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Title: 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.
16
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.

17
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).

18
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).

19
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.

20
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.

21
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.
22
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.
24
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?

25
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
40
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.

41
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.
43
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
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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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