Using Fraction Manipulatives to Reduce Learning Gaps in 4th Grade Mathematics by Bethany Warr A project submitted in partial fulfillment of the requirements for the degree of MASTER OF EDUCATION with an emphasis in CURRICULUM AND INSTRUCTION WEBER STATE UNIVERSITY Ogden, Utah May 9, 2024 USING FRACTION MANIPULATIVES 1 Table of Contents Abstract ...........................................................................................................................................3 Literature Review ..........................................................................................................................5 Missed Learning.........................................................................................................................5 Distance Learning ......................................................................................................................6 Low Socioeconomic Status ........................................................................................................7 Trauma .......................................................................................................................................8 Returning to School Protocols ...................................................................................................9 Post-pandemic Missed Learning ..............................................................................................10 Mathematics .............................................................................................................................11 Mathematical Manipulatives ....................................................................................................12 Summary ..................................................................................................................................13 Purpose..........................................................................................................................................14 Methods .........................................................................................................................................15 Participants ...............................................................................................................................15 Instruments ...............................................................................................................................16 Procedures ................................................................................................................................17 Results ...........................................................................................................................................23 Discussion......................................................................................................................................24 Manipulatives and Increased Learning ....................................................................................24 Manipulatives and Lasting Learning .......................................................................................25 USING FRACTION MANIPULATIVES 2 Limitations ...............................................................................................................................26 Further Research ......................................................................................................................27 Conclusion ....................................................................................................................................27 References .....................................................................................................................................29 Appendix A: Pre-Assessment ......................................................................................................32 Appendix B: Post-Assessment.....................................................................................................39 Appendix C: IRB Approval Letter .............................................................................................45 Appendix D: Informed Consent .................................................................................................46 USING FRACTION MANIPULATIVES 3 Abstract When students have long-term absences from instruction, due to illnesses, disasters, or pandemics like COVID-19, they experience unfinished learning that impacts the knowledge they build upon throughout their education. To assist in closing these gaps and improving content retention, a study was implemented testing the effectiveness of mathematical manipulatives in equivalent fraction instruction for fourth-grade students. This study investigated whether manipulatives led to increased learning and whether their implementation led to more enduring learning. While the initial assessment results showed the control group outperforming the treatment group with an increased mean difference in scores from the pre- to post-assessments, the treatment group's mean score after a second post-assessment dropped significantly less than the control group. While both groups experienced decreased learning, this study discusses the effectiveness of fraction manipulatives supplementing learning gaps and lending to more lasting learning. USING FRACTION MANIPULATIVES 4 Using Fraction Manipulatives to Reduce Learning Gaps in 4th Grade Mathematics Every day across the world, students attend schools to learn from qualified teachers who help them grow into well-educated citizens with a desire to better the communities around them. Unfortunately, the 2019 COVID pandemic temporarily halted children’s normal school procedures, returning students to school after the pandemic with widely varying achievements in educational development (Hanushek, 2023). Missed learning due to the COVID-19 pandemic and other school absences needs to be addressed to return students to pre-pandemic performances, which already see normal unfinished learning of 25% throughout the year when students are absent from a structured learning environment for extended periods, such as vacations and summer breaks (Karbeyaz & Kurt, 2022). Since all schools closed down and switched to various alternatives of online learning, many students were unable to complete the requirements for their grade level due to low socioeconomic status, poor internet connections, insufficient resources, or lack of interest and motivation (González et al., 2022). While most people thought the changes made to education to combat COVID-19 would be temporary and children would quickly catch up the next year, the impact on students seems to be enduring (Grawe, 2023). As students progress through their education, they work toward proficiency on state standards that build upon each other each year. Since the students entering my classroom this year had significant missed learning due to their experiences during the COVID-19 pandemic, I wanted to find a way to help them better master typically difficult math concepts. Furthermore, I wanted to implement a method of learning that would provide enduring understanding of the concepts taught, so they would have a stronger foundation to build upon when they left my classroom. Because I could only control the learning that occurs within my 4th-grade walls, and USING FRACTION MANIPULATIVES 5 because fractions tend to be difficult to grasp, I decided to implement daily use of fractional manipulatives to facilitate comprehension during the instructional module. In doing this, it was my intention to provide my students opportunities