Will a Social Culturally Contextualized Intervention Expand Students Conceptual Understanding of Fractions?

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Will a Social Culturally Contextualized Intervention Expand Students Conceptual Understanding of Fractions?

To address the research gaps noted in the literature review, the design of this qualitative study is intended to explore the effect a twelve-day intervention incorporating students own cultural backgrounds, will have on students’ fraction addition and subtraction strategies and perceived ability to solve and understand fraction concepts.

Research Questions

The driving question guiding this study, “will a social culturally contextualized intervention expand students conceptual understanding of fractions?” Specific questions addressed during this study include the following:

  1. How are student’s math discussions shaped by their experiences in a socio-cultural intervention that includes a combination of cultural contextualization, interactive conceptual scaffolding (i.e., CRA), and focused journaling?

How does the discussion contributions of students with prior experiences working in groups differ from the discussion contributions of students without prior experience?

  1. How will the socio-cultural intervention activities (cultural contextualization, interactive conceptual scaffolding, focused journaling, and student led discussions) impact students’ addition and subtraction strategies and perceived ability to solve and understand fraction concepts?
  2. What fraction misconceptions, if any, are evident in students understanding during and after the intervention as revealed by their responses to lesson prompts, journal entries, and post-test open-ended responses?

Rationale

This study is a qualitative research method approach involving a case study of fifth grade elementary students who attended a Texas urban elementary school. Fifth grade is the focus of this study as it is the first point within a student’s educational career where performance on the mathematics and reading STAAR test is used as a criterion for promotion to the next grade level. This provision is called the Student Success Initiative (SSI), which applies to all students in Texas in their fifth and eighth grade year. Students who do not master the necessary mathematical standards prior to taking the STAAR, are not promoted to the next grade level.

This study focused on the effects a social constructivist designed intervention had on students’ conceptual growth of fractions. While many research studies exist which focus on elementary students’ fraction comprehension difficulties, the incorporation of student’s linguistic and cultural background to aid in the comprehension of addition and subtraction of fractions, is far from being fully explored.

Case Study

The implementation of a qualitative case study research design is supported by researchers such as Baxter and Jack (2008) who believe, “rigorous qualitative case studies afford researchers opportunities to explore or describe a phenomenon in context using a variety of data sources” (p. 545). There is ambiguity within research as there are many definitions of what a case study is as well as the aspects within the study itself. A case study, as defined by Merriam (2009) is, “an in-depth description and analysis of a bounded system” (p. 40). An aspect of case study research is that it allows the researcher to collect and integrate quantitative survey data within the study, which enables a holistic understanding of what is being studied (Baxter & Jack, 2008).

One aspect of a case study which is agreed upon with most qualitative researchers are the characteristics of a case study. The most significant aspect of a case study is boundary of the case or its’ binding. The qualitative research design article written by Baxter and Jack (2008) shed light on the specifics fellow researchers have created suggestions of how to bind a case study, stating it should be defined by, “(a) by time and place (Creswell, 2003); (b) time and activity (Stake); and (c) by definition and context (Miles & Huberman, 1994); binding the case will ensure that your study remains reasonable in scope” (p. 547). This study will have boundaries which are specific to the student participants and excludes a proportion of the student population while remaining relevant to the national scale. As a result, this study incorporated participants who struggled with fractions the previous school year along with the current academic year. In addition, the study included students who have shown previous, and continued success in their math assessments. Overall, this study focused on the conceptual understanding of the addition and subtraction of fractions, because students must have a conceptual basis in order to build to more complex operations with fractions. For this research to be a case study, there must be specific learners selected “on the basis of typicality, uniqueness, success, and so forth” (Merriam, 2009). By setting into place specific boundaries for this study, it is intended to prevent too broad of a topic and too many objectives to be included which could prove detrimental to a research study (Yin, 2003; Stake, 1995).

Qualitative researchers consistently use case study methodology as part of their research. A case study, as defined by Merriam (2009) is, “an in-depth description and analysis of a bounded system” (p. 41). There are other researchers who choose to define case study by emphasizing different aspects. For example, Yin (2008) focuses on the research process and defines a case study, “a case study is an empirical inquiry that investigates a contemporary phenomenon within its real-life context, especially when the boundaries between phenomenon and context are not clearly evident” (p. 18). Additional researchers place emphasis on the boundaries of the case study and describe a case study as being less of a methodological choice, and more of a choice of what will be studied (Stake, 2005; Miles & Huberman, 1994). In order for the method to be considered a case study, there must be defined limits which will include specific participants, while excluding others. If the bounded system does not have a set boundary, or fence, to deviate between participants and the rest of the population, it is not considered a case study. Merriam (2009) extends on the bounded expectations, “for it to be a case study, one particular classroom of learners (a bounded system) selected on the basis of typicality, uniqueness, success, and so forth, would be the unit of analysis” (p. 41). The bounded system must be defined by specific characteristics of the participants, for example, to explain why they were included and why others were excluded.

