TIER II INTERVENTIONS:A PREVENTION STRATEGY FOR NON-PROFICIENCY AND THE BARRIERS OF APPLICATION By Wendy M. McKee A project submitted in partial fulfillment of the requirements for the degree of MASTER OF EDUCATION IN CURRICULUM AND INSTRUCTION WEBER STATE UNIVERSITY Ogden, Utah August 8, 2020 Approved ____________________________________ Sheryl J. Rushton, Ph.D. ____________________________________ Melina Alexander Ph.D. ____________________________________ Malorie Serdar M. Ed. TIER II INTERVENTIONS 2 Abstract Despite proven effectiveness of Tier II interventions in the United States’ elementary school education system, there are barriers to implementing effective Tier II interventions. General education classes contain a wide range of academic levels with some students not achieving previous grade level proficiency. Two options exist at the end of an academic year, to either socially promote or grade retain students who do not show proficiency. Studies affirm that neither social promotion nor grade retention alone are effective. The multi-tier system of support (MTSS) structural model has been put into place in education systems to support struggling students toward proficiency. Within the MTSS model, Tier II is a targeted effort to reinforce students’ education. One of the barriers to implementing effective Tier II interventions is the lack of highly trained intervention aides. For this purpose mathematics curriculum was designed to support effective instruction in Tier II mathematics interventions. TIER II INTERVENTIONS 3 Table of Contents Literature Review........................................................................................................................... 7 Grade Retention or Social Promotion and Alternatives .............................................................. 7 Grade Retention ....................................................................................................................... 8 Social Promotion ................................................................................................................... 10 Alternatives ............................................................................................................................ 13 Multi-Tiered System of Support ............................................................................................... 13 Tier I ..................................................................................................................................... 14 Tier II .................................................................................................................................... 15 Tier III ................................................................................................................................... 16 Tier II Interventions and Barriers .............................................................................................. 17 Tier II interventions ............................................................................................................... 17 Tier II barriers ........................................................................................................................ 18 Learning process ................................................................................................................ 19 Student attitude and gender bias ........................................................................................ 20 Perceived parental influence .............................................................................................. 21 Lack of teaching manipulatives ......................................................................................... 21 Time .................................................................................................................................. 22 Properly trained teachers and teacher aides ....................................................................... 23 Effective Tier II curriculum ............................................................................................... 24 Purpose ......................................................................................................................................... 25 Method ......................................................................................................................................... 26 Subject ...................................................................................................................................... 26 Context ..................................................................................................................................... 27 Educator Evaluators .................................................................................................................. 28 Procedure ..................................................................................................................................... 29 Standards .................................................................................................................................. 29 Backwards Planning .................................................................................................................. 29 Manipulatives ............................................................................................................................ 30 Gradual Release Model ............................................................................................................. 31 TIER II INTERVENTIONS 4 Scripted Lessons to support Untrained Mathematics Paraprofessionals ................................... 31 Assessment ............................................................................................................................... 32 Evaluator Findings ........................................................................................................................ 32 Accuracy................................................................................................................................... 32 Scripted Lesson Plan ................................................................................................................. 33 Supporting Understanding......................................................................................................... 34 Extra Practice ............................................................................................................................ 34 Assessment ............................................................................................................................... 34 Limitations ................................................................................................................................... 34 Recommendations ......................................................................................................................... 35 Conclusion ................................................................................................................................... 35 References .................................................................................................................................... 36 Appendix A .................................................................................................................................. 45 Appendix B .................................................................................................................................. 62 TIER II INTERVENTIONS 5 Tier II Interventions: A Prevention Strategy for Non-Proficiency and the Barriers of Application When a student fails to make grade-level progress, teachers and administrators must choose between retaining that student or socially promoting them. Grade retention is holding a student back or making a student repeat a grade when that student did not meet the academic or learning standards expectations (Doherty, 2004; Social Promotion, 2014). In contrast to grade retention is social promotion. Social promotion has been understood as advancing students to the next level or grade based on age, not proficiency, before reaching the expected learning standards (Johnson & Rudolph, 2001; McMahon, 2018; Social Promotion, n.d.). Neither grade retention nor social promotion alone, is adequate to progress students’ emotional, social, behavioral, and academic achievement (Allen, Chen, Willson, & Hughes, 2009; Doherty, 2014; Crepeau-Hobson, Nickerson, & Cook, 2016; Jimerson & Renshaw, 2012; Johnson & Rudolph, 2001; Kretschmann, Vock, Jansen, Ludtke, & Gronostaj, 2019; Martin, 2011; McMahon, 2018; Social Promotion, 2014). Given that grade retention and social promotion have been found ineffective, many Utah schools have implemented a multi-tier system of support (MTSS) structural model, known nationally as Response to Intervention (RTI), to support students who are at risk of grade retention or social promotion (Brown-Chidsey & Steege, 2010; Bruhn, Lane, & Hirsch, 2014). RTI usually is composed of three tiers for instruction; Tier I- general class, Tier II- targeted at-risk students, and Tier III- intensive instructional needs (Harlacher, Sanford, & Walker, n.d.). Research has shown Tier II interventions to be effective and is supplemental support for those who are at-risk emotionally, socially, behaviorally, and academically (Harlacher et al., n.d.; Jimerson et al., 2006). For Tier II interventions to be effective there should be; well-trained teachers, extended learning time, re-teaching of pertinent basic information, and small leveled TIER II INTERVENTIONS 6 group sizes (Johnson & Rudolph, 2001; Picklo & Christenson, 2005; Solari, Denton, Petscher & Haring, 2017). Although Tier II interventions have been shown to be effective, there are barriers that impede the quality of Tier II intervention practices (Batholomew & Jong, 2017). Some of the barriers are: the learning process, student’s attitude and gender bias, perceived parental influence, lack of teaching manipulatives, time, properly trained teachers and teacher’s aides, and effective Tier II curriculum (Allen et al., 2009; Jimerson, et al., 2006; Martin, 2011; Otaiba, et al., 2014; Picklo & Christenson, 2005). Since 70% of school districts across the country are implementing tier II interventions, educators need to find ways to overcome these barriers in order to make the quality of the interventions high (Johnson & Rudolph, 2001; Sparks, 2015). TIER II INTERVENTIONS 7 Literature Review Grade Retention or Social Promotion and Alternatives A frustration within an education system is having a wide range of academic levels within the general education class. Teachers have to spend much of their time in scaffolding to support those students who are behind. There is not a perfect solution to this dilemma. Educators are constantly looking for better ways to teach in this diverse dynamic. The two choices in the past have been to keep students back or to push them onto the next grade level. Teachers are trying to help avoid making these choices by intervening earlier. These teachers must overcome barriers to make effective interventions happen. Ideally, students should demonstrate proficiency in a class or subject area before moving onto the next grade level. This does not always happen in education systems. In 2017, the National Assessment of Education Progress claimed that only 37% of seniors in high school were able to test proficient in reading and only 25% were proficient in math (McMahon, 2018). When a student is not at grade proficiency level by the end of the school year there are two options; grade retention or social promotion. Neither option alone is adequate to progress a student’s emotional, social, behavioral, and academic achievement (Jimerson et al., 2006; McMahon, 2018). Some similarities of non-proficient students are: students in minorities such as race come from lower-income households, English is not their first language, live in metropolitan areas, their mothers’ expectation of their academic achievement is low, their fathers’ occupations are similar, they come from single-parent homes, and they are often students with disabilities (Doherty, 2004; Jimerson et al., 2006; McMahon, 2018; Social Promotion, 2014). TIER II INTERVENTIONS 8 Grade Retention. Grade retention is holding students back or making them repeat a grade or subject that they did not demonstrate the academic or learning standards expectations (Doherty, 2004; Johnson & Rudolph, 2001; Martin, 2011; Social Promotion, 2014). Retention has been documented from the early 1900s. Retention was at the highest in the 1970s, decreased in the 1980s, and then increased again in the 1990s (Allen et al., 2009). In the 1980s and early 1990s, up to 31% of students were not in the appropriate grades in terms of age (Doherty, 2004). From the early 1990s until the late 1990s retention was estimated to have been around 15% (Johnson & Rudolph, 2001; Martin, 2011). It was estimated that 30% to 50% of those retained students had been retained before ninth grade (Johnson & Rudolph, 2001). After President Clinton’s address to the union to stop social promotion an upsurge in the percentage of retention occurred (Allen et al., 2009). In 2015, 12% of 15-year-olds, stated they had been retained (Kretschmann, et al., 2019). The data show the waves of change between social promotion and retention dominate beliefs and practices (Allen et al., 2009). Proponents of grade retention believe the intention is to give the retained student time to catch up on curriculum with the student’s peers and that if the student were promoted the student would fall further and further behind (Kretschmann et al., 2019; Martin, 2011; Social Promotion, n.d.). Proponents also believe that retaining students will decrease the skill discrepancies within classrooms and allow teachers to better meet the needs of all students (Johnson & Rudolph, 2001; Kretshcmann et al.; 2019, Social Promotion, n.d.). Those who favor retention also feel it will give the student a chance to mature mentally and behaviorally (Johnson & Rudolph, 2001, Kretschmann et al., 2019). Holding