in which her practice
is embedded.
In the Fall 2006 we initiated our conversations with Ms. López about how to
incorporate a CGI approach to problem solving in her classroom. Among other topics, we
discussed how providing some context to the mathematics problems seemed to benefit
students’ understanding and their abilities to think about the mathematical situation
presented in the story (Turner et al., in press). In addition, when modeling of CGI
approach was provided by one of the researchers, the problems presented typically
involved the students as protagonists. Consequently, as part of her approach to
introducing mathematics problems for students to solve, Ms. López very often narrated
brief stories before presenting the mathematical information intended in each problem.
For instance, a week after San Valentin celebration, she starts by narrating what they did
and asking students what people usually do for this celebration. Students said they
exchange cards and candies. Then, Ms. López introduced the San Valentin math story
that involved two of the students: Juan and Clara. Ms. Lopez said: “Juan has 4 candies, 4

heart candies for San Valentin’s day, and Clara has 2 more than Juan. How many more
heart candies does Clara have?” (Translated from Spanish).
It is important to note that the way Ms. López approached this practice involved
important nuances. These mathematics stories usually convey events that had one of the
students as a protagonist, involve retelling an experience lived by all students in the class
(e.g. visit to the local Zoo), or an invented story that has students as main actors.
Accordingly, in a study of kindergarten teachers’ practices, Turner et al. (in press) found
that:
Teachers often presented stories in an informal, conversational manner, including
rich contextual information, and inviting students to respond with questions or
comments. By framing problem solving around telling and investigating stories,
teachers drew upon ways of talking and negotiating meaning that were familiar to
children.
Conceptualizing problem solving and creating meaningful stories requires Ms. López to
draw from her knowledge of students, especially appealing to students’ funds of
knowledge (González, Moll, Amanti, 2005). A typical way Ms. López starts her problem
solving lessons is by situating students in a story familiar to them. For instance, in the
following story Ms. López builds from a recent birthday party. Students and teacher are
talking about Rodolfo’s birthday. A student had commented that it was not a party and
the teacher uses this comment to anchor the story and the problem:
T: Tú me has dicho que no, pero yo creo que sí le han hecho una fiesta porque
vinieron sus tíos, sus abuelos y sus primos y para mí que eso es como una fiesta.
(You told me not, but I believe that they did a party for him because his uncles,
grandparents and cousins came and for me that it’s like a party.)
T: Vamos a suponer que en la fiesta de Rodolfo...habían 25 globos. [Underline
indicates emphasis]
(Let’s pretend that in Rodolfo’s party…there were 25 balloons.)
Ss: Veinticinco! (Twentyfive) Ohh!
T: Veinticinco globos...y que de repente vinieron varios niños entre los primos y le
dieron a cada uno de los que vinieron un globo.

(Twenty five balloons...and suddenly several children came, among the cousins
and they gave each of them that came a balloon.)
T: y vinieron diez de sus primos...y a cada uno le dieron un globo.
(and ten cousins came…and they gave each a balloon.)
T: ¿Cuántos globos quedaron al final? [Hands are up] (How many balloons were left
at the end?)
T: Primero piensen [pointing to her head] (First think)
T: ¿Cuántos globos había en la fiesta de Rodolfo? (How many balloons there were at
Rodolfo’s party?
Ss: Veinticinco. (twenty five)
T: Veinticinco. Podemos poner el número para ayudarnos. [Teacher writes the
number on white board]
(Twenty five. We can write the number to help ourselves.)
T: Había veinticinco globos...(. . .) y vinieron diez niños y le dieron a cada uno un
globo. ¿Cuántos globos quedaron al final?
(There were twenty five balloons…(. . .) and ten children came and they gave
each one a balloon. How many balloons were there at the end?)
This lesson’s excerpt shows how Ms. López presents the problems embedded in a
familiar story. Then she focuses on making sure students understand the information
provided by the story. In order to do this, she retells the story, she stresses the words that
contain relevant information, and she asks closed questions to make sure students recall
the facts. This is coherent with the way she defines how she scaffolds students to explain
their thinking, “to make sure they understand what [the story] means”:
If it’s orally (. . .), we try, with the problems [asking questions] what is the story,
tell me what is the story, what is the question. To make them say it. That’s one
way. And then when they’re reading whatever, an instruction or a problem or a
story, to make sure they understand what they are asking.., ask[ing] them
questions, by asking them a lot.(Interview, November 2007)
A very important insight underlies beneath her emphasis on providing students
with opportunities to solve problems while making sure students understand the problem
they are presented with:
These last two years, (. . .) I realized (. . .), that-before I didn’t do it. [Problem
solving] was not part of the program. But the fact that we have that hour

