texes 191 generalist ec-6 test mathematics
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Dion J. Dubois, Ed.D. 5 th Grade Teacher Stevens Park Elementary [email protected]. TExES 191 Generalist EC-6 Test Mathematics. Real Life Relationships Personal Contexts Invented Procedures Making Connections Encouraging Problem Solving - PowerPoint PPT PresentationTRANSCRIPT
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TEXES 191GENERALIST EC-6 TESTMATHEMATICS
Dion J. Dubois, Ed.D.5th Grade TeacherStevens Park [email protected]
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BIGS IDEAS IN MATHEMATICSReal Life Relationships
Personal Contexts
Invented Procedures
Making Connections
Encouraging Problem Solving
Hands-On Activities and Project-Based Learning
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COGNITIVE DEVELOPMENTSensorimotor Stage
(Infancy)
Pre-Operational Stage (Toddler to Early Childhood)
Concrete Operational Stage (Elementary)
Formal Operational Stage(Adolescence)
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COGNITIVE DEVELOPMENTSensorimotor Stage (Birth – 2 yrs old)
(Infancy)In this period, intelligence is demonstrated through
motor activity without the use of symbols. Knowledge of the world is limited (but developing)
because its based on physical interactions and experiences. Children acquire object permanence at
about 7 months of age (memory). Physical development (mobility) allows the child to begin
developing new intellectual abilities. Some symbolic (language) abilities are developed at the end of this
stage.
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COGNITIVE DEVELOPMENTPre-Operational Stage (2 – 7 yrs old)
(Toddler to Early Childhood)In this period (which has two substages),
intelligence is demonstrated through the use of symbols, language use matures, and memory and imagination are developed, but thinking is
done in a nonlogical, nonreversible manner. Egocentric thinking predominates
Can Not Think Of More Than One Thing At A Time!
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PRE-OPERATIONAL STAGEPK through 2nd Grade
CentrationTendency to Focus on One Aspect of a
Situation and Neglect the Other AspectsFocusing on Color Rather Than Shape
When Grouping Blocks or Other Shapes
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PRE-OPERATIONAL STAGEPK through 2nd GradeLack Conservation
Quantity, Length or Number of Items is unrelated to the arrangement or
appearance of items.
Nickel is more than a DimeBecause of its Size
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COGNITIVE DEVELOPMENTConcrete Operational Stage (7-11 yrs old)
(Elementary)In this stage (characterized by 7 types of conservation:
number, length, liquid, mass, weight, area, volume), intelligence is demonstrated through logical and
systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions
that are reversible). Egocentric thought diminishes.Conservation & Reverse Thinking With
Concrete Objects!
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CONCRETE OPERATIONAL STAGE2nd – 6th GradeConservation
Properties are conserved or invariant after an object undergoes
physical transformation.A Stack versus a Row of Coins
Beaker of Liquid
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CONCRETE OPERATIONAL STAGE2nd – 6th GradeDecentering
Taking into Account Multiple AspectsOf a Problem to Solve It
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CONCRETE OPERATIONAL STAGE2nd – 6th Grade
SeriationArranging Objects in an order according
To Size, Shape, Color or any other Attribute
Such as Thickness
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CONCRETE OPERATIONAL STAGE2nd – 6th GradeClassification
When a child can name and identify sets of objects
according to their appearance, size or other characteristic.
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CONCRETE OPERATIONAL STAGE2nd – 6th GradeReversibility
Objects can be Changed and thenReturned to their Original State
Fact Families4 + 5 = 9 9 – 5 = 4
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COGNITIVE DEVELOPMENTFormal Operational Stage (11+
years old)(Adolescence)
In this stage, intelligence is demonstrated through the logical use of symbols related to
abstract concepts. Early in the period there is a return to egocentric thought.
Only 35% of high school graduates in industrialized countries obtain formal
operations; many people do not think formally during adulthood.
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C13-MATHEMATICS INSTRUCTION The teacher understands how children learn mathematical skills and uses this
knowledge to plan, organize, and implement instruction and assess
learning.
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SIX STRANDS OF MATHEMATICS1. Numbers, Operations and Quantitative
Reasoning2. Patterns, Relationships and Algebraic
Thinking3. Measurement
4. Geometry and Spatial Reasoning5. Probability and Statistics6. Underlying Processes and
Mathematical Tools
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IDEAL MATHEMATICS CLASSROOM
1. Instruction is organized in Units2. Heterogeneous Groups
3. Manipulatives and Technology4. Communication
5. Challenging Activities6. Ongoing Assessment7. Parent Involvement
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CONSTRUCTIVIST APPROACH
Prior Knowledge greatly influences the learning of math and that
learning is cumulative and vertically structured.
