Running Head: SUPPORTING PROSPECTIVE ELEMENTARY ... · Supporting Prospective Elementary Mathematics Teachers’! 5 Loucks-Horsley, Stiles, Mundry, Love, and Hewson (2010 ) emphasize

Running Head: SUPPORTING PROSPECTIVE ELEMENTARY MATHEMATICS TEACHERS’

Supporting Prospective Elementary Mathematics Teachers’ Learning through Book Study

Gemma Mojica and Stephanie Wright The University if North Carolina at Chapel Hill

Chapel Hill, NC

Corresponding author: Gemma Mojica, School of Education, University of North Carolina at Chapel Hill, Campus Box 3500, Chapel Hill, NC 27599-3500, (919) 962-6608, [email protected]

Supporting Prospective Elementary Mathematics Teachers’ 2

Abstract

This design study explored a model that closely linked theory about students’ understanding of

fractions with practice in 3rd through 5th grade classrooms. Student teachers and their cooperating

teachers from the southeastern region of the US participated in professional development which

focused on linking theory in relation to children’s thinking about fractions and how to utilize this

knowledge in practice (Empson & Levi, 2011). Part of this model involved implementation of

Cognitively-Guided Instruction (CGI) word problems, selection of student work, and reflection

on interpretations of students’ understanding in discussions. Additionally, we utilized the Five

Practices (Smith & Stein, 2011) as a model for facilitating mathematically rich discussions

around CGI fraction work to connect theory taught in STs’ elementary mathematics methods

course to the Book Study. Findings indicate that the CGI Book Study supported most of the

student teachers’ understanding of children’s reasoning so that they were able to monitor, select

and sequence students’ strategies during discussions about CGI problems in their fraction

lessons.


Supporting Prospective Elementary Mathematics Teachers’ Learning through Book Study

While it is evident that teacher education programs need to address the demands of

preparing prospective teachers to design reform-oriented instruction, the preparation of

elementary teachers is often inadequate for producing the types of knowledge needed to enact

practice where all children learn meaningful and significant mathematics (National Research

Council, 2001). We argue that this is, in part, a result of the current structure of traditional

teacher preparation programs, where foundational knowledge is separated from methods courses

and coursework is disconnected from practice (Grossman, Hammerness, & McDonald, 2009).

Grossman et al. (2009) emphasize that traditional structures often separate theory from

the practical work that teachers do in the classroom, making practice a nonessential aspect of

preparation. Korthagen and Wubbels (2001) point out that student teachers (STs) rarely use

theory taught in teacher education programs and are subsequently unprepared to solve

problematic situations they encounter as novice teachers.

Another widely acknowledged challenge in preparing prospective teachers requires that

they come to understand teaching differently than what they experienced as learners in

traditional classrooms (Hammerness, Darling-Hammond, Bransford, Berliner, Cochran-Smith,

McDonald, & Zeichner, 2005). Lortie (1975) first described this dilemma as the apprenticeship

of observation, where prospective teachers’ beliefs about teaching are based on the ways in

which they were taught. At times, prospective teachers’ own experiences as students result in an

orientation to teaching mathematics that is in direct opposition to reform-oriented practice, such

as the role of the teacher is to “show and tell,” and the role of the student is to practice

procedures and learn facts (Ball, 1988; McDiarmid, 1990). For some prospective teachers, this is

only reinforced in university mathematics courses (Sowder, 2007). This may also be reinforced


by what they see in their field experiences in elementary classrooms and student teaching. If

traditional teacher education programs are not currently organized to best support teacher

learning about practice that is based on theory, to what extent do alternative models of

preparation have the potential to do so? Through our work with prospective teachers and their

cooperating teachers (CTs), we have evidence that participation in professional development

with their CTs has the potential to support student teachers (STs) in their initial teaching

experiences. Specifically, we explored the following research question: To what extent, if any,

do STs participating in a CGI Book Study utilize the Five Practices to facilitate whole class

discussions around fraction word problems?

