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Chapter 14

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University of Waterloo

Psychology

PSYCH 312

Ernie Mac Kinnon

Summer

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[ CHAPTER FOURTEEN ] MATHEMATICS DIFFICULTIES
THEORIES
MATHEMATICS DIFFICULTIES
6-7% of students in GEC show evidence of serious mathematics difficulty
Ca. 26% of WLD exhibit problems in area of math
Dylscalculia – severe disability in mathematics w/ medical connotations
EARLY NUMBER CONCEPTS AND NUMBER SENSE
Learning mathematics is a sequential process, and children must acq’ skills at an earlier stage before going
on to the next stage
Early # learning includes skills in (1) spatial rel’nshps, (2) visual-motor and visual-perception skills, and
(3) concepts of time & direction
Spatial Relationships
Play activities w/ objects help dvlp sense of space, sequence, and order
Parents of children w/ math difficulties often report their child dN enjoy/play w/ blocks, puzzles, or models
Visual-Motor and Visual-Perception Abilities
Visual-motor abilities combine motor mvmt w/ what one sees
i.e. copying figure/shape
Visual perception refers to ability to interpret what one sees
i.e. perceive geometric shape as complete and integrated entity
For a child w/ visual perception difficulty, a square may not appear as a square shape, but rather as four
unrelated lines
Concepts of Time and Direction
Have difficulty estimating time span of hr/min/week
May not be able to judge and allocate time needed to complete assignment
Sometimes forget whether it’s morning/afternoon and may even go home during recess period, thinking
school day has ended
CHARACTERISTICS OF MATHEMATICS DISABILITIES
Information-Processing Difficulties
Motor problems problems in writing #s, illegible, slow, and inaccurate
Attention problems poor attention during instruction
Problems in memory and retrieval cannot remember math facts
Problems in visual-spatial processing problems aligning #s and thus, miscalculations
Problems w/ auditory processing difficulty ―counting-on‖
Language and Reading Abilities
Lang problem may cause them to confuse math terms
i.e. plus, take away, minus, borrowing
Math Anxiety
Emotion-based reac’n to math, which causes indvdls to freeze up when they confront math problems/tests
Guidelines for dealing w/ math anxiety:
- use competition carefully - provide abundant practice w/ ~ tests
- use clear instructions
- avoid unnecessary time pressures
- try to remove pressure from test-taking situations
MATHEMATICS DISABILITIES AT THE SECONDARY LEVEL
High school math req’mts for grad becoming more rigorous (geometry, statistics, calculus)
- adolescents w/ LD and RMD cont’ to have memory deficits tht interfere w/ AT learning of
computation facts
Effective instructional strategies in math for 2 students:
- provide many examples
- provide practice in discriminating various problem types
- provide explicit instruction
MATHEMATICS STANDARDS
High Standards and Annual Testing
Scores tht students receive on these math tests affect high-stakes decisions, such as whether student will be
promoted to next grade o will receive high school diploma
In general, students w/ mathematics disabilities dN fare well under high stakes assessment approach to
math edu’ w/o sp’ considerations and accommodations
Mathematics Principles and Standards From the national Council of Teachers of Mathematics (NCTM)
NCTM principles & standards intended for all students
Table 14.2 NCTM Standards NCTM Goals
NCTM Numbers and Understand #s, ways of presenting #s, rel’nshps among #s, and # sys’s
Standards for
operations Understand meaning of op’ns and how they relate
School Compute fluently and make reasonable estimates
Mathematics Algebra Understand patterns, rel’ns, and f’ns
Rep’ and analyze mathematical situations and structures using algebraic symbols
Use mathematical models to rep’ and understand quantitative rel’nshps
Analyze change in various contexts
Geometry Analyze char’s and properties of 2D & 3D geometric rel’nshps
Specify locations and describe spatial rel’nshps using coordinate geometry and
other rep’nal sys’s
Apply transformations and use symmetry to analyze math situations
Use visualization, spatial reasoning, and geometric modeling to solve problems
Measurement Understand measurable attributes of objects and the units, sys’s and processes of
measurement
Apply appropriate tm’s, tools, and formulas to dtrmn measurements
Data analysis and Formulate q’ns tht can be addressed w/ data and collect, organize, and display
probability relevant data to answer them
Select and use appropriate statistical methods to analyze data
Dvlp and evaluate inferences and predictions tht are based on data
Understand and apply basic concepts of probability
Problem solving Build new mathematical knowledge t/ problem solving
Solve problems tht arise in mathematics and in other contexts
Apply and adapt a variety of appropriate strategies to solve problems
Monitor and reflect on process of mathematical problem solving
Reasoning and Recognize reasoning and proof as fundamental aspects of mathematics
proof Make and investigate mathematical conjectures
Dvlp and evaluate mathematical arguments and proofs
Select and use various types of reasoning and methods of proof LEARNING THEORIES FOR MATHEMATICS INSTRUCTION
Active Involvement
Manipulate materials enable students to see, touch and move objects
Chinese proverb: ―I hear and I forget. I see and I remember. I do and I understand.‖
Progression from Concrete Learning to Abstract Learning
To help progression, 3 sequential levels of mathematics instruction are suggested
1.) Concrete level: students manipulate actual materials as they work out sol’ns to # problems
2.) Semi-concrete/representational level: use pictures/tallies to rep’ concrete objects as they work on problems
3.) Abstract level: use only #s to solve mathematics problems w/o help of semi-concrete pictures or tallies
Direct Instruction of Mathematics
Comprehensive sys’ tht integrates curriculum design w/ teaching tm’s to produce instructional programs tht
are highly organized and carefully sequenced
Teachers do the following: 1.) break tasks into small steps
2.) administer probes to drmn whether students are learning
3.) supply immediate feedback
4.) provide diagrams and pictures to enhance student understanding
5.) give ample (i) practice
Learning Strategies Instruction
1. Provide elaborate explanations
2. Model learning processes
3. Provide prompts to use strategies
4. Engage in teacher-student dialogues
5. Ask processing q’ns
Problem Solving
Identified as top priority for mathematics curriculum by NCTM
Implicit in teaching of problem solving are following underlying beliefs about math:
1.) no single way to do math
2.) no single way to organize math for instructional purposes
3.) important math concepts are actually learned t/ problem solving
Most difficult area of math for many students w/ math difficulties
Req’s students know how to apply math concepts and how to use computation skills in new/diff settings
Middle-grade st

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