Research-Based Math Interventions for Students with Disabilities

1. Research-Based Math Interventions for Students with Disabilities Melissa Storm The Access Center

2. Doubly true for students with disabilities, who may also have language-based processing difficulties, in addition to insecurity in their computational and problem-solving ability. Doubly true for students with disabilities, who may also have language-based processing difficulties, in addition to insecurity in their computational and problem-solving ability.

3. Overview Math Standards Interventions for Students with Disabilities Effective Teaching Practices California Algebra I Requirement

4. NCTM Goals (1989, 2000) Learning to value mathematics Becoming confident in their ability to do mathematics Becoming mathematical problem solvers Learning to communicate mathematically Learning to reason mathematically A change in math instruction; from rote to process skills These are grade band based K-2, 3-5, 6-8, 9-12; not individual grade levels. Math is the first group to have developed standards and goals and followed through on publishing them and revising in 2000. Goals and standards are presented in broad based grade bands in order for all school districts to match to their state requirements as well.A change in math instruction; from rote to process skills These are grade band based K-2, 3-5, 6-8, 9-12; not individual grade levels. Math is the first group to have developed standards and goals and followed through on publishing them and revising in 2000. Goals and standards are presented in broad based grade bands in order for all school districts to match to their state requirements as well.

5. Equity Curriculum Effective Teaching Learning Assessment Importance of Technology Six NCTM General Principlesfor School Mathematics At the top of the list is equity: math is for all students, regardless of personal characteristics, background, or physical challenges. The standards are making an attempt to respond to SWD in a general statement to individualize appropriate strategies for each child to access knowledge of mathematics. VanDeWalle in his 4th Edition of Elementary and Middle School Mathematics-Teaching Developmentally in Chapter 23 has responded to specific ways of teaching all children mathematics. Some in special education have concerns about the inquiry or discovery approaches for students with disabilities who may have weaknesses in sight or deductive inference. Previously, special ed did have a belief on drill and skill/kill. NCTM does not mention students with disabilities specifically but addresses the equity issue. National Research Council’s Everybody Counts does mention disabilities but specific to individuals with physical disabilities. At the top of the list is equity: math is for all students, regardless of personal characteristics, background, or physical challenges. The standards are making an attempt to respond to SWD in a general statement to individualize appropriate strategies for each child to access knowledge of mathematics. VanDeWalle in his 4th Edition of Elementary and Middle School Mathematics-Teaching Developmentally in Chapter 23 has responded to specific ways of teaching all children mathematics. Some in special education have concerns about the inquiry or discovery approaches for students with disabilities who may have weaknesses in sight or deductive inference. Previously, special ed did have a belief on drill and skill/kill. NCTM does not mention students with disabilities specifically but addresses the equity issue. National Research Council’s Everybody Counts does mention disabilities but specific to individuals with physical disabilities.

6. Math Difficulties Memory Language and communication disorders Processing Difficulties Poor self-esteem Attention Organizational Skills NCTM emphasizes conceptual learning and problem solving to improve math achievement. Alternative approaches or different ways of solving problems are encouraged with an explanation or visual drawing of how the student reached the answer. NCTM emphasizes conceptual learning and problem solving to improve math achievement. Alternative approaches or different ways of solving problems are encouraged with an explanation or visual drawing of how the student reached the answer.

7. Interventions Found Effective for Students with Disabilities Manipulatives Concrete-Semi-concrete-Abstract Instruction Mnemonics Meta-cognitive strategies: Self-monitoring, Self-Instruction Computer-Assisted Instruction Explicit Instruction Deborah Ball, "Magical Hopes: Manipulatives and the Reform of Math Education", American Educator, Summer 1992 Generally on computation, some on problem-solving Mastropieri, Scruggs, & Shiah (1991), only one study on algebra; Jitendra & Xin, Handout on resource table includes various resources and research on manipulatives. Deborah Ball, "Magical Hopes: Manipulatives and the Reform of Math Education", American Educator, Summer 1992 Generally on computation, some on problem-solving Mastropieri, Scruggs, & Shiah (1991), only one study on algebra; Jitendra & Xin, Handout on resource table includes various resources and research on manipulatives.

