### Mathematics - diondubois.com

```TEXES 191
GENERALIST EC-6 TEST
MATHEMATICS
Dion J. Dubois, Ed.D.
Stevens Park Elementary
[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
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
Centration
Tendency to Focus on One Aspect of a Situation
and Neglect the Other Aspects
Focusing on Color Rather Than Shape
When Grouping Blocks or Other Shapes
PRE-OPERATIONAL STAGE
Lack Conservation
Quantity, Length or Number of Items is unrelated
to the arrangement or appearance of items.
Nickel is more than a Dime
Because 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
Conservation
Properties are conserved or invariant after an
object undergoes physical transformation.
A Stack versus a Row of Coins
Beaker of Liquid
CONCRETE OPERATIONAL STAGE
Decentering
Taking into Account Multiple Aspects
Of a Problem to Solve It
CONCRETE OPERATIONAL STAGE
Seriation
Arranging Objects in an order according
To Size, Shape, Color or any other Attribute
Such as Thickness
CONCRETE OPERATIONAL STAGE
Classification
When a child can name and identify sets of
objects
according to their appearance, size
or other characteristic.
CONCRETE OPERATIONAL STAGE
Reversibility
Objects can be Changed and then
Returned to their Original State
Fact Families
4+5=9
9–5=4
COGNITIVE DEVELOPMENT
Formal Operational Stage (11+ years old)
In this stage, intelligence is demonstrated through the
logical use of symbols related to abstract concepts. Early
Only 35% of high school graduates in industrialized
countries obtain formal operations; many people do not
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. Measurement
4. Geometry and Spatial Reasoning
5. Probability and Statistics
6. Underlying Processes and Mathematical Tools
IDEAL MATHEMATICS CLASSROOM
1.
3.
Instruction is organized in Units
2. Heterogeneous Groups
Manipulatives and Technology
4. Communication
5. Challenging Activities
6. Ongoing Assessment
7. 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
Set up learning situations
Build mathematical understanding
Provide opportunities for students to
construct their own knowledge
Provide experiences to stimulate their
thinking
5.
Encourage discovery
6.
Use divergent questions
1.
2.
3.
4.
STAGES OF MATHEMATICAL DEVELOPMENT
Concrete Stage
Representational Stages
3. Abstract Stage
1.
2.
CENTRAL TEACHING STRATEGY
Problem Solving
1.
3.
4.
2.
Make a Plan
Solve the Problem
Look for Reasonableness
PROBLEM SOLVING STRATEGIES
Act It Out
2.
Draw A Picture
3.
Find a Pattern
4.
Make a Table or List
5.
Working Backward
6.
Use Smaller Numbers
1.
MATHEMATICAL ASSESSMENT
Formative
2. Summative
3. Authentic
1.
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:
perimeter.
transformations, angles, area,
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.
B.
C.
D.
E.
Properties: Commutative, Associative and
Distributive Properties of Addition and Multiplication.
Types of Numbers: Cardinal, Ordinal, Integers,
Rational, Irrational, Real, Prime and Composite.
Ways of Writing Numbers: Whole, Decimals,
Fractions and Percent
Division
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
4X3=3X4
DISTRIBUTIVE PROPERTY
5 X (3 + 4) = 5 X 3 + 5 X 4
TYPES OF NUMBERS
Whole
Numbers
Integers
Real Numbers
Rational
Numbers
Irrational
Numbers
TYPES OF NUMBERS
Integers
-5, -3, 0, 1, 2
Rational Numbers
½ 4¾ .25 2.15 35%
Irrational Numbers
Square Roots
COMMON MATHEMATICAL DIFFICULTIES
 Place Value Difficulties






Using Zero when writing numbers
Regrouping
When numerals have a different number of digits
Multiplication/Division



Basic Facts
Distributive Property of multiplication over addition
Aligning partial products
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.
B.
C.
D.
E.
F.
Equations and Inequalities
Patterns (Repeating and Growing)
Coordinate Planes
Ordered Pairs
Functions and Input-Output Tables
Graphing Functions
LINEAR FUNCTIONS
INFORMATION ON FUNCTIONS
C016-GEOMETRY AND MEASUREMENT
The Teacher Understands Concepts and
Principles of Geometry and Measurement.
Points, Lines, Planes, Angles, Dimensions,
Solid Figures, Nets, Pyramids, Prisms
Cylinders, Spheres, Cones
Symmetry 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
Rounding
B. Estimation
C. Types of Reasoning
A.
A.
B.
Inductive- takes a series of specific observations
and tries to expand them into a more general
theory.
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 Conclusion
What do all of these shapes have in common?
 Square
 Rectangle
 Parallelogram
 Rhombus
 Trapezoid
They 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?
???
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