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INTRODUCTION

Across various stakeholders including researchers, teachers,students, and policy makers, there is broad support for making realworld connections in mathematics teaching. Beginning with Dewey'searly writings on the need for curriculum to have "real-life"relevancy (Dewey, 1902, 1938), members of the mathematics educationcommunity have repeatedly called for connections between mathematicsinstruction and real world contexts (Boaler, 1993; Gainsburg, 2008;National Governors Association Center for Best Practices & Councilof Chief State School Officers, 2010; National Council of Teachers ofMathematics, 2000). This issue has gained increasing attention in recentyears, motivated in part by statements from leading professionalassociations. For example, the National Council of Teachers ofMathematics (2000) advised that students should be able to"recognize and apply mathematics in contexts outside ofmathematics" particularly related to students' "owninterests and experiences" (p. 4). Additionally, the Common CoreState Standards for Mathematics (NGACBP & CCSSO, 2010) argued thatmathematically proficient students should be able to "apply themathematics they know to solve problems arising in everyday life,society, and the workplace" (p. 7).

Echoing these standards initiatives, students have asked for anincreased understanding of how the mathematics that they learn in schoolrelates to contexts and situation in their lives (Boaler, 2000). Middlegrades students have been found to associate "caring teachers"with the practice of connecting mathematics instruction to relevant realworld contexts (Jansen & Bartell, 2013). Middle grades teachers alsovalue real world connections in mathematics, arguing that suchconnections engage and motivate students and help to build relationshipsbetween teachers and students (Jansen & Bartell, 2013; Lloyd &Frykholm, 2000). Simic-Muller, Fernandes, and Felton-Koestler (2015)surveyed prospective mathematics teachers and found that almost allplanned to make connections to real world situations and tostudents' family backgrounds and community practices in deliveringtheir instruction. Yet other research has documented a gap betweenteachers' vision for making connections and their ability to enactthis vision in practice (Lee, 2012). Some suggest that meaningful, realworld connections are infrequent in mathematics teaching, especially atthe middle school level, and that when teachers do make theseconnections, such connections tend to be brief and to require littleresponse from students (e.g., Gainsburg, 2008; Lee, 2012).

In summary, despite widespread support for real world connectionsin mathematics teaching, little is known about how teachers actuallymake these connections, and more specifically, about the supports anddilemmas they encounter in this work. In fact, in a recent review ofresearch in the middle grades, Yoon, Malu, Schaefer, Reyes, and Brinegar(2015) highlighted a "noticeable gap" in middle gradesresearch related to teachers' beliefs and practices when connectingto diverse students, parents, and communities (p. 11). The low-incidenceof this broadly supported and potentially high-impact teaching practicesuggests the need to better understand how middle grade teachers makereal world connections in their mathematics instruction, including thechallenges, tensions, and dilemmas they experience, and the supportsthat they draw upon in response.

Mathematics and Real World Contexts

The mathematics education literature has diverged on the types ofinstructional strategies that have been classified as real worldconnections. For example, Gainsburg (2008) identified a range ofpractices, including analogies, word problems, analysis of real data,exploration of how mathematics is used in society, "hands-on"mathematical representations, and mathematical modeling of real worldphenomena. Word problems from standardized curriculum materials havefrequently been used as a means of connecting mathematics to real worldcontexts; however, word problems typically include a narrow range ofcontexts and rarely reflect the complexity of real world scenarios(Gainsburg, 2008; Lee, 2012). Recently, mathematical modeling has gainedmore prominence as a way for students to use mathematics to understand,analyze, and model real world phenomena (Asempapa, 2015; Lesh &Fennewald, 2010; NGACBP & CCSSO; 2010). For example, in the middlegrades might develop mathematical models to estimate food needed for aschool event, or to compare and select among several fundraising options(NGACBP & CCSSO; 2010). Yet the inclusion of mathematical modelingin the middle grades curriculum has been limited. Scholars have alsoemphasized different understandings of what counts as "realworld." Although simulated experiences are often regarded as realworld, some scholars advocate for authentic contexts that reflect actualscenarios from students' schools and communities (e.g., Civil,2007; Gonzalez, Andrade, Civil, & Moll, 2001), whereas others arguethat as long as scenarios are imaginable, they can support studentlearning and engagement (e.g., Van Den Heuvel-Panhuizen, 2003).

In the present study, we conceptualized real world connections toinclude flexible interconnections between (a) mathematics content(concepts and skills), (b) students' experiences and understandings(both prior experiences with mathematics and experiences outside ofschool), and (c) real world contexts, activities and scenarios. Asreflected in Figure 1, mathematics content can connect to studentsthrough students' prior knowledge and experiences with the content,and to real world contexts through ways that mathematics is used in theworld. Additionally, students' understandings and experiencesintersect with real world contexts when contexts reflect students'experiences. At the center of this tripart relationship is a nexus ofalignment where students engage in mathematics content through realworld contexts that reflect their experiential knowledge, interests, andpriorities.

Benefits of Relevant Real World Contexts

Scholars have previously outlined various arguments in support ofreal world connections in mathematics teaching. These arguments includeenhanced student motivation, engagement, and achievement (Boaler, 1993,2001; Pierce & Stacey, 2006); deepened understandings about the roleof mathematics in family, community, and cultural practices (Civil,2007; Nasir, 2002; Turner, Gutierrez, Simic-Muller, & Diez-Palomar,2009); and increased opportunities for student collaboration (Asem-papa,2015). Middle grades research has documented how real world connectionsprovide opportunities to use mathematics as a tool to critically examineissues of social justice (Aguirre et al., 2012; Gutstein, 2003, 2006),and to make connections between mathematics and students' livesoutside of school (Stevens, 2000). More generally, scholars have focusedon the benefits of connecting instruction to students' "fundsof knowledge" (Moll, Amanti, Neff, & Gonzalez, 1992; Gonzalez,Moll, & Amanti, 2005). Funds of knowledge are students' homeand community resources and experience that can be capitalized on tomake instruction more engaging and meaningful (Moll et al., 1992). Inthis article, we use two terms to connect to this work, real worldcontexts and relevant real world contexts, as the teacher in this studymade both types of connections in her mathematics instruction.Specifically, this teacher made both general real world connections aswell as real world connections that were particularly relevant to herstudents' experiences and funds of knowledge.

