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


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


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?


Trajectory Posters

Trajectory Posters

Replace with Measurement bullets

Worksheet #1


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


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


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 ?


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.


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


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


Instructional Materials

Summary Table

Instructional Materials

Summary TableSection/

Investigation

Core Mathematics Develop.

Level

Cognitive Demand

Lesson

Lesson

Lesson

Lesson

Lesson

Worksheet 2C


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?


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


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.

content trajectories, instructional materials, and curriculum decisions prom/se ohio mathematics...

Documents

mathematical concepts

big ideas trajectories

mathematical sequence

levels of cognitive

area instructional materials

mathematical reason

defensible ideas

core mathematical knowledge