Transcript Reasoning-and-proving - Teachers Development Group

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.
Supporting the Development of Students’
Capacity to Reason-and-Prove
The Task,Tools, and Talk Framework
Peg Smith
University of Pittsburgh
Teachers Development Group Leadership Seminar
February 18, 2011
Overview of Session
 Provide a rationale for focusing on reasoning-and-
proving and describe the CORP project
 Engage in two project-developed activities
 Compare and discuss three sets of tasks
 Analyze 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 for
Mathematical Practice
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of
others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Common Core State Standards for Mathematics, 2010, pp.6-7
Standards for
Mathematical Practice
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of
others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
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 activities
 Solving, analyzing, and adapting mathematical tasks
 Analyzing narrative cases
 Making 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
 Task C’ provides more
the perimeter of the figure
containing 12 trapezoids.
 Both require students to
identify the pattern of
growth of the perimeter.
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.
investigation
of whether
and why
“things work”
in mathematics
(both at the school level
and at the discipline)
reasoning-and-proving
Making
generalizations
Developing
arguments
patterns,
conjectures
proofs
(Stylianides, 2008)
19
investigation
of whether
and why
“things work”
in mathematics
(both at the school level
and at the discipline)
20
reasoning-and-proving
Making
generalizations
Developing
arguments
patterns,
conjectures
proofs
Reasoning-and-proving is defined to encompass the
breadth of the activity associated with:
 identifying patterns
 making conjectures
 providing non-proof arguments, and
(Stylianides, 2008)
 providing proofs.
Reasoning-and-Proving:
An Analytic Framework
Making Mathematical
Generalizations
Identifying a
pattern
Making a
conjecture
Providing Support to
Mathematical Claims
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
Providing a
non-proof
argument
Identifying a
pattern
Making a
conjecture
A, A’
A, A’
A’
B’
B, B’
C, C’
C’
C, C’
Providing a
proof
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., line
numbers) 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” Task
 Built on prior knowledge
 Encouraged multiple approaches
 Laid groundwork for discussing characteristics of proof
 Let Students Select Tools
 Group 1 drew pictures
 Group 6 used tiles to build a model
 Groups 3 and 4 used symbolic notations
 Let Students Do Most of the Talking and Most of the Thinking
 Asked questions to challenge students and to get them to talk to each other
 Encourage students to respond to questions posed by other students
 Instructed 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 Team
PIs:
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