Book Study Model

We propose a model for teacher preparation that closely links theory taught in methods

courses and professional development (PD) with classroom practice. Further, we argue that an

effective model of teacher preparation should emphasize a connection between theory and

classroom practice where STs and CTs build common knowledge and STs have opportunities for

enacting clinical experiences. Thus, we designed a yearlong (PD) where STs and their CTs

participated in a Book Study that culminated in STs implementing fraction units during their

student teaching. In our Book Study, we used Empson and Levi’s (2011) work as a framework

for organizing teachers’ understanding about children’s thinking about fractions. Additionally,

we utilized the Five Practices (Smith & Stein, 2011) as a model for facilitating mathematically

rich discussions around Cognitively-Guided Instruction (CGI) fraction work to connect theory

taught in STs’ elementary mathematics methods course to the Book Study.


Loucks-Horsley, Stiles, Mundry, Love, and Hewson (2010 ) emphasize that there is

consensus about what constitutes effective professional learning. Loucks-Horsley et al. identify

the components of effective PD as learning that is directly

aligned with student learning needs; is intensive, ongoing, and connected to

practice; focuses on teaching and learning of specific academic content; is

connected to other school initiatives; provides time and opportunities for teachers

to collaborate and build strong working relationships; and is continuously

monitored and evaluated (p. 5).

In addition to the first three components identified by Loucks-Horsley et al., our model

was built on designing opportunities for teachers to collaborate and build strong working

relationships. An important component of our Book Study involved ST/CT teams implementing

CGI fraction word problems in their classrooms, selecting student work for analysis to share

during sessions, and reflecting on their interpretations of students’ understanding in their

discussions with other STs and CTs. In Book Study sessions, we often discussed how the Five

Practices (Smith & Stein, 2011) could be used to organize whole class discussion after children

had solved CGI problems using their own strategies and representations. The student work

samples collected by ST/CT teams was a mechanism to support STs and CTs as they learned to

use students’ work on CGI problems as a launching point to facilitate productive discussions. An

experienced mathematics teacher educator designed and led all Book Study sessions.

Conceptual Framework

The conceptual framework for this study is comprised of two perspectives. Empson &

Levi’s (2011) extension of CGI for fractions was used as a framework for organizing teachers’


understanding of children’s thinking about fractions. Smith and Stein’s (2011) Five Practices

was utilized as a framework for organizing classroom practice.

Empson and Levi’s (2011) Extension of CGI for Fractions

Many educators and researchers recognize that an important component of effective

teacher learning is a focus on student thinking (e.g., Franke, Carpenter, Levi, and Fennema 2001;

Hammerness et al. 2005). CGI, developed by Carpenter and his colleagues, is an example of a

well-established program that was designed to assist teachers in understanding student’s informal

reasoning and to use that understanding to support students as they learn mathematics with

understanding (Carpenter, Fennema, & Franke, 1996). Levi and Empson (2011) have extended

this work beyond a focus on whole-number word problems to learning about fractions and

decimals, while staying true to the CGI philosophies. Levi and Empson’s book assists teachers in

supporting their students as they develop meaning for fraction concepts through discussing word

problems. The authors also highlight the progression of students’ strategies for solving fraction

word problems. Finally, the authors focus on helping teachers design instruction based on

students’ strategies.

Smith and Stein’s (2011) Five Practices

Recognizing that implementing pedagogical approaches that build on student thinking,

such as CGI, is challenging for teachers, Smith and Stein (2011) and colleagues (Stein, Engle,

Smith, & Hughes, 2008) have identified instructional practices that make student-centered work

manageable. According to the authors, “the purpose of the five practices is to provide teachers

with more control over student-centered pedagogy. They do so by allowing the teacher to

manage the content that will be discussed and how it will be discussed” (p. 12).


Smith and Stein (2011) describe these practices as skillful improvisation: anticipating,

monitoring, selecting, sequencing, connecting. In anticipating, teachers try to anticipate students’

likely responses to a particular task. Rather than focusing on right or wrong answers, teachers

should try to anticipate different strategies students will use to solve problems, as well as

common misconceptions or errors. Next, teachers monitor student work, paying close attention

to the ways in which students solve problems. Teachers then select students to display their work

in a specific order known as sequencing. This particular sequencing might begin with the most

popular strategy and then move to other, less popular strategies, or it may move from least

sophisticated to more sophisticated strategies. As students share their solutions, the teacher

facilitates the discussion, making connections from one strategy or explanation to another. The

teacher may also connect the solutions given to other previous tasks or build on previous

knowledge.

