Transcript CORP Advisory Board Meeting December 13

Developing Teachers’
Understanding of Proof
Peg Smith
University of Pittsburgh
Teachers Development Group Leadership Seminar
February 17, 2011
Overview of Session





Provide a rationale for focusing on reasoning
and proving and describe the CORP project
Solve and discuss the “Odd + Odd = Even” task
Engage in an analysis of student “proofs” and
discuss the opportunities for learning afforded
by such work
Discuss a framework for thinking about
reasoning and proving activities
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

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).
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
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?
Construct a proof for the
following conjecture:
The sum of two odd numbers will
always be an even number.
Private Think Time
– spend five minutes thinking
about the task individually before beginning work
with a partner or trio.
Small
Group – discuss different approaches with
your partner(s) and jointly create a proof. Once
you have proven it one way, see if you can come up
with an alternative approach.
Analyzing Student Work (Part
1)
Review the ten student responses
and
use the matrix (green) to record whether or
not each response qualifies as a proof and
provide the rationale that led you to that
conclusion.
 Discuss your ratings and rationale with
members of your group, come to a group
consensus on which responses are and are not
proofs and why. Record your groups’ decisions
on the Proof Evaluation Chart on the elmo.
 Develop a list of criteria for what
characteristics an argument must have in order
to qualify as a proof.

Criteria for Judging the Validity
of Proof

The argument must show that the conjecture or
claim is (or is not) true for all cases.

The statements and definitions that are used in the
argument must be ones that are true and accepted by
the community because they have been previously
justified.

The conclusion that is reached from the set of
statements must follow logically from the argument
made.
Analyzing Student Work (Part
2)
For each response C, E, H and I consider:

What is the limitation in the current argument that
is being made by the student? (Refer to the Criteria
for Judging the Validity of Proof list to help pinpoint
what might be missing or incorrect.)

What would it take for the current argument to be
classified as a proof?

What question(s) could you ask the student that
would help improve her argument so that it would
qualify as a proof? (How could you bridge between
where the student currently is and where you want
them to end up?)
Reasoning and Proving:
Connecting to Literature
The work in which mathematicians engage
that culminates in a formal proof involves
searching a mathematical phenomena for
patterns, making conjectures about those
patterns, and providing informal arguments
demonstrating the viability of the conjectures.
Lakatos, 1976
15
Reasoning and Proving:
An Analytic Framework
Making Mathematical Generalizations
Identifying
a pattern
Mathematical Component
Plausible
Pattern
Definite
Pattern
Making a
conjecture
Conjecture
Providing Support to Mathematical Claims
Providing a
proof
Providing a
non-proof
argument
Generic
Example
Demonstration
Empirical
Argument
Rationale
Stylianides, 2008, p. 10 16
Reasoning and Proving:
An Analytic Framework
Making Mathematical
Generalizations
Identifying
a pattern
Mathematical Component
Plausible
Pattern
Definite
Pattern
Making a
conjecture
Conjecture
Providing Support to Mathematical Claims
Providing a
proof
Generic
Example
B, D
Demonstration
A
Providing a
non-proof
argument
Empirical
Argument
C, G
Rationale
H
Stylianides, 2008, p. 10
17
Reasoning and Proving:
Connecting to Literature
By focusing primarily on the final product - that is, the proof students are not afforded the same level of scaffolding used by
professional users of mathematics to establish mathematical truth.
Therefore, reasoning and proving should be defined to
encompass the breadth of activity associated with:
•
•
•
•
identifying patterns,
making conjectures,
providing proofs, and
providing non-proof arguments.
Stylianides, 2005; Stylianides & Silver, 2004
18
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
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
Plausible Pattern
Input
Output
1
1
2
4
3
4
5
Definite Pattern
For the pattern shown below, compute the
perimeter for the first four trains, determine the
perimeter for the tenth train without constructing
it, and then write a description that could be used
to compute the perimeter of any train in the
pattern. [The edge of the hexagon has a length of
one.]