I haven’t been posting here in a long time because this year I finished my thesis for graduate school. I haven’t graduated from school for 15 years studying education. After a fifteen-year journey, I have finally reached the culmination of my graduate studies, a period marked by both academic pursuit and significant life events. Throughout this extended period of dedication, I successfully earned my special education certificate, a field that has always held a profound and unwavering passion within me. This passion manifested early in my career as a personal aide for nine formative years within a multi-handicap classroom, an experience that deeply shaped my understanding of individual needs and abilities. My professional path then led me to a school specifically catering to students with cognitive delays and emotional disabilities. It was within this environment that I truly discovered where my heart lay, a deep and abiding connection to these remarkable individuals who inspire such profound empathy within me.

Reflecting on the initial decision to embark on this graduate-level education, the path proved to be far from linear. The fifteen years it took to complete were interwoven with significant personal milestones and challenges. During this time, my life was enriched by the arrival of three children, each of whom was subsequently diagnosed with autism. This transformative experience profoundly impacted my priorities and time. Additionally, my personal life underwent significant transitions, including a separation from my husband and the subsequent joy of finding love anew, leading to a relocation from New Jersey to Indiana. Furthermore, my career journey involved transitions between three different schools, each offering unique experiences and perspectives. It is fair to say that the ebb and flow of life itself frequently intersected with the demands of graduate school, occasionally causing delays and requiring adjustments to my academic timeline.

However, looking back, I would not have navigated this journey in any other way. My current husband is an incredible source of support and love, and my three children, along with the wonderful children who are now part of our blended family, bring immense joy to my life. Together, we are a large and vibrant family of eleven children, a dynamic that I cherish deeply. Despite the length and at times challenging nature of my educational pursuit, I am immensely proud of my perseverance. There were moments when it would have been easier to give up, but I remained steadfast, chipping away at my goal by taking one class at a time, navigating the complexities of student loans for some courses and managing out-of-pocket expenses for others. Finally, I reached the point of completion, and I felt an overwhelming sense of accomplishment.

It is important to me to share this journey and to emphasize that even amidst life’s inevitable challenges and detours, achieving one’s goals is possible. While I recognize that everyone’s circumstances are unique and the obstacles faced can vary greatly, I firmly believe that giving up should never be an option. Each individual possesses the potential for greatness within their chosen field of endeavor. Personally, I have experienced the weight of loneliness, the feeling of the world crumbling underfoot during difficult times such as having essential utilities like electricity disconnected due to financial hardship. Yet, I have also experienced the profound power of love and connection.

As an educator, I have witnessed the transformative moment when a student’s eyes light up with understanding because of the instruction they have received. I have felt the warmth of a student expressing that they will miss you, a testament to the bonds formed within the classroom. I have also known the profound sorrow of losing a student, a heartbreaking experience that leaves an indelible mark. Conversely, I have experienced the immense joy of seeing a former student graduate high school and then reach out through social media, eager to reconnect and share their progress. And on a personal level, witnessing my own children grow and mature, knowing that they observed my determination to complete my education, fills me with hope that they will internalize that same resilience and drive in their own lives. It is my sincere prayer that they carry this lesson of perseverance with them, knowing that with dedication, they too can overcome obstacles and achieve their aspirations.

My deep-seated passion for poetry and creative writing initially presented a significant hurdle when I encountered the demands of academic discourse. This paper represents a considerable effort to navigate the conventions of formal writing, and I ask for your understanding as you read, keeping in mind that multiple revisions have been undertaken, and further refinement may still be necessary. My master’s thesis delves into the pervasive challenge of math word problems. It might seem paradoxical, given my current role as a math educator, but my own educational journey was marked by a significant fear of both math in general and, more specifically, word problems. This personal experience fuels my research, which seeks to understand how we, as educators, can empower students to overcome this anxiety and develop effective strategies for deconstructing and solving these problems. My focus is on identifying pedagogical approaches that foster confidence and break down the often intimidating nature of mathematical word problems, ultimately transforming a source of fear into an opportunity for growth and understanding.

 Schema-Based Instruction (SBI): in Improving Students’ Ability to Solve Math Word Problems

Abstract 

 This research, conducted within a Kentucky middle school facing socioeconomic challenges and low educational attainment, investigated the impact of schema-based instruction (SBI) on students’ mathematical word problem-solving abilities. The study sought to determine whether implementing SBI, specifically utilizing the FOPS strategy (Find the problem type, Organize information, Plan the solution, Solve the problem), enhanced students’ comprehension and problem-solving proficiency. Participants included six 7th-grade and six 8th-grade students with Individualized Education Programs (IEPs) that included mathematics goals. All students were educated in an inclusive setting, with five students receiving 20 minutes of supplemental resource math three times per week. Data collection encompassed pre- and post-math word problem-solving assessments, student work samples, observation and reflection logs, video recordings of teaching practices, and student reflections. The analysis of the collected data revealed three key themes: explicitly teaching schema-based instruction, applying schema-based instruction in practice, and the transfer of skills from the resource room to the inclusive classroom. Overall findings strongly suggested that SBI was an effective approach for improving students’ mathematics learning outcomes. 

Reflective Planning 

Demographics 

 The Henderson County School District in Kentucky served 6,955 students with a 16:1 student-teacher ratio across its 8 elementary schools, 2 middle schools, 1 high school, and alternative learning center. While the district provided a range of educational options, it faced significant socioeconomic challenges, with 39.9% of students economically disadvantaged and 17.7% of county children living below the poverty line. This disparity was reflected in the county’s educational attainment, where only 18.9% of the district population held a bachelor’s degree or higher, while 37.5% had a high school diploma or equivalent and 23.9% had some college or no degree. Alarmingly, only 3% of students received free or reduced lunch and breakfast, indicating a potential lack of support for low-income families, (KY DOE Report 2023-2024). 

South Middle School in Henderson County, KY, served 753 students and boasted a diverse student body with 73.6% Caucasian, 9.3% African American, 7.8% multiracial, 7.4% Hispanic, and 1.4% Asian students, equally split between male and female. The school excelled academically with 51% of students achieving proficiency in math and reading, exceeding both state and district averages, and earning a #21 ranking among Kentucky middle schools. In addition to its diverse demographics, South Middle School supported 85 special education students, 33 students with 504 plans, and 16 English Language Learners, demonstrating its commitment to inclusive education. 

This was my third year at South Middle School, where I had been teaching math in a co-teaching classroom and using supplemental resources for 7th and 8th graders. I was the case manager for 15 students and the service provider for 9 students, one of whom was also an English language learner. When I started developing my research goal, I knew my students were struggling in math, but I wasn’t sure what areas needed the most support. At times, it seemed like they lacked motivation, but I realized that wasn’t the case. When students struggled, giving up could seem like an easy option. When they saw word problems, they often felt defeated. This wasn’t a matter of motivation; it was about frustration and a lack of confidence in their abilities. 

The district currently utilizes C.U.B.E.S., a math strategy for tackling word problems. This acronym outlined a step-by-step process: Circle important numbers and units, Underline the question, Box math keywords, eliminate information that is not important and solve and check the answer. Although C.U.B.E.S. offered a systematic method for word problems, the district was now investigating other strategies to cultivate a more thorough grasp of the mathematical principles involved. 

Many of my students required the following accommodations: speech-to-text, text-to-speech, extended time for assignments and state assessments, manipulatives for English language arts and mathematics, calculator usage, and human scribes in case of technological difficulties. 

Many students and adults experienced math anxiety. Math required consistent practice and a willingness to persevere, and word problems often presented a significant hurdle for learners. A large percentage of our state testing (the KSA, Kentucky Summative Assessment, including NWEA MAP) focused on word problems involving various operations. Even in my classroom, the most commonly missed questions on tests and quizzes were word problems and open-response questions, which required careful analysis and problem-solving. Therefore, my goal was to equip my students with the tools they needed to conquer word problems. Mastering this skill was crucial, not only for academic success but also because word problems were fundamental to real-world applications and mathematical reasoning. 

 Goal Statement 

 This action research aimed to investigate the effectiveness of schema-based instruction (SBI) in improving 7th-grade students’ ability to solve math word problems. The research focused on enhancing students’ comprehension of word problems, problem-solving skills, and mathematical reasoning by explicitly teaching them how to identify underlying problem structures (schemas) and apply appropriate solution strategies. SBI was implemented using a combined approach that integrated various strategies, including explicit instruction and modeling. Data were collected through student work samples, class video recordings of my teaching, completed student classwork, and pre- and post-surveys to gauge students’ comfort and confidence with SBI compared to the district’s previous strategy. 

Literature Review 

 Math word problems are critical for developing students’ critical thinking skills. They require students to go beyond simply applying formulas and instead interpret real-world scenarios, translating them into mathematical equations. This translation process promotes a deeper, contextual understanding of mathematical concepts, moving beyond rote memorization to genuine comprehension of their practical use. Engaging with word problems also fosters strategic thinking and logical reasoning. Students must analyze information, identify the question, determine necessary operations, and plan a step-by-step solution. Research by Dobson et al. (2021) highlights the impact of thought patterns on learning, suggesting that the analytical processes involved in solving word problems refine these patterns, enhancing cognitive flexibility and problem-solving skills applicable beyond mathematics. 

