Effective Math Instruction for Students with High Incidence Disabilities

1. 1 Effective Math Instruction for Students with High Incidence Disabilities Joseph Calvin Gagnon, Ph.D. George Mason University

2. 2 Advanced Organizer: Math Session Educational Reform The National Council of Teachers of Mathematics Standards Characteristics of students with learning and emotional/behavioral disabilities Direct instruction (di)

3. 3 Advanced Organizer: Math Session Real world application and technology Student grouping Graduated instructional sequence Graphic Organizers Strategy instruction Instructional adaptations

4. 4 Educational Reform: Standards-driven reform is the primary approach to assuring today’s high school graduates are internationally competitive Prompted by the public dissatisfaction and poor performance by U.S. students on international assessments (McLaughlin, Shepard, & O’Day, 1995), educators, curriculum specialists, and national organizations have focused on development of challenging standards for over a decade.

5. 5 Educational Reform: Ensuring all students achieve in math is a national priority (IDEA, 1997; No Child Left Behind Act of 2001) Success in math is considered a gateway to many educational and occupational opportunities (Jetter, 1993)

6. 6 Educational Reform: Recent legislation has assisted these efforts and ensured that students with disabilities are included, to the maximum extent possible Central to this notion of reform is the assertion that all students are, “entitled to instruction that is grounded in a common set of challenging standards” (McLaughlin, 1999, p. 10)

7. 7 Educational Reform: Rigorous standards are especially crucial for students with learning disabilities (LD) and emotional disturbances (ED), who are commonly included in the general education environment. These students have historically been provided a less rigorous curriculum with IEP goals that: Focus on computation (Shriner, Kim, Thurlow, & Ysseldyke, 1993) Have minimal linkage to long-term general education outcomes (Nolet & McLaughlin, 2000; Sands, Adams, & Stout, 1995; Smith, 1990)

8. 8 Educational Reform: The NCTM Standards are a critical component of the Standards-driven reform movement At least 42 states have used the Standards as a guide to development of mathematics standards Blank and Dalkilic (1992) (as noted in Thurlow, 2000)

9. 9 Characteristics of Students with LD: On average, adolescents with LD function 2.7 grade levels below their nonlabeled peers (Wagner, 1995) Secondary teachers have noted that many of their students experience difficulty in mathematics (McLeod & Armstrong, 1982)

10. 10 Characteristics of Students with LD: Adolescents with LD have difficulty with problem application and generally perform at the 5th grade level (Cawley & Miller, 1989) Secondary students with LD experience difficulties with a range of mathematics tasks, including: Basic skills (Algozzine, O’Shea, Crews, & Stoddard, 1987) Higher-level skills/concepts and problem solving (Huntington, 1994; Hutchinson, 1993; Maccini & Hughes, 2000; Maccini & Ruhl, 2000)

11. 11 Characteristics of Students with ED Students with ED are typically 1.8 grade levels behind their nonlabeled peers (Wagner, 1995) Adolescents with ED possess characteristics hat differentiate them from nonhandicapped peers The academic success or failure of students labeled ED is greatly affected by the extent to which instruction is functional and recognized by students as relevant (Bos & Vaughn, 1994)

12. 12 Characteristics of Students with ED: These students often exhibit a general lack of persistence and concentration and have difficulties with independent class work Secondary students with ED share a common set of learner characteristics that negatively affect their academic success: motivational issues, anxiety, and impulse control (Byrne, 1984; Dweck & Elliot, 1983; Gottfied, 1985; McNeil, 1998; Patten, 1983)

13. 13 Characteristics of Students with ED: Students with ED obtain a percent correct rate between 20 and 76 on independent seatwork (Guntner & Denny, 1998) The ability to persist and work independently on open-ended mathematical tasks could greatly affect the level of success experienced in light of the more constructivist approach that guides the NCTM Standards

14. 14 Six Math Instructional Recommendations Incorporate components of direct instruction Teach strategies Embed math in real-world activities and include the use of technology into instruction

15. 15 Math Instructional Recommendations cont. Group for instruction Incorporate a graduated (i.e., Concrete-Semiconcrete-Abstract) instructional Sequence Use instructional adaptations

16. 16 Direct Instruction (di): Effective Teaching Researchers note that incorporating efficient and effective teaching components into the teaching routine promotes student learning and retention (Rosenshine & Stevens, 1986). These include: Daily review Presentation (provide overview of lesson, teach new skills at a fast rate and in small increments, model procedures, check for understanding, teach to mastery)

17. 17 Direct Instruction (di): guided practice corrective and positive feedback independent practice frequent reviews (cumulative weekly and monthly reviews)

18. 18 Direct Instruction (di): Nationally, close to 70% or more of general and special education teachers reported being prepared to use di Teachers reported using di variables frequently (i.e., 2 – 4 times per week to daily) (Gagnon & Maccini, 2004) This is promising given that use of techniques consistent with teacher-directed instruction has been empirically validated for teaching math to secondary students with LD (Kelly, Gersten, & Carnine, 1990; Moore & Carnine, 1989)

19. 19 Direct Instruction Researchers (Gagnon & Maccini, 2005) recommend providing direct instruction on a daily/weekly basis and providing weekly and monthly cumulative review.

