Mathematics Instruction

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Mathematics Instruction

• EDCI 5474

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• In this changing world, those who understand and
  can do mathematics will have significantly
  enhanced opportunities and options for shaping
  their futures. NCTM, 2000

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Guiding Questions

• What factors impair performance?
• How do standards impact students?
• What math skills are essential?
• What teaching strategies can be applied?
• What methods are effective for teaching basic
  math facts?
• What methods are effective for problem solving?

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Knowing and Doing Math

• Knowing
• Comprehend basic principles
• More than one way to explain
• More than one acceptable answer
• Difference between 2 numbers
• Doing
• Apply strategies to complete item
• take away

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Factors

• 60 of students with LD
• Reading, writing, attending, memory, language,
  motivation
• (Montague, Bos, Doucette, 1991)
• Procedural representational
• (Geary, 1993)
• Failure to understand concepts, lack
  pre-requisite skills, maintain facts
• (Cox, 1975 Miller Milam, 1987 et. al)
• Need direction instruction, strategic

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NCTM

• Problem solving
• Communication ideas
• Reasoning
• Everyday applications
• Reasonableness
• Estimation

• Computational skills
• Algebraic thinking
• Measurement
• Geometry
• Statistics
• Probablity

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Effective Instruction

• Appropriate math content
• Problem solving, communicating ideas, everyday
  applications, computation
• Establish goals expectations
• Assessment
• Systematic explicit instruction
• Demonstrate-prompt -practice sequence
• Highly organized, step by step presentation what,
  when and why use target skill
• Insure understanding and task demands
• Use learning principles
• Attention, reinforcement, varied practice
  motivation

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Efficient Teaching

• Demonstrate
• Guided practice with prompts/feedback
• Independent practice with feedback
• Advanced organizers
• Mastery of skill as generalization level

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• Teach to understand concepts
• Concrete-representational-abstract sequence
• Learning concepts and rules
• e.g. communative property of addition, inverse
  relationship, place value, time 0 0
• Monitor progress
• Check for understanding, interact, demonstrate
• Provide feedback
• Essential promote learning, mastery, independent
  learning, plan activities, motivation, confidence
  abilities
• Immediate, basic elaborate

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• Teach to mastery
• Automaticity
• Benefits retention, higher level skills,
  completion, positive feelings
• Independent practice
• Games, peers, computers, self monitor
• Teach problem solving skills
• Math knowledge, application, engage thinking
• Strategy instruction parallel

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• Teach generalization
• Perform different situations
• Motivation to apply, discuss why and how used,
  varied examples, mastery to use rather than
  remember
• Positive attitude
• Goal setting, success on prior skills, progress
  chart, high expectations, enthusiasm, reinforcing

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Math Curricula

• Distar Arithmetic SRA
• Connecting Math Concepts
• Edmark
• Project MATH
• Real Life Math
• Strategic Math
• Touch Math
• Rapid, sequenced, scripted, direct instruction
• 1 2 3 4 5 6 7 8 9

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Strategic Math

• Pretest lesson
• Teach concrete application
• Teach representational application
• DRAW (Discover, Read problem, Answer with
  drawing, Write answer)
• Teach abstract application
• Posttest
• Provide practice to fluency
• Strategic Math Series, Mercer Miller, 1991

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Math Methods for SWD

• Curriculum-based Assessment Error Analysis
• Spiral curriculum
• Fast paced, insufficient practice, developmental
  sequence gaps, order of concepts, number of
  concepts
• Performance frustration lt 70 to success 90
• Construct probes for error patterns
• Slice back to re-teach lower skill

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Concrete to Seimconcrete

• Manipulatives rather than pictures or numerals
  and symbols
• Repeated opportunities
• Relate manipulatives to numbers immediately
• Build understanding of concepts
• Questions and verbalize thinking
• Check answers with manipulatives

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Direct Instruction

• Focused, explicit, interactive
• Tasks analyzed, clear examples (models),
  questions feedback
• Hi success
• Models, demonstration, time delay
• Systematic practice in real world contexts
• Provide calculators as postsecondary level

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Sequence Rule Based

• Resequence texts
• e.g. teach zero facts earlier
• Sameness to reduce memorization
• E.g. 0 times any number 0

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Build Strategies

• Self instruction mnemonics
• Good with direct instruction
• Associate pictures, visual imagery
• Should not substitute for practice for mastery
• Good for rote memory deficits

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Motivation for Practice

• Games
• Bingo, blackjack, jeopardy, math basketball
• Real Life
• Newspapers, internet, technical courses, solve
  real life problems
• Self-Monitoring, Goal Setting
• Record, graph, choice of goals, conference,
  self-talk

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• Cooperative Learning
• Interactive practice, group rewards
• Computer Instruction
• Tutorial, drill practice, software
• With teacher directed instruction

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Teacher Student Dialogues

• Math reasoning problem solving
• Explicit modeling centered on concepts
• Verbal rehearsal, feedback, guided practice,
  check for understanding
• Think alouds, scaffolded practice

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Instructional Tips

• Number Sense numbers to objects
• One to one correspondence
• Readiness skill
• Numeration
• Number values
• Seriation
• Ordinal, less/more
• Place value
• Ones, tens, hundreds

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Basic Facts Operations

• Addition
• Sticks, counters, dots on cards, Touch Math, sums
  of ten, number lines, 1-5 w/9-5 below
• Subtraction
• Counters, sticks, number line, tally marks, Touch
  Math,
• Multiplication
• Groups in rows, Geoboards-cubes, teach rules,
  fingers
• Division
• Counters, number line to 25

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Complex Operations

• Understand place value, regrouping, state the
  operation, graph paper
• Addition with regrouping
• Concrete manipulatives, mnemonic or visual
  reminders

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Other Essential Skills

• Fractions
• Need systematic practice, examples
• Explicitly define whole, fractions equal parts
• Fair sharing
• Formal symbols, ½ or ¼
• Concrete aids before teaching rules
• Mastering Fractions interactive videodisc
• Need more time

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• Decimals
• Concrete manipulatives
• Represent fractions with denominators 10 or 100
• Visual representation less that whole
• Algebra
• Direct instruction, think alouds, concrete to
  abstract, strategy (FOIL)

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• Arithmetic Word Problems
• Group problems around familiar themes
• Pictures manipulatives of the problem
• Attack word problems
• Emphasize understanding with visuals, question
  for important information, model strategies for
  solving, give feedback during process - SOLVE

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Enhance Instruction

• Increase instructional time, reduced independent
  practice
• Review, directed instruction, guided practice,
  independent practice with feedback
• Use concrete objectives
• Teach key math terms, sum
• Small group instruction
• Real life examples
• Varied reinforcement styles, process rather
  product
• Lock, R.H. (1996). Adapting Mathematics
  Instruction in General Education for Students
  with Mathematics Disabilities, CLD.