Number Worlds Effective With Special Needs Students
The Effectiveness of SRA/McGraw-Hill Number Worlds Program on Fluency in Math Calculation for Middle School Students Identified with Special Needs By SKF Educational Services, LLC
Introduction Recent educational mandates have underscored the need for increased achievement in mathematics. The No Child Left Behind (NCLB) Act of 2001, a federally-mandated law enacted in 2002, requires public schools to make “adequate yearly progress” (AYP) in their efforts to have all students in all schools achieve proficiency in math by the 2013-14 school year. While many students will meet this goal, students with disabilities often struggle to achieve proficiency. Estimations are that between 5% and 8% of school age students have a disability in mathematics, which often manifests as difficulty in the recall of basic facts (Geary, 2004). Being fluent in basic fact computation is essential if students are to perform more complex skills as learning progresses, and is in fact a significant predictor for successful complex problem solving (Zentall & Ferkis, 1993). Mathematical fluency generally consists of speed and accuracy in performing mathematical computation, and for students with developmental or other disabilities acquiring this skill is particularly difficult. However, it is crucial that students develop mathematical fluency, as increased levels of fluency are highly correlated with increased performance on achievement tests (Skiba, Magnusson, Marston, and Erikson, 1986) and with maintenance of math skills over time (Singer-Dudeck and Greer, 2005). Current research suggests that students with disabilities often benefit from instructional programs that are explicit, provide sufficient opportunities for drill and practice, and provide frequent corrective feedback (Kroesbergen & Van Luit, 2003; Jolivette, Lingo, Houchins, Barton-Arwood & Shippen, 2006).
Purpose of Study To address student difficulties in math achievement, SRA/McGraw-Hill published Number Worlds, a Tier 2/Tier3 intervention program. Number Worlds, initially developed to target Kindergarten students who lacked the knowledge network for number sense, has since been expanded to teach a broad range of mathematical understandings for children in preschool through eighth grade (Griffin, 2004). The purpose of this study was to investigate the effects of Number Worlds on math achievement for a group of middle school students identified with special needs. Specifically, the primary research questions are: 1. What effect does Number Worlds have on the math fluency skills of a select group of students identified with special needs? 2. How does participation in the Number Worlds program affect these students’ performance on state-mandated assessments?
1
Research Design Educators and researchers must seek practical and empirically robust tools to facilitate the determination of an intervention’s presence, stability, and durability of treatment effects. Singlesubject designs, as employed in this case, allow educators to investigate the process of change for a particular child, not the average child. Unlike its correlational and descriptive cousins, single-subject design methodology is experimental; its purpose is to determine causal or functional relationships between variables (Horner, Carr, Halle, McGee, Odom, & Wolery, 2005). While there are many variants of single-subject designs, most involve only one participant or a small group of participants (3 to 8) in a single study; the dependent variables are typically observations of a target behavior; and the independent variable is a specified program or intervention procedure that is actively manipulated and carefully monitored throughout the investigation. An effect is demonstrated when the change in the target behavior covaries with manipulation of the intervention. This study utilized a multiple baseline design. Generally, multiple baseline designs contain the following elements: (a) repeated measurement of the dependent variable across at least two baselines; (b) staggered introduction of treatment across baselines, and; (c) immediate observed effects of the intervention with no observable effects in conditions in which the intervention has not been implemented. In the multiple baseline across subjects design, the same intervention is ‘staggered’ over time, and the same behavior monitored throughout the intervention.