School Context

The participating elementary school for this study consisted of 701 students for the 2015-2016 academic school year. The number of enrolled students placed the school as one of the largest school campuses in the district. The ethnic distribution of the school at the time of this study was: African American 12%, Hispanic 64.1%, White 20.5%, American Indian 0.1%, Asian 1.4%, Pacific Islander 0.4%, Two or More Races 1.4%. Over 50% of the students are economically disadvantaged. Additionally, 2% of students are English Language Learners (ELL) and 49.4% are considered At-Risk. The retention rate, specifically for fifth grade is 2.2% which is higher than the district average of 1.7% and the state rate of 0.9%. The average student to teacher class size of 20 students per teacher closely mirrors that of the district (21 students) and state (20.8). The elementary school used for this study consists of a near equal percentages of male to female: males 48% and female 52%. This student body percentage closely resembled that of the state which reports the typical elementary school in Texas consists of an approximate student body of 48.6% females and 51.4% males.

The participating elementary schools’ professional certified teacher staff consists of 97.4% females and 2.6% males. The ethnic background of the teacher staff are as follows: African American 1.0 %, Hispanic 2.0%, White 34%, American Indian, Asian, Pacific Islander, and Two or More Races are all 0.0%. The entire staff of teachers hold degrees: Bachelor’s 68.4%, Masters 28.9%, and Doctorate 2.6%. The years of teaching experience for the elementary staff include: 1-5 Years’ Experience 47.4%, 6-10 Years’ Experience 15.8%, 11-20 Years’ Experience 28.9%, and Over 20 Years’ Experience 7.9%.

The two teacher participants for the study, Mrs. Teller and Mrs. Knowles, closely reflect the demographics of professional teacher staff of the school. Mrs. Teller is a White female certified teacher who has taught elementary school for seven years. She earned a bachelor’s degree in multidisciplinary studies and a Master’s degree in math curriculum and instruction. Mrs. Knowles is a Hispanic female certified teacher who has taught elementary school for twelve years. She earned a master’s degree in technology curriculum and instruction.

According to the state of Texas Academic Performance Report (TAPR), the participating elementary school has consistently “Met Standard” in their accountability rating the past three consecutive years. In 2016, the schools’ pass rate for the STAAR assessment across all grade levels and subjects was 74%. This is below the district average of 77.3% and state average of 76.3%. Although the overall assessment scores are below the state and district average, the math scores are the highest in the school with an average of 79% average passing rate. The average for Texas on the math STAAR assessment is 78%. For the 2015 year, the elementary school had an 85% passing percentage for the STAAR math, which was a higher passing rate than the state which was 72% and the district which was 75%. The overall percentage of students in the elementary school who met the phase in standard across all tested subjects in 2015 was 76%. The district standard was 73% and the state was 77%. In 2014, the STAAR scores were relatively similar with an overall student passing rate of 77% in all subjects. This is higher than the district which reported a 70% passing rate and the state with 77%. In 2014, the elementary school reported a 91% passing rate of the STAAR math assessment, in comparison to the state of 87% and the district at 85%.

Participants

Twelve, fifth grade student participants from two classrooms were recruited for participation.  All participants were selected through purposeful sampling, as this method provides the broadest range of information as possible. Use of purposeful sampling allows the researcher to optimize the time constraints and convenience of the sample to select student participants who closely mirror the national average classroom. Patton (2002) describes the value of utilizing a purposeful sample for qualitative research stating the “information rich cases are those from which one can learn a great deal about issues of central importance to the purpose of the inquiry, thus the term purposeful sampling” (p. 230).

In addition, student participants previously completed the district assigned fraction unit during the 2016 fall semester. At the conclusion of the six weeks’ unit, all students were given a district assessment covering the fractions unit. Archival data of student performance on this district assessment were accessed to identify the participant pool for this study. The 2016 STAAR mathematics test performance was also used to determine participation eligibility. Current research utilizes samples of participants who were, for the most part, nominated by teachers or administrators (Empson, 2003; Kerslake, 1986). Using nomination as a selection criteria can result in selection bias from the teacher or administrator. To minimize selection bias, student performance on these two data sets was examined and participants selected.

The intervention was conducted at the school campus described above during regularly scheduled math intervention time. Invention activities were conducted in 30-45 minutes sessions, four times per week, for three weeks. The twelve 5th grade students were engaged in the intervention activities in groups of four, one student from each profile group. The students were identified as at or below grade level in mathematics based on state academic progress standards.