A student centered, discovery oriented approach
which promotes conceptual knowledge and independent
problem solving ability in students.
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ROLE OF THE TEACHER
1. Set up learning situations2. Build mathematical
understanding3. Provide opportunities for students to construct their own
knowledge4. Provide experiences to stimulate
their thinking5. Encourage discovery
6. Use divergent questions
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STAGES OF MATHEMATICAL DEVELOPMENT
1. Concrete Stage2. Representational
Stages3. Abstract Stage
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CENTRAL TEACHING STRATEGY
Problem Solving1. Read the Problem
2. Make a Plan3. Solve the Problem
4. Reflect on the Answer
Look for Reasonableness
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PROBLEM SOLVING STRATEGIES
1. Act It Out2. Draw A Picture3. Find a Pattern
4. Make a Table or List5. Working Backward
6. Use Smaller Numbers
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MATHEMATICAL ASSESSMENT
1. Formative2. Summative3. Authentic
Importance of Rubrics
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NCTM STANDARDS Teachers need to help students learn to value mathematics become confident in their own abilities become mathematical problem solvers learn to communicate mathematically learn to reason mathematically
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ACTIVE LEARNING ENVIRONMENT Active Learning Environments Activities should be learned centered Content must be relevant to learners Learning Centers are used to reinforce and extend learning of content Questioning strategies promote HOTS
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HIGHER ORDER THINKING SKILLS(HOTS) Knowledge Comprehension Application Analysis Synthesis Evaluation
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MANIPULATIVES IN MATHEMATICS Attribute and Base Ten Blocks Calculators Trading Chips, Counters and Tiles Cubes, Spinners, Dice Cuisenaire Rods Geoboards Pentominoes Pattern Blocks Tangrams
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MANIPULATIVES IN MATHEMATICS Attribute Blocks: sorting, comparing, contrasting, classifying, identifying, sequencing
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MANIPULATIVES IN MATHEMATICS Base 10 Blocks: addition, subtraction, number sense, place value and counting
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MANIPULATIVES IN MATHEMATICSCuisenaire Rods
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MANIPULATIVES IN MATHEMATICSGeoboards: transformations, angles, area, perimeter.
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MANIPULATIVES IN MATHEMATICSPentominoes: symmetry, area, and perimeter
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MANIPULATIVES IN MATHEMATICSTangrams: fractions, spatial awareness, geometry, area, and perimeter
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C014-NUMBER CONCEPTS AND OPERATIONS
The Teacher Understands Concepts Related To Numbers, Operations And
Algorithms, and The Properties Of Numbers.
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C14-NUMBER CONCEPTS AND OPERATIONS
A. Properties: Commutative, Associative and Distributive Properties of Addition and Multiplication.
B. Types of Numbers: Cardinal, Ordinal, Integers, Rational, Irrational, Real, Prime and Composite.
C. Ways of Writing Numbers: Whole, Decimals, Fractions and Percent
D. Operations: Addition, Subtraction, Multiplication and Division
E. Relationships between Numbers: Ratios and Proportions
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ASSOCIATIVE PROPERTY
(3 + 4) + 5 = 3 + (4 + 5)
(3 X 4) X 5 = 3 X (4 X 5)
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COMMUTATIVE PROPERTY
3 + 4 = 4 + 3
4 X 3 = 3 X 4
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DISTRIBUTIVE PROPERTY
5 X (3 + 4) = 5 X 3 + 5 X 4
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TYPES OF NUMBERS
Real Numbers
Whole Numbers Integers
IrrationalNumbers
RationalNumbers
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TYPES OF NUMBERS
Integers-5, -3, 0, 1, 2
Rational Numbers½ 4¾ .25 2.15 35%
Irrational NumbersSquare Roots
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COMMON MATHEMATICAL DIFFICULTIES Place Value Difficulties
Using Zero when writing numbers Regrouping
Addition/Subtraction Identifying addition/subtraction situations When numerals have a different number of digits
Multiplication/Division Basic Facts Distributive Property of multiplication over addition Aligning partial products
http://www.youtube.com/watch?v=e7Ult0p-uGU
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OTHER MATHEMATICAL DIFFICULTIES Greatest Common Factor Least Common Multiple Exponents (Power of Ten) - 103
Determining Events: There are four numbers (1,2,3 & 4) in a box. How many different ways can you select those numbers?
Combination: number of possible selections where the order of selection is not important : = 3 + 2 + 1
12, 13, 14, 23, 24, 34 Permutation: number of possible selections where
the order of selection IS important.: = (3 + 2 + 1) X 2 = 12, 21, 13, 14, 41, 23, 32, 24, 42, 34, 43
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COMBINATIONS AND PERMUTATIONS Combination: Order does not Matter
My fruit salad is a combination of apples, grapes and bananas
Permutation: Here the order does matter The combination to the safe was 472.
