Student Teacher 36 | - ✏ Freelance Writer
Dec 21, 2019 | #1
Programs in mathematics for children and young students vary in their usefulness and in their comprehensiveness. Some programs are very simple and require no extra time, while some are exceptionally comprehensive and include everything in the curriculum, along with the assessments to determine level placements. Each has its place, and each can be effective if it is evidence-based and supported by theory and relevant rigorous research that support its methods and strategies. This paper examines two such models and programs, placing emphasis on the program's effectiveness for special needs students, especially those in the mild or moderate categories, such as learning disabilities.
Response to Intervention (RTI) is an exceptionally comprehensive curricular program that utilized three levels, or tiers, to meet student needs. Tier 1 is what every student in the school experiences, and is the basic core instruction at all grade levels. For those students who, after being assessed, are not successful at the basic level, there is a second level which entails smaller working groups and a more intense kind of instruction for the concepts being taught. Again, after assessment, students who are still at risk of failing are moved to a third tier, again with small groupings or one-to-one instruction. At this level, the students are in an intense remediative effort designed to prevent their failure. The entire program is school-wide, with time and resources allocated for maximum success.
Everyday Mathematics is a supplemental program aimed at reinforcement of skills. It does not take the place of the regular mathematics program, nor is it as comprehensive as RTI. For special needs students, it provides a strong reinforcement of skills that may not have been learned well, and provides practice with concepts that are some times difficult. The program does not, however, provide the kinds of support that RTI is designed to provide.
Students with special needs, especially those with mild to moderate intellectual functioning, often have great difficulty with mathematical concepts, and do not typically do well with traditional instruction. Tutoring is always an option, of course, but by itself, is not a viable alternative to other kinds of instruction designed for these kinds of students. Several models of instruction have been developed suitable for special needs students, meeting their instructional needs in ways that regular program instruction cannot. Two of these models will be discussed here, along with the theoretical assumptions supporting the models, specific teaching strategies making the models efficacious, and the characteristics of the students for whom these strategies would be most effective. The first of these is Response to intervention, or RTI, a product of the National Center for Learning Disabilities, and Everyday Mathematics, a product of the Each has advantages and disadvantages for different populations of students, but both are efficacious in changing the curriculum so that positive learning takes place with the students for whom they were designed.
Response to instruction is a product of the national Center for Learning Disabilities. As the name suggests, this program is particularly effective with learning disabled students (although the program is available for everyone in the regular classroom at Tier 1), whose primary disability may be either reading or mathematics related, or both. RTI relies on what is called tiered instruction meaning that there are levels of intervention that become more intensive as students move through the curriculum. RTI maintains three tiers-the first is core instruction, which is an evidence-based, scientifically-researched core curriculum. It is typically aligned with state standards, and delivers a high-quality instructional program that for which there are known outcomes.
The curriculum is assessment-based. The assessment components of RTI are the essential elements of implementation, and the instruction that occurs are assessment-driven. The instruction is tiered because the instruction delivered to students must meet their needs and the severity of their severity of their difficulties. (ibid).
When the Tier 1 core instructional program is delivered as it should be, the expectation is that most of the students will show outcomes that are at a level of proficiency that meets minimal standards. Theoretically, 75%-85% of students should reach successful levels of competency through Tier 1. Typically, however, in the early years the percentage of successful levels of competency are about 50%-70%. At that point, those students needing more intensive instruction would move to Tier 2. (ibid).
For some students the level of instruction at Tier 1 is not sufficient, and does not help them to reach levels of minimal competence. Tier 2 represents a level of supplemental intervention for these students who are at some risk for failure but are not at the level of high risk. The ongoing assessment process identifies these students, and instruction in smaller groups that focuses on their specific needs is delivered.
Tier 3 students are at the highest risk of failure, and have been identified as having severe needs that are not satisfied by Tier 1 or Tier 2 instruction. These children receive instruction in even smaller groups, sometime one-to-one. At this point, these children may be identified as special education children, although not everyone in these groups is so identified, though their needs call for this level of intervention. (ibid).
RTI is based on the concept that evidence-based learning is appropriate and necessary to intervene in the academic performance of those at risk of failure. RTI is a school and classroom commitment to use best findings to plan, design, implement, and guide instruction. The intention of RTI is to prevent and intervene before students fail. With RTI, all students in the school get a high-quality research-based differentiated instruction in the general ed core curriculum, and all staff assume a role in student assessment and in instruction in the core program (Tier 1). All students, according to academic need, receive increasingly intense levels of targeted research-based intervention, and all of it is assessment based. Progress though the tiers is bilateral and is always based on student response to the interventions. Throughout the process, support is always given.