Methods

Context and Participants

This study took place in the southeastern United States. Eleven CTs from five different

elementary schools, in third through fifth grade classrooms, participated in this study. Six

prospective elementary teachers specializing in mathematics and science were placed in one of

the five schools for their student teaching experience. CTs were strategically chosen because we

believed STs would be more likely to see reform-oriented instruction enacted in real classroom

practice, connecting what they were learning in their methods courses and their student teaching

experiences. Many CTs either held or were working toward a Master of Education for

Experienced Teachers or other advanced degrees. Many of the CTs were familiar with CGI and

were already implementing work around addition, subtraction, multiplication and division in


their classrooms. Five of the four STs were selected to include various levels of proficiency in

implementing reform-oriented mathematics instruction. Carol, Grace, Lizzie, and Nicole were

student teaching in third grade classrooms, and Sandra was in a fourth grade classroom.

Data Collection and Analysis

The Book Study began in the fall of STs senior year and continued into their student

teaching experience in the spring. In the fall, STs were enrolled in their methods course and

participated in the Book Study as an additional experience. We studied the five STs as they

implemented their fraction lessons in the spring. The Book Study sessions and STs entire

fractions unit were video recorded. Other sources of data included the researcher’s field notes,

interviews, and children’s work from the fraction lessons. Video recordings and audio recordings

were analyzed using ATLAS.ti and coded based on a codebook (DeCuir-Gunby, Marshall, &

McCulloch, 2010) created by the research team.

Findings

Task Selection and Implementation

It was evident that all of the ST/CT teams selected CGI fraction word problems and

implemented them in their classrooms. However, the implementation of CGI fraction work

varied greatly from classroom to classroom. All of the ST/CT teams implemented the CGI

fraction word problems in the classroom at least once a week during the fall. In the spring, four

of the five STs increased the number of times they implemented the CGI fraction problems. All

of the STs structured their mathematics instruction using their CT’s structure as a model,

incorporating CGI work the same way as their CTs.

Daily implementation. Grace and Nicole implemented a CGI fraction word problem

everyday during their fraction unit in the spring. CGI word problems were a part of the daily


mathematics routine but were usually followed by other mathematical tasks. They often selected

word problems from the Empson & Levi (2011) book from the Book Study or modified

problems. In both classrooms, students solved their problems in their math journals recording at

least one strategy and an explanation of their strategy. Grace implemented the CGI problems as

morning work and monitored the children as they solved problems individually in their journals.

Grace would select three to four strategies and would to lead a whole class discussion where

children shared these strategies. Discussions usually took fifteen to twenty-five minutes.

Students would then work on multiplication and division skills brief and continue with another

forty to forty-five minutes of math instruction. Nicole implemented the word problems before

her math lesson and monitored students as they worked. Nicole typically selected one or two

students to share during the whole class discussions that would usually last about ten to fifteen

minutes. In some cases, the CGI fraction word problem was tied back into the math lesson. Both

Grace and Nicole led mathematically rich discussions based on their selection and sequencing of

students’ thinking.

Weekly implementation. Lizzie and Sandra implemented a CGI fraction word problem

at least two to three times a week during their fraction unit in the spring. Lizzie’s implemented

the word problems before her math lesson and monitored students as they worked, usually

selecting one or two students to share during the whole class discussions which usually lasted

about ten minutes. Lizzie and Sandra sometimes used CGI word problems as her lesson

facilitating whole class discussions based on student strategies. Sandra did this more frequently.

No evidence of implementation. There is no evidence that Carol implemented CGI

fraction word problems during her fraction unit in the spring. Instead, when students at her grade

level went to mathematics interventions outside of math instruction, some students were given


fraction word problems to solve. Typically, these students were those identified as Academically

or Intellectually Gifted (AIG). Students did not discuss student strategies as a whole class. The

bulk of her fraction lessons were based on direct instruction and did not incorporate CGI work.

