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Foundations of Math Skills

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Title: Foundations of Math Skills


1
Foundations of Math Skills RTI
Interventions Jim Wright www.interventioncentral.o
rg
2
Mathematics is made of 50 percent formulas, 50
percent proofs, and 50 percent imagination.
Anonymous
3
Who is At Risk for Poor Math Performance? A
Proactive Stance
  • we use the term mathematics difficulties
    rather than mathematics disabilities. Children
    who exhibit mathematics difficulties include
    those performing in the low average range (e.g.,
    at or below the 35th percentile) as well as those
    performing well below averageUsing higher
    percentile cutoffs increases the likelihood
    thatyoung children who go on to have serious math
    problems will be picked upin the screening. p.
    295

Source Gersten, R., Jordan, N. C., Flojo, J.
R. (2005). Early identification and interventions
for students with mathematics difficulties.
Journal of Learning Disabilities, 38, 293-304.
4
Profile of Students with Math Difficulties
(Kroesbergen Van Luit, 2003)
  • Although the group of students with
    difficulties in learning math is very
    heterogeneous, in general, these students have
    memory deficits leading to difficulties in the
    acquisition and remembering of math knowledge.
    Moreover, they often show inadequate use of
    strategies for solving math tasks, caused by
    problems with the acquisition and the application
    of both cognitive and metacognitive strategies.
    Because of these problems, they also show
    deficits in generalization and transfer of
    learned knowledge to new and unknown tasks.

Source Kroesbergen, E., Van Luit, J. E. H.
(2003). Mathematics interventions for children
with special educational needs. Remedial and
Special Education, 24, 97-114..
5
The Elements of Mathematical Proficiency What
the Experts Say
6
(No Transcript)
7
Five Strands of Mathematical Proficiency
  • 1. Understanding Comprehending mathematical
    concepts, operations, and relations--knowing what
    mathematical symbols, diagrams, and procedures
    mean. Understanding refers to a students grasp
    of fundamental mathematical ideas. Students with
    understanding know more than isolated facts and
    procedures. They know why a mathematical idea is
    important and the contexts in which it is useful.
    Furthermore, they are aware of many connections
    between mathematical ideas. In fact, the degree
    of students understanding is related to the
    richness and extent of the connections they have
    made. p. 10

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.
8
Five Strands of Mathematical Proficiency
  • 2. Computing Carrying out mathematical
    procedures, such as adding, subtracting,
    multiplying, and dividing numbers flexibly,
    accurately, efficiently, and appropriately. Compu
    ting includes being fluent with procedures for
    adding, subtracting, multiplying, and dividing
    mentally or with paper and pencil, and knowing
    when and how to use these procedures
    appropriately. Although the word computing
    implies an arithmetic procedure, it also refers
    to being fluent with procedures from other
    branches of mathematics, such as measurement
    (measuring lengths), algebra (solving equations),
    geometry (constructing similar figures), and
    statistics (graphing data). Being fluent means
    having the skill to perform the procedure
    efficiently, accurately, and flexibly. p. 11

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.
9
Five Strands of Mathematical Proficiency
  • 3. Applying Being able to formulate problems
    mathematically and to devise strategies for
    solving them using concepts and procedures
    appropriately. Applying involves using ones
    conceptual and procedural knowledge to solve
    problems. A concept or procedure is not useful
    unless students recognize when and where to use
    itas well as when and whether it does not apply.
    Students need to be able to pose problems,
    devise solution strategies, and choose the most
    useful strategy for solving problems.. p. 13

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.
10
Five Strands of Mathematical Proficiency
  • 4. Reasoning Using logic to explain and
    justify a solution to a problem or to extend from
    something known to something less
    known. Reasoning is the glue that holds
    mathematics together. By thinking about the
    logical relationships between concepts and
    situations, students can navigate through the
    elements of a problem and see how they fit
    together.
  • One of the best ways for students to improve
    their reasoning is to explain or justify their
    solutions to others. Reasoning interacts
    strongly with the other strands of mathematical
    thought, especially when students are solving
    problems. p. 14

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.
11
Five Strands of Mathematical Proficiency
  • 5. Engaging Seeing mathematics as sensible,
    useful, and doableif you work at itand being
    willing to do the work. Engaging in mathematical
    activity is the key to success. Our view of
    mathematical proficiency goes beyond being able
    to understand, compute, apply, and reason. It
    includes engagement with mathematics. Students
    should have a personal commitment to the idea
    that mathematics makes sens and thatgiven
    reasonable effortthey can learn it and use it
    both in school and outside school.. p. 15-16