Despite these benefits, many students, even those without identified learning disabilities, struggle significantly with math word problems. This widespread difficulty indicates the inherent complexity of these problems, requiring skills beyond basic computation. This challenge is amplified for students with difficulties in English Language Arts (ELA) due to the text-based nature of word problems, which heavily relies on reading comprehension. Students need strong skills in decoding, vocabulary, identifying key information, and understanding context to extract the mathematical essence. Weaknesses in reading fluency, vocabulary, and comprehension can create major obstacles. The interaction between language processing and mathematical reasoning is crucial, highlighting the interconnectedness of academic domains. 

Recognizing the importance of word problems and the challenges they pose, this chapter will explore effective instructional strategies and tools to support mastery. Drawing on principles from mathematics education and ELA instruction, we will examine techniques to equip students with the skills and confidence to solve word problems. We will focus on strategies for students with learning difficulties and general comprehension challenges. Our investigation will also consider the cognitive processes involved, including the role of schema theory, which emphasizes the importance of prior knowledge in comprehension and problem-solving. Furthermore, we will explore the interplay between reading comprehension and mathematical abilities, aiming to understand how strengths in one area can support the other and how weaknesses can create barriers. By examining these cognitive underpinnings, we aim to inform the development of targeted and effective instructional interventions. 

Theoretical Framework 

This study is grounded in cognitive theory, a psychological perspective emphasizing the role of internal mental processes—beliefs, desires, interpretations, and memory—in how individuals respond to the world and construct their actions, emotions, and learning (Dobson et al., 2021). Cognitive theory posits that the interaction between an individual’s cognitive structure and external stimuli is key to understanding behavior and knowledge acquisition. It highlights the learner’s active role in processing information, organizing it into meaningful structures, and using these structures to guide future thought and action. This theoretical framework underscores the need to examine the specific cognitive processes students use when tackling math problems and how these processes influence their problem-solving strategies. Understanding how students mentally represent problems, access prior knowledge, and monitor their progress is vital for designing effective instruction. 

A rigorous research methodology was employed to provide a strong empirical foundation for this investigation. The initial data collection phase involved extensive Internet searches across academic databases such as Google Scholar, JSTOR, EBSCOhost, and ERIC, selected for their comprehensive coverage of educational research. Strict selection criteria were applied to ensure the relevance and rigor of the literature. Only full-text, peer-reviewed articles directly addressing schema-based instruction, their implementation in ELA and mathematics, and specific schema-based strategies for solving math word problems were included. This systematic approach to the literature review ensured the inclusion of high-quality, pertinent research findings, strengthening the study’s validity and reliability. The focus on peer-reviewed articles guaranteed that the foundational knowledge had undergone critical scrutiny and met established academic standards. 

The Intricate Challenges of Math Word Problems for Students 

Math word problems present a unique and often significant challenge for students across all educational levels. Their complexity arises from a combination of factors beyond numerical manipulation, demanding a sophisticated interaction of comprehension, interpretation, and strategic thinking. Research consistently shows that students and adults alike experience difficulties with these narratives embedded in mathematical contexts (Boonen et al., 2016; Daroczy et al., 2015). While they offer the potential to connect abstract math concepts to real-world applications, their multifaceted nature often leads to anxiety, decreased confidence, and hindered mathematical progress (Khoshaim, 2020). 

The difficulty with word problems stems from several key areas. Primarily, they require a cognitive shift from rote application of algorithms to a deep understanding of the problem’s context. Unlike isolated calculations that focus on procedural fluency, word problems necessitate actively engaging with the narrative, analyzing information, identifying relevant elements, and strategically planning a solution. This multi-step process requires explicit instruction and significant support to guide students through each stage and foster independent problem-solving skills (Pongsakdi et al., 2019). The transition from recognizing a mathematical operation in symbolic form to identifying the appropriate operation within a narrative is a significant cognitive demand for many learners. 

Furthermore, success in tackling word problems heavily relies on strong language proficiency. Students must decode and interpret the written text, navigate potentially complex sentence structures, and identify key information within the narrative. A strong understanding of mathematical vocabulary, such as “sum,” “difference,” “product,” and “quotient,” is essential for accurately translating linguistic information into mathematical representations. This complex interplay between language comprehension and mathematical reasoning can be particularly challenging for students with weaker reading skills (Wakhata et al., 2019). Consequently, research often indicates a strong correlation between reading comprehension scores and performance on math word problems, highlighting the fundamental role of linguistic abilities in this area. Ambiguous phrasing or extraneous information can further complicate these linguistic challenges, requiring students to critically evaluate the relevance and meaning of each element in the problem statement. 

The persistent difficulties encountered with word problems can have significant negative impacts on students’ motivation and their self-perception as mathematicians. Repeated struggles and feelings of inadequacy can lead to the development of a fixed mindset, where students believe their mathematical abilities are unchangeable and that they are inherently “bad at math” (Dweck, 2006). This detrimental mindset can create a substantial psychological barrier to future learning and engagement with mathematical concepts. Additionally, anxiety and frustration often associated with attempting and failing to solve word problems can further impair cognitive function, hindering students’ ability to focus, persevere, and ultimately progress (Khoshaim, 2020). This negative emotional cycle can establish a self-perpetuating pattern of avoidance and underachievement in mathematics. 

To effectively address these multifaceted challenges, educators must recognize the inherent complexities of math word problems and implement targeted instructional strategies. Explicit and systematic instruction in various problem-solving strategies, such as drawing diagrams, identifying keywords (with the caveat that keyword reliance can be misleading), working backwards, and using manipulatives, can provide students with a range of approaches for different problem types. Simultaneously, a focused emphasis on vocabulary development, specifically targeting mathematical terms and their contextual meanings, can improve students’ ability to accurately interpret problem statements. Moreover, integrating strategies to enhance reading comprehension skills, such as identifying the main idea, summarizing information, and making inferences, can directly support students in extracting the necessary mathematical information from the narrative. By providing this comprehensive and targeted support, educators can foster greater confidence, reduce anxiety, and ultimately empower students to view math word problems not as insurmountable obstacles, but as opportunities to apply and deepen their mathematical understanding within meaningful real-world contexts. 

Math and Comprehension 

Several pedagogical strategies show promise for improving students’ ability to solve algebraic word problems, which fundamentally involves translating real-world situations into algebraic language (Huda & Sutiarso, 2023). While these problems often present a significant challenge, adapting successful instructional techniques from other disciplines, such as language arts, can offer valuable pathways to improvement. 

One such strategy is adapting schema-based instruction, a well-established method in reading comprehension and writing. In language arts, schema theory suggests that readers and writers use existing knowledge structures, or schemas, to understand and create text. This same principle can be applied to mathematics. By explicitly teaching students to identify common problem types or schemas in algebraic word problems, educators can provide them with a framework for understanding the problem’s underlying structure. For example, students can learn to recognize “equal groups” problems, “comparison” problems, or “rate-time-distance” problems. Once a student identifies the schema, they can access prior knowledge about how to approach that type of problem, facilitating the selection of appropriate equations and solution strategies (Powell & Fuchs, 2018). This explicit instruction in recognizing and using mathematical schemas can help students move beyond simply manipulating numbers in isolation towards a deeper understanding of the problem’s inherent relationships. 

Furthermore, the power of visualization, a cornerstone of effective reading and writing instruction, can be used to improve mathematical comprehension. Just as readers create mental images to understand narratives and writers use visual descriptions to engage their audience, students can benefit from visualizing the scenarios presented in algebraic word problems (Dwivedi et al., 2017). Encouraging students to create mental images, draw diagrams, or use manipulatives can make abstract mathematical concepts more concrete and accessible. By visually representing the quantities and relationships described in the problem, students can develop a more intuitive understanding of the problem’s structure. This visual approach can be particularly helpful for students who struggle with abstract reasoning or who are visual learners. For instance, in a word problem involving distance and speed, a student might draw a timeline or a diagram showing the movement of objects to clarify the relationships between the variables. 

The “3 Reads” protocol, a valuable tool for fostering deep reading comprehension, offers another promising approach for enhancing mathematical problem-solving skills (Fisher & Frey, 2012). This strategy involves having students read a word problem three times, with a specific focus for each reading. The first read aims at understanding the overall context of the problem – what is the situation about. 

Action Plan and Data Sources 

Implementation of Action Research Plan  

Reflecting on my teaching career, I taught a variety of subjects, though math had consistently been a source of apprehension for me. As a middle school student myself, I personally struggled with mathematics, particularly word problems. Unfortunately, this challenge was also evident among many of my students, posing a significant issue given that word problems constituted approximately 70% of state assessments and placement tests within our district. 

Observing my students as they tackled word problems, I noticed a recurring pattern: they lacked the fundamental skills required to deconstruct the problem, arrive at a solution, and genuinely understood the question being asked. Following research into effective instructional strategies, I decided to implement a schema-based approach to enhance my students’ learning outcomes. Specifically, our focus was on multiplicative schema-based instruction, recognizing its particular relevance for middle school students who frequently encountered multi-step problems involving multiple operations. 

In addition to schema-based instruction, we incorporated the use of various math manipulatives, including dice, algebra tiles, fraction tiles, number lines, counters, geometric shapes, cubes, place value disks, fraction bars, and Geo-boards for exploring area and perimeter. 

Ultimately, we meticulously analyzed the data collected before, during, and after the intervention to evaluate its effectiveness and identify any emerging trends. A detailed timeline outlining the data collection process was included in Appendix B. 

