Embedding Language Support in Developmental Mathematics Lessons

Abstract

Three community college faculty members used improvement science techniques to design, develop, and refine contextualized developmental mathematics lessons, where language and literacy pedagogy and related supports figured prominently in these instructional materials. This article reports on the role that their design experiences played in professional learning. The article uses a model of professional learning developed by Clarke and Hollingsworth (2002) as the lens to describe and analyze their experiences. Results indicated theoretically noteworthy variation among the faculty. The results highlight a strong connection between faculty willingness to experiment, during trial enactment of these lessons, in their classrooms and faculty growth in knowledge and belief structures about the importance of language and literacy to mathematics teaching and learning. Implications for design-based development, as an important ongoing professional development activity for mathematics instructors, are discussed.

Keywords

design-based research teacher leadership community colleges literacy/reading teacher education

Introduction

Developmental mathematics courses have been called “the graveyard of dreams and aspirations” (Merseth, 2011). More than 60% to 70% of U.S. community college students must take at least one developmental mathematics class before they can take credit-bearing college courses (Bailey, Jeong, & Cho, 2010); yet, only 30% of these students pass their developmental mathematics courses (Levin & Calcagno, 2007). It is not unusual for students to spend years repeating developmental mathematics courses and to ultimately leave college without graduating or earning a credential. Over the past three decades, researchers (Celedon-Pattichis & Gomez, 2010) and practitioners (Hite, 2009) alike have pointed to the language demands of mathematics as an important area of concern in students’ mathematics achievement.

This need is felt perhaps most starkly among those who learn in the high-stakes world of developmental mathematics courses where recent immigrants, career changing professionals, working parents, and new high school graduates often converge to pursue their academic and career goals and where these dreams can come to a halt (Cohen & Brawer, 2008). Recently, Gomez, Rodela, Lozano, and Mancevice (2013) explored how students’ difficulty in developmental mathematics frequently relates to mathematics texts’ language and literacy demands rather than the students’ mathematical ability. In recent years, the link between high-quality professional development for teachers and improved student outcomes has become increasingly clear (Borko, 2004; Boylan, 2002; Desimone, 2009). We need to understand much more about how mathematics instructors learn about students’ language and literacy needs and ways they may leverage language to support mathematics teaching and learning (Gomez, Rodela, et al., 2013). One approach to addressing this problem is to improve instructor’s ability to support students’ language and literacy needs through professional development and, in turn, improve student outcomes. Similar to the other contributions in this special issue, the field of improvement science has inspired the work we report here.

Following Bryk, Gomez, Grunow, and LeMahieu (2015), we posit two key elements of research centered in a continuous improvement ethic, which include attending to (a) the user’s voice in working toward useful and usable solutions to problems and (b) disciplined inquiry and iterative testing in problem solving. Bryk et al. suggest that user centeredness (i.e., voice) is important to making progress in improvement, as well as having specific techniques to work toward refinement of solutions (i.e., disciplined inquiry). These are among the key elements of improvement science. In the work reported here, we probe how engagement with these design-laden aspects of improvement can serve as valuable professional development opportunities for faculty who take part in them.

This article’s guiding hypothesis is that using improvement science techniques to engage community college faculty in collaborative curriculum design and development may be a key site for faculty learning. In particular, we focus here on faculty opportunities to learn as they create learning environments for students to use language to improve both their understanding and ability to represent their understanding of developmental mathematics lessons. We argue that design experiences help instructors address their students’ language and literacy needs and contribute to improved student outcomes. Most importantly, given the focus of the article, we argue that participation in design-based improvement cycles has the potential to (a) increase teacher knowledge, beliefs, and skills about the value of literacy in math learning and (b) evolve their classroom practice to better integrate literacy-centered activities into their routines.

Design as Professional Development

Our study has been influenced by a number of previous studies that suggest the potential of collaborative curriculum design as an effective and powerful tool for teacher professional development. Previous efforts have included the use of teacher design teams to develop and implement new biology curricula (Simmie, 2007), to design communicative English language lessons (Anto, 2013; Voogt et al., 2015), to align curricula with the needs of local industry (Bakah, 2011;Voogt et al., 2015), and to create technology-integrated lessons (Kafyulilo, 2013; Voogt et al., 2015). Others have used teams of teachers and external experts to design participatory professional development (Itow & Hickey, 2012; Voogt et al., 2015), curricula enriched with information and communication technologies (Voogt et al., 2015), the design of a new chemistry curriculum (Coenders, Terlouw, Dijkstra, & Pieters, 2010), and early elementary science curriculum (Goodnough, 2005).

Design as professional development aligns with numerous efforts to transform professional development. Traditional profressional development is typically passive, episodic, and disconnected from deep issues of student learning and classroom practice (Ball & Cohen, 1999; Little, 1990). Recent efforts call for professional development that is continuous, active, collaborative, and situated in the on-going work of the teacher (Borko, 2004; Little, 1990; Mindich & Lieberman, 2012). Involving teachers in collaborative design can be a powerful tool to foster meaningful teacher learning and development (Simmie, 2007; Voogt et al., 2015; Voogt et al., 2011). Collaborative design involves the systematic development of curriculum, technological tools, and artifacts, in a team, as a means of making improvements against a specific problem of practice (Barab & Squire, 2004; Voogt et al., 2011). Typically, design teams may consist solely of teachers or include educational researchers, discipline experts, or experienced designers (Voogt et al., 2011). Design work, situated in real-world contexts, engages instructors as active learners in their own classrooms and provides critical opportunities for reflective discourse with colleagues that is centered on practice (Koehler & Mishra, 2005). The collaborative design work reported here involved three community college instructors representing colleges in the Northeast and Midwest and four university researchers.