separated for that, it’s like we are working, a little bit every Wednesday, and the
kids are getting used to talking more, (. . .) explaining more, being more clear… I
think it’s crucial. And this is connected to what they are being asked, not only in
math but in reading too. They need to make connections in reading. The story they
read how does it connect to their own life. (Interview, November 2007).
Ms. López is operating from and with the belief that it is important to provide Latino
students with opportunities for meaningful and situated learning that “connect[s] to their
own life.” In this regard, research has explained the role of context-embedded learning
situations to support comprehension (Celedón-Pattichis, 2004).
During our debriefing meetings, we discussed at length the importance of
providing students with opportunities to verbalize their thinking and to explain the
strategies they use to solve the ‘mathematical stories.’ The modeling sessions also
provided examples of different uses of questioning to help students to explain how they
solve the problems. Ms. López constructed overtime her own approach at using
questioning to scaffold students’ explanations. In doing so, she is working on establishing
classroom norms of interaction that are specifically relevant for when they engage in
problem solving. From a sociocultural perspective, Ms. López is providing an external
scaffold for students to discuss their solutions that ultimately allows them to construct an
internal understanding of the mathematical ideas. McClain & Cobb (2001) explained the
importance of teacher’s role in the construction of sociomathematical norms that define
the classroom discourse and the conditions of possibility for teaching and learning. The
following classroom episode from February 2007 (Field notes) illustrates how the teacher

uses questioning and how this becomes part of a pattern of interaction between teacher
and students through which she actively guides students’ mathematical thinking.
The teacher presented a compare problem to students (See Table 1 for examples
of the different types of CGI problems). They have to establish who has more pencils and
how many more. Students have been struggling to understand the concept behind
comparing two quantities. Their tendency was to add the numbers even before they
listened to the problems’ question. Ms. López had been consistently working on focusing
students’ attention on the problem information, especially she wanted students to be able
‘to listen’ to the question. In this particular episode, Alan and Belen are modeling for the
group the problem posed by the teacher using pencils.
T: Belen tiene 8 lápices y Alan tiene 3 menos que Belen.
(Belen has 8 pencils, and Alan has 3 pencils less than Belen.)
[Alan takes pencils, he thinks for a while, then gets 5.]
T: Belen tiene… ¿cuántos lápices? [Teacher directs the question to the class.]
(Belen has, how many?)
Ss: Ocho! (Eight)
T: Alan tiene 3 lápices menos que Belen. ¿Quién tiene más lápices?
(Alan has 3 pencils less than Belen. Who has more pencils?)
Ss Belen.
T: ¿Cuántos más? (How many more?)
Ss: Tres más. (Three more)
T: ¿ Cómo lo explicamos...que ella tiene más? (How do we explain that she has
more?)
[A student (S1) is called to the front. Teacher repeats the problem and asks him to
show how he solves it.]
T: ¿Cuántos más tiene él que ella? (How many more does he have than her?)
S1: Tres. (Three)
T: ¿Tres más? Vamos a ver...cuéntalos. (Three more? Let’s see...count them.)
[S1 counts pencils.]
S1: El tiene 5…..ella tenía 8 y tiene 3 menos. (He has 5. She had 8 and he has 3 less.)
T: ¿Cuántos tiene Alan? [To the class] (How many pencils does Alan have?
Ss: Cinco. (Five)
T: ¿Quién tiene menos lápices? (Who has less pencils?)
Ss: Alan.
T: ¿Cuántos menos? (How many less?)
Ss: Tres menos. (Three less)

