TEXES 191GENERALIST EC-6 TESTMATHEMATICS

Dion J. Dubois, Ed.D.5th Grade TeacherStevens Park Elementaryddubois@dallasisd.org

BIGS IDEAS IN MATHEMATICSReal Life Relationships

Personal Contexts

Invented Procedures

Making Connections

Encouraging Problem Solving

Hands-On Activities and Project-Based Learning

COGNITIVE DEVELOPMENTSensorimotor Stage

(Infancy)

Pre-Operational Stage (Toddler to Early Childhood)

Concrete Operational Stage (Elementary)

Formal Operational Stage(Adolescence)

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.

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!

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

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

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!

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

CONCRETE OPERATIONAL STAGE2nd – 6th GradeDecentering

Taking into Account Multiple AspectsOf a Problem to Solve It

CONCRETE OPERATIONAL STAGE2nd – 6th Grade

SeriationArranging Objects in an order according

To Size, Shape, Color or any other Attribute

Such as Thickness

CONCRETE OPERATIONAL STAGE2nd – 6th GradeClassification

When a child can name and identify sets of objects

according to their appearance, size or other characteristic.

CONCRETE OPERATIONAL STAGE2nd – 6th GradeReversibility

Objects can be Changed and thenReturned to their Original State

Fact Families4 + 5 = 9 9 – 5 = 4

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.

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 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

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 MATHEMATICSCuisenaire Rods

MANIPULATIVES IN MATHEMATICSGeoboards: transformations, angles, area, perimeter.

MANIPULATIVES IN MATHEMATICSPentominoes: symmetry, area, and perimeter

MANIPULATIVES IN MATHEMATICSTangrams: 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 ALGEBRAThe 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, PrismsCylinders, 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 STATISTICSThe 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 PROCESSESThe 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 REASONINGGoing from the General to the Specific

A Quadrilateral has four sides. What other figures has four sides? Square Rectangle Parallelogram Rhombus Trapezoid

INDUCTIVE REASONINGSpecific Examples – General Conclusion

What 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????