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.
12
Three General Levels of Math Skill Development
(Kroesbergen Van Luit, 2003)
  • As students move from lower to higher grades,
    they move through levels of acquisition of math
    skills, to include
  • Number sense
  • Basic math operations (i.e., addition,
    subtraction, multiplication, division)
  • Problem-solving skills The solution of both
    verbal and nonverbal problems through the
    application of previously acquired information
    (Kroesbergen Van Luit, 2003, p. 98)

Source Kroesbergen, E., Van Luit, J. E. H.
(2003). Mathematics interventions for children
with special educational needs. Remedial and
Special Education, 24, 97-114..
13
What is Number Sense? (Clarke Shinn, 2004)
  • the ability to understand the meaning of
    numbers and define different relationships among
    numbers. Children with number sense can
    recognize the relative size of numbers, use
    referents for measuring objects and events, and
    think and work with numbers in a flexible manner
    that treats numbers as a sensible system. p. 236

Source Clarke, B., Shinn, M. (2004). A
preliminary investigation into the identification
and development of early mathematics
curriculum-based measurement. School Psychology
Review, 33, 234248.
14
What Are Stages of Number Sense? (Berch, 2005,
p. 336)
  1. Innate Number Sense. Children appear to possess
    hard-wired ability (neurological foundation
    structures) to acquire number sense. Childrens
    innate capabilities appear also to be to
    represent general amounts, not specific
    quantities. This innate number sense seems to be
    characterized by skills at estimation
    (approximate numerical judgments) and a
    counting system that can be described loosely as
    1, 2, 3, 4, a lot.
  2. Acquired Number Sense. Young students learn
    through indirect and direct instruction to count
    specific objects beyond four and to internalize a
    number line as a mental representation of those
    precise number values.

Source Berch, D. B. (2005). Making sense of
number sense Implications for children with
mathematical disabilities. Journal of Learning
Disabilities, 38, 333-339...
15
The Basic Number Line is as Familiar as a
Well-Known Place to People Who Have Mastered
Arithmetic Combinations
16
Internal Numberline
  • As students internalize the numberline, they are
    better able to perform mental arithmetic (the
    manipulation of numbers and math operations in
    their head).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 19 20 21 22 23 24 25 26 27 28 29
17
Mental Arithmetic A Demonstration
  • 332 x 420 ?

Directions As you watch this video of a person
using mental arithmetic to solve a computation
problem, note the strategies and shortcuts that
he employs to make the task more manageable.
18
\Mental Arithmetic Demonstration What Tools Were
Used?
19
Math Computation Building Fluency Jim
Wright www.interventioncentral.org
20
"Arithmetic is being able to count up to twenty
without taking off your shoes." Anonymous
21
Benefits of Automaticity of Arithmetic
Combinations ( (Gersten, Jordan, Flojo, 2005)
  • There is a strong correlation between poor
    retrieval of arithmetic combinations (math
    facts) and global math delays
  • Automatic recall of arithmetic combinations frees
    up student cognitive capacity to allow for
    understanding of higher-level problem-solving
  • By internalizing numbers as mental constructs,
    students can manipulate those numbers in their
    head, allowing for the intuitive understanding of
    arithmetic properties, such as associative
    property and commutative property

Source Gersten, R., Jordan, N. C., Flojo, J.
R. (2005). Early identification and interventions
for students with mathematics difficulties.
Journal of Learning Disabilities, 38, 293-304.
22
Associative Property
  • within an expression containing two or more of
    the same associative operators in a row, the
    order of operations does not matter as long as
    the sequence of the operands is not changed
  • Example
  • (23)510
  • 2(35)10

Source Associativity. Wikipedia. Retrieved
September 5, 2007, from http//en.wikipedia.org/wi
ki/Associative
23
Commutative Property
  • the ability to change the order of something
    without changing the end result.
  • Example
  • 23510
  • 25310

Source Associativity. Wikipedia. Retrieved
September 5, 2007, from http//en.wikipedia.org/wi
ki/Commutative
24
How much is 3 8? Strategies to Solve
Source Gersten, R., Jordan, N. C., Flojo, J.
R. (2005). Early identification and interventions
for students with mathematics difficulties.
Journal of Learning Disabilities, 38, 293-304.
25
Math Skills Importance of Fluency in Basic Math
Operations
  • A key step in math education is to learn the
    four basic mathematical operations (i.e.,
    addition, subtraction, multiplication, and
    division). Knowledge of these operations and a
    capacity to perform mental arithmetic play an
    important role in the development of childrens
    later math skills. Most children with math
    learning difficulties are unable to master the
    four basic operations before leaving elementary
    school and, thus, need special attention to
    acquire the skills. A category of interventions
    is therefore aimed at the acquisition and
    automatization of basic math skills.