Recent research with middle and high school students found thatwhen tasks reflected familiar contexts, students drew on their knowledgeof the situation to support successful problem solving (Jansen &Bartell, 2013; Walkington, Petrosino & Sherman, 2013). Thesebenefits were particularly pronounced for challenging tasks and forstudents who previously struggled with problem solving (Walkington etal., 2013). Jansen and Bartell's (2013) framework for caringmathematics instruction in the middle grades found that one aspect of a"caring" teacher's practice was the selection of tasksthat were "relevant and interesting" and "related to thereal world." One student reported that he appreciated teachers whoused problems that "give you a story in the background... [because]it's more interesting" (p. 44). In short, the increasedcognitive demand and situatedness of relevant, real world tasks benefitstudents both in mathematics classrooms and outside of school(Gainsburg, 2008).

Challenges for Teachers

Connecting mathematics instruction with real world contexts,particularly contexts relevant to students' own experiences, can bechallenging. However, research related to teachers' perspectives onthese challenges is limited, as studies have tended to focus onstudents. These studies have established both students' tendency toignore real world considerations when solving textbooklikecontextualized mathematics tasks (Greer, 1997; Verschaffel, De Corte,& Lasure, 1994), as well as the potential for more authentic,relevant tasks that connect to students' experiences to supportstudent sense making (Nesher & Hershkovitz, 1997; Verschaffel etal., 1994).

Teachers report they have limited planning and instructional timeas well as scarce resources (e.g., curriculum or assessments) for makingreal world connections (Asempapa, 2015; Depaepe, De Corte, &Verschaffel, 2009; Gainsburg, 2008). The challenges may be particularlypronounced for middle grades teachers who may only see students for oneperiod per day, and have limited opportunities to learn aboutstudents' out-of-school interests and experiences. Moreover,classroom management concerns when implementing real world lessons, aswell as high stakes testing constraints have been found to createchallenges for teachers (Gainsburg, 2008).

Yet there is a notable gap in the research related to dilemmas thatteachers navigate when implementing real world connections inmathematics classrooms. As da Ponte (2009) noted in his discussion ofresearch on teachers' conceptions of real world connections,"we are lacking refined models that account for the relations ofcontextual variables and teachers' conceptions and practice"(p. 289). In an attempt to narrow this gap, the present study wasdesigned to explore this underresearched teacher practice. Specifically,we examined the challenges that one middle grades teacher encounteredand the supports she drew upon as she strove to make relevant, realworld connections in her mathematics instruction.

CONCEPTUAL FRAMEWORK: DILEMMATIC SPACES

Individuals negotiate dilemmas on a daily basis. Dilemmas have beenconceptualized as instances where "two values, obligations, orcommitments conflict and there is no right thing to do" (Honig,1994, p. 568). These decisions are not always unambiguous; ratherindividuals make decisions in the "grey zone" where aclear-cut distinction between a right and wrong choice does not exist(Kakabadse, Korac-Kakabadse & Kouzmin, 2003). The nature andconstruction of ethical dilemmas has been explored in the humanitiesliterature; however, education scholars have begun to recognize thatteachers also face dilemmas of practice (e.g., Ehrich, Kimber,Millwater, & Cranston, 2011).

Dilemmas in teaching include those related to high-stakes testingand accountability measures (Singh, Martsin, & Glasswell, 2015),collaboration among colleagues and mentors (Orland-Barak, 2006; M.Turner, 2016), conflicting policy mandates and pedagogical values(Jonasson, Makitalo, & Nielsen, 2015), as well as ethical dilemmasof practice (Shapira-Lishchinsky, 2011). Research has begun to explorehow dilemmas are constructed, understood, and navigated by teachers. Forexample, Ehrich and colleagues (2011) argued that ethical dilemmas arerealized as teachers have to make decisions based on identified choices.Multiple forces shape these dilemmas, including political and societalcontexts, professional ethics, organizational cultures, institutionalcontexts, teachers' beliefs and values, and the beliefs and valuesof trusted confidants (Ehrich et al., 2011). All of these impingingforces shape how teachers conceptualize dilemmas, as well as thepossible choices that are conceived and the ultimate decision that ismade.

One particularly illuminating lens for exploring pedagogicaldilemmas is Fransson and Grannas' (2013) conceptual framework ofdilemmatic spaces. Whereas in Ehrich et al.'s model ethicaldilemmas stemmed from a "critical incident," Fransson andGrannas conceptualized dilemmatic spaces as being "everpresent" and not necessarily limited to ethical issues. Byincluding the relational category of space within their framework,Fransson and Grannas (2013) argued that a dilemmatic space could beconceptualized as occurring within the relationships of "two ormore positions." In other words, the concept of space allowsscholars to explore how dilemmas are created in relationships between anindividual and larger contextual factors (e.g., policy or schoolclimate), as well as relationships between various individuals (e.g.,teachers, parents, students, or colleagues). These relationships ofteninvolve positioning, because teachers position themselves in relation toothers (Fransson & Grannas, 2013). Additionally, the negotiation ofdilemmatic spaces leads to the constitution and reconstitution ofteachers' identities as they react in relation to ever-presentdilemmas. Fransson and Grannas (2013) argued that the conceptualframework of dilemmatic spaces allows scholars to understand "thecomplexity and dynamics of teachers' work and how teachers aredefined, positioned, and related to others" (p. 9) as well as howinteractions with ever-present dilemmatic spaces influenceteachers' evolving professional identities.

The conceptual framework of dilemmatic spaces is particularlysuited to exploring the case of one middle school's teacherattempts to connect her mathematics instruction to relevant real worldcontexts. Making these connections involves negotiating multipledilemmas, such as whether and when to veer from the designatedcurriculum. In the present study, we sought to unpack the variousdilemmas contributing to this larger dilemmatic.