Participation in the design process is an opportunity for learning by doing (Edelson, 2002; Krajcik et al., 1998). Numerous studies in the learning sciences (Bransford, Brown, & Cocking, 1999; Collins, Brown, & Newman, 1989; Greeno, 1998) and teacher education literature (Ball & Cohen, 1999; Lieberman, 1995; Lytle & Cochran-Smith, 1990) suggest that meaningful and deep learning is best promoted by learning-by-doing or active inquiry approaches. Designers must identify problems with their designs to understand how to improve them (Edelson, 2002). This iterative process of digging deeply into a classroom problem is an effective way to advance understanding of recurring and difficult teaching problems (Krajcik et al., 1998), which in turn provides the foundation for evolving knowledge, beliefs, and skills (Goodnough, 2005; Itow & Hickey, 2012; Voogt et al., 2015) as well as developing improved classroom practices (Coenders et al., 2010; Goodnough, 2005; Itow & Hickey, 2012; Kafyulilo, 2013; Bakah, 2011; Voogt et al., 2015).

A core component of design—as an element of improvement work—is collaboration. Teacher education scholars have long offered evidence of the potential of collaboration to foster teacher professional growth (Grossman, Wineburg, & Woolworth, 2001; Hord & Sommers, 2008; Little, 1990; Penuel, Fishman, Yamaguchi, & Gallagher, 2007; Putnam & Borko, 2000). Designers work collaboratively to problem-solve and critically reflect on their work (Battista & Clements, 2000). Design provides a venue for assembling distributed knowledge where participants bring their expertise and engage in reflective discourse. This allows participants to examine, make sense of, and evolve their own beliefs and practices (Koehler & Mishra, 2005; Lieberman, 1995; Shrader et al., 1999; Voogt et al., 2015). Shrader et al. (1999) argued that such design collaborations can be transformative while producing useful and usable products. Furthermore, collaborations, wherein clinical experts (e.g., classroom teachers) and scholarly experts (e.g., university-based researchers) are working together, provide the opportunity for participant growth beyond their local network of colleagues (Ball & Cohen, 1999).

In the work reported here, we engaged with mathematics instructors to improve the usability and utility of mathematics story problems that were designed for developmental mathematics students. Specifically, our collaborative design work focused on story problem design and iterative refinement. Story problems are particularly challenging for many math learners at every stage of schooling, especially, many developmental math students whose less than optimal (for college-level work) reading comprehension skills render it difficult to identify and extract important information from text. Story problems contain content that is typically represented in narrative form with numerical information variously presented in multiple symbol systems: words, numbers, and graphics (e.g., charts, tables). To solve problems, learners must recognize the issue represented by the words and numbers, see the connections between the context and numerical information, and engage in the required mathematical manipulations as appropriate. In addition, over the past two decades, students have been increasingly asked to justify their answers. For example, being asked to “show their work” (i.e., calculations), explain the reasoning behind predictions made, and provide interpretations of solutions given to word problems. This shift is important, because it creates more of a peerage between the calculations and the language used to explain the calculations.

The challenge is to design story problem context lessons that instructors can effectively use to engage their students in mathematics learning. Zawaiza and Gerber (1993) pointedly argued that students must know that numbers are relevant and understand how to relate them. Relevance here means that the numbers and mathematical manipulations they imply are coupled to situations in the problem. Relating to the numbers means that the learner must appreciate that it is part of the problem-solving task to understand how numbers in abstract representations are explicitly reified in the words of the problem context. We extend this claim to suggest that in mathematics story-based lessons, students must know that the numbers and the words that describe problem situations are relevant in themselves and to each other. Students must be able to identify this relevance, connect the relevance to larger lesson concepts, use computation to solve the problem, and finally must use written language (sometimes integrated with other representational systems) to present and explain the application of the word–number relevance to address the problem context issue at hand.

We sought the collaboration of college instructors to design and develop a suite of 12 story lessons. Our primary and most pragmatic aim was the completion of useful and usable lessons. In this article, we take up a subsidiary aim. This article focuses on the role that engaging in the use of improvement science, and improvement-science-inspired processes, in the designing of these lessons has in contributing to faculty learning. Three domains of professional learning are highlighted in this article: changes in instructors’ beliefs in the value of language-focused instruction in mathematics, changes in instructors’ ability to execute instruction that uses language in effective ways, and changes in the value instructors’ place on student learning spurred by language and literacy-infused lessons. In what follows, we describe some of the aspects of faculty learning about the role of language in mathematics teaching that are linked to their engagement in design-based activities.

To design and test the contextualized lessons, we use an Agile Improvement Process (Agile Process). The Agile Process is a method to engage in improvement-centered iterative refinement of designed artifacts. The Agile Process involves collaborative lesson design followed by practice-oriented, rapid feedback to improve that design. During the Agile Process, testing occurs in iterative cycles of lesson enactment, analysis, and refinement. Again, in this article, we aim to understand the role of collaborative design of mathematics lessons in spurring mathematics instructors’ learning about the role of language in mathematics teaching. One way to answer this question is to explore connections between iterative design processes and professional growth.

Examining the Underlying Processes of Teacher Change

We were guided by the Interconnected Model of Professional Growth (IMPG) to analyze teacher change (see Figure 1). This model identifies four domains of teacher change: “the personal domain (teacher knowledge, beliefs and attitudes), the domain of practice (professional experimentation), the domain of consequence (salient outcomes), and the external domain (sources of information, stimulus or support)” (Clarke & Hollingsworth, 2002, p. 950). The IMPG suggests that changes occur in the four domains through the mediating processes of reflection and enactment. Through the processes of reflection and enactment, growth in one domain often leads to growth in others (Clarke & Hollingsworth, 2002).

Figure 1.

The Interconnected Model of Professional Growth.

Most studies attempt to measure the impact of professional development on teacher learning in specific domains; however, like Voogt et al. (2015), we aim to better understand how the underlying processes that lead to teacher growth might be coupled to the activities of iterative design. Specifically, we examine how engaging in design involves reflection and enactment, as well as how these might be connected to change along the axis of two domains: (a) knowledge, beliefs, and attitudes and (b) professional experimentation (see Figure 1). Although Clarke and Hollingsworth (2002) restricted their description, of this domain, to knowledge, beliefs, and attitudes, we extend this domain to include the notion of skill. Much of changing pedagogy is about changing practice, which to us means translating knowledge, beliefs, and attitudes into action.