Intervention Plan   

The skill that this action research study aimed to improve was my students’ ability to comprehend and solve math word problems. I used a schema-based, multi-operational intervention model known as the combined approach. This approach was particularly suitable for middle school math, as students at that level encountered multi-step and multi-operational problems. 

Schema-based instruction utilized concrete examples to help students understand problems. The FOPS strategy, a multi-step process, guided students through problem identification, analysis, and resolution. The first crucial step was to read the problem carefully. Students read the problem at least twice, with some strategies suggesting three readings. The next step was to identify the type of problem and determine the necessary operation to solve it. Many students rushed to solve without understanding the operation, which often led to errors. 

The “F” in FOPS stood for “figure out,” where students determined what the problem was asking. This involved identifying the problem type and organizing the information, often using diagrams or manipulatives. The “P” in FOPS was for “plan,” where students developed a strategy to solve the problem. Once the plan was in place, they could then “solve” the problem (“S” in FOPS). 

Schema-based instruction employed concrete examples to help students grasp problem-solving concepts. The FOPS strategy, a multi-step approach, guided students through the problem-solving process. The first crucial step was careful reading. Students read the problem at least twice, or even three times for complex problems. Next, they identified the problem type to determine the necessary operation. Many students rushed to solve without fully understanding the problem, which often led to errors. 

Steps  

Read the problem (2-3 times)  

F – Figure out the operation  

O – Organize the information  

P – Plan an attack  

S – Solve and show work  

Does it make sense? Check your answer  

Each week, I introduced students to new manipulatives and other ways to make the problem more concrete. They used drawings and models to work out each problem. During the first week, using our manipulatives, we worked out a plan. I worked with each student to help them use these skills to develop a plan to solve the problem. 

In Week 1, I introduced the idea of reading the problem twice. 

Week 2 focused on the “F” in FOPS. We spent that week reading and figuring out what the problem was asking and what operation we needed to use to solve it. 

Week 3 involved focusing on different math manipulatives such as algebra cubes and geometric shapes to help make each problem more concrete. 

In Week 4, I continued to introduce students to new manipulatives and other ways to make the problem more concrete. They used drawings and models to work out each problem. That week, using our manipulatives, we worked out a plan. I worked with each student on how we could use these skills to develop a plan to solve the problem. 

Week 5 was dedicated to solving the problem, but that week we specifically focused on solving, showing our work, and checking if the answer made sense. 

After students solved the problem, we reread the problem and asked if their answer made sense in relation to the question. If they said yes, I asked them to explain why. If they said no, they also had to explain why. If the answer was indeed incorrect, we focused on identifying why it was wrong and how to correct it. 

Weeks 6 through 10 were centered around the principle that math thrives on repeated practice. The more one practiced, the better they became. We continued to practice, adjusting our approach based on student work samples, observations, recordings, and assessments to pinpoint areas where students needed more support and the specific parts of the problem-solving method they were struggling with. 

Data Sources  

 For this research, I will be working with both 7th and 8th-grade special needs students who exhibit math deficiencies. To gather relevant data, I will employ a diagnostic assessment using multiple math probes aligned with the Kentucky State Curriculum for each student’s grade level.  

Bi-Weekly Probes: Student work samples and associated teacher feedback were collected. Documentation included specific interventions, manipulatives, student performance, and required support for subsequent probes. Math word problem probes aligned with IEP goals were administered twice weekly. 

 Audio and Video Recordings: To identify my research goal and pinpoint the specific area within it that my students struggled with most, I utilized audio and video recordings of my teaching practices. As I continue with my research, I will use this method to track my students’ understanding as they progress through the intervention.  

 Observation and Reflection Logs: Multiple reflection logs will be collected throughout the data collection process. These logs will focus on student progress, including descriptions of work samples, areas requiring further improvement, common mistakes, and notable successes. This analysis will help me refine my interventions and determine the appropriate focus for each student.  

 Pre-tests and Post-tests: Pre-tests were used to establish a baseline for students before the intervention. I will be administering several assessments throughout the process to monitor their progress. At the end of the data collection period, I will administer a post-test to measure their overall progress through the intervention.  

NWEA Math Maps tests- A final data point will be their overall performance on the NWEA Math Maps test administered in April 2025. The initial scores used for this research were derived from their performance on the fall and winter 2024 assessments.  

Analysis of the Data and Evidence 

 Through detailed and comprehensive analysis, the effectiveness of a schema-based instructional intervention, specifically focusing on the FOPS strategy, was evaluated to enhance seventh-grade students’ comprehension and problem-solving abilities in mathematics word problems. This action research aimed to address the prevalent nature of word problems, constituting approximately 75% of the NWEA MAP Growth assessment and the end-of-year state testing, and to improve the performance of students in a co-teach classroom who faced challenges in mathematics and had math-related goals within their Individualized Education Programs (IEPs). The intervention, conducted between December 2024 and late March 2025, utilized a multifaceted instructional approach, including small group work, one-on-one instruction, and whole-group lessons, to provide tailored support for individual student needs. 

To rigorously analyze the intervention’s impact, a multi-faceted approach was employed, encompassing a deep dive into a reflective log, close examination of recorded classroom videos, and detailed review of collected student artifacts. The reflective log served as a rich source of qualitative data, capturing ongoing observations and reflections throughout the intervention. Systematic analysis involved thorough reading, chunking into meaningful sections, and implementing a color-coded system to identify and track emerging categories and patterns. Classroom videos provided insights into actual classroom practices and student interactions, with attention paid to engagement, instructional strategies, and overall dynamics. The same color-coding system facilitated direct comparison between reflections and observed events. 

The coding process revealed initial connections, leading to the synthesis of related codes into broader themes. This iterative process identified key factors influencing student learning and teaching practices. Further refinement involved detailed video analysis with focused viewing segments and meticulous coding. While initial video analysis was exploratory, later analysis became more focused on research aims, particularly the implementation of the FOPS strategy. A tally chart tracked FOPS usage, providing quantitative data to complement qualitative insights. 

The culmination of this analysis yielded three key themes: Explicit Instruction and Schema Identification with special education students, Schema Application and Problem Solving, and Transfer and Generalization. Within the theme of Schema Application and Problem Solving, a significant instructional component involved integrating drawings and diagrams to enhance understanding. When addressing the “Organize” step of the FOPS strategy, students were guided to translate problem information into visual representations such as bar models for part-part-whole and comparison problems, arrays for equal groups, and before-and-after models for change scenarios. These visuals not only helped organize the information but also made the necessary operations in the “Plan” step more apparent. During the “Solve” and “Check” phases, these diagrams served as a tangible reference for evaluating the reasonableness of answers. The explicit teaching of how to create and interpret these visuals for each schema, directly linked to the FOPS steps through modeling, guided practice, and discussion, emerged as a crucial element in supporting students’ comprehension and problem-solving abilities. The subsequent sections will delve deeper into the evidence supporting these themes, drawing upon the analyzed reflective logs, classroom videos, and student artifacts to provide a comprehensive understanding of the intervention’s effects. 

This is the baseline data that I took.  There will always be some outliers, in any data. I have a few students who scored higher than most. The mean of this data is 18.8%, the median is 10% and the range is 60.  

Table 1. 

Explicit Teaching Schema-Based Strategy 

 The absence of prior exposure to schema-based instruction among the student participants in this research study constituted a noteworthy initial condition. Within the school district where the study was conducted, a distinct pedagogical methodology had been the exclusive mode of mathematics instruction. Consequently, the introduction of a schema-based approach engendered certain immediate difficulties as students navigated this novel framework. This unfamiliarity with the fundamental principles and application of schema-based problem-solving represented a salient potential impediment that necessitated careful consideration and proactive strategies throughout the research endeavor. Recognizing this initial lack of experience was crucial for interpreting student performance and for the effective implementation and evaluation of the schema-based intervention. The research design incorporated measures to mitigate this challenge, including explicit instruction on schema identification and application, ample opportunities for guided practice, and ongoing assessment of student understanding of the schema-based strategies. The potential influence of this initial unfamiliarity on student learning trajectories and overall outcomes remained a key variable under observation and analysis as the study unfolded. 

My reflection logs indicate that students with math deficits, especially those on my caseload, struggle with word problems more than other students. While scheme-based instruction incorporates visual aids and drawings, I found that my students required a more concrete approach. Therefore, I integrated manipulatives into my instruction along with drawings (Observation/Reflection log #6) 

 To comprehensively address the varied learning styles and ensure the mathematical success of all my students, a multifaceted approach involving a range of accommodations was intentionally integrated into my instructional practices throughout the academic year. Recognizing that an overemphasis on rote calculation can sometimes obscure the deeper conceptual understanding of mathematics, the consistent and flexible use of calculators was permitted in various activities, from daily practice exercises to formal assessments. This strategic allowance aimed to empower students to concentrate on the reasoning and problem-solving processes inherent in mathematics, fostering a more profound grasp of the underlying principles rather than getting bogged down in lengthy and potentially error-prone manual computations. The calculator became a tool for exploration and verification, allowing students to test conjectures and analyze results more efficiently. 

