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### Math Reasoning: Assisting the Struggling Middle and High School LearnerJim Wrightwww.interventioncentral.org

### Intervention Research & Development: A Work in Progress School Learner

### Intervention: Key Concepts Interventions for Secondary Schools?

### Strands of Math Proficiency Symbolic Format Are More Difficult

### RTI & Secondary Literacy: Symbolic Format Are More DifficultExplicit Vocabulary Instruction

### RTI & Secondary Literacy: Vocabulary Terms, DefinitionsExtended Discussion

### RTI & Secondary Literacy: DiscussionsReading Comprehension

### Secondary Group-Based Math Intervention Example Students With Mathematical Problem Solving

Download PowerPoint from this workshop at: School Learnerhttp://www.interventioncentral.org/SSTAGE.php

Georgia ‘Pyramid of Intervention’ School Learner

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… Curriculum’?: More Research Needed

“…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 the ‘Instructional Hierarchy’

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.

Intervention Script Builder the ‘Instructional Hierarchy’ Form

Increasing the Intensity of an Intervention: Key Dimensions the ‘Instructional Hierarchy’

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 the ‘Instructional Hierarchy’ 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 the ‘Instructional Hierarchy’ ,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, the ‘Instructional Hierarchy’ 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, the ‘Instructional Hierarchy’ 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, the ‘Instructional Hierarchy’ 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’ Intervention Plan…

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 Recommendations

- 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: RecommendationsDefine ‘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 Procedural Knowledge When Solving Math Problems

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 When Solving Math Problems

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 Procedural Knowledge When Solving Math Problems Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.

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

Math Problem Scaffolding Examples Procedural Knowledge When Solving Math Problems(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 Procedural Knowledge When Solving Math Problems(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 Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.

“Past research on fraction learning indicates that food contexts are particularly meaningful contexts for students (Mack, 1990, 1993).” p. 319

Math Problem Scaffolding Examples Problems: Use Familiar Student Contexts(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 Problems: Use Familiar Student Contexts(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 Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.

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

- 5 Strands of Mathematical Proficiency Symbolic Format Are More Difficult
- 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 Symbolic Format Are More Difficult

- 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.) Symbolic Format Are More Difficult

- 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 Symbolic Format Are More DifficultKnowledge

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… Symbolic Format Are More Difficult

- 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.

- 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 Symbolic Format Are More Difficult

“…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 Symbolic Format Are More Difficult

- 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 Symbolic Format Are More Difficult

- 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 Symbolic Format Are More Difficult

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 Organizers or Displays: A Sampling

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 Organizers or Displays: A Sampling

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).

Word Definition Map Example Organizers or Displays: A Sampling

Semantic Feature Analysis Organizers or Displays: A Sampling

A target vocabulary term is selected for analysis in this grid-like graphic display. Possible features or properties of the term appear along the top margin, while examples of the term are listed ion the left margin. The student considers the vocabulary term and its definition. Then the student evaluates each example of the term to determine whether it does or does not match each possible term property or element.

Semantic Feature Analysis Example Organizers or Displays: A Sampling

- VOCABULARY TERM: TRANSPORTATION

Comparison/Contrast (Venn) Diagram Organizers or Displays: A Sampling

Two terms are listed and defined. For each term, the student brainstorms qualities or properties or examples that illustrate the term’s meaning. Then the student groups those qualities, properties, and examples into 3 sections:

- items unique to Term 1
- items unique to Term 2
- items shared by both terms

Provide Regular In-Class Instruction and Review of Vocabulary Terms, Definitions

Present important new vocabulary terms in class, along with student-friendly definitions. Provide ‘example sentences’/contextual sentences to illustrate the use of the term. Assign students to write example sentences employing new vocabulary to illustrate their mastery of the terms.