Sample The middle school selected for this study is in a relatively low-income school district located in central Ohio. According to the 2000 census, the town’s population is 13,500 and the median household income is about $26,700. The city school district, with an average daily enrollment of 2,338 students, is predominantly Caucasian (94.3%) and economically disadvantaged (39.4%). The special education population approximates 16%, and is the highest of surrounding districts. The district’s subpopulation of middle school students with special needs was selected for the intervention program, as this particular group experienced a failure to meet AYP in reading and mathematics. According to the information provided on the Ohio Department of Education website, during the 2008 school year 70.9% of the district’s middle school students with disabilities scored in the ‘Limited’ or ‘Basic’ range on the math portion of the Ohio Achievement Test (OAT). The original sample selected for this study included 27 middle school students ranging in grade from the 6th to the 8th grade. All students were identified with special needs early in elementary school and had a history of receiving special needs services, primarily in a resource room setting. Shortly after program inception, one student was removed from the school and subsequently placed on home instruction. Three students were excluded from the analysis based on poor attendance/frequent removals from class. The final sample yielded 23 students available for analysis. All students except one are on free/reduced lunch. The student sample contains 24 Caucasian students, 1 African American student, and 1 Hispanic student, and slightly more girls (n = 13) than boys (n = 10). All students are eligible for special needs services, according to the state of Ohio’s criteria for program eligibility. Figure 1 provides a distribution summary of student by special needs 2
placement. Approximately 61% of students (n = 14) are identified as ‘cognitively disabled’ (CD) based on significant weaknesses in cognitive functioning and adaptive behavior skills. Roughly 26% of students (n = 6) are identified with a learning disability. Two students are identified as Other Health Impaired (OHI), based on a diagnosis of ADD or ADHD and associated learning difficulties. One student is identified as having an emotional disturbance (ED), meaning that the student’s behavior significantly impedes learning. The mean IQ for this sample is 73, considered in the ‘Low’ or ‘Borderline’ range.
Figure 1. Distribution of Sample by Special Needs Status
All students participating in the program demonstrated specific weaknesses in math, and received core math instruction in a resource room setting. Prior to administration of the Number Worlds program, students were placed into homogenous groupings based on their performance on the Number Worlds Placement Tests. The placement tests, consisting of 20 to 25 items arranged in a multiple-choice format, are administered to students to determine which level of instruction is most appropriate. For the group of students in this study, 14 placed at a Number Worlds level ‘C’ and 9 placed at Number Worlds level ‘D’. Program instruction occurred in the resource room setting, with roughly 5 to 8 students per grouping. Instruction was provided by an Ohio-certified teacher with special needs training and licensure. The Number Worlds program began in September of 2009 and will continue until May of 2010.
3
Data Preliminary findings presented in this report reflect the first four months of Number Worlds instruction. Data collected over the course of the intervention is in the form of 60-second fluency probes, administered monthly. The 60-second fluency assessments have practical and functional utility in the classroom setting and have demonstrated reliability and validity (Axtell, McCallum, Bell, and Poncy, 2009; Miller, Hall, and Heward, 1995;). The probes consist of an array of 100 addition, subtraction, multiplication, and division facts arranged in a random order. Probes were generated using a math probe worksheet creator, easily accessed on www.interventioncentral.com . Students were administered three baseline probes at the beginning of September and prior to program administration. After implementation of Number Worlds, student progress was monitored via the monthly administration of math fluency probes. On the baseline and intervention assessments, students were instructed to work as quickly as they could and complete as many items as possible until the examiner said ‘stop’. Students were informed that they could skip items they did not know. After administration of the fluency probes, the number of items answered correctly was tallied and recorded.
Results The following results reflect the first four months of Number Worlds instruction. Level C Figures 2 though 6 represent the calculation fluency scores for the 14 students participating in Number Worlds, Level C. For ease of interpretation, each graph contains the performance of three students. The first data point for each student represents the median score of the baseline assessments. The vertical line demarks the boundary between the baseline data and the intervention data. Each data point to the left of the vertical line reflects the student’s performance on each subsequent progress monitoring assessment. The x-axis represents the number of math calculation items correctly answered by the student. The y-axis reflects month of program instruction, where baseline represents ‘September’, ‘1’ represents ‘October’, and so forth.
4
Figure 2. Calculation Fluency Scores for Students A, B, and C
Figure 3. Calculation Fluency Scores for Students D, E, and F
5
Figure 4. Calculation Fluency Scores for Students G, H, and I
Figure 5. Calculation Fluency Scores for Students J, K, and L
6
Figure 6. Calculation Fluency Scores for Students M and N
Level D Figures 7, 8, and 9 represent calculation fluency for the 9 students participating in Number Worlds level D. For ease of interpretation, each figure presents the performance of three students.