to establish lasting learning of fraction concepts and lessen learning gaps. Literature Review To find a solution to help students overcome COVID-19 and other missed learning opportunities, this literature review begins with background information. Next, data specific to situations during the pandemic will be discussed, followed by available suggestions for returning to normal education, and the missed learning assessed after the pandemic. Finally, additional measures will be discussed as a solution to help students return to pre-pandemic educational achievement levels and a strategy to supplement gaps in learning. Missed Learning Missed learning, also known as unfinished learning or learning loss, is a commonly used term in education that describes diminishing skills learned by students in a school year (Donnelly & Patrinos, 2022). Donnelly and Patrinos (2022) further explained by saying, Historic data provides researchers with information regarding where student learning should be year over year and is often measured through regular testing. Learning loss occurs when educational progress does not occur at the same rate at which it has historically compared to previous years. (Abstract section, Para. 3) Moscoviz and Evans (2022) elaborated that unfinished learning can be considered in two different categories, learning that was forgotten during periods of absence and that which was never learned but should have been. It is important to remember when examining data, however, that it is not a linear progression, but constantly ebbing and flowing as students progress through USING FRACTION MANIPULATIVES 6 their educational careers, so long-term data is more accurate to analyze (Harmey & Moss, 2021). Estimates of average missed learning can be 25% (Karbeyaz & Kurt, 2022), but global studies show that declines from the break of summer vacation can reach about 65% in reading and between 37-50% in math (Maldonado & De Witte, 2022). A final consideration in the general unfinished learning that occurs as children move from one grade to another is the power of an effective teacher, especially in lower elementary grades, and the correlation to future success in life (Hanushek, 2023). Hanushek’s study (2023) showed that amongst Black students with low income, the typical growth made in reading in the 1990s ranged from a low half-a-year’s progress to a full year and a half of progress, dependent upon the teacher’s prowess. Due to normal absences from school, extended absences for holidays and breaks, and varying teacher effectiveness, missed learning in education is already a common obstacle educators must overcome each new school year with their classes. Distance Learning Once the COVID-19 pandemic became prevalent, the best way to combat the highly contagious nature of the disease was to close schools and all places that encouraged a high density of people, switching students to long-distance learning through online mediums (Molnár, & Hermann, 2023). Unfortunately, distance learning education was not uniform throughout school districts because families had internet connectivity issues, a lack of appropriate hardware to facilitate learning, and differing time, money, or knowledge to assist their students’ education (Karbeyaz & Kurt, 2022). Furthermore, students found online classes less engaging than face-toface instruction, not to mention that pre-pandemic, children were often encouraged to spend less time online to promote mental health. Additionally, Leeuw et al. (2023) pointed out that children experience learning from two main sources in their lives, school and home, and it has been USING FRACTION MANIPULATIVES 7 difficult to know if one has a greater effect over the other since children spend a significant time being influenced by both. Peers could also be added to the factors of education for children, especially older students, but COVID-19 restrictions also removed all unnecessary socialization (Zhang & Storey, 2022). Hanushek (2023) argued that the “biggest problem of education during the pandemic was depriving students of the full abilities of their most effective teachers, and recovery from the damage of these years can come only from an expanded role for these teachers" (para. 2). Low Socioeconomic Status Unfortunately for students with low socioeconomic status (SES), losing in-person education created missed learning. It increased the already large gap in comparison to higher SES students (Maldonado & De Witte, 2022). A review by Moscoviz and Evans (2022) found unfinished learning was prevalent in all the studies, with the most occurring in low SES populations and high dropout rates in secondary schools. A contributing factor to the widening gap could be linked to teachers at higher SES schools being able to teach more effectively online than their lower SES counterparts, due to a variety of factors that could be remedied with more resources readily available to the high SES schools (Molnár & Hermann, 2023). In addition to variations in teacher effectiveness, the parents of students with lower SES tend to not have adequate education to support their children's at-home learning (Molnár & Hermann, 2023). Instances of the lack of support from parents and navigating technology are shown in a study by Chetty et al. (2020) in which students with a high SES background took a temporary dip in successfully implementing an online math platform for learning. Their low SES contemporaries suffered a 50% drop below baseline levels, and did not recover to pre-COVID levels of use. A study from the Netherlands showed that students were better supported in high SES two-parent USING FRACTION MANIPULATIVES 8 homes with fewer than three siblings in comparison to students in one-parent homes with many siblings and low SES (Leeuw et al., 2023). Moreover, students in such situations before the pandemic were experiencing greater declines in math than their high SES stable family counterparts, most likely because they were not receiving much at-home help to begin with. Upon observing children in Uruguay, Gonzalez et al. (2022) found that children from low SES homes “experience overcrowding in small houses, food insecurity, unstable home life, and mental health issues, including trauma, anxiety, and depression. These children suffer more from the impact of school closures, thus increasing achievement disparities" (p. 911). Overall, students in high SES home situations had improved resources to assist them in effectively completing distance learning while low SES students did not. Trauma In addition to removing children from the close interactions of friends and the stable relationships with teachers at school, students had a variety of traumatic factors thrust upon them during the pandemic that were not necessarily addressed when schools resumed the following years (Kerr et al., 2021). There were a myriad of interactive factors contributing to hardships among young learners related to the impact of the pandemic on children's families (e.g., job loss, financial losses, remote working, illness, death, stress, mental health, and improvised parenting practices), teachers (e.g., stress, the sudden switch to online learning), social lives (e.g., loss of social contact), and access to services (e.g., daycare, delayed healthcare visits). (González et al., 2022, p. 910) Additionally, the children’s parents were experiencing great amounts of stress due to illnesses, job loss, lowered incomes, increased prices, emotional stressors, and burnout, all of which can USING FRACTION MANIPULATIVES 9 combine to create