RTI is based in part on Carroll's Theory of School Learning. Carroll believed that time needed to learn was a function of individual difference and teacher or school mediated variables. Carroll also believed that there was a difference between aptitude and ability; aptitude can be quantified, but ability is something not quantifiable, although all students possess it. In speaking of time to learn, Carroll posited that the interaction of individual differences and the actual conditions of teaching and learning was of crucial importance. Time to learn is influenced, he believed, by the opportunity to learn and quality of instruction.
Opportunity to learn represents an allocation of time and resources that are sufficient for each student. Quality of instruction is more than just effective instruction-it is also efficient instruction. High quality instruction, therefore, is a combination of the opportunity to learn for each student and the effective and efficient delivery of instruction. These are the hallmarks of RTI-a comprehensive program of instruction that accounts for instructional quality by managing the effectiveness of instruction and the efficiency of instruction by utilizing several tiers of instruction that increase in intensity as student need indicates, freeing other to devote more effective time and more efficient time to further tasks. (Carroll, 1963).
Everyday Mathematics is a program developed by the University of Chicago School Mathematics Project (UCSMP). It is, according to McGraw-Hill (n.d). a structured, research-based program that helps to develop mathematical reasoning and strong math skills. The authors of the program examined how other nations teach math and effective classroom practice. It was field tested and reviewed by mathematicians, educational specialists, and classroom teachers, and revised per their comments and concerns before being released for publication. It has been used now in classrooms for more than twenty years, and is based on research that shows that children learn best hen they are presented with topics quickly, and revisit them frequently for review and practice. The sequence of instruction is carefully orchestrated to maximize the conditions for learning and retaining what is learned. New concepts and skills are introduced informally, and examined again in different contexts over several grade levels. The program encourages the use of manipulatives and skill-based mathematical games, along with pencil-and-paper activities.
Students exposed to Everyday Mathematics have shown themselves to be mathematically literate on several standardized assessments, including state-mandated tests and commercially-available tests. The National Academy of Sciences (National Research Council, 2004) asserts that Everyday mathematics has been scrutinized and researched more than any other mathematics curriculum. (McGraw Hill, n.d).
The US Department of Education, Institute of Educational Sciences (2007) reviewed sixty one studies of the program. Four of the studies met the standards of the IES, and those are reviewed below.
Carroll (1998) examined 76 fifth graders in four classrooms from four school districts and compared the progress of these students using Everyday Mathematics with 91 students in four districts matched for demographics and geographic location. The Everyday Mathematics group met Extent of Evidence criteria for successful progress. Riordan and Noyce (2001) examined 3,781 fourth graders at 67 schools and compared them with 5, 102 fifth graders at 78 similar schools. The results were the same as the Carroll study. Waite (2000) looked at 732 third-, fourth-, and fifth-graders in six schools using Everyday Mathematics, and compared them with 2, \704 third-, fourth-, and fifth-graders in twelve similar schools, matched on baseline math achievement scores, and the results here were the same as in the previously mentioned studies. Woodward and Baxter (1997) looked at 104 third grade students in five classrooms and compared them with 101 third graders in four classrooms in the same school. Again, the results of the study met the criteria for Extent of Evidence. WWC (What Works Clearinghouse) applies Extent of Evidence each study in terms of its effectiveness and its rigor. The Extent of Evidence for Everyday Mathematics is medium to large for math achievement.
Many teachers use these activities as a warming up activity when students arrive to school each morning. The exercises are short, mostly reviews of previous material. Students are encouraged to collaborate on these activities, and they are generally short enough and accessible enough (they do not usually require any kind of research) for special needs students to use them and complete them in a reasonable amount of time. A typical activity will use manipulatives for counting or for graph-building, and some require calculator use. Others are puzzles that involve mathematical reasoning. When special needs children are collaborating with their peers, the work generally goes smoothly and the special needs children seem to understand most of the work, and with some prodding, can do the same concepts again, and with repeated practice over time, become independently able to do the activities. The theoretical foundation is clearly correct here-short introductions to concepts, repeated practice and multiple exposures work best for special needs children, who usually do not have the attention span or intellectual capability to immerse themselves in mathematical concepts for long periods of time. Though the activities do require subskills such as measuring with a ruler, how to make simple graphs, and other basics of mathematical inquiry, special needs children seem able to learn some of this "on the fly" and complete their activities successfully.
