Math Reasoning: Assisting the Struggling Middle and High School Learner Jim Wright www.interventioncentral.org. Download PowerPoint from this workshop at: http://www.interventioncentral.org/SSTAGE.php. Intervention Research & Development: A Work in Progress. Georgia ‘Pyramid of Intervention’.
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Math Reasoning: Assisting the Struggling Middle and High School LearnerJim Wrightwww.interventioncentral.org
Intervention Research & Development: A Work in Progress
Georgia ‘Pyramid of Intervention’
Source: Georgia Dept of Education: http://www.doe.k12.ga.us/ Retrieved 13 July 2007
“… 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).
“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.
“…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.
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.
“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
Intervention: Key Concepts
Student learning can be thought of as a multi-stage process. The universal stages of learning include:
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.
Interventions should be written up in a ‘scripted’ format to ensure that:
Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools. Routledge: New York.
Interventions can move up the RTI Tiers through being intensified across several dimensions, including:
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.
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.
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:
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.
Important elements of math instruction for low-performing students:
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..
Source: Rourke, B. P. (1993). Arithmetic disabilities, specific & otherwise: A neuropsychological perspective. Journal of Learning Disabilities, 26, 214-226.
Team Activity: Define ‘Math Reasoning’…
“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.
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.
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.
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.
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.
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
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.
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
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem-solving. Cognition and Instruction, 23(3), 313–349.
Strands of Math Proficiency
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.
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.
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
Motivation
Table Activity: Evaluate Your School’s Math Proficiency…
RTI & Secondary Literacy:Explicit Vocabulary Instruction
“…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.
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.
Source: Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading Teacher, 56(8), 786-795.
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.
Teachers can use graphic displays to structure their vocabulary discussions and activities (Boardman et al., 2008; Fisher, 2007; Texas Reading Initiative, 2002).
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).
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).
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.
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:
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.
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’.
The student is trained to use an Internet lookup strategy to better understand dictionary or glossary definitions of key vocabulary items.
RTI & Secondary Literacy:Extended Discussion
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.
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…
RTI & Secondary Literacy:Reading Comprehension
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).
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).
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:
Teach Students to Identify Underlying Structures of Math Problems
Most algebra problems contain predictable elements:
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
Developing Student Metacognitive Abilities
“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.
“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.
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).
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:
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:
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
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:, Discuss 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.
Team Activity: Favorite Math Websites…
Secondary Group-Based Math Intervention Example