Transcript I& 2451 Special Topics in Mathematics

Building Teachers Capacity to
Create and Enact Tasks that
Engage Students in Challenging
Mathematical Activity
Peg Smith
University of Pittsburgh
Teachers’ Development Group
Leadership Seminar on Mathematics Professional Development
February 16, 2007
Overview
 Solve and discuss a series of multiplication tasks
 Consider the role of representation in developing
students’ understanding of mathematics and
increasing the cognitive demands of a task
 Read and discuss the Case of Monique Butler
 Discuss the factors that support or inhibit students
engagement in high level thinking and reasoning
 Discuss the knowledge needed for teaching that
teachers might learn from these experiences
Task1:
Multiplying Binomials
Directions: Find the product of each
expression in its simplest form:
1. 2x (x – 1)
2. (x + 1) (x + 2)
3. (x - 2) (3x + 3)
4. (x – 3) (x + 3)
5. (2x + 2) (2x – 2)
6. (x + 3) (x + 3)
Making Sense of Multiplication
Using any of the tools available at your
table:
• Model 3 x 4
• Model 3 x n
Making Sense of Multiplication
Using any of the tools available at your
table:
• Model 6 x 13
• Model 6(n + 3)
Making Sense of Multiplication
Using any of the tools available at your
table:
• Model 26 x 12
• Model (2x + 6)(x + 2)
Multiplying Binomials:
Task 2
Directions: Use algebra tiles to show the
product of these binomials. Make a sketch of
your model.
1. (x + 3) (x + 3)
2. 2x (x - 1)
3. (x - 3) (x + 3)
4. (x - 2) (3x + 3)
(Note that these four problems also appeared in Task 1.)
Comparing Task 1 & Task 2
How are Tasks 1 and 2
the same and how are they different?
Comparing Task 1 & Task 2
In what ways did we “make
connections to meaning” in our work
on Task 2?
Ways of Representing
Mathematical Ideas
Pictures
Written
Symbols
Manipulative
Models
Real-World
Situations
Oral
Language
Van De Walle, 2004, p. 30
Linking Multiple Representations
The learner who can, for a particular
mathematical problem, move fluidly among
different mathematical representations has
access to a perspective on the mathematics in
the problem that is greater than the perspective
any one representation can provide.
Driscoll, 1999
Linking Multiple Representations
Research has shown that children who have
difficultly translating a concept from one
representation to another are the same children who
have difficulty solving problems and understanding
computations. Strengthening the ability to move
between and among these representations
improves the growth of children’s concepts.
Lesh, Post, Behr, 1987
Linking to Research/Literature
If we want students to develop the
capacity to think, reason, and problem
solve then we need to start with highlevel, cognitively complex tasks.
Stein and Lane, 1996
Consider…
What if the textbook teachers are using doesn’t
have many high-level, cognitively complex
tasks?
How can you do help teacher learn to modify
tasks so that they will provide opportunities for
students to develop the capacity to think,
reason, and problem solve?
General Techniques for
Modifying Tasks
Ask students to create real world stories for “naked
number” problems
Use an additional representation and make
connections between two (or more) representations
Solve an “algebrafied” version of the task
Use a task “out of sequence” before students have
memorized a rule or have practiced a procedure
that can be routinely applied
Eliminate components of the task that provide too
much scaffolding
General Techniques for
Modifying Tasks
Ask students to create real world stories for “naked
number” problems
Use an additional representation and make
connections between two (or more) representations
Solve an “algebrafied” version of the task
Use a task “out of sequence” before students have
memorized a rule or have practiced a procedure
that can be routinely applied
Eliminate components of the task that provide too
much scaffolding
Task A
Larson, R., Boswell, L., Kanold, T., & Stiff, L. (2001). CA Algebra 1: Concepts and Skills, McDougal Littell, 213.
Task A
Adaptations
For the following equations:
1. Graph on a coordinate
plane.
2. Identify 3 solutions for each
equation.
3. Are your solutions unique?
Explain your reasoning.
a. y= 3x - 5
b. 5x + 2y = 10
c. -5x -3y = 12
3x - 5y = 15
(i)
Find 3 solutions to the
equation
(ii) Plot the points and describe
all the patterns you see.
(Group extension: plot all
points on group graph)
(iii) How many solutions do you
believe exist for this
function? Explain your
reasoning.
Task B
Smith, S., Charles, R., Dossey, J., & Bittinger, M. (2001). Algebra 1, California Edition, Prentice Hall, 321.
Task B
Adaptations
Find the slope of the
lines containing these
points using a graph.
11. (4, 0) (5, 7)
13. (0,8) (-3,10)
17, (0,0) (-3,-9)
Can you write a rule that will
always work for finding the
slope? (Or explain why the
rule you already know WORKS?)
Original Problem +
How can you use the slope to
find other points on the same
line?
Use a graph to explain why the
rule you used to find the slope
works.
Final Reflections on Task
Adaptation
 The one lesson I found particularly enlightening was the
modification of tasks. I used this myself in the PTR assignment,
and see the benefits of doing so more in the future. The goal of
maintaining the original intent of the problem, was an important
consideration that I will remember.
 …I learned more about how to create tasks with a certain goal in
mind and how to improve my questioning…I also felt more
inspired to design more tasks.
 I learned how to adapt an activity to raise the cognitive demand
to challenge my learning support students and still foster
achievement. I feel it is possible to do more open-ended tasks
even though they struggle with computation and procedures to
build their mathematical knowledge.
The Case of Monique Butler
Analyzing the Case of
Monique Butler
Discuss with colleagues at your table:
• What did Monique Butler want her students
to learn?
• What did Monique Butler’s students learn?
Cite evidence from the case (i.e., paragraph
numbers) to support your claims.
Analyzing the Case of
Monique Butler
What prevented Monique Butler’s students
from learning what she wanted them to
learn?
(Cite evidence from the case to support your
claims.)
Analyzing the Case of
Monique Butler
In terms of the Mathematical Tasks
Framework, what is Monique Butler a
case of?
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000, p. 4
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000, p. 4
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000, p. 4
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000, p. 4
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000, p. 4
Factors Associated with the Maintenance and
Decline of High-Level Cognitive Demands
Decline