20. 20 Direct Instruction Example: Gagnon, J. C., & Maccini, P. (2005). Direct instruction in mathematics for youth with learning disabilities in middle school. Washington, DC: The Access Center: Improving Outcomes for all Students K-8. http://www.k8accesscenter.org/training_resources/directinstructionmath.asp

21. 21 Real World Problem Solving and Technology: Technology-based instructional approaches can significantly affect student learning and acquisition of higher-level math concepts; particularly when embedded within real-world problem solving tasks (Maccini & Gagnon, 2005) This approach relies on the use of a computer, calculator, or other specialized systems as the mode of instruction (Vergason & Anderegg, 1997)

22. 22 Real World Problem Solving and Technology: Technology-based instruction can: Assist teachers in moving away from a focus on memorization and routine manipulation of numbers in formulas and toward instruction and activities embedded in real-world problems (Bottge & Hasselbring, 1993) Promote active student learning (Kelly, Gersten, & Carnine, 1990)

23. 23 Real World Problem Solving and Technology: Embedding problem solving information within a real world context helps: Activate student conceptual knowledge when presented with a real-life problem solving situation (Gagne, Yekovich, & Yekovich, 1993) Improve student motivation, participation, and generalization (Palloway & Patton, 1997)

24. 24 “Rather than capitalizing on the insights and motivation that students bring to the classroom, schools may actually be wasting valuable time by withholding more authentic and motivating problems until ‘prerequisite’ skills are acquired” (Bottge et al., 2001, p. 312) It is effective to use videodisc-based interventions that embed interesting and age-appropriate problem-solving situations (Bottge, 1999; Bottge & Hasselbring, 1993; Bottge et al., 2001; Bottge et al., 2002)

25. 25 Real World Problem Solving and Technology: Recommendations: Incorporate di (e.g., model, guided practice, review, feedback) within technology-based interventions Incorporate effective instructional design variables (examples follow) within technology-based instruction to reduce student confusion and mathematical errors

26. 26 Real World Problem Solving and Technology: Discrimination: Skills are introduced, practiced, and mixed with other types of problems. Specific instruction and remediation provide for discrimination Range of Examples: Students introduced to fractions less than one, improper fractions, and provided strategies for reading and writing both

27. 27 Real World Problem Solving and Technology: Explicit Strategy Teaching: Students provided explicit problem solving strategies Computer software should incorporate a wide range of examples and nonexamples into instruction for discrimination practice and generalization

28. 28 Real World Problem Solving and Technology: Recommendations: Incorporate technology-based tutorial programs that embed basic math skills and higher order thinking within problem-solving situations This allows students to practice remedial skills within context For example, it is recommended that computers be available to students with LD for tutorial assistance

29. 29 Real World Problem Solving and Technology: Limitations to the use of technology: The review was limited to 11 published articles that met all criteria Although 73% (n = 8) of the studies determined significant treatment effects, three of the studies noted that the proficiency levels of students with disabilities fell below the established criterion for learning of 80%

30. 30 Real World Problem Solving and Technology: Further, of the articles that obtained significant findings, only 45% (n = 5) of the interventions directly programmed for maintenance and 55% (n = 6) programmed for generalization The generalizability of the findings may also be of concern because no information was available on new technologies (e.g., DVD and streaming video)

31. 31 Technology and Real-World Activities: Research Recommendations: It is recommended (Gagnon & Maccini, 2005) to provide technology-based learning activities real-world activities that incorporate effective teaching variables on a daily/weekly basis

32. 32 Real World Problem Solving and Technology: Calculators: In one study, calculator use was the most prevalent adaptation noted by teachers Maccini & Gagnon, 2002) Consistent with Etlinger and Ogletree (1982), teacher responses involved two primary categories:

33. 33 Real World Problem Solving and Technology: The "practical" function: The use of calculators to complete tedious calculations, save time, increase student motivation, and to decrease math anxiety The "pedagogical" function: Relates to similarities between calculators, textbooks, and manipulatives in that each enhances student understanding and competence in mathematics

34. 34 Real World Problem Solving and Technology: These classifications are consistent with the five primary functions of calculators as stated by the NCTM Within the practical classification, NCTM identifies the use of calculators to Perform tedious computations that arise when working with real data in problem solving situations Concentrate on the problem-solving process rather than calculations associated with problems Gain access to mathematics beyond their level of computational skill