Generate ‘Possible Sentences’ Vocabulary Terms, Definitions

The teacher selects 6 to 8 challenging new vocabulary terms and 4 to 6 easier, more familiar vocabulary items relevant to the lesson. Introduce the vocabulary terms to the class. Have students write sentences that contain at least two words from the posted vocabulary list. Then write examples of student sentences on the board until all words from the list have been used. After the assigned reading, review the ‘possible sentences’ that were previously generated. Evaluate as a group whether, based on the passage, the sentence is ‘possible’ (true) in its current form. If needed, have the group recommend how to change the sentence to make it ‘possible’.

Provide Dictionary Training Vocabulary Terms, Definitions

The student is trained to use an Internet lookup strategy to better understand dictionary or glossary definitions of key vocabulary items.

- The student first looks up the word and its meaning(s) in the dictionary/glossary.
- If necessary, the student isolates the specific word meaning that appears to be the appropriate match for the term as it appears in course texts and discussion.
- The student goes to an Internet search engine (e.g., Google) and locates at least five text samples in which the term is used in context and appears to match the selected dictionary definition.

Extended Discussions: Why This Instructional Goal is Important

Extended, guided group discussion is a powerful means to help students to learn vocabulary and advanced concepts. Discussion can also model for students various ‘thinking processes’ and cognitive strategies (Kamil et al. 2008, p. 22). To be effective, guided discussion should go beyond students answering a series of factual questions posed by the teacher: Quality discussions are typically open-ended and exploratory in nature, allowing for multiple points of view (Kamil et al., 2008).

When group discussion is used regularly and well in instruction, students show increased growth in literacy skills. Content-area teachers can use it to demonstrate the ‘habits of mind’ and patterns of thinking of experts in various their discipline: e.g., historians, mathematicians, chemists, engineers, literacy critics, etc.

Use a ‘Standard Protocol’ to Structure Extended Discussions

Good extended classwide discussions elicit a wide range of student opinions, subject individual viewpoints to critical scrutiny in a supportive manner, put forth alternative views, and bring closure by summarizing the main points of the discussion. Teachers can use a simple structure to effectively and reliably organize their discussions…

‘Standard Protocol’ Discussion Format Discussions

- Pose questions to the class that require students to explain their positions and their reasoning .
- When needed, ‘think aloud’ as the discussion leader to model good reasoning practices (e.g., taking a clear stand on a topic).
- Supportively challenge student views by offering possible counter arguments.
- Single out and mention examples of effective student reasoning.
- Avoid being overly directive; the purpose of extended discussions is to more fully investigate and think about complex topics.
- Sum up the general ground covered in the discussion and highlight the main ideas covered.

Reading Comprehension: Why This Instructional Goal is Important

Students require strong reading comprehension skills to succeed in challenging content-area classes.At present, there is no clear evidence that any one reading comprehension instructional technique is clearly superior to others. In fact, it appears that students benefit from being taught any self-directed practice that prompts them to engage more actively in understanding the meaning of text (Kamil et al., 2008).

Assist Students in Setting ‘Content Goals’ for Reading Important

Students are more likely to be motivated to read--and to read more closely—if they have specific content-related reading goals in mind. At the start of a reading assignment, for example, the instructor has students state what questions they might seek to answer or what topics they would like to learn more about in their reading. The student or teacher writes down these questions. After students have completed the assignee reading, they review their original questions and share what they have learned (e.g., through discussion in large group or cooperative learning group, or even as a written assignment).

Teach Students to Monitor Their Own Comprehension and Apply ‘Fix-Up’ Skills

Teachers can teach students specific strategies to monitor their understanding of text and independently use ‘fix-up’ skills as needed. Examples of student monitoring and repair skills for reading comprehension include encouraging them to:

- Stop after every paragraph to summarize its main idea
- Reread the sentence or paragraph again if necessary
- Generate and write down questions that arise during reading
- Restate challenging or confusing ideas or concepts from the text in the student’s own words

Algebra Word Problems: Predictable Elements Problems Students may fail to discriminate relevant from irrelevant information. Students may commit translation errors when processing relational statements.

