STRENGTHENING MATHEMATICS INSTRUCTION

Cognitive Complexity and Instructional Practices

Characteristics of the Workshop

18-24 hours of professional development; 8 modules to allow for flexibility in scheduling

Standards based and tied to the CSTs and CSU placement standards

Includes content and activities for teachers of Algebra 1 Geometry, Algebra 2, Pre-Calculus

Draws on problems and lessons from the major textbooks

Designed for teacher practice and implementation between workshop sessions based on lesson study model

Reflective of the recently adopted national mathematics standards

No cost to the school(s) for workshop and materials

Workshop Outcomes

Why?

= confused students

What are some of the causes that lead to students being confused about mathematical concepts and content?

Cognitively Complex Problems

• Extend previously encountered tasks

• Integrate several topics and/or concepts

• Recognize and use underlying mathematical structures

• Use multiple representations

• Consider multiple approaches to the problem

• Identify patterns

• Be flexible and strategic in their mathematical thinking

These types of problems require students to

Causes of Low Proficiency Levels

Activity

1. Think about things that you believe contribute to low proficiency levels in students’ work.

1. Write each idea on a separate post-it note.

Example 3 – The Real Numbers

Arrange the numbers in increasing order from smallest to largest

If 0 < x < 1, arrange the terms in increasing numerical order from smallest to largest

1

x x2

1x

x2 1

x x

53

1 3 2 1 0

Locating Cognitively Complex Problems

1. Choose a section or chapter in your textbook that you will be teaching in the next few weeks.

2. Use post-it notes to indicate any problems that are cognitively complex.

3. At your table, discuss the following questions:• Where did you find these problems?• Compare the number of complex problems to the

number of standard problems in your textbook.• How often do you assign these problems

for homework?• How often do you include these problems

in your section/chapter assessments?

Activity

Geometry – Extension #3 (Problem)

A square is inscribed in a circle of radius 3 units. What is the total area enclosed within the circle but outside the square?

A circle of radius 3 units is inscribed in an equilateral triangle. Find the length of the side of the triangle.

Motivating and Making Sense of Definitions

The Definition

x x if x 0 x if x 0

The Context

It’s Your Turn to Identify Structures!

Discuss:• Have I provided my

students with these types of problems? If not, why?

• How would I begin to incorporate more of these types of problems in my teaching?

• What are some challenges I might face in developing these types of problems?

As a Learner

1 2

As a Teacher

Partner Up with someone you haven’t worked with before.

Using the activity page:

• Determine the basic structure for each of the problems.

• Determine which problems were easier and harder for you and why.

• Share your “AHA’s” with each other.

It gave me a starting point to improve instruction…Working with my fellow teachers and having time

to explore complexity was most valuable…Learning about cognitive layering in problems is

very important…I learned to ask more open-ended questions and

use “what if” to explore mathematical ideas without fear

This workshop showed me strategies to helpstudents think mathematically…

What teachers said about a pilot workshop