Source Kroesbergen, E., Van Luit, J. E. H.
(2003). Mathematics interventions for children
with special educational needs. Remedial and
Special Education, 24, 97-114.
26
Big Ideas Learn Unit (Heward, 1996)
  • The three essential elements of effective student
    learning include
  • Academic Opportunity to Respond. The student is
    presented with a meaningful opportunity to
    respond to an academic task. A question posed by
    the teacher, a math word problem, and a spelling
    item on an educational computer Word Gobbler
    game could all be considered academic
    opportunities to respond.
  • Active Student Response. The student answers the
    item, solves the problem presented, or completes
    the academic task. Answering the teachers
    question, computing the answer to a math word
    problem (and showing all work), and typing in the
    correct spelling of an item when playing an
    educational computer game are all examples of
    active student responding.
  • Performance Feedback. The student receives timely
    feedback about whether his or her response is
    correctoften with praise and encouragement. A
    teacher exclaiming Right! Good job! when a
    student gives an response in class, a student
    using an answer key to check her answer to a math
    word problem, and a computer message that says
    Congratulations! You get 2 points for correctly
    spelling this word! are all examples of
    performance feedback.

Source Heward, W.L. (1996). Three low-tech
strategies for increasing the frequency of active
student response during group instruction. In R.
Gardner, D. M.S ainato, J. O. Cooper, T. E.
Heron, W. L. Heward, J. W. Eshleman, T. A.
Grossi (Eds.), Behavior analysis in education
Focus on measurably superior instruction
(pp.283-320). Pacific Grove, CABrooks/Cole.
27
Math Intervention Tier I or II Elementary
Secondary Self-Administered Arithmetic
Combination Drills With Performance
Self-Monitoring Incentives
  1. The student is given a math computation worksheet
    of a specific problem type, along with an answer
    key Academic Opportunity to Respond.
  2. The student consults his or her performance chart
    and notes previous performance. The student is
    encouraged to try to beat his or her most
    recent score.
  3. The student is given a pre-selected amount of
    time (e.g., 5 minutes) to complete as many
    problems as possible. The student sets a timer
    and works on the computation sheet until the
    timer rings. Active Student Responding
  4. The student checks his or her work, giving credit
    for each correct digit (digit of correct value
    appearing in the correct place-position in the
    answer). Performance Feedback
  5. The student records the days score of TOTAL
    number of correct digits on his or her personal
    performance chart.
  6. The student receives praise or a reward if he or
    she exceeds the most recently posted number of
    correct digits.

Application of Learn Unit framework from
Heward, W.L. (1996). Three low-tech strategies
for increasing the frequency of active student
response during group instruction. In R. Gardner,
D. M.S ainato, J. O. Cooper, T. E. Heron, W. L.
Heward, J. W. Eshleman, T. A. Grossi (Eds.),
Behavior analysis in education Focus on
measurably superior instruction (pp.283-320).
Pacific Grove, CABrooks/Cole.
28
Self-Administered Arithmetic Combination
Drills Examples of Student Worksheet and Answer
Key
Worksheets created using Math Worksheet
Generator. Available online at http//www.interve
ntioncentral.org/htmdocs/tools/mathprobe/addsing.p
hp
29
Self-Administered Arithmetic Combination Drills
30
How to Use PPT Group Timers in the Classroom
31
Math Shortcuts Cognitive Energy- and Time-Savers
  • Recently, some researchershave argued that
    children can derive answers quickly and with
    minimal cognitive effort by employing calculation
    principles or shortcuts, such as using a known
    number combination toderive an answer (2 2 4,
    so 2 3 5), relations among operations (6 4
    10, so 10 -4 6), n 1, commutativity, and so
    forth. This approach to instruction is consonant
    with recommendations by the National Research
    Council (2001). Instruction along these linesmay
    be much more productive than rote drill without
    linkage to counting strategy use. p. 301

Source Gersten, R., Jordan, N. C., Flojo, J.
R. (2005). Early identification and interventions
for students with mathematics difficulties.
Journal of Learning Disabilities, 38, 293-304.
32
Math Multiplication Shortcut The 9 Times
Quickie
  • The student uses fingers as markers to find the
    product of single-digit multiplication arithmetic
    combinations with 9.
  • Fingers to the left of the lowered finger stands
    for the 10s place value.
  • Fingers to the right stand for the 1s place
    value.