Figure 7. Calculation Fluency Scores for Students AA, BB, and CC
7
Figure 8. Calculation Fluency Scores for Students DD, EE, and FF
Figure 9. Calculation Fluency Scores for Students GG, HH, and II
8
Data analysis was conducted using the Percent of Non-overlapping Data technique (PND). First described by Cohen (1988), the PND is a commonly-used method for analyzing data in singlesubject designs where performance after treatment is compared to performance before treatment (Parker and Hagan-Burke, 2007). The PND is calculated by first determining the number of data points in the intervention phase that exceeds the highest data point in the baseline phase. This value is divided by the total number of data points in the intervention phase, and multiplied by 100, yielding a percentage score. Higher percentage scores reflect more efficacious interventions: values of 90% or higher reflect ‘highly effective’ interventions; values of 70% to under 90% reflect ‘moderately effective’ interventions; values from 50% to under 70% reflect ‘mild’ or ‘questionably effective’ interventions; and values below 50% reflect an ‘ineffective intervention’ (Ma, 2006). Table 1 presents the PND by category for the two instructional levels of Number Worlds. Table 1. Percent of Non-Overlapping Data for Students, Number Worlds Levels C and D
Level C Category
n Highly Effective 9 Moderately Effective 3 Mildly Effective 0 Ineffective 2
D % 64 21 0 14
n 5 2 0 2
% 56 22 0 22
Table 2 presents the PND by category for the sample as a whole (not aggregated by instructional level). Preliminary results suggest that Number Worlds instruction is ‘highly effective’ in increasing math fluency for 61% of students identified with special needs. For 22% of students, Number Worlds is considered ‘moderately effective’. Roughly 17% of students (n = 4) did not significantly increase their fluency in basic math calculation above baseline levels. Table 2. Percent of Non-Overlapping Data for Students (Total)
Category
n Highly Effective 14 Moderately Effective 5 Mildly Effective 0 Ineffective 4
% 61 22 0 17
9
Teacher Interview Prior to the administration of the January progress monitoring assessment, a semi-structured interview was conducted with the intervention specialist responsible for administering the Number Worlds program. The purpose of the interview was to gain the teacher’s perception on the overall utility of the program, to provide student feedback, and to obtain suggestions for improvement after four months of using the program. The intervention specialist stated that she most likes the ‘booklets the students work with…they are simple and yet cover topics in a way that make students think’. She further states that the ’leveling’ of the books and the weekly testing ‘keeps them on their toes’. Student feedback has been consistent, as most students state they enjoy ‘the workbooks and the practice on Building Blocks’(the accompanying software). The intervention specialist indicated that the Number Worlds program has been particularly helpful in unanticipated ways, in that ‘it has helped with my organization and to be able to differentiate instruction more’. This last comment is of particular importance as she reported: I have many students at different levels-mixed levels, based on grade or reading levels. This makes it difficult to teach a lesson. However, I have been able to work with students one-on-one more often with this program. Once they have the routine down, they are doing much better.
Discussion Preliminary results reveal that the Number Worlds program increases calculation fluency in a sizeable percentage of students identified with special needs. For approximately 83% of students in this sample, Number Worlds is highly to moderately effective. There exists a slight discrepancy in performance between students in Level C and Level D; Number Worlds was highly to moderately effective for increasing math fluency skills for slightly more students in the Level C group (85%) than for students in the Level D group (78%). A possible explanation for these findings might reflect strategies students used to solve the problems. On several occasions, many students (e.g., Students K, L, and BB) used time-consuming strategies (e.g., counting on fingers, making ‘dots’ or ‘tick marks’ on paper) to assist with calculation. While considered acceptable strategies, they are time-consuming and had a detrimental effect on performance. Further observation reveals that several low-performing students ( Students D, G, and J) have difficulty with fine motor control and tend to write slowly and deliberately. Student performance may be a reflection of poor fine motor control rather than lack of knowledge of basic facts. Alternative means of assessment will be explored for these students as the study progresses over the remaining four months of the school year.