Most algebra problems contain predictable elements:

- Assignment statements: Assign a “particular numerical value to some variable.”
- Relational statements: “Specify a single relationship between two variables.”
- Questions: “Involve the requested solution (e.g.,’ What is X?’).”
- Relevant facts: “Any other type of information that might be useful for solving the problem.”
Problem translation: The process of “taking each of these forms of information and using them to develop corresponding algebraic equations.”

The most common problems were noted:

Source:National Mathematics Advisory Panel. (2008). Foundations for success: The final Report of the National Mathematics Advisory Panel: Chapter 4: Report of the Task Group on Learning Processes. U.S. Department of Education: Washington, DC. Retrieved from http://www.ed.gov/about/bdscomm/list/mathpanel/reports.html

Importance of Metacognitive Strategy Use… Problems

“Metacognitive processes focus on self-awareness of cognitive knowledge that is presumed to be necessary for effective problem solving, and they direct and regulate cognitive processes and strategies during problem solving…That is, successful problem solvers, consciously or unconsciously (depending on task demands), use self-instruction, self-questioning, and self-monitoring to gain access to strategic knowledge, guide execution of strategies, and regulate use of strategies and problem-solving performance.” p. 231

Source: Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.

Elements of Metacognitive Processes Problems

“Self-instruction helps students to identify and direct the problem-solving strategies prior to execution. Self-questioning promotes internal dialogue for systematically analyzing problem information and regulating execution of cognitive strategies. Self-monitoring promotes appropriate use of specific strategies and encourages students to monitor general performance. [Emphasis added].” p. 231

Source: Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.

Combining Cognitive & Metacognitive Strategies to Assist Students With Mathematical Problem Solving

Solving an advanced math problem independently requires the coordination of a number of complex skills. The following strategies combine both cognitive and metacognitive elements (Montague, 1992; Montague & Dietz, 2009). First, the student is taught a 7-step process for attacking a math word problem (cognitive strategy). Second, the instructor trains the student to use a three-part self-coaching routine for each of the seven problem-solving steps (metacognitive strategy).

Cognitive Portion of Combined Problem Solving Approach Students With Mathematical Problem Solving

In the cognitive part of this multi-strategy intervention, the student learns an explicit series of steps to analyze and solve a math problem. Those steps include:

- Reading the problem. The student reads the problem carefully, noting and attempting to clear up any areas of uncertainly or confusion (e.g., unknown vocabulary terms).
- Paraphrasing the problem. The student restates the problem in his or her own words.
- ‘Drawing’ the problem. The student creates a drawing of the problem, creating a visual representation of the word problem.
- Creating a plan to solve the problem. The student decides on the best way to solve the problem and develops a plan to do so.
- Predicting/Estimating the answer. The student estimates or predicts what the answer to the problem will be. The student may compute a quick approximation of the answer, using rounding or other shortcuts.
- Computing the answer. The student follows the plan developed earlier to compute the answer to the problem.
- Checking the answer. The student methodically checks the calculations for each step of the problem. The student also compares the actual answer to the estimated answer calculated in a previous step to ensure that there is general agreement between the two values.

Metacognitive Portion of Combined Problem Solving Approach Students With Mathematical Problem Solving

The metacognitive component of the intervention is a three-part routine that follows a sequence of ‘Say’, ‘Ask, ‘Check’. For each of the 7 problem-solving steps reviewed above:

- The student first self-instructs by stating, or ‘saying’, the purpose of the step (‘Say’).
- The student next self-questions by ‘asking’ what he or she intends to do to complete the step (‘Ask’).
- The student concludes the step by self-monitoring, or ‘checking’, the successful completion of the step (‘Check’).