Source Russell, D. (n.d.). Math facts to learn
the facts. Retrieved November 9, 2007, from
http//math.about.com/bltricks.htm
33
Students Who Understand Mathematical Concepts
Can Discover Their Own Shortcuts
  • Students who learn with understanding have less
    to learn because they see common patterns in
    superfically different sicuations. If they
    understand the general principle that the order
    in which two numbers are multiplied doesnt
    matter3 x 5 is the same as 5 x 3, for
    examplethey have about half as many number
    facts to learn. p. 10

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.
34
Application of Math Shortcuts to Intervention
Plans
  • Students who struggle with math may find
    computational shortcuts to be motivating.
  • Teaching and modeling of shortcuts provides
    students with strategies to make computation less
    cognitively demanding.

35
Math Computation Motivate With Errorless
Learning Worksheets
  • In this version of an errorless learning
    approach, the student is directed to complete
    math facts as quickly as possible. If the
    student comes to a number problem that he or she
    cannot solve, the student is encouraged to locate
    the problem and its correct answer in the key at
    the top of the page and write it in.
  • Such speed drills build computational fluency
    while promoting students ability to visualize
    and to use a mental number line.
  • TIP Consider turning this activity into a
    speed drill. The student is given a kitchen
    timer and instructed to set the timer for a
    predetermined span of time (e.g., 2 minutes) for
    each drill. The student completes as many
    problems as possible before the timer rings. The
    student then graphs the number of problems
    correctly computed each day on a time-series
    graph, attempting to better his or her previous
    score.

Source Caron, T. A. (2007). Learning
multiplication the easy way. The Clearing House,
80, 278-282
36
Errorless Learning Worksheet Sample
Source Caron, T. A. (2007). Learning
multiplication the easy way. The Clearing House,
80, 278-282
37
Math Computation Two Ideas to Jump-Start Active
Academic Responding
  • Here are two ideas to accomplish increased
    academic responding on math tasks.
  • Break longer assignments into shorter assignments
    with performance feedback given after each
    shorter chunk (e.g., break a 20-minute math
    computation worksheet task into 3 seven-minute
    assignments). Breaking longer assignments into
    briefer segments also allows the teacher to
    praise struggling students more frequently for
    work completion and effort, providing an
    additional natural reinforcer.
  • Allow students to respond to easier practice
    items orally rather than in written form to speed
    up the rate of correct responses.

Source Skinner, C. H., Pappas, D. N., Davis,
K. A. (2005). Enhancing academic engagement
Providing opportunities for responding and
influencing students to choose to respond.
Psychology in the Schools, 42, 389-403.
38
Math Computation Problem Interspersal Technique
  • The teacher first identifies the range of
    challenging problem-types (number problems
    appropriately matched to the students current
    instructional level) that are to appear on the
    worksheet.
  • Then the teacher creates a series of easy
    problems that the students can complete very
    quickly (e.g., adding or subtracting two 1-digit
    numbers). The teacher next prepares a series of
    student math computation worksheets with easy
    computation problems interspersed at a fixed rate
    among the challenging problems.
  • If the student is expected to complete the
    worksheet independently, challenging and easy
    problems should be interspersed at a 11 ratio
    (that is, every challenging problem in the
    worksheet is preceded and/or followed by an
    easy problem).
  • If the student is to have the problems read aloud
    and then asked to solve the problems mentally and
    write down only the answer, the items should
    appear on the worksheet at a ratio of 3
    challenging problems for every easy one (that
    is, every 3 challenging problems are preceded
    and/or followed by an easy one).