Furthermore, to bridge the gap between abstract mathematical concepts and concrete understanding, a rich assortment of math manipulatives was thoughtfully incorporated into the curriculum. These tangible resources, such as base-ten blocks, fraction bars, algebra tiles, and geometric solids, provided students with opportunities to physically interact with mathematical ideas. This hands-on engagement facilitated the visualization of abstract concepts, making them more accessible and meaningful. By manipulating these tools, students could construct their own understanding of mathematical relationships, build intuitive insights, and develop a stronger foundation in key mathematical domains. The use of manipulatives moved mathematical learning beyond the purely symbolic and into the realm of active exploration and discovery. 

Acknowledging that certain physical or learning differences might present barriers to demonstrating mathematical understanding through traditional methods, additional supports were provided to ensure equitable participation. For students who experienced difficulties with the physical act of writing, scribes were made available to record their mathematical thinking and solutions. This accommodation ensured that their cognitive abilities and mathematical reasoning were accurately represented, without being limited by fine motor skills or processing speed related to writing. The scribe acted as a facilitator, allowing students to fully articulate their mathematical understanding and problem-solving strategies. 

Recognizing that mathematical texts often contain complex language and specialized vocabulary that can pose challenges for some learners, the provision of readers was also implemented. This accommodation was particularly beneficial for students who might struggle with decoding or comprehending written mathematical problems and instructions. By having access to readers who could orally present the written material, these students could focus on the mathematical content itself, rather than being hindered by reading difficulties. This ensured that all students had equitable access to the information necessary to engage in mathematical tasks and demonstrate their problem-solving abilities. 

The overarching goal in implementing these diverse accommodations was to cultivate a truly inclusive learning environment within the mathematics classroom. By proactively addressing potential barriers to learning and providing individualized support, I aimed to create a space where every student felt valued, supported, and empowered to actively participate in mathematical learning and ultimately achieve their full mathematical potential. This commitment to differentiated instruction and the provision of appropriate accommodation reflects a belief that all students can succeed in mathematics when provided with the necessary tools and support tailored to their individual needs. 

Figure 1 

Data from Video Recording of Lesson # 1 Before Intervention 

This video, captured prior to the implementation of my intervention, offered a valuable glimpse into my students’ problem-solving processes when tackling mathematical word problems. My initial intention in recording this session was purely exploratory, aiming to observe their natural inclinations and strategies. Upon reviewing the footage, before solidifying my research topic, I was struck by a significant discrepancy in their approach depending on the level of guidance provided. Specifically, while engaged in a whole class setting where we collaboratively read the word problems aloud, their participation suggested a degree of comprehension. However, their independent problem-solving behaviors, as clearly documented in the video, painted a different picture. When left to their own devices, a concerning pattern emerged. Instead of engaging in a careful and thoughtful reading of the problem statement, their primary focus appeared to be the identification of numerical values. They would swiftly circle these numbers and underline the question being asked, seemingly as a perfunctory step. This surface-level engagement with the text bypassed the critical stage of truly understanding the context, the relationships between the quantities, and the underlying requirements of the problem. Consequently, their selection of a mathematical operation seemed to be driven by the mere presence of certain numbers or keywords, rather than a logical deduction based on a comprehensive understanding of the problem’s narrative. This observation underscored a significant deficiency in their ability to thoroughly read and interpret word problems independently, a realization that ultimately became the central focus of my research investigation. The video served as a crucial piece of evidence, highlighting the need for an intervention designed to cultivate deeper reading comprehension skills as a foundational element of mathematical problem-solving. 

 Initially, upon reviewing my first video recording, a significant insight emerged in my reflective log concerning my methodology for introducing word problems to my students. Although a specific lesson topic had not yet been determined, the primary focus of my pedagogical approach, I observed, was to cultivate in my students the indispensable practice of meticulously reading and comprehending the entirety of a word problem before attempting any solution strategy. This foundational step, I recognized, was absolutely critical for fostering deep understanding and ultimately leading to successful problem-solving outcomes. Without this initial careful reading, students often jump to calculations based on keywords, missing the underlying mathematical relationships and the specific question being asked. This premature engagement with numbers can lead to frustration and incorrect answers, hindering the development of true problem-solving skills. 

As I began to delve deeper into the principles of schema-based instruction and explore its potential applications within my classroom, the reasons behind my students’ struggles with mathematical word problems became increasingly apparent. I realized that their difficulty stemmed not just from a lack of computational fluency, but from an inability to identify the underlying problem structure and connect it to relevant prior knowledge. 

 In my second video lesson, I intentionally and explicitly taught my students using the initial phase of SBI, specifically a schema application strategy. This strategy is known as FOPS, which is an adaptation of a common mnemonic and stands for Find the answer, Organize information, explain your work, and double-check your answer. 

As a schema application strategy employed in the early stages of instruction, FOPS provides a systematic procedure for students to follow. It consists of two main parts: “The Approach” and “The Drawing Part 2.” “The Approach” outlines the cognitive sequence embodied in the FOPS acronym: Finding the unknown and crucial information, Organizing this information, Explaining the steps or thought process, and double-checking the solution. This equips students with a structured method for addressing problems. 

Supplementing this approach is “The Drawing Part 2,” which underscores the vital role of visual aids like illustrations and diagrams. Drawing offers a concrete means to arrange information, visualize the problem’s structure (schema), and make abstract ideas more tangible. This supports both the planning stages and the explanation of the solution within the broader FOPS framework. Together, the methodical steps and the visual element of FOPS provide students with a robust instrument for applying their schemas to effectively solve word problems. 

Furthermore, we actively incorporated the use of visual aids, such as drawings and illustrations, as tools to represent and math word problems. As evidenced by the data captured in the second video, a notable shift occurred in my students’ engagement and understanding. The data reveals a greater number of “categories” of student interaction, indicating a richer and more nuanced problem-solving process. Significantly, prior to the explicit instruction on using drawings, my students primarily focused on extracting numbers and performing operations without a clear understanding of the problem’s context. However, after being introduced to visual representations, they began to ask more pertinent and insightful questions, demonstrating an emerging ability to see the connections between different pieces of information within the problem. This indicated a developing comprehension of the underlying schema. My deliberate use of questioning techniques and the provision of more targeted teacher feedback, facilitated by the use of this instructional strategy, proved to be invaluable for my own professional growth. It pushed me to become a more interactive educator, prompting me to formulate and pose more thought-provoking questions designed to stimulate deeper cognitive processing in my students. This shift from simply telling students how to solve a problem to guiding them through the process of understanding marked a significant evolution in my teaching practice. 

Figure 2 

Data from Video Recording of Lesson During Intervention 

Table 2 

Data from Video Recording of Lesson # 2 Indicating Schema-based Instruction Steps  

Column 1	Yes	No
Did I go over FOPS?		NO
Did I use Manipulatives?	YES
Did I use Drawings?		NO
Did I go over F Figure Out		NO
Did I go over To Organize Data		NO
Did I go over P Planning the Attack		NO
Did I go over S Solve the problem?		NO
Did we read the problem at least 3 times?		NO

In the second video, we delved into the realm of scale drawings and dimensions, which were predominantly presented as word problems. This focus on word problems aligned with my interest in working with them, although at that juncture, I had not yet determined the specific approach I would take. I did start drawing due to the topic, however I did not do this because of schema.   

We tried something new called 3 reads which is something my students do in ELA class. I wanted to focus on reading the problem and actually figuring out what the problem is asking.  

In a subsequent teaching experience, I discovered that I was intuitively drawn to a scheme-based instructional approach without deliberately identifying it as such. This realization occurred during our unit on percentages, a topic we emphasized due to its practical relevance and real-world applications. To effectively teach this concept, I employed explicit instruction, which involved a clear and detailed explanation of simple interest. Recognizing the potential challenges students might face with word problems, I provided them with a structured problem-solving approach. This approach included breaking down the problem into manageable steps and guiding students through each step, ensuring a thorough understanding of the process. 

The primary distinction between my initial two videos and my subsequent observations lies in my focus. Observing my teaching practices and gaining deeper insights into my students’ learning styles have guided me towards my action research objective. I was particularly struck by my students’ improved comprehension of word problems when they engaged in repeated readings, utilized drawings, and employed manipulatives to illustrate the problems more effectively. This realization highlighted the significance of incorporating diverse learning strategies to cater to the varied needs of my students. 

Furthermore, my evolving understanding of my students’ learning processes has prompted me to reflect on my instructional methods and adapt them accordingly. By recognizing the diverse ways in which my students learn, I can create a more inclusive and supportive learning environment that fosters their academic growth and development. 

Figure 3 

Video 3 Instruction on Direct Instruction on FOPS and Drawing and Modeling 

Table 3 

 Video 3 Schema-based Instruction Steps

Video 3	Yes	No
Did I go over FOPS?	YES
Did I use Manipulatives?	YES
Did I use Drawings?	YES
Did I go over F Figure Out	YES
Did I go over To Organize Data	YES
Did I go over P Planning the Attack	YES
Did I go over S Solve the problem?	YES
Did we read the problem at least 3 times?	YES
Did they read the problem?	YES

Video number 3 was recorded while my class was on NTI (Non-Traditional Instruction) due to winter weather. Unfortunately, the first video recording did not capture what I had intended, which highlights one of the limitations of this research. I followed up by recording another video where I worked with a small group. In this session, I explicitly reviewed the concept of FOPS (Figure out the Problem, Organize data, Plan your attack and solve) I also, incorporating drawing and manipulatives into the lesson to enhance understanding. 

