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  1. Learner Profile Card Gender Stripe Auditory, Visual, Kinesthetic Modality Analytical, Creative, Practical Sternberg Student’s Interests Multiple Intelligence Preference Gardner Array Inventory

  2. Array Interaction Inventory • Directions: • Rank order the responses in rows below on a scale from 1 to 4 with 1 being “least like me” to 4 being “most like me”. • After you have ranked each row, add down each column. • The column(s) with the highest score(s) shows your primary Personal Objective(s) in your personality.

  3. Array Interaction Inventory, cont’d

  4. Personal Objectives/Personality Components Teacher and student personalities are a critical element in the classroom dynamic. The Array Model (Knaupp, 1995) identifies four personality components; however, one or two components(s) tend to greatly influence the way a person sees the world and responds to it. A person whose primary Personal Objective of Production is organized, logical and thinking-oriented. A person whose primary Personal Objective is Connection is enthusiastic, spontaneous and action-oriented. A person whose primary Personal Objective is Status Quo is insightful, reflective and observant. Figure 3.1 presents the Array model descriptors and offers specific Cooperative and Reluctant behaviors from each personal objective.

  5. Keys to Differentiation There are two keys to differentiation: • Know your kids • Know your content

  6. “In times of change, the learners inherit the earth while the learned find themselves beautifully equipped to deal with a world that no longer exists.” Eric Hoffer

  7. It Begins with Good Instruction The greatest enemy to understanding is coverage. Howard Gardner

  8. Know Know These are the facts, vocabulary, dates, places, names, and examples you want students to give you. The know is massively forgettable. “Teaching facts in isolation is like trying to pump water uphill.” -Carol Tomlinson Facts 2X3=6, Vocabulary numerator, slope

  9. KNOW (Facts, Vocabulary, Definitions) • Definition of numerator and denominator • The quadratic formula • The Cartesian coordinate plane • The multiplication tables

  10. Able to Do Skills • Basic skills of any discipline • Thinking skills • Skills of planning, independent learning, etc. • The skill portion encourages the students to “think” like the professionals who use the knowledge and skill daily as a matter of how they do business. This is what it means to “be like” a mathematician, an analyst, or an economist.

  11. Research about teaching suggests that learning by struggling at first with a concept enables students to benefit from an explanation that brings the ideas together (Schwartz & Bransford, 2000).

  12. BE ABLE TO DO (Skills: Basic Skills, Skills of the Discipline, Skills of Independence, Social Skills, Skills of Production) • Describe these using verbs or phrases: • Analyze, test for meaning • Solve a problem to find perimeter • Generalize your procedure for any situation • Evaluate work according to specific criteria • Contribute to the success of a group or team • Use graphics to represent data appropriately

  13. Juicy Verbs

  14. Certain methods of teaching, particularly those that emphasize memorization as an end in itself tend to produce knowledge that is seldom, if ever, used. Students who learn to solve problems by following formulas, for example, often are unable to use their skills in new situations. (Redish, 1996)

  15. It Begins with Good Instruction Adding It Up (National Research Council) – Rule-based instructional approaches that do not give students opportunities to create meaning for the rule, or to learn when to use them, can lead to forgetting, unsystematic errors, reliance on visual clues, and poor strategic decisions.

  16. Research about teaching suggests learning may be hindered by • isolated sets of facts that are not organized and connected or organizing principles without sufficient knowledge to make them meaningful (NRC, 1999) • Students have become accustomed to receiving an arbitrary sequence of exercises with no overarching rationale.”(Black and Wiliam, 1998))

  17. Understand Major Concepts and Subconcepts These are the written statements of truth, the core to the meaning(s) of the lesson(s) or unit. These are what connect the parts of a subject to the student’s life and to other subjects. It is through the understanding component of instruction that we teach our students to truly grasp the “point” of the lesson or the experience. Understandings are purposeful. They focus on the key ideas that require students to understand information and make connections while evaluating the relationships that exist within the understandings.

  18. UNDERSTAND (Essential Truths That Give Meaning to the Topic) Begin with I want students to understand THAT… • Multiplication can have different meanings in different contexts, including repeated addition, groups and creation of area. • Fractions always represent a relationship of parts and wholes. • Addition and subtraction show a final count of the same thing. • Functions can be represented in many ways (graphs, words, tables, equations) but all representations are of the same function.

