Content Trajectories, Instructional Materials, and Curriculum Decisions PROM/SE Ohio Mathematics...
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Transcript of Content Trajectories, Instructional Materials, and Curriculum Decisions PROM/SE Ohio Mathematics...
Content Trajectories, Instructional
Materials, and Curriculum Decisions
Content Trajectories, Instructional
Materials, and Curriculum Decisions
PROM/SE Ohio Mathematics Associates
Institute
Spring 2005
PROM/SE Ohio Mathematics Associates
Institute
Spring 2005
PROM/SE Ohio 2005 Spring Mathematics Associates Institute
Agenda for 6-8Agenda for 6-8
Identifying “Big Ideas” Trajectories for Measurement: Area, Perimeter and Area
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
Characteristics of a Coherent Mathematical
Trajectory
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 otherMathematical sequence and connections
are defensibleIdeas become increasingly more
sophisticated
Handout #1
PROM/SE Ohio 2005 Spring Mathematics Associates Institute
Brainstorming the “Big Ideas” Trajectory
Brainstorming the “Big Ideas” Trajectory
Measurement: Area, Perimeter and Volume What are the “big ideas” in
measurement for area? Perimeter? Volume?
How would you organize these ideas to form a trajectory of mathematical content?
PROM/SE Ohio 2005 Spring Mathematics Associates Institute
Trajectory Posters
Trajectory Posters
Replace with Measurement bullets
Worksheet #1
PROM/SE Ohio 2005 Spring Mathematics Associates Institute
Low Cognitive Demand
Low Cognitive Demand
Tasks rely heavily on memorization or following a routine procedure
Require little thinking or reasoning
Focused on correct answersExplanations 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
Middle Grades Task Set
Middle Grades Task Set
Sort the middle grades tasks for grade 6-8 by levels of cognitive demand
Record the task number and indicate the level (low, moderate, or high) on Worksheet 2A
Share your classifications with your team members
Discuss any differences and why they may have occurred
Try to resolve any disagreements about levels
Middle Grades Task Set
a. What is the perimeter of a rectangle with dimensions 24 and 15?
b. The Buckeye Drink Company is marketing a new soft drink. The drink will be sold in a new “Fun Size” cylindrical can which holds 200 cm3.
a. Joe suggested that the can have a radius of 2 cm. What would the height need to be?
b. What would be the surface area of Joe’s can?
Volume of a cylinder V = πr2h
Surface Area of a cylinder S = 2πr2+ 2πrh
3. What is the area of a square with perimeter of 36 meters? 4. A box of popcorn sells for $4.00. If you want to sell a smaller box, one-half as large
for each dimension, what is a fair price to charge for the smaller box? A. 50 cents B. $1.00 C. $2.00 D. $2.50
5. Which of the following parallelograms has a perimeter of 20 inches?
Base Side 10 10 8 12 4 6 4 5
Fun Size
Drink
7. Find the area of the following triangle. 8. You need to build a rectangular pen for a rabbit. You decide to build the pen using the
back of the house as one side and enclosing the other 3 sides with 24 feet of fencing. If you want the rabbit to have as much room as possible, what should be the length and width of the pen?
9. What is the surface of a cube with an edge of 12 centimeters?
10. You are given several 60-inch strips of framing material. Each strip is to be used to make a frame that encloses 1-inch square tiles. Give the dimensions of all the possible rectangular frames that can be made using the strips.
11. Susan has a sheet of cardboard that measures 9 feet by 6 feet. She uses scissors
and tape to make the entire sheet of cardboard into a closed box that is a cube. A.) Wha t is t he su rf ace ar ea of t he box?
B.) Wha t is t he length of e ach edg e of t he b ox? Exp lain y ou r reasoning.
C.) Wha t is t he volum e o f th e box ?
PROM/SE Ohio 2005 Spring Mathematics Associates Institute
Moderate Cognitive Demand
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
High Cognitive Demand
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
Instructional Materials & Content
Trajectories
Instructional Materials & Content
Trajectories
Individually or in PairsIdentify and record the core
mathematical knowledge by lesson on Worksheet 2c
Indicate the developmental level (I, D, S, A)
Indicate the cognitive demand for each lesson (low, moderate, high)
Worksheet 2B
PROM/SE Ohio 2005 Spring Mathematics Associates Institute
Instructional Materials
Summary Table
Instructional Materials
Summary TableSection/
Investigation
Core Mathematics Develop.
Level
Cognitive Demand
Lesson
Lesson
Lesson
Lesson
Lesson
Worksheet 2C
PROM/SE Ohio 2005 Spring Mathematics Associates Institute
Summary of Instructional
Materials Review
Summary of Instructional
Materials Review
As a teamWhat 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
Mapping to Benchmarks &
Indicators
Mapping to Benchmarks &
IndicatorsIdentify 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 materialsWhich 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
Building New Tasks from OldBuilding 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.