Teaching from the back table, I projected my Chromebook onto the Smart Board. We began with a review activity using a handout I created based on the “FOPS” acronym. Discussing each letter reinforced concepts from the previous lesson. Following this, we tackled several word problems together, utilizing both drawings and manipulatives. (Observation Log #7) 

We dedicated significant time to working through a variety of word problems, specifically chosen to align with our state’s educational standards. This approach was taken to ensure that students are well-prepared for the diverse range of mathematical challenges they’ll encounter, as word problems truly permeate every facet of mathematics. 

 We explicitly went over what the FOPS acronym means. We wrote FOPS on the board. Each time we went through a word problem, I had them write FOPS on the side of the paper, and as we went through each step, I had them cross out the letter that we went through. Time, we started a set of word problems, we went over FOPS and then started working through the word problems as with anything that is new you need to review it several times and with special ed students you probably need to review it even more.   

 Effective learning requires repetition, as “the brain needs to hear an idea at least three times” (Ryshke, 2011, para. 5). 

As we tackled these problems, we made a concerted effort to explicitly address common errors, taking the time to reinforce fundamental math concepts. I leveraged technology, using my computer and projector to visually display the drawings and diagrams as I modeled them for the class. To further support their learning, each student was provided with a set of manipulatives to use alongside the visual instruction. 

When we explored geometry, I incorporated 3D shapes that could be opened and examined, facilitating a deeper understanding of both volume and surface area. We also utilized an array of other tools, including counters, rulers, and decimal circles, to provide concrete, hands-on learning experiences. 

Recognizing that some students needed more than just abstract representations, we went beyond drawings and incorporated manipulatives into our lessons. This approach aligns with a core principle of human-based instruction: presenting schemas visually is crucial for comprehension. By combining visual and tactile elements, we created a multi-faceted learning experience designed to cater to a variety of learning styles and needs. 

Figure #4 

 Video 4 Instruction on Direct Instruction on FOPS and Drawing and Modeling 

Table 4 

Video 4 Schema-based Instruction Steps 

Video  4	Yes	No
Did I go over FOPS?	YES
Did I use Manipulatives?	YES
Did I use Drawings?	YES
Did I go over F Figure Out	YES
Did I go over To Organize Data	YES
Did I go over P Planning the Attack	YES
Did I go over S Solve the problem?	YES
Did we read the problem at least 3 times?	YES
Did they read the problem?	YES

The fourth video involved revisiting the three-read method with FOPS, specifically focusing on word problems. We continued to emphasize the use of drawings and manipulatives to aid in understanding and problem-solving. After working through five-word problems together, the students were given a separate worksheet to practice independently. 

I had been providing weekly “probes,” which were assessments comprised of various word problems aligned with their IEP goals and current content. Following the review with FOPS, students worked independently on a new set of word problems, utilizing both drawings and the manipulatives available on their desks. Additional manipulatives were available on a back table, but the students seemed to find the ones on their desks sufficient, as these were the materials they had been consistently using throughout the lessons. 

The class began with the setup of technology. I turned on my Chromebook and connected the doc cam. Meanwhile, A powered up the smart board and ensured the Google Meet was operational before starting the Google Classroom and then the Google Meet on the smart board, muting the volume. (Observation Log #8) 

Technology plays a crucial role in enhancing the learning experience within the classroom. Tools such as Chromebook, Google Meet, and the doc cam have become indispensable in creating a more interactive and engaging environment. For instance, Google Meet and the doc cam enable me to share my screen with students, allowing them to simultaneously view the drawings and annotations I make in relation to word problems. This real-time visualization helps students grasp the concepts being taught and fosters a deeper understanding of the problem-solving process. Additionally, I can effectively demonstrate the use of manipulatives and drawings through the doc cam, providing students with a visual model that they can emulate. By incorporating technology into the classroom, I can create a more dynamic and collaborative learning experience that caters to the diverse needs of my students. 

Technology is integral to improving classroom learning experiences. Tools like Chromebooks, Google Meet, and the doc cam have become essential for fostering a more interactive and engaging learning environment. For example, Google Meet and the doc cam allow me to share my screen with students, enabling them to see the drawings and annotations I make in relation to word problems in real-time. This live visualization not only aids students’ comprehension of the concepts being taught but also encourages a deeper understanding of the problem-solving process. 

Furthermore, I can effectively demonstrate the use of manipulatives and drawings through the doc cam, providing students with a visual model that they can reference and apply in their own work. By incorporating technology into the classroom, I can create a more dynamic and collaborative learning experience that caters to the diverse needs of my students and helps to create a more inclusive learning environment. 

Technology also allows for differentiated instruction, as students can work at their own pace and receive individualized support. Educational software and online resources can provide students with additional practice and reinforcement, while also allowing them to explore topics of interest in greater depth. Additionally, technology can facilitate communication and collaboration among students, both within and outside of the classroom. Students can use online tools to work together on projects, share ideas, and provide feedback to one another. 

Overall, technology has the potential to transform the classroom into a more active and student-centered learning environment. By using technology effectively, teachers can create a more engaging and personalized learning experience that meets the needs of all students. As technology continues to evolve, it is important for educators to stay informed about new developments and to find ways to integrate them into their teaching practices in a meaningful and effective way. 

Regarding instruction and student adaptation to this new concept, it’s been difficult, especially at first.  

Moving to FOPES has presented challenges. One difficulty is how many times students should read the problem. With CUBES, they’d often circle numbers without reading and just underline the last sentence. This is a problem because they don’t know if the numbers are important or if the last sentence is the actual question. So, we’ve gone from not reading at all, to reading at least twice (in this lesson, we read one problem four times!). This is very different for them. Another struggle is using manipulatives. I’ve had to be creative in finding them, but they’re really important. On the practice page, we used a spinner and the “Wheel of Names” app. There are many other math manipulatives out there that I need to find. I’ve been told I can request more, so I’m hoping to find more to use. This research for my master’s degree will help me later. I’ve noticed an improved student understanding when I use manipulatives, even simple ones like dice and colored cubes. It helps them be more hands-on and really see what the problem is asking. (Observation Log #9) 

Explicitly implementing schema-based instruction involved making abstract mathematical concepts more tangible for students. Using manipulatives and drawings allowed students to physically represent problems, aiding their understanding of problem structures essential for schema application. A key component of this implementation was emphasizing the importance of carefully reading the word problem. Students were taught strategies like reading the problem three times or more, a practice they articulated and adopted. To reinforce this, a connection was drawn to the practice of rereading in English Language Arts. Students who embraced the schema-based approach demonstrated a deeper understanding, recognizing that merely circling numbers was insufficient and instead focusing on grasping the problem’s essence through these explicit strategies. This focus on structured representation and careful problem analysis formed the core of the schema-based instruction. 

Table 5 

Pre-Test and Post-Test

Applying Schema-Based Strategy 

Before the targeted schema-based instruction and intervention, students demonstrated significant challenges in mastering math word problems, particularly in the process of Schema Application. There was a general difficulty in activating and applying mental frameworks to recognize patterns, understand structures, and select appropriate solution pathways. Educators faced challenges accurately gauging students’ prior knowledge, especially for middle school students who often had significant learning gaps, potentially missing foundational knowledge from earlier grades. For special education students with accommodations and modified coursework, traditional assessments frequently did not accurately reflect their true understanding. Students often struggled specifically with converting between fractions, decimals, and percentages, noting particular confusion with the correct direction for the decimal shift. Many students, especially general education students, admitted to relying on less effective methods like simply circling numbers in a problem to solve it. For many students, manipulatives were essential for their understanding, indicating a need for concrete representations prior to the intervention’s focused use of them. 

During the implementation of the schema-based strategy instruction, students began to respond to the targeted approaches and explicit teaching. As familiar contexts were leveraged in word problems, students showed increased accessibility and connection of mathematical concepts to real-world situations. In units like probability, students reportedly “grasped easily due to our geographical location” when concepts were tied to weather forecasts, showing engagement with real-world examples. Specific strategies were introduced and practiced, and students demonstrated adoption and understanding.  

The teacher’s script, as recorded in Observation Log #9, clearly illustrates this strategy: “Everyone, raise your right hand. Think of it this way: we say the Pledge of Allegiance every morning, right? Stand up and place your right hand on your heart like you’re saying the Pledge. See? The hand you use for the Pledge is your right hand. (The other one is your left.) Everyone here is right-handed; you can also think, ‘I hold my pen with my right hand.’ Don’t get discouraged; this is something a lot of people struggle with. I often struggle with it myself, and I often think of my writing hand to help me.” This personalized and relatable approach aimed to reduce anxiety and provide a concrete reference point for a potentially confusing concept.  (Observation #9) 

Similarly, when introducing the principles of probability, the teacher utilized a short clip from The Weather Channel, a familiar source of information for many students. This real-world example served as a springboard for engaging students in active discussion and critical thinking, as evidenced by the teacher’s prompting questions recorded in Observation Log #9 “Okay, so when you look at the Weather Channel, it usually gives you the day’s weather and the percent chance of precipitation. What’s precipitation?… Look, right now it says a 100% chance of rain. What does that mean?” These examples highlight the intentional efforts made to connect abstract mathematical concepts to students’ existing knowledge and everyday experiences, fostering deeper understanding and engagement. (Observation 9) 

Students actively adopted the taught strategies, like reading problems multiple times; when asked how many times a problem should be read, they would confidently reply, “Three times or more.” They embraced the FOPS strategy, recognizing its value over just circling numbers, with students countering that the old method “wouldn’t lead to understanding the problem’s essence.” Students demonstrated their understanding by guiding peers, engaging in the FOPS process, articulating problem-solving methods, and coming to the board to show illustrations of their solutions. The use of drawings and manipulatives was effective, with students physically representing problems, making abstract concepts more tangible and proving essential for many students’ understanding, showing their responsiveness to concrete support. 