Understanding how teachers negotiate dilemmas as they make relevantreal world connections has the potential to productively inform teachereducation efforts and to support teachers in this practice. With thisaim in mind, we addressed the following research questions:

1. What, if any, dilemmas did this teacher negotiate whileconnecting her mathematics instruction to relevant real world contexts?and

2. What, if any, supports did this teacher draw upon when facedwith said dilemmas?

METHODOLOGY

Overview of Larger Project

The present study is part of a multiuniversity, longitudinalresearch project called Teachers Empowered to Advance Change inMathematics (TeachMATH) (Turner et al., 2012, 2014) in which we followedprospective elementary teachers from teacher preparation mathematicsmethods courses, through student teaching, and into their first years asearly career teachers. Our research focused broadly on teachers'understandings and practices for incorporating children'smathematical thinking (Carpenter, Fennema, Franke, Levi, & Empson,1999), and children's home and community-based funds of knowledge(Gonzalez et al., 2005) in mathematics instruction.

Case Study Methods

In the present study, we used case study methods (Stake, 2006/2013;Yin, 2013) to investigate one early career teachers' utilization ofreal world connections in her mathematics teaching. Case study methodswere appropriate for the present study because they facilitated an"extensive and in-depth description" of a phenomenon understudy, in this case, the complexities involved in teachers' makingreal world connections in mathematics teaching (Yin, 2013, p. 4). Wesought to understand how the teacher under study implemented andmaintained her commitment to making relevant real world connections inher mathematics instruction despite the presence of evolving dilemmas.

Participant Selection

Ms. K was purposefully selected (Creswell, 2013) as our case studybecause previous analyses (Turner, Sugimoto, Stoehr, & Kurz, 2016)established that she (more so than other participants in the largerresearch project) frequently enacted and reflected on real worldconnections in her instruction. In effect, Ms. K was a case ofpersistent intent to connect mathematics teaching to relevant, realworld contexts. That said, dilemmas often arose, and thus selecting Ms.K as a case provided insight into the complexities of this challengingteaching practice (Stake, 2006/2013). Ms. K identified as a biculturalLatina/White European descent woman in her mid-20s. Ms. K'smother's family was originally from Mexico and spoke Spanish. Ms.K's father was White, of European descent. Although Ms. K couldexchange everyday Spanish phrases with family members, she did notconsider herself to be a Spanish/English bilingual. Ms. K attended aprivate catholic school for her K--12 education and noted that herfamily struggled financially when she was growing up.

Teaching Contexts

For this analysis, we followed Ms. K from student teaching throughthe end of her second year of full-time teaching. During this time, Ms.K worked at various Title I schools in the same urban public schooldistrict. The schools were located in the same region of the city andserved similar student communities (i.e., student populations rangedfrom 80% to 95% Latino/a, and from 74% to 80% free/reduced lunch)(National Center for Education Statistics, 2013, 2014). During her firstyear of teaching, Ms. K taught in a seventh-grade self-containedclassroom and was responsible for teaching all subjects. The school useda "highly contextualized" mathematics curriculum entitledInvestigations in Number, Data, and Space (Technical Education ResearchCenters, 2014). During her second year, Ms. K moved to a science,technology, engineering, and mathematics focused magnet middle schoolwhere she taught seventh grade mathematics and had a grade levelmathematics team with whom she collaborated and planned. The school useda traditional mathematics text (Holt McDougal, 2011), which includedfewer real world connections.

Data Collection and Analysis

Data Sources

Consistent with case study methodology, there was prolonged contactwith Ms. K using various data collection tools, including observations,interviews, artifact analysis (see Table 1) (Creswell, 2013; Stake,2006/2013; Yin 2013). We conducted beginning, middle, and end of schoolyear interviews (3 interviews per year), or in the case of studentteaching, an end of student teaching interview. These interviews focusedon various topics, including understandings about children's homeand community experiences, conceptualization and utilization of realworld connections, and the supports and challenges that Ms. K.encountered when making connections. Additionally, across the 3-yearperiod, Ms. K was observed teaching 20 mathematics lessons. Each studentteaching observation was a single mathematics lesson, whereas eachobservation during early career teaching included two or threesuccessive lessons. Each set of lessons observed was accompanied by pre-and post-lesson interviews that explored the topics outlined above.During observations, we scripted all interactions, and then usedfieldnotes to produce a detailed lesson summary. Photos were taken todocument images on the board, handouts, and student work samples.Secondary data collection tools consisted of transcripts of early careerteachers' study group conversations.

Data Analysis

Multiple iterative cycles of analysis (Creswell, 2013) wereemployed. A preliminary analysis was conducted with first-cycle codingto demarcate segments of data relative to our research foci (Miles,Huberman, & Saldana, 2013). Specifically, we coded all data sourcesusing the single code "connection to real world and/orchildren's experience" in the qualitative data analysissoftware Hyper-Research (Researchware, 2011). This code was defined asany instance when Ms. K reflected on or made connections to the realworld and/or students' experiences or opened space for her studentsto make their own connections during instruction. Discrete instanceswere demarcated from the beginning of the narration about a connectionto the end; any time the topic or connection was changed we counted thisas a new instance. In all, 338 discrete instances were identified.

We completed our second cycle of coding using this refined dataset. We began with open coding related to our research questions. First,we coded for any challenges, tensions, or perceived difficulties thatarose in our data set because we anticipated that these difficultieswould help us better understand the dilemmatic space. Second, we codedfor anything that seemed to support the teacher as she negotiatedidentified challenges in order to better understand how she negotiatedthe larger dilemmatic space. An initial set of codes was drawn fromliterature (e.g., curricular challenges and supports, challenges withand support from colleagues); then, through open coding, new codesemerged (e.g., supports and challenges related to personal experiencesand interactions with students and parents). This resulted in a set ofcodes that was then used to code all data instances. To ensureconsistency in coding data, two of us independently coded a randomlyselected subset of the instances (approximately one third), and then metto discuss and resolve any discrepancies.

In the third cycle of analysis, we created analytic memos (Marshall& Rossman, 2014) for each code. Memos identified themes related tochallenges or supports, and included representative examples of Ms.K's understandings and practices. Themes were established byreviewing all data instances that were assigned the relevant code andnoting patterns, contrasts, and consistencies. For example, for the"supports: students" code, themes were identified related toinformal conversations with students and eliciting students'connections during lessons. We met regularly to discuss and refineemerging themes using representative examples from the coded data.