In summary, this article aims to explore whether engaging in design activity, such as iterative testing, influences the domain of practice and the personal domain for participating teachers. We want to understand whether engaging in design influences teachers’ repertoires of practice (including new ways to represent mathematics) as they relate to language and literacy in mathematics (the domain of practice), and their beliefs about the efficacy of language and literacy in helping students come to understand mathematics conceptually (the domain of beliefs). Similar to Clarke and Hollingsworth (2002), the study presented here posits that participation in professional learning opportunities results in professional growth. In exploring design as an effective form of professional development, we hypothesize that increased engagement in design activities (including participation in initial lesson development, lesson reviews, and lesson testing) will lead to increased changes in knowledge, beliefs, attitudes, and increased willingness to experiment in the classroom.

Method

Data Sources

This study analyzes design work with 12 Quantway® contextualized developmental mathematics lessons (see Figure 2). Quantway is a community college pathway through developmental mathematics, focusing on quantitative reasoning. The pathway aims to prepare students for success in college-level mathematics and to develop quantitatively literate students. In Quantway, story problems (called “problem situations” in the lesson text) frame the lesson and allow the mathematics to unfold throughout the lesson. Problem situations use real-world data (e.g., census figures) to be as authentic and relevant as possible. For the research reported here, the 12 contextualized Quantway lessons were explicitly connected to the health care, information technology, and environmental science fields. For example, one lesson on dimensional analysis (unit conversion) was originally anchored by a problem about gas prices and calculating mileage. This lesson was changed to a problem about acetaminophen (Tylenol®) overdoses in children and engaged students in careful calculations of medicine.

Figure 2.

Contextualized lessons.

The data also include a total of 36 interviews representing 1 interview, following each lesson enactment, per instructor. We used a semi-structured, open-ended interview protocol (Seidman, 2006; Stake, 1995). The interviews aimed to query instructors about their experiences teaching the contextualized lessons and their recommendations for refinements. Also, we analyzed 10 hr of videotaped observational data of classroom lessons. The study includes reflective field notes created during each interview. Finally, from the video of each lesson, we developed ethnographic fieldnotes, analytic memos of classroom instruction, and in these, made judgments about student engagement from analysis.

Participants

For this study, we chose to examine data from 3 of the 16 instructors who participated in lesson testing in Summer 2014 (Phase 1), Fall 2014 (Phase 2), and Spring 2015 (Phase 3). The instructors were selected based on their varying degrees of participation in the lesson design process. All instructor names (Nick, Kevin, and Jane) are pseudonyms to protect the identity of participants while allowing the analyses (and the reader) to link data and findings by case.

Each of the instructors volunteered to test new, contextualized versions of lessons, and more importantly, to be interviewed following the testing of each lesson. The interviews had a specific design purpose. First, interviews were established with the intent to provide the authors with feedback on the lessons to support future refinements. The interviews also gave instructors an opportunity within 1 to 2 days after lesson enactment to reflect on their pedagogy, their students, and the newly designed contextualized lessons.

Nick was the instructor most heavily involved in lesson design. He co-designed the first iteration of the lessons, serving as the mathematics expert on the design team. He tested and provided feedback about the environmental science lessons during Phase 3 of lesson testing (Spring 2015). Our second instructor, Kevin, was moderately involved in lesson design. Although he did not participate in the initial design of the lessons, he participated in a mathematics review of all the lessons prior to lesson testing. During the testing process, he made extensive revisions to the lessons and shared them with other participating colleagues at his institution prior to their teaching of each lesson. He tested the health care lessons (Fall 2014, Phase 2) and the environmental science lessons (Spring 2015, Phase 3). Jane, the instructor least involved in design, was involved in lesson design only to the extent that she tested the health care lessons (Fall 2014, Phase 2), providing lesson feedback during the individual instructor phone interviews after lesson testing. Instruction feedback, provided, informed lesson revisions for future cycles of testing.

Lesson Design

The collaborative design process between developmental mathematics faculty and educational researchers focused on two overarching design goals: (1) reducing language and literacy barriers in lessons and (2) contextualizing lessons to align with health care, environmental science, and information technology fields.

Aim 1: Reducing language and literacy barriers in the lesson

The lessons were written specifically to support students who struggle with reading and writing. We used the language and literacy design principles outlined in Gomez, Lozano, and Rodela’s (2013) guidebook, Designing and Instructing Mathematics and Statistics Curriculum With Language and Literacy Needs as an Important Feature for Consideration. First, the contextualized lessons were written avoiding common language and literacy barriers for developmental mathematics students. We avoided using unfamiliar problem contexts, idiomatic phrases, vocabulary with multiple meanings, unclear directions, complex and compound sentences, and passive voice. We embedded the Comprehension and Synthesis chart (CaS chart; see Figure 2) into each contextualized lesson. The CaS chart is a language and literacy comprehension tool developed by the authors (see Figure 2) and inspired by double-entry journals (Hughes, Kooy, & Kanevsky, 1997) and the Cornell notes system (Pauk & Owens, 1962/2010). The CaS chart provided a systematic way for students to organize information in a story problem or problem situation.

Each contextualized lesson includes at least one writing task (the modal number of writing tasks was two) that requires students to explain their mathematical response using contextual information provided in the lesson. Using relevant evidence to support answers, and using complete sentences, is also a struggle for most students. For this reason, writing tasks included writing tips and sometimes model sentences to provide additional support for struggling students.

Aim 2: Contextualizing lessons for the health care, environmental science, and information technology

Our second aim was threefold. First, we sought to align the problem stories in the lessons with contexts and situations that students may encounter as they enter the health care, environmental science, and information technology fields. Second, we aimed to use contexts that are compelling, authentic, and relevant for all students. Finally, we focused on promoting students valuing of this learning by conveying to them that the mathematics they are learning in Quantway lessons really matter.

Agile Process

One of the basic tenants of our work, and of improvement science generally, is to leverage and integrate expertise from a diverse colleagueship of practitioners and scholars to solve problems of educational practice. To this end, we engaged improvement researchers; health care, environmental science, and information technology experts; mathematics faculty; and community college developmental mathematics students in a unique collaborative process to design and test the contextualized lessons, called the Agile Process (see Figure 3).

Figure 3.

Comprehension and synthesis chart.