Combined Cognitive & Metacognitive Elements of Strategy Students With Mathematical Problem Solving

Combined Cognitive & Metacognitive Elements of Strategy Students With Mathematical Problem Solving

Combined Cognitive & Metacognitive Elements of Strategy Students With Mathematical Problem Solving

Combined Cognitive & Metacognitive Elements of Strategy Students With Mathematical Problem Solving

Combined Cognitive & Metacognitive Elements of Strategy Students With Mathematical Problem Solving

Combined Cognitive & Metacognitive Elements of Strategy Students With Mathematical Problem Solving

Combined Cognitive & Metacognitive Elements of Strategy Students With Mathematical Problem Solving

Applied Problems: Pop Quiz Students With Mathematical Problem Solving

Directions:As a team, read the following problem. At your tables, apply the 7-step problem-solving (cognitive) strategy to complete the problem. As you complete each step of the problem, apply the ‘Say-Ask-Check’ metacognitive sequence. Try to complete the entire 7 steps within the time allocated for this exercise.

7-Step Problem-Solving:Process

- Reading the problem.
- Paraphrasing the problem.
- ‘Drawing’ the problem.
- Creating a plan to solve the problem.
- Predicting/Estimat-ing the answer.
- Computing the answer.
- Checking the answer.

Q: “To move their armies, the Romans built over 50,000 miles of roads. Imagine driving all those miles! Now imagine driving those miles in the first gasoline-driven car that has only three wheels and could reach a top speed of about 10 miles per hour.

For safety's sake, let's bring along a spare tire. As you drive the 50,000 miles, you rotate the spare with the other tires so that all four tires get the same amount of wear. Can you figure out how many miles of wear each tire accumulates?”

A:“Since the four wheels of the three-wheeled car share the journey equally, simply take three-fourths of the total distance (50,000 miles) and you'll get 37,500 miles for each tire.”

Source: The Math Forum @ Drexel: Critical Thinking Puzzles/Spare My Brain. Retrieved from http://mathforum.org/k12/k12puzzles/critical.thinking/puzz2.html

At your tables:, Students With Mathematical Problem SolvingDiscuss these key ‘next steps’ for moving forward with RTI & Math Reasoning in your school or district.

Begin to draft an action plan to implement each of these steps.

Be prepared to report out on your work.

RTI & Math Reasoning: Key Next Steps…- Define ‘math reasoning’ and task-analyze to create a checklist of subskills that make up that term. (This checklist can be framed as student ‘look-for’ behaviors and adjusted to each grade level).
- Develop school-wide screening measures to identify students at-risk for math computation and math reasoning skills. Also develop the capacity to complete diagnostic math assessments for students with more severe math deficits.
- Set up ‘knowledge brokers’ in your school who will monitor math instructional and intervention programs and research findings by attending workshops, visiting websites, reading professional journals, etc.—and give them opportunities to share these updates with school staff.

Team Activity: Favorite Students With Mathematical Problem SolvingMath Websites…

- At your table:
- Discuss math websites that you have used and have found to be helpful.
- Be prepared to report out on your favorite math websites.

Math Mentors: Training Students to Independently Use On-Line Math-Help Resources

- Math mentors are recruited (school personnel, adult volunteers, student teachers, peer tutors) who have a good working knowledge of algebra.
- The school meets with each math mentor to verify mentor’s algebra knowledge.
- The school trains math mentors in 30-minute tutoring protocol, to include:
- Requiring that students keep a math journal detailing questions from notes and homework.
- Holding the student accountable to bring journal, questions to tutoring session.
- Ensuring that a minimum of 25 minutes of 30 minute session are spent on tutoring.

- Mentors are introduced to online algebra resources (e.g., www.algebrahelp.com, www.math.com) and encouraged to browse them and become familiar with the site content and navigation.

Math Mentors: Training Students to Independently Use On-Line Math-Help Resources

- Mentors are trained during ‘math mentor’ sessions to:
- Examine student math journal
- Answer student algebra questions
- Direct the student to go online to algebra tutorial websites while mentor supervises. Student is to find the section(s) of the websites that answer their questions.

- As the student shows increased confidence with algebra and with navigation of the math-help websites, the mentor directs the student to:
- Note math homework questions in the math journal
- Attempt to find answers independently on math-help websites
- Note in the journal any successful or unsuccessful attempts to independently get answers online
- Bring journal and remaining questions to next mentoring meeting.

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