Selection of Participants. Test scores from the STAAR math assessment administered in the 2015-2016 school year and the district benchmark assessment for the fractions unit were used to identify potential student participants. The results from the STAAR mathematics test are divided into three performance standards. The standards are scheduled to follow a predetermined annual phase in allowing for, “consistent, incremental improvements toward the final recommended Level II performance standards in 2021-2022” (TEA, 2011). The “Phase In” standards are important to understand as they are established to require continual improvement and growth for both students and educators. The performance standards are as follows: Level I: Unsatisfactory, Level II: Satisfactory, and Level III: Advanced. The Scale scores are divided as follows: Level I: 868-1453, Level II: 1467-1657, and Level III: 1670-2068. Within Level II is a subcategory of “Recommended” with the scale scores of 1657-1589. The subcategory of Recommended within Level II performance standard is where the state of Texas aspires to have the majority of students by the 2021-2022 academic school year.

Reporting data from the fourth-grade mathematics STAAR test administered in March of 2016, report 21% of students in the state Texas tested with a raw score in the Level 1: Unsatisfactory category. The percentage was calculated based on a scale score. Students who obtained a score within the Level I performance standard are considered below grade level and did not pass the March 2016 Math STAAR.

Using STAAR mathematics assessment scores from students’ 2015 in conjunction with the 2016 district assessment scores, parameters were created for student participants. To establish student placement, a table with four profile groups were created to categorize students. Profile Group 1 consists of students who met both STAAR and district assessment standards. Students in Profile Group II comprises of students who did not meet state standards on the STAAR assessment, but did pass the district assessment. Profile Group III encompasses students who successfully met the state assessment standards but did not meet district standards. Lastly, Profile Group IV is comprised of students who did not pass the STAAR assessment and did not pass the district assessment. Table 3.1 illustrates the four profile groups.

The participants of this study have completed the district fractions unit, taught, during the Fall of 2016. For the district, students who score 40% or higher on the fraction unit assessment are considered passing. For the participating district, achieving a score of 40% or above is acceptable, as it aligns with state phase in standards. The study included a purposeful sampling of students who have not passed the STAAR math assessment or district assessment. In addition, participants will consist of students who have passed both the STAAR and district assessment, as well as students who have passed one but not the other assessment.

Profile Group: II

Failed STAAR

Passed District

Profile Group: I

Passed STAAR

Passed District

Profile Group: III

Failed STAAR

Failed District

Profile Group: IV

Passed STAAR

Failed District

Table 3. 1 Student Profile Group Assignment

Including past progress measures will provide information about the students’ past and perceived current number sense ability level. The intention of this study is to include participant students who have a history of struggling with fractions and do not currently indicate future growth progression. As described by Patton (2002), “when a typical site sampling strategy is used the site is specifically selected because it is not in any major way atypical, extreme, deviant, or intensely unusual” (p. 236). LeCompte & Preissle (1993) use the term criterion-based selection in place of the term purposeful; the researchers believe when using criterion based selection, “you create a list of the attributes essential to your study and then proceed to find or locate a unit matching the list” (p. 70).

Intervention Framework

Over the course of twelve days, this study followed a social constructivist intervention framework. Participating students worked through a myriad of activities with fractions in small groups. The cultural contextualization, interactive experiences, focused journaling were aspects of the conceptual scaffolding. These allowed for focused student led discussions to occur throughout the intervention. The result was anticipated to be for students to achieve a greater conceptual understanding of the addition and subtraction of fractions. In addition, record evidence of an increase in student reflection of perceptions in their ability levels in reference to addition and subtraction of fractions. To achieve these goals, the framework was cyclical in incorporating social interactions amongst students, their interests, while providing exposure to multiple models of fractions.

The intervention comprised of precise mathematical language throughout, beginning with the definition of a fraction. Wu (2011) explains that using the, “precise definitions as a starting point, logical progression from topic to topic, and most importantly, explanations that accompany each step (p. 371). Once this foundation is set, the intervention will then build upon it. The CRA model is applied to each activity as it allows for students to be exposed to several methods of not only visually viewing the fraction, but to manipulate it and understand the way it looks in various forms. Petit and colleagues (2016) explain that by, “intentionally varying the structures of the problems that students interact with can help deepen student understanding” (p. 197). Students who are in the Spring semester of their fifth-grade year have been exposed to fractions in different forms, such as concrete and abstract. The intervention is not intended to expose students to an unfamiliar representational model, but incorporate the models they have been taught thus far in their academic career to help in their conceptualization of fractions.