In the initial stages of design, we consulted texts specific to health care, environmental science, and information technology to identify situations and contexts that students may encounter as they enter these studies and careers. We then worked collaboratively with experienced Quantway instructors and industry-specific community college faculty (e.g., Emergency Medical Technician [EMT] faculty) to align mathematical objectives with potential health care, environment, and information technology contexts. Using this planning and analysis, designers, experienced Quantway faculty, and educational researchers collaboratively developed an initial draft of each new lesson.

Drafts then underwent an extensive review and refinement process. First, language and literacy experts reviewed the lessons. Next, three experienced Quantway faculty members who were not involved in initial lesson design reviewed and edited the lessons. Following this, the lessons were again refined and sent to industry-specific community college faculty (e.g., EMT faculty) for review, which was followed by a final round of refinement before lesson testing.

Using Plan-Do-Study-Act (PDSA) Cycles to Test and Improve Lessons

To test the lessons, we use PDSA cycles (Langley et al., 2009). The PDSA cycle is a widely used tool in improvement research. PDSA cycles create disciplinary ways of engaging in activity aimed toward testing ideas about and approaches to practice. PDSA activities may be happening simultaneously with different instructors but organized in ways that allow evidence to accumulate (Bryk et al., 2015). The steps in the PDSA cycle include planning (plan), running the test (do), summarizing results (study), and deciding the next steps (act). The ACT stage can result in adopting for further testing, adapting, or abandoning the tested idea. This process allowed us to address new ideas and learn quickly, fail fast, and improve rapidly (see Figure 4).

Figure 4.

Agile improvement process cycle.

Figure 5.

PDSA cycles (Diagram designed by team, using vision of PDSA cycles from Bryk et al., 2015).

For example, we began by asking one or two instructors to teach one of the lessons. We collected and analyzed data, summarized results, and made immediate refinements to that lesson for a new testing by the next set of instructors. At this point, one to three instructors tested the refined lesson (the average lesson was tested by two instructors). These cycles continued until each participating instructor had taught the lesson. Midway through the PDSA cycles, we analyzed data in greater depth for themes and insights that emerged across instructors and made medium-level refinements to the lesson. At the end of the agile process of all the lessons, we conducted in-depth analysis of the data and made long-term level refinements to the lessons. This iterative process developed lessons that fulfill their design principles and can be used reliably at scale.

Analysis

Iterative refinement data sources

For each lesson, ethnographic field notes and analytic memos were used to capture instruction and student engagement in the videos. Guided by a data management process for transcribing and analyzing interviews and reflective notes (Halcomb & Davidson, 2006), instructor interviews were transcribed and researchers’ reflective notes were reviewed and expanded. The transcriptions and notes were later analyzed in two coding cycles. In the first coding cycle, interview transcripts were reviewed and emerging themes were noted. During the second coding cycle, we reread the ethnographic video notes and added analytic memos to include information found in the interviews. We triangulated data across interview and video data sources.

Each instructor was treated as an individual case. For each case, following Clarke and Hollingsworth’s (2002) Interconnected Model of Teacher Professional Growth, we looked for evidence of changes. Clarke and Hollingsworth use the term “change sequence” to name evidence-based connections that specify growth within and across domains. One way to see change is to code groupings of instructors’ activity as change sequences. In addition, change sequences allow us to see elements of the PDSA cycle, in action, and its impact on instructors’ professional learning. For example, if a PLAN called for an instructor to use the CaS chart, as we will see shortly, sometimes even though a PLAN was specified, instructors reported that the PLAN had no real impact on their belief structure.

In accordance with the Clarke and Hollingsworth model, we identified changes in the following categories: (a) changes in knowledge, belief, attitudes, or skill; (b) changes in classroom practice; and (c) changes in student outcomes. We also examined the domains within which these changes occurred. A domain in this model refers to the realms that encompass an instructor’s world. The personal domain describes instructor knowledge, skills, attitudes, and beliefs. Based on emerging themes in the instructor interviews, we created the following sub-codes to categorize instructor knowledge, skills, attitudes, and beliefs:

beliefs about student knowledge, ability, or needs;

beliefs about pedagogy, including beliefs about proper or improper teaching practices; and

beliefs about curriculum, where curriculum includes both the broader Quantway curriculum and the specific contextualized lesson.

The domain of practice pertains to all forms of professional experimentation, including the ways instructors create new curriculum materials and test new practices. For example, if an instructor who usually writes notes on a blackboard decides instead to give students individual handouts, we would identify this change as taking place in the domain of practice. The domain of consequences involves the outcomes of new practices for instructors and students. For example, we looked for new practice changes connected to student feedback to the instructors. The fourth domain, the external domain, is where instructors become acquainted with new ideas, practices, or strategies introduced by others. Instructors were introduced to the CaS chart (Figure 2). Changes in use of, or beliefs about, the CaS chart would be considered changes in the external domain.

Results

Our findings lend support to the perspective taken in the Clarke and Hollingsworth’s (2002) professional growth model. The results reported here suggest that the domains of professional development conceptualized in the model appear to act in concert. Findings suggest that each of the three instructors showed evidence of a relationship between changes in the domain of practice, the external domain, the domain of consequences, and in the personal domain. As Clarke and Hollingsworth and Opfer and Pedder (2013) have reported, participation in professional development has more than a strictly linear impact on changes in belief or practice. Change here was instead reciprocal and could begin in any domain. However, the data do suggest that participation in design, regardless of extent of involvement, led to instructor growth in specific domains for each of the faculty involved, though the form and apparent direction of change varied in ways, that we did not hypothesize, and will describe in more detail, shortly.

Of the three instructors (Nick, Kevin, and Jane) who participated in the design and enactment effort, Nick’s actions, reflections, and public design activity made it easier for us to see the relationship between the domains. Nick was involved in the initial design of the contextualized lessons. In interviews, he often explained his thinking in ways that reflected heavily upon his practice. He also openly shared his opinions of the curriculum, students, and pedagogy. This openness made it easier to observe and identify changes in his knowledge and belief, practice, and salient outcomes.

