peg smith university of pittsburgh tasks, tools, and talk: a framework for enacting the ccss...

PEG SMITHUNIVERSITY OF PITTSBURGH

Tasks, Tools, and Talk:A Framework for Enacting the CCSS

Mathematical Practices

North Carolina Council of Teachers of Mathematics Leadership SeminarOctober 24, 2012

Position

Developing students’ capacity to engage in the mathematical practices specified in the Common Core State Standards will ONLY be accomplished by engaging students in solving challenging mathematical tasks, providing students with tools to support their thinking and reasoning, and orchestrating opportunities for students to talk about mathematics and make their thinking public. It is the combination of these three dimensions of classrooms, working in unison, that promote understanding.

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively3. Construct viable arguments and critique the

reasoning of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

Common Core State Standards for Mathematics, 2010, pp.6-7

Overview

Discuss the task, tools, and talk frameworkReview and discuss examples of tasks that

support engagement in the mathematical practices

Analyze and discuss a narrative case with respect to the task, tools, and talk

Discuss the potential of the task, tools, and talk framework for supporting your work with teachers related to the CCSS.

Tasks, Tools, and TalkFramework

the tasks or activities in which students engage should provide opportunities for them to “figure things out for themselves” (NCTM, 2009, p.11), and to justify and communicate the outcome of their investigation;

tools (i.e., language, materials, and symbols) should be available to provide external support for learning (Hiebert, et al, 1997); and

productive classroom talk should make students’ thinking and reasoning public so that it can be refined and/or extended (Chapin, O’Conner, & Anderson, 2009).

Comparing Two Versions of a Task

Compare the two versions of the Adding Odd Numbers Task and consider how they are the same and how they are different

Consider the opportunities each task provides to engage in the Standards for Mathematical Practice


Adding Odds - Version 1

MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases.

29. Conjecture: The sum of any two odd numbers is ______?

1 + 1 = 2 7 + 11 = 181 + 3 = 4 13 + 19 = 323 + 5 = 8 201 + 305 = 506

30. Conjecture: The product of any two odd numbers is ____?

1 x 1 = 1 7 x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = 15 201 x 305 = 61,305

Adding Odds - Version 2

For problems 29 and 30, complete the conjecture based on the pattern you observe in the examples. Then explain why the conjecture is always true or show a case in which it is not true.

MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases.

29. Conjecture: The sum of any two odd numbers is ______?

1 + 1 = 2 7 + 11 = 181 + 3 = 4 13 + 19 = 323 + 5 = 8 201 + 305 = 506

30. Conjecture: The product of any two odd numbers is ____?1 x 1 = 1 7 x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = 15 201 x 305 = 61,305


Same

Both ask students to complete a conjecture about odd numbers based on a set of finite examples that are provided

Different

V2 asks students to develop an argument that explains why the conjecture is always true (or not)

V1 can be completed with limited effort; V2 requires considerable effort – students need to figure out WHY this conjecture holds up

The number of ways to enter and solve the problem

Comparing Two Versions of a TaskOpportunities to Engage in the Mathematical Practices

Version 1

MP 7 – look for and make use of structure

Version 2

MP 7 – look for and make use of structure

MP1 – Make sense of problems and persevere in solving them

MP3 – Construct viable arguments and critique the reasoning of others

MP5 – Use appropriate tools strategically

Characteristics of Tasks Aligned with SMP

High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008)

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

Require justification or explanation (Boaler & Staples, 2008)

Make connections between two or more representations (Lesh, Post & Behr, 1988)

Open-ended (Lotan, 2003; Borasi & Fonzi, 2002)Multiple ways to enter the task and to show

competence (Lotan, 2003)


Compare the two versions of the Tiling a Patio Task and consider the extent to which each exemplifies the characteristics of tasks that align with the Standards for Mathematical Practice.

Tiling a Patio

Alfredo Gomez is designing patios. Each patio has a rectangular garden area in the center. Alfredo uses black tiles to represent the soil of the garden. Around each garden, he designs a border of white tiles. The pictures shown below show the three smallest patios that he can design with black tiles for the garden and white tiles for the border.

Patio 1 Patio 2 Patio 3

Tiling a Patio: Aligned with SMP?

High cognitive demand - no specified pathway to follow, requires students to explore relationships Significant content - equivalence, rate of change

Require justification or explanation - explain in d and e

Make connections between two or more representations - connect rule to visual; could also connect with tables and graphs

Open-ended - different descriptions and rules can be written and in different forms

Multiple ways to enter the task and to show competence (Lotan, 2003)-- build patios, draw pictures, make tables, write equations, draw graphs

Mathematical Tasks:A Critical Starting Point for Instruction

Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.

Stein, Smith, Henningsen, & Silver, 2000

The level and kind of thinking in which students engage determines what they will learn.

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997


If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks.

Stein & Lane, 1996


There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.

Lappan & Briars, 1995


If we want students to develop the capacity to think, reason, and problem solve then we need

to start with high-level, cognitively complex tasks.