Source Hawkins, J., Skinner, C. H., Oliver, R.
(2005). The effects of task demands and additive
interspersal ratios on fifth-grade students
mathematics accuracy. School Psychology Review,
34, 543-555..
39
How to Create an Interspersal-Problems Worksheet
40
Additional Math Interventions Jim
Wright www.interventioncentral.org
41
Math Instruction Unlock the Thoughts of
Reluctant Students Through Class Journaling
  • Students can effectively clarify their knowledge
    of math concepts and problem-solving strategies
    through regular use of class math journals.
  • At the start of the year, the teacher introduces
    the journaling weekly assignment in which
    students respond to teacher questions.
  • At first, the teacher presents safe questions
    that tap into the students opinions and
    attitudes about mathematics (e.g., How important
    do you think it is nowadays for cashiers in
    fast-food restaurants to be able to calculate in
    their head the amount of change to give a
    customer?). As students become comfortable with
    the journaling activity, the teacher starts to
    pose questions about the students own
    mathematical thinking relating to specific
    assignments. Students are encouraged to use
    numerals, mathematical symbols, and diagrams in
    their journal entries to enhance their
    explanations.
  • The teacher provides brief written comments on
    individual student entries, as well as periodic
    oral feedback and encouragement to the entire
    class.
  • Teachers will find that journal entries are a
    concrete method for monitoring student
    understanding of more abstract math concepts. To
    promote the quality of journal entries, the
    teacher might also assign them an effort grade
    that will be calculated into quarterly math
    report card grades.

Source Baxter, J. A., Woodward, J., Olson, D.
(2005). Writing in mathematics An alternative
form of communication for academically
low-achieving students. Learning Disabilities
Research Practice, 20(2), 119135.
42
Math Review Incremental Rehearsal of Math Facts
Step 1 The tutor writes down on a series of
index cards the math facts that the student needs
to learn. The problems are written without the
answers.
43
Math Review Incremental Rehearsal of Math Facts
KNOWN Facts
UNKNOWN Facts
Step 2 The tutor reviews the math fact cards
with the student. Any card that the student can
answer within 2 seconds is sorted into the
KNOWN pile. Any card that the student cannot
answer within two secondsor answers
incorrectlyis sorted into the UNKNOWN pile.
44
Math Review Incremental Rehearsal of Math Facts
45
Math Review Incremental Rehearsal of Math Facts
46
Applied Math Helping Students to Make Sense of
Story Problems Jim Wright www.interventioncentra
l.org
47
Advanced Math Quotes from Yogi Berra
  • Ninety percent of the game is half mental."
  • Pair up in threes."
  • You give 100 percent in the first half of the
    game, and if that isn't enough in the second half
    you give what's left.

48
Applied Math Problems Rationale
  • Applied math problems (also known as story or
    word problems) are traditional tools for having
    students apply math concepts and operations to
    real-world settings.

49
Sample Applied Problems
  • Once upon a time, there were three little pigs -
    ages 2, 4, and 6. Are their ages even or odd?
  • Every day this past summer, Peter rode his bike
    to and from work. Each round trip was 13
    kilometers. His friend Marsha rode her bike18
    kilometers' each day, but just for exercise. How
    much further did Marsha ride her bike than Peter
    in one week?
  • Suzy is ten years older than Billy, and next year
    she will be twice as old as Billy. How old are
    they now?

50
Applied Math Problems Some Required Competencies
  • For students to achieve success with applied
    problems, they must be able to
  • Comprehend the text of written problems.
  • Understand specialized math vocabulary (e.g.,
    quotient).
  • Understand specialized use of common vocabulary
    (e.g., product).
  • Be able to translate verbal cues into a numeric
    equation.
  • Ignore irrelevant information included in the
    problem.
  • Interpret math graphics that may accompany the
    problem.
  • Apply a plan to problem-solving.
  • Check their work.

51
Potential Blockers of Higher-Level Math
Problem-Solving A Sampler
  • Limited reading skills
  • Failure to master--or develop automaticity in
    basic math operations
  • Lack of knowledge of specialized math vocabulary
    (e.g., quotient)
  • Lack of familiarity with the specialized use of
    known words (e.g., product)
  • Inability to interpret specialized math symbols
    (e.g., 4 lt 2)
  • Difficulty extracting underlying math
    operations from word/story problems or
    identifying and ignoring extraneous information
    included in word/story problems

52
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.
53
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.
54
Math Intervention Tier I High School Peer
Guided Pause
  • Students are trained to work in pairs.
  • At one or more appropriate review points in a
    math lecture, the instructor directs students to
    pair up to work together for 4 minutes.
  • During each Peer Guided Pause, students are
    given a worksheet that contains one or more
    correctly completed word or number problems
    illustrating the math concept(s) covered in the
    lecture. The sheet also contains several
    additional, similar problems that pairs of
    students work cooperatively to complete, along
    with an answer key.
  • Student pairs are reminded to (a) monitor their
    understanding of the lesson concepts (b) review
    the correctly math model problem (c) work
    cooperatively on the additional problems, and (d)
    check their answers. The teacher can direct
    student pairs to write their names on the
    practice sheets and collect them to monitor
    student understanding.