unsettling home environments and even increased child abuse and neglect (Kerr et al., 2021). The stress and uncertainty of a parent proportionately affect the stress of the child, who might lash out with bad behaviors in response, thus increasing the stress of the parent and escalating a negative home environment. In Hungary, Molnár and Hermann (2023) found that a combination of these factors resulted in both long- and short-term unfinished learning in students. Students experienced a great amount of trauma alongside their parents and teachers when the pandemic closed down schools. Returning to School Protocols Storey and Zhang (2021) conducted a literature review during the shutdown to provide a comprehensive review of the COVID-19 slide, or decrease in student scores and performance as a result of the long absences from normal school schedules due to the pandemic, as schools were about to reopen, so educators would know how best to assist students. Additional studies showed that students would be returning with varying degrees of successful home education, and instruction would need to adapt (Molnár & Hermann, 2023), with younger students needing more support and elementary teachers needing to take note and make changes to counteract missed learning (Donnelly & Patrinos, 2022). Some achievement differences among returning students amount to multiple academic years’ worth of learning loss (Hanushek, 2023). In Tűrkiye, administrators quickly implemented a program to increase student socialization and provide mental care opportunities, resulting in teachers reporting that “students were happy, their social communication skills and academic knowledge increased, they developed a positive attitude towards the lesson and their self-confidence increased" (Karbeyaz & Kurt, 2022, p. 70) by the end of the program. Harmey and Moss (2021) methodically reviewed studies of students returning to school after disasters, concluding three important factors that facilitated smooth USING FRACTION MANIPULATIVES 10 transitions back to school. First, local school leaders and administrators were the best sources not only for information about the students in their areas, but also crucial in quickly dispersing necessary information and how best to transition everyone back. Second, the curriculum needed to be adjusted to fit individual student needs and pacing, not the prescribed advice of lawmakers trying to achieve quick catch-up. Third, schools are a great resource to support the mental and emotional health of students and should factor in time to allow students to both learn and reflect on the traumatic events experienced. Unfortunately, in the rush and excitement to reopen schools, as well as the additional protocols that had to be implemented to safely bring back students, the missed learning data does not show that the advice was heeded. Post-pandemic Missed Learning Missed Learning due to the COVID-19 pandemic and the changes made in education were thought to be temporary, but seemed to be more lasting (Grawe, 2023). Instead of seeing scores recover from the pandemic, Asadullah et al. (2023) found unprecedented declines lower than the previous three decades, essentially voiding all prior learning achievements before COVID-19. Hanushek (2023) warned The best estimates place learning losses at the equivalent of a year or more of schooling, resulting in 6 to 9 percent lower lifetime earnings for the average student and much more for disadvantaged students. The country as a whole will face a less well-prepared workforce, with enormous cumulative losses to GDP over the coming decades. (para. 1) The effects were widespread. Preschool-aged children suffered the most in cognitive and motor development, with a decreased positive attitude toward learning following closely behind (González et al., 2023). Kindergarteners were not as ready in reading to begin school as they were in previous years, while 2nd- through 8th-grade students experienced unfinished learning in USING FRACTION MANIPULATIVES 11 math, reading, and science post-remote learning, and gaps for all ages continued to widen throughout the year (Molnár & Hermann, 2023). Marginalized groups may never catch up on the foundational learning that was missed without a new program implementation (Zhang & Storey, 2022). Implementing a new program to correct and fill in learning gaps would also benefit students in higher grades, thus helping avoid their high rates of dropping out (Moscoviz & Evans, 2022). Overall, missed learning occurs over varying subjects, grades, and countries, with younger students affected more than students in secondary school (Donnelly & Patrinos, 2022). Mathematics As students returned to school, experts predicted that there would not only be a decrease in learning gains, but teachers would also be faced with classrooms filled with students at various skill levels, all of which might take years to recover from (Kuhfeld et al., 2020). Bielinski et al. (2020) found that elementary teachers should be prepared to see many students returning to school with both reading and math gaps big enough to receive remedial education. From typical summer studies, it is seen that learning forgotten over the break occurs up to amounts equivalent to one month of education, but it has been found to have an even more significant effect on math (Cooper et al., 1996). When Lewis and Kuhfeld (2021) continued their study of six million postpandemic students, they discovered math fared worse than reading. In the second school year post-COVID-19, math scores had dropped an additional 8 to 12 percentile points compared to pre-pandemic levels, while reading was only 3-6 points lower. The evidence holds true in other countries as well. In Italy, students from various backgrounds and SES statuses experienced missed learning, with girls doubling that of boys (Contini et al., 2022). In general, the United States has done poorly in mathematics in comparison to its international counterparts, perhaps because countries that were doing well in mathematics had a nationwide standard in place for USING FRACTION MANIPULATIVES 12 teaching math (Vernille, 2001). This encouraged the United States to follow suit with national standards after the third international math test heading into the 1990s, as individual states had previously shouldered the role of deciding educational standards. Experts agree that the best way to help students close the gaps is to adjust math instruction to meet students where they are, helping to build on concepts they firmly understand while expanding their abilities to new concepts (Rose & Watson, 2022). One way to meet students at their level is for teachers to consult across grades, asking for tips, suggestions, and support for students with unfinished learning, as suggested by Kuhfeld et al. (2020). As Borko et al. (1992) illustrated, a teacher who does not have a solid understanding of the whys behind common mathematical algorithms will not be able to help students understand the reasons for how problems are solved, and coordinating with other grade levels can help alleviate this issue. Another common approach to help students overcome the abstractness of mathematics is to utilize manipulatives, assisting students to solidly connect what they are learning to real-life problems (Boggan, et al., 2010). Mathematical Manipulatives Kelly (2006) stated, “If we want children to learn to think deeply and ponder real mathematics..., we must teach them and assess their knowledge in ways that will allow them to show us what they really understand