Conclusion
Both systems work very well for all students, including mildly or moderately disabled children. RTI is much more comprehensive and intense than is Everyday Mathematics. RTI is a total curricular effort, while EM is a supplemental activity regimen that reinforces skills. RTI requires a commitment from all teachers in the school, while EM is normally a simple part of each elementary teacher's morning list of tasks to complete, with no particular commitment required, and no allocation of resources beyond the ordinary. RTI is designed to identify and intervene with at-risk students so that they do not fail. EM is designed to supplement practice and does not distinguish students who are at risk of failure, nor does it provide progressively more intense instruction to those who are at risk. RTI is a structured tier approach that progressively supports those students who are not minimally competent at basic instructional levels. EM offers no such support.
For special needs students, RTI is an extensively supporting system taking into account quality of instruction with effectiveness and efficiency. Though EM is used daily, and has shown to be effective at reinforcing mathematical skills, it is not an extensive use of time and though effective, is not necessarily efficient, unless sit is used by the classroom teacher to identify specific skill needs that need remediation. It is likely that students who would qualify for Tier 3 of RTI would find little success with EM when working independently.
REFERENCES
US Department of Education, Institute of Education Sciences. (2007). Intervention: Everyday Mathematics.
Carroll, J. B. (1963). A model of school learning. Teachers College Record, 64, 723-733. National Center for Learning Disabilities. (2010). Tiered instruction and intervention in a Response-to-Intervention Model.
Carroll, W. M. (1998). Geometric knowledge of middle school students in a reform-based mathematics curriculum. School Science and Mathematics, 98(4), 188-197. Everyday Mathematics. (n.d). McGraw Hill.
Haley, C. (2007). Response to Intervention: Overview. PDE/BSE.
Riordan, J. E., & Noyce, P. E. (2001). The impact of two standards-based mathematics curricula on student achievement in Massachusetts. Journal for Research in Mathematics Education, 32(4), 368-398.
US Department of Education, Institute of Education Sciences. (2007). Intervention: Everyday Mathematics.
Waite, R. D. (2000). A study of the effects of Everyday Mathematics on student achievement of third-, fourth-, and fifth-grade students in a large north Texas urban school district. Dissertation Abstracts International, 61(10), 3933A. (UMI No. 9992659).
Woodward, J., & Baxter, J. (1997). The effects of an innovative approach to mathematics on academically low-achieving students in inclusive settings. Exceptional Children, 63(3), 373-388.
Response to Intervention (RTI) is an exceptionally comprehensive curricular program that utilized three levels, or tiers, to meet student needs. Tier 1 is what every student in the school experiences, and is the basic core instruction at all grade levels. For those students who, after being assessed, are not successful at the basic level, there is a second level which entails smaller working groups and a more intense kind of instruction for the concepts being taught. Again, after assessment, students who are still at risk of failing are moved to a third tier, again with small groupings or one-to-one instruction. At this level, the students are in an intense remediative effort designed to prevent their failure. The entire program is school-wide, with time and resources allocated for maximum success.Everyday Mathematics is a supplemental program aimed at reinforcement of skills. It does not take the place of the regular mathematics program, nor is it as comprehensive as RTI. For special needs students, it provides a strong reinforcement of skills that may not have been learned well, and provides practice with concepts that are some times difficult. The program does not, however, provide the kinds of support that RTI is designed to provide.
Math Interventions
Students with special needs, especially those with mild to moderate intellectual functioning, often have great difficulty with mathematical concepts, and do not typically do well with traditional instruction. Tutoring is always an option, of course, but by itself, is not a viable alternative to other kinds of instruction designed for these kinds of students. Several models of instruction have been developed suitable for special needs students, meeting their instructional needs in ways that regular program instruction cannot. Two of these models will be discussed here, along with the theoretical assumptions supporting the models, specific teaching strategies making the models efficacious, and the characteristics of the students for whom these strategies would be most effective. The first of these is Response to intervention, or RTI, a product of the National Center for Learning Disabilities, and Everyday Mathematics, a product of the Each has advantages and disadvantages for different populations of students, but both are efficacious in changing the curriculum so that positive learning takes place with the students for whom they were designed.