Source Hawkins, J., Brady, M. P. (1994). The
effects of independent and peer guided practice
during instructional pauses on the academic
performance of students with mild handicaps.
Education Treatment of Children, 17 (1), 1-28.
55
Applied Problems Encourage Students to Draw
the Problem
  • Making a drawing of an applied, or word,
    problem is one easy heuristic tool that students
    can use to help them to find the solution and
    clarify misunderstandings.
  • The teacher hands out a worksheet containing at
    least six word problems. The teacher explains to
    students that making a picture of a word problem
    sometimes makes that problem clearer and easier
    to solve.
  • The teacher and students then independently
    create drawings of each of the problems on the
    worksheet. Next, the students show their drawings
    for each problem, explaining each drawing and how
    it relates to the word problem. The teacher also
    participates, explaining his or her drawings to
    the class or group.
  • Then students are directed independently to make
    drawings as an intermediate problem-solving step
    when they are faced with challenging word
    problems. NOTE This strategy appears to be more
    effective when used in later, rather than
    earlier, elementary grades.

Source Hawkins, J., Skinner, C. H., Oliver, R.
(2005). The effects of task demands and additive
interspersal ratios on fifth-grade students
mathematics accuracy. School Psychology Review,
34, 543-555..
56
Interpreting Math Graphics
57
Housing Bubble Graphic New York Times 23
September 2007
58
Classroom Challenges in Interpreting Math Graphics
  • When encountering math graphics, students may
  • expect the answer to be easily accessible when in
    fact the graphic may expect the reader to
    interpret and draw conclusions
  • be inattentive to details of the graphic
  • treat irrelevant data as relevant
  • not pay close attention to questions before
    turning to graphics to find the answer
  • fail to use their prior knowledge both the extend
    the information on the graphic and to act as a
    possible check on the information that it
    presents.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).
Using QARs with charts and graphs. The Reading
Teacher, 56, 2127.
59
Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics
  • Students can be more savvy interpreters of
    graphics in applied math problems by applying the
    Question-Answer Relationship (QAR) strategy. Four
    Kinds of QAR Questions
  • RIGHT THERE questions are fact-based and can be
    found in a single sentence, often accompanied by
    'clue' words that also appear in the question.
  • THINK AND SEARCH questions can be answered by
    information in the text but require the scanning
    of text and making connections between different
    pieces of factual information.
  • AUTHOR AND YOU questions require that students
    take information or opinions that appear in the
    text and combine them with the reader's own
    experiences or opinions to formulate an answer.
  • ON MY OWN questions are based on the students'
    own experiences and do not require knowledge of
    the text to answer.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).
Using QARs with charts and graphs. The Reading
Teacher, 56, 2127.
60
Applied Problems Individualized Self-Correction
Checklists
  • Students can improve their accuracy on
    particular types of word and number problems by
    using an individualized self-instruction
    checklist that reminds them to pay attention to
    their own specific error patterns.
  • The teacher meets with the student. Together they
    analyze common error patterns that the student
    tends to commit on a particular problem type
    (e.g., On addition problems that require
    carrying, I dont always remember to carry the
    number from the previously added column.).
  • For each type of error identified, the student
    and teacher together describe the appropriate
    step to take to prevent the error from occurring
    (e.g., When adding each column, make sure to
    carry numbers when needed.).
  • These self-check items are compiled into a single
    checklist. Students are then encouraged to use
    their individualized self-instruction checklist
    whenever they work independently on their number
    or word problems.

Source Pólya, G. (1945). How to solve it.
Princeton University Press Princeton, N.J.
61
Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence
  1. DISTINGUISHING DIFFERENT KINDS OF GRAPHICS.
    Students are taught to differentiate between
    common types of graphics e.g., table (grid with
    information contained in cells), chart (boxes
    with possible connecting lines or arrows),
    picture (figure with labels), line graph, bar
    graph. Students note significant differences
    between the various graphics, while the teacher
    records those observations on a wall chart. Next
    students are given examples of graphics and asked
    to identify which general kind of graphic each
    is. Finally, students are assigned to go on a
    graphics hunt, locating graphics in magazines
    and newspapers, labeling them, and bringing to
    class to review.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).
Using QARs with charts and graphs. The Reading
Teacher, 56, 2127.
62
Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence
  1. INTERPRETING INFORMATION IN GRAPHICS. Students
    are paired off, with stronger students matched
    with less strong ones. The teacher spends at
    least one session presenting students with
    examples from each of the graphics categories.
    The presentation sequence is ordered so that
    students begin with examples of the most concrete
    graphics and move toward the more abstract
    Pictures gt tables gt bar graphs gt charts gt line
    graphs. At each session, student pairs examine
    graphics and discuss questions such as What
    information does this graphic present? What are
    strengths of this graphic for presenting data?
    What are possible weaknesses?

