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 DevelopmentFebruary 15, 2007

Overview

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 TasksFencing TaskMs. Brown’s class will raise rabbitsfor their spring science fair. Theyhave 24 feet of fencing with whichto build a rectangular rabbit pen inwhich 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.

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?

Comparing Two Mathematical Tasks

Think privately about how you would go about solving each task (solve them if you have time)

Talk with you neighbor about how you did or could solve the task

Martha’s CarpetingThe 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 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)

Significant content (i.e., they have the potential to leave behind important residue) (Hiebert et. al, 1997)

Multiple ways to show competence (Lotan, 2003)

Require justification or explanation (Boaler & Staples, in press)

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

To what extent do students appear to have entry into the task?

To what extent are students grappling with significant content?

Conclusions

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

How do we help teachers become connoisseurs of mathematical tasks that are equitable?

Martha’s Carpeting TaskUsing the Area Formula

A = l x w

A = 15 x 10

A = 150 square feet

Martha’s Carpeting TaskDrawing a Picture

10

15

The Fencing TaskDiagrams on Grid Paper

The Fencing TaskUsing 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 TaskGraph of Length and Area

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Length

Area

The Fencing TaskGraph of Length and Area

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Length

Area

The Fencing TaskEquation and Graph

P = 2l + 2w

24 = 2l + 2w

12 = l + w

l = 12 - w

A = l x w

A = l(12 – l)

A = 12l – l2

The Fencing TaskEquation and Calculus

A = 12l – l2

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 – 2l12 – 2l = 0l = 6

If l is 6, then the width is 12 – 6 or 6. Thus, theconfiguration 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 DealsObservations 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