Foundations of Math Skills

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Foundations of Math Skills RTI
InterventionsJim Wrightwww.interventioncentral.o
rg
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Elbow Group Activity What are common student
math concerns in your school?

• In your elbow groups
• Discuss the most common math problems that you
  encounter in your school(s). At what grade level
  do you typically encounter these problems?
• Be prepared to share your discussion points with
  the larger group.

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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..
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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..
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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.
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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...
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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.
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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
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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
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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.
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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).

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16 17 18 1920 21 22 23 24 25 26 27 28 29
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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.
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Think-aloud and the Internal Numberline in
Action What is 37 multiplied by 46?
Well, lets see. First, I know that 30 times 46
would be like multiplying 46 times 10 three
times in a row. That would be, um, 460 times 3.
Three times zero is zero ones place value, 6
times 3 is 18 tens place valuecarry a one and
add it to 4 times 3 to give you 13. So 460
times 3 is 1380. And that takes care of 30 times
46. Now I have to solve for 7 times 46.
Hmmm7 times 40 would be 280I know that because
7 times 4 is 28just add another zero. I can
then add 1380 and 280and that would be 1660. I
knew that because 1380 plus 300 is 1680 and then
I just subtracted 20. Whats left? Um7 times
6. That would be 42. So 1660 and 42 would be,
uhsubvocally 1670, 1680, 1690, 1700and two.
Aloud The answer is 1702.
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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.
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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.
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Self-Administered Arithmetic Combination
DrillsExamples of Student Worksheet and Answer
Key
Worksheets created using Math Worksheet
Generator. Available online athttp//www.interve
ntioncentral.org/htmdocs/tools/mathprobe/addsing.p
hp
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Self-Administered Arithmetic Combination Drills
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Math Intervention Tier I or II Elementary
Middle School Cover Copy Compare

• The student is given a math worksheet with 10
  number problems and answers on the left side of
  the page.
• For each problem, the student
• Studies the correctly completed problem on the
  left side of the page.
• Covers the problem with an index card.
• Copies the problem from memory on the right side
  of the page.
• Solves the problem.
• Uncovers the correct model problem to check his
  or her work.
• If the students problem was done incorrectly,
  the student repeats the process until correct.

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.
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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.
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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.

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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.

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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.

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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.

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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
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Tier 2 Phase 1 Math Interventions for Jared
Progress-Monitoring
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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,
  Cover-Copy-Compare, 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.

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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.
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Tier 2 Phase 2 Math Interventions for Jared
Progress-Monitoring
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Tier 2 Math Interventions for Jared

• Cover-Copy-Compare 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.

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The application to create CBM Early Math Fluency
probes online
http//www.interventioncentral.org/php/numberfly/
numberfly.php
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Examples of Early Math Fluency (Number Sense)
CBM Probes
Sources Clarke, B., Shinn, M. (2004). A
preliminary investigation into the identification
and development of early mathematics
curriculum-based measurement. School Psychology
Review, 33, 234248. Chard, D. J., Clarke, B.,
Baker, S., Otterstedt, J., Braun, D., Katz, R.
(2005). Using measures of number sense to screen
for difficulties in mathematics Preliminary
findings. Assessment For Effective Intervention,
30(2), 3-14
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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

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CBM Math Computation
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CBM Math Computation Probes Preparation
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CBM Math Computation Sample Goals

• Addition Add two one-digit numbers sums to 18

• Addition Add 3-digit to 3-digit with regrouping
  from ones column only

• Subtraction Subtract 1-digit from 2-digit with
  no regrouping

• Subtraction Subtract 2-digit from 3-digit with
  regrouping from ones and tens columns

• Multiplication Multiply 2-digit by 2-digit-no
  regrouping

• Multiplication Multiply 2-digit by 2-digit with
  regrouping

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CBM Math Computation Assessment Preparation

• Select either single-skill or multiple-skill math
  probe format.
• Create student math computation worksheet
  (including enough problems to keep most students
  busy for 2 minutes)
• Create answer key

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CBM Math Computation Assessment Preparation

• Advantage of single-skill probes
• Can yield a more pure measure of students
  computational fluency on a particular problem type

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CBM Math Computation Assessment Preparation

• Advantage of multiple-skill probes
• Allow examiner to gauge students adaptability
  between problem types (e.g., distinguishing
  operation signs for addition, multiplication
  problems)
• Useful for including previously learned
  computation problems to ensure that students
  retain knowledge.

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CBM Math Computation Probes Administration
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CBM Math Computation Probes Scoring
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CBM Math Computation Assessment Scoring

• Unlike more traditional methods for scoring
  math computation problems, CBM gives the student
  credit for each correct digit in the answer.
  This approach to scoring is more sensitive to
  short-term student gains and acknowledges the
  childs partial competencies in math.

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Math Computation ScoringExample
12 CDs
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Math Computation Scoring
Numbers Above Line Are Not Counted
Placeholders Are Counted
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CBM Math Computation Activity

• Score the number of correct digits on your math
  probe.

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Trainer Question What objections or concerns
might teachers have about using CBM math
computation probes? How would you address these
concerns?