Mathematics instruction
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Mathematics Instruction . EDCI 5474. In this changing world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures. NCTM, 2000. Guiding Questions. What factors impair performance? How do standards impact students?

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Mathematics instruction

Mathematics Instruction

EDCI 5474


Mathematics instruction

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


Guiding questions

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?


Knowing and doing math

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”


Factors

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


  • Mathematics instruction

    Problem solving

    Communication ideas

    Reasoning

    Everyday applications

    Reasonableness

    Estimation

    Computational skills

    Algebraic thinking

    Measurement

    Geometry

    Statistics

    Probablity

    NCTM


    Effective instruction

    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


    Efficient teaching

    Efficient Teaching

    • Demonstrate

    • Guided practice with prompts/feedback

    • Independent practice with feedback

    • Advanced organizers

    • Mastery of skill as generalization level


    Mathematics instruction

    • 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


    Mathematics instruction

    • 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


    Mathematics instruction

    • 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


    Math curricula

    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


    Strategic math

    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


    Math methods for swd

    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 < 70 to success 90%

      • Construct probes for error patterns

      • “Slice back” to re-teach lower skill


    Concrete to seimconcrete

    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


    Direct instruction

    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


    Sequence rule based

    Sequence & Rule Based

    • Resequence texts

      • e.g. teach “zero” facts earlier

    • Sameness to reduce memorization

      • E.g. 0 times any number = 0


    Build strategies

    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


    Motivation for practice

    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


    Mathematics instruction

    • Cooperative Learning

      • Interactive practice, group rewards

    • Computer Instruction

      • Tutorial, drill & practice, software

      • With teacher directed instruction


    Teacher student dialogues

    Teacher Student Dialogues

    • Math reasoning & problem solving

    • Explicit modeling centered on concepts

    • Verbal rehearsal, feedback, guided practice, check for understanding

    • Think alouds, scaffolded practice


    Instructional tips

    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


    Basic facts operations

    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


    Complex operations

    Complex Operations

    • Understand place value, regrouping, state the operation, graph paper

    • Addition with regrouping

      • Concrete manipulatives, mnemonic or visual reminders


    Other essential skills

    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


    Mathematics instruction

    • 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)


    Mathematics instruction

    • 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


    Enhance instruction

    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.


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