Jane, although minimally involved in the early conceptual design work, seemed to also benefit from this experience. Her design contributions drew heavily from students’ responses to the new curriculum. In the classroom, she constantly paced the room asking students about their opinions of the lesson, and particularly, the language and literacy tools. In interviews, she often justified recommendations, with references to individual students’ backgrounds and experiences. For example, Jane had 10 international students in her classroom and expressed concern about the CaS chart’s (in)ability to meet her students’ language and literacy needs. Jane’s student-centered approach appeared to be connected to changes in the domain of consequences (student outcomes). This was frequently followed by changes in the personal domain (beliefs).

Kevin, by contrast, though heavily involved in curriculum design, provided the least rationale, for his decision making. Specifically, he did not articulate his justification for changes as often or as clearly as the other instructors. The limited communication about his rationale for his decision making rendered it difficult to discern the direction of the relationship between the domains. Of all the participants, Kevin was the most willing to experiment with new materials, whether he was a co-designer or trying out curricular materials created by others. Kevin often made changes to lessons without being prompted and shared his revised versions of lessons with his colleagues. Despite his flexibility and high level of participation, Kevin did not exhibit any clear changes in the personal domain of knowledge, beliefs, or attitudes, although changes across other domains were apparent from observational and interview data. He experienced the weakest connection between domains.

In what follows, we describe, in more detail, the reciprocal nature of changes in instructors’ domains. When referencing interview transcripts or classroom observation notes, we will use the following system: “quote” (Instructor Name, Data Source, Lesson Context, and Unit). For example, the notation (Nick Interview, Health Care 3.1) would follow a quote from an interview with Kevin after he taught Lesson 3.1 in the health care context. We highlight the nature of the change and the relationship of experimentation, reflection, and direction of domain changes. The opportunities for change occurred as faculty worked their way through several PDSA cycles. The PDSA process maps onto instructor experience in the following ways: “PLAN” occurs as instructors worked out the execution of the lesson, “DO” occurs as instructors enact the lessons, “STUDY” includes instructor’s participation in interviews, and “ACT” occurs as faculty, along with the design team, work out the plan for the next testing cycle.

There is evidence to suggest that all three instructors experienced changes in the domain of practice as a consequence of their engagement in the design process. As we encounter each change sequence, we underscore how instructors’ actions are indicative of connections across the domains of practice. We were informed by instructor interviews, classroom video observations, and field notes to identify change sequences.

Change Sequence: Changes in Domain of Practice Lead to Changes in Personal Domain

It was through these interview-based reflections that evidence of changes in knowledge and beliefs began to unfold. In accordance with the Interconnected Model of Teacher Professional Growth (Clarke & Hollingsworth, 2002), reflection mediated change between domains. As Ertmer and Newby (1996) have claimed, reflection is a critical component in learning from experience (van de Wiel, Szegedi, & Weggeman, 2004). One example, from Nick, illustrates the importance of reflection. In one interview, Nick, who was involved in designing the original iteration of the lessons, described new knowledge about curriculum design while reflecting on his practice. In this reflection, he recalls the timing of the lesson enactment and notes that the experience, and his reflection about the experience, will inform design suggestions about the structure of the lesson:

. . . I was worried about time so I basically did not do the CaS chart. I’m glad I didn’t, because if I had, we wouldn’t have gotten very far . . . we finished the first problem situation . . . upon extended reflection I think the lesson is probably a little too long. I think problem situation two could be put in further applications. (Nick Interview, Environmental Science 2.3)

In this example, Nick’s beliefs about the way the lesson should be reorganized are related to his experimentation practice, that is, the impromptu decision not to use the CaS chart. In the lesson enactment, Nick decided that one way to get through the lesson in the time allotted for enactment was to eliminate the CaS chart. Here, we see Nick engaging in a “plan,” with a prediction that it will lead to executing the lesson in the appropriate amount of time, and “do,” executing the lesson. However, upon reflecting on this practice during the interview, he realized that by moving the second problem situation into the further applications section (for homework or further practice in class), he could shorten the lesson while retaining the language support (the CaS chart), which reflects the “act” phase.

In other instances, it was enactment, rather than reflection, that mediated the relationship between changes in the domain of practice and changes in the personal domain. In some cases, instructors involved in curriculum design seem to be unwilling to experiment with the CaS chart language support tool in the classroom. In our analysis, we viewed the decision not to use the chart as increased experimentation in the external domain, meaning instructors were experimenting to see when and how the CaS chart would work for their instruction. We viewed this as “increased experimenting” and not abandonment of the tool as long as the instructors cotinued to use the tool long-term. For example, although Nick taught three contextualized environmental science lessons, the CaS chart was a part of this enactment only in the first and the third enactments. During the interview, that is, in the “study” phase, following the first contextualized lesson, Nick reflected on his belief about the CaS chart’s lack of usability:

The CaS chart doesn’t work . . . it gets them thinking about it and I guess that’s good . . . I think a different way to approach language and literacy is maybe to have them write just a few sentences summarizing the problem—a summary. That at least allows them to process and understand the situation. Understanding the situation itself is a reasonable goal, and articulating that in writing is potentially useful. (Nick Interview, Environmental Science 1.8)

In this interview, Nick’s comments offer a healthy skepticism about the value of the CaS chart in supporting reading comprehension than more traditional language approaches. In fact, he offers a design suggestion and prediction, specifically that having students summarize the information in the problem situation may be more effective than the CaS chart. In essence, this is an example of Nick’s engagement in the “act” phase because he is suggesting the structure of the next PDSA cycle, that is using summary writing as literacy support rather than the CaS chart.

After teaching the third lesson and using the CaS chart for a second time “do,” Nick’s perspective changed. In his comments, he reported a new sense of the tool’s usability:

I did the CaS chart [in the first lesson] just because I was supposed to. This time when I did the CaS chart, it really occurred to me the use of this idea. What it really can do . . . it’s an opportunity to make them just sort of process the basic information in the problem. It helps them to absorb it. (Nick Interview, Environmental Science 2.3)

Nick’s comment, “ . . . When I did the CaS chart, it really occurred to me . . . ” is strongly suggestive of changes in knowledge and beliefs about curriculum following his engagement in professional experimentation. We suggest that in this example, Nick’s beliefs did not change merely by reflecting on his practice, but instead, change occurred because he enacted the third lesson. In short, the act of teaching the lesson was what moved his beliefs.