Following the period of explicit instruction and practice, students were tasked with applying schema-based strategies and utilizing their learned tools more independently. The data collected through pre- and post-test analysis provides clear evidence of their ability to do so successfully. A consistent and statistically significant increase was observed in every student’s scores from the initial pre-test to the final post-test. This significant improvement is directly correlated with the implementation of the practiced skills within the context of repeated application opportunities, including assessment probes. Over the duration of the study, students actively engaged in practicing the target schema-based skills and subsequently applied these strategies and utilized tools like FOPS and visual aids to solve problems on their own, a clear and measurable pattern of score escalation emerged. This demonstrates that consistent practice and the opportunity for independent application of learned schema-based skills and tools were key factors in fostering student growth and mastery of word problem solving. The NWEA MAP data comparing fall and spring scores further supports this, indicating overall learning and growth over the school year where the intervention occurred. While the ultimate goal is transfer beyond school, the test results confirm students successfully applied the intervention’s strategies when assessed. 

Transfer of Skill from Resource Room to Inclusive Classroom 

The deliberate implementation of schema-based strategy instruction, incorporating the FOPS Application (a structured thought process) and the strategic use of visual aids alongside drawings, marked a significant pedagogical shift within the resource classroom setting. This dedicated instruction occurred for 40-minute blocks, three times per week, providing focused opportunities for students to internalize these strategies. Furthermore, to foster wider application and integration, these schema-based techniques were woven into daily co-teaching sessions, taking place twice daily for 40 minutes each. This dual approach aimed to provide both explicit, targeted instruction and consistent reinforcement within the inclusive general education environment. 

A central and overarching goal of this instructional approach was to demonstrably improve students’ comprehension and problem-solving abilities when faced with mathematical word problems. The schema-based framework was specifically chosen for its potential to move beyond rote memorization and cultivate a deeper, structural understanding of problem types. Crucially, a parallel objective was to actively facilitate the transfer of this learned instruction from the specialized resource classroom into the mainstream general education classroom. This emphasis on transfer recognized that true mastery lies in the ability to apply knowledge and skills across diverse contexts. A fundamental aim underpinning this initiative was the standardization of this powerful strategy within the co-teaching setting. The ultimate vision was to empower special education students to such an extent that they could confidently and effectively instruct their general education peers in the application of these schema-based problem-solving methods. This peer-to-peer instruction not only benefits the general education students but also serves to solidify the understanding and metacognitive skills of the special education students acting as instructors. 

The primary objective of introducing schema-based instruction within the resource setting and its subsequent integration into collaborative co-teaching sessions was firmly rooted in the desire to seamlessly facilitate the transfer of these newly acquired cognitive strategies to the less-supported environment of the general education classroom. A core aim was to cultivate a learning environment where special education students became active participants in sharing their learning, capable of instructing their general education peers in the practical application of these effective problem-solving methods. This proactive approach was designed to foster a sense of shared learning and to standardize the application of these strategies across the entire inclusive educational landscape. 

During the course of implementation, an encouraging phenomenon emerged: the transfer of these strategies began to manifest organically, without explicit prompting. As instruction progressed and students gained familiarity with the schema-based approaches – including the structured FOPS Application and the use of visual aids to represent problem structures – they naturally began to take on the role of guides for their classmates. For instance, a seemingly simple yet telling example occurred when students were questioned about the optimal number of times a word problem should be read to ensure full comprehension. Those students who had internalized the schema strategies confidently responded, often in unison, with “Three times or more,” clearly articulating a key strategy they had not only learned but had also made their own. Their engagement went beyond recitation; they would actively assist their peers in navigating the steps of the FOPS process, patiently explaining the purpose of each stage and demonstrating their own problem-solving approaches through clear illustrations on the whiteboard or on their individual papers. This act of teaching and demonstrating the strategies to others provided compelling and powerful evidence of their comprehensive and nuanced understanding of the underlying concepts. Furthermore, this peer-led instruction significantly facilitated the generalization of the schema-based approach throughout the co-taught classroom, creating a more collaborative and supportive learning atmosphere 

Central to achieving mastery in navigating and solving mathematical word problems is the fundamental cognitive process of Schema Application. This process inherently involves building upon the foundation of students’ existing prior knowledge and their accumulated problem-solving experiences. Learners actively activate and strategically apply schemas – these are the organized mental frameworks that represent their understanding of the world – to efficiently recognize recurring patterns within problems, understand the underlying structures of different problem types, and ultimately select the most appropriate and effective solution pathways. This is not a static process; rather, it is a cumulative one, where each successfully solved problem serves to further strengthen and refine the relevant schemas. Facilitating this crucial process effectively involves strategically leveraging familiar contexts that students can readily relate to. By grounding abstract mathematical concepts in relatable scenarios, educators can make problems feel less abstract and more accessible, thereby significantly aiding students in the application of relevant pre-existing schemas. This deliberate connection of mathematical principles to real-world situations that students recognize and understand is a key element in fostering genuine and lasting comprehension. 

Recognizing the inherent diversity in students’ learning needs, particularly within an inclusive classroom, specific and focused attention must be directed towards students with special education designations. Their journey of schema development and application often necessitates the implementation of differentiated instruction strategies. This may include the explicit and direct teaching of underlying problem structures, the provision of visual scaffolding to support understanding, the use of graphic organizers to help structure their thinking, and the provision of additional individualized guided support tailored to their specific learning profiles. Ensuring equitable access to these powerful problem-solving strategies requires a flexible and adaptive approach to instruction. Educators frequently encounter challenges in accurately gauging students’ prior knowledge, especially in the middle school setting where students, particularly those with special education needs and those following modified coursework, may present significant gaps in their foundational understanding. Traditional assessment methods alone may not always provide a complete or accurate reflection of these students’ true understanding and abilities. Therefore, the utilization of alternative assessment methods, such as comparing fall and spring administrations of standardized assessments like the NWEA MAP Growth Math scores, can provide valuable insights for informing instructional decisions and effectively targeting individualized interventions to address specific learning needs. 

The systematic implementation of schema-based strategy instruction, encompassing the structured FOPS Application (as a guide for their internal thought process) and the purposeful integration of visual aids such as drawings to represent problem elements, was deliberately initiated within the dedicated resource classroom environment. This focused instruction was delivered in 40-minute sessions, conducted three times per week, allowing for in-depth exploration and practice of these techniques. Moreover, to ensure broader application and reinforcement, these strategies were also actively integrated into the daily co-teaching sessions that occurred within the general education classroom, taking place twice each day for 40-minute durations. During other instructional periods throughout the school day, students actively participated in the co-teach inclusive classroom, providing further opportunities to apply and refine their understanding within a less specialized setting. 

The overarching objective driving this research-informed instructional approach is to significantly enhance all students’ comprehension of mathematical word problems through the consistent and explicit application of schema-based instruction. Furthermore, a critical component of this objective is to thoroughly examine the extent to which this instruction effectively transfers from the more supportive resource setting to the less structured environment of the general education classroom. The primary and key aim of this endeavor is to establish and standardize the use of this powerful strategy within the co-teach setting. The ultimate goal is to empower special education students to develop sufficient mastery that they can confidently and effectively instruct their general education peers in the correct and strategic application of these schema-based problem-solving techniques, fostering a more collaborative and supportive learning environment for all. 

I completed a similar problem with them in the supplemental class using math manipulatives.  

During a shopping example, manipulatives and drawings were used: “You bought a magazine for $5 and 8 erasers,” (read 3 times). The problem was then modeled with physical objects, asking, “Okay, great. Thank you. So, what I’m going to do is take out eight of those erasers that you guys borrow a lot and put them right here. And I don’t have a magazine, but I do have a book, so let’s just assume I bought this book for $5. And I’m going to put that here. Okay, so now I have my book and my eight erasers. Now, who would like to read this word problem again for the second time?” (Observation Log #11). 

During a recent math lesson, one of my students took the lead in demonstrating how to solve a word problem on the whiteboard. To effectively illustrate his process, he utilized the document camera, projecting his handwritten work for the class to follow. 

 The problem stated, “You bought a magazine for $5 and eight erasers.” At a certain point in his explanation, he paused and posed a question to the entire class. “Class,” he inquired, “how many times have we met?” He patiently waited for their responses. It was the students who attended his supplemental instruction class who confidently answered, “Three times.” Acknowledging their correct response, he affirmed, “Good. Now, we’re going to write ‘FOPS’ on the side of the paper.” Following this, he meticulously guided the class through each subsequent step of solving the word problem. A similar work sample from this student is found in Appendix D.  

The effectiveness of drawings and manipulatives in enhancing schema-based instruction was observed, proving valuable for visual and tactile representation, especially for students for whom manipulatives are essential for understanding abstract concepts. Adapting strategies through scaffolding and using manipulatives as key support created a more inclusive environment. 