RTI - Response to Intervention
Response to instruction is a product of the national Center for Learning Disabilities. As the name suggests, this program is particularly effective with learning disabled students (although the program is available for everyone in the regular classroom at Tier 1), whose primary disability may be either reading or mathematics related, or both. RTI relies on what is called tiered instruction meaning that there are levels of intervention that become more intensive as students move through the curriculum. RTI maintains three tiers-the first is core instruction, which is an evidence-based, scientifically-researched core curriculum. It is typically aligned with state standards, and delivers a high-quality instructional program that for which there are known outcomes.
The curriculum is assessment-based. The assessment components of RTI are the essential elements of implementation, and the instruction that occurs are assessment-driven. The instruction is tiered because the instruction delivered to students must meet their needs and the severity of their severity of their difficulties. (ibid).
When the Tier 1 core instructional program is delivered as it should be, the expectation is that most of the students will show outcomes that are at a level of proficiency that meets minimal standards. Theoretically, 75%-85% of students should reach successful levels of competency through Tier 1. Typically, however, in the early years the percentage of successful levels of competency are about 50%-70%. At that point, those students needing more intensive instruction would move to Tier 2. (ibid).
For some students the level of instruction at Tier 1 is not sufficient, and does not help them to reach levels of minimal competence. Tier 2 represents a level of supplemental intervention for these students who are at some risk for failure but are not at the level of high risk. The ongoing assessment process identifies these students, and instruction in smaller groups that focuses on their specific needs is delivered.
Tier 3 students are at the highest risk of failure, and have been identified as having severe needs that are not satisfied by Tier 1 or Tier 2 instruction. These children receive instruction in even smaller groups, sometime one-to-one. At this point, these children may be identified as special education children, although not everyone in these groups is so identified, though their needs call for this level of intervention. (ibid).
RTI is based on the concept that evidence-based learning is appropriate and necessary to intervene in the academic performance of those at risk of failure. RTI is a school and classroom commitment to use best findings to plan, design, implement, and guide instruction. The intention of RTI is to prevent and intervene before students fail. With RTI, all students in the school get a high-quality research-based differentiated instruction in the general ed core curriculum, and all staff assume a role in student assessment and in instruction in the core program (Tier 1). All students, according to academic need, receive increasingly intense levels of targeted research-based intervention, and all of it is assessment based. Progress though the tiers is bilateral and is always based on student response to the interventions. Throughout the process, support is always given.
RTI is based in part on Carroll's Theory of School Learning. Carroll believed that time needed to learn was a function of individual difference and teacher or school mediated variables. Carroll also believed that there was a difference between aptitude and ability; aptitude can be quantified, but ability is something not quantifiable, although all students possess it. In speaking of time to learn, Carroll posited that the interaction of individual differences and the actual conditions of teaching and learning was of crucial importance. Time to learn is influenced, he believed, by the opportunity to learn and quality of instruction.
Opportunity to learn represents an allocation of time and resources that are sufficient for each student. Quality of instruction is more than just effective instruction-it is also efficient instruction. High quality instruction, therefore, is a combination of the opportunity to learn for each student and the effective and efficient delivery of instruction. These are the hallmarks of RTI-a comprehensive program of instruction that accounts for instructional quality by managing the effectiveness of instruction and the efficiency of instruction by utilizing several tiers of instruction that increase in intensity as student need indicates, freeing other to devote more effective time and more efficient time to further tasks. (Carroll, 1963).
Everyday Mathematics
Everyday Mathematics is a program developed by the University of Chicago School Mathematics Project (UCSMP). It is, according to McGraw-Hill (n.d). a structured, research-based program that helps to develop mathematical reasoning and strong math skills. The authors of the program examined how other nations teach math and effective classroom practice. It was field tested and reviewed by mathematicians, educational specialists, and classroom teachers, and revised per their comments and concerns before being released for publication. It has been used now in classrooms for more than twenty years, and is based on research that shows that children learn best hen they are presented with topics quickly, and revisit them frequently for review and practice. The sequence of instruction is carefully orchestrated to maximize the conditions for learning and retaining what is learned. New concepts and skills are introduced informally, and examined again in different contexts over several grade levels. The program encourages the use of manipulatives and skill-based mathematical games, along with pencil-and-paper activities.
Students exposed to Everyday Mathematics have shown themselves to be mathematically literate on several standardized assessments, including state-mandated tests and commercially-available tests. The National Academy of Sciences (National Research Council, 2004) asserts that Everyday mathematics has been scrutinized and researched more than any other mathematics curriculum. (McGraw Hill, n.d).
The US Department of Education, Institute of Educational Sciences (2007) reviewed sixty one studies of the program. Four of the studies met the standards of the IES, and those are reviewed below.