students back, because they have not proven to understand the curriculum, also creates student responsibility and accountability for the learning (McMahon, 2018). Almost 80% of students interviewed, in one study, thought that retention should be in TIER II INTERVENTIONS 9 place if a class is failed (McMahon, 2018). Proponents support the research that students who are retained earlier in their academic career, before third grade, have substantial academic gains (short-term and long-term), decreased likelihood of being retained in later grades, and does not affect their academic achievement or motivation (Martin, 2011; McMahon, 2018). Those in favor of retention also believe that retention will help give the students an extra chance to develop the life and academic skills needed to be successful later in life (Doherty, 2004). Eighty-seven percent of parents interviewed in a study said they would support policies where students had to pass a test to move on, even if it meant grade retention (Doherty, 2004). This goes along with what President Clinton called for in his State of the Union Address to require assessments to measure student achievements (Allen et al., 2009). Although there is research showing the ills of retention, which will be discussed later, some of the newest research show that there is very little connection to the emotional consequences of peer relationships when retained (Martin, 2011). In a meta-analysis study by Allen and colleagues, they found that their research challenged the commonly held view that grade retention is negative for academic or social advancement (Allen et al., 2009). On the opposite side, the majority of studies point out that retention is ineffective on academic achievement or personal adjustment (Crepeau-Hobson et al., 2016). Research showed that retention, especially repeated retention, increased dropout rates before graduating from high school, and students not attending post-secondary education (Doherty, 2004; Johnson & Rudolph, 2001; Martin, 2011; McMahon, 2018, Social Promotion, n.d.). Because studies have shown retention leads to drop out, it is suggested that students are left with inadequate knowledge and skills for life (Doherty, 2004). Retention is also related to poor school attendance, poor attitude, negative self-esteem, low homework completion rate, and low TIER II INTERVENTIONS 10 motivation (Doherty, 2004; Kretschmann et al., 2019; Martin, 2011). A connection with greater emotional anxiety and disruptive classroom behavior is found with retention (Martin, 2011). Retention is associated with poorer academic achievement when compared to groups who have similarities but were promoted, and the students tend to never catch up to even an “average” student, the retained students show to have a slower achievement rate (Crepeau-Hobson et al., 2016; Johnson & Rudolph, 2001, Martin, 2011; Social Promotion, n.d.). Even when students show gains when retained, the gains seem to diminish within three years (Allen et al., 2009; Crepeau-Hobson et al., 2016; Johnson & Rudolph, 2001; Martin, 2011). Retention is a very expensive option, repeated retention can cost states and school districts quadruple of a child’s education (Martin, 2011; Social Promotion, n.d.). Social Promotion. Social promotion is the advancement of students to the next level or grade, based on age, not proficiency, progressing students without their comprehending the grade level material or expected learning standards (Johnsons & Rudolph, 2001; Social Promotion, 2014; McMahon, 2018). The practice of social promotion has been traced back to the 1930s when it was noticed that grade retaining, was not always positive (Johnson & Rudolph, 2001). The rates of social promotion have varied over the years depending on the theories of policymakers. History shows that policy switches between favoring social promotion or retention about every decade (Allen et al., 2009; Social Promotion, n.d.). Social promotion is not something that is recorded or kept track of by school systems nor do schools like to admit to using it. This makes social promotion hard to get exact rates of how prevalent it is (Doherty, 2004). When interviewed, the majority of teachers report that they have socially promoted within the past year (Doherty, 2004). There was a drop in social promotion when President Clinton urged to end social promotion in his 1998 and 1999 State of the Union Addresses. President TIER II INTERVENTIONS 11 Clinton wanted standardized tests to show if a student was ready to be promoted (Allen et al., 2009). It was estimated that in the 1990s the rate of social promotion was higher than 31% (Doherty, 2004). Subsequent research conducted in the 2000s illuminated the ills of grade retention, bringing social promotion into current favor (Allen et al., 2009; Martin, 2011). There are several reasons why some policymakers, districts, administrators, and community members are in favor of social promotion. One idea is the consideration for non-academic factors such as societal pressures and expectations (Social Promotion, n.d.). Those who choose social promotion, for this reason, were concerned about separating a student from their peer group or friends (Martin, 2011; Social Promotion, n.d.). Proponents of social promotion worry about retained students’ ability to interact with the new cohort of students they would be with if held back (Social Promotion, n.d.). Supporters worry retention will also damage a child’s self-esteem or cause other emotional and psychological problems (Social Promotion, n.d.). Advocates further fear retainment of students might bring students to struggle more academically because they will label themselves as “dumb” (Martin, 2011). Social promotion is thought to help students from creating further behavioral problems (Martin, 2011; Social Promotion, n.d.). Those in favor of social promotion also cite studies that recognize socially promoted students tend to be able to be more employable (Martin, 2011). The schools themselves might want to avoid negative attention or consequences for reporting too much retention especially if significant amounts within one school are held back (Social Promotion, n.d.). Often considered in social promotion decisions are factors such as attendance, teacher endorsement, test conclusions, and overall peer involvement (Doherty, 2004). One parent, who was interviewed after their child had been promoted three times, explained that social promotion undermined both the parent’s and teacher’s credibility by not TIER II INTERVENTIONS 12 holding the student accountable (McMahon, 2018). The overall value of the student’s education is downscaled, they lose motivation to learn the material because if it was important they would be required to learn and understand. When no consequence is perceived the less motivation will be apparent (McMahon, 2018). One student was quoted as saying, “Who cares? If you do the work you are moving on, if you do not do the work, you are moving on. The work is not important; we cannot care if you learn it” (McMahon, 2018 p. 512). A socially promoted student could face failure with consequences for the first time and are unprepared for the overall high school experience (Doherty, 2004; McMahon, 2018). This enhances the chances of socially promoted students to drop out or fail to graduate (McMahon, 2018). In one research study, 96% of students who had been socially promoted three times ended up dropping out of high school (McMahon, 2018). Another shocking statistic is that since promotion is intended to keep students with their peers, 90% of those who were socially promoted twice, no matter at what grade, were still in high school after their peer group had graduated (McMahon, 2018). Critics of social promotion also point out that social promotion changes the education system from emphasizing education to emphasizing efficiency (McMahon, 2018). Each class becomes burdened with students who are underprepared. Socially promoted students require more of the teacher’s and classes’ time and attention because there is an added dissonance between levels of students (Martin, 2011; Social Promotion, n.d.). Because social promotion is not well tracked, it can mislead parents, policymakers, and the public to think that all students are making adequate progress and that classes are running well (Social Promotion, 2014). This could specifically hurt those promoted students who could get passed over for extra help if a learning need was misdiagnosed because of social promotion (Social Promotion, n.d.). McMahon stated, “By socially promoting students, schools are essentially doing nothing.” (McMahon, 2018 p. 513). TIER II INTERVENTIONS 13 Alternatives. The decision to retain or to promote is perhaps one of the greatest divided stances within the public education forum (Jimerson et al., 2006). From examining the research of social promotion and grade retention, it is clear to see that neither, by themselves, is truly effective in fostering and supporting needs to improve desired educational outcomes (Crepeau- Hobson et al., 2016; Jimerson et al., 2006; Johnson & Rudolph, 2001). Many researchers agree that the best strategy would be to prevent and eliminate the need for a decision between social promotion and grade retention because both of these methods focus on the end of school results (Johnson & Rudolph, 2001; McMahon, 2019; Picklo & Christenson, 2005). Prevention is key to eliminate the need to deal with failure or non-proficiency in the first place (Johnson & Rudolph, 2001). All educators are encouraged to find those evidence-based programs, policies, and strategies that will be effective (Jimerson et al., 2006; McMahon, 2018,). Research has shown that the Multi-Tiered System of Support’s Tier II Interventions are effective to help advance non-proficient students (Bruhn, Woods-Grove, Fernando, Choi, & Troughton, 2017; Jimerson et al., 2006). Multi-Tiered System of Support In 2004, Congress updated the Individuals with Disabilities Education Act (IDEA), this update included endorsing an instructional framework model of tiered supports to help identify students who would need more intensive support (Harlacher, Walker, & Sanford, 2010). This model of tiered support is known as Response to Intervention (RtI) (Burns, Appleton, & Stehouwer, 2005; Rosen, n.d.). The RtI model generally involves; First, having a whole-class lesson. Next, monitoring students closely for academic deficits. Placing those with a found deficit into an intervention, continuing to progress monitor in order to consider if the student is progressing or not. Finally, if found to not be progressing considering if the student is in need of TIER II INTERVENTIONS 14 special education services (Burns et al., 2005, Kearney & Graczyk, 2014; Utley & Obiakor, 2015). This model was to separate those students who truly had a learning disability from those who had just been poorly taught, known as “instructionally” disabled (Fuchs & Fuchs, 2006). The RtI framework has been drawn out of the special education field and used often by forty-seven of the states in the United States, as a method to improve not only academics of all general education classes but also behaviors (Charlton et al., 2018; Samuels, 2016). The term Multi-Tiered System of Supports (MTSS) is used to cover both the academic and behavior framework models (Samuels, 2016). A school or district implementing MTSS is commonly trying to handle both the behavior and academic interests. To help make sense of why both academics and behavior should be addressed, Samuels (2016) gives the example that if a student does not understand what is being taught then the student is more likely to act out behaviorally and a student who has behaviors that restrict them from being able to focus will drop academically. The Elementary and Secondary Education Act was updated on December 10, 2015, by President Obama. This Act is known as the Every Student Succeeds Act (ESSA; Charlton et al., 2018). The ESSA included MTSS as a framework model to follow to support exceptional student growth based on supporting evidence of the effectiveness of tiered research (Charlton et al., 2018). The tiered levels within MTSS are organized from least intensive to the most intensive level. Tier I. The most primary or comprehensive tier level, known as Tier I, is the general education class. The Tier I class should be designed to be able to support academic growth for about 80% of students (Charlton et al., 2018). To accomplish reaching 80% of student’s academic needs, general education classes should be of effective, high-quality, strong instruction. Research has demonstrated that the proper training of general educators, through TIER II INTERVENTIONS 15 professional development, to be important (Charlton et al., 2018). Professional development such as live training, webinars, handbooks, coaching, modeling, and consulting were all found beneficial (Charlton et al., 2018). Thorough Tier I level universal screening of all students is important (Samuels, 2016). These screenings allow the education team members to detect those students who are falling behind early (Samuels, 2016). Tier II. After looking at the data from the universal screenings and discovering those students who need extra support, the education team moves to the next tier, Tier II. Tier II is intended to provide extra support for 10% to 15% of students (Charlton et al., 2018). Tier II is for those students who are identified to be “at-risk” with not being able to keep up with their peer group academically or behaviorally (Fuchs & Fuchs, 2006). These “at-risk” students are monitored more closely by the general education teacher. If these students are shown to still be unresponsive to classroom instruction, showing no improvement in catching up with their peers, they will be given more intensive instruction (Fuchs & Fuchs, 2006). These instruction sessions can be given within the classroom time or outside the classroom time and are often known as interventions (Fuchs & Fuchs, 2006). Tier II instruction is given in a different method than Tier I whole group