The concepts of transfer and generalization of knowledge represent fundamental and critical components of a holistic and effective educational experience. It is imperative that students develop the ability to actively connect newly acquired information with their pre-existing base of knowledge and, crucially, to flexibly apply this integrated understanding across a diverse range of contexts and situations. This ability to transfer and generalize transcends the limitations of rote memorization, which often results in brittle and context-dependent knowledge. A persistent challenge faced by educators is the common tendency among students to compartmentalize their understanding, leading to a failure to recognize and apply skills and knowledge learned in one specific context to seemingly related situations encountered elsewhere. Therefore, it becomes imperative to explicitly and intentionally instruct for transfer. This involves providing students with numerous and varied opportunities to apply their learning in diverse settings, actively fostering metacognitive reflection on their learning processes, and explicitly connecting seemingly disparate concepts to build a more cohesive and interconnected understanding of the subject matter. 

The development of robust schemas is not merely a superficial shortcut to problem-solving; rather, it represents the cultivation of a powerful cognitive tool that elevates students’ thinking to significantly higher levels of abstraction and complexity. While rote learning and the memorization of specific procedures certainly have their place in foundational skill development, their utility is inherently limited when faced with novel or complex problems. In contrast, the cultivation of higher-order cognitive skills, such as the ability to recognize underlying problem schemas, develop deep conceptual understanding, engage in effective problem modeling, and implement flexible problem-solving strategies, proves to be far more transferable and ultimately more valuable in the long run. These higher-order skills empower students to extrapolate beyond specific examples, adapt their knowledge and strategies to navigate novel and unfamiliar situations, and approach problems with a flexible and creative mindset, ultimately transforming knowledge from a collection of isolated facts into an enduring and adaptable asset that serves them well beyond the confines of the classroom. 

During the active implementation phase of this schema-based instruction, students not only readily adopted the taught strategies but also proactively began to facilitate the process of knowledge transfer, naturally stepping into the role of guides and mentors for their peers. Initially, as instruction progressed within the classroom, these students would spontaneously offer assistance and help lead class discussions related to problem-solving approaches. For a concrete example, when the question arose as to how many times a complex word problem should be thoughtfully read before attempting a solution, the students who had internalized the schema strategies would confidently reply, often in unison, with a clear directive: “At least three times.” More specifically, they would actively engage in assisting and guiding other students through each of the structured steps of the FOPS process as the class worked collaboratively on various problems. They would patiently remind their peers what each letter in the acronym stood for, help them systematically Find the necessary information embedded within the problem statement, guide them in how to effectively Organize this information in a meaningful way, collaborate on Planning the logical sequence of steps required to arrive at a solution, and provide. 

Conclusion 

 Schema-Based Instruction (SBI), exemplified by the FOPS approach, presents a robust and effective pedagogical framework with the significant potential to deepen students’ mathematical understanding and cultivate higher-level problem-solving skills, especially for those who experience difficulties in mathematics. This instructional method moves beyond the limitations of superficial strategies that rely on rote memorization of keywords or isolated “tricks,” which often fail to foster true comprehension. Instead, SBI directly confronts the fundamental challenge many struggling math students encounter: the inability to recognize and apply underlying mathematical structures or schemas as a means to approach and solve problems effectively and meaningfully. 

The research findings strongly suggest that explicit and systematic instruction in identifying various problem schemas, such as combine, compare, and change problems, is a cornerstone of effective SBI. This explicit teaching should be thoughtfully coupled with the use of visual representations, such as diagrams, charts, and manipulatives, to make abstract mathematical structures more concrete and accessible to learners. Furthermore, ample opportunities for students to practice applying these identified schemas across a diverse range of problem contexts are absolutely crucial for solidifying their understanding. This carefully structured and iterative approach empowers students to build progressively upon their prior problem-solving experiences, actively connecting new mathematical concepts and problem types to their existing knowledge base. This process of connecting new learning to prior understanding is not only essential for the successful application of schemas but is also critically important for the broader transfer and generalization of learned skills to entirely novel problem-solving situations that they may encounter in the future. To further fortify this instructional framework, it is vital to address foundational mathematical basics proactively and to explicitly teach and clarify common misconceptions that students may hold, thereby creating a more stable and accurate foundation for schema acquisition and application. 

A central and compelling theme that emerges from this body of work is the stark contrast between the transient and often misleading reliance on memorized “tricks” and the development of durable, higher-level thinking capacities that are intentionally fostered by the implementation of SBI and the FOPS approach. While tricks might offer the illusion of quick solutions to specific problem types, their utility is often limited to those exact scenarios, and they frequently lead to significant errors when students encounter variations or more complex problems. More concerningly, a reliance on tricks typically hinders the development of genuine conceptual understanding, leaving students without a flexible or adaptable framework for approaching mathematical challenges. Conversely, SBI is specifically designed to cultivate durable and transferable skills that go far beyond rote procedures. These skills include the ability to effectively model mathematical problems using appropriate schemas, engage in strategic planning before attempting a solution, and engage in critical reflection on the problem-solving process after a solution has been reached. Evidence meticulously gathered through the detailed analysis of student work samples and observations of their problem-solving approaches consistently indicates a noticeable and positive shift away from the use of ineffective, trick-based strategies towards the adoption of more robust, schema-driven methods when SBI is implemented thoughtfully and effectively in the classroom. 

The successful implementation of SBI and the FOPS methodology necessitates a careful and nuanced consideration of the diverse characteristics of the student population, with particular attention to the needs of diverse learners and those students who require specialized education supports. Adaptations to instruction, the strategic use of differentiation techniques to meet individual learning needs, the provision of appropriate instructional pacing that allows sufficient time for understanding, and the incorporation of effective scaffolding strategies are all absolutely necessary during both the initial phases of explicit schema teaching and the subsequent phases of independent schema application. These considerations ensure that all students, regardless of their prior knowledge or learning styles, can meaningfully access and derive benefit from the instruction. Furthermore, paying close attention to the affective domain of learning reveals that building a strong sense of competence in mathematics through the development of robust schema understanding can have a profound and positive impact on students’ overall confidence in their mathematical abilities and their intrinsic motivation to engage with and persevere through challenging mathematical tasks. This boost in confidence and motivation can create a positive feedback loop, further enhancing their learning and achievement in mathematics. 

In conclusion, Schema-Based Instruction, particularly as it is operationalized through structured and explicit methods like the FOPS approach, stands out as a particularly powerful and promising pedagogical approach in mathematics education. Its benefits extend beyond merely improving students’ accuracy in solving mathematical problems; it fundamentally enhances their underlying mathematical reasoning abilities. By explicitly focusing on the development and application of problem schemas, SBI equips students with the crucial transferable, higher-level thinking skills that are not only essential for their ongoing academic success in mathematics but also for their ability to effectively apply mathematical thinking and problem-solving skills to real-world situations they will encounter throughout their lives. The compelling evidence generated by this research strongly supports prioritizing the widespread adoption and effective implementation of SBI as a primary means of fostering genuine and lasting mathematical proficiency in students, moving away from an over-reliance on temporary procedural shortcuts that often fail to build a solid foundation of conceptual understanding. 

Table 7 

 Pre and Post-Assessments 

 The data derived from the pre- and post-assessments indicates a marked enhancement in student achievement following the implementation of the pedagogical intervention. A significant proportion of students exhibited elevated scores on the post-assessment relative to the pre-assessment. This advancement is substantiated by the increased mean score of 57% on the post-assessment, as opposed to the considerably lower mean score of 12% on the pre-assessment. The post-assessment score range was 100, with a median of 53%, thereby underscoring the variability in student performance. 

 The data derived from the pre- and post-assessments indicates a marked enhancement in student achievement following the implementation of the pedagogical intervention. A significant proportion of students exhibited elevated scores on the post-assessment relative to the pre-assessment, demonstrating improvement across the group. This advancement is substantiated by the increased mean score of 57% on the post-assessment, as opposed to the considerably lower mean score of 12% on the pre-assessment. While all students showed progress, the degree of improvement varied, as underscored by the post-assessment score range of 100 and a median of 53%. For some students, the magnitude of improvement was not as substantial as ideally desired, suggesting varying needs or factors influencing the extent of their progress. 

Nonetheless, the aggregate findings unequivocally suggest that the instructional strategy utilized was efficacious in augmenting overall student learning outcomes. The substantial augmentation in mean and median scores offers compelling substantiation for the intervention’s success in promoting student growth. Subsequent analysis could explore the factors contributing to the differing levels of improvement among students and examine avenues to further adapt instruction to accommodate the heterogeneous needs of all, thereby striving to maximize every student’s opportunity to achieve significant success. 

Reflections 

  This action research investigated the efficacy of Schema-Based Instruction (SBI), specifically the FOPS strategy (Find the Problem type, Organize information, Plan the solution, Solve the problem), in enhancing students’ comprehension and problem-solving proficiencies with respect to mathematical word problems. 

The central finding of this investigation was the demonstrably positive effect of explicit schema identification and application instruction through the FOPS strategy. Pre- and post-assessment data corroborated this finding, revealing a considerable increase in the mean score from 12% to 57%. 

Although a generally positive trend was observed, individual student progress varied. The majority of students exhibited improved mathematical word problem-solving abilities. Some students displayed substantial progress, with score increases reaching 100%, while others manifested more modest gains. Conversely, a minority of students experienced a marginal decline (up to 20%), underscoring the heterogeneity in individual responses to the intervention despite the aggregate positive outcome. This variability in student advancement emphasized the necessity of continuous formative assessment and adaptable grouping within instructional practices. Initially, SBI was implemented through a standardized approach; however, the disparate rates of learning prompted the incorporation of differentiated support and opportunities for collaborative, self-paced learning. This pedagogical adaptation, moving toward a more responsive and personalized implementation of SBI, facilitated a more effective accommodation of the diverse learning requirements within the classroom. 