T: ¿Cómo sabemos? (How do we know?)
S2: If she has 8 and we take away 3, then he has five. [Student answers in English]
(Field Notes, 2/28/07)
Even though a first look at these data could indicate that there is too much teacher
talk and too many closed questions3
, a more in depth and contextualized analysis shows
that this approach initially might be effective to scaffold students to notice relevant
information in the story, to establish a correlation between the action modeled and the
verbalization of those actions. There is a clear sequence and rationale in the questioning
that this teacher uses consistently. Initially, she asks students to retell the story. This
seems to be important to situate students’ mathematical thinking in a concrete and
familiar context. She follows with a series of closed questions (e.g. How many more?
How many less? ) that require students to provide factual information. The teacher
believes that this type of questions seems to help them recover the information they need
in order to solve the problem. As the example shows, she interjects open-ended questions
to elicit students’ explanations about how they came up with the answer (e.g. How do we
know? What did you do? How did you think?). Our prolonged involvement in Ms.
López’s classroom permits to infer that her use of questioning provides students with an
external scaffold to their thinking that eventually could become internalized as tools that
students can use while thinking about a problem’s solution.
Learning While Adapting Curriculum
We contend that professional development needs to be grounded in teachers’
practice and curriculum implementation in order to impact students’ achievement. As
Drake & Sherin (2006) affirm, “teachers must be considered as critical agents in their
3
We understand closed questions as those that require one-word answers or fill-in-the blank type of
response.

own learning from and implementation of curriculum” (p.154). This is particularly
important when trying to make sense of bilingual teachers’ growth as they engage in
adapting a mathematics reform curriculum to fit Latino/a students’ learning needs,
especially considering different aspects of language and culture that become intertwined
with the teaching and learning process.
We approached our collaboration with Ms. López with an emphasis on infusing a
CGI perspective on problem solving into the reform curriculum the school was
implementing. However, considering issues of language and culture in relation to
Latino/a students’ mathematical understanding was also a central focus of our joint work.
Our conversations with Ms. López involved discussing how this particular approach to
problem solving in first grade articulated with her curriculum. Ms. López’s reflections
illustrate her insights on this matter.
She saw integrating contextualized problem solving into her reform curriculum as
a positive and necessary addition because she considered it lacking in this area in terms
of the quality and quantity of problems offered for students to solve. She commented:
[The mathematics curriculum] does not work a lot with story problems. They give
one example which is very simple and then the kids get hooked on that example.
They all do the same example with different animals and different people but the
same thing. (Interview, April 2007)
Her years of experience teaching and using this reform curriculum, together with
the trainings she had participated in order to learn about its implementation provide her
with the tools to assess it in relation to what she perceives are the learning needs of her
students. These experiences have impacted her self-perception, allowing to define herself
as authority in relation to curriculum implementation. Her words are eloquent:

When I first started, I would follow what the book says and as I was learning
more, I realized what was more important. It's like, "Oh, this is really not that
important, it is OK" and I will touch on it, but it's not crucial, so I made my own
decisions of what I think is more important. (Interview, November 2006)
We acknowledge that her stance is not only the outcome of our collaboration, but
it is an essential element in this bilingual teacher’s growth path and in her decision to
participate in learning about how an instructional emphasis on problem solving and
communication can affect Latino/a students’ mathematical understanding. Ms. López
understood the importance and challenges of linguistically and culturally contextualized
mathematical learning and recognized the importance of scaffolding children’s thinking
so that they value mathematics in their everyday lives. This understanding allowed her to
see a disconnect between certain tasks proposed by the actual reform curriculum and
Latino/a parents’ background knowledge (Civil, 2002). She provided countless examples
that illustrates the disconnect. For instance, a homework activity asked parents and
students to measure elements using cooking measurement tools such as measurement cup
and the scale. Students came back the next day with the assignment incomplete due to the
fact that it is not a common practice in Mexican families to use measurement tools. They
rely on estimating the needed quantities for each recipe –referred as calculating “a ojo” or
eyeballing. This type of experiences provided Ms. López with an important insight into
her curriculum that despite its Spanish translated version still presents students and
parents with linguistic and cultural barriers that she needs to mediate. In this process, we
found that she is progressively reconceptualizing her teaching role as mediator between
home culture and curriculum. Her attempts at bridging the distance between family funds
of knowledge and curriculum requirements translated in different ways to promote more
parent involvement. For instance, she carefully looks over the homework designed by the

