Warm-Up

Begin reading The Case of Nancy Edwards on the green handout.

As you read, consider the following questions:Did the teacher's decision to have students

actually work on creating a proof -- rather than just fill in the blank with the word "even" -- appear to be a good use of class time? Why or why not?

What did the teacher do to support her students as they worked on this task?

Be sure to cite evidence from the case (i.e., line

numbers) to support your claims.

The Task, Tools, and Talk Framework

Peg SmithUniversity of Pittsburgh

Supporting the Development of Students’ Capacity to Reason-and-Prove

Teachers Development Group Leadership SeminarFebruary 18, 2011

Overview of Session

Provide a rationale for focusing on reasoning-and-proving and describe the CORP project

Engage in two project-developed activitiesCompare and discuss three sets of tasksAnalyze a narrative case

Consider the potential of the activities to foster teacher learning and discuss situations in which the materials might be used

Why Reasoning-and-Proving?

Core practice in mathematics that transcends content areas

Often conceptualized as a particular type of exercise exemplified by the two-column form used in high school geometry rather than as a key mathematical practice

Difficult for students (and teachers)Growing consensus in the community

that it should be “a natural, ongoing part of classroom discussions, no matter what topic is being studied” (NCTM, 2000, p.342)

Standards forMathematical 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

Standards forMathematical 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

CORP: Cases of Reasoning-and-Proving

Focuses on reasoning-and-proving across content areas

Supports the development of mathematical knowledge needed for teaching (see Ball, Thames, & Phelps, 2008)

Features different types of practice-based activitiesSolving, analyzing, and adapting

mathematical tasksAnalyzing narrative casesMaking sense of student work samples

Provides opportunities for teachers to apply what they are learning to their own practice

Three Guiding Questions

What is reasoning-and-proving?

How do high school students benefit from engaging in reasoning-and-proving?

How can teachers support the development of students’ capacity to reason-and-prove?

Three Guiding Questions

What is reasoning-and-proving?

How do high school students benefit from engaging in reasoning-and-proving?

How can teachers support the development of students’ capacity to reason-and-prove?

How can teachers support the development of students’ capacity to reason-and-prove?

select high-level (Stein & Smith, 1998) tasks that require students to reason about and make sense of the mathematics, and have the potential to leave behind important residue about the structure of mathematics (Hiebert, et al, 1997);

encourage the use of tools (i.e., language, materials, and symbols) to provide external support for learning (Hiebert, et al, 1997); and

engage students mathematical discourse or talk in order to make students’ thinking and reasoning public so that it can be refined and/or extended. This includes student-to-student exchanges as well as teacher-to-student exchanges.

Connecting to Literature:Mathematical Reasoning

…it’s important for students to gain experience using the process of deduction and induction. These forms of reasoning play a role in many content areas. Deduction involves reasoning logically from general statements or premises to conclusions about particular cases. Induction involves examining specific cases, identifying relationship among cases, and generalizing the relationship. Productive classroom talk can enhance or improve a person’s ability to reason both deductively and inductively.

Chapin, O’Connor, & Anderson, 2003, p. 78

Connecting to Literature:Mathematical Reasoning

…both plausible and flawed arguments that are offered by students create an opportunity for discussion. As students move through the grades, they should compare their ideas with others’ ideas, which may cause them to modify, consolidate, or strengthen their arguments or reasoning. Classrooms in which students are encouraged to present their thinking, and in which everyone contributes by evaluating one another’s thinking, provide rich environments for learning mathematical reasoning.

NCTM, 2000, p. 58

Activity 1: Compare and Discuss Three Sets of Tasks

Compare each task to its adapted version (A to A’, B to B’, C to C’)

Determine how each original task is the same and how it is different to its adapted version

Look across the three sets and consider: Do the differences between a task and

its modified version matter? What were the modifications in the tasks trying to accomplish?

Task A and A’Same Both ask students

to complete a conjecture about odd numbers based on a set of given examples

Different Task A’ asks

students to develop an argument

Task A can be completed with limited effort; Task A’ requires considerable effort to explain WHY this conjecture holds up

Task B and B’Same Same geometry

content Based on the same

construction Relate to a specific

theorem about the diagonals of a parallelogram

Different Task B asks students to

perform the construction once and then explain why the figure is a parallelogram; Task B’ asks students to perform the construction several times and make a conjecture about the type of figure produced.

The way Task B’ is phrased makes it more likely for students to engage in trying to develop a proof (“make an argument that explains why the same figure is produced each time” vs. “state a theorem”).