about the tasks being tested” (p. 185). Manipulatives provide an additional medium for students to actively participate and better understand what they are learning, and deeper understanding leads to more lasting knowledge (Moyer, 2001). Mathematical manipulatives are solid objects that can be rearranged and handled by students while being adaptable to the student’s individual needs and understanding as math is both taught and reviewed (Boggan, et al., 2010). Virtual manipulatives are also widely available, which are USING FRACTION MANIPULATIVES 13 the same as their physical counterpart, but can be accessed through any internet browser and often have more manipulative features due to their digital nature (Reimer & Moyer, 2005). As students learn mathematics, providing the opportunity to use these diverse tools helps them develop flexibility in their thinking, leading to increased success in problem-solving (Reimer & Moyer, 2005). The manipulation of objects can allow students to more closely mimic the way human brains process information and problem-solve, leading to an increase in creative thinking and allowing students a method to demonstrate that thinking to teachers by repeating the manipulation for them (Suh, et al., 2005). “Elementary teachers who use manipulatives to help teach math can positively affect student learning. Students at all levels and of all abilities can benefit from manipulatives” (Boggan, et al., 2010, p. 5). As teachers compete with stimulating video games and media for the attention of their students, manipulatives provide a fun hands-on feel to learning that not only captures curiosity, but also engages investigational inquiry (Jones & Tiller, 2017). While all manipulative use is valuable, they are especially useful in fraction instruction as many studies have found that students struggle to understand fractions and have difficulties when performing fraction computations (Wilkie & Roche, 2023). Fraction strips, like those used in this study, are recommended for manipulation in solving fraction addition and subtraction problems, as well as determining equivalence (Boggan, et al., 2010). Students can add and remove pieces when performing calculations, and line up different fractions to compare equivalency. Summary Even though missed learning is common when students are absent from school, the COVID-19 pandemic, combined with unprecedented school closures, increased trauma, reduced mental and emotional health, and trials of distance learning merged in an extraordinary situation USING FRACTION MANIPULATIVES 14 that left students worldwide struggling to return to pre-pandemic levels of education. Without adjustments to the curriculum and pacing, considering varying and differing levels of missed learning for students, they will never be able to catch up to peers, or, even worse, not achieve proficiency in basic skills that will allow them to become valued members of society. It is imperative to find ways to help support students academically and emotionally when they return from disruptions to their education, recognizing that not all weathered the experiences in the same manner, and differentiating curriculum to meet each student’s learning where they currently are, then elevate them to the potential they can achieve. While teachers cannot control other factors contributing to the unfinished learning of their students, they can control the learning opportunities available in the classroom. Targeted math instruction utilizing manipulatives could help close the gaps for students who experienced missed learning due to school closures, distance learning, trauma, varying SES conditions, and extended absences. Purpose With two module units on fractions upcoming in my 4th-grade classroom, I decided to implement an action research project that would deepen student learning. Since utilizing manipulatives more closely mimics the way a human brain thinks (Suh, et al., 2005) and allows students to better understand the abstract concept of fractions, I decided to use manipulatives during daily instruction to help students build capacity in working with fractions. This method of instruction was an intentional decision to assist students to learn the content related to state standards and retain that knowledge over a longer period of time. In an effort to close learning gaps for students who have missed large amounts of school due to COVID-19 and other factors, as well as help students who continue to have absences and miss instructional time, this study will investigate two questions. First, to what extent does the USING FRACTION MANIPULATIVES 15 use of manipulatives while studying fractions in fourth-grade mathematics lead to increased success? In this study, success is defined as an increase in students' scores from the preassessment to the T1 assessment. Second, to what extent does the use of manipulatives while studying fractions in fourth-grade mathematics lead to an endurance of the knowledge? Methods In this action research project, a three-week quasi-experimental study was conducted which applied a targeted treatment to increase fraction comprehension and mastery. By utilizing fraction manipulatives daily during the 1-hour instructional block, students had an opportunity to demonstrate the abstract concepts they learned as they broke down fractions in their unit fractions and compared equivalent and nonequivalent fractions. Students had access to fraction tiles that represented the different ways a whole number could be divided into parts. When classroom instruction occurred, students were shown how to manipulate the fraction tiles as they compared two values with different denominators. They were also encouraged to use the tiles on their own to solve problems individually by physically laying the problem out on their desk with the manipulatives. This visualization was designed to help students notice that fractions with different values are equivalent, such as 1/3 and 4/12, while they worked independently on practice problems. Participants Thirty-two 9- to 10-year-old fourth-grade students in two classrooms participated in this study. These fourth-grade students experienced the effects of the COVID-19 pandemic when they were in kindergarten. Their school shut down three months before completing the year and switched to online learning. Since they were 5 years old, many failed to complete online assignments and teachers were ill-prepared to facilitate online learning. Upon returning to in- USING FRACTION MANIPULATIVES 16 person school in first grade, they experienced a year filled with unusual factors related to the aftermath of the pandemic, including a school day that was shortened by 1 hour. In subsequent years, students were often absent more due to quarantine concerns, illnesses, and extended vacation habits formed during the pandemic. Recruitment for this study was performed based on convenience sampling from an average-achieving, local elementary school. This Title 1 school was located in a rural community with about 20% of the students qualifying for free and reduced lunch. Students with Individualized Education Plans (IEPs), 504 plans, and other issues that may not yet have been documented were included in both groups. The researcher was the instructor of the treatment group consisting of 19 students, nine boys and ten girls from various socioeconomic backgrounds. The parents of one-third of the students in the control group chose to opt out of the study, so the control group consisted of 13 students with a similar makeup and distribution as the treatment