curriculum and adapts it when necessary to facilitate parents’ understanding and making
possible for them to help their children to complete the task. She also encourages parents
to participate in early morning activities, and welcomes them to stay to observe or
collaborate with different tasks. During Fall 2007, she decided to design and implement a
mathematics workshop for parents to demonstrate for them how they can create problems
for children to solve at home. She presented this idea to us explaining that she realized
how different the actual approach to teaching mathematics is from what Mexican parents
had experienced as learners. Her goal was to provide them with tools to understand the
importance of mathematical reasoning over procedural learning.
We believe that central to bilingual teachers development is the understanding of
students’ culture and the validation of families’ knowledge and language. Research has
demonstrated that central to enhancing students’ learning is the instructional integration
of cultural practices, and family and community knowledge (e.g., Civil, 2006; González,
Moll, & Amanti, 2005). Ms. López understood that in order to empower her students to
learn mathematics she also needs to include their parents acknowledging what they know
but, at the same time, providing them with resources that can support students’ academic
achievement.
Teaching in Spanish and Building Academic Language
We initiated our collaboration with Ms. López with the idea of exploring how
language plays a role in learning and understanding mathematics. Overtime, language
and oral communication became a frequent topic raised by Ms. López as a central area
for students to develop. Discussing the characteristics of teacher discourse as an avenue
for Latino/a students’ mathematical learning, Khisty (1995) explained that “Talk (. . . ) is

the critical vehicle by which an individual internalizes meanings” (p.290). The teacher
clearly understood that even speaking in the same language, she needed to make sure that
her language of instruction was comprehensible and, at the same time, a source of
enrichment of students’ linguistic resources. Meanwhile, her instruction became
progressively more informed with the importance of introducing specific mathematical
language while encouraging students’ oral or written representation of their reasoning.
Supporting Latino students’ academic language development, even while teaching
in students’ native language is challenging but crucial to their future academic success.
Research has reported that Latino students’ low graduation rate (45% of Mexican
American students do not complete high school) is due not only to their English
proficiency but to the lack of opportunities they are afforded to develop a strong
academic discourse in Spanish (López-Bonilla, 2002). There is no doubt that minority
students who have the opportunity to learn in the context of a bilingual program transfer
the skills learned in their first language to the second language (Baker, 2006). Ms. López
reflected about the difference between everyday language and academic language
(Bielenberg & Wong Filmore, 2004/2005; López-Bonilla, 2002; Valdés, 2004). This
teacher often expresses her concerns about the academic and testing demands these
students will be exposed to in the future, and the need to generate instructional moves
that will improve their academic language. Many of our conversations focused on how to
build on the differences between academic vs. everyday language, and since the
beginning of our collaboration she has been grappling with understanding the
interconnection of both in the context of instruction:
Well, I look at the academic language, native language _ Spanish _ and the
everyday language. And it is not the same. And the majority of the kids that I

have here, their level _ it's not limited language, but it's daily everyday common
language. At home, they don't receive - the majority - do not receive academic
language. (Interview, November 2006)
The notion of academic language should be defined not only in terms of the
specific structures and a specialized lexicon but with reference to the sociocultural
elements that integrate any socially accepted discourse (López-Bonilla, 2002). Initially,
Ms. López’s conception of academic language seemed to be restricted to the introduction
of the specialized vocabulary.
If you are not used to explaining your reasoning, in Spanish, or whatever
language you talk, if you don't know the vocabulary, you are not going to be able
to explain. (…) when we look at specific words that they use, specific vocabulary
that we use in math, (…) it's not (…) a vocabulary, that you use everyday, so it
has to be taught (…) _ and practiced. (Interview, November 2006)
However, as our work with her progressed, and we observed her teaching and her
interaction with students, we conjectured that her emphasis on improving students’
vocabulary was not construed with disregard to other linguistic and cultural elements and
students’ meaning construction in the context of mathematics learning. On the contrary,
her vision of academic language seemed to be embedded with a sociocultural
understanding of learning and teaching. Clearly, she understands the nuances of language
and how words’ meaning are constructed in context and immersed in culture. In a recent
conversation, she brought up the mathematical term “pattern” that is translated in her
actual curriculum as “patrón.” As she explained, the word ‘patrón’ has different meanings
in Spanish, depending on the context of use it could mean boss or pattern. However, she
reflected on the important social and cultural connotations of this word. The ‘patrón’ is a
figure of authority, typically a respected authority, and a broadly used concept among
Mexican American families. The mental representation that the use of this word entails