10
References Axtell, P. K., McCallum, S., Bell., S. M., & Poncy, B. (2009). Developing math automaticity using a class-wide fluency building procedure for middle school students: A preliminary study. Psychology in the Schools, 46(6), 526 – 538. Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum. Geary, D. C., (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4 – 15. Griffin, S. (2004). Building number sense with Number Worlds: A mathematics program for young children. Early Childhood Research Quarterly, 19 (1), 173-180. Horner, R.H., Carr, E.G., Halle, J., McGee, G., Odom, S., and Wolery, M. (2005). The use of single-subject research to identify evidence-based practice in special education. Council for Exceptional Children, 71(2), 165-179. Jolivette, K., Lingo, A. S., Houchins, D. E., Barton-Arwood, S. M., & Shippen, M. E. (2006). Building math fluency for students with developmental disabilities and attentional difficulties using Great Leaps Math. Education and Training in Developmental Disabilities, 41(4), 392- 400. Kroesbergen, E., & Van Luit, J. (2003). Mathematics interventions for children with special education needs. Remedial and Special Education, 24, 97-114. Ma, Hsen-Hsing (2006). An alternative method for quantitative synthesis of single-subject researches: Percentage of data points exceeding the mean. Behavior Modification, 30 (5), 598-617. Miller, S. P., Hall, S. W., & Heward, W. L. (1995). Effects of sequential 1-minute time trials with and without inter-trial feedback and self-correction on general and special education students’ fluency with math facts. Journal of Behavioral Education, 5, 319 – 345. Parker, R. I., & Hagan-Burke, S. (2007). Useful effect size interpretations for single case research. Behavior Therapy, 38, 95 – 105. Singer-Dudeck, J., & Greer, R. D. (2005). A long-term analysis of the relationship between fluency and the training and maintenance of complex math skills. The Psychology Record, 55, 361 – 376. Skiba, R., Magnusson, D., Marston, D., & Erikson, K. (1986). The assessment of mathematics performance in special education: Achievement tests, proficiency tests, and formative evaluation. Minneapolis: Special Services, Minneapolis Public Schools. Zentall, S. S., & Ferkis, M. A. (1993). Mathematical problem solving for youth with ADHD, with and without learning disabilities. Learning Disability Quarterly, 16, 6 – 17.
11
12
Introduction Recent educational mandates have underscored the need for increased achievement in mathematics. The No Child Left Behind (NCLB) Act of 2001, a federally-mandated law enacted in 2002, requires public schools to make “adequate yearly progress” (AYP) in their efforts to have all students in all schools achieve proficiency in math by the 2013-14 school year. While many students will meet this goal, students with disabilities often struggle to achieve proficiency. Estimations are that between 5% and 8% of school age students have a disability in mathematics, which often manifests as difficulty in the recall of basic facts (Geary, 2004). Being fluent in basic fact computation is essential if students are to perform more complex skills as learning progresses, and is in fact a significant predictor for successful complex problem solving (Zentall & Ferkis, 1993). Mathematical fluency generally consists of speed and accuracy in performing mathematical computation, and for students with developmental or other disabilities acquiring this skill is particularly difficult. However, it is crucial that students develop mathematical fluency, as increased levels of fluency are highly correlated with increased performance on achievement tests (Skiba, Magnusson, Marston, and Erikson, 1986) and with maintenance of math skills over time (Singer-Dudeck and Greer, 2005). Current research suggests that students with disabilities often benefit from instructional programs that are explicit, provide sufficient opportunities for drill and practice, and provide frequent corrective feedback (Kroesbergen & Van Luit, 2003; Jolivette, Lingo, Houchins, Barton-Arwood & Shippen, 2006).
Purpose of Study To address student difficulties in math achievement, SRA/McGraw-Hill published Number Worlds, a Tier 2/Tier3 intervention program. Number Worlds, initially developed to target Kindergarten students who lacked the knowledge network for number sense, has since been expanded to teach a broad range of mathematical understandings for children in preschool through eighth grade (Griffin, 2004). The purpose of this study was to investigate the effects of Number Worlds on math achievement for a group of middle school students identified with special needs. Specifically, the primary research questions are: 1. What effect does Number Worlds have on the math fluency skills of a select group of students identified with special needs? 2. How does participation in the Number Worlds program affect these students’ performance on state-mandated assessments?
