Transcript Engaging Teachers and Students in Sustained Mathematics

Providing All Students with Access to High
Quality Mathematics Instruction:
The Role of Tasks in Achieving Equity
Peg Smith
University of Pittsburgh
Teachers’ Development Group
Leadership Seminar on Mathematics Professional Development
February 15, 2007
Overview
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Discuss what it means for a task to promote
equity
Compare and discuss two tasks
Consider features of equitable tasks
Analyze a classroom episode and consider the
learning opportunities afforded by the task
Relate the discussions of equitable tasks to the
knowledge needed for teaching
Tasks and Equity:
What’s the relationship?
How can a mathematical task promote or
inhibit equity?
Comparing Two Mathematical Tasks
Martha’s Carpeting Task
Martha was recarpeting her
bedroom which was 15 feet long
and 10 feet wide. How many
square feet of carpeting will she
need to purchase?
Fencing Task
Ms. Brown’s class will raise rabbits
for their spring science fair. They
have 24 feet of fencing with which
to build a rectangular rabbit pen in
which to keep the rabbits.
1. If Ms. Brown's students want their
rabbits to have as much room as
possible, how long would each of the
sides of the pen be?
2. How long would each of the sides of
the pen be if they had only 16 feet of
fencing?
3. How would you go about determining
the pen with the most room for any
amount of fencing? Organize your
work so that someone else who reads
it will understand it.
Comparing Two Mathematical Tasks
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Think privately about how you would go
about solving each task (solve them if you
have time)
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Talk with you neighbor about how you did
or could solve the task
Martha’s
Carpeting
The Fencing Task
Comparing Two Mathematical Tasks
How are Martha’s Carpeting Task and
the Fencing Task the same and how
are they different?
Similarities and Differences
Similarities
 Both require prior
knowledge of area
 Area problems
Differences
 Way in which the area
formula is used
 The need to generalize
 The amount of thinking
and reasoning required
 The number of ways the
problem can be solved
 The range of ways to
enter the problem
Characteristics of Tasks That
Promote Equity
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Allow entry to students with a range of skills and
abilities
Open-ended (Lotan, 2003; Borasi & Fonzi, 2002)
High cognitive demand (Stein et. al, 1996; Boaler &
Staples, in press)
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Significant content (i.e., they have the potential to
leave behind important residue) (Hiebert et. al, 1997)
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Multiple ways to show competence (Lotan, 2003)
Require justification or explanation (Boaler &
Staples, in press)
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Make connections between two or more
representations (Lesh, Post & Behr, 1988)
Cal’s Dinner Card Deals
Is CDCD an equitable task?
Why or why not?
Students at Work on Cal’s Dinner
Card Deals
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To what extent do students appear to
have entry into the task?
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To what extent are students grappling with
significant content?
Conclusions
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The narrowness by which success in
mathematics class is often judged means that
some students will rise to the top whilst others
sink to the bottom.
When there are many ways to be successful,
many more students are successful.
Tasks that are multidimensional provide all
students with the opportunity to engage in
mathematical work.
Boaler & Staples, in press
Considering the Knowledge Needed
for Teaching
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How do we help teachers become
connoisseurs of mathematical tasks that
are equitable?
Martha’s Carpeting Task
Using the Area Formula
A=lxw
A = 15 x 10
A = 150 square feet
Martha’s Carpeting Task
Drawing a Picture
10
15
The Fencing Task
Diagrams on Grid Paper
The Fencing Task
Using a Table
Length
Width
Perimeter
Area
1
11
24
11
2
10
24
20
3
9
24
27
4
8
24
32
5
7
24
35
6
6
24
36
7
5
24
35
The Fencing Task
Graph of Length and Area
40
35
30
Area
25
20
15
10
5
0
0
1
2
3
4
5
6
7
Length
8
9
10
11
12
13
The Fencing Task
Graph of Length and Area
40
35
30
Area
25
20
15
10
5
0
0
1
2
3
4
5
6
7
Length
8
9
10
11
12
13
The Fencing Task
Equation and Graph
P = 2l + 2w
24 = 2l + 2w
12 = l + w
l = 12 - w
A=lxw
A = l(12 – l)
A = 12l – l2
The Fencing Task
Equation and Calculus
2
A = 12l – l
This is a quadratic equation of a parabola that has a maximum.
Finding the derivative of the equation, then setting that derivative
equal to zero, will give us the l value for the maximum.
A(l) = 12l – l2
A’(l) = 12 – 2l
12 – 2l = 0
l=6
If l is 6, then the width is 12 – 6 or 6. Thus, the
configuration with the maximum area is 6 x 6.
Open-Ended Tasks
An open-ended task offers many more
opportunities for success for all students
than traditional tasks that recognize only
one correct solution and one way to
achieve it.
Borasi & Fonzi, p.20
Multidimensional Tasks
Different resources and hands-on
materials attract more students and entice
them to participate, thus opening
additional avenues for students to
understand the learning task.
Lotan, 2003
Cal’s Dinner Card Deals
Observations
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Plan B costs $12 even if you don’t have any dinners
Each graph has a different symbol
Each graph is a line
Each graph goes up from the lower left to the upper
right
Plan A keeps on raising by $8 as you go up
The graphs seem to cross at certain places
One of the lines crosses zero, the Regular Price, but the
other two don't
No matter which plan you have, you can get nine
dinners for less than $100
Ways to Solve (or begin to solve)
Cal’s Dinner Card Deals
Build a table for the data on the graph and
look for a pattern
 Use the graph itself to find a way to
describe what changes and what stays
the same for each plan
 Write an equation from the graph or from
the table
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