for Latino students and families is usually one of a male figure. Her awareness of how
culture impacts meaning construction provides her with important resources to integrate
in the construction of the mathematical discourse in her classroom, helping students
differentiate the nuances of language while they internalize the mathematical concept of
pattern.
Overtime, Ms. López became actively engaged in discussing alternatives to
extend the mathematics language in the classroom so that her students can communicate
mathematically and explain their thinking (Khisty & Chval, 2002; Moschkovich, 2007).
Concurrently, Ms. López grappled with deconstructing her view of students as lacking
vocabulary and understanding students’ linguistic resources to express their thinking.
I've noticed the language part, the fact that the kids, I don't [want to] say that they
have a limited language ability, but the fact they cannot explain, verbally, many
things, you know. They call things, "esto, esto, lo otro" [this, this, or that. They
don't have a specific vocabulary for some things. (Interview, April 2007)
Ms. López movement to a more comprehensive view of academic language is a key
component of her ongoing growth and her in depth understanding of what entails to learn
mathematics in the actual reform context. As Moschkovich (2007) explain, focusing on
vocabulary development narrows the view of mathematical communication. “The narrow
view can have a negative impact on assessment and instruction for bilingual learners” (p.
5). Another important element in Ms. López’s approach was the way she prioritizes
teaching mathematics concepts in the native language, Spanish (Cummins, as cited in
Baker, 2006). In a recent interview, we explored with her the issue of language as an
intervening agent in mathematical learning:
T: Because I teach in Spanish, I’m addressing them in what they know…but
even though it’s their native language for the majority, some words and

concepts that they do not know for whatever reason, and I can tell you
some of them: ‘antes y después’ (before and after).
R: They don’t know the words?
T: They know the words but they don’t know what they mean. . . .If you ask
them: ¿Qué viene antes..? (What is before..?)And they’re not sure what is
antes o después (before and after). Many times they say ‘ayer’ (yesterday)
instead of mañana (tomorrow). They get confused with those terms. And
it’s not all of them, it’s some of them.
R: What other words do they have problems with?
T: In those problems that we do…the compare [CGI problems]: ¿Cuál es la
diferencia? (What is the difference?)
. . .
T: The way they understand better, es ‘cuanto le falta a él para que tenga
tanto como yo’ (How much is needed so he has the same than me?) Esa es
la manera que lo entienden. (That is the way they understand it)
. . .
T: And I wonder if it is because they don’t use these terms that often in their
language or [she didn’t finish this sentence].
Ms. López’s unfinished sentence hints on her search for a broader and deeper answer to
her question on the meaning and impact of academic language development. This excerpt
shows new elements in her thinking about language, not just as words but also as
meaning making. She seems to comprehend that in order for students to make meaning
of her words she needs to build from what they understand (e.g. How much is needed so
he has the same than me?) to take them further in the use of a specific mathematics
register (e.g. What is the difference?). The importance of instructional strategies that
foster the development of students’ mathematical register has been documented
extensively (Celedón-Pattichis,2004). The challenge for us and for Ms. López is to
continue elaborating our conceptualization of academic language beyond the specific
language required for a subject matter “that will automatically be acquired through
content-based instruction” (Valdés, 2004, p. 121). As Valdés (2004) contends, there is no
consensus among stakeholders about the meaning of academic language. In the meantime