Source Mesmer, H.A.E., Hutchins, E.J. (2002).
Using QARs with charts and graphs. The Reading
Teacher, 56, 2127.
63
Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence
  1. LINKING THE USE OF QARS TO GRAPHICS. Students are
    given a series of data questions and correct
    answers, with each question accompanied by a
    graphic that contains information needed to
    formulate the answer. Students are also each
    given index cards with titles and descriptions of
    each of the 4 QAR questions RIGHT THERE, THINK
    AND SEARCH, AUTHOR AND YOU, ON MY OWN. Working
    in small groups and then individually, students
    read the questions, study the matching graphics,
    and verify the answers as correct. They then
    identify the type question being asked using
    their QAR index cards.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).
Using QARs with charts and graphs. The Reading
Teacher, 56, 2127.
64
Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence
  • USING QARS WITH GRAPHICS INDEPENDENTLY. When
    students are ready to use the QAR strategy
    independently to read graphics, they are given a
    laminated card as a reference with 6 steps to
    follow
  • Read the question,
  • Review the graphic,
  • Reread the question,
  • Choose a QAR,
  • Answer the question, and
  • Locate the answer derived from the graphic in the
    answer choices offered.
  • Students are strongly encouraged NOT to read the
    answer choices offered until they have first
    derived their own answer, so that those choices
    dont short-circuit their inquiry.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).
Using QARs with charts and graphs. The Reading
Teacher, 56, 2127.
65
Mindful Math Applying a Simple Heuristic to
Applied Problems
  • By following an efficient 4-step plan, students
    can consistently perform better on applied math
    problems.
  • UNDERSTAND THE PROBLEM. To fully grasp the
    problem, the student may restate the problem in
    his or her own words, note key information, and
    identify missing information.
  • DEVISE A PLAN. In mapping out a strategy to solve
    the problem, the student may make a table, draw a
    diagram, or translate the verbal problem into an
    equation.
  • CARRY OUT THE PLAN. The student implements the
    steps in the plan, showing work and checking work
    for each step.
  • LOOK BACK. The student checks the results. If the
    answer is written as an equation, the student
    puts the results in words and checks whether the
    answer addresses the question posed in the
    original word problem.

Source Pólya, G. (1945). How to solve it.
Princeton University Press Princeton, N.J.
66
Applied Problems Timed Quiz
  • 4-Step Problem-Solving
  • UNDERSTAND THE PROBLEM.
  • DEVISE A PLAN.
  • CARRY OUT THE PLAN.
  • LOOK BACK.
  • Suppose 6 monkeys take 6 minutes to eat 6
    bananas.
  • How many minutes would it take 3 monkeys to eat
    3 bananas?
  • How many monkeys would it take to eat 48
    bananas in 48 minutes?

Source Puzzles Brain Teasers Monkeys
Bananas. (n.d.). Retrieved on October 22, 2007,
from http//www.syvum.com/cgi/online/serve.cgi/tea
sers/monkeys.tdf?0
67
Applied Problems Timed Quiz
  • 4-Step Problem-Solving
  • UNDERSTAND THE PROBLEM.
  • DEVISE A PLAN.
  • CARRY OUT THE PLAN.
  • LOOK BACK.
  • Mr. Brown has 12 black gloves and 6 brown gloves
    in his closet. He blindly picks up some gloves
    from the closet. What is the minimum number of
    gloves Mr. Brown will have to pick to be certain
    to find a pair of gloves of the same color?

Source Puzzles Brain Teasers Monkeys
Bananas. (n.d.). Retrieved on October 22, 2007,
from http//www.syvum.com/cgi/online/serve.cgi/tea
sers/monkeys.tdf?0
68
Math Computation Fluency RTI Case Study
69
RTI Individual Case Study Math Computation
  • Jared is a fourth-grade student. His teacher,
    Mrs. Rogers, became concerned because Jared is
    much slower in completing math computation
    problems than are his classmates.