  19. Some questions for identifying truly “big ideas” • Does it have many layers and nuances, not obvious to the naïve or inexperienced person? • Do you have to dig deep to really understand its meanings and implications even if you have a surface grasp of it? • Is it (therefore) prone to misunderstanding as well as disagreement? • Does it yield optimal depth and breadth of insight into the subject? • Does it reflect the core ideas as judged by experts?

  20. Hints for Writing Essential Understandings • Essential understandings synthesize ideas to show an important relationship, usually by combining two or more concepts. • For example: • People’s perspectives influence their behavior. • Time, location, and events shape cultural beliefs and practices. • Tips: • When writing essential understandings, verbs should be active and in the present tense to ensure that the statement is timeless. • Don’t use personal nouns- they cause essential understanding to become too specific, and it may become a fact. • Make certain that an essential understanding reflects a relationship of two or more concepts. • Write essential understandings a complete sentences. • Ask the question: What are the bigger ideas that transfer to other situations.

  21. Concepts Some concepts • span across several subject areas • represent significant ideas, phenomena,intellectual process, or persistent problems • Are timeless • Can be represented though different examples, with all examples having the same attributes • And universal For example, the concepts of patterns, interdependence, symmetry, system and power can be examined in a variety of subjects or even serve as concepts for a unit that integrates several subjects.

  22. Discipline-based Concepts • Art-color, shape, line, form, texture, negative space • Literature-perception, heroes and antiheroes, motivation, interactions, voice • Mathematics-number, ratio, proportion, probability, quantification • Music-pitch, melody, tempo, harmony, timbre • Physical Education-movement, rules, play, effort, quality, space, strategy • Science-classification, evolution, cycle, matter, order • Social Science- governance, culture, revolution, conflict, and cooperation

  23. Mortimer Adler’s List of the Most Important Concepts in Western Civilization • Angel • Animal • Aristocracy • Art • Astronomy • Beauty • Being • Cause • Chance • Change • Citizen • Constitution • Courage • Custom and convention • Definition • Democracy • Desire • Dialectic • Duty • Education • Element • Emotion • Eternity • Evolution • Experience • Family • Fate • Form • God • Good and Evil • Government • Habit • Happiness • History • Honor • hypothesis • 37. Idea • Immortality • Induction • Infinity • Judgment • Justice • Labor • Language • Law • Liberty • Life and death • Logic • Love • Man • Mathematics • Matter • Mechanics • Medicine • Memory/Imagination • Metaphysics • Mind • Monarchy • Nature • Necessity • Oligarchy • One and Many • Opinion • Opposition • Philosophy • Physics • Pleasure and Pain • Poetry • Principle • Progress • Prophecy • Prudence • Punishment • Quality • Quantity • Reasoning • Relation • Religion • Revolution • Rhetoric • Same/Other • Science • Sense • Sign/Symbol • Sin • Slavery • Soul • Space • State • Temperance • Theology • Time • Truth • Tyranny • Universe • Virtue/Vice • War & Peace • Wealth • Will • Wisdom • World

  24. Mathematical Concepts Number Error / Uncertainty Ratio Measurement Proportion Behavior Symmetry Relationships Pattern Probability Function Truth Order Problem solving Change System Quantification Prediction Representation • Mathematical Understandings Our number system maintains order and is rich with patterns. Mathematicians quantifydata in order to establish real-world probabilities. All measurement involves error and uncertainty.

  25. Strands of Mathematical ProficiencyAdding It Up, 2001 • Conceptual Understanding • Procedural Fluency • Strategic Competence • Adaptive Reasoning • Productive Disposition

  26. Adding It Up: Helping Children Learn Mathematics, NRC, 2001

  27. Strands of Mathematical Proficiency: Adding It Up, 2001 • Conceptual Understanding - Comprehension of mathematical concepts, operations and relations

  28. Strands of Mathematical Proficiency: Adding It Up, 2001 • Conceptual Understanding - “Refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which it is useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention.” P. 118

  29. Strands of Mathematical Proficiency: Adding It Up, 2001 • Procedural Fluency - Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

  30. Strands of Mathematical Proficiency: Adding It Up, 2001 • “Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. By the same token, a certain level of skill is required to learn many mathematical concepts with understanding.” Page 122

  31. Strands of Mathematical Proficiency: Adding It Up, 2001 • Strategic Competence - Ability to formulate, represent, and solve mathematical problems, especially with multiple approaches.