class and received instruction as normal without the use of manipulatives. Instruments The instruments for this study included a pre-assessment (see Appendix A) given before the module instruction commenced and the same post-assessment administered twice (see Appendix B) to observe the growth of mathematical knowledge and application as well as to determine whether there was a significant increase in the treatment group compared to the control group. The initial post-assessment (referred to as Test 1, or T1) was used to show whether there would be a difference in the treatment immediately, while the second postassessment (referred to as Test 2, or T2) administered 6 weeks later would show whether the manipulatives aided in a more lasting learning process. A previous instructor created the assessments used and are currently used as a common 4th-grade assessment in conjunction with USING FRACTION MANIPULATIVES 17 the prescribed school curriculum. To maintain consistency with the 4th-grade team and avoid over-testing the student, the researcher decided to use this required assessment as the study instrument. The pre-test, used to establish a baseline, and the T2 test, which evaluated retention, were additional assessments, not required within the normal school curriculum. Procedures Before the study began, the students were randomly assigned to new teachers for the 2023-2024 school year. An institutional review board, or IRB, was obtained through the participating university (see Appendix C). Informed consent (see Appendix D) was given to each participant, explaining the procedures and providing students and parents the opportunity to opt out. The group assigned to my class participated in the treatment, and a group assigned to my colleague was the control. The study began when teachers reached the 5th module of the curriculum and began instruction about equivalent fractions. For all assessments in this study, students were given as much time as needed, but most completed the assessments within 30 minutes. To gather data, both groups of students were given a pre-assessment, with questions similar to those on the postassessment. This assessment provided a foundation to illustrate where gaps in knowledge existed and how teachers could accommodate instruction accordingly, as well as a baseline to measure the learning that took place during the duration of the study. Students in the control group continued to receive mathematics instruction for this module in line with the curriculum program used in the school, the same that they have received up to this point in the school year. This instruction included an online teaching platform that students complete daily before teacher instruction. This online instruction was a front-loading of the day’s content and included virtual instruction with practice problems and limited interactive USING FRACTION MANIPULATIVES 18 components. Next, students received direct instruction, and then completed a problem set with sample questions written in a symbolic format (e.g. ⅖ and ⅝) or displayed pictorially (e.g. partially shaded in circles) for the day’s information in an “I do, We do, You do” scaffolded format. Then, students were released to work on problems independently to practice new skills, receiving additional support as needed. Finally, students completed a homework assignment and an exit ticket on that information the following morning. It is important to note that while these students had opportunities to interact with pictorial and symbolic representations of fractions, they did not have exposure to physical manipulatives. Students in the treatment group received the developed program, incorporating manipulatives daily, online and virtually, during the 1-hour mathematics block. During teacherled instructional time, students had fraction tiles available for use (see Figure 1). Figure 1. Fraction tiles used for hands-on manipulation in the treatment group. Students were free to use the available fraction manipulatives at any time, including during individual or group work. Virtual manipulatives were provided by a digital platform that offered USING FRACTION MANIPULATIVES 19 many interactive components in a different format than the standard manipulative tiles (see Figure 2). Figure 2. An example of the screen on the online learning platform. The first standard for this module, 4.NF.3, focused on understanding the meaning of a unit fraction. We wanted students to be able to move flexibly between a given fraction and its corresponding unit fractions with the same denominator. For example, we wanted students to recognize that a fraction like 3/8 is the same as 1/8 + 1/8 + 1/8 and as 3 x 1/8. This concept is particularly tricky to grasp since students always equate a larger number to mean a greater value. Seeing the individual unit fractions helped students recognize that a whole divided into more equal-sized pieces results in a smaller-sized piece of the whole than when that same whole is divided into fewer equal-sized pieces (for example, 1/4 has a larger unit fraction than 1/12). Students also needed to achieve proficiency on standard 4.NF.1, which requires them to explain, recognize, and generate equivalent fractions. The students used fraction tiles, tape diagrams, and fraction strips to identify equivalent fractions. When students replicated textbook problems on their desks using concrete manipulatives, they could see how two seemingly USING FRACTION MANIPULATIVES 20 different numbers could represent the same amount. For example, it is difficult for most 4thgrade students to understand on paper that 2/3, 4/6, and 6/9 are equivalent fractions until they have the opportunity to examine the relationship between the numbers and their pictorial representations, as in the figure below. Figure 3. An example of turning an abstract question from the text into a concrete visualization. As students became more familiar with equivalent fractions through these simple manipulations, they were able to transition to mathematical algorithms to calculate larger equivalent fractions, like those seen on the assessments. The final standard, 4.NF.2, required students to compare fractions with different denominators or numerators. From their previous work with unit fractions, students realized that when denominators were the same, the larger numerator was greater. To confirm, they could lay out the two different fractions and compare their sizes. Again, as they felt more comfortable in USING FRACTION MANIPULATIVES 21 their understanding, the transition to algorithms was easier. Additionally, they were able to rationalize with benchmark fractions, like 1/2, to compare two fractions. When placing the fraction tiles on their desks, they saw whether the fraction was greater or less than one-half and used that information in their comparisons. Other instruction continued to follow the curriculum and was the same as the control group, taking into account variations among the different teachers’ instructional styles. After three weeks, post-assessments were administered to both groups. In another 6 weeks, a second post-assessment was again administered to compare the lasting effects of the treatment. Scoring of the assessments was performed on a 3-point scale by the researcher, with a total of 96 possible points. The names of all students were on the papers and the same scoring method was applied to all problems. If the student did not attempt anything on the problem, a 0 was scored. Students who attempted some work, but showed to not yet be proficient scored a 1. When