By fifth grade, students are expected to have moved away from the concrete and representations models of fraction and be comfortable with abstract representations of fractions. This study incorporates fifth grade students of all mathematic ability levels, which means there was a probability some or all participants would struggle with activities based solely on abstract representations. As a result, the CRA model was used in a cyclical manner throughout the intervention. Similarly, the linear, set, and area models were used in a recurrent manner as well. This method of intervention is based in researcher studies, such as Lesh, Landau, and Hamilton (1993) who emphasized that mathematics should be taught and represented in multiple ways, and students who are exposed to a variety of fraction representations, required to move back forth between representations, deepen their understanding. Precise mathematical language was used for the definition of the set, area, and linear models to help students gain an understanding of the models as well as have a point of reference throughout the intervention. Inclusion of the above-mentioned definitions were written into the students’ intervention journal. Definitions provided by Petit and colleagues (2016) were used for the models. The area model is when, “the fraction indicates the covered part of the whole unit of area (Petit et al., 2016, p. 10). Set model incorporates an, “equal number of objects, the number of objects in the subset of the defined set of objects” (Petit et al., 2016, p. 10). The linear model is a, “unit of distance or length, equal distance, the location of a point in relation to the distance from zero with regard to the defined unit” (Petit et al., 2016, p.10).

Definitions were recorded by students into their math intervention journal. The researcher collected the journals daily and read student entries daily, as students were observed taking notes in their journals that were not given by the researcher. For example, Rose would regularly open her journal to write down visual models to help her use the picture she drew to explain to her peers how to work through a problem. When prompted by the researcher, “I notice you are very comfortable with using the area model with rectangles to help you. Is that how you view fractions most often in your mind?” Rose replied, “yeah, I don’t know why, but any time I do things with fractions, I see rectangles, different sizes, connected together, sometimes not. Like I don’t really see the number, you know ½, but I see a rectangle cut in half.” Observing the students, incorporating a journaling with specific prompts, and allowing students to take notes they felt necessary during the activity allowed for a more in-depth understanding of the students’ conceptualization of fractions. Carpenter and colleagues (1999) substantiate this claim by describing how the understanding of student’s thinking allows teachers to, “use that understanding to select problems that challenge children to engage in problem solving and that children are willing to work at to solve” (p.104).

Interactive Experiences

The incorporation of social cultural contextualization intervention

through the duration of the intervention permitted students to scaffold one another’s learning while providing insight into their true personal interests outside of the classroom and their understanding of the task at hand. Students were encouraged to discuss how to solve work through problems and share their strategies. While there was reluctance in the first meeting to share their strategies, through encouragement and support of their ability level, students routinely shared without needing to be asked or prompted. When a strategy works and the student can explain their process, even if the strategy is different from any that have been taught, it is important to let the student know they are correct (Carpenter et a., 1999). As a result, “no one’s solution strategy is any better than anyone else’s, and each child’s thinking becomes important to everyone” (Carpenter et a., 1999). Once students realize their voice and explanations will be valued, not ridiculed, they are more willing to share their thought process.

Lesson Example

A general outline of lessons was created to provide a basic foundational map of the activities the students would complete daily. It is important to note, the researcher worked with each group for only 30 minutes a day. The lessons were created in such a way to allow students ample time to discuss their work, strategies, each other’s questions, and complete the task within the allotted timeframe. Table 3.2 is an example of a lesson which required students to use manipulatives and the linear model. During the lessons, the researcher did not intervene with students work or conversations, but observed. By doing so, the researcher was able to generate a holistic understanding of the participants and their fraction conceptual understandings. The social interactions as well as the student answers to the writing prompts influenced the next day’s activity.

Table 3. 2 Lesson Example

Date:
Activity Description Manipulatives
Spiral review Activity:

Area model

Using the geoboard (students can use other manipulatives if necessary), count how many ways you and your partner can create fourths. Geoboard

Dry erase board/markers

Intervention Journal

Today’s Focus:

Linear Model; correctly placing a fraction on a number line

Reiteration:

Linear “in the case of the number line, for example 1/3 is represented by a location on the line or scale indicating the distance from zero, not 1/3 of the whole line” (Petit, et al., 2016, p. 11).

“How far did Lexi go?”

-students are first given a paper with several questions about a trip that Lexi went on in her neighborhood.

-questions include: “Lexi walked 3/8 of a mile to her Bestie’s house. Create a number line that would accurately represent this distance.”

-students work together to answer questions by building the number lines using any means they are comfortable with.

-Students share their work to be checked by the researcher to ensure its’ accuracy.

Number line

Cuisenaire rods

Linking Cubes

Dry erase board/markers

Intervention Journal

-no distinguishing marks on manipulatives, such as fraction tiles.

Writing Prompt: Did you get frustrated trying to build multiple number lines to solve the problems today? Why or why not? What helped me get past it?

