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=EDUC 509A Fall 2011 Unit #1 MathLab Assignment= Geetha Jayaraman, Roger Migdow

Introduction
This page provides the lesson plan, demonstration of usage of mathematical language and facilitation of assimilation of such language by students, brief history and research applications of the learning theory, and few relevant examples of the demonstration of assigned learning theory to solving problems. The learning theory that we have been assigned to work on is "inductive and deductive reasoning" for solving algebraic problems. The rest of the document is organized as follows: brief definition and description of inductive and deductive reasoning methods, description of the algebraic problem chosen for the assignment, detailed lesson plan, historical context of selected activity, explanation of the usage of mathemtical language and the facilitation of assimilation of such language by the students, brief history and research applications of inductive v. deductive teaching methods, and finally, few relevant examples and resources demonstrating the application of the theory in problem solving.

Basics and Definitions of Inductive and Deductive Reasoning
Induction and deduction are elements of critical thinking and can be used as a basis for mathematical instruction, deductive being the most commonplace, albeit not the most effective method for learning new math concepts.

Inductive reasoning is best used for arguments based on experience or observation while deductive reasoning is best used for laws or rules. Inductive reasoning moves from specific details and observations to the more general principles and processes that explain them (e.g., Newton experienced an apple falling from a tree onto his head which inspired his Law of Gravity). The Inductive method is also known as the “scientific method” in which observation of nature is key. It is open-ended and exploratory initially and notions developed with an inductive argument support a conclusion called a hypothesis.

Deductive reasoning is narrower and moves from general truths with an complete explanation to specific conclusions and predictions for the observations supporting it. It is concerned with testing a hypothesis and is dependent on its premises since a false or inconclusive premise can lead to a false or inconclusive conclusion. Deductive reasoning leads to a confirmation of the original hypothesis and is based on logic.

To recap: Inductive reasoning is employed when adding another piece of information can strengthen the argument. If one cannot improve an argument by adding more evidence, one is employing deductive reasoning. The following table is reproduced from: The Guide To Inductive & Deductive Reasoning**, ** Induction vs. Deduction, October, 2008, by //The Critical Thinking Co.™// Staff.

//Description:// Given a specific hypothesis or observation, one combines observation and prior knowledge to reach a general conclusion. ||< //Example//: In //Dr. DooRiddles//, the student must generate answers in many categories in order to find the right answer to a riddle. //A page from a book,// //A thing you think with be;// //You can put it in your table,// //Or see it on a tree.// //Conclusions:// A leaf is a page in a book; lief is a syllable in the word be-lief; a leaf is part of a table; a leaf is part of a tree. The answer to the riddle is leaf. ||< //Recommended Products// //for Practice//: __Developing Critical Thinking through Science __ __Sciencewise __ __Editor in Chief® __ __Dr. DooRiddles __ __Infusing the Teaching of Critical Thinking into Content __ || || //Description:// Given a general conclusion or principle, one determines specific consequences or applications. ||< //Example//: To solve a //Mind Benders//® activity, the student is given a body of general information with certain clues and must then deduce the answers to specific questions. //Davis, Edwards, and Jones are an astronaut, a computer programmer, and a skin diver. Davis is not the astronaut or programmer. Jones is not the astronaut.// //What does each person do?// //Conclusions//: Davis is a skin diver, so Jones is the programmer, and Edwards is the astronaut. ||< //Recommended Products// //for Practice//: __Mind Benders® __ __Red Herring Mysteries __ __Word Roots __ __Reading Detective® __ __Science Detective® __ __Math Detective® __ ||
 * < **Quick Reference Guide** ||
 * < //Skill:// **Induction**
 * < //Skill:// **Deduction**
 * < //Skill:// **Deduction**

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Description of Algebraic Problem
The problem chosen for this exercise has been picked from the algebraic activities sampler (Activity #8, Brahier, p. 213). Specifically, it has to do with finding the sum of an infinite series and prove that 1/2 + 1/4 + 1/8 + 1/16 +..... = 1.

**Zeno's (Aristotle's) Dichotomy Paradox**
This paradox is known as the ‘dichotomy’ paradox because it involves repeated division into two, similar to the problem we chose to present. This description requires one to complete an infinite number of tasks, which Zeno maintained is an impossibility. Before one can reach any particular place, (e.g., a stationary bus) s/he must reach half of the distance to it. But, before reaching the last half, s/he must complete the next quarter of the distance. Reaching the next quarter, s/he must then cover the next eighth of the distance, then the next sixteenth, and so on. Thus, there are an infinite number of steps that must first be accomplished before s/he could reach the bus, with no way to establish the size of any "last" step. As well, there seems to be another problem because since any possible finite first distance could be divided in half, there is no first distance to run, so the trip cannot even begin! One might include then that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. Of course, the solution provided in Activity #8 (Brahier, p. 213) of having the students cut a whole piece of paper in half, and then half again, etc., is an excellent way of showing that indeed the sum of all the parts of this series is in fact equal to 1!

Usage of and Facilitation of the Assimilation of Mathematical Language
We are making the assumption that the students have been exposed to Algebra concepts and symbols so that they are prepared to tackle a problem involving infinite series and convergence. A review of these concepts and symbols follows:

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 * A series is the sum of the terms of a sequence.
 * A sequence is an ordered list of objects, with a recognizable pattern that can be written as a formula to arrive at the next term in the sequence.
 * Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
 * Given an infinite sequence of numbers, a series is the result of adding all those terms together.
 * Convergence refers to the notion that a sequence approaches a limit.
 * Divergence refers to the notion that a sequence never approaches a limit.
 * Symbolically, this can be represented with the summation symbol ∑ which represents the task of adding all of the terms of the sequence together.
 * Each fraction to be added together is descibed by dividing 1 by 2 raised to the 'n' power where 'n' represents all the integers from 1 to infinity. The entire concept can then be compactly represented as follows:
 * As students learn to seek out answers on the Web, the input for determining the sum of this infinite series appropriate for search engine requests and forum Q&A posts should be as follows: sum_{n=1}^infty frac{1}{2^n} = frac{1}{2}+ frac{1}{4}+ frac{1}{8}+cdots.

**Sociocultural Theory and Inductive/Explicit vs. Deductive/Implicit Reasoning**
In sociocultural theory, Vygotsky's zone of proximal development (ZPD) construct, which measures the distance between what a learner can handle independently and what s/he can do with assistance from a more knowledgeable educator, which begins with modeling (examples) rather than explaining the rules of the concept, is clearly an inductive-explicit trajectory. "As Vygotsky (1986) stated, 'Direct teaching of concepts is impossible and fruitless. A teacher who tries to do this usually accomplishes nothing but empty verbalism, a parrot-like repetition of words by the child, simulating a knowledge of the corresponding concepts but actually covering up a vacuum.'" See: Explaining to Exemplifying, Exploring, and Extending L2 Grammar: the ZPD-4Ex Axiom.

Examples of Application of Inductive and Deductive Teaching Methods
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An Example of Application of Inductive Reasoning for Problem Solving

An Example of Application of Deductive Reasoning for Problem Solving

Peer review article relating Inductive and Deductive Teaching Methods and Vygotsky's Sociocultural Learning theory