students showed some related work that was approaching proficiency, they scored a 2. Students with correct answers and visible thinking for the problem scored a 3. For example, in Figure 4 below, the student was asked to explain how they arrived at the answer for each part of question 6. In 6a, the student received a 3, since the answer was correct, but a 0 for the second part of that problem, since there was no work shown. The same occurred for 6b. In 6c, the student received a 3 for correctness as well as a 3 for the work that was provided to explain how the answer was reached. The same is true for 6d and 6e. Once totals were calculated and the class means were found for each test for the treatment and control groups, the mean differences were calculated between the pre- and T1 assessments and the T1 and T2 assessments and compared. USING FRACTION MANIPULATIVES Figure 4. An example of student work. 22 USING FRACTION MANIPULATIVES 23 Results Independent t-tests were used to determine whether there were statistical differences between the means of growth and retention between tests. Independent t-tests are robust enough to handle the unequal sample size. A Levene’s test was used to determine that there was an equal variance assumed. There was a statistically significant difference in the means of the differences of the scores for the pre- and T1 assessments for the treatment (M=41.47, SD=15.35) and control (M=61.77, SD=17.37) groups; t(30)=3.48, p=.002. These results suggest that the control group had greater growth from the pre-assessment to the T1 assessment compared to the treatment group. Specifically, the control group increased their scores more on average. There was a significant difference in the scores for the T1 and T2 assessments for the treatment (M=-7.58, SD=9.39) and control (M=-20.46, SD=15.61) groups; t(30)=2.92, p=.007. The results for the second assessment suggest that the treatment group had a greater retention of knowledge compared to the control group. Specifically, the results suggest that using manipulatives during mathematics instruction increases retention. The total possible on all tests was 96. The mean scores of the treatment group on the pre-, T1, and T2 assessments were 54.3, 95.8, and 88.2 respectively. The mean scores of the control group on the pre-, T1, and T2 assessments were 32.2, 94, and 73.5 respectively (see Figure 5). USING FRACTION MANIPULATIVES 24 Figure 5. The score means for each group, out of 96 points. Discussion Manipulatives and Increased Learning The first question of this study investigated whether manipulatives would increase learning in a fourth-grade classroom studying fractions. The results indicated that the control group, who did not use manipulatives during instruction, actually had a greater mean growth in the scores from their pre- to T1 assessments. The control group's mean growth was almost 62 points, while the treatment group increased by 41 points. These results help illustrate that an effective teacher has the power to facilitate student learning, as found in Hanushek’s (2023) study. It emphasizes the importance of having students in school to avoid the effects of missed and unfinished learning, which can reach up to 50% in math over summer breaks (Maldonado & De Witte, 2022). One possible reason for the increased success of the control group is the initial scores of the pre-assessment. The mean score for the pre-assessment in the treatment group was USING FRACTION MANIPULATIVES 25 54 points, while the control mean was 32 points. Since the treatment group was initially higher and the instrument used had a point ceiling, there was not as much room available for statistical growth, leading to the control group outperforming them. Manipulatives and Lasting Learning The study’s second query examined if implementing fraction manipulatives in a fourthgrade classroom would lead to more lasting learning for the students. Both the treatment and control groups performed well on the T1 assessment, but both mean differences dropped between T1 and T2. This showed that unfinished learning happens even when students are present in school and supported the findings of Donnelly and Patrinos (2022), which emphasized that elementary teachers needed to provide extra support to their younger students. In regards to manipulative use, the treatment group had a mean loss of 7.58 points from T1 to T2, while the control group scores lost a mean of 20.46 points. This was similar to the findings of Moyer (2001) who concluded that active participation in learning led to a deeper understanding that endured. Since math scores are dropping (Asadullah et al.,2023; Lewis & Kuhfeld, 2021), incorporating manipulatives into instruction could help combat the problem. Results of this study showed a drop in scores over time, but to a lesser degree when manipulatives were involved, which gives additional credibility to the conclusion of Boggan, et al. (2010) that manipulatives gave a concrete connection to the abstract learning of mathematics. Harmey and Moss (2021) called for adjusted pacing and curriculum in the transition back to school after the pandemic. Rose and Watson (2022) suggested a change in pacing and for teachers to meet students where they are as they returned from COVID-19 closures. Unfortunately, due to the missed classroom time and varying levels of students, combined with residual protocols and unexpected absences, teachers felt panic over the missed learning and the USING FRACTION MANIPULATIVES 26 pressure to “catch up” in a short period. Manipulative use takes more time to implement and manage, and many teachers do not want to allocate resources to something that requires more of this precious commodity. This study showed spending extra time on manipulative instruction in the classroom could pay off in the long run with concept retention. Additionally, the SES of students could be found irrelevant from the study of Chetty et al. (2020) when all students are given manipulatives to work with leading to improved student understanding, thus lessening the burden on parents at home who possibly have insufficient education or time to assist their children. Finally, utilizing digital manipulatives could allow students to complete assignments effectively even when they have missed school due to various factors, keeping them from missing too much learning during their absence. Limitations This study had limitations in the small sample size that was tested. In the treatment group, all students opted in, while in the control group, one-third of the students opted out. Recognizing that the sample size initially was small, these changes could have led to inconsistent data. Since it was a quasi-experimental study, there was no guarantee that the groups were completely random or had equivalent capabilities, causing some underlying discrepancies in the study. Additionally, differences in teaching styles between the two classroom instructors might have created unpredictable data. This is evident in the baseline data from the pre-assessment, showing students were initially at different levels of understanding. Variations in teaching styles could be overcome by studying two groups of students who share the same teacher. Since the researcher graded all assessments from both groups and names were on all papers, there is a possibility of bias in the grading. Although the researcher applied the same scoring method to all tests, grading the tests blind and by more than one party could have helped control for USING FRACTION MANIPULATIVES 27 inconsistencies. Another