SEMINAR PRESENTATION:HEURISTICS AND DEDUCTIVE REASONING IN

PROBLEM SOLVINGPrajish Prasad

under the guidance of Prof. Sridhar Iyer

OUTLINE

★ Heuristics in Mathematical Problem Solving

○ WISE Methodology

★ Deductive Reasoning

★ Extension of seminar to PhD

HEURISTICS IN MATHEMATICAL

PROBLEM SOLVING

HEURISTICS IN MATHEMATICAL PROBLEM SOLVINGDefinition of Mathematical Problem Solving, Heuristics, Examples

Literature Survey

Limitations of Heuristic Strategies

IntroductionTeaching-Learning

of Heuristic Strategies

Does teaching heuristic strategies improve problem solving

WISE Methodology

Application of WISE to various types of problems

MATHEMATICAL PROBLEM SOLVING

Two Definitions - [1]

Definition 1: "In mathematics, anything required to be done, or requiring the doing of something." - Problem solving as routine exercises

Definition 2: "A question... that is perplexing or difficult." - Problem solving as an art

THE ART OF MATHEMATICAL PROBLEM SOLVING

○ “As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth ” [1]

○ Mathematics involves guessing, intuition and discovery similar to the physical sciences - Heuristics

HEURISTICS

Definition -

“Heuristic is any approach to problem solving, learning, or discovery that employs a

practical method not guaranteed to be optimal or perfect, but sufficient for the immediate

goals”

Introduced by George Polya - “How to Solve It”

HEURISTICS STRATEGIES

Analogous Problem -

To solve a complicated problem, it often helps to examine and solve a simpler analogous problem. Then exploit your solution

Other examples - Draw a figure. Introduce suitable notation.

Solve a part of the problemLook for a patternConsider special cases

WISE METHODOLOGY

WISE METHODOLOGY

Weaken - Weaken the given problem P by weakening the instance or the objective

Identify - Identify a candidate problem P’ based on the weakening

Solve - Solve P’

Extend - Extend the solutions of P’ towards solving problem P

(Operationalized to solve certain problems in graph theory)

Figure taken from [2]

Algorithm of WISE Methodology

CANDIDATE PROBLEMS WHICH CAN USE WISE METHODOLOGYMath Puzzles -

Example - There are 100 light switches, all of them are off. First, you walk by them, turning all of them on. Next, you walk by them turning every other one off. Then, you walk by them changing every third one. On your 4th pass, you change every 4th one. You repeat this for 100 passes. At the end, how many lights will be on?

Use of WISE Methodology - Use WISE to weaken the instance, of upto 5 nos? Extend for 10 nos. Check for prime nos, then for perfect primes

CANDIDATE PROBLEMS WHICH CAN USE WISE METHODOLOGYBasic Counting Problems (Permutations and Combinations) -

Example - How many words of length 8 can you form, where the first letter is the same as the last letter?

Use of WISE Methodology - First weaken the instance to 2 letters and weaken the objective to any two letters - 26^2Now extend to 8 numbers with the above objective - 26^8Extend the objective - 1st and last can be chosen in 26 ways, the remaining 6 letters in 26^6 ways. = 26*26^6 = 26^7

CANDIDATE PROBLEMS WHICH CAN USE WISE METHODOLOGYRecursive Algorithms

Example - Write a recursive algorithm to find the factorial of a given number

Use of WISE Methodology - Weaken the instance to 1 or 2 numbers. Then weaken the objective to calculate for any 2 consecutive numbers.

INSIGHTS FROM CANDIDATE PROBLEMSProblem Type Example Insights

Math Puzzles There are 100 light switches, all of them are off. First, you walk by them, turning all of them on. Next, you walk by them turning every other one off. Then, you walk by them changing every third one. On your 4th pass, you change every 4th one. You repeat this for 100 passes. At the end, how many lights will be on?

Good candidate problems are those in which we can weaken the instance

Permutations and Combinations How many words of length 8 can you form, where the first letter is the same as the last letter?

Good candidate problems to use WISE since both instances and objectives can be weakened

Recursive Algorithms Write a recursive algorithm to find the factorial of a given number

Good candidate problems to use WISE since instances can be weakened

CAN HEURISTICS STRATEGIES IMPROVE PROBLEM SOLVING

Experiment -[3]○ Two groups of students - same problem-

solving training○ Heuristic strategies were explicitly mentioned

to only one of the groups○ Each student - worked on 20 problems, then

saw solutions. 5 heuristic strategies○ “Heuristic” students outscored non-heuristic

students in post-test. Significant difference○ Transcripts of the solutions show that explicit

use of the strategies accounted for differences between the two groups.

Solution sheet of “Heuristic” students

Figure taken from [3]

LIMITATIONS OF HEURISTICSToo many heuristic strategies

Polya’s Book - “How to Solve It” - 40 heuristics

Set of keys. But deciding which to use for a particular problem is difficult.

LIMITATIONS OF HEURISTICSDescriptive nature makes it hard to directly apply it to the problem

Example - Analogous problem

Identifying that the particular problem indeed can use the "analogous problem" heuristic

Generate analogous problemsChoose the appropriate analogous problemSolve the analogous problemExtract important information from the problem i.e either the solution or the method.

DEDUCTIVE REASONING

Definition, Motivation and Examples of Deductive Reasoning

Literature Survey

Teaching-Learning of Deductive Reasoning

DEDUCTIVE REASONING

IntroductionProcesses of Deductive

Reasoning

Literature survey of how reasoning is done by individuals

Literature survey of existing strategies and solutions. Proposed solution

DEDUCTIVE REASONING - INTRODUCTIONDefinition, Motivation and Examples of Deductive Reasoning

Literature Survey

Teaching-Learning of Deductive Reasoning

IntroductionProcesses of Deductive

Reasoning

DEDUCTIVE REASONING - INTRODUCTION

Example -

I have to present my seminar at 9.30 amIt takes me half an hour to reach IITTherefore, I have to leave at 9 am

It takes me an hour to reach IIT if I leave between 8am and 10amTherefore, I have to leave at 8.30am

DEDUCTIVE REASONING - INTRODUCTION

Example -

No one at the country house mentioned that the guard dogs barked the night of the crime.The victim was alone.Guard dogs bark at strangers.

Therefore, the suspect was known to the guard dogs

DEFINITION -“The process of reasoning from one or more statements

(premises) to reach a logically certain conclusion”

DOMAINS OF DEDUCTIVE REASONING

Relational reasoning

○ Based on the logical properties of relations as greater than, on the right of, and after.

○ Example - The cup is on the right of the saucer.

The plate is on the left of the saucer.The fork is in front of plate.The spoon is in front of the cup. What is the relation between the fork and the spoon?

DOMAINS OF DEDUCTIVE REASONING

Propositional reasoning

○ Based on negation and connectives if, or, and.

○ Example - If the ink cartridge is empty then the printer won’t work.

The ink cartridge is empty.So, the printer won’t work.

DOMAINS OF DEDUCTIVE REASONING

Syllogistic Reasoning

○ Based on pairs of premises.Each contain a single quantifier, such as all or some.

○ Example - All artists are bakers.

Some bakers are chemists.Therefore, some artists are chemists

WHY IS IT IMPORTANT TO IMPROVE DEDUCTIVE REASONING

Competitive Exams -

○ Exams like GRE, GMAT, CAT have sections on logical reasoning

○ Good reasoning ability is an essential requirement for doing well in grad school Taken from http://barronstestprep.com/blog/tag/logical-reasoning-questions/

WHY IS IT IMPORTANT TO IMPROVE DEDUCTIVE REASONING

For researchers

○ Defend methods of conducting research○ Find flaws/limitations in existing research/theories○ Argumentation - reasoning systematically (for anyone in general)

PROCESSES OF DEDUCTIVE REASONINGDefinition, Motivation and Examples of Deductive Reasoning

Literature Survey

Teaching-Learning of Deductive Reasoning

IntroductionProcesses of Deductive

Reasoning

Literature survey of how reasoning is done by individuals

PROCESS OF DEDUCTIVE REASONING -FORMAL SYNTACTIC PROCESS○ Underlies several theories - Most prominent - Lance Rips

○ Reasoners extract the logical forms of premises

○ Use rules(similar to logic) to derive conclusions

○ Example rule - modus ponens rule -If A then B

ATherefore B

PROCESS OF DEDUCTIVE REASONING -FORMAL SYNTACTIC PROCESS○ Example -

If the ink cartridge is empty the printer won’t work. (P1)The printer is working (P2)Can we conclude that the ink cartridge is not empty?The ink cartridge is empty (Supposition)The printer won’t work (P3 - Modus ponens on P1 and Supposition)Contradiction between P2 and P3Therefore, our supposition is wrong.