(students have been prompted at the beginning of the intervention to write several sentences to answer, not overgeneralized “yes, I liked it” form of responses.)

Quantitative Instrumentation

Archival Data

State testing scores from the students’ fourth grade mathematics STAAR test were collected for identification of eligible student participants as well as to document the consistency of difficulties the students have faced with number sense over time. The use of official state documentation records provides, “both historical and contextual dimensions to your observations and interviews” (Glesne, 1999).

Attitude Scale

The goal of the study it to deepen students’ conceptual understanding of addition and subtraction of fractions. In addition, determine if the intervention altered students’ perceived mastery of the topic. For this reason, students completed a student math survey. The survey targets students’ belief about mathematics, their perceived understanding of fractions and mathematics in general, as well as their comfort level of working in a social math environment. The survey for this study was based on the student attitude survey (SAS) created by members of the Dartmouth research and innovation in STEM education. The SAS was constructed with the incorporation of prior research surveys, such as research on affect and attitude by Golding, Epstein, & Schorr (2007). The work of Locke (2000) of interpersonal values and supporting research of enjoyment of mathematics by Ma (1997) and Thorndike-Christ (1991) was also incorporated into the creation of the SAS. After its’ creation, “an external advisory board helped direct the design of the instrument, refine it, and validate it by piloting it with students and teachers in their own graduate teacher education classrooms” (Brookstein et al., 2011, p. 2). The SAS consisted of twenty-seven questions. It was modified to fewer questions to allow fifth grade students to have an appropriate amount of time to answer the questions during interventions. Modifications of the SAS were minimal but necessary. For example, question number two on the SAS asks students to reflect on their middle school experience. In its place, the survey asked students to answer how they feel their confidence level in their abilities to solve addition and subtraction fraction problems. Questions which were not applicable to the target participant audience were replaced with appropriate questions. As a result, the survey is aligned with the needs of this study to render data on student perceived ability level with fractions and interactions in the math classroom.

Qualitative Data Sources and Collection Methods

Small Group Intervention

The researcher met separately with the student groups representing the four types of student profiles described above to engage in intervention activities. Working with small groups of students allowed for a unique community of learners’ dynamic to be created. It was the intention of the researcher to create a safe learning environment for the students to feel comfortable in various social interactions with one another and the researcher.

The twelve-day intervention for this study the course of this study, were implemented with three separate small groups of students at varying ability levels. The study followed a cognitively guided instructional approach in order to break from the mainstream teaching of process standards with fractions. For that reason, the suggestions of Carpenter and colleagues (1999) were utilized as a framework for the interventions. Based on students’ initial responses in the pre-assessment, it was determined to begin with defining a fraction. Students were introduced to explicit terminology in the definition of what a fraction is, and wrote the definition down in their math intervention journal. After which, instruction moved to unit fractions, equivalence, then to addition and subtraction of fractions.

In support of the interventions curriculum, questions about the students’ interests, hobbies, and life outside of school were asked and documented. This allowed for a lens into the students’ life in which to pull information to then incorporate into the interventions themselves. By doing so, it was anticipated students would not only have more interest into the math problem they were posed with solving, but that the scenario would make sense as it would then be relevant to that students’ life. In addition, each lesson began with a concrete activity for students to work together to solve. The lessons then moved to picture representations to abstract, consistently moving through the CRA model.

Interviews

Interviews were conducted with two teacher participants. Interviews occur in qualitative research to, “find out from them those things we cannot directly observe, we cannot observe feelings, thoughts, and intentions” (Patton, 2002). The questions within the interviews were semi-structured. Based on the answers given by the participants, follow up questions, or probes, were asked. By utilizing a semi-structured interview framework, the researcher is able to, “respond to the situation at hand, to the emerging worldview of the respondent, and to new ideas on the topic” (Merriam, 2009). Furthermore, the semi-structured interview design permitted the researcher to capitalize on the consistency afforded by a highly-structured interview, yet include details offered by open-ended or emergent interviews. The interviews were audio recorded; this practice ensures that everything said is preserved for analysis (Merriam, 2009). Each interview was then transcribed verbatim to provide a record for analysis (Merriam, 2009). Interview of teachers were conducted at a time of convenience for them; before school, during their conference period, or after school.

Student Survey

The first intervention activity for students to complete was the student perceived fraction ability survey. The survey used was based on the “Attitude Survey” created by Dartmouth Research and Innovation in STEM Education. Alterations to the survey were necessary for both the participant demographic it was administered to as well as for the research study context. To complete the survey, students were circled the most appropriate response they felt best reflected their personal feelings about the question. The survey consisted of 10 statements. Students circled corresponding numbers of 1: Agree, 2: Neutral/Unsure, 3: Agree. Students completed the survey twice: once at the beginning of the intervention and a second time at its’ conclusion. The survey template is located in Appendix D.