Monitoring, Selecting, and Sequencing

Four of the five STs made explicit connections to using the Five Practices to organize

whole class discussions as students’ shared their strategies and thinking for fraction word

problems, to varying degrees. Some STs made these connections while observing their CTs.

Others made this connection while reflecting on their own student teaching. For example, during

a Book Study session, the facilitator asked a third grade ST to share her reflections about

connecting practice that she observed in her CT’s CGI lesson with theory discussed in her

methods course from the Five Practices (Smith & Stein, 2011).

F: ST was in there watching and was noticing how CT made her choices to pick the

person she had actually worked with, but you were noticing that she was doing what

Dr. M had just told you about in methods class.

ST: We had talked about it in class that Monday.

F: How you choose people to present. And, so, you were very on top of why she [CT]

was doing what she was doing.

ST: Yeah, it was interesting cause we had talked in class on Monday about how it’s not

just even how the kids see it this way. It’s not just, ‘Oh why don’t you share? Why

don’t you share?’ Like there is a method to the madness of, uhm, you know, as

you’re going around watching what the kids are doing and thinking ahead if this

would be a good strategy to have, like, shared first and then this one because this one

builds off the first one or whatever. And, so, like hearing it in [methods] class and


turning around two days later and seeing it in action it was good to see.

CT: Good planning.

ST: Yes.

It is evident that this CT is making connections between observations of her CT’s practice and

how monitoring, selecting, sequencing, and connecting can be used to organize decision-making

about facilitating classroom discussions.

Other STs were able to articulate how they used the practices of selecting and sequencing

in their own practice. For example, Nicole explained how she selected students to share

strategies after solving fraction word problems:

Usually I’d pick a student that had like a pretty clear drawing that would be like

easy for all the students to understand. There were some students that had their

own like strategies that worked, but I, like, that might confuse students that aren’t

quite there yet.

Similarly, Grace, commented:

Often I would choose, and I had to be careful with this, which students I chose, if

they did it a certain way and made a really common mistake. I might call them up

to the board to explain, ‘This is ok, like, don’t get discouraged, but this is a

common mistake that all of you could make. Uhm, I’d probably do something like

that first if it happened, and, again, I would have to kinda figure out is the student

really gonna be ok, me telling them you’re wrong, uhm, kinda in a nicer way. Are

they going to be ok with that? So, I would do that if it was something common.

Uhm, but, I would choose kinda the most basic strategy first and then get higher


and higher with more complex. If they explained it in a different way, or if they

were doing the most complex, uhm, I would save that to the end.

It is important to note that both STs were able to explicitly reason about using the Five

Practices as a framework to guide decision-making in whole class discussions. In observations

of their lessons, after monitoring students working on fraction word problems, both Nicole and

Grace intentionally selected and sequenced strategies that were most accessible to the majority of

the students in the class. Grace also selected strategies from less sophisticated to more

sophisticated, often connecting strategies like repeated addition and repeated subtraction through

discussions of students’ work.

Vignette of a CGI discussion in Grace’s classroom. In addition to the rational

described above, Grace also selected student strategies to be shared in whole class discussions

based on common misconceptions. The following discussion from Grace’s classroom illustrates

how she selected and sequenced student work. On the first day of Grace’s fraction unit in the

spring, children solved the following word problem: At Olivia’s party, there were 20 children.

There were also 90 party favors. How many party favors did each child get, if each child got the

same amount? There should be no “left overs.” Students were under the assumption that the

party favors could be split. As students solved the word problem in their mathematics notebooks,

Grace monitored their work. Students came to the carpet, and Grace asked Morton to share his

strategy first (see Figure 1.1).


Morton’s Strategy Lily’s Strategy Dedra’s Strategy

Figure 1.1. Grace’s sequence of 3rd grade students’ strategies

As Morton began drawing his diagram from his mathematics notebook on the SMART

Board, Grace told the class, “Now, if Morton’s way of solving this problem works for you, you

should write that in your math notebook, ok. You should always be paying attention to different

ways that you can do problems.” After Morton completed his diagram the following whole class

discussion occurred:

Grace: So, Morton, explain to us what you did.