1
Research Design Educators and researchers must seek practical and empirically robust tools to facilitate the determination of an intervention’s presence, stability, and durability of treatment effects. Singlesubject designs, as employed in this case, allow educators to investigate the process of change for a particular child, not the average child. Unlike its correlational and descriptive cousins, single-subject design methodology is experimental; its purpose is to determine causal or functional relationships between variables (Horner, Carr, Halle, McGee, Odom, & Wolery, 2005). While there are many variants of single-subject designs, most involve only one participant or a small group of participants (3 to 8) in a single study; the dependent variables are typically observations of a target behavior; and the independent variable is a specified program or intervention procedure that is actively manipulated and carefully monitored throughout the investigation. An effect is demonstrated when the change in the target behavior covaries with manipulation of the intervention. This study utilized a multiple baseline design. Generally, multiple baseline designs contain the following elements: (a) repeated measurement of the dependent variable across at least two baselines; (b) staggered introduction of treatment across baselines, and; (c) immediate observed effects of the intervention with no observable effects in conditions in which the intervention has not been implemented. In the multiple baseline across subjects design, the same intervention is ‘staggered’ over time, and the same behavior monitored throughout the intervention.
Sample The middle school selected for this study is in a relatively low-income school district located in central Ohio. According to the 2000 census, the town’s population is 13,500 and the median household income is about $26,700. The city school district, with an average daily enrollment of 2,338 students, is predominantly Caucasian (94.3%) and economically disadvantaged (39.4%). The special education population approximates 16%, and is the highest of surrounding districts. The district’s subpopulation of middle school students with special needs was selected for the intervention program, as this particular group experienced a failure to meet AYP in reading and mathematics. According to the information provided on the Ohio Department of Education website, during the 2008 school year 70.9% of the district’s middle school students with disabilities scored in the ‘Limited’ or ‘Basic’ range on the math portion of the Ohio Achievement Test (OAT). The original sample selected for this study included 27 middle school students ranging in grade from the 6th to the 8th grade. All students were identified with special needs early in elementary school and had a history of receiving special needs services, primarily in a resource room setting. Shortly after program inception, one student was removed from the school and subsequently placed on home instruction. Three students were excluded from the analysis based on poor attendance/frequent removals from class. The final sample yielded 23 students available for analysis. All students except one are on free/reduced lunch. The student sample contains 24 Caucasian students, 1 African American student, and 1 Hispanic student, and slightly more girls (n = 13) than boys (n = 10). All students are eligible for special needs services, according to the state of Ohio’s criteria for program eligibility. Figure 1 provides a distribution summary of student by special needs 2
placement. Approximately 61% of students (n = 14) are identified as ‘cognitively disabled’ (CD) based on significant weaknesses in cognitive functioning and adaptive behavior skills. Roughly 26% of students (n = 6) are identified with a learning disability. Two students are identified as Other Health Impaired (OHI), based on a diagnosis of ADD or ADHD and associated learning difficulties. One student is identified as having an emotional disturbance (ED), meaning that the student’s behavior significantly impedes learning. The mean IQ for this sample is 73, considered in the ‘Low’ or ‘Borderline’ range.
Figure 1. Distribution of Sample by Special Needs Status
All students participating in the program demonstrated specific weaknesses in math, and received core math instruction in a resource room setting. Prior to administration of the Number Worlds program, students were placed into homogenous groupings based on their performance on the Number Worlds Placement Tests. The placement tests, consisting of 20 to 25 items arranged in a multiple-choice format, are administered to students to determine which level of instruction is most appropriate. For the group of students in this study, 14 placed at a Number Worlds level ‘C’ and 9 placed at Number Worlds level ‘D’. Program instruction occurred in the resource room setting, with roughly 5 to 8 students per grouping. Instruction was provided by an Ohio-certified teacher with special needs training and licensure. The Number Worlds program began in September of 2009 and will continue until May of 2010.