Routinizing problematic aspects of
the task
Shifting the emphasis from meaning,
concepts, or understanding to the
correctness or completeness of the
answer
Providing insufficient time to wrestle
with the demanding aspects of the
task or so much time that students
drift into off-task behavior
Engaging in high-level cognitive
activities is prevented due to
classroom management problems
Selecting a task that is inappropriate
for a given group of students
Failing to hold students accountable
for high-level products or processes
Maintenance







Scaffolding of student thinking and
reasoning
Providing a means by which
students can monitor their own
progress
Modeling of high-level performance
by teacher or capable students
Pressing for justifications,
explanations, and/or meaning
through questioning, comments,
and/or feedback
Selecting tasks that build on
students’ prior knowledge
Drawing frequent conceptual
connections
Providing sufficient time to explore
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A
casebook for professional development. New York, NY: Teachers College Press.
Analyzing the Case of
Monique Butler
How was your experience multiplying
binomials (Task 2) similar or different from
Monique Butler’s students’ experience
multiplying binomials?
Analyzing the Case of
Monique Butler
What lessons can we learn from the
Case of Monique Butler?
The Role of the Teacher
Worthwhile tasks alone are not sufficient for effective
teaching. Teachers must also decide what aspects
of a task to highlight, how to organize and
orchestrate the work of the students, what questions
to ask to challenge those with varied levels of
expertise, and how to support students without
taking over the process of thinking for them and
thus eliminating the challenge.
NCTM, 2000, p. 19
Considering the Knowledge
Needed for Teaching
 What knowledge needed for teaching might
teachers develop through experience such
as those we discussed today?
 How will know if teachers learned what you
intended for them to learn?
Mathematical Knowledge for
Teaching
Common
Content
Knowledge
Knowledge at
the mathematical
horizon
Specialized
Content
Knowledge
Knowledge of
Content and
Students
Knowledge of
Content and
Teaching
Knowledge
of
Curriculum
Ball, Bass, Hill, and Thames, 2006
Making Sense of Multiplication
Using an area model to calculate 6 x 13:
13
6
6 x 13 = 78
Making Sense of Multiplication
Using an area model to calculate 6 x 13:
10
6
10 x 6
3
3x6
6 x 13 = 6 x (10 + 3) =
(6 x 10) + (6 x 3) =
60 + 18 = 78
Making Sense of Multiplication
Using an area model for 6(n + 3):
n
6
6xn
3
6x3
6(n + 3) =
(6 x n) + (6 x 3) =
6n + 18
Making Sense of Multiplication
Using an area model to calculate 26 x 12:
20
10
2
6
20 x 10
6 x 10
20 x 2
6 x 2
26 x 12 = (20 + 6) (10 + 2) =
(20 x 10) + (20 x 2) + (6 x 10) + (6 x 2) =
200 + 40 + 60 + 12 = 312
Making Sense of Multiplication
Using an area model for (2x + 6)(x + 2):
2x
6
x
2
4x
(2x + 6)(x + 2) =
(2x)(x) + (2x)(2) + (6)(x) + (6)(2) =
2x2 + 4x + 6x + 12 =
2x2 + 10x + 12
“Algebrafying” a Task
Existing arithmetic activities and word
problems are transformed from problems
with a single numerical answer to
opportunities for pattern building,
conjecturing, generalizing, and justifying
mathematical facts and relationships.
Blanton & Kaput, 2003, p.71
“Algebrafying”
the Sweater Sale Task
Original
The cost of a sweater at
J. C. Penney's was
$45.00.
At the "Day
and Night Sale" it was
marked 30% off of the
original price.
What
was the price of the
sweater
during
the
sale?
Modified
The cost of a sweater at J. C.
Penney's was $45.00. At the "Day
and Night Sale" it was marked 30%
off of the original price. What was
the price of the sweater during the
sale?
What would the sale price be if the
original price was $46? $47? $48?
$100?
What is the relationship between
the original price of the sweater
and the sale price?
Task C
Larson, R., Boswell, L., Kanold, T. D., & Stiff, L. (2001). CA Algebra 1: Concepts and Skills, McDougal Littell, 273.