8. Research on Using Manipulatives The use of concrete materials – Can produce meaningful use of notational systems Can increase student concept development Is positively related to increases in student mathematics achievement Is positively related to improved attitudes towards mathematics. In a comprehensive review of activity based learning in mathematics in kindergarten through grade eight, Suydam and Higgins concluded that using manipulative materials produces greater achievement gains than not using them. In a more recent meta-analysis of sixty studies (kindergarten through post-secondary) that compared the effects of using concrete materials with the effects of more abstract instruction, Sowell concluded that the long-term use of concrete instructional materials by teachers knowledgeable in their use improved student achievement and attitudes. For example, in studies by Resnick and Omanson and by Labinowicz, the use of base-ten blocks showed little impact on children’s learning. In contrast, both Fuson and Briars and Hiebert and Wearne reported positive results from the use of base-ten blocks. The differences in results among these studies might be due to the nature of the students’ engagement with the concrete materials and their orientation towards the materials in relation to notation and numerical values. In general, however, the ambiguities in some of the research findings do not undermine the general consensus that concrete materials are valuable instructional tools. In a comprehensive review of activity based learning in mathematics in kindergarten through grade eight, Suydam and Higgins concluded that using manipulative materials produces greater achievement gains than not using them. In a more recent meta-analysis of sixty studies (kindergarten through post-secondary) that compared the effects of using concrete materials with the effects of more abstract instruction, Sowell concluded that the long-term use of concrete instructional materials by teachers knowledgeable in their use improved student achievement and attitudes. For example, in studies by Resnick and Omanson and by Labinowicz, the use of base-ten blocks showed little impact on children’s learning. In contrast, both Fuson and Briars and Hiebert and Wearne reported positive results from the use of base-ten blocks. The differences in results among these studies might be due to the nature of the students’ engagement with the concrete materials and their orientation towards the materials in relation to notation and numerical values. In general, however, the ambiguities in some of the research findings do not undermine the general consensus that concrete materials are valuable instructional tools.

9. Issues with Manipulatives Teachers may not trust the usefulness or efficiency of manipulative objects for higher-level algebra. Classroom limitations: Rigid schedules; movement of students and teachers; organization and supply of manipulatives. Dominance of textbook lessons Confidence of teachers in their mathematics knowledge compared to confidence in the use of manipulatives One study (Howard & Perry) secondary teachers used manipulatives once a month; primary teachers used daily.

10. Concrete-Semi-concrete-Abstract (C-S-A) Phase of Instruction C-S-A is an instructional sequence supporting students’ understanding of mathematical concepts. In the concrete phase, students represent the problem with concrete objects - manipulatives. In the semi-concrete or representational phase, students draw or use pictorial representations of the quantities During the abstract phase of instruction, students involve numeric representations, instead of pictorial displays. C-S-A is often integrated with meta-cognitive instruction, i.e. mnemonics

11. Mnemonics Star Strategy Search the word problem Translate the word into an equation in picture form Answer the problem Review the solution (Maccini & Gagnon’s article, “Preparing Students with Disabilities for Algebra”) STAR strategy refers to Maccini & Gagnon’s article, “Preparing Students with Disabilities for Algebra.” The strategy is outlined on the next few slides. STAR strategy refers to Maccini & Gagnon’s article, “Preparing Students with Disabilities for Algebra.” The strategy is outlined on the next few slides.

12. Using the Star Strategy Search the word problem Students read the problem carefully, Regulate their thinking through self-questions, “What facts do I know? “What do I need to find?” and, Write down facts. Translate the words into an equation in picture form Students choose a variable for the unknown Identify the operation (s) Represent the problem using CONCRETE APPLICATION of CSA. Draw a picture of the representation (SEMI-CONCRETE) Write an algebraic equation (ABSTRACT application)

13. Using the Star Strategy Answer the Problem Use the appropriate operations (+, -, x or / ) Use rules of solving simple equations Use rules to add/subtract positive and negative numbers Review the solution Reread the problem Check the reasonableness of the answer Check the answer.

14. Meta-cognitive Strategies Self-Instruction Strategies include: Advanced or Graphic Organizers Support from structured worksheets and strategy instruction General guidelines to direct themselves: 1. Re-read information for clarity; 2. Diagram representation of the problems before solving them; 3. Write algebraic equations for solving the problems. Many studies found that prior to instruction many students bypassed problem representation and started with trying to solve the problems.Many studies found that prior to instruction many students bypassed problem representation and started with trying to solve the problems.

