Mathematical Explanations and Arguments Number Theory for Elementary School Teachers: Chapter 1 by...
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Mathematical Explanations and Arguments Number Theory for Elementary School Teachers: Chapter 1 by Christina Dionne
Mathematical Explanations and Arguments Number Theory for Elementary School Teachers: Chapter 1 by Christina Dionne Number Theory for Elementary School
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
Slide 2
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 (1893-1903)
4Frege David Hilbert's old axiomatic proof theoryDavid Hilbert's
old axiomatic proof theory (1903) 4 Failure of the aims of Hilbert
through Gdel'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
Slide 3
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 : 11-13 years:
able to handle certain formal operations -- implication and
exclusion but cant do a proof by exhaustion 14-15 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
Slide 4
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
Slide 5
Types of Proof Proof by exhaustion Postulational proofs Proofs
by induction Proofs by contradiction Commonalities: Notice
systematic pattern Make a conjecture Defend using logic
Slide 6
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
Slide 7
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?
Slide 8
Extend What They Already Know Probing questions: Can previous
knowledge be applied? Is there a different way to approach it?
Slide 9
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?
Slide 10
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
Slide 11
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
Slide 12
Defend or Modify Arguments Group discussion: Different
approaches Simplest way? Most convincing way? Student concerns
Slide 13
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.
Slide 14
Helpful and Guiding Activities Writing Activities Cooperative
Learning Groups Visual Aids Manipulatives Dont just use one!
Slide 15
Writing Activities Can be used at all stages of a proof Types:
Journals and Learning logs Think sheets KWL Words to math
Slide 16
Cooperative Learning Groups Develops reasoning skills Provides
different insight Different group types offer different
opportunities: http://edtech.kennesaw.edu/intech/co
operativelearning.htm
Slide 17
Visual Aids Great for Visual learners Identify whats known and
what needs to be found. Can increase motivation Break down
processes into steps
Slide 18
Manipulatives Concrete reasoning to abstract Provides a base
for different approaches. Motivational opportunities...Time to
play!
Slide 19
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
Slide 20
Different Approaches Informally: Adding the steps (Arithmetic
to algebraic) Creating squares (Pictorial to algebraic) Formally: n
(Proof by induction)
Slide 21
A Look Into a Classroom... 9th grade class, with previous
knowledge on finding patterns, measurement, estimation, evaluation
of algebraic expressions.
Slide 22
Questions to Consider Was this engaging to the students? How
did the teacher respond to the different strategies? Were
objectives met?
Slide 23
References 1. Wall, Edward. Number Theory for Elementary School
Teachers: The Practical Guide Series. New York: McGraw-Hill, 2010.
Print. 2. Chapter 7: Standards for Grades 9-12." Principals and
Standards for School Mathematics. Comp. David Barnes. Reston, VA:
NCTM, 2000. 287-363. Electronic Principals and Standards. NCTM.
Web. Oct.
2011..http://www.usi.edu/science/math/sallyk/Standards/document/chapter7/reas.htm
3. "Teaching Math: Grades 9-12: Reasoning and Proof." Learner.org.
Annenberg Foundation, 2011. Web. Oct.
2011..http://www.learner.org/courses/teachingmath/grades9_12/session_04/index.html
4. von Plato, Jan, "The Development of Proof Theory", The Stanford
Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta
(ed.),.http://plato.stanford.edu/archives/fall2008/entries/proof-theory-development/
5. Lester, Frank K. "Developmental Aspects of Children's Ability to
Understand Mathematical Proofs." Journal for Research in
Mathematics Education 6.1 (1975): 14-25. Www.jstor.org. National
Council of Teachers of Mathematics (NCTM). Web. Oct.
2011..Www.jstor.orghttp://www.jstor.org/stable/748688