Strengthening mathematics instruction

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?
What are
some of the
causes that
lead to
students
being

confused
about
mathematical
concepts and
content?

= confused students


Cognitively Complex Problems
These types of problems require students to
• 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


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
5
3

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

− −1

3

−1

2

0

x

x2

1

x

− x2

−

1
x

x


Locating Cognitively Complex Problems
Activity
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?


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 if x ≥ 0
x =
 −x if x < 0

The
Context


It’s Your Turn to Identify Structures!
1

As a
Learner

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.

2
As a
Teacher

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?


What teachers said about a pilot workshop

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 help
students think mathematically…