In the fourth cycle of analysis, we looked across the memos toidentify larger dilemmas of practice. The dilemmas lifted off theparticulars to synthesize discrete challenges into more comprehensivedilemmas of practice. For example, Ms. K repeatedly referred to resourcerelated challenges (e.g., curriculum, grade level team, and personalexperience), and these individual challenges were combined into thelarger dilemma of how to address her lack of resources. Additionally, welooked across the code memos for specific supports that this teacherdrew upon when negotiating identified dilemmas. We then used theanalytic memos to select three lessons that represented the patterns ofdilemmas and supports established across the larger data set. Wereviewed detailed lesson summaries for each of the selected lessons toconfirm, refine, and contextualize our interpretations of the dilemmasas they unfolded in practice. In the next section, we use these lessonexemplars to present our findings.

FINDINGS

Ms. K expressed a consistent commitment to making relevant, realworld connections in her mathematics teaching. This commitment and hernegotiation of multiple dilemmas related to this practice contributed toan ever-present dilemmatic space for Ms. K. Before describing thesedilemmas, we discuss Ms. K's rationale for relevant real worldconnections.

Commitment to Making Real-World Connections

Ms. K's commitment to real world connections in hermathematics teaching was rooted in (a) her own schooling experience and(b) her desire to engage students with the content. Ms. K recalled thather own teachers did not connect their mathematics instruction to realworld contexts even when prompted by students. She noted, "I thinkthat was always the question I was afraid of the question, 'When amI ever going to use this?' I remember hearing that all the time [asa student]. Our teachers never really had good answers." (Year 2,Middle of the Year Interview). Ms. K explained that she strove to makeconnections between mathematics and real world contexts explicit so thatstudents understood that mathematics had real meaning in their livesbeyond the classroom. Additionally, Ms. K emphasized during theinterviews that real world connections, particularly connections relatedto their own home and community experiences, engaged her middle schoolstudents. She explained:

If [the lesson] is not something that they've done or that they canconnect to... first, they're not as interested, and second, they'renot trying to learn it They're just, "get through this 50 minutes, andthen I'll move on to social studies." (Year 2, End of year interview)

Ms. K felt that by connecting to real world contexts that wererelevant to her students, she was able to engage and motivate herseventh-grade students even when they preferred other subjects. Ms.K's commitment to connecting her mathematics instruction torelevant contexts helped sustain her practice even as she negotiatedvarious dilemmas, as shown in the following lesson exemplars.

Lesson Exemplar 1:

Creating Real World Inequalities

During her second year of teaching, Ms. K negotiated severalchallenges as she drew upon real world contexts to deepen herstudents' understandings of linear inequalities. The mathematicalgoals for this set of lessons were: (a) construct linear inequalities torepresent real world situations, and (b) accurately solve inequalitiesusing addition and subtraction. During the lessons, students createdinequalities to represent a given real world situation as well assituations in their own lives. Ms. K shared that generating relevantexamples was challenging because of her own lack of experience with howthe content connected to real world scenarios. She reflected, "Ihadn't taught inequalities before, and I was struggling with how amI going to make this relevant to them? I never saw relevance in iteither" (Year 2, Postlesson interview). Ms. K explained that shefelt more confident teaching equations because "with theequations... I've had a lot of context to draw from because I wasable to teach it last year with a different curriculum that was a littlebit easier to make those connections" (Year 2, Postlessoninterview). However, while her previous curriculum included multipleexamples of how equations can model real world situations, it lackedexamples for inequalities. This led to her first challenge of how torelate the content to a relevant real world example with littlecurricular support.

Ms. K started planning the lessons with her curriculum, which had aPowerPoint that defined key vocabulary words (i.e., inequality, equal,compound inequality). However, she opted to discard the PowerPoint aftertalking with her mother about the lesson. She realized that her motherwas confused by the terms inequality and equal because she thought thatthe terms represented complete opposites (i.e., equal or not equal)instead of seeing that inequalities could also represent values that aregreater than or less than. Reflecting on how this conversation supportedher decision to veer from the text, and the teacher-directed teaching ofvocabulary, Ms. K noted:

There's a whole PowerPoint for me to go through that just tellsstudents [definitions], but I want to... have them generate thedefinitions first so that it's more memorable.... [Students] needsomething where they're looking at something that isn't equal... but aninequality is still this greater than, less than type of thing. I needto tie them both [equal and inequality] together. (Year 2, Pre- andPostlesson interviews)

Ms. K felt that by explicitly contrasting the terms, students wouldbetter understand how equal and inequality were related but different.She also planned to elicit definitions that were relevant to thestudents in an effort to make the terms more understandable.

Ms. K began by eliciting students' definitions of equal andunequal, and asking students to give an example from "inside andoutside of math" (Year 2, Fieldnotes). Students generated variousreal world contexts including differences in prices of cereals, weightsor heights of various objects, and the minimum age required to obtain ajob. During this initial discussion, Ms. K encountered a second dilemmaof how to honor students' ideas and also align them with hermathematical goals. For example, one student suggested that aninequality could be written for the "comparison of [the cost of]two different cereals" in order to identify the cheapest brand.This context focused students on direct comparisons of two prices. Ms. Kresponded by skillfully reframing the student's idea to expand theconversation to inequalities. She suggested, "we are buying cerealand the Target brand is $3 per box and then everything else is more(i.e., Price of other brands > $3/box), so we know we are going tospend $3 or more" (Year 2, Fieldnotes). Instead of abandoning thisstudent's suggestion, Ms. K was able to productively rework theexample to create an inequality.