instruction. Often the Gradual Release Method is used to focus on the process explicitly. The “I do, We do, You do” Gradual Release model is effective within explicit Tier II instruction (Fisher & Frey, 2014). In the “I do, We do, You do” model, the instructor shows explicitly how to complete a skill, this is the “I do” portion. The students and instructor then complete the skill explicitly together, this is the “We do”. Lastly, the student completes the skill independently, the “You do”. This gradual release removes scaffolding for the process to be taught. The students continue to TIER II INTERVENTIONS 16 be closely monitored to detect growth or lack of responding to interventions, if they fail to respond they will enter the next tier. Tier III. Tier III is restricted to about 5% of students who need more exclusive or individualized assistance (Charlton et al., 2018). At this stage of the tier system very small groups between 1-3 students are used to intensify the learning opportunities. These students are often monitored daily to see if they are progressing or showing signs of the remediation working. If the students fail to show signs of improvement or are nonresponsive, learning disabilities are much likely the cause. The MTSS model allows teachers, special educators, school psychologists, and principals to rule out other causes of the student not showing progress (Charlton et al., 2018). Each state and district defines what they will use as their own MTSS model. The Utah State Board of Education published their own Utah’s Multi-Tiered System of Supports (UMTSS; Utah State Board of Education, 2018). The publication states in its introduction, “A school’s fundamental purpose is student learning. As such, some students may need extended time and support to learn critical skills and knowledge at high levels” (Utah State Board of Education, 2018, p. 3). The UMTSS stresses that education should be based on equity. Each student should receive an equal opportunity to achieve, no matter the student’s background is within ethnicity, religion, gender, or disabilities (Utah State Board of Education, 2018). The UMTSS specifies that all three tiers should be based on high-quality instruction. Within the UMTSS there is a team-based approach to help all students, this brings together resources that will make learning more effective for individuals and groups (Utah State Board of Education, 2018). These teams determine what is the most important items to master in different subjects, share strategies to teach, help understand what background knowledge is needed, connect standards to other TIER II INTERVENTIONS 17 standards especially those in the future, and how they will assess the skills taught (Utah State Board of Education, 2018). Once skills have been assessed, educators should be using the data to make decisions for each student. It is paramount to offering the best education for each individual student. Data are collected continuously through progress monitoring to ensure the needs are met to move students fluently between the tiers (Utah State Board of Education, 2018). Tier II Interventions and Barriers Tier II interventions. As stated above, all students receive whole group core instruction as a tier I class. Within the MTSS system, Tier II interventions have been shown to be effective in helping advance students who show non-proficient assessments within the tier I level (Bryant et al., 2008; Harlacher et al., n.d.). Interventions are the first level of supplemental support. These interventions are intended to be a dedicated time outside of the core Tier I instruction time. Effective interventions through studies have shown to have the following components; intervene early, explicit instruction, smaller groups, extended time to practice and find answers, and repetitive times throughout the week (Foorman, Herrerar, & Dombeck, 2018). Interventions are meant to be more intensive, this usually means smaller groups of students, around 5-8 students. The interventions principally focus on added occasions to practice and fully understand skills taught from the core Tier I class (Harlacher et al., n.d.). Interventions allow flexibility to group students for different demands. The groups can be fluid with students moving in and out according to the individual student’s needs. The length of time spent in the intervention can vary as well. Some schools and districts choose a daily intervention while others will hold interventions two to three times a week (Cary et al., 2017). The length of each intervention also can vary (Brown-Chidsey, Bronaugh, & McGraw, 2009). Educators must understand how to intensify instruction to students’ needs either through time every day or extended time two to TIER II INTERVENTIONS 18 three times a week. The average time is thirty minutes three times a week (Foorman et al., 2018). Early intervention is of the utmost importance because studies show that at-risk students can have considerable gains academically when instruction is given at the correct intensity and done early in the student’s educational career (Foorman et al., 2018; Harlocher et al., n.d.). Tier II interventions are taught by the general education teacher or an intervention specialist. Students in Tier II interventions should be assessed in their progress at least bimonthly, but usually weekly, to see if the students are making gains in the academic gap between themselves and their peers. This monitoring allows educators to have data-driven decisions for students. Effectively run Tier II interventions can help those who are behind who might otherwise fall further behind and start to exhibit scores comparable with other students who have learning disabilities, showing a false positive for a learning disability (Otaiba et al., 2014). Some critics of the MTSS model have noted that the system struggles to work effectively if there is a competition between areas of funding, personnel, legislative support, training, and time (Charlton et al., 2018). Critics have also cited that as educators try to implement the MTSS model, such as RTI, students have to wait to fail to get the help at a Tier III level. Some proponents suggest that students who are at the lowest overall in testing should be put immediately into the Tier III group, instead of getting further behind (Otaiba et al., 2014). Tier II barriers. Despite research showing the effectiveness of Tier II interventions, there are barriers to implementing effective Tier II Interventions (Batholomew & Jong, 2017; Stephens et al., 2012). Some of the barriers to implementing Tier II Interventions are; the learning process, student’s attitude and gender bias, perceived parental influence, lack of teaching manipulatives, time, properly trained teachers and teacher’s aides, and effective Tier II curriculum. Within the research, for this paper, it was difficult to find information for TIER II INTERVENTIONS 19 interventions in mathematics. Most intervention literature is directed at the literacy needs of the child. Interventions in reading are usually the primary concern of elementary schools, mathematics is secondary but still very important that students understand basics to build on (McLeod, Fisher, & Hoover, 2003). In discussing barriers as they pertain to mathematics, Pia (2015) points out, our world depends greatly on science and science depends on mathematics. Mathematics is what allows our practical life to exist as it does today, understanding mathematics will allow the world’s economy to flourish in the future (Pia, 2015). Mathematics is vital within our educational classrooms because of its ability to expand reasoning processes in children (Pia, 2015). A teacher must be able to overcome barriers to help facilitate this learning process in mathematics. Learning process. Learning has been defined by Banks (2000), as a process of taking in information that will change behavior or understanding. To learn mathematics efficiently, students must be able to not only follow procedures but understand the reasons for the procedures. Stuart (2000) stated that “Mathematics is like a sport: 90 percent mental – one’s mathematics confidence - - and 10 percent physical – one’s mathematics competence in performing mathematical skills” (pp.330-331). Having a negative attitude toward mathematics can cause an antipathy to learn (Davadas & Lay, 2017). Children who feel they have not been successful in the past with mathematics can develop this negative attitude and mathematical anxiety. These feelings can stop the student’s learning process and cause performance problems (Pia, 2015; Stuart, 2000). A student’s negative attitude can also be formed because of a lack of connection to how mathematics is relevant in daily life. This lack of connection encourages a lack of motivation of the student and an attitude of, why learn if it will not be needed (Pia, 2015). A teacher must combat this barrier by making mathematics interesting and personal to students. TIER II INTERVENTIONS 20 The teacher must show how what is being taught will be used to solve problems within the students’ own lives (Pia, 2015). Other common barriers for teaching mathematics through the learning process are; the different paces at which students learn, the differences of current mastery levels of the students in a class, and the number of students within a classroom (Pia, 2015). Teachers must be able to plan for the class in general but also serve the individual learning needs of each student’s learning process (Pia, 2015). Student attitude and gender bias. There still exists a barrier in gender bias within the mathematics learning process, which often affects girls’ attitudes toward mathematics (Levin, 2019; Pia, 2015). There have been no differences found in cognitive abilities between genders (Levin, 2019). Stereotypical social outlooks and beliefs make it more difficult for women to seek mathematics for careers (Levin, 2019). Often mathematics is looked at as having an inborn talent to be successful, this discourages women more than men (Levin, 2019). The student’s attitude toward mathematics and their confidence in learning mathematics determine their achievement level more than other factors (Levin, 2019). In younger years, such as middle school or younger, the stigma seems to affect female students less, revealing that 45% of Olympiad teams, a group of students who compete in mathematics contests, are females. As they age the numbers who participate in Olympiad teams drop to 7% in high schools (Levin, 2019). Studies have also found that textbooks in mathematics classes reinforce gender patterns, which encourages boys to participate more (Levin, 2019). When it comes to teachers showing a bias toward gender, male teachers are more likely to favor boys while female teachers do not exhibit bias (Levin, 2019). Teachers must combat this through examples of breaking down barriers of the stigma that mathematics is a “difficult subject” that only boys can tackle (Pia, 2015). TIER II INTERVENTIONS 21 Perceived parental influence. Perceived parental influence can be a barrier to teaching mathematics as well (Davadas & Lay, 2017). Parents who do not help in direct or indirect ways will create an impact on students. Parents who help directly with homework or test reviews at home allow their students to feel and understand the importance of learning mathematics. Those parents who do not help are exemplifying a sense of unimportance in the learning of mathematics (Davadas & Lay, 2017). Parents who show involvement, or lack of involvement, in indirect ways by encouraging, or discouraging, will leave an impact on the student’s attitude. Parents verbalizing their expectations and attitudes toward mathematics will impact the student’s attitude toward mathematics (Davadas & Lay, 2017). If parents show a strong dislike or anxiety towards mathematics their child will also tend to exhibit these same feelings (Pia, 2015). Lack of teaching manipulatives. Lack of access to manipulatives, as teaching aides, can be a barrier to teaching mathematics. Studies have shown that those students who use manipulatives usually outperform those who do not, also students who used manipulatives showed increase retention and problem-solving during a test (Sarama & Clements, 2009). Mathematics can seem very abstract to students who are just listening to mathematics being explained. Students learn more by using their senses of touch, sight, smell, and investigating more than just listening (Pia, 2015). Using math manipulatives and teaching aides make learning pertinent (Pia, 2015). Good manipulatives support in assisting students in constructing, enlarging, and bridging various symbols of mathematical ideas (Sarama & Clements, 2009). Manipulatives do not always need to be something a student holds in their hands. Computers can bring images that are just as individually purposeful to students as a physical object (Sarama & Clements, 2009). Computer manipulatives can also be quicker and easier to manipulate (Sarama & Clements, 2009). With either sort of manipulative, the students are more overall engaged TIER II INTERVENTIONS 22 within each mathematics activity when paired with well-planned instruction (Pia, 2015; Sarama & Clements, 2009). Time. The limited amount of time teachers have to teach can be restricting to implementing Tier II interventions. Teachers are required to teach, depending on the state and district, a certain amount of allocated time for each of the basic subjects (Cotton, 1989; Marzano, 2000). The largest amount of time in elementary schools is dedicated to reading and language arts, the second-largest amount in mathematics (McLeod et al., 2003; Utah State Board of Education, 2019). Core curriculum standards require a certain amount of content coverage of all subjects in each grade. Often teachers find it difficult to find the time to be able to teach the required core in Tier I, general education class (Bolden & Newton, 2009; Wilson, 2015). As teachers are trying to cover all this content, finding extra time for Tier II interventions can be restricted. Tier II interventions, as