Math Reasoning: Assisting the Struggling Middle and High School Learner Jim Wright www.interventioncentral.org

1
Math Reasoning Assisting the Struggling Middle
and High School LearnerJim Wrightwww.interventi
oncentral.org
2
Download PowerPoint from this workshop
athttp//www.interventioncentral.org/SSTAGE.php
3
Intervention Research Development A Work in
Progress
4
Georgia Pyramid of Intervention
Source Georgia Dept of Education
http//www.doe.k12.ga.us/ Retrieved 13 July 2007
5
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).
6
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.
7
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.
8
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 processcriteria 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 shouldnt be afraid to ask)..
Learning Disabilities Research Practice,
22(2),129136.
9
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.betterhighschool
s.org/pubs/ p. 9
10
Intervention Key Concepts
11
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.
12
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 andusing
  the script as a checklistverify 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.
13
Intervention Script Builder Form
14
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.
15
Research-Based Elements of Effective Academic
Interventions

• Correctly targeted The intervention is
  appropriately matched to the students 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.
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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.

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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).

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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 students 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).

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Core Instruction, Interventions, Accommodations
Modifications Sorting Them Out

• Modification. A modification changes the
  expectations of what a student is expected to
  know or dotypically 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.

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Intervention Footprint 7-Step Lifecycle of an
Intervention Plan

1. Information about the students academic or
   behavioral concerns is collected.
2. The intervention plan is developed to match
   student presenting concerns.
3. Preparations are made to implement the plan.
4. The plan begins.
5. The integrity of the plans implementation is
   measured.
6. Formative data is collected to evaluate the
   plans effectiveness.
7. The plan is discontinued, modified, or replaced.

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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 students response to the intervention
• Clearly defined problem. The students 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 students 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.
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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 childrens 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..
23
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 numberminuendfrom 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 students 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.
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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?

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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), 313349.
26
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), 313349.
27
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), 313349.
28
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), 313349.
29
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), 313349.
30
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), 313349.
31
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), 313349.
32
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), 313349.
33
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), 313349.
34
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), 313349.
35
Strands of Math Proficiency
36

• 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.
37
Five Strands of Mathematical Proficiency

1. Understanding Comprehending mathematical
   concepts, operations, and relations--knowing what
   mathematical symbols, diagrams, and procedures
   mean.
2. Computing Carrying out mathematical procedures,
   such as adding, subtracting, multiplying, and
   dividing numbers flexibly, accurately,
   efficiently, and appropriately.
3. 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.
38
Five Strands of Mathematical Proficiency (Cont.)

1. Reasoning Using logic to explain and justify a
   solution to a problem or to extend from something
   known to something less known.
2. Engaging Seeing mathematics as sensible, useful,
   and doableif you work at itand 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.
39
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 doableif you work at itand being willing to
  do the work.

Motivation
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Five Strands of Mathematical Proficiency (NRC,
2002)

• Table Activity Evaluate Your Schools 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.

• 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 doableif you work at itand being willing to
  do the work.

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RTI Secondary LiteracyExplicit Vocabulary
Instruction
42
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 childs 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/a
uthor/chard_hmm05.pdf.
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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
  vocabularywhile 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/a
uthor/chard_hmm05.pdf.
44
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 dont
  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 assessmentrather than treating
  as an afterthought.

Source Adams, T. L. (2003). Reading mathematics
More than words can say. The Reading Teacher,
56(8), 786-795.
45
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.

46
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).

47
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).

48
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49
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).

50
Word Definition Map Example
51
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52
Semantic Feature Analysis

• 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.

53
Semantic Feature Analysis Example

• VOCABULARY TERM TRANSPORTATION

54
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55
Comparison/Contrast (Venn) Diagram

• Two terms are listed and defined. For each term,
  the student brainstorms qualities or properties
  or examples that illustrate the terms 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

56
(No Transcript)
57
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.

58
Generate Possible Sentences

• 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.

59
Provide Dictionary Training

• 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.

60
RTI Secondary LiteracyExtended Discussion
61
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.

62
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

63
Standard Protocol Discussion Format

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

64
RTI Secondary LiteracyReading Comprehension
65
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).

66
Assist Students in Setting Content Goals for
Reading

• Students are more likely to be motivated to
  read--and to read more closelyif 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).

67
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 students own words

68
Teach Students to Identify Underlying Structures
of Math Problems
69
Algebra Word Problems Predictable Elements

• 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
• Students may fail to discriminate relevant from
  irrelevant information.
• Students may commit translation errors when
  processing relational statements.

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/a
bout/bdscomm/list/mathpanel/reports.html
70
Developing Student Metacognitive Abilities
71
Importance of Metacognitive Strategy Use

• 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 solvingThat 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.
72
Elements of Metacognitive Processes

• 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.
73
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).

74
Cognitive Portion of Combined Problem Solving
Approach

• 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.

75
Metacognitive Portion of Combined Problem Solving
Approach

• 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).

76
Combined Cognitive Metacognitive Elements of
Strategy
77
Combined Cognitive Metacognitive Elements of
Strategy
78
Combined Cognitive Metacognitive Elements of
Strategy
79
Combined Cognitive Metacognitive Elements of
Strategy
80
Combined Cognitive Metacognitive Elements of
Strategy
81
Combined Cognitive Metacognitive Elements of
Strategy
82
Combined Cognitive Metacognitive Elements of
Strategy
83
Applied Problems Pop Quiz

• 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?

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-SolvingProcess
• 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.

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 _at_ Drexel Critical
Thinking Puzzles/Spare My Brain. Retrieved from
http//mathforum.org/k12/k12puzzles/critical.think
ing/puzz2.html
84
RTI Math Reasoning Key Next Steps
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.

1. 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).
2. 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.
3. 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.

85
Team Activity Favorite Math 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.

86
Secondary Group-Based Math Intervention Example
87
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
  mentors 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.

88
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.

89
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