and despite the context of instruction, teachers are expected to teach academic language
and students are expected to learn it.
Understanding Student Thinking
Ms. López repeatedly reflected on her new insights regarding students’ thinking
and mathematical understanding as a result of collective (teacher-researchers) inquiry on
students’ work. As we explained before, analyzing student work was a reflective tool that
we introduced in our debriefing sessions and that overtime became like a natural source
we drew from in our conversations. Often times, Ms. López brings up a piece of student
work (produced during problem solving sessions or as homework) to illustrate her
thoughts and insights on students’ mathematical learning. We believe that this reflective
process is essential for teacher development, especially in the context of mathematics
reform. As research has contended, one of the biggest challenges of changing the way
mathematics is taught relates to providing teachers with learning experiences that can
help them change their perceptions of the nature of mathematics and the way it should be
taught (Crockett, 2002).
Ms. López gained awareness of the different problem solving strategies students
were developing:
They are using that, and so the counting on, let's say that they're counting
whether it's crayons or the coins or the number line I can see that they can count
by tens and that they can, do make the jump to, to switch to the one, to the fives
it's a little more difficult. (April 2007)
The awareness of the different strategies students were using to solve the different
problems she presented impacted her insights about students’ representations of their
solutions to the problems. Ms. López developed an increasing understanding of how
students’ pictorial representations and verbalizations of their solutions to problems gave

insight into students’ thinking about mathematical problems. Collectively, we found that
central to our conversations was the examination of how students’ mental images or their
internal cognitive representations translated into external representations, either verbal or
pictorial (Goldin & Shteingold, 2001). In February 2007, after a problem solving session
on compare problems with a set unknown (See Table 1), Ms. López observed how
students struggled representing in drawing their thinking. One of the problems posed to
students was: “Ms. Mary (a researcher) has 11 pencils and Ms. Sandra (another
researcher) has 7 pencils less than Ms. Mary. How many pencils does Ms. Sandra have?”
The teacher noticed students were able to solve the problem mentally but had trouble
finding a way to represent it. “I've seen that they can reason it in their heads, they still
have a hard time to put it, to show it in paper or even to explain it. They still have a hard
time explaining how do you do it, how do you solve this” (April 2007). The following
example of Jenna’s drawing and writing illustrates aspects of our conversation:
Figure 1. Jenna’s drawing of the pencil comparison problem
Jenna was able to solve the problem and find the right answer (4 pencils), but her
drawing does not show how she came up with that answer either does the equation she
chose to include. Situations like this one pushed Ms. López thinking forward as she

engaged in trying to make sense of these discrepancies while reflecting on instructional
moves that could help students construct representational tools.
Gaining insight on how drawing is “a window into the mind of a child” (Woleck,
2001, p. 215) and how it can become a mathematical tool to represent and think about a
particular problem is an important outcome of our collaboration. According to the NCTM
Standards, “Representing ideas and connecting the representations to mathematics lies at
the heart of understanding mathematics” (NCTM, 2000, p. 136). Clearly, the process of
finding ways to support students’ verbal and pictorial representational skills is not an
easy one. Ms. López grappled with finding a balance between allowing students to solve
and represent problems on their own, and demonstrating for them effective problem
solving strategies. While she understood the importance that students develop linguistic
and representational tools to share their thinking, she questioned to what extent her
interventions support their capacity to explain their thinking. Many times during our
conversations she raised the question to what extent she should demonstrate or not
different ways to solve the problems. It seems clear to us that this type of questioning is
what generates opportunities for learning in and from practice (Ball & Cohen, 1999).
Incorporating CGI problem solving into her mathematics instruction afforded Ms. López
with a significant opportunity to learn from practice. Especially in what relates to the
impact of teacher’s careful scaffolding of oral and written communication in students’
native language to support the development of mathematical process skills fundamental
to success in reform mathematics (NCTM, 2000).