discussed earlier are recommended to be twenty to thirty minutes in length most days of the week. With the amount of curriculum to be taught in each subject allocating extra time for interventions will need to be taken either from the subject itself or from another subject (Bolden & Newton, 2009). Finding time for interventions can be frustrating to teachers who often already feel they cannot do an adequate job of teaching the curriculum within the allotted time of Tier I (Wilson, 2015). Other time restrictions can cause barriers for implementing Tier II interventions as well, such as preparation time. Preparation time, or planning time, is essential to having effective, successful learning (Jarmolowski, 2017). There is a saying often quoted by Benjamin Franklin “By failing to prepare, you are preparing to fail.” Quality lessons in education have to be painstakingly thought out and written. Some interviewed elementary school teachers stated that the amount of time they are given for planning is not near enough time (Kominiak, 2016). An TIER II INTERVENTIONS 23 average workday for an elementary teacher is seven and a half hours (Nittler, 2016). The average amount of planning time is forty-five minutes a day (Nittler, 2016). Most elementary teachers are teaching four or five subjects each day. For each of those subjects, they have to prepare a lesson. Those lessons take a lot of time to create, especially for new teachers. Even for schools that have a scripted curriculum, it takes time to rehearse and practice the lessons to understand what is to be taught. If a teacher is creating their lesson plans it often takes longer to develop and write the lesson than to administer the lesson (Avery, Beach, & Coler, 2003). To have effective Tier II interventions it should be properly planned. Since teachers are already struggling to have enough time to plan the Tier I lessons, Tier II interventions might not be properly prepared for. Properly trained teachers and teacher aides. Another barrier to teaching mathematics is instructor related. Some teachers who teach mathematics do not have a strong background in mathematics (Pia, 2015). Teachers from other backgrounds are required to teach mathematics because of a shortage of teachers (Pia, 2015). These teachers can lack the motivation, self-confidence, and may have a negative attitude about mathematics themselves (Bolden & Newton, 2009; Pia, 2015). Teacher affective support during math class time is necessary to foster positive attitudes in the students (Davadas & Lay, 2017). If a teacher shows respect, interest, fair treatment, caring, encouragement, and high expectations towards students, then the students will have higher motivation and a positive attitude toward mathematics (Davadas & Lay, 2017). Teachers need to explore the best possible ways to introduce mathematical skills. Today’s students are exposed to technology at a very young age so traditional methods of teacher lecture may not be able to grab and hold the student’s attention (Davadas & Lay, 2017; Pia, 2015). Untrained, or improperly trained, teachers or teacher’s aides, can also be a barrier to implementing quality Tier II interventions. Improved professional development leads to TIER II INTERVENTIONS 24 improved learning (Kassissieh & Barton, 2009). There should be tremendous levels of in-depth and continuous professional development for teachers and teacher’s aides (Fuchs & Vaughn, 2012). With the shortage of teachers, often those who are teaching may not be fully qualified to teach. Knowing an education subject and knowing how to teach it to others is very different. Often teacher’s aides are not educated on how to teach at all. They are individuals just looking for temporary work. They show up each day to fulfill the time required to assist without knowing any other strategies of how to help someone else learn a certain subject or skill other than what they learned while attending grade school (Gerber, Finn, Achilles & Boyd-Zaharias, 2001). To make Tier II interventions truly effective, all teachers and teacher’s aides should have professional development (Newcomer; Freeman & Barrett, 2013). Without the training, the teacher’s aides will have little to no positive effect on the academic performance of the students (Gerber et al., 2001). Effective Tier II curriculum. Curriculum is described as a path or course of study (Dewinter & Rumbley, 2010). With time being a limited commodity within Tier I teaching time and preparation time, teachers certainly have very little time to find or create the lessons, or curriculum, for Tier II interventions. Even with evidence-based, prewritten curriculum lessons it takes time for teachers to make the lessons work, or diversify the lessons, for their students (Gresham & Little, 2012; Teachers need better resources to plan great lessons, 2020). It would seem almost manageable if every student who needed intervention time also needed the same skill but that rarely happens (Gresham & Little, 2012). Usually, there will be multiple skills that need to be covered and lessons needed to be planned for each student. Even if intervention students can be grouped with several students who need similar skills, the individual groups will still require different lessons for each group (Gresham & Little, 2012). The curriculum needs to TIER II INTERVENTIONS 25 be designed in a way that eases learning challenges and could be easily implemented by a paraprofessional or volunteer (Fuchs, n.d; Kelley & Goldstein, 2015). All of this planning and designing of curriculum lessons is left to be planned by the general education teacher (“Teachers need better resources,” 2020). The general education teacher tries to ensure that all students have enough, different, instructional strategies to be demonstrated through high-quality instruction (Gresham & Little, 2012). With prewritten curriculum lessons, the cost can be more than the school district or teacher can afford to purchase (Berger, 2018). This again leaves the teacher to develop the curriculum lessons for interventions for themselves and any teaching-aides. This can be a huge consumption of time when they have to be planned and adjusted daily throughout the week. For Tier II interventions to be effective barriers need to be overcome. Barriers such as; the learning process, student’s attitude, gender bias, perceived parental influence, lack of teaching manipulatives, time restraints, properly trained teachers and teacher aides, and effective Tier II curriculum. Purpose General education teachers find it frustrating to teach within classes where there is a large gap in academic needs within a classroom. Research has shown that neither grade retention nor social promotion is adequate to progress students to grade-level proficiency (Jimerson & Renshaw, 2012). In an attempt to find a better way to progress students toward proficiency, many schools have implemented a Multi-tiered System of Support (MTSS). The MTSS supports those students who are at risk of grade retention or social promotion (Brown-Chidsey & Steege, 2010). Research has shown that, within the MTSS, Tier II interventions are an effective support to help advance non-proficient students (Jimerson et al., 2006). Despite research showing the TIER II INTERVENTIONS 26 effectiveness of Tier II interventions, there are barriers for teachers to implement effective Tier II interventions (Batholomew & Jong, 2017). One particular barrier is lack of time to properly train teacher’s aides or math aides who come in to help administer Tier II interventions. As such, the general education teacher has to take time to plan curriculum lessons for each of the Tier II intervention groups and quickly try to describe to the teacher’s aide or math aides what is to be done. The objective of this project was to create math curriculum packets for math aides. These curriculum packets could be handed to math aides so they could follow from one math skill or concept to the next math skill or concept within a math unit. These packets included instructions, manipulatives, practice equations, and assessments for the math aide to be able to carry on a twenty to thirty-minute Tier II intervention without a lot of training or instruction from the teacher. The primary objectives for the packet are to 1) have an effective prewritten Tier II curriculum lesson ready for a math aide to start teaching without training, 2) have progressive Tier II curriculum lessons which lead to understanding an entire mathematics unit, 3) have all materials needed for; instructions, manipulatives (to make concepts more concrete), practice equations, and assessments (to check if the students understand the concept or skill taught), available within a packet. Method Subject Effective and easy to implement curriculum was needed for educators to implement Tier II interventions. Utah State Core Standards for Mathematics and Box Elder Learning Standards were used to develop lesson plans that could be incorporated within a fifth grade, Tier II, TIER II INTERVENTIONS 27 mathematics interventions (Utah State Board of Education, 2016; Box Elder School District, n.d.). The targeted standards were from the Utah State Core Standard in Mathematics (2016), fifth grade, Standard one: “Students will apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They will develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them” (p. 37) and the Box Elder Learning Standard (n.d), fifth grade, Mathematics Target one: “Add and Subtract fractions and mixed numbers with unlike denominators and solve real-world problems” (p. 1). Context The literature review above stated that it is important to have properly trained paraprofessionals and curricular aides (see literature review). The need for this curriculum project was based on the context of the school district it was created for. Within the targeted school district, mathematics teacher aides were hired based on the need to limit the size of Tier II interventions to five to six students in a group. There were no previous qualifications required to be hired as a mathematics aide. The economics of the school district did not allow for the quality training of the mathematics aides. Many of the mathematics aides had not received any training before the first day of serving as a Tier II instructor. The mathematics aides were hired to instruct thirty-minute Tier II intervention lessons, four days a week, kindergarten through fifth grade. The scripted lesson plans would help provide consistency and quality of instruction. There were no previous curricular lessons to help instruct the mathematics aides in the targeted school district. TIER II INTERVENTIONS 28 Educator Evaluators Intervention lesson plans were evaluated by five educators. The evaluative educators were chosen as follows; Educator one was an instructional paraprofessional mathematics aide for one year. She was chosen based on the desire to have a novice mathematics aide to evaluate the ease of use of the scripted curriculum. Educator two was a fifth-grade general education teacher having taught three years within the fifth grade. She was chosen as an educator who served most of her time teaching language arts subjects but also supported mathematics instruction during intervention time. This would allow the scripted lessons to be viewed from someone who did not teach mathematics all the time. Educator three was an experienced teacher of twelve years, all within the first grade, and two years as an instructional coach. She was selected having been heavily involved in Tier II interventions with scripted lessons and evaluations in reading within grades from kindergarten to fifth grade. She also helped to train and organize instructional aides in reading and mathematics. Educator four was an elementary general education teacher for twenty years. She had taught within; resource, behavior units, as well as general education class in grades Kindergarten, second, third, and fifth grades. She specialized in mathematics for seven years in fifth grade. She became an instructional coach working with intervention specialists and aides for the past two years. Educator five was an elementary general education teacher with eighteen years of experience. She had taught in second, third, and fifth grades. She had English Language Learner and mathematics endorsements. She also had specialized in Dual Language Instruction that gives narrowed time to mathematics instruction and Tier II mathematics interventions. She had expertise in being able to condense lessons down into confined time limits to be the most effective. The educators were asked to look for specific items to be critiqued (see TIER II INTERVENTIONS 29 appendix A). The intervention curriculum was revised based on the feedback received (see Evaluators Findings section). Procedure Standards The curriculum project was based on the Utah State Board of Education’s (2016) Core Curriculum standards 5.NF.1, “Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.” and 5.NF.2., “Solve real-world problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators” (p. 40) and the Box Elder Learning Standards (B.E.L.S.) number 1, “Add and Subtract Fractions and mixed numbers with unlike denominators and solve real-world problems.” (Box Elder School District, n.d. p. 1). The adding and subtracting fractions standard was chosen for the curriculum project because fifth grade is the first public educational exposure to this mathematics skill. Students who obtain a strong understanding of fraction mathematics skills and principles will have a strong foundation for future growth. Backwards Planning Backwards planning was chosen to examine first the end goal of what students should be able to do to be considered proficient at the end of the unit (Isecke, 2011). The end goal was for students to be able to fluently add or subtract mixed numbers that had unlike denominators and would need to be regrouped. Backwards planning allowed for skills to be accurately broken down and decided if a particular skill should be taught before or after the succeeding lesson (Isecke, 2011). TIER