Morton: I put them into groups of 10, then I made groups of 20. Then, I realized

there was one group of 10 left, and I realized there was 4 groups [pointing

to each group of 10]. I split them into 4 groups and passed them out.

Grace: Good. Then how many party favors does each child get?

Morton writes 4 ¼.

Grace: Good. Very good. Does everyone understand what Morton did?

Students: I got 4 ½.

Joanie: Yeah, if you do that, and you have 10 left so you split them all in half.

Grace: Right, ok. So, it is supposed to be 4 ½, but let’s see what Morton did, and

lets see if we can try and figure out kinda what happened.


Grace went on to lead a discussion about Morton’s strategy. After Morton shared, Lily

and Dedra shared their strategies, respectively (see Figure 1.1). Lily used repeated addition, drew

the10 remaining party favors, and split them in half. Similarly, Dedra used repeated subtraction,

drew the10 remaining party favors, and split them in half. Both girls’ strategies resulted in 4 ½.

After Dedra shared her strategy on the SMART Board, Grace asked the class, “How is Dedra’s

way similar, but also different to Lily’s?”

In this snapshot of Grace’s teaching, it is evident that Grace engaged in monitoring,

selecting, and sequencing. She also encouraged students to make explicit connections between

Lily and Dedra’s strategies and explain the similarities and differences. In retrospect, Grace

indicated she would not have chosen to sequence Morton’s strategy first. As described earlier,

she selected Morton’s strategy because she perceived it to be a common error and thought it

would be a good learning opportunity for others. While she valorized Morton’s process, it is

questionable as to whether her discussion clarified the error regarding the unit, as this was quite a

sophisticated strategy. Grace’s CT, supervisor, and the researcher were in the classroom when

the discussion occurred and used this as a opportunity to discuss sequencing with Grace. After

this coaching moment, Grace consistently sequenced strategies by choosing the most accessible

to be shared first.

Discussion and Conclusions

After participating in a CGI Book Study about the teaching and learning of fractions with

opportunities for enactment, all but one ST used their understanding of students’ reasoning so

that they were able to select and implement appropriate tasks to engage students’ thinking around

important mathematics relating to fractions during whole class discussions. Two STs, Grace and

Nicole, were able to lead mathematically rich discussions to develop students’ thinking. Both of


these STs incorporated CGI work as part of their instruction on a daily basis, where substantial

classroom time was devoted to students presenting and discussing explanations of their

strategies.

Even with an intensive supplemental experience, such as the Book Study, one ST did not

select or implement appropriate mathematical tasks for learning about fractions. There was no

evidence that she selected and implemented CGI work in a meaningful way. In fact, only

engaging AIG students in meaningful, significant mathematics is in direct contradiction to CGI

and reform-oriented philosophies about teaching and learning mathematics. This indicates that

some STs will need more support outside of their university experiences and supplemental

experiences in order to enact theory that they are learning into practice. Additionally, factors that

lead to this decision may need to be considered by professional developers and teacher

educators.

STs often selected the most accessible student strategies to share during class discussions.

All but one of the STs was able to sequence students’ strategies to lead productive discussions

about CGI word problems in their fraction lessons. In some instances, as in the case of Grace,

STs made explicit connections among student strategies in discussions. Most of the STs were

able to move beyond “show and tell” and use student responses productively.

We agree with Smith and Stein (2011) that the art of teaching is improvisational. Using

the Five Practices and Empson and Levi’s (2011) CGI work assisted most of the STs in making

good decisions in the moment-to-moment aspects of teaching. For STs, these frameworks have

the potential to be powerful tools in grounding their instruction in student thinking and

organizing their practice to make the best use of students’ ideas to develop mathematical

understanding through discourse.


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Running Head: SUPPORTING PROSPECTIVE ELEMENTARY ... · Supporting Prospective Elementary Mathematics Teachers’! 5 Loucks-Horsley, Stiles, Mundry, Love, and Hewson (2010 ) emphasize