Change Sequence: Changes in External Domain Leads to Change in Domain of Consequences (Salient Outcomes) Leading to Changes in Personal Domain

We found evidence that pointed to changes in the external domain leading to impact in the domain of consequences. This, in turn, led to changes in the personal domain. Two of the three instructors persisted in using the CaS chart despite initially disliking the language support tool. For example, Jane, who had had minimal involvement in the lesson design, engaged in the “do” phase by enacting the first contextualized lesson. During her interview (the “study” phase), she expressed her disappointment with the CaS chart and its suitability for placement in a lesson:

It bothers me. I have question. If you decide to go forward with this lesson, this chart is gonna remain here? It bothers me. It’s really good. It should be in a strong package [of language support information]. At the beginning when we are preparing students. If [the] problem statement is too long we should teach them to do this. It didn’t help for the lesson today . . . it was like a disconnect. What you’re asking in question C. Here it should be something different. Here they spent time. Maybe it’s me. I went around asking does it help, do you like [it]? They say “yes, yes” but I believe they say [that] because that’s what I want to hear. They were not honest. (Jane Interview, Health Care 1.4)

After her third “do” phase, an enactment of the CaS chart, Jane told us in the interview,

For the first time I found it useful. This time it was different. It’s like the students know. You put some information. Like a guideline. So then that they give you the first step. What to look for. I like that. They were able to fill out [Column] B and they were able to find out quantitative information without my help. (Jane Interview, Health Care 3.1)

Jane’s reflection offers evidence of a change in her domain of consequences or salient outcomes. In contrast to their experience during the first application of the CaS chart, during the third enactment of a contextualized lesson, Jane’s students were able to fill out part of the CaS chart without instructor assistance. This was an important outcome in that students appeared to be making the CaS chart their own. However, this change was not due to instructor experimentation, but rather the changes are a result of design changes researchers and other instructors made to the CaS chart (external domain) during the “act” phase.

In what follows, during the interview following her third CaS chart enactment, Jane not only recognized the design change (offering sentence starters as suggestions) in the CaS chart but also offered an endorsement of the design for future lessons.

. . . what happen is that you did something different with this one? You put some information to fill out the rest. I think if you do it that way with the others [CaS charts] it will be helpful . . . this way you set it up different, “Amanda and Andy go to the doctor because . . . ” and the students follow the idea. I found that they like it. Even myself . . . keep sentence starters . . . it worked this time. (Jane Interview, Health Care 3.1)

Jane’s interview comments provide an instance of changes in the external domain (CaS chart providing students with sentence starters) that led to change in the domain of consequences (i.e., students could work independently on CaS chart), which led to change in personal domain (i.e., Jane now likes using the CaS chart). Enactment mediated the relationship between changes in the external domain and in the domain of consequences. Jane’s reflection on these outcomes mediated the process of her arriving at a conclusion about the usability of CaS chart. That is, she found value in the updated CaS chart design. This reflection about the value of the CaS chart is an instance of a change in the personal domain.

We also saw support of Jane’s reflection about the value of the CaS chart and the tool’s role in the lesson in a video observation of her third classroom enactment of the CaS chart. In that enactment, Jane, when introducing the day’s lesson, described the CaS chart as a tool needed to solve a problem. Specifically, in the “do” phase, Jane announced that the CaS chart is mainly used to find and gather all the information needed to solve problems (Jane Observation notes, Health Care 3.1). We believe her announcement signaled that she had come to see the CaS chart as a valuable mathematics instructional tool.

Change Sequence: Unclear Direction but a Relationship Exists

Earlier, we noted that all three instructors showed evidence of relationships in the domains of study. However, with one of the instructors, Kevin, the evidence regarding the direction of the relationship was unclear. In the example to follow, Kevin made a change in the planned lesson, which could be evidence of a commitment to a particular focus on literacy use in instruction in mathematics. From the perspective of Clarke and Hollingsworth’s (2002) model, we would hope to establish what in Kevin’s planning experiences or in-the-moment student reaction prompted that change. However, as you will see, in the following description, neither planning experiences nor student reactions appeared to lead to Kevin’s action.

Kevin was heavily involved in curriculum design and provided valuable support for design; however, Kevin’s rationales for changes were often unarticulated or unclear. During an interview, in the “study” phase, after teaching the second lesson, Kevin reported that he had made a decision to use more than one language and literacy tool: annotation of the lesson text (e.g., circling key words). It was not clear what led Kevin to select the annotation approach rather than the original CaS chart design. There are three potential explanations for his selection: he believed that annotation was a more effective approach to helping students solve the mathematics problem (personal domain), he was driven by potential student dislike of the CaS chart (domain of consequences), or he wanted to try a new approach to supporting language in mathematics learning (domain of practice). For whatever reason, he selected the annotation approach. When Kevin reflected about which tools he would like to see designed into the four lessons, he noted, “I like to have two CaS charts, an annotation, and a free choice” (Kevin Interview, Health Care 3.1). In his reflection, Kevin described how the use of annotation, as a language support tool in mathematics, had unfolded in the classroom:

They didn’t all [of the students] do all of the things I had in my annotation scheme, but they did all do something. Uhh . . . I asked later if they thought the annotation helped them find the information. One of the students said yes so I guess that’s ok. I think they liked—well I know—they liked not having to do the CaS chart. They were obviously very pleased to do something different. I think they liked the idea of annotation. It seemed easy. (Kevin Interview, Health Care 3.1)

There is evidence of change in Kevin’s domain of practice. In an earlier interview, Kevin reported that he planned to allow students to freely select a language support approach for the last lesson. In the final lesson as in the other three lessons, the CaS chart had a prominent role in supporting students reading comprehension in mathematics. However, Kevin reported during the interview following the final lesson enactment that, in a “plan” phase, he removed the CaS chart from the lesson and, reflecting the “do” phase, replaced it with annotation notes that had been shared with him by some of the authors during a design activity. Kevin had previously been unfamiliar with annotation as an approach to supporting reading comprehension in mathematics. Perhaps he hoped to expand his repertoire of language support tool use, or he may have felt that the approach was accessible. In the subsequent “study” phase interview, Kevin reported that the students liked reading using annotation: “They [the students] liked it. There was a uniform relief to not have to do the CaS chart again” (Kevin Interview, Health Care 4.3).

