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Dion J. Dubois, Ed.D. 5 th Grade Teacher Stevens Park Elementary

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Real Life Relationships Personal Contexts Invented Procedures Making Connections Encouraging Problem Solving Hands-On Activities and Project-Based Learning

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Sensorimotor Stage (Infancy) Pre-Operational Stage (Toddler to Early Childhood) Concrete Operational Stage (Elementary) Formal Operational Stage (Adolescence)

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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.

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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!

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PK through 2 nd Grade 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

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PK through 2 nd Grade 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

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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!

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2 nd – 6 th Grade Conservation 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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2 nd – 6 th Grade Decentering Taking into Account Multiple Aspects Of a Problem to Solve It

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2 nd – 6 th Grade Seriation Arranging Objects in an order according To Size, Shape, Color or any other Attribute Such as Thickness

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2 nd – 6 th Grade Classification When a child can name and identify sets of objects according to their appearance, size or other characteristic.

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2 nd – 6 th Grade Reversibility Objects can be Changed and then Returned to their Original State Fact Families = 9 9 – 5 = 4

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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.

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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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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

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1. Instruction is organized in Units 2. Heterogeneous Groups 3. Manipulatives and Technology 4. Communication 5. Challenging Activities 6. Ongoing Assessment 7. Parent Involvement

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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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1. Set up learning situations 2. Build mathematical understanding 3. Provide opportunities for students to construct their own knowledge 4. Provide experiences to stimulate their thinking 5. Encourage discovery 6. Use divergent questions

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1. Concrete Stage 2. Representational Stages 3. Abstract Stage

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Problem Solving 1. Read the Problem 2. Make a Plan 3. Solve the Problem 4. Reflect on the Answer Look for Reasonableness

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1. Act It Out 2. Draw A Picture 3. Find a Pattern 4. Make a Table or List 5. Working Backward 6. Use Smaller Numbers

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1. Formative 2. Summative 3. Authentic Importance of Rubrics

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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 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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Knowledge Comprehension Application Analysis Synthesis Evaluation

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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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Attribute Blocks: sorting, comparing, contrasting, classifying, identifying, sequencing

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Base 10 Blocks: addition, subtraction, number sense, place value and counting

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Cuisenaire Rods

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Geoboards: transformations, angles, area, perimeter.

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Pentominoes: symmetry, area, and perimeter

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Tangrams: fractions, spatial awareness, geometry, area, and perimeter

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The Teacher Understands Concepts Related To Numbers, Operations And Algorithms, and The Properties Of Numbers.

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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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(3 + 4) + 5 = 3 + (4 + 5) (3 X 4) X 5 = 3 X (4 X 5)

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3 + 4 = X 3 = 3 X 4

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5 X (3 + 4) = 5 X X 4

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Real Numbers Whole Numbers Integers Irrational Numbers Rational Numbers

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Integers -5, -3, 0, 1, 2 Rational Numbers ½ 4¾ % Irrational Numbers Square Roots

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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

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Greatest Common Factor Least Common Multiple Exponents (Power of Ten) 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 : = 12, 13, 14, 23, 24, 34 Permutation: number of possible selections where the order of selection IS important.: = ( ) X 2 = 12, 21, 13, 14, 41, 23, 32, 24, 42, 34, 43

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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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The Teacher Understands Concepts Related To Patterns, Relations, Functions, And Algebraic Reasoning.

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A. Equations and Inequalities B. Patterns (Repeating and Growing) C. Coordinate Planes D. Ordered Pairs E. Functions and Input-Output Tables F. Graphing Functions

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https://www.youtube.com/watch?feature=player_embedded&v=AZroE4fJqtQ

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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, Cones Symmetry and Transformations

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Cubes Spheres Cones (Circular Prism) Tetrahedron (Triangular Prism)

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Line, Ray, Line Segment Circle Triangle Quadrilateral (square, rhombus or diamond, parallelogram, trapezoid) Pentagon Hexagon Octagon

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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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Congruent – same size/same shape Similar – same shape – not the same size

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Angle Acute Right Obtuse Sides Equilateral Scalene

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Translations Reflections Glide-Reflections Rotations Dilations (expansions and contractions) Tessellations

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Temperature Money Weight, Area, Capacity, Density Percent Speed and Acceleration Pythagorean Theory Right Angle Trigonometry

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Customary and Standard (Metric) Units Length Temperature Capacity Weight Perimeter Area Volume

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The Teacher Understands Concepts Related to Probability and Statistics and Their Applications.

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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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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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Mode = Most Often Mean = Average Median = Middle Number Range Normal Distribution

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The 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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A. Rounding B. Estimation C. 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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Going 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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Specific Examples – General Conclusion What do all of these shapes have in common? Square Rectangle Parallelogram Rhombus Trapezoid They All Have Four Sides

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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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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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