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

2. Welcome Investigations & 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. PROM/SE Ohio 2005 Spring Mathematics Associates Institute
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. PROM/SE Ohio 2005 Spring Mathematics Associates Institute
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. PROM/SE Ohio 2005 Spring Mathematics Associates Institute
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. PROM/SE Ohio 2005 Spring Mathematics Associates Institute
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. PROM/SE Ohio 2005 Spring Mathematics Associates Institute
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. PROM/SE Ohio 2005 Spring Mathematics Associates Institute
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

17. 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

18. Instructional Materials Fraction Summary Table
Section/
Investigation
Core Mathematics
Develop.
Level
Cognitive Demand
Lesson
Worksheet 2C
PROM/SE Ohio 2005 Spring Mathematics Associates Institute

19. 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

20. 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

21. 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