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Content Trajectories, Instructional Materials, and Curriculum Decisions

Explore fractions trajectories, instructional materials, and curriculum decisions. Identify big ideas and examine cognitive demand levels in tasks.

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Content Trajectories, Instructional Materials, and Curriculum Decisions

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  1. Content Trajectories, Instructional Materials, and Curriculum Decisions PROM/SE Ohio Mathematics Associates Institute Spring 2005

  2. WelcomeInvestigations & Everyday Math Users • Please sit with no more than 6 people at a table. • There should only be one person from each district at a table. PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  3. Agenda for K-5 • Identifying “Big Ideas” Trajectories for Fractions: • Concepts, Comparing, Ordering, and Equivalence • Instructional Materials and Content Trajectories • Lunch--12:00 p.m. • Mapping Benchmarks & Indicators to Instructional Materials • Reflections & Next Steps PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  4. Identifying the “Big Ideas” Trajectory for Fractions Individually • Make a concept map for the fractions: • concepts, comparing, ordering, and equivalence In teams • Look across the maps: • Describe similarities and differences. • What might account for the differences? PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  5. Looking Across Our Maps for “Big Ideas” • Concepts • Part to whole • Parts of sets • Numerator and denominator • Equal parts • Models: measurement, linear, area, volume • Representations: words , picture, number, physical objects • Relationships between numerator & denominator • Comparing • Benchmark fractions, 1/2, 1/4, … • Greater than one, less than one • Comparing two fractions • Symbols, >, <, = PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  6. More of Our Big Ideas • Ordering • Numberline • Estimation closer to 0, 1/2, 1 • Greatest to least • Improper fractions & mixed numbers • More than two fractions • Equivalence • Factors & multiples • Same value • Multiplying by forms of 1 (e.g., 2/2, 3/3) to obtain equivalent fraction • Simplest form/ lowest terms PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  7. BREAK • Place posters on wall • 10:15 to 10:30 • After break, do a gallery walk to examine different posters. • What did you notice that was interesting or unique? • How are the sequences similar? How are they different? • Sit by curriculum programs • Mixed grade level groups (K-6) • Different districts represented PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  8. Characteristics of a Coherent Mathematical Trajectory • Every component has a mathematical reason for being included • Designed with awareness of students’ understandings and misunderstandings • Sequence developed with clear sense of developmental levels • Ideas build on each other • Mathematical sequence and connections are defensible • Ideas become increasingly more sophisticated Handout #1 PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  9. Reflecting on “Big Ideas” Trajectories In teams • Create a poster that outlines your content trajectory • What models should be used to represent fractions • Parts of a unit whole • Parts of a set • Location on a number line • Division of whole numbers • When should equal parts be introduced? Fraction notation? • Does your trajectory include fractions greater than one? • When should equivalence of fractions be introduced? Worksheet #1 PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  10. Low Cognitive Demand • Tasks rely heavily on memorization or following a routine procedure • Require little thinking or reasoning • Focused on correct answers • Explanations focus solely on how a procedure was used and lack a connection to concepts or meaning Handout #2 PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  11. Moderate Cognitive Demand • Tasks require several different processes and relate two or more mathematical concepts (e.g., multi-step problems) • Procedures are connected to underlying concepts and meanings and cannot just be followed mindlessly • Students are asked to make connections among representations and may be asked to give some explanations. PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  12. High Cognitive Demand • Tasks require significant analysis and reasoning • Students have to put ideas together in ways they have not seen before in a lesson or in ways that make connections to other previously learned mathematical concepts • There is no predictable rehearsed approach suggested by the task or example Handout #2 PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  13. Elementary Task Set Individually • Solve the problems in the set • Sort the tasks by levels of cognitive demand • Record the task number and indicate the level (low, moderate, or high) on Worksheet 2A In Teams • Discuss ways of reasoning on a few problems of interest • Share your classifications • Discuss any differences and why they may have occurred • Try to resolve any disagreements about levels PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  14. Instructional Materials & Content Trajectories Individually or in Pairs • Identify and record the core mathematical knowledge by lesson on Worksheet 2C • Indicate the developmental level (I, D, S, A)--see worksheet 2B • Indicate the cognitive demand for each lesson (low, moderate, high) Worksheet 2B PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  15. Instructional Materials Fraction Summary Table Worksheet 2C PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  16. Summary of Instructional Materials Review As a team • What are some areas your materials handled well? • Describe any gaps that you identified. • Identify overlaps and decide upon the importance. • What mathematical content seems to be irrelevant and doesn’t appear to fit? • What issues did you find with developmental levels? • What issues emerged regarding the cognitive demands of tasks? PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  17. Mapping to Benchmarks & Indicators • Identify the appropriate Benchmark or Indicator for each idea you listed on Worksheet 2C • Code indicators • Black - at expected grade level • Red - expected at higher grade • Blue - expected at lower grade • Yellow - not addressed at all in instructional materials • Which indicators occur in multiple grade levels? Why? • Where do gaps exist and how might you address them? Worksheet 3a PROM/SE Ohio 2005 Spring Mathematics Associates Institute

  18. Building New Tasks from Old • Select 2-3 tasks/problems from your instructional materials that you classified as low cognitive demand tasks. • Identify the mathematics in the task/problem and describe how it relates to the mathematical goals of the lesson. • Modify the problem so that is has a moderate or high cognitive demand • Record problem on chart paper to post • Describe how the revised task pushes students thinking. PROM/SE Ohio 2005 Spring Mathematics Associates Institute

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