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C015-PATTERNS AND ALGEBRAThe Teacher Understands Concepts
Related To Patterns, Relations, Functions, And Algebraic Reasoning.
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C015-PATTERNS AND ALGEBRA
A. Equations and InequalitiesB. Patterns (Repeating and
Growing)C. Coordinate PlanesD. Ordered PairsE. Functions and Input-Output
TablesF. Graphing Functions
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COORDINATE PLANE-QUADRANTS
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LINEAR FUNCTIONS
https://www.youtube.com/watch?feature=player_embedded&v=AZroE4fJqtQ
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INFORMATION ON FUNCTIONS
www.khanacademy.org
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C016-GEOMETRY AND MEASUREMENT
The Teacher Understands Concepts and Principles of Geometry and Measurement.
Points, Lines, Planes, Angles, Dimensions,Circles, Triangles, Quadrilaterals,
Solid Figures, Nets, Pyramids, PrismsCylinders, Spheres, Cones
Symmetry and Transformations
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SOLIDS (THREE-DIMENSIONAL FIGURES) Cubes Spheres Cones (Circular Prism) Tetrahedron (Triangular Prism)
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NETS (TWO-DIMENSIONAL FIGURES) Line, Ray, Line Segment Circle Triangle Quadrilateral (square, rhombus or
diamond, parallelogram, trapezoid) Pentagon Hexagon Octagon
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PERIMETER, AREA AND VOLUME Perimeter – outside of a two-
dimensional figure Area – inside of a two-dimensional
figure Surface Area - outside of a three-
dimensional figure Volume – inside of a three-dimensional
figure
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SIMILARITY AND CONGRUENCE Congruent – same size/same shape Similar – same shape – not the same
size
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ANGLES Angle
Acute Right Obtuse
Sides Equilateral Scalene
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TRANSFORMATIONAL GEOMETRY Translations Reflections Glide-Reflections Rotations Dilations (expansions and contractions) Tessellations
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TRANSLATION
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REFLECTION
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ROTATION
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GLIDE REFLECTION
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DILATION
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TESSELLATION
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MEASUREMENT Temperature Money Weight, Area, Capacity, Density Percent Speed and Acceleration Pythagorean Theory Right Angle Trigonometry
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MEASUREMENT Customary and Standard (Metric) Units
Length Temperature Capacity Weight
Perimeter Area Volume
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C017-PROBABILITY AND STATISTICSThe Teacher Understands Concepts Related to Probability and Statistics
and Their Applications.
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PROBABILITY Probability is the likelihood or chance
that something is the case or that an event will occur. Probability theory is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
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PROBABILITY In mathematics, a probability of an
event A is represented by a real number in the range from 0 to 1 and written as P(A).
An impossible event has a probability of 0, and a certain event has a probability of 1.
Outcome = any possible result Event = group of outcomes Combinations= list of all possible
outcomes
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STATISTICS Mode = Most Often Mean = Average Median = Middle Number Range Normal Distribution
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NORMAL DISTRIBUTION
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STEM AND LEAF PLOT
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HISTOGRAMS-CONTINUOUS DATA
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C18-MATHEMATICAL PROCESSESThe Teacher Understands
Mathematical Processes And Knows How To Reason Mathematically,
Solve Mathematical Problems, And Make Mathematical Connections
Within And Outside Of Mathematics.
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C018-MATHEMATICAL PROCESSES
A.RoundingB.EstimationC.Types of Reasoning
A. Inductive- takes a series of specific observations and tries to expand them into a more general theory.
B. Deductive - starting out with a theory or general statement, then moving towards a specific conclusion
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DEDUCTIVE REASONINGGoing from the General to the Specific
A Quadrilateral has four sides. What other figures has four sides? Square Rectangle Parallelogram Rhombus Trapezoid
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INDUCTIVE REASONINGSpecific Examples – General Conclusion
What do all of these shapes have in common?
Square Rectangle Parallelogram Rhombus TrapezoidThey All Have Four Sides
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HOW CHILDREN LEARN MATH Theories and Principles of Learning Using prior mathematical knowledge Mathematics manipulatives Motivate students Actively engagement Individual, small-group, and large-group
setting
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ASSESSMENT Purpose, characteristics, and uses of various
assessments (Formative/Summative) Consistent assessments Scoring procedures Evaluation of a variety of assessment methods
and materials for reliability, validity, absence of bias, clarity of language, and appropriateness of mathematical level.
Relationship between assessment and instruction Modification of assessment for ELL students
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QUESTIONS????