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TEXES 191 GENERALIST EC-6 TEST MATHEMATICS Dion J. Dubois, Ed.D. 5th Grade Teacher 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 (Adolescence) 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 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 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 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 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 2nd – 6th Grade Decentering Taking into Account Multiple Aspects Of a Problem to Solve It CONCRETE OPERATIONAL STAGE 2nd – 6th Grade Seriation Arranging Objects in an order according To Size, Shape, Color or any other Attribute Such as Thickness CONCRETE OPERATIONAL STAGE 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 2nd – 6th Grade 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) (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 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. Read the Problem 2. Make a Plan Solve the Problem Reflect on the Answer 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 Operations: Addition, Subtraction, Multiplication and 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 Addition/Subtraction Using Zero when writing numbers Regrouping 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 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 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, 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? ???