Stein & Lane, 1996


Tools

Tools can be thought of as “amplifiers of human capacities” (Brunner, 1966, p.81).

“Tools should help students do things more easily or help students do things they could not do alone” (Hiebert, et al, 1997, p.53).

Representations as Tools

Pictures

Written Symbols

Manipulative Models

Real-worldSituations

Oral Language

Lesh, Post, and Behr, 1987

Tools

Adding Odds Task Square tiles that can be used to build the rectangular

model

Drawing of dots that can be group by two

Use of symbolic notation2x is even; 2x + 1 odd

The 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.

Stein, Smith, Henningsen, & Silver, 2009, p. xvii

The Fencing Task

What tools could you provide that would help students engage in this task?

What difference do you think the tools would make?

Fencing Task Approaches

Build pens with physical materialsDraw pens on grid paperMake a table of the dimensions of possible

pensMake a graph that shows the relationship

between one linear dimension and the areaSet up an algebraic equation and solve

Fencing Task Approaches

Build pens with physical materials (linear and area pieces)

Draw pens on grid paper (grid paper)Make a table of the dimensions of

possible pens Make a graph that shows the relationship

between one linear dimension and the area (graph paper or graphing calculator)

Set up an algebraic equation and solve

Talk

Students must talk, with one another as well as in response to the teacher. When the teacher talks most, the flow of ideas and knowledge is primarily from teacher to student. When students make public conjectures and reason with others about mathematics, ideas and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community.

NCTM, 1991, p.34

The Case of Darcy Dunn

Read the Case of Darcy Dunn

Consider the way tasks, tools, and talked supported students engagement in the Standards for Mathematical Practice.


Teacher selected a task that had the potential to engage students in SMP (e.g., 1, 2, 3, 4, 5, 7)

Teacher provided students with tools they could use to explored the problem (tiles, grid paper, colored pencils, calculators)

Teacher provided students with diagrams of the patios that helped them explain their reasoning to the class

Teacher pressed for explanations and encouraged students to questions each other

Teacher engaged the class in creating a mathematical model that was consistent with the verbal description given by a student and the diagram of the patio

Teacher gave homework that required providing and justifying a conclusion to the question, “Can they all be right?”


Evidence that students were engaged in the mathematical practices MP1 – Students were able to connect verbal descriptions with the

diagram and with the equation. MP2 – Beth, Faith, and Devon were able to make sense of

quantities and their relationships in problem situations (Tamika’s table that didn’t get shared yet is another example)

MP3 – Beth, Faith, and Devon justified their conclusions and communicated them to others; all students were asked to consider the equivalence of the 3 equations for homework and justify their conclusions)

MP4- The class was able to write algebraic equations for the situations described by Beth, Faith, and Devon.

MP7 – Beth, Faith, Devon, and others identified the underlying structure of the pattern that they used to generalize

Reflect

In what ways might the Task, Tools, and Talk framework help you in your work with teachers?

THANK YOU!

Draw a Picture

Every odd number (like 11 and 13) has one loner number. Add the two loner numbers and you will get an even number (24). Now add all together the loner numbers and the other two (now even) numbers.

Build a Model

If I take the numbers 5 and 11 and organize the counters as shown, you can see the pattern.

You can see that when you put the sets together (add the numbers), the two extra blocks will form a pair and the answer is always even. This is because any odd number will have an extra block and the two extra blocks for any set of two odd numbers will always form a pair.

Use Algebra

If a and b are odd integers, then a and b can be written a = 2m + 1 and b = 2n + 1, where m and n are other integers.

If a = 2m + 1 and b = 2n + 1, then a + b = 2m + 2n + 2.

If a + b = 2m + 2n + 2, then a + b = 2(m + n + 1).

If a + b = 2(m + n + 1), then a + b is an even integer.

Logical Argument

An odd number = [an] even number + 1. e.g. 9 = 8 + 1

So when you add two odd numbers you are adding an even no. + an even no. + 1 + 1. So you get an even number. This is because it has already been proved that an even number + an even number = an even number.

Therefore as an odd number = an even number + 1, if you add two of them together, you get an even number + 2, which is still an even number.

Tiling a Patio Using a Visual Model to Find a Pattern of

Growth

T = 2p + 6

T = 2(p + 2) + 2

T = 3(p + 2) - p

Tiling a Patio Making a Table to Find the Pattern of Growth

Tiling a Patio Using a Graph to Determine the Pattern of Growth

The Fencing TaskBuilding Pens

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

The Fencing TaskEquation and Graph

P = 2l + 2w24 = 2l + 2w12 = l + w l = 12 - w

A = l x wA = 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,

Task Analysis Guide

peg smith university of pittsburgh tasks, tools, and talk: a framework for enacting the ccss...

Documents

task opportunities

odd numbers task

mathematical practice

patio task

problem slide

appropriate tools

challenging mathematical

mathematical practices