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**Math Reasoning: Assisting the Struggling Middle and High**School LearnerJim Wrightwww.interventioncentral.org**Download PowerPoint from this workshop**at:http://www.interventioncentral.org/SSTAGE.php**Georgia ‘Pyramid of Intervention’**Source: Georgia Dept of Education: http://www.doe.k12.ga.us/ Retrieved 13 July 2007**An RTI Challenge: Limited Research to Support Evidence-Based**Math Interventions “… in contrast to reading, core math programs that are supported by research, or that have been constructed according to clear research-based principles, are not easy to identify. Not only have exemplary core programs not been identified, but also there are no tools available that we know of that will help schools analyze core math programs to determine their alignment with clear research-based principles.” p. 459 Source: Clarke, B., Baker, S., & Chard, D. (2008). Best practices in mathematics assessment and intervention with elementary students. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp. 453-463).**Tier 1: What Are the Recommended Elements of ‘Core**Curriculum’?: More Research Needed “In essence, we now have a good beginning on the evaluation of Tier 2 and 3 interventions, but no idea about what it will take to get the core curriculum to work at Tier 1. A complicating issue with this potential line of research is that many schools use multiple materials as their core program.” p. 640 Source: Kovaleski, J. F. (2007). Response to intervention: Considerations for research and systems change. School Psychology Review, 36, 638-646.**Limitations of Intervention Research…**“…the list of evidence-based interventions is quite small relative to the need [of RTI]…. Thus, limited dissemination of interventions is likely to be a practical problem as individuals move forward in the application of RTI models in applied settings.” p. 33 Source: Kratochwill, T. R., Clements, M. A., & Kalymon, K. M. (2007). Response to intervention: Conceptual and methodological issues in implementation. In Jimerson, S. R., Burns, M. K., & VanDerHeyden, A. M. (Eds.), Handbook of response to intervention: The science and practice of assessment and intervention. New York: Springer.**Schools Need to Review Tier 1 (Classroom) Interventions to**Ensure That They Are Supported By Research There is a lack of agreement about what is meant by ‘scientifically validated’ classroom (Tier I) interventions. Districts should establish a ‘vetting’ process—criteria for judging whether a particular instructional or intervention approach should be considered empirically based. Source: Fuchs, D., & Deshler, D. D. (2007). What we need to know about responsiveness to intervention (and shouldn’t be afraid to ask).. Learning Disabilities Research & Practice, 22(2),129–136.**What Are Appropriate Content-Area Tier 1 Universal**Interventions for Secondary Schools? “High schools need to determine what constitutes high-quality universal instruction across content areas. In addition, high school teachers need professional development in, for example, differentiated instructional techniques that will help ensure student access to instruction interventions that are effectively implemented.” Source: Duffy, H. (August 2007). Meeting the needs of significantly struggling learners in high school. Washington, DC: National High School Center. Retrieved from http://www.betterhighschools.org/pubs/ p. 9**Big Ideas: The Four Stages of Learning Can Be Summed Up in**the ‘Instructional Hierarchy’ (Haring et al., 1978) Student learning can be thought of as a multi-stage process. The universal stages of learning include: • Acquisition: The student is just acquiring the skill. • Fluency: The student can perform the skill but must make that skill ‘automatic’. • Generalization: The student must perform the skill across situations or settings. • Adaptation: The student confronts novel task demands that require that the student adapt a current skill to meet new requirements. Source: Haring, N.G., Lovitt, T.C., Eaton, M.D., & Hansen, C.L. (1978). The fourth R: Research in the classroom. Columbus, OH: Charles E. Merrill Publishing Co.**Scripting Interventions to Promote Better Compliance**Interventions should be written up in a ‘scripted’ format to ensure that: • Teachers have sufficient information about the intervention to implement it correctly; and • External observers can view the teacher implementing the intervention strategy and—using the script as a checklist—verify that each step of the intervention was implemented correctly (‘treatment integrity’). Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools. Routledge: New York.**Increasing the Intensity of an Intervention: Key Dimensions**Interventions can move up the RTI Tiers through being intensified across several dimensions, including: • Type of intervention strategy or materials used • Student-teacher ratio • Length of intervention sessions • Frequency of intervention sessions • Duration of the intervention period (e.g., extending an intervention from 5 weeks to 10 weeks) • Motivation strategies Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools. Routledge: New York. Kratochwill, T. R., Clements, M. A., & Kalymon, K. M. (2007). Response to intervention: Conceptual and methodological issues in implementation. In Jimerson, S. R., Burns, M. K., & VanDerHeyden, A. M. (Eds.), Handbook of response to intervention: The science and practice of assessment and intervention. New York: Springer.