Subsequent lesson tasks were designed to relate inequalities toreal world contexts. Ms. K shared that the examples in the textbook were"really lame" (e.g., amount of fabric needed to make a quilt),further expanding her dilemma of how to relate the content to relevantcontexts. Therefore, Ms. K looked to outside resources as a support. Forinstance, she had recently received an email about a video news storyfeaturing a 12-year-old youth activist lobbying to reinstate a state lawthat allowed 16- and 17-year-olds to preregister to vote so that theycould vote as soon as they turned 18 (Byler & Park, 2013). Ms. Kthought students would be interested in the age-related issue because oftheir interest in other age-related activities like driving. Aftershowing the video, Ms. K asked students to create inequalities thatrepresented the current voter registration age (i.e., a [greater than orequal to] 18) and the activist's desired age (i.e., a [greater thanor equal to] 16), where a refers to the age required to register.

Next, students wrote inequalities related to situations in theirown lives that they wanted to change, including reducing the cost ofcollege tuition or single-family homes, reducing homework assignments,increasing the number of chips in a bag, and several examples related tothe cost of items like shoes or manicures. However, some cases werechallenging to mathematize using inequalities. For example, one studentdecided to mathematize her current number of homework assignments andher desired number of homework assignments, using two equations torepresent the current (H = 4) and desired (H = 2) number of homeworkassignments. This created a dilemma for Ms. K of whether to disregardthis student's ideas in favor of other examples that aligned withthe mathematical goals, or to work with the student's ideas,pulling them toward the intended content. Eventually, Ms. K decided toguide the student to consider how she could use inequalities instead.

Ms. K: You get four [assignments] everysingle day?Student: Well, no, sometimes it's one.Ms. K: Okay, well, so it can be less thanfour. You're not stuck at four all the time.So you want the most to be two, but areyou okay with one assignment?Student: Yeah.Ms. K: Then you're fine with less than 2. Itdoesn't have to be equal.

Ms. K focused the students' attention on the concept of equaland unequal, and whether the number of homework assignments couldfluctuate. This discussion prompted the student to create an inequalityto more accurately represent the scenario (i.e., H [less than or equalto] 2).

Other challenges that students faced during the lesson includedaccurately writing inequalities to represent current and desiredsituations and correctly solving inequalities and interpreting theresults (Turner et al., 2016). These challenges contributed to a thirddilemma of how to respond when the real world contexts added complexity,but did not seem to support students' conceptual understanding. Forexample, some students wanted to represent cheaper gas prices usinginequalities. They started with the current scenario (e.g., Gas (G)costs $2.99/gallon). To represent their desired scenario (i.e., that gasprices should decrease at least $0.30 per gallon), they wrote aninequality involving subtraction (e.g., G - .30 [less than or equal to]2.99), which would result in an increase not a decrease in price (G[less than or equal to] 3.29). In the moment, Ms. K had to decidewhether to abandon the context to focus students on symbolic work withinequalities, or to use the context to help facilitate students'understandings. To support students' understanding of their ownmisconceptualization, Ms. K asked students to solve their inequality andto use the context to reason about the result. Students realized thatthis inequality increased the cost, which was the opposite of theirdesired change, and revised the inequality accordingly (e.g., G + .30[less than or equal to] 2.99).

Lesson Exemplar 2: Graphing and Scaling Video Game Characters

During her first and second year of teaching, Ms. K taught lessonsfocused on the following related mathematical concepts: (a) plottingcoordinates, (b) scaling plotted figures, and (c) determining if figureswere mathematically similar. In the lessons, students graphed points ona coordinate plane to create a figure and scaled said figure inmathematically similar and nonsimilar ways. Specifically, students usedsimple linear equations or "rules" to multiply or divide thefigures' coordinates in order to enlarge or shrink their figures(i.e., 1.5x, 1.5y or 1/3x, 1/3y).

During her first year, Ms. K used a set of unmodified lessons fromthe Investigations curriculum (Technical Education Research Centers,2014) to teach these concepts. The lessons focused on a fictionalcomputer game involving the mathematically similar Wump family whosemembers were the "same shape" but of "varying sizes"(Technical Education Research Centers, 2014). Students were given thecoordinates of the main video game character (Mug Wump) and were askedto enlarge the character using given rules (i.e., 2x, 2y). Next,students determined whether or not the newly scaled figures weremathematically similar to Mug Wump. After the lessons, Ms. K reflectedon students' reactions to the context:

There's always some storyline there [in the curriculum], and I feellike they're interested in the video game, but there's not a whole lotof opening up into their world necessarily or what they're doing withtheir parents at home. (Year 1, Postlesson interview)

Although Ms. K felt that her students were "interested in thevideo game," she did not know if the context related tostudents' experiences because students did not "open up."Ms. K explained that she "always tries to think about howthey're connecting [what they're learning] with their...experiences... but that hasn't been as apparent in thislesson." In other words, Ms. K tried to prioritize connectionsbetween her mathematical teaching and students' experiences but wasunsure if this prepackaged lesson accomplished this goal. Thisprofessional pondering contributed to her first dilemma related to howto use her curriculum to make these connections.

At the end of her second year, Ms. K taught a similar set oflessons focused on coordinate graphing and mathematical similarity;however, her school adopted a new curriculum that lacked contextualizedproblems. The limited curricular resources added to her dilemma of howto address the mathematics content in ways that were relevant tostudents' interests and experience. Ms. K decided to use the Wumplesson as a starting point, but found a real world context she feltwould be more relatable to her students, namely characters from popularanimated movies. In her words:

I saw this video about how Pixar uses math in movies. He [animator]goes through and it's coordinate geometry and... he talks abouttranslations and resizing images, and he just uses one of the moviecharacters. I'm like, that's such a good idea. (Year 2, Prelessoninterview)

Ms. K felt the animated movie context would be particularlyengaging because students often discussed the recently released movie,Frozen (Year 2, Postlesson interview).

Ms. K introduced the lesson by showing students the video clip andthen used the movie animation context as a springboard for students todraw and manipulate their own characters through scaling andtranslations. Ms. K described the lesson's tasks as:

I'm going to let them [students] draw whatever [character] they want.I'm going to tell them to make simple figures because they're going tobe moving them a lot... [I'm] going to let them play around with itand let them resize them, and eventually they're going to start movingthem around the grid, and working on translations. (Year 2, Prelessoninterview)

Ms. K planned to give students the freedom to draw any characterthey chose, with the caveat that "simple figures" with alimited number of vertices might be easier to scale and manipulate. Thismodified set of lessons also incorporated an expanded set ofmathematical goals (i.e., translations of figures, compared to theprevious year's lessons).