Unlike in previously mentioned examples of the direction of change in instructors’ domain, Kevin’s rationale for his decision to select annotation rather than the previously planned CaS chart design is not clear. Kevin’s comments seem to indicate a change in the external domain leading to change in his practice. In addition, in Kevin’s interview reflection, his comments suggest a change in student outcomes (i.e., that students like annotation better than the CaS chart). When asked during the interview whether he would be more likely to use annotation than the CaS chart, Kevin’s comments suggest a desire to provide students with an array of language supports in mathematics.

Yeah I think so. But I would want to cover them [annotation and the CaS chart] both. I think the CaS chart is a good idea. I think they need exposure to more than one tool. I think in this class annotation is more likely to help but I think they’re both good tools. (Kevin Interview, Health Care 4.3).

We speculate that Kevin, as a creative designer and mathematics instructor, may have an unstated belief in the value of variety in language support in mathematics teaching and learning. Of course, this is a post hoc characterization of what we see in Kevin’s choices. In the moment, we did not probe Kevin further, so lacking more detail we cannot know his reasoning with more certainty.

During an earlier interview, in a “do” phase enactment of the first contextualized lesson, Kevin’s comments indicate a willingness to experiment with design by voluntarily making changes to lessons. He described a change he made to the lesson. In his view, the change resolved a language issue for students who might confuse one variable (e.g., the proportion of paper that is recycled) with another (e.g., the proportion of recycled materials that are paper). Although left unstated, Kevin’s decision to make the change to the lesson suggests that he has developed in his teaching practice a new sensitivity to the subtleties that language can communicate to buttress mathematics understanding and was willing to experiment to make progress.

In my edited version for #3, I had proportion 1- paper that is recycled and proportion 2- recycled that is paper. I designed that right into the question. That took care of it. They all understood how they were comparing proportions 1 and 2. (Kevin Interview, Environmental Science 1.8)

Although we are left to speculate on the direction of the changes in Kevin’s domains, what seems clear is that Kevin was influenced by his underlying assumptions, beliefs about the language support that students needed, and students’ feedback about the annotation versus CaS language support tools. Kevin’s decision to make the change to the lesson suggests that he may have had underlying reticence about the CaS chart, which then made it likely that he would use annotation when student responses gave him opening to do so. Also, Kevin may have always intended to introduce multiple tools during lesson enactment. An alternative hypothesis is that when Kevin found that his students seemed to like annotation better than the CaS chart, this may have led him to replace a “free choice” lesson with a second annotation.

Discussion

Many mathematics instructors rarely come to mathematics instruction prepared to teach with language and literacy issues in mind (Gomez, Rodela, et al., 2013). They are primarily prepared to teach, and see, mathematics as formalisms and mathematical manipulations. Sociocultural studies of teaching and learning in mathematics (cf. Celedon-Pattichis, 2004), studies of language and its use in mathematics (Donlan, 2009; Edwards, Maloy, & Anderson, 2009; Moschkovich, 2010), and studies of strategic pedagogical practices aimed at supporting language in mathematics (Blessman & Myszczak, 2001; Schleppegrell, 2007; Zawaiza & Gerber, 1993) have found that community college and K-12 mathematics faculty have limited experience with supporting language in mathematics. By and large, mathematics instructors look to other disciplines such as English Language Arts to support students’ language needs. Yet we know that facility with language, especially with mathematical vocabulary and contextual language in story problems, can support student learning and success. Too often when mathematics faculty are consulted about language and its role in mathematics instruction, faculty argue that it is not their role to teach language, as if language is a skill separate from mathematics. There is the common phrase “but I’m not an English teacher.” This perspective is particularly troubling in community college contexts where students are trying to acquire the mathematics necessary to support transitions into careers and are frequently taught mathematics by instructors who hold advanced degrees in mathematics, but not necessarily in mathematics education. Mathematics in career settings such as nursing or environmental science is inextricably coupled to being able to use and express mathematical reasoning and understanding through language.

Why Engagement in Design Activity Matters for Professional Development

In our findings, some of the faculty comments underscore the tension in teaching language to support mathematics learning of highly contextualized material. The results presented here provide a small glimpse into the potential of design settings to surface this tension for faculty, as well as to begin to alleviate it through teacher learning and growth. Our results suggest that design may have potency here for at least three reasons. First, traditional professional development, as it relates to practice (whether focused on language or some other problem), often asks teachers to suspend disbelief in a context outside their usual realm of practice. Learners are asked to carry out the practice because some authoritative resource like an expert or a journal source says it is important to do. Design differs from traditional professional development in that the design settings we constructed allow faculty specific opportunities to engage in these practices in their classrooms, to reflect on the efficacy of their practices, and their role in them as design team members in their own right. The practices come to work because the teachers as improvement researchers make them work. In making them work, they build understanding of, and commitment to, them as practices. This reflection, based on action, gives design its power to both change belief and knowledge structures and to encourage faculty to experiment.

Second, and related, design settings are explicitly constructed to provide team members with voice and authority to change, supplemented by support to do so. In the cases we observed here, faculty opinion about the structures around the CaS charts and the routines coupled to those structures influence subsequent iterations of the lesson. Design differs from more traditional forms of professional development even those that occur on multiple occasions within schools. In the case of the design process we examined in this article, there are specific improvement objectives and tasks for instructors, such as executing a lesson and being interviewed about the execution. Importantly, this process is closely coupled in time and subsequent enactments and revisions. Instructors know their comments and feedback contribute to this process. This makes design as professional development perhaps a particularly effective venue to both change knowledge and beliefs and encourage experimentation. In this case, design as a setting allowed faculty to see the impact of their ideas and have support to bring their ideas into reality.

Third and finally, the design team that we built as a part of this effort lasted longer than traditional professional development episodes. Faculty had an opportunity to reflect and enact lessons over multiple occasions. The extended nature of design as professional development allows faculty belief structures to grow and change slowly, rather than the more common, but naïve, expectation that a brief and well-structured professional development encounter will have the power to change knowledge structures, beliefs, and create in faculty an appetite to try new things.