Task C and C’

Same Different

Both ask students to find the perimeter of the figure containing 12 trapezoids.

Both require students to identify the pattern of growth of the perimeter.

Task C’ provides more scaffolding by first asking students to find the perimeter of the first four figures.

Finding the perimeter of the 12th figure comes in different places: after finding the general rule in Task C vs. before the rule in Task C’.

Task C’ asks students to explain why the generalization always works.

Activity 1: Compare and Discuss Three Sets of Tasks

Compare each task to its adapted version (A to A’, B to B’, C to C’)

Determine how each original task is the same and how it is different to its adapted version

Look across the three sets and consider: Do the differences between a task and

its modified version matter? What were the modifications in the tasks trying to accomplish?

Do the differences matter? What were the modifications trying to accomplish?

1. Press students to do more than in the original versions of the task.

2. Engage students more in the development of arguments, including proofs (without actually saying “prove”).

3. Give students the opportunity to do more investigation.

4. Give students more access to the task.Underlying idea that runs across the

modifications: To engage students in a broader range of

activities as they investigate whether and why “things work” in

mathematics.

The notion of Reasoning-and-Proving

19

Makinggeneralizations

proofs

Developingarguments

investigation of whether

and why“things work”

in mathematics (both at the school level

and at the discipline)

reasoning-and-proving

(Stylianides, 2008)

patterns,conjectures

The notion of Reasoning-and-Proving

20

Makinggeneralizations

proofs

Developingarguments

investigation of whether

and why“things work”

in mathematics (both at the school level

and at the discipline)

reasoning-and-proving

(Stylianides, 2008)

patterns,conjectures

Reasoning-and-proving is defined to encompass the breadth of the activity associated with:identifying patternsmaking conjecturesproviding non-proof arguments, andproviding proofs.

Reasoning-and-Proving: An Analytic Framework

Making Mathematical Generalizations

Providing Support to Mathematical Claims

Identifying a pattern

Making a conjecture

Providing a

non-proof argument

Providing a proof

Stylianides, 2008, p. 10

Reasoning-and-Proving: An Analytic Framework

Making Mathematical Generalizations

Providing Support to Mathematical Claims

Identifying a pattern

Making a conjecture

Providing a non-proof

argument

Providing a proof

A, A’

C, C’

A, A’

B’

C, C’

A’

B, B’

C’

Stylianides, 2008, p. 10

Activity 2:Analyze a Narrative Case

Read The Case of Nancy Edwards on the green handout.

As you read, consider the following questions:Did the teacher's decision to have students actually

work on creating a proof -- rather than just fill in the blank with the word "even" -- appear to be a good use of class time? Why or why not?

What did the teacher do to support her students as they worked on this task? (Pay particular attention to how the task, tools and talk supported students’ learning.)

Be sure to cite evidence from the case (i.e., linenumbers) to support your claims.

Was this time well spent?Sharpened students’ skills related to proof in a

familiar domain. This might support later use of these skills in a less familiar domain.

Made it clear that examples are helpful but not sufficient to prove. This idea will come up repeatedly and this task can be referred to.

Introduced a variety of approaches that can be used to solve this task. This might help focus future discussions of proof less on form and more on substance.

Surfaced many issues that could be used to develop criteria for proof that could be used throughout the year.

What did the teacher do….?

Selected a “Good” TaskBuilt on prior knowledgeEncouraged multiple approachesLaid groundwork for discussing characteristics of proof

Let Students Select ToolsGroup 1 drew picturesGroup 6 used tiles to build a modelGroups 3 and 4 used symbolic notations

Let Students Do Most of the Talking and Most of the ThinkingAsked questions to challenge students and to get them to talk to

each otherEncourage students to respond to questions posed by other

studentsInstructed each new group to relate their approach to the other

approaches that had been discussed

Take a few minutes to consider…

The learning opportunities afforded by activities such as those we discussed today

The situations in which the activities might be used

The ways in which the Tasks, Tools, and Talk framework might support teachers’ development as well as students’ learning

CORP Project TeamPIs: Peg Smith and Fran Arbaugh

Senior Personnel: Gabriel Stylianides, Mike Steele, Amy Hilllen Jim Greeno, Gaea Leinhardt

Graduate Students: Justin Boyle, Michelle Switala, Adam Vrabel, Nursen Konuk

Advisory Board: Hyman Bass, Gershon Harel, Eric Knuth, Bill McCallum, Sharon Senk, Ed Silver