Ms. K's concern about students drawing "simplefigures" was particularly perceptive as evidenced by how the lessonunfolded. When she introduced the lesson, Ms. K noted that her students"were excited" (Year 2, Postlesson interview). Yet as thelessons progressed, some students were so caught up in designing oftheir figures that they neglected the mathematics of the task (i.e.,determining coordinates, scaling and translating figures). Ms. K shared:

What I'm having a little trouble with is they're so excited to justget on the design that they're forgetting the coordinate piece.... Iwant to give them that freedom to create what they want and to drawsomething cool to them. At the same time, I don't want them to havesomething so detailed that they're proud of it, and then I tell them,"Okay. Move it," ... and they're spending hours doing it again. (Year2, Postlesson interview)

Some students were so focused on making detailed characters (seeFigure 2) that several days into the lesson, they were still working oninitial drawings instead of plotting the coordinates and scaling andtranslating the figures. Ms. K anticipated that the detailed figureswould be challenging for students to scale and translate because itwould take "hours" to redraw the figures. Ms. K was faced withthe dilemma of how to honor students' contributions while advancingher mathematical goals. Ultimately, she decided to have students frametheir complex figures with a rectangle, and then use the coordinates ofthe four corners to resize the frame. Once the frame was redrawn,students could redraw their figures within the new frame so that theywould not have to simplify their drawings.

Lesson Exemplar 3: Scaling Recipes to Make Tortilla Soup

During her second year of teaching, Ms. K taught a set of lessonson multiplication of fractions. Following one textbook-based lesson thatincluded decontextualized computation tasks, Ms. K realized manystudents lacked a conceptual understanding of the meaning ofmultiplication by a fraction. She explained, "they know what to do,but they're still not understanding why their product is smaller.They don't understand the concepts behind it, basically, and sothey're apprehensive about following the procedure" (Year 2,Prelesson interview). At the same time, Ms. K felt pressure from hergrade level team to continue moving through the curriculum, noting"my team teachers are moving on to dividing. They're a lotmore, 'Let's go by the book and teach them the process, andthen if we have time, we'll bring in all this otherstuff'" (Year 2, Postlesson interview). The textbook featureda follow up "hands on activity" in which students drewfractional parts and then split the parts into smaller portions tovisually represent multiplying fractions (Year 2, Prelesson interview).Yet the texbook's tasks lacked a connection to students'experiences or to relevant contexts, which Ms. K felt was an importantcomponent of building understanding.

Thus Ms. K was faced with the dilemma of whether to revisitfraction multiplication via the text's "hands onactivity" or modify the lesson. She opted to veer from the text anddesigned a set of tasks around scaling a recipe for tortilla soup. Ms.K's choice of context was informed by a survey in which studentsreported using mathematics in family cooking activities. She explained,

What's missing [in the book] is that connection to real life or anyrelevance to them.... I did a survey toward the beginning of the yearabout where they see math in their homes, and the majority of it wascooking or baking, so I thought that the recipe would be a goodidea--something they're familiar with and they already see math in.(Year 2, Prelesson interview)

Ms. K's hope was that connecting fraction multiplication toscaling down a recipe would help students understand how multiplying bya proper fraction results in a smaller quantity. She noted:"I'm hoping they'll see--to serve less people, it'smultiplying by a half" (Year 2, Prelesson interview).

Ms. K acknowledged that generating lessons that connected torelevant contexts "completely on my own" was challenging. Shefound that leveraging her own understanding of the context (i.e.,scaling recipes) supported her in making the connection as authentic aspossible. She explained:

How can I make it the most realistic? I thought about getting one of mynana's recipes... [and] what her recipes look like, they're allhand-written, and a lot of them are "to taste"... I wanted it to lookfamiliar. (Year 2, Postlesson interview)

Not only did Ms. K leverage her own family experience to identify arelevant real world context, but she also decided to make the context as"realistic" as possible by creating a handwritten recipe thatwould seem more "familiar" to students (see Figure 3).

To launch the lesson, Ms. K shared about her own experience cookingtortilla soup, including how she often had to adjust the recipe forgroups of different sizes so that she knew how much of each ingredientto purchase at the store. In response, students discussed food theyenjoyed making with their families, and asked Ms. K if the recipe wasauthentic. Ms. K explained during interviews that she often beganlessons by telling relevant stories about herself and her family becausethis practice invited students to share their own experiences, whichhelped Ms. K build connections:

I get [to work with students for only] an hour each day, and it's beena lot harder to make those connections. ... When I have something like[my family recipe] that's sharing a part of me, they're willing toshare a part of themselves. .... Everybody is a little bit more open(Year 2, Postlesson interview).

Next, Ms. K drew students' attention to the ingredients andservings (16) in the recipe, and said that she "needed help.... tofigure out how much of each ingredient to purchase" when she makesthe soup for different numbers of people. The task read: Adjust therecipe for smaller and larger groups of family and friends. How much ofeach ingredient is needed to serve 8, 32, 40, 4 or 82 people?

As students worked, Ms. K probed their thinking with questions suchas, "How did you get this number?" or "What happened whenyou were feeding more people?" Students used multiple strategies toadjust the recipe, envisioning themselves cooking the soup, andquestioning if their adjusted ingredients made sense. Some students madeconnections between multiplying by a fraction and their priorunderstanding of operations with percents (i.e., multiplying by 1/2 istaking 50% of a quantity). Ms. K noted that for some students, thelesson enhanced their confidence toward fraction multiplication, becausethey could connect the procedures to the act of adjusting the amount ofeach ingredient in the recipe. She stated, "they had this hugeexperience that they could draw from to help them work with thenumbers" (Year 2, Postlesson interview). However, not all studentsembraced the context as a tool for making sense of the mathematics. Someremained focused on the procedures and determining which operations andquantities to use so they could efficiently complete the calculations.Ms. K reflected on the dilemmas she experienced as she tried to leveragethe context to engage these students in the mathematics: "I wastrying to explain it to them in the real world context, and they justwanted, "What operation do I use?" They didn't want tohave to think about why they're using the operations" (Year 2,Postlesson interview).