  32. Strands of Mathematical Proficiency: Adding It Up, 2001 • Adaptive Reasoning - Capacity for logical thought, reflection, explanation, and justification

  33. Strands of Mathematical Proficiency: Adding It Up, 2001 • Productive Disposition - Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy

  34. Research suggests that learning is enhanced by providing opportunities for • Struggling • Choosing and evaluating strategies • Contrasting cases • Organizing information • Making connections (NRC, 1999)

  35. Dividing FractionsDemonstrating the 5 Strands • What does it mean to divide? What meanings does division have? • Repeated subtraction • Partitioning or dividing up into groups • Measurement (fits into) • Of these meanings, which one works with dividing fractions?

  36. Dividing FractionsDemonstrating the 5 Strands • Measurement Model 6 Divided by 2 Can you think of examples where you would need to divide fractions?

  37. Dividing FractionsDemonstrating the 5 Strands • Dividing fractions with fraction strips

  38. Dividing FractionsDemonstrating the 5 Strands • Try dividing some fractions with like denominators on your own using the fraction strip model • Share your findings. • Do you see a pattern in dividing fractions with like denominators?

  39. Dividing FractionsDemonstrating the 5 Strands • Do you know a rule that can help speed up the process for dividing fractions without strips? • Can you think of a way to use the pattern discovered with dividing common denominators to make sense of this rule?

  40. Dividing FractionsDemonstrating the 5 Strands Write problem with common denominators Divide the numerators

  41. Dividing FractionsDemonstrating the 5 Strands • Now relate the pattern to the algorithm of invert and multiply… • Where does the common denominator come from? Invert and multiply!

  42. Dividing FractionsDemonstrating the 5 Strands • How can knowing how to divide fractions help you in your life? • Think of as many ideas as you can for the benefits of knowing how to divide fractions!

  43. Fraction Activity • What went well for you? • What was a challenge for you? • What did you learn from this activity?

  44. USE OF INSTRUCTIONAL STRATEGIES. The following findings related to instructional strategies are supported by the existing research: • Techniques and instructional strategies have nearly as much influence on student learning as student aptitude. • Lecturing, a common teaching strategy, is an effort to quickly cover the material: however, it often overloads and over-whelms students with data, making it likely that they will confuse the facts presented • Hands-on learning, especially in science, has a positive effect on student achievement. • Teachers who use hands-on learning strategies have students who out-perform their peers on the National Assessment of Educational progress (NAEP) in the areas of science and mathematics. • Despite the research supporting hands-on activity, it is a fairly uncommon instructional approach. • Students have higher achievement rates when the focus of instruction is on meaningful conceptualization, especially when it emphasizes their own knowledge of the world.

  45. Make Card Games!

  46. Make Card Games!

  47. m=3 b=6 -2/3 Build – A – Square • Build-a-square is based on the “Crazy” puzzles where 9 tiles are placed in a 3X3 square arrangement with all edges matching. • Create 9 tiles with math problems and answers along the edges. • The puzzle is designed so that the correct formation has all questions and answers matched on the edges. • Tips: Design the answers for the edges first, then write the specific problems. • Use more or less squares to tier. • Add distractors to outside edges and “letter” pieces at the end. Nanci Smith

  48. Flippers! • You will need 2 sheets of construction paper, of different colors. (You’ll only use ½ a sheet of the second color though.) • Fold the frame color into fourths horizontally (hamburger folds). • Back-fold the same piece in the opposite directions so that it is well creased and flexible. • Fold the frame at the center only, and make cuts from the fold up to the next fold line. 7 cuts for 8 sections is easy to do, but cut as many as you like. • Fold the second color of paper into fourths as well. Cut these apart. You will only use 2 of the strips. • Basket-weave the two strips into the cut strips of the frame. The two sides need to be woven in opposite directions. • To use the flipper, write questions on the woven colors. To find the answers, fold the flipper so that the center is pointed at you, then pull the center apart to reveal answer spaces. • Flipper works in this way on both sides! Nanci Smith, 2004

  49. Angles Relationship RAFT ** This last entry would take more time than the previous 4 lines, and assesses a little differently. You could offer it as an option with a later due date, but you would need to specify that they need to explain what the angles are, and anything specific that you want to know such as what is the angle’s complement or is there a vertical angle that corresponds, etc.