**Research-Based Elements of Effective Academic Interventions**• ‘Correctly targeted’: The intervention is appropriately matched to the student’s academic or behavioral needs. • ‘Explicit instruction’: Student skills have been broken down “into manageable and deliberately sequenced steps and providing overt strategies for students to learn and practice new skills” p.1153 • ‘Appropriate level of challenge’: The student experiences adequate success with the instructional task. • ‘High opportunity to respond’: The student actively responds at a rate frequent enough to promote effective learning. • ‘Feedback’: The student receives prompt performance feedback about the work completed. Source: Burns, M. K., VanDerHeyden, A. M., & Boice, C. H. (2008). Best practices in intensive academic interventions. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp.1151-1162). Bethesda, MD: National Association of School Psychologists.**Core Instruction,Interventions, Accommodations &**Modifications: Sorting Them Out • Core Instruction. Those instructional strategies that are used routinely with all students in a general-education setting are considered ‘core instruction’. High-quality instruction is essential and forms the foundation of RTI academic support. NOTE: While it is important to verify that good core instructional practices are in place for a struggling student, those routine practices do not ‘count’ as individual student interventions.**Core Instruction, Interventions, Accommodations &**Modifications: Sorting Them Out • Intervention. An academic intervention is a strategy used to teach a new skill, build fluency in a skill, or encourage a child to apply an existing skill to new situations or settings. An intervention can be thought of as “a set of actions that, when taken, have demonstrated ability to change a fixed educational trajectory” (Methe & Riley-Tillman, 2008; p. 37).**Core Instruction,Interventions, Accommodations &**Modifications: Sorting Them Out • Accommodation. An accommodation is intended to help the student to fully access and participate in the general-education curriculum without changing the instructional content and without reducing the student’s rate of learning (Skinner, Pappas & Davis, 2005). An accommodation is intended to remove barriers to learning while still expecting that students will master the same instructional content as their typical peers. • Accommodation example 1: Students are allowed to supplement silent reading of a novel by listening to the book on tape. • Accommodation example 2: For unmotivated students, the instructor breaks larger assignments into smaller ‘chunks’ and providing students with performance feedback and praise for each completed ‘chunk’ of assigned work (Skinner, Pappas & Davis, 2005).**Core Instruction,Interventions, Accommodations &**Modifications: Sorting Them Out • Modification. A modification changes the expectations of what a student is expected to know or do—typically by lowering the academic standards against which the student is to be evaluated. Examples of modifications: • Giving a student five math computation problems for practice instead of the 20 problems assigned to the rest of the class • Letting the student consult course notes during a test when peers are not permitted to do so • Allowing a student to select a much easier book for a book report than would be allowed to his or her classmates.**‘Intervention Footprint’: 7-Step Lifecycle of an**Intervention Plan… • Information about the student’s academic or behavioral concerns is collected. • The intervention plan is developed to match student presenting concerns. • Preparations are made to implement the plan. • The plan begins. • The integrity of the plan’s implementation is measured. • Formative data is collected to evaluate the plan’s effectiveness. • The plan is discontinued, modified, or replaced.**Interventions: Potential ‘Fatal Flaws’**Any intervention must include 4 essential elements. The absence of any one of the elements would be considered a ‘fatal flaw’ (Witt, VanDerHeyden & Gilbertson, 2004) that blocks the school from drawing meaningful conclusions from the student’s response to the intervention: • Clearly defined problem. The student’s target concern is stated in specific, observable, measureable terms. This ‘problem identification statement’ is the most important step of the problem-solving model (Bergan, 1995), as a clearly defined problem allows the teacher or RTI Team to select a well-matched intervention to address it. • Baseline data. The teacher or RTI Team measures the student’s academic skills in the target concern (e.g., reading fluency, math computation) prior to beginning the intervention. Baseline data becomes the point of comparison throughout the intervention to help the school to determine whether that intervention is effective. • Performance goal. The teacher or RTI Team sets a specific, data-based goal for student improvement during the intervention and a checkpoint date by which the goal should be attained. • Progress-monitoring plan. The teacher or RTI Team collects student data regularly to determine whether the student is on-track to reach the performance goal. Source: Witt, J. C., VanDerHeyden, A. M., & Gilbertson, D. (2004). Troubleshooting behavioral interventions. A systematic process for finding and eliminating problems. School Psychology Review, 33, 363-383.**How Do We Reach Low-Performing Math Students?: Instructional**Recommendations Important elements of math instruction for low-performing students: • “Providing teachers and students with data on student performance” • “Using peers as tutors or instructional guides” • “Providing clear, specific feedback to parents on their children’s mathematics success” • “Using principles of explicit instruction in teaching math concepts and procedures.” p. 51 Source:Baker, S., Gersten, R., & Lee, D. (2002).A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103(1), 51-73..