Other students embraced the real world nature of the task, and usedtheir understandings to argue for answers that while reasonable from areal world perspective, did not align with the mathematical goals of thelesson. For instance, to adjust the original recipe (which served 16) toserve 40 guests, one group decided to triple all the quantities so thatthe recipe would serve 48, "[the students] were just, like,"Can we just do three times more? ... [It will serve] 48;we'll have enough." Ms. K acknowledged the reasonableness ofthe solution, but also wanted to push students toward a more preciseanswer, because scaling the recipe for 40 servings provided additionalopportunities to reason about multiplying by a fraction. She explained,"You'll have leftovers. I think that could be an option,too... but then what if I don't want leftovers? How do we cut itback?" (Year 2, Postlesson interview). The students' solutionreflected a tension between realistic, real world solutions (which maynot always require precision), and the mathematical goals of aparticular lesson (which in this case, benefited from precision).

Another dilemma was related to balancing the authenticity of thetask presentation with the scaffolding students needed to supportunderstanding. The hand-written recipe, followed by general questionsabout how to adjust the recipe for different numbers of people resultedin a task that was more open-ended than those students typicallyencountered in textbooks. For example, recipe scaling tasks in problemsolving oriented curricula (Abels, Wijer, Pligge, & Hedges, 2006)often include scaffolds such as ratio tables that help students keeptrack of quantities and their operations on those quantities. While Ms.K's task presentation was potentially "more authentic,"the lack of scaffolding was challenging for some students who struggledto keep track of adjustments to ingredients on their own. When Ms. Knoticed that students' challenges organizing their work preventedthem from using the results of one adjustment to the recipe to makefurther adjustments, she decided (based on a suggestion from one of theresearchers), to provide students with a blank ratio table (see Figure4). As she introduced the tool, she again shared about her experiences:

My weakness in math has always been that I'm not organized... I usuallyget the right answer because I'm really good at math, like you guys arereally good at math, but I make little mistakes because my work's allover the place.... This is going to help you, and it's going to helpme, and we're going to work on it together." (Year 2, Postlessoninterview)

Students connected with Ms. K's experiences, responding,"Oh, that's me too, miss" (Year 2, Lesson fieldnotes).Ms. K later noted that using the ratio table to adjust the recipe, whilepotentially less authentic, supported students' success with thetas k. "I t was s o easy f or them, all of a sudden, to see how thenumbers related because they were lined up" (Year 2, Postlessoninterview).

FIGURE 4Ratio Table Given to Students During Tortilla Soup Lesson.Serving SizeOilTomatoesOnionGarlicCartons ofChickenBruthMincedCilantroSlicedJalapenosRotisserieChickenAvocadoesPacks ofQuesoFried CornTortillasLimeWedgesNote: This table was given to students to help them organize their workfor the lesson.

DISCUSSION

In discussing the study's findings, we focus on three dilemmasthat Ms. K negotiated in the larger dilemmatic space of connectingmathematics instruction to relevant real world contexts.

Dilemma 1: Lack of Curricular, Relational, and Personal Resourcesto Support Connections

Ms. K experienced several resource-related challenges in her effortto connect mathematical content to relevant real world contexts, whichcontributed to her first dilemma. Specifically, for the inequalities andcoordinate graphing lessons, although the curriculum provided real worldcontexts, the contexts were not relevant to her students'experiences and understandings. In the multiplying fractions lesson, Ms.K's curriculum focused more on computational and visualrepresentations rather than connections to real world contexts. In fact,across the lessons observed, Ms. K was challenged by curricular tasksthat focused on abstract mathematics content or generic real worldconnections rather than connections that foregrounded students'experiences. Ms. K was also challenged by a lack of relational resourcesfrom her grade level team. For example, although Ms. K wanted to pauseto carefully reflect on how real world connections might deepen herstudents' understanding of multiplying fractions, her colleagueswanted to move through the curriculum. Ms. K ultimately chose to createher own lessons, which led to an additional, personal challenge, namely,Ms. K's limited knowledge of how specific mathematics content(e.g., linear inequalities) connected to real world scenarios. Thevarious components of this resource-related dilemma are represented inFigure 5 by solid arrows that connect the challenges with correspondingcircles in the diagram. For instance, Ms. K's curriculum focused ondecontextualized mathematical content (math content/standards) orgeneralized real world connections (mathematics in the real world),while her grade level team and her own knowledge focused solely on themathematical content (math content/standards). Taken together, the lackof curricular, relational, and personal resources contributed to thedilemma of whether to follow her curriculum and/or team's plan orto veer from these plans and look to outside resources for support.

Returning to the framework of dilemmatic spaces, Ms. K faced adilemma of how to position her curriculum and her team, as well as howto negotiate her developing identity as a teacher. Ultimately, Ms. Kpositioned herself as a teacher with the authority to veer from the textand her team, but she was sometimes challenged by a limited knowledge ofrelevant connections. Therefore, she identified outside supports thatshe could draw upon in her negotiation of this larger dilemma, includinglooking to her students to generate real world connections (e.g., linearinequalities in their own lives), her experiences with real worldapplications of content (e.g., cooking), templates from prior lessons(e.g., the Wump lesson), and a broader relational network (e.g., hermother or emailed videos from friends). Ms. K leveraged these supportsto move her instruction from the mathematics content and mathematics inthe real world spaces into the center of Figure 5 where contentconnected to relevant real world experiences.

Although broad challenges in planning real world connections havebeen well documented (Asempapa, 2015; Depaepe, De Corte, &Verschaffel, 2009; Gainsburg, 2008), less understood is how teachersovercome these challenges. Our detailed analysis contributes to ourunderstanding of what a successful negotiation of such challenges mightlook like and identifies specific supports that teachers may draw uponwhen negotiating resource-related dilemmas. Specifically, our studyshows that Ms. K was able to look outside of traditional resources(i.e., textbook or grade level team), to find support for negotiatingthis dilemma from her own students, social network, and her ownexperiences.