70
Tier 1 Math Interventions for Jared
  • Jareds school uses the Everyday Math curriculum
    (McGraw Hill/University of Chicago). In addition
    to the basic curriculum the series contains
    intervention exercises for students who need
    additional practice or remediation. The
    instructor, Mrs. Rogers, works with a small group
    of children in her roomincluding Jaredhaving
    them complete these practice exercises to boost
    their math computation fluency.

71
Tier 2 Standard Protocol (Group) Math
Interventions for Jared
  • Jared did not make sufficient progress in his
    Tier 1 intervention. So his teacher referred the
    student to the RTI Intervention Team. The team
    and teacher decided that Jared would be placed on
    the schools educational math software, AMATH
    Building Blocks, a self-paced, individualized
    mathematics tutorial covering the math
    traditionally taught in grades K-4. Jared
    worked on the software in 20-minute daily
    sessions to increase computation fluency in basic
    multiplication problems.

72
Tier 2 Math Interventions for Jared (Cont.)
  • During this group-based Tier 2 intervention,
    Jared was assessed using Curriculum-Based
    Measurement (CBM) Math probes. The goal was to
    bring Jared up to at least 40 correct digits per
    2 minutes.

73
Tier 2 Math Interventions for Jared (Cont.)
  • Progress-monitoring worksheets were created using
    the Math Computation Probe Generator on
    Intervention Central (www.interventioncentral.org)
    .

Example of Math Computation Probe Answer Key
74
Tier 2 Phase 1 Math Interventions for Jared
Progress-Monitoring
75
Tier 2 Individualized Plan Math Interventions
for Jared
  • Progress-monitoring data showed that Jared did
    not make expected progress in the first phase of
    his Tier 2 intervention. So the RTI Intervention
    Team met again on the student. The team and
    teacher noted that Jared counted on his fingers
    when completing multiplication problems. This
    greatly slowed down his computation fluency. The
    team decided to use a research-based strategy,
    Explicit Time Drills, to increase Jareds
    computation speed and eliminate his dependence on
    finger-counting. During this individualized
    intervention, Jared continued to be assessed
    using Curriculum-Based Measurement (CBM) Math
    probes. The goal was to bring Jared up to at
    least 40 correct digits per 2 minutes.

76
Explicit Time Drills Math Computational
Fluency-Building Intervention
  • Explicit time-drills are a method to boost
    students rate of responding on math-fact
    worksheets.
  • The teacher hands out the worksheet. Students
    are told that they will have 3 minutes to work on
    problems on the sheet. The teacher starts the
    stop watch and tells the students to start work.
    At the end of the first minute in the 3-minute
    span, the teacher calls time, stops the
    stopwatch, and tells the students to underline
    the last number written and to put their pencils
    in the air. Then students are told to resume work
    and the teacher restarts the stopwatch. This
    process is repeated at the end of minutes 2 and
    3. At the conclusion of the 3 minutes, the
    teacher collects the student worksheets.

Source Rhymer, K. N., Skinner, C. H., Jackson,
S., McNeill, S., Smith, T., Jackson, B. (2002).
The 1-minute explicit timing intervention The
influence of mathematics problem difficulty.
Journal of Instructional Psychology, 29(4),
305-311.
77
Cover-Copy-Compare Math Computational
Fluency-Building Intervention
  • The student is given sheet with correctly
    completed math problems in left column and index
    card. For each problem, the student
  • studies the model
  • covers the model with index card
  • copies the problem from memory
  • solves the problem
  • uncovers the correctly completed model to check
    answer

Source Skinner, C.H., Turco, T.L., Beatty, K.L.,
Rasavage, C. (1989). Cover, copy, and compare
A method for increasing multiplication
performance. School Psychology Review, 18,
412-420.
78
Tier 2 Phase 2 Math Interventions for Jared
Progress-Monitoring
79
Tier 2 Math Interventions for Jared
  • Explicit Timed Drill Intervention Outcome
  • The progress-monitoring data showed that Jared
    was well on track to meet his computation goal.
    At the RTI Team follow-up meeting, the team and
    teacher agreed to continue the fluency-building
    intervention for at least 3 more weeks. It was
    also noted that Jared no longer relied on
    finger-counting when completing number problems,
    a good sign that he had overcome an obstacle to
    math computation.

80
Group Activity Tier I Math Interventions
  • Look at the math intervention ideas in your
    handout.
  • Based on these ideas and other intervention
    strategies that you may have used as an educator,
    generate a list of up to FIVE Tier I
    (classroom-based) interventions you would
    recommend to teachers in your school.
  • Be prepared to share your results.
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