Assessment

To assess student fraction ability, a pre-and post-assessment were given to each of the student participants. The pre-assessment data was used as a guide for where the interventions would begin in relation to fractions. Students completed a pre-assessment survey, yet a pre-assessment was administered as part of the conceptualization as data collected strictly from the student survey was not considered to be as accurate as an actual assessment. The assessments were not multiple choice and required students to write their answer or explanations which provides the researcher with a more accurate understanding of student’s conceptual knowledge of fractions. Questions number 1-3 attempted to establish which models of fractions students are accustomed to and recognize (Kerslake, 1986). Numbers four through six attempted to discover whether students could acknowledge fractions as numbers. Questions seven through nine sought to assess students’ ability to recognize equivalent fractions. Finally, questions ten through twelve established students’ ability to add and subtract fractions.

Questions were structured in a manner students might not have been familiar with, for example question number two provided eight pictures with variations of ¾ represented. Students were asked to circle each picture they believed represented ¾ accurately. The assessments were based off the work of Kerslake (1986) and rephrased to meet the needs of this study. The assessments were created in such a way because the, “the structure of a problem impacts the nature of the evidence of student understanding that is elicited by that problem” (Petit et al., 2016).

The first question asked was “How would you explain to someone, who didn’t know, what a fraction is?” When administered, all 12 student participants skipped this question and came back to answer it after they had completed the remainder of questions. Responses to the question included: “it is a number with a line in between”, “it is when you have a cake or pizza and you split it up, and then you have 3/8”, “I don’t understand what a fraction is”, “a fraction is a number that has a numerator and denominator”, while another student stated, “I’d have to give them a fraction set and tell them to study it.” Based on the evidence within the assessment, the intervention activities began with students defining a fraction in their intervention journal in conjunction with providing hands on activities and social interactions to create a more solid conceptualization of a fraction.

Teacher Interviews

Interviews of teachers provided information and insight pertaining to the method of instruction student participants have been exposed to within their mathematics classroom. Considering the nature of interviewing, questions were semi-structured allowing the researcher to ask additional clarifying or follow-up questions. Teacher interview questions targeted five main topics: 1) individual math career background and experiences, 2) curriculum design the teacher implements or follows, 3) individual teaching practices, 4) teacher beliefs of students in the mathematics classroom, and 5) practices teacher uses to embrace culturally and linguistically diverse students. The questions served as an outline of the interview, as probing questions were asked for clarification and to gain a better understanding of the answer given as well as additional context. The interviews were designed to provide in depth information about the dynamics of the classroom during the spring data collection, in addition to how the classroom functioned previously in the fall semester when teachers introduced the fractions unit to students. For this situation, the interview, “allowed us to enter into the other person’s perspective” (Patton, 2002). An area of interest is the method of teaching both math teachers utilized when introducing fractions. Interviewing the teachers allowed the researcher to gain insight into the pedagogical methods used with the student participants in this study. In addition, the interview attempted to gain insight into how the teachers felt overall of how their fraction lesson instruction went, what they noticed students struggled with, and which aspects or interventions benefited students most

Observations

Observations of the participants setting is a common method used in qualitative research (Berg, 2004). Obtaining observational data allowed for a, “firsthand encounter with the phenomenon of interest rather than a secondhand account of the world obtained in an interview” (Merriam, 2009). An observation is defined as a “detailed notation of behaviors, events and the contexts surrounding the events and behaviors” (Best & Khan, 1998). Utilizing observation allowed the researcher to observe how the teachers teach, their pedagogy, social interactions amongst students, as well as how this specific group of learners’ functions within their learning community. Observing the two classrooms allowed for the participant students to be observed in the classroom. The researcher was able to observe student participants and how they interacted with their fellow community of learners. Observational notes include whether student participants are interacting, active participants of their learning community, keep to themselves, as well as the frequency of such interactions. Merriam (2009) describes how, “observation makes it possible to record behavior as it is happening” (p. 118). In addition, the inclusion of observation allowed for the triangulation of the emerging findings (Merriam, 2009).

For this study, it was determined that the relationship between the researcher and student participants would implement an observer as participant framework for data intervention data collection. It is believed that this would allow for a more accurate and informative firsthand account of the interactions of the participants. Highly detailed field notes were taken immediately following each intervention time. Worked into the intervention schedule were ten minutes intervals where the researcher did not have students. During this time, highly descriptive field notes were recorded in the researchers’ journal. Bogdan and Biklen (2007) reflect a key aspect between observing and recording is the time frame in which it is done, stating “the more time that passes between observing and recording…the poorer your recall will be…less likely you will ever get to record your data” (p. 127). In reflection, each day following the conclusion of the interventions, the researcher would add details or thoughts to the field notes for that day. Included in the notes are direct quotes of students, description of the setting and activities, as well as reflective comments from the observer.

