TExES 191 Generalist EC-6 Test Mathematics

1. Dion J. Dubois, Ed.D. 5th Grade Teacher Stevens Park Elementary ddubois@dallasisd.org TExES 191Generalist EC-6 TestMathematics

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

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

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. Cognitive Development

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! Cognitive Development

6. PK through 2nd 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 Pre-Operational Stage

7. PK through 2nd 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 Pre-Operational Stage

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! Cognitive Development

9. 2nd – 6th Grade 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

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

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

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

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

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. Cognitive Development

15. The teacher understands how children learn mathematical skills and uses this knowledge to plan, organize, and implement instruction and assess learning. C13-Mathematics Instruction

16. Numbers, Operations and Quantitative Reasoning Patterns, Relationships and Algebraic Thinking Measurement Geometry and Spatial Reasoning Probability and Statistics Underlying Processes and Mathematical Tools Six Strands of Mathematics

17. Instruction is organized in Units Heterogeneous Groups Manipulatives and Technology Communication Challenging Activities Ongoing Assessment Parent Involvement Ideal mathematics classroom

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. Constructivist Approach

19. Set up learning situations Build mathematical understanding Provide opportunities for students to construct their own knowledge Provide experiences to stimulate their thinking Encourage discovery Use divergent questions Role of the Teacher

20. Concrete Stage Representational Stages Abstract Stage Stages of mathematical Development

21. Problem Solving Read the Problem Make a Plan Solve the Problem Reflect on the Answer Look for Reasonableness Central Teaching Strategy

22. Act It Out Draw A Picture Find a Pattern Make a Table or List Working Backward Use Smaller Numbers Problem Solving Strategies

23. Formative Summative Authentic Importance of Rubrics Mathematical Assessment

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 NCTM Standards

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 Active Learning Environment

26. Knowledge Comprehension Application Analysis Synthesis Evaluation Higher Order Thinking Skills(HOTS)

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

28. Attribute Blocks: sorting, comparing, contrasting, classifying, identifying, sequencing Manipulatives in Mathematics

29. Base 10 Blocks: addition, subtraction, number sense, place value and counting Manipulatives in Mathematics

30. Cuisenaire Rods Manipulatives in Mathematics

31. Geoboards: transformations, angles, area, perimeter. Manipulatives in Mathematics

32. Pentominoes: symmetry, area, and perimeter Manipulatives in Mathematics

33. Tangrams: fractions, spatial awareness, geometry, area, and perimeter Manipulatives in Mathematics

34. The Teacher Understands Concepts Related To Numbers, Operations And Algorithms, and The Properties Of Numbers. C014-Number Concepts and Operations

35. 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 Operations: Addition, Subtraction, Multiplication and Division Relationships between Numbers: Ratios and Proportions C14-Number Concepts and Operations

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

37. Commutative Property 3 + 4 = 4 + 3 4 X 3 = 3 X 4

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

39. Types of Numbers Real Numbers Whole Numbers Integers Rational Numbers Irrational Numbers

40. Types of Numbers 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 Common Mathematical Difficulties

42. 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 Other Mathematical Difficulties

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. Combinations and Permutations

44. The Teacher Understands Concepts Related To Patterns, Relations, Functions, And Algebraic Reasoning. C015-Patterns and Algebra

45. Equations and Inequalities Patterns (Repeating and Growing) Coordinate Planes Ordered Pairs Functions and Input-Output Tables Graphing Functions C015-Patterns and Algebra

46. Coordinate plane-Quadrants

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

48. www.khanacademy.org Information on Functions

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 C016-Geometry and Measurement

50. Cubes Spheres Cones (Circular Prism) Tetrahedron (Triangular Prism) Solids (three-dimensional figures)