II INTERVENTIONS 30 The individual skills were built progressively into the succeeding lesson. Each lesson focused on just one skill. The lesson plans started with basic fraction knowledge. Even though basic fraction knowledge was taught within previous grade levels, this knowledge of understanding of fractions has sometimes been a missed starting point in fifth grade due to educators assuming it has been thoroughly covered in earlier grades. Lesson plans were then ordered; basic combining of unit fractions, adding non-unit fractions, comparing fractions, subtracting unit and non-unit fractions, understanding and creating equivalent fractions, adding and subtracting fractions with unlike or uncommon denominators, improper fractions, mixed numbers, and regrouping principles in adding and subtracting mixed numbers. Manipulatives Often misunderstandings in mathematics concepts are due to high levels of abstract thinking or the inability to make a concrete connection (Pia, 2015). To truly develop concreteness of math concepts manipulatives needed to be added to the curriculum (Pia, 2015). The lesson plans were created to provide concrete, engaging, objective connections. The inclusion of modeling clay created tangible, concrete, engaging, and objectifying manipulative. Modeling clay gives students the ability to make many different shapes and equations representing fractions. This would allow students to build fractions instead of just seeing pictures or visualizing abstract ideas. Although using manipulatives tend to take up time through having to teach how to use them effectively, modeling equations or concepts, and sometimes distracting, they are an absolute necessity for many students to fully understand and grasp concepts (Sarama & Clements, 2009). The manipulatives may take longer to model a practice problem but the understanding will help translate to other future equations without using manipulatives. TIER II INTERVENTIONS 31 Manipulatives build fluency in future mathematical problem-solving (Sarama & Clements, 2009). Gradual Release Model The lesson plans were created as a re-teach intervention. The students had been previously taught through a Tier I Inquiry method. The gradual release model of “I do, We do, You do” fits well with Tier II explicit instruction, as stated in the literature review (Fisher & Frey, 2014). The “I do, We do, You do” teaching model gives scaffolding support that students need to be successful in the short amount of time allotted for Tier II interventions. The teaching model also fit the Tier II small group set up of five to six students to instructor ratio (Fisher & Frey, 2014). There was substantial verbal discussion to model the thinking process out loud for mathematics concepts within the lesson plans. Scripted Lessons to Support Untrained Mathematics Paraprofessionals Teaching paraprofessionals or teaching aides often are hired without teaching degrees or teaching experience. Educational funding, or lack of funding, does not give enough paid time for proper training of educational paraprofessionals (Kassissieh & Barton, 2009). Every year mathematics paraprofessionals arrive the first day to assist in teaching without having any idea how to instruct a certain mathematics concept. Neither the paraprofessional nor teacher are given time to meet, go over what will be taught, nor how the mathematics concept should be taught (Jarmolowski, 2017). Left to their understanding most paraprofessionals will teach the way they were taught, which is usually outdated or ineffective (Gerber, Finn, Achilles & Boyd-Zaharias, 2001). Scripted lessons were decided to keep lessons taught in an updated and corresponding format with Tier I instruction. The lesson plans needed to be easy to follow for someone teaching TIER II INTERVENTIONS 32 the concept for the first time and could be easily reproduced within a school budget. Using color to differentiate between script and instruction was not feasible. To distinguish between script and instruction, boldface type was designed for the script and italicized format for instructions. Assessment Adding an assessment element within each lesson plan was an important component to access if the lesson plan was being effective. A quiz to assess was included as the final portion of the lesson plans. Evaluator Findings Once the lesson plans were written, the evaluator’s questionnaire was designed. There were four questions for each lesson plan (see appendix A). The questionnaire asked for evaluations of accuracy of the information, ease in which lessons could be followed, helpfulness for student understanding, and suggestions for improvement. Each evaluator was assigned four lesson plans to evaluate. Three lesson plans assigned were different for each individual for overall analysis. One lesson plan, number eight, was the same for control to check for inter-rater reliability with the same lesson plan. Each evaluator filled out the questionnaire for each lesson plan. Changes were made based on their feedback. The evaluator’s questionnaire helped to create a credible curriculum and changes to enhance the curriculum. Accuracy The first question served to create the accuracy of the information within the lesson. There was very little feedback that needed to be changed in the accuracy of the curriculum. One equation had the wrong answer within one of the lesson plans. Most other accuracy was about grammar corrections. Changes were made accordingly. TIER II INTERVENTIONS 33 Scripted Lesson Plan Most evaluators stated that the lessons could be followed easily with the bold script and italicized instructions. Changes in spacing between paragraphs to give natural breaks between script and instruction were added. One evaluator thought it would be hard to teach any scripted lesson, stating it would be better to break the lesson into three different parts of rules/steps, procedures, and problems to solve. Further clarification led to a discussion of different formats and a concession to stick with the script because of the lack of time an untrained paraprofessional, as mentioned in the method context section, will have to read through rules/steps or procedures while students wait. An instruction-based curriculum would mimic a group sitting down to play a game and everyone watching one person read the rules to a game. The instructor then trying to explain the rules while the students started to try and play the first round. This would take up too much of the instructional Tier II intervention time. With the limited time in Tier II interventions and lack of pre-training, a script was thought best to ensure proper instruction. The added suggestion that the front cover should give a clear description or example of what was to be taught within the lesson plan. Changes to the front page of each lesson plan were made to show an example of what the lesson plan is teaching, to give the instructor a foresight into the lesson from the beginning. Page numbers were added to the lesson plans in case pages became unorganized. More explicit written examples of the mathematics equations were written in as well to help with the flow of writing example to the students for the paraprofessionals. TIER II INTERVENTIONS 34 Supporting Understanding The next survey question asked if the lesson plan would help support the understanding of the students during interventions. All evaluators agreed that it would support understanding and gave insights into which parts were particularly strong. Some evaluators also gave suggestions on making the actual equations being used easier or different. Some of the equations were changed to the evaluators’ suggestions. All evaluators felt the lessons would be beneficial. Specifically, the paraprofessional acknowledged how helpful this would be for new mathematics paraprofessionals coming in each year. Extra Practice One instructional coach suggested giving enough practice equations to extend the lessons as long as needed for repeated exposure to concepts. Several practice problems were added to each lesson plan. Assessment The instructional coaches also wanted clarification if the quiz would be given to all students regardless if they seemed to understand the content. This prompted changes to the instructions for the quiz to be given when the students show an understanding of the concept. Also, clarification with the instructional coach that the students would stay at the same level until mastery was met before progressing because the lesson plans were based on progressive skills. Limitations Several limitations are noted within this curriculum design. Within the questionnaire, the first question asks for accuracy to be assessed by the educator evaluators. Educator number one is a paraprofessional mathematics aide. The ability for this evaluator to assess accuracy may not be fully equitable, as she has not studied the math concepts in depth. Educator Evaluator number TIER II INTERVENTIONS 35 three has never taught fifth-grade mathematics as well. This evaluator may not fully be able to access the quality of the curriculum. Another limitation is that evaluator number two does not consistently instruct in mathematics. For this evaluator to conclude the effectiveness of supporting students learning may be limited. The overall small sample of evaluators was within the same district and state could also be considered a limitation of the evaluation of the curriculum. The non-diversity of all-female evaluators is a limitation as well. The most considerable limitation is that the curriculum has not been used or proved within an actual setting. Recommendations The strong recommendation within this curriculum project will be to use the curriculum to find its effectiveness (see appendix B). Another recommendation would be setting up control groups to see if this curriculum is any more or less effective than other Tier II mathematics fraction intervention curriculum. Conclusion Tier II mathematics intervention curriculum was created to help support students who struggle to gain proficiency within Tier I mathematics instruction. Tier II interventions, within the MTSS instructional model, is a supportive prevention strategy for students who are non-proficient within their current grade level. Effective Tier II interventions may help to alleviate the ineffectual decision to socially promote or grade retain students. 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Essential systems for sustainable implementation of Tier 2 supports. Journal of Applied School Psychology 29(2), 126-147. DOI: 10.1080/15377903.2013.778770. Nittler, K. (2016). How much time do teachers get to plan and collaborate? National Council on Teacher Quality. Retrieved from https://www.nctq.org/blog/July-2016:-How-much-time-do- teachers-get-to-plan-and-collaborate TIER II INTERVENTIONS 42 Otaiba, S. A., Connor, C. M., Folsom, J.S., Wanzek, J., Greulich, L., Schatschneider, C., & Wagner, R.K. (2014). To wait in Tier I or intervene immediately: a randomized experiment examining first-grade response to intervention in reading. Exceptional Children 81(1), 11-27. doi:10.1177/0014402914532234. Pia, K. F. (2015). Barriers in teaching learning process of mathematics at secondary level: a quest for quality improvement. American Journal of Education Research 3(7). 822-831. doi:10.12691/education-3-7-5. Picklo, D. & Christenson, S. (2005). Alternatives to retention and social promotion; the availability of instructional options. Remedial and Special Education 26(5), 258-268. Rosen, P (n.d.). MTSS: What you need to know. Educational Strategies. Retrieved from https://www.understood.org/en/learning-thinking-differences/treatments-approaches/ educational-strategies/mtss-what-you-need-to-know Samara, J. & Clements D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives 3(3), 145-150. https://doi.org/10.1111/j.1750-8606.2009.00095.x Samuels, C. A. (2016). What are multitierd systems of supports? Education Week 365 (15), 58- 59. Social Promotion. (2014, August 14). The Glossary of Education Reform. Retrieved from https://www.edglossary.org/social-promotion/ Social Promotion. (n.d.) In Merriam-Webster’s online dictionary (11th ed). Retrieved from https://www.merriam-webster.com/dictionary/social%20promotion Solari, E. J., Denton, C. A., Petscher, Y. & Haring C. (2017). Examining the effects and feasibility of a teacher-implemented tier 1 and tier 2 intervention in word reading, fluency and comprehension. Journal of Research on Educational Effectiveness, 1-29. DOI: 10.1080/19345747.2017.1375582 TIER II INTERVENTIONS 43 Sparks, S. D. (2015) First graders who were identified for more help fell further behind. Education Week 35(12), 1-12. Retrieved from https://www.edweek.org/ew/articles/2015/11/11/study-rti-practice-falls-short-of-promise. html Stephens, D., Cox, R., Downs, A., Goforth, J., Jaeger, L., Matheny, A., Plyler, K., Ray, S., Riser, L., Sawyer, B., Thompson, T., Vickio, K., & Wilcox, C. (2012). I know there ain’t no pigs with wigs: Challenges to tier 2 intervention. The Reading Teacher, 66(2), 93-103. Retrieved from www.jstor.org/stable/23322716 Stuart, V. B. (2000). Math curse or math anxiety? Teaching Children Mathematics, 6, 330-336. Retrieved March 29, 2020 from http://driscollmathathome.weebly.com/uploads/1/9/8/5/19850427/tcm2000-01-330a.pdf Teachers need better resources to plan great lessons. (2020, February 24). Sydney Morning Herald [Sydney, Australia], p.14. Retrieved from https://go-gale-com. hal.weber.edu/ps/i.do?p=STND&u=ogde72764&v=2.1&it=r&id=GALE%7CA6148 13720&inPS=true&linkSource=interlink&sid=STND Utah State Board of Education. (2019). Utah’s K-12 Literacy Framework. Retrieved from https://www.uen.org/k12educator/downloads/LitFramework.pdf Utah State Board of Education. (2018). Utah’s multi-tiered system of supports for mathematics. Retrieved from https://www.schools.utah.gov/file/ff0a8f0c-66ce-486d-b51e- 0509ff33f533 Utah State Board of Education. (2016). Utah core state standards for mathematics: Elementary levels (K-5) Retrieved from https://www.schools.utah.gov/file/8686fa6b-4ded-4a33-92a5- e444a5029f48 TIER II INTERVENTIONS 44 Utley, C. A. & Obiakor, F. E. (2015). Special issue: research perspectives on multi-tiered system of support. Learning Disabilities: A Contemporary Journal, 13(1) 1-2. Wilson, R. (2015, February 27). Education researcher says we’re teaching too much to be good at anything. School News Network. Retrieved February 27, 2020 from https://www.schoolnewsnetwork.org/2015/02/27/education-researcher-robert-marzano-says- we-re-teaching-too-much-be-good-anything/ TIER II INTERVENTIONS 45 Appendix A Math Aide Lesson Plan Survey Name: Educator #1 Lesson plan #:_1______________________________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Everything looks accurate. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? The format was very easy to follow, especially the way the scripted words are emphasized in bold. It is always nice to have extra info that can save time, like the reminder about what numbers are easy to divide shapes into. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? This lesson plan would be very valuable to help the student visualize the basics of fractions and what they mean when they say “5/5 = 1 whole”. Maybe the only thing I would change would be possibly taking a minute to emphasize and explain why the pieces should be equal sizes. 4. Any other suggestions for improvement? TIER II INTERVENTIONS 46 Math Aide Lesson Plan Survey Name: Educator #1 Lesson plan #:__2____________________________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Everything looks accurate. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? The term “unit fraction” was a term I was not familiar with (or may have forgotten) until I started working in math classes. So it is nice that it is clearly defined here for those aides or subs that may not know or remember that particular term. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Most students will grasp the concept quickly with this lesson as written. It may be helpful for some students to explain what is NOT a unit fraction. So, for example, take two pieces of the whole, write the fraction 2/6, and ask if this is a unit fraction and why or why not. 4. Any other suggestions for improvement? A triangle may be a little hard for the helper or students to make equal parts out of (unless there is a trick that I don’t know). I have had some students spend an unnecessary amount of time on details like this. TIER II INTERVENTIONS 47 Math Aide Lesson Plan Survey Name: Educator #1 Lesson plan #:_3_____________________________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Everything looks accurate. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? Instructions are clear and written in a way that explains the basic principle on a students’ level. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes, this is a good and simple visual example to show how adding fractions only changes the numerator, but the denominator will remain the same. This is a concept that some students struggle with at first. 4. Any other suggestions for improvement? These are just very small points, but on the final section of each lesson (quiz section), most of the lessons say to have the student hand in the paper, one just says to sit quietly (#1), and another ends saying to collect them (#8). Might be helpful to have a standard wording on each one. Also, some supply lists say “whiteboard markers” and others just say “markers”. For a sub or new aide it might be helpful to note “whiteboard markers” every time. TIER II INTERVENTIONS 48 Math Aide Lesson Plan Survey Name: Educator #1 Lesson plan #:_8_____________________________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Everything looks accurate. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? This lesson is a little more complicated to explain in written words alone. Although I understood the concept well, it may be helpful to a sub or a new aide to draw a simple picture with the fractions lined up and show where you would write the multipliers, so that they are lined up similar to how their teacher might write equivalent fractions. Maybe even draw a simple picture of the clay model as well. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes, this is a good and simple visual example to explain the concept of equivalent fractions. I have noticed that some kids are scared away by the larger terms like “equivalent”, so using the clay to model that they are simply equal sized fractions will be very helpful. 4. Any other suggestions for improvement? TIER II INTERVENTIONS 49 Math Aide Lesson Plan Survey Name: Educator #2 Lesson plan #: 7 Equivalent fractions____________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Yes, everything looks accurate. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? I think it was simple and easy to follow. I like that you gave examples of fractions that could be used to practice, so that the aides didn’t have to come up with the numbers. Visually, I like how the script is bold and the actions for the aide to do are italicized. That makes it easy for aides’ eyes to find their place. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? I think using the playdoh and having something tangible that the students are working with and creating fractions with is fun and will help students understand and remember the experience and thus help them understand and remember equivalent fractions. I like that they also write the numbers down on their whiteboards. All of this will help support their understanding and retention of equivalent fractions. 4. Any other suggestions for improvement? Typo-2nd page, 2nd paragraph “an = sign” instead of “and = sign.” TIER II INTERVENTIONS 50 Math Aide Lesson Plan Survey Name: Educator #2 Lesson plan #:8 Creating equivalent fractions unsimplified____________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Yes, everything is accurate. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? Yes, everything was very easy to follow. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes, I think this intervention lesson will support student’s understanding. My only thought was that ⅓ and 3/9 might be harder to cut into a circle than maybe ½ and 2/4. Or any fraction with 2, 4, 8 in the denominator would be easier for students to cut. Then they could practice ⅓ and 3/9 on the whiteboard. 4. Any other suggestions for improvement? At this point, would you want to tell students that the multiplier 3/3 is equal to 1? And that 1 multiplied by any number keeps its value. Or maybe that’s something that they’ve already learned. TIER II INTERVENTIONS 51 Math Aide Lesson Plan Survey Name: Educator #2 Lesson plan #:9 Create Equivalent Fractions through simplifying 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Yes, everything is accurate. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? I had to read this section a few times because it seems like it’s saying the same thing. “We can divide 6 by 3 and we get 2. We divide 9 by 3 and we get 3. So our new equivalent simplified fraction is ⅔. Looking at our fractions 6 divided by 3 is 2 for the numerator and 9 divided by 3 equals 3 in the denominator. So our end equivalent fraction is ⅔.” Everything else was easy to follow. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes, I love the visuals and tangibles of working with the playdoh. I also love how that is supplemented and reinforced with writing on the whiteboards. 4. Any other suggestions for improvement? I think it would be super fun, inexpensive, and memorable to give the students an actual piece of gum to practice the re-teach example. Grammar from 1st paragraph under “8 minute practice together.”— “If we listed numbers that can be multiplied to equal 6 or factors it (use “they” instead, since “numbers” is plural) would be.” TIER II INTERVENTIONS 52 Math Aide Lesson Plan Survey Name: Educator #2 Lesson plan #: 10 Comparing unlike fractions_______________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Correction on 3rd page 2nd paragraph. It says: “We can now easily compare 3/6 to 2/6. 3/6 is less than (<) 2/6.” 3/6 is greater than 2/6. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? Yes, it was easy to follow. I like that the answers were also included in the “7 minute practice section.” 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes, I think it would support student understanding. Would the T chart idea be helpful when the denominators are not multiples of each other? 4. Any other suggestions for improvement? TIER II INTERVENTIONS 53 Math Aide Lesson Plan Survey Name: Educator #3 Lesson plan #4: Adding non unit fractions______________________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Yes 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? It was easy for me to follow- a teacher’s aide will easily be able to follow it. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes 4. Any other suggestions for improvement? No, I think your lesson plans look great. I do have a couple of other questions/ thoughts. I am assuming this lesson plan is just for 1 day of interventions. Will you give each student the assessment at the end, even if they are showing that they still do not understand? If they do not understand (based off of the quiz data), will they just move on to the next lesson or will additional lesson plans be provided on this particular math skill? TIER II INTERVENTIONS 54 Math Aide Lesson Plan Survey Name: Educator #3 Lesson plan #:_5_____________________________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Yes 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? Yes 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes 4. Any other suggestions for improvement? It looks great. Easy to follow. TIER II INTERVENTIONS 55 Math Aide Lesson Plan Survey Name: Educator #3 Lesson plan #:__6_____________________________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? All information is correct. I did notice one small typo. The 3rd paragraph down under the 5 minute reteach- an “r” needs to be added to the word partner. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? Yes. The bold and italicized words made it easy to follow. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes. I love how each step is broken down to really help set up the students for success. 4. Any other suggestions for improvement? No TIER II INTERVENTIONS 56 Math Aide Lesson Plan Survey Name: Educator #3 Lesson plan #:_8_______________________________________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? Yes 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? Yes 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? Yes, the hands on materials and the I do, we do, you do model set up the students for success. 4. Any other suggestions for improvement? no TIER II INTERVENTIONS 57 Math Aide Lesson Plan Survey Name: Evaluator #4 Lesson plan #:______________8 - 14____________________________ 1. Is the information provided accurate? If not, what needs to be changed to make it accurate? I thought the lesson plan was very accurate and easily presented. 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? The scripting was good, with easy to use wording and instructions. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? I think it is a great idea for interventions. My only suggestion would be to use this lesson plan for multiple days. Having the students here the same wording and practicing the same skill for several days would be helpful so making sure that there are enough samples and practice problems to last for more than one day. This would also help so much in the prep for the para with a week or so worth of problems to work through with the kids. 4. Any other suggestions for improvement? TIER II INTERVENTIONS 58 Math Aide Lesson Plan Survey Name: Educator #5 Lesson plan #8 1. Is the information provided accurate? Yes. It’s very clear and concise. If not, what needs to be changed to make it accurate? 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? To me, the script feels a bit overwhelming. I found it difficult to imagine reading aloud from the script while teaching a small group of students, especially while modeling manipulatives and using a whiteboard. My suggestion is to take all the great information you have and separate the lesson plan into three parts:  rules/steps  procedures (highlighted script)  problems to solve The aide can independently read and learn the math procedures, review the rules/steps with the students (and have the copy as a reference guide for all of them), and use his/her own words to teach the lesson. 3. Will this intervention help support understanding for a student during interventions? Yes. I think the review and quiz are especially strong parts of this lesson. If not, what will be a suggestion that would help? 4. Any other suggestions for improvement? Because it’s an intervention group, I suggest you review basic vocabulary at the beginning of the lesson: numerator, denominator, whole fraction, and equivalent fractions TIER II INTERVENTIONS 59 Math Aide Lesson Plan Survey Name: Educator #5 Lesson plan #11 1. Is the information provided accurate? Yes If not, what needs to be changed to make it accurate? 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? See comments in lesson plan #8. 3. Will this intervention help support understanding for a student during interventions? Yes If not, what will be a suggestion that would help? Instead of using a whiteboard, I suggest the students also model with clay the first example (like fractions). I think it would be clearer for the students to compare the similarities and differences between like and unlike fractions if their models are the same--clay and clay (manipulative and manipulative) rather than whiteboard and clay (numbers and manipulative). 4. Any other suggestions for improvement? I suggest that upfront you clearly review or explain the denominator rules: Look at the denominators. 1 – Are they the same? Begin your math operation. 2 – Are they different, but one is a multiple of the other? Multiply the smaller denominator by the number of times it can be multiplied to equal the larger denominator. Now begin your math operation. 