Limitations or Drawbacks 

  The implementation of a novel instructional strategy invariably faces initial obstacles and resistance. This was indeed observed in my educational institution, where the pre-existing pedagogical method was firmly established, rendering it challenging for students to acclimate to the transition to schema-based instruction. The shift necessitated a period of adaptation, yet over time, the students progressively embraced the innovative methodology, which encompassed learning to discern the foundational schema, including the diagrammatic representation of its components, to formulate and resolve mathematical problems. 

A further impediment was the restricted timeframe available for direct student engagement on schema-based instruction. The institution’s continued adherence to the conventional approach mandated that students receive supplementary instruction in the innovative method separately. Within the co-teaching classroom environment, adherence to the established institutional methodology was obligatory. Nevertheless, as progress was made in schema-based instruction, students increasingly integrated its constituent aspects, particularly the utilization of schemata (diagrammatic components), into classroom activities, albeit within the confines of the institution’s established framework. 

Regrettably, several setbacks were encountered due to adverse weather conditions. Inclement weather events, including snowfall, severe storms, and flooding, precipitated the implementation of Non-Traditional Instruction (NTI) days, wherein the institution transitioned to virtual learning modalities. Consequently, the opportunity to conduct small group instructional sessions on schema-based instruction, focusing on the identification and diagrammatic representation of schemata, was precluded during these NTI days. This interruption in the regular instructional schedule impinged upon the capacity to consistently practice and reinforce schema-based instructional skills. 

Summary of My Journey 

    Initially, my research proposal centered on co-teaching; however, I subsequently revised this focus. The primary reason for this alteration was the recognized complexity inherent in researching co-teaching practices. The interdependence of two instructors within this pedagogical framework presents challenges to maintaining consistency, thereby impeding the identification and analysis of replicable patterns or trends. 

Observations during the academic year underscored this complexity. The observed co-teaching arrangement was characterized by a marked degree of variability, thereby demonstrating the difficulty in conducting research within contexts where pedagogical practices are highly variable. 

The preliminary phase of my research, inclusive of initial video recordings and attendant observations, diverged from the intended examination of schema-based instruction. This deviation resulted from engagements in Professional Learning Community (PLC) meetings with mathematics educators of seventh and eighth grades, wherein a pertinent research focus was sought. During these deliberations, it was determined that the majority, specifically seventy-five percent, of assessments utilized for student placement, namely state-administered examinations and the NWEA MAP test, consisted of word problems. 

Subsequent inquiry corroborated the prevalence of word problems within the mathematics curriculum. These problem types were identified as integral components of each mathematical unit. This insight prompted a revision of the research design, resulting in a concentration on effective strategies for instruction and learning related to word problems. 

Dissemination of research findings to the Professional Learning Community (PLC) and departmental leadership is intended. The pedagogical strategy devised demonstrates greater efficacy when contrasted with the current district-adopted methodology. Nevertheless, further research is necessary to evaluate the strategy with students who do not present with mathematical deficiencies. This would enhance the generalizability of the research findings and provide corroborative data regarding the utility of the strategy for a broader cohort of students. 

References 

Chen, J., Wang, J., Liu, S., & Zhang, Y. (2024). Figuring figures: An assessment of large language models on different modalities of math word problems. In Proceedings of the 2024 9th International Conference on Machine Learning Technologies (ICMLT 2024) (pp. 1-10). Association for Computing Machinery. https://doi.org/10.1145/3674029.3674041 

Dobson, K., Fernandez, A., & Hofmann, S. G. (2021). Theoretical framework. In A. Wenzel (Ed.), Handbook of cognitive behavioral therapy: Overview and approaches (pp. 31–50). American Psychological Association. https://doi.org/10.1037/0000218-002 

Dwivedi, U., Rajput, N., Dey, P., & Varkey, B. (2017). VisualMath: An automated visualization system for understanding math word-problems. In Proceedings of the Companion on the 22nd International Conference on Intelligent User Interfaces (IUI ’17 Companion) (pp. 105-108). Association for Computing Machinery. https://doi.org/10.1145/3030024.3040989 

Elliott, E. C. (2023). Why Word Problems are Hard for High School Math Students: Problem Formulation and Disciplinary Literacy (Senior Theses, 586). [Note: University information is missing from the provided source]. 

Gattuso, J. (2015). Problem solving strategies: Helping students develop a conceptual understanding of word problems (Student Research Submissions, 182). [Note: University information is missing from the provided source]. 

Hott, B. L., Peltier, C., Heiniger, S., Palacios, M., Le, M. T., & Chen, M. (2021). Using schema-based instruction to improve the mathematical problem-solving skills of a rural student with EBD. Learning Disabilities: A Contemporary Journal, 19(2), 127–142. [Note: DOI was mentioned as added but not provided in the input]. 

Khoshaim, H. B. (2020). Mathematics teaching using word-problems: Is it a phobia! International Journal of Instruction, 13(1), 855-868. https://doi.org/10.29333/iji.2020.13155 

Math Coach’s Corner. (2022, April 4). The Three-Reads Protocol. https://www.mathcoachscorner.com/2022/04/the-3-reads-protocol-for-solving-word-problems/ 

McCoy, G. D. (2023). The Effect of the Three Reads Strategy on the Story Problem Solving Skills of Middle School Students Who are Deaf/Hard of Hearing (Publication No. 30310843) [Doctoral dissertation, Minot State University]. ProQuest Dissertations & Theses Global. 

Nandhini, K., & Balasundaram, S. R. (2011). Math word question generation for training students with learning difficulties. In Proceedings of the International Conference & Workshop on Emerging Trends in. [Note: Conference name and publication details are incomplete in the provided source]. 

Pongsakdi, N., Kajamies, A., Veermans, K., Lertola, K., Vauras, M., & Lehtinen, E. (2019). What makes mathematical word problem solving challenging? Exploring the roles of word problem characteristics, text comprehension, and arithmetic skills. Frontiers in Psychology, 10, Article 2565. https://doi.org/10.3389/fpsyg.2019.02565 

Powell, S. R. (2011). Solving word problems using schemas: A review of the literature. Learning Disabilities Research & Practice, 26(2), 94-108. [Note: DOI not provided in the input]. 

Powell, S. R., & Fuchs, L. S. (2018). Effective word-problem instruction: Using schemas to facilitate mathematical reasoning. Teaching Exceptional Children, 51(1), 31-42. https://doi.org/10.1177/00400599187772 

Vanderbilt University. (2024). High-Quality Mathematics Instruction: What Teachers Should Know [IRIS Module]. IRIS Center. https://iris.peabody.vanderbilt.edu/module/math/cresource/q2/p06/ 

Wakhata, R., Mutarutinya, V., & Balimuttajjo, S. (2019). Secondary school students’ attitude towards mathematics word problems. International Journal of Educational Research, 16, 1-11. https://doi.org/10.17762/ijer.v16i0.310 

Wakhata, R., Balimuttajjo, S., & Mutarutinya, V. (2023). Relationship between students’ attitude towards, and performance in mathematics word problems. Frontiers in Psychology, 14, Article 1290126. https://doi.org/10.3389/fpsyg.2023.1290126 

Appendices 

Appendix A Timeline for action research study  

  

Time Frame	Focus	Description of Research Component	Action Required	Data Notes
Week 1  12/2-12/6	Pre-test	Students will be given several different types of math word problems.	Administer word problems to students in various math forms	Analyze results to identify student needs and baseline data.     Create a word problem goal for each student.
Week 2-10  12/9-2/26	Strategy interventions	Students will be instructed on specific strategies.	Instruct students using schema-based instruction (FOPSFOPS) and concrete manipulatives to help them solve math word problems.	This instructional approach will provide explicit instruction on schema-based multiplicative strategies, incorporating the use of math manipulatives and diagrams to enhance student understanding and problem-solving abilities.
Week 2-10  12/9-2/26	Audio and Video Recordings	The teacher will observe and reflect on lessons and student feedback.	Record and reflect on videos.	Analyze observations to determine the effectiveness of interventions.
Week 2-10  12/9-2/26	Reflection Logs	The teacher will observe and reflect on lessons and student feedback.	Observe and reflect on lessons.	Analyze observations to determine the effectiveness of interventions.
Week 2-10  12/9-2/26	Formative/Summative  assessments	Students will demonstrate proficiency in using the strategy on various assessments such as CFA’s and CSA’s.	Administer formative and summative assessments.
Week 2-10  12/9-2/26	Student work samples	Students will demonstrate proficiency in using the strategy	Students will complete math probes using the intervention and math manipulatives.	Analyze student work samples to determine the effectiveness of the interventions.
Week 14  3/24-3/28	Post-tests	Students will be administered the post-test. Students will be administered the post-test.	Administer the post-test.
Week 15   4-14-4/18	NWEA Math Maps Test.	Students will be administered the NWEA Math Maps Test	Administer the NWEA Math Maps test	Analyze the results to determine the effectiveness of the research goal.

   

  

Appendix B Sample Student Work (Pre-Test) 

Appendix C  Student Work Sample (Post-Test) 

Appendix D   NWEA Math Maps Scores 

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