35. 35 Real World Problem Solving and Technology: The pedagogical function coincides with two other uses identified by NCTM To explore, develop, and reinforce concepts including estimation, computation, approximation, and properties To experiment with math ideas and discover patterns

36. 36 Real World Problem Solving and Technology: Salend and Hoffstetter (1996) asserted the importance of: Training students to use calculators Using an overhead projector to teach this skill Locating and describing the function of each key to students Providing examples of calculator use

37. 37 Real World Problem Solving and Technology: Students should be provided opportunities to practice calculations, including estimation skills and reviewing answers obtained through calculator use

38. 38 Real World Problem Solving and Technology: The following recommendations for teachers are noted: Model calculator application Use calculators in computation, problem solving, concept development, pattern recognition, data analysis, and graphing Integrate calculator use in assessment and evaluation

39. 39 Real World Problem Solving and Technology: Remain current with state-of-the-art technology Explore and develop new ways to use calculators to support instruction and assessment

40. 40 Resource Maccini, P., & Gagnon, J. C. (2005). Mathematics and technology-based interventions for secondary students with learning disabilities. In D. Edyburn, K. Higgins, & R. Boone, The handbook of special education technology research and practice (pp. 599-622). Winston-Salem, NC: Knowledge By Design, Inc.

41. 41 Grouping for Instruction Grouping for instruction involves cooperative group activities and peer tutoring. Cooperative learning refers to an “instructional arrangement for teaching academic and collaborative skills to a small, heterogeneous group of students” (Rivera, 1996, p. 1) Peer tutoring is a systematic tutoring arrangement in which peers rotate assisting one another, where one acts as the tutor (coach) and the other as the tutee (learner)

42. 42 Grouping for Instruction: Grouping for instruction involves the use of small group instruction, one-on-one support, cooperative group activities, individualized instruction, and peer tutoring Research: Grouping adaptations reduce occurrences of behavioral problems (Penno, Frank, & Wacker, 2000) Peer-assisted learning promotes computational skills (Calhoon & Fuchs, 2003) Classwide peer tutoring is effective in strengthening basic math skills (Allsopp, 1997)

43. 43 Grouping for Instruction: Peer tutoring has several benefits including: Promoting active student responding Providing student’s opportunities to correct errors Providing students with immediate feedback Teaching self-management Providing a structured, task-focused opportunity for positive social interaction

44. 44 Grouping for Instruction: Classwide peer tutoring (CWPT): Typically, the entire class participates in CWPT simultaneously First, students are paired and the pairs are separated into two groups within the classroom Each session last approximately 30 minutes and can be implemented from two to five days per week

45. 45 Grouping for Instruction: Within a session, each student Spends 10 minutes acting as the tutor Spends 10 minutes as the tutee Students are then provided approximately 5 minutes to record their individual points

46. 46 Grouping for Instruction: Points may be earned individually for: Correct responses Error correction Following the tutoring procedures

47. 47 Grouping for Instruction: To increase self-management and positive student interactions, teachers may designate certain instances where the tutor provides the points to the tutee At the end of the week, teams meeting a certain criteria level may earn a special reinforcing activity

48. 48 Grouping for Instruction: To implement CWPT: The teacher must identify the specific procedural steps and expectations for students to follow Students should be taught the exact procedures through a simple three-step process First, the teacher explains and posts the exact list of procedures

49. 49 Grouping for Instruction: Then, the teacher models the peer tutoring process and allows students to participate in role playing Lastly, the teacher provides an opportunity for the students to use the process and receive feedback on their correct use of the format

50. 50 Grouping for Instruction: Recommendations for Practice: Cooperative learning strategies Classwide peer-tutoring Use of teaching assistants to support learning

51. 51 Grouping for Instruction: Research Research Recommendations: It is recommended (Allsopp, 1997; O’Melia & Rosenberg, 1994) to provide cooperative learning and peer tutoring arrangements 2-4 times per week Resource: Allsopp, D. H. (1997). Using classwide peer tutoring to teach beginning algebra problem-solving skills in heterogeneous classrooms. Remedial and Special Education, 18, 367-379.

52. 52 Concrete-Semiconcrete-Abstract Instructional Sequence (C-S-A) Bruner’s structure-oriented theory of learning: Enactive mode (e.g., the “doing” phase” - using concrete objects to represent problems - concrete representations) Iconic mode (e.g., the “seeing phase” visualizing representations of the problem - semiconcrete representations) Symbolic mode (e.g., using abstract symbols to represent the problem - abstract representations)

53. 53 C-S-A: Empirical studies have validated CSA use with students with high incidence disabilities for: Whole number operations Word problems Place value Introductory algebra skills

54. 54 C-S-A: Implications: It is recommended (Gagnon & Maccini, 2001) to provide instruction using the graduated instructional sequence on a daily/weekly basis when introducing new math concepts and advancing to more abstract ideas. Resources: (Handout) Gagnon, J. C., & Maccini, P. (2001). Preparing students with disabilities for algebra: Kindergarten through secondary school. TEACHING Exceptional Children, 33(2), 8-15.

