1. Dion J. Dubois, Ed.D. 5 th Grade Teacher Stevens Park Elementary ddubois@dallasisd.org

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

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

4. 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.

5. 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!

6. 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

7. 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

8. 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!

9. 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

10. 2 nd – 6 th Grade Decentering Taking into Account Multiple Aspects Of a Problem to Solve It

11. 2 nd – 6 th Grade Seriation Arranging Objects in an order according To Size, Shape, Color or any other Attribute Such as Thickness

12. 2 nd – 6 th Grade Classification When a child can name and identify sets of objects according to their appearance, size or other characteristic.

13. 2 nd – 6 th Grade Reversibility Objects can be Changed and then Returned to their Original State Fact Families 4 + 5 = 9 9 – 5 = 4

14. 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.

15. The teacher understands how children learn mathematical skills and uses this knowledge to plan, organize, and implement instruction and assess learning.

16. 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

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

18. 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.

19. 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

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

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

22. 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

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

24. 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

25. 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

26.  Knowledge  Comprehension  Application  Analysis  Synthesis  Evaluation

27.  Attribute and Base Ten Blocks  Calculators  Trading Chips, Counters and Tiles  Cubes, Spinners, Dice  Cuisenaire Rods  Geoboards  Pentominoes  Pattern Blocks  Tangrams

28.  Attribute Blocks: sorting, comparing, contrasting, classifying, identifying, sequencing

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

30.  Cuisenaire Rods

31.  Geoboards: transformations, angles, area, perimeter.

32.  Pentominoes: symmetry, area, and perimeter

33.  Tangrams: fractions, spatial awareness, geometry, area, and perimeter

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

35. 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

36. (3 + 4) + 5 = 3 + (4 + 5) (3 X 4) X 5 = 3 X (4 X 5)

37. 3 + 4 = 4 + 3 4 X 3 = 3 X 4

38. 5 X (3 + 4) = 5 X 3 + 5 X 4

39. Real Numbers Whole Numbers Integers Irrational Numbers Rational Numbers

40. Integers -5, -3, 0, 1, 2 Rational Numbers ½ 4¾.25 2.15 35% Irrational Numbers Square Roots

41.  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

42.  Greatest Common Factor  Least Common Multiple  Exponents (Power of Ten) - 10 3  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

43.  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.

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

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

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

48. www.khanacademy.org

49. 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

50.  Cubes  Spheres  Cones (Circular Prism)  Tetrahedron (Triangular Prism)

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

52.  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

53.  Congruent – same size/same shape  Similar – same shape – not the same size

54.  Angle  Acute  Right  Obtuse  Sides  Equilateral  Scalene

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

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

63.  Customary and Standard (Metric) Units  Length  Temperature  Capacity  Weight  Perimeter  Area  Volume

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

65.  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.

66.  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

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

71. The Teacher Understands Mathematical Processes And Knows How To Reason Mathematically, Solve Mathematical Problems, And Make Mathematical Connections Within And Outside Of Mathematics.

72. 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

73. Going from the General to the Specific  A Quadrilateral has four sides. What other figures has four sides?  Square  Rectangle  Parallelogram  Rhombus  Trapezoid

74. Specific Examples – General Conclusion What do all of these shapes have in common?  Square  Rectangle  Parallelogram  Rhombus  Trapezoid They All Have Four Sides

75.  Theories and Principles of Learning  Using prior mathematical knowledge  Mathematics manipulatives  Motivate students  Actively engagement  Individual, small-group, and large-group setting

76.  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