limitation is that this study was performed with two fourth-grade classes from a rural elementary school that experienced the effects of the COVID-19 pandemic, and the results for these students may not apply to others. Finally, though the long-term effects of the manipulatives were tested after six weeks, that may not have been enough time to test the duration of the gains achieved through using manipulatives. Further Research There are many avenues to pursue for future research. First, additional investigation can be completed on this topic looking into the interaction of the analytical left brain with the visual right brain as students learn mathematics with manipulatives. Next, this study utilized both physical and digital manipulatives during instruction. A study investigating the benefits and restrictions of each version could be useful to discover whether they are equally effective. Furthermore, investigation into lasting learning could be performed again by administering another post-assessment (T3) at the end of the school year, fifteen weeks after the initial treatment assessment. Alternatively, with knowledge constantly ebbing and flowing (Harmey & Moss, 2021), commencing a longitudinal study on the use of manipulatives in mathematics for fractions as well as expanding into other topics could determine the effectiveness of concrete representations to assist in understanding abstract mathematical concepts and determining the lasting effects of the learning. Finally, a deep dive into individual questions to find trends and decide whether students did better on certain question types might also yield helpful information. Conclusion Manipulatives themselves are not a magic wand making mathematical concepts clear for students, but can provide a platform for experiencing educational insights (Moyer, 2001). While the use of manipulatives during a mathematics lesson takes more time and does not seem to show USING FRACTION MANIPULATIVES 28 beneficial results initially, the lasting learning that resulted from their utilization is a strong argument for incorporation into instruction. With so many teachers concerned about their students' performance on state-wide mandated testing, this lasting learning could lead to improved test scores. In addition, students would better understand and recall the concepts they learned and be more equipped to solve new problems they faced. In the post-pandemic world that continues to deal with the aftermath of COVID-19, extended student absences, illnesses, traumas, low SES, and other difficult life factors, the incorporation of manipulatives by educators in fraction learning, as well as mathematics in general, may lessen unfinished learning and close learning gaps. USING FRACTION MANIPULATIVES 29 References Asadullah, M. N., Bouhlila, D. 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Examining technology uses in the classroom: Developing fraction sense using virtual manipulative concept tutorials. Journal of Interactive Online Learning, 3(4), 1-21. Vernille, K. (2001). Why are US mathematics students falling behind their international peers? University of Maryland, College Park, MD. Wilkie, K., & Roche, A. (2023). Primary teachers’ preferred fraction models and manipulatives for solving fraction tasks and for teaching. Journal of Mathematics Teacher Education, 26(6), 703-733. Zhang, Q. & Storey, N. (2022) Controversies behind COVID learning loss: Historical issues, current measurements, and future strategies. Theory Into Practice. 61(3), 300-311. DOI: 10.1080/00405841.2022.2096380 USING FRACTION MANIPULATIVES Appendix A: Pre-Assessment Equivalent and Comparing Fractions: Pre-Assessment 1a. Draw a model that shows 3/6. 1b. Represent the 3/6 above as a sum of unit fractions. 1c. Rewrite the equation you made in part b as the multiplication of a whole number by a unit fraction. 1d. Break down the 3/6 above using only 2 addends. 33 USING FRACTION MANIPULATIVES 2a. Draw a model that shows 2/8. 2b. Represent the 2/8 above as a sum of unit fractions. 2c. Rewrite the equation you made in part b as the multiplication of a whole number by a unit fraction. 2d. Break down the 2/8 above using only 2 addends. 34 USING FRACTION MANIPULATIVES 3a. Draw a model that shows 6/5. 3b. Represent the 6/5 above as a sum of unit fractions. 3c. Rewrite the equation you made in part b as the multiplication of a whole number by a unit fraction. 3d. Break down the 6/5 above using only 2 addends. 35 USING FRACTION MANIPULATIVES 36 4. Using the fractional units shown, identify the fraction of the rectangle that is shaded: a. ____________________ b. ____________________ c. ____________________ Use multiplication to explain why the fractions shown in 4b and 4c are equivalent. USING FRACTION MANIPULATIVES 37 5. Cross out the fraction that is NOT EQUIVALENT to the other three. a. 8/10 40/50 5/20 ⅘ 5/15 120/80 24/21 8/7 Show how you know: b. 3/2 30/20 Show how you know: c. 32/28 16/12 Show how you know: USING FRACTION MANIPULATIVES 38 6. Fill in the blank with <, =, or > to make a true number sentence. Justify each response by drawing a model, creating common denominators or numerators, or explaining a comparison to a benchmark fraction. For proficiency 4, justify each response using TWO OR MORE of the listed strategies. a. 7/5__________13/5 b. 3/4__________6/8 Show how you know: Show how you know: c. d. 9/9__________99/99 Show how you know: e. 1/9__________2/9 Show how you know: 5/2__________3/2 Show how you know: USING FRACTION MANIPULATIVES Appendix B: Post-Assessment Equivalent and Comparing Fractions: Common Assessment 1a. Draw a model below that shows ¾. 1b. Represent the ¾ above as a sum of unit fractions. 1c. Rewrite the equation you made in part b as the multiplication of a whole number by a unit fraction. 1d. Break down the ¾ above using only 2 addends. 39 USING FRACTION MANIPULATIVES 2a. Draw a model below that shows 2/4. 2b. Represent the 2/4 above as a sum of unit fractions. 2c. Rewrite the equation you made in part b as the multiplication of a whole number by a unit fraction. 2d. Break down the 2/4 above using only 2 addends. 40 USING FRACTION MANIPULATIVES 3a. Draw a model below that shows 5/4. 3b. Represent the 5/4 above as a sum of unit fractions. 3c. Rewrite the equation you made in part b as the multiplication of a whole number by a unit fraction. 3d. Break down the 5/4 above using only 2 addends. 41 USING FRACTION MANIPULATIVES 42 4. Using the fractional units shown, identify the fraction of the rectangle that is shaded: a. ____________________ b. ____________________ c. ____________________ d. Use multiplication to explain why the fractions shown in 4a and 4b are equivalent. USING FRACTION MANIPULATIVES 43 5. Cross out the fraction that is NOT EQUIVALENT to the other three. a. 3/5 60/100 6/10 6/5 12/8 8/4 9/6 3/2 Show how you know: b. 6/4 3/2 Show how you know: c. 6/4 16/12 Show how you know: USING FRACTION MANIPULATIVES 44 6. Fill in the blank with <, =, or > to make a true number sentence. Justify each response by drawing a model, creating common denominators or numerators, or explaining a comparison to a benchmark fraction. For proficiency 4, justify each response using TWO OR MORE of the listed strategies. a. 6/5__________14/5 b. ⅗__________2/15 Show how you know: Show how you know: c. d. 6/6__________12/12 Show how you know: e. 3/9__________4/9 Show how you know: 8/2__________6/2 Show how you know: USING FRACTION MANIPULATIVES 45 Appendix C: IRB Approval Letter January 8, 2024 Sheryl Rushton Bethany Warr WSU, Teacher Education Re: Exempt - Initial - IRB-AY23-24-198 Supplementing Opportunity Gaps in Mathematics with Fractional Manipulatives Dear Sheryl Rushton: The Weber State University Institutional Review Board has rendered the decision below for Supplementing Opportunity Gaps in Mathematics with Fractional Manipulatives. Decision: Exempt Approval: January 8, 2024 Selected Category: Category 1. Research, conducted in established or commonly accepted educational settings, that specifically involves normal educational practices that are not likely to adversely impact students’ opportunity to learn required educational content or the assessment of educators who provide instruction. This includes most research on regular and special education instructional strategies, and research on the effectiveness of or the comparison among instructional techniques, curricula, or classroom management methods. Findings: Research Notes: Subjects are considered children, signatures/consent and parental assent are required, and they may choose not to participate. Anonymity and confidentiality are addressed appropriately, and the type of information gathered could not "reasonably place the subjects at risk of criminal or civil liability or be damaging to the subjects' financial standing, employability, or reputation" (Code of Federal Regulations 45 CFR 46, Subpart D). The research is taking place in an educational setting. You have one year to complete the study. Please remember that any anticipated changes to the project and approved procedures must be submitted to the IRB prior to implementation. Any unanticipated problems that arise during any stage of the project require a written report to the IRB and possible suspension of the project. If you have any questions please contact your review committee chair or irb@weber.edu. Sincerely, Daniel Hubler, Ph.D. Interim Chair, College of Education IRB Sub-committee USING FRACTION MANIPULATIVES 46 Appendix D: Informed Consent IRB STUDY # IRB-AY23-24-198 WEBER STATE UNIVERSITY INFORMED CONSENT Supplementing Opportunity Gaps in Mathematics with Fractional Manipulatives You are invited to participate in a research study of using manipulatives to decrease opportunity gaps in learning fractions. You were selected as a possible subject because your student is in a 4th grade class at the local elementary school. We ask that you read this form and ask any questions you may have before agreeing to be in the study. The study is being conducted by Bethany Warr, a Graduate Student at Weber State University. STUDY PURPOSE The purpose of this study is to identify whether the daily use of fractional manipulatives will facilitate the learning process of 4th grade students and provide opportunities to use concrete materials to better understand abstract concepts. Additionally, researchers are investigating whether the use of manipulatives will provide more enduring learning, thus helping students overcome opportunity gaps that may have occurred during absences from in-person instruction. NUMBER OF PEOPLE TAKING PART IN THE STUDY: If you agree to participate, you will be one of 40 subjects who will be participating in this research. PROCEDURES FOR THE STUDY: If you agree to be in the study, you will do the following things: ● ● ● ● ● Attend school daily and participate in mathematical learning and discussions, including assigned homework. Utilize fraction manipulatives on a daily basis. Take a pre-test to assess your knowledge base. Take a post-test immediately after the end of the fraction module. Take another post-test 4 weeks after the initial post-test. USING FRACTION MANIPULATIVES 47 RISKS OF TAKING PART IN THE STUDY: For face-to-face research, the risks include the possibility of being infected by the novel coronavirus 2019 (COVID-19) or other communicable diseases. The risks of participating in this study include: ● Feeling uncomfortable when completing the pre- and post-tests. ● Feeling uncomfortable when using manipulatives. ● A possible loss of confidentiality. ● A possibility of risks that are currently unforeseen. BENEFITS OF TAKING PART IN THE STUDY You will not receive payment for taking part in this study. ALTERNATIVES TO TAKING PART IN THE STUDY: Instead of being in the study, you have the option to not use manipulatives when the rest of the class is utilizing them. You also have the option to not take part in the assessments for this study. COSTS/ COMPENSATION FOR INJURY In the event of physical injury resulting from your participation in this research, necessary medical treatment will be provided to you and billed as part of your medical expenses. Costs not covered by your health care insurer will be your responsibility. Also, it is your responsibility to determine the extent of your health care coverage. There is no program in place for other monetary compensation for such injuries. However, you are not giving up any legal rights or benefits to which you are otherwise entitled. If you are participating in research which is not conducted at a medical facility, you will be responsible for seeking medical care and for the expenses associated with any care received. CONFIDENTIALITY Efforts will be made to keep your personal information confidential. We cannot guarantee absolute confidentiality. Your personal information may be disclosed if required by law. Your identity will be held in confidence in reports in which the study may be published and databases in which results may be stored. Organizations that may inspect and/or copy your research records for quality assurance and data analysis include groups such as the study investigator and his/her research associates, the Weber State University Institutional Review Board or its designees, and (as allowed by law) state or federal agencies, specifically the Office for Human Research Protections (OHRP) and the Food and Drug Administration (FDA) [for FDA-regulated research and research involving positron-emission scanning], the National Cancer Institute (NCI) [for research funded or supported by NCI], the National Institutes of Health (NIH) [for research funded or supported by NIH], etc., who may need to access your medical and/or research records. USING FRACTION MANIPULATIVES 48 CONTACTS FOR QUESTIONS OR PROBLEMS For questions about the study, contact the researcher Bethany Warr at (801) 928-8180 or the researcher’s mentor Dr. Sheryl Rushton at (801) 917-4191. For questions about your rights as a research participant or to discuss problems, complaints or concerns about a research study, or to obtain information, or offer input, contact the Chair of the IRB Committee IRB@weber.edu. VOLUNTARY NATURE OF STUDY Taking part in this study is voluntary. You may choose not to take part or may leave the study at any time. Leaving the study will not result in any penalty or loss of benefits to which you are entitled. Your decision whether or not to participate in this study will not affect your current or future relations with Morgan Elementary School. SUBJECT’S CONSENT In consideration of all of the above, I give my consent to participate in this research study. I will be given a copy of this informed consent document to keep for my records. I agree to take part in this study. Subject’s Printed Name: Subject’s Signature: (must be dated by the subject) Date: Printed Name of Person Obtaining Consent: Signature of Person Obtaining Consent: Date: If the study involves children who will be providing their assent on this consent document, rather than on a separate assent document, use the following signatures: Printed Name of Parent: Signature of Parent: Date: