Dion J. Dubois, Ed.D. 5 th Grade Teacher Stevens Park Elementary [email protected].
-
Upload
isaac-bridges -
Category
Documents
-
view
213 -
download
1
Transcript of Dion J. Dubois, Ed.D. 5 th Grade Teacher Stevens Park Elementary [email protected].
TEXES 191GENERALIST EC-6 TESTMATHEMATICS
Dion J. Dubois, Ed.D.5th Grade TeacherStevens Park [email protected]
BIGS IDEAS IN MATHEMATICS
Real Life Relationships
Personal Contexts
Invented Procedures
Making Connections
Encouraging Problem Solving
Hands-On Activities and Project-Based Learning
COGNITIVE DEVELOPMENT
Sensorimotor Stage (Infancy)
Pre-Operational Stage (Toddler to Early Childhood)
Concrete Operational Stage (Elementary)
Formal Operational Stage(Adolescence)
COGNITIVE DEVELOPMENT
Sensorimotor 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.
COGNITIVE DEVELOPMENT
Pre-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!
PRE-OPERATIONAL STAGE
PK through 2nd GradeCentration
Tendency to Focus on One Aspect of a Situation and Neglect the Other Aspects
Focusing on Color Rather Than ShapeWhen Grouping Blocks or Other Shapes
PRE-OPERATIONAL STAGE
PK 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
COGNITIVE DEVELOPMENT
Concrete 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!
CONCRETE OPERATIONAL STAGE
2nd – 6th GradeConservation
Properties are conserved or invariant after an object undergoes
physical transformation.A Stack versus a Row of Coins
Beaker of Liquid
CONCRETE OPERATIONAL STAGE
2nd – 6th GradeDecentering
Taking into Account Multiple AspectsOf a Problem to Solve It
CONCRETE OPERATIONAL STAGE
2nd – 6th GradeSeriation
Arranging Objects in an order accordingTo Size, Shape, Color or any other
AttributeSuch as Thickness
CONCRETE OPERATIONAL STAGE
2nd – 6th GradeClassification
When a child can name and identify sets of objects
according to their appearance, size or other characteristic.
CONCRETE OPERATIONAL STAGE
2nd – 6th GradeReversibility
Objects can be Changed and thenReturned to their Original State
Fact Families4 + 5 = 9 9 – 5 = 4
COGNITIVE DEVELOPMENT
Formal 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.
C13-MATHEMATICS INSTRUCTION
The teacher understands how children learn mathematical skills and uses this
knowledge to plan, organize, and implement instruction and assess
learning.
SIX STRANDS OF MATHEMATICS
1. Numbers, Operations and Quantitative Reasoning
2. Patterns, Relationships and Algebraic Thinking
3. Measurement4. Geometry and Spatial Reasoning
5. Probability and Statistics6. Underlying Processes and
Mathematical Tools
IDEAL MATHEMATICS CLASSROOM
1. Instruction is organized in Units2. Heterogeneous Groups
3. Manipulatives and Technology4. Communication
5. Challenging Activities6. Ongoing Assessment7. Parent Involvement
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.
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
STAGES OF MATHEMATICAL DEVELOPMENT
1. Concrete Stage2. Representational
Stages3. Abstract Stage
CENTRAL TEACHING STRATEGY
Problem Solving1. Read the Problem
2. Make a Plan3. Solve the Problem
4. Reflect on the Answer
Look for Reasonableness
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
MATHEMATICAL ASSESSMENT
1. Formative2. Summative3. Authentic
Importance of Rubrics
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
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
HIGHER ORDER THINKING SKILLS(HOTS) Knowledge Comprehension Application Analysis Synthesis Evaluation
MANIPULATIVES IN MATHEMATICS
Attribute and Base Ten Blocks Calculators Trading Chips, Counters and Tiles Cubes, Spinners, Dice Cuisenaire Rods Geoboards Pentominoes Pattern Blocks Tangrams
MANIPULATIVES IN MATHEMATICS
Attribute Blocks: sorting, comparing, contrasting, classifying, identifying, sequencing
MANIPULATIVES IN MATHEMATICS
Base 10 Blocks: addition, subtraction, number sense, place value and counting
MANIPULATIVES IN MATHEMATICS
Cuisenaire Rods
MANIPULATIVES IN MATHEMATICS
Geoboards: transformations, angles, area, perimeter.
MANIPULATIVES IN MATHEMATICS
Pentominoes: symmetry, area, and perimeter
MANIPULATIVES IN MATHEMATICS
Tangrams: fractions, spatial awareness, geometry, area, and perimeter
C014-NUMBER CONCEPTS AND OPERATIONS
The Teacher Understands Concepts Related To Numbers, Operations And
Algorithms, and The Properties Of Numbers.
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
ASSOCIATIVE PROPERTY
(3 + 4) + 5 = 3 + (4 + 5)
(3 X 4) X 5 = 3 X (4 X 5)
COMMUTATIVE PROPERTY
3 + 4 = 4 + 3
4 X 3 = 3 X 4
DISTRIBUTIVE PROPERTY
5 X (3 + 4) = 5 X 3 + 5 X 4
TYPES OF NUMBERS
Real Numbers
Whole Numbers
Integers
IrrationalNumbers
RationalNumbers
TYPES OF NUMBERS
Integers-5, -3, 0, 1, 2
Rational Numbers½ 4¾ .25 2.15 35%
Irrational NumbersSquare Roots
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
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
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.
C015-PATTERNS AND ALGEBRA
The Teacher Understands Concepts Related To Patterns, Relations,
Functions, And Algebraic Reasoning.
C015-PATTERNS AND ALGEBRA
A. Equations and InequalitiesB. Patterns (Repeating and
Growing)C. Coordinate PlanesD. Ordered PairsE. Functions and Input-Output
TablesF. Graphing Functions
COORDINATE PLANE-QUADRANTS
LINEAR FUNCTIONS
https://www.youtube.com/watch?feature=player_embedded&v=AZroE4fJqtQ
INFORMATION ON FUNCTIONS
www.khanacademy.org
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, Prisms
Cylinders, Spheres, ConesSymmetry and Transformations
SOLIDS (THREE-DIMENSIONAL FIGURES) Cubes Spheres Cones (Circular Prism) Tetrahedron (Triangular Prism)
NETS (TWO-DIMENSIONAL FIGURES)
Line, Ray, Line Segment Circle Triangle Quadrilateral (square, rhombus or
diamond, parallelogram, trapezoid) Pentagon Hexagon Octagon
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
SIMILARITY AND CONGRUENCE
Congruent – same size/same shape Similar – same shape – not the same
size
ANGLES
Angle Acute Right Obtuse
Sides Equilateral Scalene
TRANSFORMATIONAL GEOMETRY
Translations Reflections Glide-Reflections Rotations Dilations (expansions and contractions) Tessellations
TRANSLATION
REFLECTION
ROTATION
GLIDE REFLECTION
DILATION
TESSELLATION
MEASUREMENT
Temperature Money Weight, Area, Capacity, Density Percent Speed and Acceleration Pythagorean Theory Right Angle Trigonometry
MEASUREMENT
Customary and Standard (Metric) Units Length Temperature Capacity Weight
Perimeter Area Volume
C017-PROBABILITY AND STATISTICS
The Teacher Understands Concepts Related to Probability and Statistics
and Their Applications.
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.
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
STATISTICS
Mode = Most Often Mean = Average Median = Middle Number Range Normal Distribution
NORMAL DISTRIBUTION
STEM AND LEAF PLOT
HISTOGRAMS-CONTINUOUS DATA
C18-MATHEMATICAL PROCESSES
The Teacher Understands Mathematical Processes And Knows
How To Reason Mathematically, Solve Mathematical Problems, And Make Mathematical Connections
Within And Outside Of Mathematics.
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
DEDUCTIVE REASONING
Going from the General to the Specific A Quadrilateral has four sides. What
other figures has four sides? Square Rectangle Parallelogram Rhombus Trapezoid
INDUCTIVE REASONING
Specific Examples – General ConclusionWhat do all of these shapes have in common?
Square Rectangle Parallelogram Rhombus TrapezoidThey All Have Four Sides
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
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
QUESTIONS????