55. 55 Teach Strategies A strategy refers to, “a plan that not only specifies the sequence of needed actions but also consists of critical guidelines and rules related to making effective decisions during a problem solving process” (Ellis & Lenz, 1996, p. 24). A number of features help to make strategies effective for students, including:

56. 56 Teach Strategies Use strategy steps that include familiar words, are stated simply and concisely, and begin with action verbs to facilitate student involvement (e.g., Read the problem carefully), and that are sequenced appropriately Use strategy steps that remind students to read the problem carefully, to obtain a whole picture of the problem (problem representation), to solve the problem, and to check their answers (problem solution)

57. 57 Strategy Instruction: Strategy Instruction Can Include: Structured worksheets/cue cards to help students remember problem solving steps or strategies for solving problems Mnemonics to help students recall problem solving steps or important facts Research: Strategy instruction that incorporated a first-letter mnemonic and structured worksheets helped students with LD learn prealgebra skills and concepts (Maccini & Hughes, 2000)

58. 58 Strategy Instruction: Implications Need to use: On a daily/weekly basis, use strategies that incorporate memory devices, sequenced strategy steps, and both problem representation and solution Resource: Maccini, P., & Gagnon, J. C. (in press). Math strategy instruction for middle school students with learning disabilities. Washington, DC: The Access Center: Improving Outcomes for all Students K-8

59. 59 Instructional Adaptations Instructional adaptations include structured worksheets/graphic organizers, self-monitoring devices, and advance organizers. Provide graphic organizer/structured worksheets to help students remember and recall information (e.g., steps to a strategy). Incorporate self-monitoring to help students monitor their problem solving behavior

60. 60 Instructional Adaptations Use advance organizers to help students identify, organize, understand, and retain information (Lenz, Bulgren, & Hudson, 1990).

61. 61 Instructional Adaptations: Organizers Students with disabilities have difficulties: Remembering and recalling information (Olson & Platt, 1996) Identifying relevant information Organizing information Using visual organizers, such as structured worksheets, prompt cards, or graphic organizers helps students analyze and solve math problems (Gagnon & Maccini, 2001)

62. 62 Instructional Adaptations: Organizers Graphic organizers should be taught to students using di, used when introducing new material, and used during instruction to help students organize the information (Maccini & Gagnon, 2005) Self-monitoring or individualized self-instruction checklists should be used to help prompt students to use the correct steps/procedures (Dunlap & Dunlap, 1989)

63. 63 Instructional Adaptations: Organizers For examples of organizes, key components, ways to develop them and instruct students in using organizers, see: Maccini, P., & Gagnon, J. C. (2005). Math graphic organizers for students with disabilities. Washington, DC: The Access Center: Improving Outcomes for all Students K-8. Available at http://www.k8accesscenter.org/training_resources/documents/MathGraphicOrg.pdf

64. 64 Instructional Adaptations: Recommendations for Practice: Include assignment adaptations to maintain student attention: Examples: Salend (1990) supports the adaptation of assignments through A decrease in the number of problems assigned and includes three related suggestions Reviewing previously mastered skills Dividing a task or worksheet in to smaller tasks or sections

65. 65 Instructional Adaptations: Inappropriate student behavior decreases when students are presented with a sequence of shortened assignments versus one long assignment (Dunlap et al., 1993)

66. 66 Instructional Adaptations: Meese (1994) identifies several effective assignment modifications: Divide assignments into chunks and have timelines for each chunk Extend time for completing assignments Encourage the use of calculators and computers Allow groups to complete some written assignments

67. 67 Instructional Adaptations: 5. Reduce the amount of copying needed throughout the assignment (e.g., from board, notetaking) 6. Require students to paraphrase an assignment's tasks (p. 350-351) 7. A reduction in the number of problems assigned to students (Salend, 1994)

68. 68 Instructional Adaptations: Research It is recommended to use these instructional adaptations daily (advance organizer), or on an as needed basis (graphic organizer, self-monitoring devices). For example, it is recommended to provide an advance organizer to help orient students to the lesson-of-the-day or the new topic.

69. 69 Resources For more information on teaching reading and math to secondary students with emotional and behavioral disorders, see: Gagnon, J. C., Wehby, J., Strong, A., & Falk, K. B. (2005). Research-based reading and math interventions for youth with emotional disturbance. In L. M. Bullock, R. A. Gable, & K. J. Melloy (Eds.), Sixth CCBD mini-library series. Arlington, VA: Council for Children with Behavioral Disorders. Available: http://www.cec.sped.org/