Mathematical Explanations and Arguments Number Theory for Elementary School Teachers: Chapter 1 by Christina Dionne Number Theory for Elementary School Teachers: Chapter 1 by Christina Dionne

History of Reasoning and Proof Indian proof (upapattis) vs. Greek proof (apodeixis) 1 The development of proof theory can be naturally divided into 4 : The prehistory of the notion of proof in ancient logic and mathematics, largely thanks to Euclid and his Elements (~300 BC) 4 The discovery by Gottlob Frege that mathematical proofs, and not only the propositions of mathematics, can (and should) be represented in a logical system ( ) 4Frege David Hilbert's old axiomatic proof theoryDavid Hilbert's old axiomatic proof theory (1903) 4 Failure of the aims of Hilbert through Gödel's incompleteness theorems (~1930) 4 Gentzen's creation of the two main types of logical systems of contemporary proof theory, natural deduction and sequent calculus (1935) 4 Applications and extensions of natural deduction and sequent calculus, up to the computational interpretation of natural deduction and its connections with computer science. 4

Developmental Perspective of Reasoning and Proof Elementary years- They have the notion of proof, but usually only through thoughtful trial and error. 1 According to Piaget 5 : years: able to handle certain formal operations -- implication and exclusion but cant do a proof by exhaustion years: able to deal with premises that require hypothetico- deductive reasoning However, problem-solving processes are employed by children at all age levels, just the degree of complexity being the key factor. This may be because of just a difference in the lack of experience. 5 the ability of children to create the essence of mathematical proofs may be superior to their ability to write proofs. Young children may be unable to demonstrate their ability to produce proofs because of a lack of mathematical experience and sophistication. 5

Importance of Proof "Through the classroom environments they create, mathematics teachers should convey the importance of knowing the reasons for mathematical patterns and truths. In order to evaluate the validity of proposed explanations, students must develop enough confidence in their reasoning abilities to questions others' mathematics arguments as well as their own. In this way, they rely more on logic than on external authority to determine the soundness of a mathematical argument." 2

Types of Proof Proof by exhaustion Postulational proofs Proofs by induction Proofs by contradiction Commonalities: Notice systematic pattern Make a conjecture Defend using logic

Objectives of Proof 2 Reason about a problem Extend what they already know Make a conjecture Provide a convincing argument Refine their thinking Defend or modify their arguments

Reason About a Problem Ask them probing questions: What is known? definitions, properties, patterns? What needs to be known? Are there (usable) theorems leading to it?

Extend What They Already Know Probing questions: Can previous knowledge be applied? Is there a different way to approach it?

Make a Conjecture Probing questions: What pattern are you trying to show? Is it general, or specific? What approach is easiest? What approach is hardest? Any useful previous knowledge?

Provide a Convincing Argument Probing questions: Does it convince you? Will it convince a friend?..a skeptic? Are properties and theorems used correctly? Did you prove your objective? Or something else? Writing activities: Makes thought visible Easier to manipulate and analyze logic

Refine Their Thinking Teacher guided discussion Individual focus: Simpler and easier way? Can it be in math terms? Group focus: Redirect when necessary Introduce more information if necessary

Defend or Modify Arguments Group discussion: Different approaches Simplest way? Most convincing way? Student concerns

In General... Let students play with the problem. Guide them through what they know, what they want to know, and what they need to know. (Writing activities) Have them find their argument, and work in groups to develop them. Work with students to turn their thinking into a formal proof if appropriate.

Helpful and Guiding Activities Writing Activities Cooperative Learning Groups Visual Aids Manipulatives Don’t just use one!

Writing Activities Can be used at all stages of a proof Types: Journals and Learning logs Think sheets KWL Words to math

Cooperative Learning Groups Develops reasoning skills Provides different insight Different group types offer different opportunities: operativelearning.htm

Visual Aids Great for Visual learners Identify whats known and what needs to be found. Can increase motivation Break down processes into steps

Manipulatives Concrete reasoning to abstract Provides a base for different approaches. Motivational opportunities...Time to play!

The Staircase Problem How many blocks do you need to build a staircase with 1 step? 3 steps? 10 steps? 100 steps? n steps? 1 step2 steps3 stepsn steps

Different Approaches Informally: Adding the steps (Arithmetic to algebraic) Creating squares (Pictorial to algebraic) Formally: ∑ n (Proof by induction)

A Look Into a Classroom... 9th grade class, with previous knowledge on finding patterns, measurement, estimation, evaluation of algebraic expressions.

Questions to Consider Was this engaging to the students? How did the teacher respond to the different strategies? Were objectives met?

References 1. Wall, Edward. Number Theory for Elementary School Teachers: The Practical Guide Series. New York: McGraw-Hill, Print. 2. Chapter 7: Standards for Grades 9-12." Principals and Standards for School Mathematics. Comp. David Barnes. Reston, VA: NCTM, Electronic Principals and Standards. NCTM. Web. Oct "Teaching Math: Grades 9-12: Reasoning and Proof." Learner.org. Annenberg Foundation, Web. Oct von Plato, Jan, "The Development of Proof Theory", The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.),. 5. Lester, Frank K. "Developmental Aspects of Children's Ability to Understand Mathematical Proofs." Journal for Research in Mathematics Education 6.1 (1975): National Council of Teachers of Mathematics (NCTM). Web. Oct