Final Considerations and Implications
Ongoing reflection, collegial conversations with researchers, and a focus on
analysis of student work contributed to enhance teacher’s understandings of students’
mathematical learning, especially in relation to practices that provide students with
opportunities to solve contextualized mathematics problems, to communicate their
solutions, and to represent their thinking as a pathway to the development of higher
thinking skills (NCTM, 2000). Ms. López’s experiences as learner and teacher of
mathematics (Drake & Sherin, 2006), her beliefs (Aguirre & Speer, 2000) and
understandings of bilingual education (Varghese, 2004) shaped her implementation of
CGI as a reform-oriented approach to mathematics instruction.
The outcomes of our collaboration with Ms. López provide evidence of the
relevance of creating ‘situated’ professional development communities “that promote the
practice of shared inquiry grounded in teachers’ work” (Crockett, 2002, p. 609). In
addition, focusing on reflecting about teacher’s practice from the perspective of student
work is an effective strategy to situate professional development in the context of the
classroom while keeping in perspective the wider educational context defined by the
adopted reform curriculum, the school’s approach to bilingual education, and the Latino
community in which it is embedded. The importance and uniqueness of this type of
approach to professional development lies in our focus on building it from within,
meaning that we initiated our work with Ms. López upon her agreement and we
constructed it with her classroom as the ‘center’. We genuinely believe in creating a

community of learners and practice in our work with the teachers (Wenger, 1998), we
learn in the process as much as we hope Ms. López learns with us.
Traditional explanations to the poor performance of Spanish-speaking Latino
students from low socioeconomical background have focused on individual and cultural
deficits. These misconceptions sometimes hinder teachers’ best teaching intentions as
they assess students’ struggles with learning any subject matter. In this case, Ms. López
worked at deconstructing the assumption that students lack the language needed for
learning, even though this assumption is still permeating the actual interpretation of the
achievement gap. Professional development should afford bilingual teachers the
opportunity to experience, in the context of their practice, that Latino students can
successfully develop and communicate complex mathematical thinking (Khisty, 1997;
Turner et al., in press). Granting Latino students with opportunities for quality
mathematics learning experiences requires bilingual teachers prepared to comprehend
how subject matter learning is entangled with language development and cultural
background. We believe that more research is needed that explores how professional
development can better support bilingual teachers in the process of adapting reform
curriculum for Latino/a students.
Our work with Ms. López illuminated the potential of professional development
initiatives that validates bilingual teachers’ agency in terms of enacting curriculum and
language policies (Varghese, 2004). Ms. López actively reflected about her curriculum,
and how she perceived it addressed the needs of her students. In addition, she searched
for avenues to improve her use of language in the classroom to make sure she was
maximizing (to the best of her possibilities) the quality of her students’ learning

experiences. Based on our understanding of what is effective professional development,
we believe this is a crucial outcome. Teacher change does not occur as a consequence of
a one-day training, or a month long series of workshops. Teacher change is the result of
the ongoing and complex interplay of teaching and learning. It has been widely argued
that teacher development is an ongoing process that requires the creation of “school
cultures where serious discussions of educational issues occur regularly, and where
teachers’ professional communities become productive places for teacher learning”
(Franke et al., 2001). Ms. López’s committed quest for understanding and for practices
that best fit the learning needs of her Spanish-speaking students probes the importance of
searching for effective ways to support bilingual teachers that are committed to improve
the quality of Latino students’ learning experiences.

Table 1. Selected CGI Problem Types (English Version)
Problem Structure Problem
Join Result Unknown a. María has 6 candies. Her sister gives her 6 more candies.
How many candies does María have now?
Separate Result Unknown b. Corina had 14 cookies. She ate 6 of them. How many
cookies does Corina have left?
Join Change Unknown c. Karla wants to buy a toy plane that costs 11 dollars.
Right now, she only has 7 dollars. How many more dollars does
Karla need so that she can buy the toy plane?
Multiplication d. Antonia has 4 bags of marbles. There are 5 marbles in each
bag. How many marbles does Antonia have altogether?
Partitive Division e. Marcos had 15 marbles. He shared the marbles with 3 friends
so that each friend got the same number of marbles. How
many marbles did each friend get? (Marcos did not keep any
marbles for himself.)
Measurement Division f. Diego had 10 cookies, and some little bags. He wants to put 2
cookies in each bag to give to his friends. How many bags can
he make?
Compare g. Mario has 12 toy cars. His sister Rebecca has 9 toy cars.
How many more toy cars does Mario have than Rebecca?

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