3
Data Preliminary findings presented in this report reflect the first four months of Number Worlds instruction. Data collected over the course of the intervention is in the form of 60-second fluency probes, administered monthly. The 60-second fluency assessments have practical and functional utility in the classroom setting and have demonstrated reliability and validity (Axtell, McCallum, Bell, and Poncy, 2009; Miller, Hall, and Heward, 1995;). The probes consist of an array of 100 addition, subtraction, multiplication, and division facts arranged in a random order. Probes were generated using a math probe worksheet creator, easily accessed on www.interventioncentral.com . Students were administered three baseline probes at the beginning of September and prior to program administration. After implementation of Number Worlds, student progress was monitored via the monthly administration of math fluency probes. On the baseline and intervention assessments, students were instructed to work as quickly as they could and complete as many items as possible until the examiner said ‘stop’. Students were informed that they could skip items they did not know. After administration of the fluency probes, the number of items answered correctly was tallied and recorded.
Results The following results reflect the first four months of Number Worlds instruction. Level C Figures 2 though 6 represent the calculation fluency scores for the 14 students participating in Number Worlds, Level C. For ease of interpretation, each graph contains the performance of three students. The first data point for each student represents the median score of the baseline assessments. The vertical line demarks the boundary between the baseline data and the intervention data. Each data point to the left of the vertical line reflects the student’s performance on each subsequent progress monitoring assessment. The x-axis represents the number of math calculation items correctly answered by the student. The y-axis reflects month of program instruction, where baseline represents ‘September’, ‘1’ represents ‘October’, and so forth.
4
Figure 2. Calculation Fluency Scores for Students A, B, and C
Figure 3. Calculation Fluency Scores for Students D, E, and F
5
Figure 4. Calculation Fluency Scores for Students G, H, and I
Figure 5. Calculation Fluency Scores for Students J, K, and L
6
Figure 6. Calculation Fluency Scores for Students M and N
Level D Figures 7, 8, and 9 represent calculation fluency for the 9 students participating in Number Worlds level D. For ease of interpretation, each figure presents the performance of three students.
Figure 7. Calculation Fluency Scores for Students AA, BB, and CC
7
Figure 8. Calculation Fluency Scores for Students DD, EE, and FF
Figure 9. Calculation Fluency Scores for Students GG, HH, and II
8
Data analysis was conducted using the Percent of Non-overlapping Data technique (PND). First described by Cohen (1988), the PND is a commonly-used method for analyzing data in singlesubject designs where performance after treatment is compared to performance before treatment (Parker and Hagan-Burke, 2007). The PND is calculated by first determining the number of data points in the intervention phase that exceeds the highest data point in the baseline phase. This value is divided by the total number of data points in the intervention phase, and multiplied by 100, yielding a percentage score. Higher percentage scores reflect more efficacious interventions: values of 90% or higher reflect ‘highly effective’ interventions; values of 70% to under 90% reflect ‘moderately effective’ interventions; values from 50% to under 70% reflect ‘mild’ or ‘questionably effective’ interventions; and values below 50% reflect an ‘ineffective intervention’ (Ma, 2006). Table 1 presents the PND by category for the two instructional levels of Number Worlds. Table 1. Percent of Non-Overlapping Data for Students, Number Worlds Levels C and D
Level C Category
n Highly Effective 9 Moderately Effective 3 Mildly Effective 0 Ineffective 2
D % 64 21 0 14
n 5 2 0 2
% 56 22 0 22
Table 2 presents the PND by category for the sample as a whole (not aggregated by instructional level). Preliminary results suggest that Number Worlds instruction is ‘highly effective’ in increasing math fluency for 61% of students identified with special needs. For 22% of students, Number Worlds is considered ‘moderately effective’. Roughly 17% of students (n = 4) did not significantly increase their fluency in basic math calculation above baseline levels. Table 2. Percent of Non-Overlapping Data for Students (Total)
Category
n Highly Effective 14 Moderately Effective 5 Mildly Effective 0 Ineffective 4
% 61 22 0 17
9
Teacher Interview Prior to the administration of the January progress monitoring assessment, a semi-structured interview was conducted with the intervention specialist responsible for administering the Number Worlds program. The purpose of the interview was to gain the teacher’s perception on the overall utility of the program, to provide student feedback, and to obtain suggestions for improvement after four months of using the program. The intervention specialist stated that she most likes the ‘booklets the students work with…they are simple and yet cover topics in a way that make students think’. She further states that the ’leveling’ of the books and the weekly testing ‘keeps them on their toes’. Student feedback has been consistent, as most students state they enjoy ‘the workbooks and the practice on Building Blocks’(the accompanying software). The intervention specialist indicated that the Number Worlds program has been particularly helpful in unanticipated ways, in that ‘it has helped with my organization and to be able to differentiate instruction more’. This last comment is of particular importance as she reported: I have many students at different levels-mixed levels, based on grade or reading levels. This makes it difficult to teach a lesson. However, I have been able to work with students one-on-one more often with this program. Once they have the routine down, they are doing much better.