3 – If 1 and 2 are not true, then multiply the two denominators by each other. (Because this is an intervention group, I think teaching them to find the lowest common denominator is a big diversion from the initial concept being taught.) TIER II INTERVENTIONS 60 Math Aide Lesson Plan Survey Name: Educator #5 Lesson plan #12 1. Is the information provided accurate? Yes. It is very explicit. If not, what needs to be changed to make it accurate? 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? See comments in lesson plan #8. 3. Will this intervention help support understanding for a student during interventions? Yes. The introduction and reteach is extremely clear. I like that you have outlined the specific math steps. If not, what will be a suggestion that would help? 4. Any other suggestions for improvement? TIER II INTERVENTIONS 61 Math Aide Lesson Plan Survey Name: Educator #5 Lesson plan #13 1. Is the information provided accurate? Yes If not, what needs to be changed to make it accurate? 2. Is the lesson plan easy to follow for a teacher’s aide? If not, what could improve the lesson plan? See comments in lesson plan #8. Because this is an intervention lesson, I would move on to the step of dividing as the method to change an improper fraction to a mixed number. I think it’s a tangent to go into a mini lesson on the four different ways to write the division sign, what they mean, where to find them, etc. I suggest using a different manipulative than clay because it’s time consuming. Perhaps tile blocks, paper strips, or connecting color cubes would be effective. 3. Will this intervention help support understanding for a student during interventions? If not, what will be a suggestion that would help? I suggest you explicitly review vocabulary, perhaps giving the students examples for them to sort or identify: proper fractions, improper fractions, and mixed numbers. (I realize this is a separate lesson, but the skill needs to be mastered before moving ahead.) 4. Any other suggestions for improvement? Another strategy to change an improper fraction (5/3) to a mixed number is to take out the whole fraction/s (3/3) and then what is left (2/3) is the proper fraction: 5/3 = 3/3 + 2/3 = 1 2/3 TIER II INTERVENTIONS 62 Appendix B #1 What is a fraction? What does the numerator and denominator represent? TIER II INTERVENTIONS 63 #1 What is a fraction? Purpose: Students will understand that a fraction is part of a whole. Materials needed:  Modeling clay  whiteboards  Quiz Paper (4 x 6)  stick or items to cut clay  Dry erase markers  Pencils Instructions: Words in Bold are a script that can be read. Re-Teach Instructor: The purpose of today is to realize that fractions are part of a whole. Write “Fractions are part of a whole” on your whiteboard. Get out a piece of clay for yourself. I have 1 whole clay. I will mold it into a pizza shape. I will cut it into 2 pieces. These pieces are each part of my 1 whole clay. I can write this fraction on my whiteboard. I now have 2 equal pieces of clay and I cut my whole clay into 2 equal parts. The top number, or numerator, of the fraction tells us how many pieces I have. The bottom number, or denominator, tells us how many parts our whole was cut up into. Write 2/2 on your whiteboard and explain it again. The 2 in the numerator represents 2 pieces. The denominator represents how many parts my whole was cut into. When we divide our whole into parts they must be equally sized to be a fraction. Practice TOGETHER with students Give each student a whiteboard, dry erase marker, piece of clay and stick or object to cut with. This section is done together with the instructor. You should mold your own clay and model fractions on the instructor whiteboard. Please mold your clay into a flattened circle or pizza shape. Now cut your clay into 4 even parts. TIER II INTERVENTIONS 64 On your whiteboard write down how many pieces you have in the numerator or top of your fraction. Give time for students to complete. Write down how many parts you cut your clay into in the denominator or bottom of the fraction. Give time for students to complete. Tell your shoulder partner what you notice about the numbers. Give them about 30 sec to partner talk. You noticed that the numerator and the denominator have the same number. When we have the same number of pieces and the same number of parts we have 1 whole or we just say it is equal to 1. Please shape your clay into a rectangle. Give time for them to complete. Cut your clay into 5 parts, making them equal as best as you can. Give time for them to complete. Write your fraction on your whiteboard that represents how many pieces and how many parts in your clay. Give time for them to complete. Tell your shoulder partner how you cut your rectangle and why you wrote your fraction on your whiteboard. Give time to complete. Listen for their answers for understanding. I see we wrote 5/5 on our whiteboards. The numerator shows the 5 pieces we have. The Denominator (5) shows that the whole was cut into 5 parts or 5ths. Practice. In this section students work independently. Support where needed. This section can be used for multiple days for practice, until the instructor feels they are ready to be assessed. This is time for them to practice at their own pace. You can have the students stay together or work individually. Please roll your clay back into one whole again. I will give you a shape to mold your clay into and a number to cut your whole into equal parts. Then write the fraction that represents the clay on your whiteboard. You can change the shape of the clay examples; rectangle, line, square, pizza, rope etc. TIER II INTERVENTIONS 65 Give out numbers that can easily be cut like: 2,3,4, 6, 8, 9, 10, 12. Check to see their fraction they write is correct. Quiz Give this quiz to students when you feel they are ready or understand what has been taught. Students work alone, No support for assessing. Have the students turn in whiteboards and markers. Pass out quiz paper and pencils. Please put your name on the front of your paper and do your work on the back. Please show all your work. When you are done please turn it into me and sit quietly for others. I will roll my clay into a shape and cut it into equal parts. You will write the fraction I modeled with the clay on your paper. Instructor will roll clay into a pizza shape and cut into 5 pieces. Ask the students to write down the fraction Roll your clay into a rectangle. Cut your clay into 8 pieces. Ask the students to write down the fraction TIER II INTERVENTIONS 66 #2 Unit Fractions A Unit fraction is 1 piece of the parts of a fraction. Examples ⅓, ¼, ⅛. = ¼= unit fraction TIER II INTERVENTIONS 67 #2 unit Fractions Purpose: Students will understand what a unit fraction is and how to write a unit fraction. Materials  Modeling clay  Stick or object to cut clay  Whiteboards  Dry erase Marker  pencils  4 x 6 paper for quiz Re-teach: Bold words can be used as a script to teach. Today we will be making different fractions with clay and learning how to write them in fraction form. Remember a fraction is part of 1 whole. Take a piece of clay. If I divide this clay up into 6 equal pieces- cut your piece into 6. I have 6 individual pieces so that would be my numerator and my denominator is 6 because I divided my whole into 6 equal parts. Write 6/6 on your whiteboard. If I am only looking at 1 of the pieces of my clay- Hold one piece up- I would have one of 6 parts. Or 1/6. Write ⅙ on the whiteboard. My numerator is 1 because I am only showing 1 of the pieces. My denominator is 6 because I divided my whole into 6 equal parts or 6ths. 1 piece of a whole is called a unit fraction. Write “unit fraction” down on the whiteboard. A unit fraction is 1 piece of a whole. Write 3/6 on the whiteboard and show 3 of 6 pieces of clay. 3/6 is not a unit fraction because it is more than 1 piece of the parts. Put your clay back together and make it into a rectangle. Cut it into 8 parts. I have divided my rectangle into 8 parts. If I look at just one piece like this - hold one piece- this would be 1 piece of 8 parts or ⅛. - Write ⅛ on the whiteboard. TIER II INTERVENTIONS 68 1/8 is the unit fraction for something divided into 8 parts. 1 is the numerator because it is 1 piece. 8 is the denominator because I divided my whole into 8 parts or 8ths. Practice TOGETHER Give each student a piece of clay, object to cut with, whiteboard, and dry erase marker. We are going to make unit fractions together. Please make a rectangle with your clay and cut it into 4 even parts. Instructor also shapes clay and cuts it into 4 parts. Give time to complete Hold 1 piece in your hand. This is 1 unit of the parts. Please write the fraction for this piece on your whiteboard. Give time to complete. Write ¼ on your whiteboard. It is 1 of 4 parts or ¼. Please turn and teach your partner what a unit fraction is. Give them time to complete. Listen to their explanations for their understanding. Let’s roll our clay out again into a square shape. Please cut it into 3 parts. Instructor also shapes clay and cuts it into 3 parts. Give time to complete. Hold up 1 piece. This is 1 unit of the parts. We would call this the unit fraction. Please write this fraction value on your whiteboards. Give time to complete. Write 1/3 on your whiteboard. Our answer is ⅓. 1 piece of 3 parts. I will write down different fractions on my board. Please give me a thumbs up if it is a Unit Fraction or thumbs down if it is not. On your board write these different fractions waiting in between to assess if the students understand that a unit fraction is only 1 piece of the parts. Write down ¼, ¾, ⅖, 1/9, ⅘, ⅜, 1/1000 Practice In this section students work independently. Support where needed. This section can be used for multiple days for practice, until the instructor feels they are ready to be assessed. TIER II INTERVENTIONS 69 Give the students different shapes and number of parts to divide such as: worm, square, cube, pizza, sphere, rectangle. Have them pull one piece out and write that unit fraction on their whiteboards. Easier numbers to divide shapes are, 2, 3, 4,6,8,9,12,16 Quiz Give this quiz to students when you feel they are ready or understand what has been taught. Have the students turn in clay, cutting object, whiteboards, and dry erase markers. Pass out quiz paper and pencils. Please put your name on the front and do your work on the back. Please show all your work. When you are done please turn it into me and sit quietly for others. I will be cutting my clay into parts. Please write down the Unit Fraction. Roll your clay into a long “worm” shape and cut it into 6 parts. After they have finished the first, do one more. Shape your clay into a pizza and cut it into 8 pieces. Have them write the unit fraction. TIER II INTERVENTIONS 70 #3 Combining Unit fractions Adding Unit fractions can create a larger fraction. = ⅛ + ⅛ + ⅛ + ⅛ + ⅛ = ⅝ TIER II INTERVENTIONS 71 #3 Combining unit fractions Purpose: Students will be able to create fractions from unit fractions. Materials:  Clay  Stick or item to cut Clay  Whiteboards  Dry erase Markers  Quiz paper (4x6)  Pencils Re-teach: Bold words can be used as a script to teach Get out one piece of clay. Shape it into a pizza shape. Cut the dough into 8 parts. Instructor: Show with your fingers what my denominator will be for fractions I create from these parts…Give time to see hands Correct there are 8 equal parts. This would be our denominator. Hold up on your fingers how many would be used for a Unit fraction…Give time to see hands 1 piece of the whole is a Unit fraction, you should be holding up 1 finger. If I write this fraction it would be ⅛. Write ⅛ on your whiteboard. When I take more than one piece of the 8, like 2 (take 2 pieces of the pizza) I would have ⅛ + ⅛. Write ⅛ +⅛ on the whiteboard. ⅛ +⅛ is equal to 2/8. I can create different fractions by adding unit fractions together. The size of the parts has not changed so my denominator will stay the same number, 8 in this case. The numerator changes to how many pieces I have added together. Practice Together Pass out a piece of clay, object to cut clay, whiteboard, and dry erase markers for each student. In this section the instructor will model along with the students. TIER II INTERVENTIONS 72 Please mold your clay into a rectangular shape and cut your rectangle into 6 equal parts. Give time for students to complete. Tell your shoulder partner what our denominator will be for this fraction line… Give time to respond Yes, our denominator will be 6. If we take 1 unit piece and 1 unit piece tell your partner how many pieces of our whole we have…Give time to respond Write the equation of adding the fractions with me. Write on your whiteboard while the students also write this equation ⅙ +⅙ = 2/6. Explain to your partner what you notice about the fractions and the answer…Give time to respond The denominator digit did not change because we still have the same size of equal parts; each part is still ⅙ of the whole rectangle. The numerator digit changed because we were looking at 2 pieces of the whole. When we add fractions with the same denominator the denominator stays the same and we add the numerators. Let's do another example. Shape your clay into a pizza… Give time for students to complete Now cut your pizza shape into 4 equal parts. Give time for students to complete. Let’s say we were going to give away 3 pieces. Write down with me how we would write that. Write on your whiteboard 1/4 +1/4+1/4 = ¾. Notice again how the numerators add together but the denominators stay the same because we still have the same size of parts that we divided our whole into. Practice - In this section student’s work independently. Support where needed. This section can be used for multiple days for practice, until the instructor feels they are ready to be assessed Have the students create different shapes and numbers of parts to divide such as: worm, square, cube, pizza, and rectangle. Easier numbers to cut are 2,3,4,6,8,9,12,16. TIER II INTERVENTIONS 73 Have them pull a few different amounts of pieces out and write that equation with adding unit fractions on their whiteboards. 