**Profile of Students With Significant Math Difficulties**• Spatial organization. The student commits errors such as misaligning numbers in columns in a multiplication problem or confusing directionality in a subtraction problem (and subtracting the original number—minuend—from the figure to be subtracted (subtrahend). • Visual detail. The student misreads a mathematical sign or leaves out a decimal or dollar sign in the answer. • Procedural errors. The student skips or adds a step in a computation sequence. Or the student misapplies a learned rule from one arithmetic procedure when completing another, different arithmetic procedure. • Inability to ‘shift psychological set’. The student does not shift from one operation type (e.g., addition) to another (e.g., multiplication) when warranted. • Graphomotor. The student’s poor handwriting can cause him or her to misread handwritten numbers, leading to errors in computation. • Memory. The student fails to remember a specific math fact needed to solve a problem. (The student may KNOW the math fact but not be able to recall it at ‘point of performance’.) • Judgment and reasoning. The student comes up with solutions to problems that are clearly unreasonable. However, the student is not able adequately to evaluate those responses to gauge whether they actually make sense in context. Source: Rourke, B. P. (1993). Arithmetic disabilities, specific & otherwise: A neuropsychological perspective. Journal of Learning Disabilities, 26, 214-226.**Team Activity: Define ‘Math Reasoning’…**• At your table: • Appoint a recorder/spokesperson. • Discuss the term ‘math reasoning’ at the secondary level. Task-analyze the term and break it down into the essential subskills. • Be prepared to report out on your work. • What is the role of the Student Support Team in assisting teachers to promote ‘math reasoning’?**Assisting Students in Accessing Contextual, Conceptual, &**Procedural Knowledge When Solving Math Problems “Well-structured, organized knowledge allows people to solve novel problems and to remember more information than do memorized facts or procedures... Such well-structured knowledge requires that people integrate their contextual, conceptual and procedural knowledge in a domain. Unfortunately, U.S. students rarely have such integrated and robust knowledge in mathematics or science. Designing learning environments that support integrated knowledge is a key challenge for the field, especially given the low number of established tools for guiding this design process.” p. 313 Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Types of Knowledge: Definitions**Conceptual Knowledge: “…integrated knowledge of important principles (e.g., knowledge of number magnitudes) that can be flexibly applied to new tasks. Conceptual knowledge can be used to guide comprehension of problems and to generate new problem-solving strategies or to adapt existing strategies to solve novel problems.” p. 317 Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Types of Knowledge: Definitions**Procedural Knowledge: “…knowledge of subcomponents of a correct procedure. Procedures are a type of strategy that involve step-by-step actions for solving problems, and most procedures require integration of multiple skills. For example, the conventional procedure for adding fractions with unlike denominators requires knowing how to find a common denominator, how to find equivalent fractions, and how to add fractions with like denominators.” 318 Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Types of Knowledge: Definitions**Contextual Knowledge: “…our knowledge of how things work in specific, real-world situations, which develops from our everyday, informal interactions with the world. Students’ contextual knowledge can be elicited by situating problems in story contexts.” p. 316 Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Math Problem Scaffolding Examples (Modeled after**Rittle-Johnson & Koedinger, 2005) Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Math Problem Scaffolding Examples (Modeled after**Rittle-Johnson & Koedinger, 2005) Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Leveraging the Power of Contextual Knowledge in Story**Problems: Use Familiar Student Contexts “Past research on fraction learning indicates that food contexts are particularly meaningful contexts for students (Mack, 1990, 1993).” p. 319 Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Math Problem Scaffolding Examples (Modeled after**Rittle-Johnson & Koedinger, 2005) Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Math Problem Scaffolding Examples (Modeled after**Rittle-Johnson & Koedinger, 2005) Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**Research is Unclear Whether Math Problems in Story or**Symbolic Format Are More Difficult The reason for contradictory findings about the relative difficulty of math problems in story or symbolic format may be explained by grade-specific challenges in math. “First, young children have pervasive exposure to single-digit numerals, but some words and syntactic forms are still unknown or unfamiliar [explaining why younger students may find story problems more challenging]. In comparison, older children have less exposure to large, multidigit numerals and algebraic symbols and have much better reading and comprehension skills [explaining why older students may find symbolic problems more challenging].” p. 317 Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.**5 Strands of Mathematical Proficiency**• Understanding • Computing • Applying • Reasoning • Engagement Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.**Five Strands of Mathematical Proficiency**• Understanding: Comprehending mathematical concepts, operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean. • Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately. • Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately. Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.**Five Strands of Mathematical Proficiency (Cont.)**• Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known. • Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work. Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.**Conceptual Knowledge**Procedural Knowledge Metacognition Synthesis Five Strands of Mathematical Proficiency(NRC, 2002) • Understanding: Comprehending mathematical concepts, operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean. • Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately. • Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately. • Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known. • Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work. Motivation**Table Activity: Evaluate Your School’s Math Proficiency…**• As a group, review the National Research Council ‘Strands of Math Proficiency’. • Which strand do you feel that your school / curriculum does the best job of helping students to attain proficiency? • Which strand do you feel that your school / curriculum should put the greatest effort to figure out how to help students to attain proficiency? • Be prepared to share your results. Five Strands of Mathematical Proficiency(NRC, 2002) • Understanding: Comprehending mathematical concepts, operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean. • Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately. • Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately. • Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known. • Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work.**Comprehending Math Vocabulary: The Barrier of Abstraction**“…when it comes to abstract mathematical concepts, words describe activities or relationships that often lack a visual counterpart. Yet studies show that children grasp the idea of quantity, as well as other relational concepts, from a very early age…. As children develop their capacity for understanding, language, and its vocabulary, becomes a vital cognitive link between a child’s natural sense of number and order and conceptual learning. ” -Chard, D. (n.d.) Source: Chard, D. (n.d.. Vocabulary strategies for the mathematics classroom. Retrieved November 23, 2007, from http://www.eduplace.com/state/pdf/author/chard_hmm05.pdf.**Math Vocabulary: Classroom (Tier I) Recommendations**• Preteach math vocabulary. Math vocabulary provides students with the language tools to grasp abstract mathematical concepts and to explain their own reasoning. Therefore, do not wait to teach that vocabulary only at ‘point of use’. Instead, preview relevant math vocabulary as a regular a part of the ‘background’ information that students receive in preparation to learn new math concepts or operations. • Model the relevant vocabulary when new concepts are taught. Strengthen students’ grasp of new vocabulary by reviewing a number of math problems with the class, each time consistently and explicitly modeling the use of appropriate vocabulary to describe the concepts being taught. Then have students engage in cooperative learning or individual practice activities in which they too must successfully use the new vocabulary—while the teacher provides targeted support to students as needed. • Ensure that students learn standard, widely accepted labels for common math terms and operations and that they use them consistently to describe their math problem-solving efforts. Source: Chard, D. (n.d.. Vocabulary strategies for the mathematics classroom. Retrieved November 23, 2007, from http://www.eduplace.com/state/pdf/author/chard_hmm05.pdf.**Promoting Math Vocabulary: Other Guidelines**• Create a standard list of math vocabulary for each grade level (elementary) or course/subject area (for example, geometry). • Periodically check students’ mastery of math vocabulary (e.g., through quizzes, math journals, guided discussion, etc.). • Assist students in learning new math vocabulary by first assessing their previous knowledge of vocabulary terms (e.g., protractor; product) and then using that past knowledge to build an understanding of the term. • For particular assignments, have students identify math vocabulary that they don’t understand. In a cooperative learning activity, have students discuss the terms. Then review any remaining vocabulary questions with the entire class. • Encourage students to use a math dictionary in their vocabulary work. • Make vocabulary a central part of instruction, curriculum, and assessment—rather than treating as an afterthought. Source: Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading Teacher, 56(8), 786-795.**Vocabulary: Why This Instructional Goal is Important**As vocabulary terms become more specialized in content area courses, students are less able to derive the meaning of unfamiliar words from context alone. Students must instead learn vocabulary through more direct means, including having opportunities to explicitly memorize words and their definitions. Students may require 12 to 17 meaningful exposures to a word to learn it.**Enhance Vocabulary Instruction Through Use of Graphic**Organizers or Displays: A Sampling Teachers can use graphic displays to structure their vocabulary discussions and activities (Boardman et al., 2008; Fisher, 2007; Texas Reading Initiative, 2002).**4-Square Graphic Display**The student divides a page into four quadrants. In the upper left section, the student writes the target word. In the lower left section, the student writes the word definition. In the upper right section, the student generates a list of examples that illustrate the term, and in the lower right section, the student writes ‘non-examples’ (e.g., terms that are the opposite of the target vocabulary word).**Semantic Word Definition Map**The graphic display contains sections in which the student writes the word, its definition (‘what is this?’), additional details that extend its meaning (‘What is it like?’), as well as a listing of examples and ‘non-examples’ (e.g., terms that are the opposite of the target vocabulary word).