15. Examples ofSelf-Monitoring Strategies Cue cards to ask themselves while representing problems (card is eventually withdrawn) Structured worksheet to help organize their problem-solving activities that contained spaces for goals, unknowns, knowns, and visual representations. Questions as prompts for students while solving problems Results - students’ representation of the algebraic word problems were similar to those of experts (Hutchinson, 1993). Sample questions: Have I read and understood each sentence? Any words whose meaning are unknown? Have I got the whole picture? A representation of the problem? Have I written down my representation? – goal, unknowns, known, type of problem, equation? Is this new problem the same type of problem previously worked? Results - students’ representation of the algebraic word problems were similar to those of experts (Hutchinson, 1993). Sample questions: Have I read and understood each sentence? Any words whose meaning are unknown? Have I got the whole picture? A representation of the problem? Have I written down my representation? – goal, unknowns, known, type of problem, equation? Is this new problem the same type of problem previously worked?

16. Maccini, P., & Hughes, C. A. (2000). Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research & Practice, 15, 10-21.Maccini, P., & Hughes, C. A. (2000). Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research & Practice, 15, 10-21.

17. Computer Aided Instruction Programs for remediation and instruction Demonstration of concepts visually and with online manipulatives Games Spreadsheets Math computer programs demonstrate concepts, instruct, and remediate student errors and misunderstandings from preschool through college levels. Some programs are useful for basic skills instruction. There are many good computer math games that encourage students to learn while enjoying the experience. Other programs may be used for instruction or remediation. These present problems that students answer. If the answer is correct, the student is usually rewarded with a “Great Job!” or an animated response on the computer screen. If the answer is wrong, the computer will demonstrate the correct way to answer the problem. Immediate feedback helps reinforce the correct process. Finally, programs are available that demonstrate mathematical concepts that are better explained through visual and/or manipulative resources. Spreadsheets are another computer application that has many uses in math classes. Students can use spreadsheets to complete calculations, look for patterns, and explore probability and statistics. Math computer programs demonstrate concepts, instruct, and remediate student errors and misunderstandings from preschool through college levels. Some programs are useful for basic skills instruction. There are many good computer math games that encourage students to learn while enjoying the experience. Other programs may be used for instruction or remediation. These present problems that students answer. If the answer is correct, the student is usually rewarded with a “Great Job!” or an animated response on the computer screen. If the answer is wrong, the computer will demonstrate the correct way to answer the problem. Immediate feedback helps reinforce the correct process. Finally, programs are available that demonstrate mathematical concepts that are better explained through visual and/or manipulative resources. Spreadsheets are another computer application that has many uses in math classes. Students can use spreadsheets to complete calculations, look for patterns, and explore probability and statistics.

18. Use Explicit Instruction Begin lesson by Tapping prior knowledge Modeling how to solve problems while thinking aloud Prompting students when they needed assistance in the activity. Also engage students in dialogue that promotes the development of student-generated problem-solving strategies and reflective thinking (students self-evaluate while they are solving problems). Also engage students in dialogue that promotes the development of student-generated problem-solving strategies and reflective thinking (students self-evaluate while they are solving problems).

19. Empirically Validated Components of Effective Instruction Teacher-based activities – C-S-A (Manipulatives) Direct/Explicit instruction Teaching Prerequisite Skills Computer Assisted Instruction Strategy Instruction Structured Worksheets; Diagramming Graphic organizers Maccinni, McNaughton, & Ruhl (1999), Montague and Applegate (2000); ;Witzell, Smith & Brownell, Feigenbaum (2000), Maccinni, (1989), Maccinni, McNaughton, & Ruhl (1999), Gagnon & Maccinni (2001), Van Garderen & Montague (2003), Miller and Strawser, 1996), Ives and Hoy (2003), Howard, Perry, & Tracey (2004) Maccinni, McNaughton, & Ruhl (1999), Montague and Applegate (2000); ;Witzell, Smith & Brownell, Feigenbaum (2000), Maccinni, (1989), Maccinni, McNaughton, & Ruhl (1999), Gagnon & Maccinni (2001), Van Garderen & Montague (2003), Miller and Strawser, 1996), Ives and Hoy (2003), Howard, Perry, & Tracey (2004)

