A Different'iated Mathematics Classroom

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A Different.iated Mathematics Classroom

• February 7, 2007
• Presented By Dr. Laura Rader

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Agenda

• Welcome and Opening Remarks
• PART I PREPARE YOURSELF
• Who are our struggling
• learners? (activity)
• Operational Definitions
• Principles of a
• Differentiated
• Mathematics Classroom
• BREAK (15 minutes)

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Agenda Continued

• PART II MATH, MAKING A
• DIFFERENCE- YOUR STUDENTS CAN DO IT!
• Taking the magic and mystery out of
  math

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The river-crossing problem.

• Nineteen campers are hiking through Acadia
  National Park when they come to a river. The
  river moves too rapidly for the campers to swim
  across it.

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The campers have 1 canoe, which holds 3 people.
On each trip across the river, 1 of the 3 canoe
riders must be an adult. There is only 1 adult
among the 19 campers. How many trips across the
river are necessary to get all the children to
the other side?
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Are any of these middle school students in your
mathematics classes?

• Suzie gets the assignment wrong because sloppy
  writing led to misreading and misalignment of
  numbers
• Jeff has not yet memorized the multiplication
  tables
• Juan reverses numbers when copying from the book
  or the chalkboard
• Ashley cannot decide what to do when solving
  math word problems
• Alfredo cannot remember algebraic formulas
• Gerard cannot remember the procedural sequence
  for division computation

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Reality

• The reality is that approximately 5-8 of
  school-age students have memory or other
  cognitive deficits that interfere with their
  ability to acquire, master, and apply
  mathematical concepts and skills (Geary, 2004).
  These students with mathematical learning
  disabilities (MLD) are at risk for failure in
  middle school mathematics because they generally
  are unprepared for the rigor of the middle school
  mathematics curriculum.

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DyscalculiaSpecific Types

• Verbal Dyscalculia (oral language)- a math
  disorder in retrieving mathematics labels, terms,
  and symbols
• Practognostic Dyscalculia (Practo doing,
  gnostic knowing, i.e. knowing by doing) a math
  disorder in applying math concepts when using
  manipulative objects in the environment (either
  visual or three-dimensional)

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DyscalculiaSpecific types Continued

• Lexical dyscalculia (reading)- a math disorder
  that involves impaired reading of math vocabulary
  and symbols
• Graphical Dyscalculia (writing) a math disorder
  that is an impairment in the writing of
  mathematics symbols, equations, and other
  relevant language terms

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DyscalculiaSpecific Types Continued

• Ideognostical Dyscalculia (ideas) a math
  disorder that centers on impaired mathematical
  thinking or impaired conceptualizations (the
  ideas) in mathematics
• Operational Dyscalculia (operations) a math
  disorder focusing on impaired applications of
  algorithms to the four basic math operations

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ACTIVITY

• Please take 10 minutes to work through the
  problems with your colleagues.
• Determine the error pattern
• Determine the specific type/s of dyscalculia
• Discussion to follow

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Rationale for Differentiation

• Instruction can be a one size fits all approach
• Gives students an opportunity to express
  themselves
• Gives students ownership for their learning

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Differentiated Instruction

• suggests that you can challenge all learners by
  providing materials and tasks at varied levels
  of difficulty, with varying degrees of
  scaffolding, through multiple instructional
  groups, and with time variations...

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Differentiated Instruction

• Further, differentiation suggests that teachers
  can craft lessons in ways that tap into multiple
  student interest to promote heightened learner
  interest.
  
• Carol Ann Tomlinson

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Characteristics

• Teachers begin where the students are
• Engages students through different learning
  modalities
• Students compete against themselves
• Teachers use classroom time flexibly
• Teachers are diagnosticians, prescribing the best
  possible instruction for each student.

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3 Key Questions

• WHAT IS THE TEACHER DIFFERENTIATING?
• HOW IS HE DIFFERENTIATING?
• WHY IS HE DIFFERENTIATING?

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Tomlinsons 8 Strategies

• Compacting the curriculum
• Independent study
• Interest groups
• Tiered assignments
• Flexible grouping
• Learning centers
• Adjusting questions
• Mentorships

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Strategy 1

• Compacting the curriculum

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Green Contract

• A- or higher.
• May skip all odd problems from assignments.
• May loop out of class lectures.
• Choose a project
• Green projects are more in depth.

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Blue Contract

• B or higher
• May skip every third problem in assignments
• May loop in and out of class lectures.
• Choose a project

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Results

• Increase value of mathematics for students who
  chose to contract
• Decreased motivation for the whole class
• Increased motivation for individual students

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Strategy 2

• Independent study

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Strategy 3

• Interest groups-multiple intelligence survey

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Strategy 4

• Tiered assignments

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Circle one

• GREEN I know and can use the distributive
  property.
• YELLOW I have heard of the distributive property
  before and vaguely remember it.
•  
• RED I do not know what the distributive property
  is or I do not understand it.
•  
• Solve 9 2(2x 2) 2

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Green

• Do 2 problems from each section in the assignment
  from the book AND choose 1 of the following
  projects
• Research the history of the distributive
  property and give a report or presentation
• Research the applications of the distributive
  property and demonstrate or give a report

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Green

• Be a student aid to others in the classroom
  (limit one aid per day)
• Select a project from the end of the chapter in
  the book.
• Propose another idea. Include your timeline.
  