Student Perception

Prior to the beginning the work of fraction interventions, students were given a ten-question survey. The survey required students to self-reflect on how they view their ability to solve fractions. Students checked a box for each question which were: I don’t agree, I feel ok about it, and I agree. Along with the answer box to check were face emoji’s which coincide with the answer to help students who were unsure of the meaning of the answer. In conjunction with the survey, students were given an open-ended response (OER) to answer. Students answered an OER about their feelings of fractions at the beginning, middle, and end of the study. The first OER asked students to describe their own confidence level when working with fraction in the math classroom. Similar OER’s were given to students to answer toward the middle and then again at the end of the study. See Appendix for survey.

Coding

Once data was collected, the vast amount of text must be reduced to the data which is most significant and of interest (McCracken, 1988; Wolcott, 1990). To reduce the data to a manageable level, themes and patterns were created through coding of the data. Considering the researcher is open to the information the data will provide, open coding was used in this study. Under open coding, the data was thoroughly reviewed and analyzed. Assigning codes to the data allowed the researcher to observe patterns within the data and lead to the construction of themes, or categories (Merriam, 2009). Based on open coding, the preliminary themes derived from the data, were fleshed out which required categories to be revised (Merriam, 2009). After open coding was conducted, the next step was axial coding. The categories created from open coding were then grouped together, called axial coding (Corbin & Strauss, 2007). Subcategories were created under each theme which allowed for a more in depth understanding of the data.

Collection Methods

Triangulation of Data

Triangulation of data occurs with the use of various forms of data within the study. The incorporation of triangulation means the, “refining, broadening, and strengthening conceptual linkages” (Berg, 2004). Triangulation of the data ensured validity of the findings. The collection of data from multiple sources to compare enhanced data quality based on the principles of validation of the findings (Knafl & Breitmayer, 1989).

The study included various open-ended response questions as part of an exit ticket for students to complete in their intervention journal after the daily activity. The open-ended response consisted of questions pertaining to that specific activity completed, reflection, or explanation of strategy. Questions, for example, required students to answer whether they felt the intervention tasks were beneficial to their learning, extended their knowledge of fractions, gave them a voice in their mathematics studies, and if working more constructively within their community of learners empowers them to take ownership of their learning. In addition, students answered mathematical problems focused on the specific mathematical content from several lesson activities. Student participants completed a formative pre-and post-assessment, which included parallel questions between the two tests. The pre-assessment was given prior to the interventions beginning, while the post assessment was given at the conclusion of the twelve-day intervention.

The formative pre- and post-assessment consisted of twelve questions, none of which were multiple choice. Each question required the student to develop their answer based off the information given to them. Using the work of Kerslake (1986) and Petit et al., (2016) a formative assessment was created which comprised of four separate sections assessing the students’ ability to recognize models of fractions, fractions on a number line, equivalent fractions, and addition and subtraction of fractions.

Timeframe and Chronological Sequence of Study

The research for this study was conducted during Spring semester of 2017 over the course of three weeks. During this time, the researcher remained cognizant of the impending STAAR retest specific fifth grade students were preparing to take in May. The observations were the initial activity for this study. By conducting the observations first, the student participants were able to become familiar with the researcher within a comfortable environment prior to interventions. The setting for student observations were conducted in their normally scheduled classroom, time, and with their current math teacher. Following the observations, teacher interviews were conducted in mid-April prior, to the intervention being conducted.

The setting for the twelve-day intervention was a hallway within the school. The daily math lessons occurred during the students regularly scheduled school day. In preparation for the impending retest of the STAAR for students who did not pass the first administration in March, the fifth-grade team created an alternate student schedule which allows for specific interventions times for students throughout the day every week. The math lessons for this study were conducted during this intervention time. Student participants had an alternate daily schedule which included meeting with the researcher for additional math lessons.

Location

The primary location of data collection is an elementary school located on the South Plains in Lubbock, Texas. A pseudonym for the school is being used to ensure confidentiality. The elementary school is the largest elementary school in the district. The population size allowed for a larger participant draw, which provided student and teacher participants needed for this study.

There was variation of class, mathematic achievement, gender, and class among participants. However, the study assessed students’ perceived fraction ability after the implementation of a social culturally contextualized intervention.