20. Reinforce strategy application through corrective positive feedback Examine students’ math work noting patterns and evidence of strategy. Meet with students individually or in small groups. Makes one positive statement about students’ work or thinking. Specify error patterns. Demonstrate how to complete the problem using one of the strategies. Provide an opportunity to practice the strategy on a similar problem type (guided practice). End with a positive comment . Follow these steps to provide reinforcement of a strategy application. 1. While noting error patterns, the teacher looks for evidence related to the presence or absence of strategy use. 2. Once this is completed, the teacher meets with students individually or in small groups. The teacher can note the error patterns show, then use a combination of demonstration and guided practice until strategy is implemented. Follow these steps to provide reinforcement of a strategy application. 1. While noting error patterns, the teacher looks for evidence related to the presence or absence of strategy use. 2. Once this is completed, the teacher meets with students individually or in small groups. The teacher can note the error patterns show, then use a combination of demonstration and guided practice until strategy is implemented.

21. Recommendations and Conclusions Provide instruction in basic arithmetic. Use think-aloud techniques Allot time to teach specific strategies. Provide guided practice before independent practice Provide a physical and pictorial model Relate to real-life events Let students practice, practice, practice Poor arithmetic background will make some algebraic questions cumbersome and difficult. when modeling steps to solve equations - Demonstrate the steps to the strategy while verbalizing the related thinking. Students will need time to learn and practice the strategy on a regular basis. When constructing their interpretation of steps under teacher guidance, students need to understand why they are solving equations. When students build their own proper understanding of how to solve equations, it is less likely that they will forget the steps. so that students can first understand what to do for each step and then understand why. 5. This aids the process for solving equations. , such as hands-on materials, illustrations, or diagrams. 6. Students can understand problems if they can visualize or have experience with the situation Poor arithmetic background will make some algebraic questions cumbersome and difficult. when modeling steps to solve equations - Demonstrate the steps to the strategy while verbalizing the related thinking. Students will need time to learn and practice the strategy on a regular basis. When constructing their interpretation of steps under teacher guidance, students need to understand why they are solving equations. When students build their own proper understanding of how to solve equations, it is less likely that they will forget the steps. so that students can first understand what to do for each step and then understand why. 5. This aids the process for solving equations. , such as hands-on materials, illustrations, or diagrams. 6. Students can understand problems if they can visualize or have experience with the situation

22. California Algebra I Requirement In 2000 legislation was enacted to require students – as a condition of receiving a high school diploma – to complete Algebra I. The requirement applied beginning with students graduating in 2003-04 Algebra I seen as a life skill needed to be successful in the 21st century Means that students need to have met or exceeded the rigor of the content standards for Algebra IAlgebra I seen as a life skill needed to be successful in the 21st century Means that students need to have met or exceeded the rigor of the content standards for Algebra I

23. Algebra I Requirement Waivers General waivers offered 2003-04 A student with a disability who has an IEP can request a waiver if: They are enrolled in Algebra They are provided with accommodations as specified by the IEP They are still unable to pass the course The state board of ed. Offered general waivers in 2003-04 because districts were not up to speed No more general waivers will be granted – all students are required to take Algebra It’s important to have all students enrolled in Algebra, but a student with a disability who has an IEP can request a waiver if they stay enrolled and have accommodations per the IEP, but are still unable to pass the courseThe state board of ed. Offered general waivers in 2003-04 because districts were not up to speed No more general waivers will be granted – all students are required to take Algebra It’s important to have all students enrolled in Algebra, but a student with a disability who has an IEP can request a waiver if they stay enrolled and have accommodations per the IEP, but are still unable to pass the course

24. Algebra I Requirement and Students with Disabilities Algebra I can and should be taught to all students, including students with disabilities May need more than one class May need practical ways of demonstrating skills and competencies May need supplementary materials May need more than one class or an extended period of time to complete the course of study and standards required May need to use manipulatives and/or calculators or other practical ways to show that they have mastered a skill or competency May need supplementary instructional materials that help address their learning needs May need more than one class or an extended period of time to complete the course of study and standards required May need to use manipulatives and/or calculators or other practical ways to show that they have mastered a skill or competency May need supplementary instructional materials that help address their learning needs

25. For More Information: http://www.cde.ca.gov/be/st/ss/mthalgebra1.asp Algebra content standards grades 8-12 http://www.cde.ca.gov/sp/se/fp/algebra1.asp Guidance and resources for teaching algebra concepts to students with disabilities http://www.cde.ca.gov/re/lr/wr/specialedauthority.asp Information about the Algebra I waiver for students with disabilities