Carroll (1998) examined 76 fifth graders in four classrooms from four school districts and compared the progress of these students using Everyday Mathematics with 91 students in four districts matched for demographics and geographic location. The Everyday Mathematics group met Extent of Evidence criteria for successful progress. Riordan and Noyce (2001) examined 3,781 fourth graders at 67 schools and compared them with 5, 102 fifth graders at 78 similar schools. The results were the same as the Carroll study. Waite (2000) looked at 732 third-, fourth-, and fifth-graders in six schools using Everyday Mathematics, and compared them with 2, \704 third-, fourth-, and fifth-graders in twelve similar schools, matched on baseline math achievement scores, and the results here were the same as in the previously mentioned studies. Woodward and Baxter (1997) looked at 104 third grade students in five classrooms and compared them with 101 third graders in four classrooms in the same school. Again, the results of the study met the criteria for Extent of Evidence. WWC (What Works Clearinghouse) applies Extent of Evidence each study in terms of its effectiveness and its rigor. The Extent of Evidence for Everyday Mathematics is medium to large for math achievement.
Many teachers use these activities as a warming up activity when students arrive to school each morning. The exercises are short, mostly reviews of previous material. Students are encouraged to collaborate on these activities, and they are generally short enough and accessible enough (they do not usually require any kind of research) for special needs students to use them and complete them in a reasonable amount of time. A typical activity will use manipulatives for counting or for graph-building, and some require calculator use. Others are puzzles that involve mathematical reasoning. When special needs children are collaborating with their peers, the work generally goes smoothly and the special needs children seem to understand most of the work, and with some prodding, can do the same concepts again, and with repeated practice over time, become independently able to do the activities. The theoretical foundation is clearly correct here-short introductions to concepts, repeated practice and multiple exposures work best for special needs children, who usually do not have the attention span or intellectual capability to immerse themselves in mathematical concepts for long periods of time. Though the activities do require subskills such as measuring with a ruler, how to make simple graphs, and other basics of mathematical inquiry, special needs children seem able to learn some of this "on the fly" and complete their activities successfully.
Conclusion
Both systems work very well for all students, including mildly or moderately disabled children. RTI is much more comprehensive and intense than is Everyday Mathematics. RTI is a total curricular effort, while EM is a supplemental activity regimen that reinforces skills. RTI requires a commitment from all teachers in the school, while EM is normally a simple part of each elementary teacher's morning list of tasks to complete, with no particular commitment required, and no allocation of resources beyond the ordinary. RTI is designed to identify and intervene with at-risk students so that they do not fail. EM is designed to supplement practice and does not distinguish students who are at risk of failure, nor does it provide progressively more intense instruction to those who are at risk. RTI is a structured tier approach that progressively supports those students who are not minimally competent at basic instructional levels. EM offers no such support.
For special needs students, RTI is an extensively supporting system taking into account quality of instruction with effectiveness and efficiency. Though EM is used daily, and has shown to be effective at reinforcing mathematical skills, it is not an extensive use of time and though effective, is not necessarily efficient, unless sit is used by the classroom teacher to identify specific skill needs that need remediation. It is likely that students who would qualify for Tier 3 of RTI would find little success with EM when working independently.
REFERENCES
US Department of Education, Institute of Education Sciences. (2007). Intervention: Everyday Mathematics.
Carroll, J. B. (1963). A model of school learning. Teachers College Record, 64, 723-733. National Center for Learning Disabilities. (2010). Tiered instruction and intervention in a Response-to-Intervention Model.
Carroll, W. M. (1998). Geometric knowledge of middle school students in a reform-based mathematics curriculum. School Science and Mathematics, 98(4), 188-197. Everyday Mathematics. (n.d). McGraw Hill.
Haley, C. (2007). Response to Intervention: Overview. PDE/BSE.
Riordan, J. E., & Noyce, P. E. (2001). The impact of two standards-based mathematics curricula on student achievement in Massachusetts. Journal for Research in Mathematics Education, 32(4), 368-398.
US Department of Education, Institute of Education Sciences. (2007). Intervention: Everyday Mathematics.
Waite, R. D. (2000). A study of the effects of Everyday Mathematics on student achievement of third-, fourth-, and fifth-grade students in a large north Texas urban school district. Dissertation Abstracts International, 61(10), 3933A. (UMI No. 9992659).
Woodward, J., & Baxter, J. (1997). The effects of an innovative approach to mathematics on academically low-achieving students in inclusive settings. Exceptional Children, 63(3), 373-388.