In the language of Clarke and Hollingsworth (2002), the external resources that faculty have, to bring to instructional situations, rarely include specific support and training, particularly in the use of language as a teaching tool in mathematics. Furthermore, one might argue that because few specific resources exist, instructors have little knowledge and not a high degree of belief in language’s efficacy in teaching mathematics. Both of these conspire to depress willingness to experiment with language as a teaching tool in mathematics. This experience with design-based development as professional learning opportunities helps ameliorate these factors.

Challenges to Design as Professional Development

Although design may be a potent venue for professional development and teacher learning, we recognize that providing broad-based opportunities like these for many instructors in community colleges is not easy. Design presents a challenging problem of spread. How do you create local venues in community college organizations on a wide scale to engage in design opportunities? The work that we have presented here had few instructors (n = 16) and we only presented the analyses of three. The design activities were highly supported by our team. Are community college organizations capable of presenting design activities like those presented here with much less support than was presented to the faculty who took part in the design work we described? We also wonder how to keep design a fresh activity. The lessons faculty members worked on are new and, therefore, obviously great candidates for vibrant design activity. What happens when lessons like these have existed in an organization for many years? Will faculty still be interested in them as targets for design opportunities?

Although we do not have an answer to these questions, right now we think problems like these can be addressed. Lesson study as it is carried out in Japan (Lewis, 2000) is such an example. Lesson study regimes as described by Lewis (2000) and Lewis, Perry, and Murata (2006) often take up the improvement of long-standing lesson artifacts. The responsibility for improvement of these artifacts belongs to many faculty members who are a part of teaching and learning organizations. Although the design work described here surely is not as articulate as lesson study in Japan, the elements of iterative refinement and professional commitment to improvement are something that one can imagine collections of community colleges and mathematics professionals can adopt in ways that fit their local organizations. Indeed, that is one of the central arguments offered by Bryk et al. (2015) in their recent volume on Improvement Science in education. Following their thinking, we can envision a world in education in which community colleges and other educational organizations adopt a commitment to improvement and approach day-to-day work differently. Such a world would understand and invest in networked improvement communities. In such a world, it would be part of the regularly compensated work of faculty members to keep lessons and curricula relevant and vibrant to students. Instructors would share their work and view the continuous improvement and support of lessons as just the norm of working in community colleges.

The experiences we have recounted here are encouraging. They suggest that the design-based aspects of improvement efforts can be beneficial to teacher learning and how faculty and community colleges think about their practice. They also highlight instructors’ willingness to change their practice based on the experience of opportunities to engage in iterative refinement. We hope this work and the work undertaken by others will further specify the ways that faculty can engage in design to produce changes in knowledge and beliefs and deepen a willingness to engage in experimentation and practice.

Footnotes

Acknowledgements

The authors thank Sarah Mejia for her assistance in the early analysis of the data reported here. They also thank Paul LeMahieu and Ann Edwards for very thoughtful comments that improved this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors thank the Carnegie Foundation for the Advancement of Teaching for their generous support for this work through a contract from the Department of Labor.

Author Biographies

Kimberley Gomez is an associate professor of education and information studies at the University of California, Los Angeles (UCLA). She is also a senior fellow at the Carnegie Foundation for the Advancement of Teaching. Her research seeks to increase underrepresented students’ access to Science, Technology, Engineering and Mathematics (STEM) learning and STEM opportunities in middle, high school, and community college settings. Her work has emphasized the design, and support, of language and literacy-rich mathematics and science teaching and learning and has helped to advance instructional approaches and learning outcomes at the intersection of STEM and literacy. She holds a PhD in educational psychology from The University of Chicago.

Louis M. Gomez is a professor of urban schooling and information studies and holds the MacArthur Chair in digital media and learning in the Graduate School of Education and Information Studies at the UCLA. He is also a senior fellow at the Carnegie Foundation for the Advancement of Teaching. His research interests encompass improvement science in education, the application of computing and networking technology to teaching and learning, applied cognitive science, and human–computer interaction. He holds a PhD in cognitive psychology from UC Berkeley.

Katherine C. Rodela is an assistant professor of educational leadership at Washington State University, Vancouver. Her research seeks to connect theoretical analyses of culture, language, learning, and leadership with practical applications for increased educational equity, particularly for students of color, low-income students, and other underrepresented communities in our schools and universities. In her research, she examines alternative forms of educational leadership and its impact on K-12 school reform and educational processes with a focus on the roles of Latina mothers in community-based educational movements through ethnographic research and she engages in improvement science research to increase access and retention in community colleges for underrepresented students in the STEM fields. She holds a PhD in education from Stanford University.

Emily S. Horton is a doctoral student in the urban schooling division at the Graduate School of Education and Information Studies, UCLA. Her research interests focus on addressing issues of equity and access in urban and under-served communities at all levels of education. She is specifically interested in design of learning environments, language and literacy practices in mathematics teaching and learning, and contextualization of mathematics curricula. Prior to pursuing doctoral work, she worked as a middle grades teacher and a digital media librarian. She earned her bachelor of arts degree in education, and her master of science in library science degree from the University of North Carolina at Chapel Hill.

Jahneille Cunningham is a graduate student researcher in the Graduate School of Education and Information Studies at the UCLA. Her research is focused on improving STEM education for under-served students, with an emphasis on mathematics education. Through her work in curriculum design, she hopes to equip students from marginalized groups with the tools they need to pursue their academic and career goals. She received her BA in sociology from the University of Miami.

Rocio Ambrocio (UCLA, 2016) is an undergraduate research assistant with the Improvement by Design research group at the UCLA, and is a former research assistant with the UCLA Department of Civil and Environmental Engineering. Her research interests are specifically focused on STEM opportunities for underrepresented minorities and for women. Her volunteer efforts involve social action in support of educational opportunities for all. She has been a mathematics tutor for the UCLA Community Based After-School Program and is currently a tutor and activities director with the UCLA project, Working for Immigrant Literacy Development (WILD).

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