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Transcript of Developing Teachers’ Understanding of Proof Developing Teachers’ Understanding of Proof Peg...
Developing Teachers’ Developing Teachers’ Understanding of ProofUnderstanding of Proof
Peg SmithUniversity of Pittsburgh
Teachers Development Group Leadership Seminar
February 17, 2011
Overview of SessionOverview 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?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:Connecting to Literature:Mathematical ReasoningMathematical 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:Connecting to Literature:Mathematical ReasoningMathematical 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 forStandards forMathematical PracticeMathematical 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 forStandards forMathematical PracticeMathematical 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: CORP: Cases of Reasoning and ProvingCases 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 QuestionsThree 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 QuestionsThree 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 Construct a proof for the following conjecture:following conjecture:
The sum of two odd numbers The sum of two odd numbers will always be an even number.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.
Criteria for Judging the Criteria for Judging the Validity of ProofValidity 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 Analyzing Student Work (Part 2)(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?)
15
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
Reasoning and Proving:Connecting to Literature
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Reasoning and Proving:An Analytic Framework
Making Mathematical Generalizations
Providing Support to Mathematical Claims
Math
em
atical Co
mp
on
ent
Identifying a pattern
Making a conjecture
Providing a proof
Providing a non-proof argument
Plausible Pattern
Definite Pattern
Conjecture Generic Example
Demonstration
Empirical Argument
Rationale
Stylianides, 2008, p. 10
17
Reasoning and Proving:An Analytic Framework
Making Mathematical Generalizations
Providing Support to Mathematical Claims
Math
em
atical Co
mp
on
ent
Identifying a pattern
Making a conjecture
Providing a proof
Providing a non-proof argument
Plausible Pattern
Definite Pattern
Conjecture Generic Example
B, DDemonstration
A
Empirical Argument
C, G
Rationale
H
Stylianides, 2008, p. 10
18
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
Reasoning and Proving:Connecting to Literature
Take a few minutes to Take a few minutes to consider…consider…
The learning opportunities afforded by activities such as those we discussed today
The situations in which the activities might be used
CORP Project TeamCORP 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
Definite PatternDefinite 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.]