Dilemma 2: Aligning Students' Real World Connections toMathematical Content

In this second dilemma, Ms. K was challenged when the real worldconnections students generated based on their own experiences andunderstandings did not align with her mathematical goals. For example,during the inequalities lesson, students' examples lent themselvesto equations rather than inequalities. During the graphing lesson,students drew complex characters that were difficult to resize andtranslate. In other words, students contributed ideas that reflectedtheir own interests and real world understandings (represented in Figure5 by the arrow from Dilemma 2 to the students' experiences in thereal world space), but these ideas did not always align with themathematical goals of the lesson (i.e., they did not intersect with themathematical content space).

Ms. K was faced with the dilemma of whether she should (a) dismissstudents' ideas and insert ideas that more explicitly orefficiently addressed her mathematical goals and thereby risking studentdisengagement, (b) accept students' ideas against the background ofher mathematical goals, or (c) reframe students' ideas to betteralign with the intended mathematics. In negotiating this dilemmaticspace, Ms. K positioned her students as important contributors to thelessons as she honored and yet guided their ideas so that the ideasconnected with her mathematical goals. Ms. K leveraged an in-the-momentsupport of reframing that allowed her to take students'contributions and with minor suggestions or task adaptations align themwith her content focus (i.e., students drew a frame around their complexcharacters to make resizing easier.) In other words, Ms. K found ways topull students' ideas into the center space (see Figure 5). Animportant contribution of this finding is that it draws attention tochallenges that can occur when students' ideas do not match themathematical goals of a lesson, a tension that has not been explicitlynoted in prior research. Moreover, this dilemma highlights the import ofin-the-moment supports that teachers can draw upon in order to addressthis dilemma.

Dilemma 3: Using Real World Connections to Efficiently SupportStudent Understandings

In this final dilemma, the complexity of real world connectionscontributed to students' confusion or distraction from themathematics. For example, in the inequalities lesson, some studentsfound the real world context of gas prices confusing and wroteinequalities that did not accurately reflect the situation. In themultiplying fractions lesson, some students struggled when they weregiven an authentic recipe card and asked to scale the ingredientswithout any structure for organizing their work. A few studentspreferred to ignore the recipe context altogether, and others used realworld considerations to argue for solutions that lacked mathematicalprecision. These challenges led to the larger dilemma of whether Ms. Kshould consider abandoning real world contexts in instances when theyconfused students, continue with the contexts even if students resistedor when contexts seemed to detract from mathematical goals, or adapt thecontexts and tasks to maintain a dual focus on students'mathematical understanding and mathematics in the real world. As shownin Figure 5, by the arrow extending from Dilemma 3 to the mathematics inthe real world space, real world connections did not always intersectwith students' understandings and experiences with the content. Inthese instances, Ms. K had to find ways to pull the real worldmathematics into the center of the diagram in order to supportstudents' developing understandings.

In negotiating this dilemmatic space, Ms. K positioned herself asan agentive, resourceful teacher while prioritizing both students'mathematical understandings and real world connections. To achieve thisbalance, Ms. K drew upon several in-the-moment supports, including toolsto scaffold and organize students' work (e.g., the ratio tabletool). Moreover, instead of abandoning contexts in the face of studentconfusion or distraction, Ms. K decided to push students deeper into thecontext to support their understanding. While the potential confusion ordistraction that real world contexts can create for students has beennoted (Greer, 1997; Verschaffel et al., 1994), our findings move beyondthe challenge itself to document potential in-the-moment supports thatthis teacher leveraged to address student confusion.

IMPLICATIONS

The present study documented the dilemmas that one middle schoolmathematics teacher faced and the supports she drew upon in her effortto connect her instruction to real world contexts that were relevant toher students' understandings and experiences. An importantcontribution of this work is a more nuanced understanding of how thesedilemmas and supports manifested in this teacher's practice. Inturn, this work points to new directions for how teachers can beprepared to take up this high-impact practice in middle schoolmathematics classrooms. Specifically, we found that planning forrelevant real world connections is only one piece of this practice eventhough it is the piece that is often discussed in the teacherpreparation literature (e.g., Julie, 2002; Simic-Muller et al., 2015).Another critically important component for this teacher wasin-the-moment moves for effectively negotiating the dilemmas that arosein practice, which were quite challenging. Our findings demonstrate thatteachers need to be supported not just in planning for real worldconnections, but also in responding in productive ways to the dilemmasthat arise during enactment. We found "in-the-moment"adjustments, like reframing students' contributions to align withmathematical goals and leveraging the context to help students makesense of the mathematics, to be particularly supportive and efficientstrategies. Effective use of these strategies appears to be especiallyconsequential for middle school mathematics teachers as they havelimited time to work with multiple groups of students. Ultimately, notpreparing teachers for these in-the-moment adaptations risks underminingthe practice, as these adaptations are critical for maintaining atripart focus on student experiences and understandings, real worldcontexts, and the mathematical content.

Acknowledgment: This study is partially supported by the NationalScience Foundation under Grant Number 1228034. Any opinions, findings,conclusions, or recommendations expressed are those of the authors anddo not necessarily reflect the views of the National Science Foundation.

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Amanda T. Sugimoto

Portland State University

Erin E. Turner

University of Arizona

Kathleen J. Stoehr

Santa Clara University

* Correspondence concerning this article should be addressed to:Amanda T. Sugimoto, [emailprotected]

TABLE 1Summary of Data Collection ToolsData Collection Tool Total CollectedObservational fieldnotes 20 days of observationsPre- and postlesson 22 interviewsinterviewsBeginning, middle, and 6 interviewsend-of-year interviewsTranscripts of teacher 11 transcriptsstudy groupconversationsData Collection Tool FocusObservational fieldnotes Lesson task, goals, launch, closure, and teacher-student interactionsPre- and postlesson Goals and considerations in planning lesson;interviews connecting to children's mathematical thinking, to children's home and community knowledge, and real world contexts during lessonBeginning, middle, and Supports and challenges related to connectingend-of-year interviews to children's mathematical thinking, children's home and community knowledge, and real world contexts in mathematics teachingTranscripts of teacher Planning and debriefing lessons with peersstudy groupconversations

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