Data Analysis

Continual analysis of data as they are collected enables focusing and refining the study (Glesne, 1999). The ongoing analysis of data allowed for the data to remain organized, focused, and not as overwhelming to the researcher as the sheer volume of material collected for this study was large (Merriam, 2009). By analyzing the data as it is collected, the researcher attempted to discover themes in the data while making sense of the data (Strauss & Corbin, 1990). The data analysis was cyclical in manner as to not only make sense of the data but to understand the dynamic of the study. A risk associated with the data analysis, is the mistake of treating each data source independently and its findings conveyed separately. To avoid this, “the researcher must ensure that the data are converged in an attempt to understand the overall case, not the various parts of the case, or the contributing factors that influence the case” (Baxter & Jack, 2008).

The initial step of data analysis was the examination of the individual interviews. After the data was transcribed, teacher interview responses were analyzed for language which depicted the intricacies of the classroom dynamics and student interactions. The data was grouped by similar expressions or phrasing. Focus words included group activity, student centered, differentiation, and personalities. When these words were found, they were marked. Data which provided understanding into the research questions were also evaluated. Furthermore, teacher responses were used to develop not only an understanding of the teaching methods implemented prior to the research study being conducted, but also the data in relation to student responses to methods of learning and interacting with their peers when learning in the math classroom. Data reduction and analysis products

Open coding is utilized as well, as it allowed for the exploration of the data and opportunity of themes to emerge. Implementing open coding occurred by, “assigning some sort of shorthand designation to various aspects of your data so that you can easily retrieve specific pieces of data” (Merriam, 2009). Once broad themes were created, they were placed together to determine an overarching theme for the study (Berg, 2004). Throughout the research study, the data, themes, and categories created were regrouped and reorganized as emergent new themes were realized or redefined.

Data Management Plan

Prior to this study being conducted, the researcher followed proper protocol and requested permission from the Institutional Review Board at Texas Tech University. In addition, authorization from Lubbock Independent School District was requested. Upon approval, an additional request was sent to the principal of the participating elementary school. A copy of the review board letter is included in Appendix A of this study.

Upon approval from the Texas Tech Institutional Review Board, a letter packet was sent home to the parents of the adolescent fifth grade students. Each packet contained a research study information page, parent consent form, a student assent form, as well as meeting information. In addition, the researcher offered times to meet with the parents face to face or over the phone, at their convenience, to answer any questions they might have about the study and their students’ participation. Included with the consent letter sent home is an invitation to attend an informal meeting held after school at the participating elementary school. This meeting allowed for an open forum for parents and their students to meet the researcher, ask questions, as well as learn about the study being conducted.

Qualitative research generates a great deal of data, which if not consistently and continually organized and archived, can overwhelm the researcher. To prevent such an instance, in this study, data is collected and stored electronically. Student work samples were scanned and saved within an assigned document file. Interviews were recorded then transcribed electronically and stored in a word document. The utilization of the computer to store and maintain data for this study allowed for ease of access to the data as well as necessary organization.

Validity and Transferability

Threats to the reliability and validity of a qualitative study are discussed in past and present research. The burden of proof for transferability of this study lies with the researcher, as Lincoln and Guba (1985) explain that, “the investigator needs to provide sufficient descriptive data” in order for transferability to be possible (p. 298). A rich thick description, a phrase created by Gilbert Ryle (1973), and defined today as a “highly descriptive, detailed presentation of the setting and in particular, the findings of the study” (Merriam, 2009). Incorporating a rich thick description of the setting, participants, and findings will help enable transferability. Lincoln and Guba (1985) explain, “the best way to ensure the possibility of transferability is to create a thick description of the sending context so that someone in a potential receiving context may assess the similarity between them and the study” (p. 125).

As previously stated, careful consideration and attention was given when selecting the study sample to allow for maximum variation to help enhance the transferability of this study by creating the possibility of, “a greater range of application by readers or consumers of the research” (Merriam, 2009). When interacting with the participants, whether through the interview or intervention, member checking was consistently incorporated as a means of clarifying and interpreting the conversations and statements made (Baxter & Jack, 2008; Merriam, 2009). Additionally, an audit trail is created by the researcher which will include a “detailed account of the methods, procedures, and decision points in carrying out the study” (Merriam, 2009).

Summary

The use of qualitative research approach is used as it enables the researcher to obtain the information desired for this study. The use of interviews, observation, previous state assessment scores, previous district scores, provided sufficient data needed for this study. As stated by Stake (1995), Simons (1980), and Yin (1984,) collecting data from multiple sources and utilizing various techniques of gathering data contributes to the strength of case study method. Baxter and Jack (2008) go further as to describe that a case study, “is an excellent opportunity to gain tremendous insight into a case; it enables the researcher to gather data from a variety of sources and to converge the data to illuminate the case” (p. 556). The use of a qualitative approach in this research permitted the researcher to not only implement specific intervention activities to help students progress with fraction addition and subtraction strategies, but also obtain insight of understanding from the student perspective.

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