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Yellow

• 1.a) Calculate mentally Using the distributive
  property how much do 5 tapes cost if they sell
  for 8.97 each?
• b) Monicas hourly wage is 12.00. If she
  receives time and a half for overtime, what is
  her overtime- hourly wage?

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Yellow

• 2. Write five problems similar to the above
  examples that can be solved mentally using the
  distributive property. Exchange your five
  problems with another group and solve them.
  Compare your answers.
• 3. Complete the assignment at the end of the
  lesson. You may work in groups if you desire.

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Red

• 1. Work with Mrs. Z.
•  
• 2. Do the first five problems of the assignment.
• 3. Complete the assignment. Feel free to work
  with a neighbor.

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Strategy 5

• Flexible grouping

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Strategy 6

• Learning centers and assessment stations

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Strategy 7

• Adjusting questions/prompting a student with a
  question ahead of time (helps with verbal
  dyscalculia)

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Strategy 8

• Mentorships

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Differentiated instruction has as many faces as
it has practitioners and as many outcomes as
there are learners. Kim Pettig
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Part IIActivity

• Why are students with MLD such poor mathematical
  problem solvers?
• Take a moment to solve the following problem
• Caroline owns a dog kennel. She usually has 15
  dogs to care for every week. Each dog eats about
  10 pounds of food per week. She pays 1.60 per
  pound for food. How much does Caroline pay to
  feed 15 dogs each week?

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Example continued

• Now, stop and make a list of the cognitive
  processes and metacognitive strategies you used
  to solve the problem.

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Process Suggestions

• Rereading the problem or parts of the problem
• Identifying the important information
• Asking yourself questions
• Putting the problem in your own words
• Visualize or draw a picture or diagram of the
  problem
• Telling yourself what to do
• Estimating the outcome
• Working backward and forward
• Checking that the process and the product are
  correct

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How can we teach students with MLD to be better
math problem solvers?

• Verbal Rehearsal- acronym RPV-HECC
• R Read for understanding
• P Paraphrase - in your own words
• V Visualize draw a picture or diagram
• H Hypothesize make a plan
• E Estimate predict the answer
• C Compute do the arithmetic
• C Check make sure everything is right

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Process Modeling

• Thinking aloud while demonstrating an activity

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Visualization- the basis for understanding the
problem

• Students with MLD need to be taught how to select
  the important information in the problem and
  develop a schematic representation
• Drawing a picture is not enough- students must be
  able to visualize the relationships among the
  pieces of information in the problem.

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Role Reversal

• Have students change places and become the
  teacher
• Fosters students to become independent rather
  than dependent

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Peer Coaching

• Gives students opportunities to see how other
  students approach mathematical problems
  differently, how they use cognitive processes and
  self-regulation strategies differently and how
  they represent and solve problems differently.

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Peer Coaching Example

• Example Your parents want to buy new school
  clothes for you and they said you could spend
  150.00. Make a list of items you would buy.
  Use newspaper ads to find prices. Then,, decide
  which items you will actually purchase. Work
  with your group to complete your list. Compare
  your final purchases with the purchases of the
  other group members.

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Performance Feedback

• Immediate, corrective and positive
• Allow students to graph their own progress

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Distributed Practice

• To maintain high performance students with MLD
  need to practice intermittently over time

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POSSIBLE ASSESSMENT STRATEGIES

• Portfolios
• Two grades personal grade grade compared to
  class
• Give superscripts A1, A2, or A3
• Replacement grade

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TIPS FOR SUCCESS

• Clearly express criteria for success
• For projects, stress planning and check-in dates.
• Provide choice for your students
• Use task cards or assignment sheets
• Give students as much responsibility for their
  learning as possible

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TIPS FOR SUCCESS

• Begin with a familiar topic
• Take small steps
• Gather various resources
• Clearly express criteria for success
• Have a plan to help students when you are busy.

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If you finish early

• Choose another project
• Do Math Stumpers
• Explore math web sites
• Try to solve Tangrams
• Try to solve wooden puzzles
• Play a 2 person game
• Help your neighbor

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Possible Positives

• Teachers are partners with their students
• Student interest is tapped
• Greater retention
• Choice is motivating
• Allows students to learn at different paces

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Possible Positives

• Allows for multiple forms of intelligence
• Gives teachers a different view of students
• Challenges all students

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FINAL THOUGHTS

• In the end, all learners need your energy, your
  heart and your mind. They have that in common
  because they are young humans. How they need you
  however, differs. Unless we understand and
  respond to those differences, we fail many
  learners
• Tomlinson, C.A. (2001). How to differentiate
  instruction in mixed ability classrooms (2nd
  Ed.). Alexandria, VA ASCD

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References

• Geary, D.C. (2004). Mathematics and learning
  disabilities. Journal of Learning Disabilities,
  37, 4-15.
• Montague, M. Jitendra, A. (2007) Teaching
  mathematics to middle school students. New York
  The Guilford Press.
• Tomlinson, C.A. (2003). Fulfilling the promise of
  the differentiated classroom. Alexandria,
  Virginia ASCD.
• lrader_at_ccny.cuny.edu