5 min Quiz Give this quiz to students when you feel they are ready or understand what has been taught. Students work alone, No support for assessing. Have the students turn in clay, cutting objects, whiteboards, and dry erase markers. Pass out quiz paper and pencils. Please put your name on the front and do your work on the back. Please show all your work. When you are done please turn it into me and sit quietly for others. I will mold my clay into a shape and take some pieces out. Write down the unit fraction equation that I show that represents my pieces. Mold your clay into a worm shape and cut it into 8 equal parts. Take 5 pieces out. TIER II INTERVENTIONS 74 #4 Adding non- unit fractions with like denominators Sharing a licorice rope, Suzanne cut her rope into 8 parts. She shared 3 pieces. Then she shared 2 pieces. Write down the fractions that she shared and the total amount she shared. ⅜ + 2/8 = ⅝ TIER II INTERVENTIONS 75 #4 Adding non- unit fractions with like denominators Purpose: Students will be able to add non-unit fractions accurately and repeatedly Materials  Clay  Object to cut clay  Whiteboard  Markers  Quiz Papers  Pencils Re-teach - Bold words can be used as a script to teach. Instructor: Use a piece of clay to illustrate the real world problem. Mold the clay into a longer rectangular shape. Cut it into 6 equal parts. I am going to show you an example of how we could use fractions in our lives. Mr. Nelson is building a sidewalk. He built 2/6 of the sidewalk yesterday. Lay down 2 of your pieces. Write 2/6 on your whiteboard. Today he built 3/6 of the sidewalk. Lay down 3 more of your pieces. Write down 3/6 on your whiteboard. Tell your shoulder partner what fraction of the total sidewalk he has built… Listen for answers… Correct he has built ⅚ of the sidewalk. Write down an addition sign between your 2/6 and 3/6 on your whiteboard. Put and equals sign and write ⅚. Teach your shoulder partner what you notice from my whiteboard on how to add fractions… Listen for answers. You’ve noticed that we add the numerators but the denominators do not change. This happens because we did not change the amount of equal parts the sidewalk was broken up into just how many of the pieces he completed each day. TIER II INTERVENTIONS 76 Practice together- In this section model with your clay and whiteboard along with the students. Give each student a piece of clay, object to cut the clay, a whiteboard and dry erase marker. Please make your clay into a pizza shape. Cut it into 8 equal pieces. Give time to complete. Yesterday Jordan ate 2/8 of the pizza. Lay out those pieces. Watch as students lay out 2 of their pieces. Then lay yours out. Write down this fraction on your whiteboard. Let them write it first on their board then write it on yours. 2/8 Today Jordan at 4/8 of the pizza. Lay these pieces out and then write the fraction on your whiteboard. Let them do it first then follow with your clay and on your board. 2/8 4/8 Tell your shoulder partner how many pieces of the whole pizza he ate. Listen for understanding. Write the fraction of how much pizza Jordan ate on your whiteboard. Write 6/8 on your board. 2/8 4/8 6/8 Now let's turn this into an equation by adding the addition and equals signs. Write in the addition sign and equals sign. 2/8 + 4/8 = 6/8 Notice the numerator adds together but the denominator stays the same digit. This is because the denominator explains how many PARTS there are to the whole pizza. Let’s do another one. Please roll your clay into a long line. Cut it into 12 pieces. This is easiest if you cut it in half, cut the half pieces in half, and then cut each of those pieces into 3 equal pieces. Give time to complete. Today Sharon wanted to give pieces of her licorice rope to her friends. She gave 5/12 of the licorice rope to some friends first. Show how much she gave with your clay then write it on your whiteboards. Show with your clay and whiteboard after they do. TIER II INTERVENTIONS 77 Then she gave 3/12 of the licorice rope away. Show how much she gave with your clay and then write it on your whiteboard. Do the same with your clay and whiteboard. What was the total fraction she gave away? Write that Fraction on your board. Then add the appropriate math function forms. On your whiteboard you should have 5/12 + 3/12 = 8/12. Notice again that the numerators are added together and the denominators stay the same because we didn’t change the amount of parts the licorice rope was divided into. Practice- In this section students work independently. Support where needed. This section can be used for multiple days for practice, until the instructor feels they are ready to be assessed. Have the students create different shapes and numbers of parts to divide such as: worm, square, cube, pizza, sphere, rectangle. Have them pull a few different amounts of pieces out and write that equation with fractions on their whiteboards. Some ideas to shape their clay into are:  Sharing a piece of gum- Divide into 5 parts, share with 2 then 1 friend  Taking turns on an Xbox - shape clay into clock shape divide it into equal parts of 6. First person uses 2/6 of an hour, the second person uses 3/6 of an hour.  Building a fence- make a long string to act as a measuring tape, divide into 9 parts. Build 4/9 then build 3/9  Walking to a friend’s house. Shape into line or rectangle. Divide into 8. One friend walks ⅜ of the way, the other friend walks ⅝ of the way.  Dividing a piece of bread - make a square divided into 7 parts. Give 2/7 then 4/7 away.  Making a fruit by the foot last longer by dividing it into sections. - Mold into a long flat tape. Divide into 8 parts. You ate 2 pieces then ate 4 pieces. What fraction of the total did you eat? 5 min Quiz- Give this quiz to students when you feel they are ready or understand what has been taught. Students work alone, No support for assessing. Have the students turn in whiteboards and markers. Pass out quiz paper and pencils. Please put your name on the front and do your work on the back. Please show all your work. When you are done please turn it into me and sit quietly for others. TIER II INTERVENTIONS 78 Mold your clay into a long string shape and cut it into 8 equal parts. Sadie wants to make bracelets for 8 of her friends from a piece of string. She used 3 pieces to make bracelets on Monday and 5 pieces on Tuesday. How much of the whole string has she used? Please write down the equation that shows how much of the string she used. TIER II INTERVENTIONS 79 #5 Comparing fractions with the Same Denominator TIER II INTERVENTIONS 80 #5 Comparing fractions with the Same Denominator Purpose: Students will be able to compare fractions with the same denominator and show which is the largest. Materials  Clay  Object to cut Clay  Whiteboards  Dry Erase Markers  Quiz Paper  Pencil Re-Teach: Bold words can be used as a script to teach. Today we are going to be exploring how to recognize comparing numbers and recognizing which is the larger value. On your whiteboard write the numbers 9 and 5. Tell your shoulder partner which of these numbers is larger and explain how you know. Listen for explanations. I heard you say that 9 is the larger number. Write a > between the 9 and 5. 9>5 Let’s do another one. Write the numbers 10 and 17. Tell your shoulder partner again which is larger and explain how you know. Listen for explanations. I heard you say that 17 is the larger number. Write a < between 10 and 17. 10<17 Fractions are numbers too. Remember fractions are parts of a whole. Take a piece of clay, split it in half, and roll it into 2 equally sized pizza shapes. Then cut it into 8 parts. Write the fraction ⅜ and ⅝ on your whiteboard. In these two fractions we can compare them easily because they have been divided into the same sized parts or we can say they have the same denominators. When comparing fractions it is easier to tell which fraction is larger if they have the same denominator. TIER II INTERVENTIONS 81 If I use my clay as models I have two equal pizza shapes that have been divided into 8 parts. If I take 3 from the first pizza, take 3 pieces and put them forward on the table. Then take 5 from the second pizza, take 5 from the second clay model and put them on the table. I can compare to see which fraction is the greater value or the larger number. I can see that ⅜ is a smaller amount than ⅝. I can also compare by looking just at the fractions. Use the whiteboard to demonstrate. The fractions both have 8 parts, both have 8 as the denominator, Point to the denominators, so they are equal to compare. The numerators are different, Point to the numerators. To tell which fraction is larger when they have the same denominator, it will be the one with the larger digit in the numerator. Looking at our model of the clay, pointing to the clay, we can see that the fraction with 5 pieces has more clay than the fraction with only 3 pieces. On the written fractions, the 5 in the numerator is larger than the 3 in the other numerator. So I can tell from the fractions that ⅝ is larger than ⅜. I will write a greater than sign in between my fractions ⅝ > ⅜ Let’s do some modeling and writing of equations to compare together. Practice together- Model with the students with clay and whiteboard as they do also. Give each student a whiteboard, Dry erase Marker, Clay and object to cut the clay. Please split your clay into two equal size pieces. Then roll it into equal size ropes about a foot. Show with your hands the length you would like. Give time to complete. Now cut it into 6 equal parts for each rope. On our boards we will write 2 different fractions. Each fraction will have a 6 in the denominator because we have cut them each into 6 parts. Write 6 in the TIER II INTERVENTIONS 82 denominator of the two fractions to compare. Wait until all students have put the fractions on their boards. Now we are going to compare 4 pieces from this first rope and 3 pieces from this other rope. This is the amount of pieces of the parts so that will be our numerator. Write our numerators in the fractions. Write the 4 and 3 on the whiteboard in the numerators. Watch the students do theirs. Because we have the same denominator or the same amount of parts in each rope we can compare and see which one is larger by looking at the numerators or the amount of pieces. Tell your shoulder partner which rope fraction is larger. Listen as the say which is larger. You are right the 4/6 fraction is larger because four pieces of 6 parts is more than 3 pieces of 6 parts or 3/6. Place a greater than sign between the 2 fractions. 4/6 > 3/6 Let's do another comparison. Erase your boards and mold your clay back into separate pieces of equal size. Give time to complete This time roll your clay out into equal pizza shapes. Give time to complete Cut the shapes into 12 parts. Give time to complete Since we cut our clay into 12 parts we can compare because all the parts are the same size. Write down what our denominator will be. Watch the students to see what number they put in the denominators to compare. Write a 12 in the denominator on your whiteboard. Our denominator should be 12. Now we will take 7 pieces from our first pizza and 9 from our second pizza. Write what our numerators will be in our fractions. Watch as they fill in the numerators. Write in 7 and 9 on your own numerator. Now that we have our fractions we can compare which is larger. Tell your shoulder partner which group of clay is larger. Listen to hear answers. Looking at our written fractions, tell your shoulder partner which fraction is larger. Listen for answers. TIER II INTERVENTIONS 83 Correct 9/12 is larger than 7/12 because 9 pieces of 12 parts is larger than 7 pieces of 12 parts. Write the correct greater than or less than sign in between the fractions. Have the students follow. Practice- Students work independently. Support as needed. This section could be expanded to multiple lessons for repeated practice. Have the students mold their clay pieces into different shapes such as: worm, square, cube, pizza, sphere, and rectangle. Have them cut them into different amounts of equal parts. Have them pull 2 different amounts of pieces out. Have them write the fractions and determine which fraction has more. Easier to divide clay numbers are 2,3,4,6,8,9,12,16 Move into just writing different fractions on whiteboards without using the clay if possible, to practice looking at fractions and comparing them. 5 min Quiz- Give this quiz to students when you feel they are ready or understand what has been taught. Have the students turn in Clay, cutting objects, whiteboards and dry erase markers. Pass out quiz paper and pencils. Please put your name on the front and do your work on the back. Please show all your work. When you are done please turn it into me and sit quietly for others. Have the students write the fractions 8/13 and 5/13 on their paper. Have the students put a <,>,= sign in between the fractions to show which is greater. Have the students also write the fractions 4/9 and 6/9 on their paper. Have the students put a <,>,= sign in between the fractions to show which is greater. TIER II INTERVENTIONS 84 #6 Subtracting fraction with like denominators. TIER II INTERVENTIONS 85 #6 Subtracting fraction with like denominators. Purpose: Students will be able to correctly subtract fractions through modeling and writing fractions Materials  Clay  Object to cut Clay  Whiteboards  Dry Erase Markers  Quiz paper 4x6  Pencils Re-teach- Bold words can be used as a script to teach. Today we will be working on subtracting fractions. Let me start with an example. Mold your clay into a long flat rectangle about 12 inches. Pretend my clay is a piece of bubble tape bubble gum and I wanted it to last for 10 different times. Cut it into 10 equal parts, this is easiest if you divide it in half then divide each half into 5 pieces. I cut it into 10 parts. Right now I have 10 pieces of 10 parts (write the fraction 10/10 on your whiteboard). I can write this with the 10 in the numerato