Discussion Preliminary results reveal that the Number Worlds program increases calculation fluency in a sizeable percentage of students identified with special needs. For approximately 83% of students in this sample, Number Worlds is highly to moderately effective. There exists a slight discrepancy in performance between students in Level C and Level D; Number Worlds was highly to moderately effective for increasing math fluency skills for slightly more students in the Level C group (85%) than for students in the Level D group (78%). A possible explanation for these findings might reflect strategies students used to solve the problems. On several occasions, many students (e.g., Students K, L, and BB) used time-consuming strategies (e.g., counting on fingers, making ‘dots’ or ‘tick marks’ on paper) to assist with calculation. While considered acceptable strategies, they are time-consuming and had a detrimental effect on performance. Further observation reveals that several low-performing students ( Students D, G, and J) have difficulty with fine motor control and tend to write slowly and deliberately. Student performance may be a reflection of poor fine motor control rather than lack of knowledge of basic facts. Alternative means of assessment will be explored for these students as the study progresses over the remaining four months of the school year.
10
References Axtell, P. K., McCallum, S., Bell., S. M., & Poncy, B. (2009). Developing math automaticity using a class-wide fluency building procedure for middle school students: A preliminary study. Psychology in the Schools, 46(6), 526 – 538. Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum. Geary, D. C., (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4 – 15. Griffin, S. (2004). Building number sense with Number Worlds: A mathematics program for young children. Early Childhood Research Quarterly, 19 (1), 173-180. Horner, R.H., Carr, E.G., Halle, J., McGee, G., Odom, S., and Wolery, M. (2005). The use of single-subject research to identify evidence-based practice in special education. Council for Exceptional Children, 71(2), 165-179. Jolivette, K., Lingo, A. S., Houchins, D. E., Barton-Arwood, S. M., & Shippen, M. E. (2006). Building math fluency for students with developmental disabilities and attentional difficulties using Great Leaps Math. Education and Training in Developmental Disabilities, 41(4), 392- 400. Kroesbergen, E., & Van Luit, J. (2003). Mathematics interventions for children with special education needs. Remedial and Special Education, 24, 97-114. Ma, Hsen-Hsing (2006). An alternative method for quantitative synthesis of single-subject researches: Percentage of data points exceeding the mean. Behavior Modification, 30 (5), 598-617. Miller, S. P., Hall, S. W., & Heward, W. L. (1995). Effects of sequential 1-minute time trials with and without inter-trial feedback and self-correction on general and special education students’ fluency with math facts. Journal of Behavioral Education, 5, 319 – 345. Parker, R. I., & Hagan-Burke, S. (2007). Useful effect size interpretations for single case research. Behavior Therapy, 38, 95 – 105. Singer-Dudeck, J., & Greer, R. D. (2005). A long-term analysis of the relationship between fluency and the training and maintenance of complex math skills. The Psychology Record, 55, 361 – 376. Skiba, R., Magnusson, D., Marston, D., & Erikson, K. (1986). The assessment of mathematics performance in special education: Achievement tests, proficiency tests, and formative evaluation. Minneapolis: Special Services, Minneapolis Public Schools. Zentall, S. S., & Ferkis, M. A. (1993). Mathematical problem solving for youth with ADHD, with and without learning disabilities. Learning Disability Quarterly, 16, 6 – 17.
11
12