PROCESS OF DEDUCTIVE REASONING -MENTAL MODELS

Theory of Mental Models state

“Reasoning is based not on syntactic derivations from logical forms but on manipulations of mental models representing situations”

Example - The ink cartridge is empty and the printer is not working i ̴p

The ink cartridge is empty and the printer is not working

i ̴p

The ink cartridge is empty or the printer is not working

i ̴p

i ̴p

If the ink cartridge is empty, then the printer is not working

i ̴p ...

The ink cartridge is empty, if and only if the printer is not working

i ̴p ...

The printer is working (P2)

p

Reason - The Principle of Truth“Individuals tend to minimise the load on working memory by representing explicitly only what is true, and not what is false” [4]

Can we conclude that the ink cartridge is not empty - NO

If the ink cartridge is empty, then the printer is not working(P1)

i ̴p ...

ILLUSIONS IN PROPOSITIONAL REASONING

Mental Model Fully Explicit Model

If the ink cartridge is empty, then the printer is not working(P1)

i ̴p ̴i ̴p ̴i p

If the ink cartridge is empty, then the printer is not working(P1)

i ̴p ...

Can we conclude that the ink cartridge is not empty -YES

If the ink cartridge is empty, then the printer is not working(P1)

i ̴p ̴i ̴p ̴i p

The printer is working (P2)

p

EXPERIMENTAL RESULTS - ITwo experiments

Score on modus ponens was significantly higher than modus tolens

If the ink cartridge is empty, then the printer is not working

i ̴p ̴i ̴p ̴i p

Conclusion - Fallacies result due to construction of mental models and not fully explicit mental models

EXPERIMENTAL RESULTS - II

Score on modus tollens was significantly higher in biconditional than modus tolens with a conditional

If the ink cartridge is empty, then the printer is not working

i ̴p ̴i ̴p ̴i p

The ink cartridge is empty, if and only if the printer is not working

i ̴p ̴i p

Conclusion - Greater the number of models, greater is the difficulty in performing deductions

DEDUCTIVE REASONINGDefinition, Motivation and Examples of Deductive Reasoning

Literature Survey

Teaching-Learning of Deductive Reasoning

IntroductionProcesses of Deductive

Reasoning

Literature survey of how reasoning is done by individuals

Literature survey of existing strategies and solutions. Proposed solution

TEACHING-LEARNING OF DEDUCTIVE REASONINGTitle Pedagogy

FeaturesMethodology Parameters to

measure effectiveness

Results

Tarski's World(1993)[5]

Interactivity Creation of 3-D objects and checking propositions

Hyperproof(HP)(1994)[6]

Interactivity Extension to Tarski’s World. Control group - HP without graphics

Transfer of learning, scores in post test

Good transfer of learning,but strong interactions between pre-existing individual differences and methods of teaching

TEACHING-LEARNING OF DEDUCTIVE REASONINGTitle Pedagogy

FeaturesMethodology Parameters to

measure effectiveness

Results

MIZAR-MSE,WINKE(1993)[7]

Proof checker - check assignments

Students given assignments. Use tool to complete assignments

Syllog[8]

Interactive Proofs, Gamification

4 day course on logic. Fourth day use of system

Scores in post test Significant difference between pre and post test scores

PROPOSED SOLUTIONMental Model Theory consistent with experimental results.

Reasoning is based on manipulations of mental models representing situations

Providing a TEL environment which will allow learners to manipulate explicit models while reasoning and arrive at a conclusion

Choice of TEL environment - Scratch

Toy Example

REFERENCES[1] A. H. Schoenfeld, Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), New York: MacMillan, Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). .

[2] M. Jagadish, “A Problem-Solving Methodology Based on Extremality Principle and its Application to CS Education,” PhD. thesis, CSE. Dept., IIT Bombay.,Mumbai., 2015.

[3] A. H. Schoenfeld, "Teaching Problem-Solving Skills", The American Mathematical Monthly.,ser. 10, vol. 87, pp. 794-805, Dec 1980

[4] P. N. Johnson-Laird, "Deductive Reasoning", Annu. Rev. Psychol., vol. 50, pp. 109–135, 1999

REFERENCES[1] A. H. Schoenfeld, Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), New York: MacMillan, Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). .

[2] M. Jagadish, “A Problem-Solving Methodology Based on Extremality Principle and its Application to CS Education,” PhD. thesis, CSE. Dept., IIT Bombay.,Mumbai., 2015.

[3] A. H. Schoenfeld, "Teaching Problem-Solving Skills", The American Mathematical Monthly.,ser. 10, vol. 87, pp. 794-805, Dec 1980

[4] P. N. Johnson-Laird, "Deductive Reasoning", Annu. Rev. Psychol., vol. 50, pp. 109–135, 1999