Foundations of Math Skills

1
Foundations of Math Skills RTI
InterventionsJim Wrightwww.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
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(No Transcript)
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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...
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The Basic Number Line is as Familiar as a
Well-Known Place to People Who Have Mastered
Arithmetic Combinations
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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).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 1920 21 22 23 24 25 26 27 28 29
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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.
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\Mental Arithmetic Demonstration What Tools Were
Used?
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Math Computation Building FluencyJim
Wrightwww.interventioncentral.org
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"Arithmetic is being able to count up to twenty
without taking off your shoes." Anonymous
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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.
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
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
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 InterventionsJim
Wrightwww.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 ProblemsJim Wrightwww.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 GraphicNew York Times23
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.