SEMINAR PRESENTATION:
HEURISTICS AND DEDUCTIVE REASONING IN
PROBLEM SOLVING
Prajish Prasad
under the guidance of
Prof. Sridhar Iyer
★ Heuristics in Mathematical
Problem Solving
OUTLINE
○ WISE Methodology
★ Deductive Reasoning
★ Extension of seminar to PhD
HEURISTICS IN
MATHEMATICAL
PROBLEM SOLVING
HEURISTICS IN MATHEMATICAL PROBLEM SOLVING
Definition of Mathematical
Problem Solving, Heuristics,
Examples
Introduction
Does teaching heuristic
strategies improve problem
solving
WISE Methodology
Teaching-Learning
Literature
of Heuristic
Survey
Strategies
Application of WISE to
various types of problems
Limitations of Heuristic
Strategies
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 problem
Look for a pattern
Consider 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)
Algorithm of WISE Methodology
Figure taken from [2]
CANDIDATE PROBLEMS WHICH CAN USE WISE
METHODOLOGY
Math 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
METHODOLOGY
Basic 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 - 26^2
Now extend to 8 numbers with the above objective - 26^8
Extend 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
METHODOLOGY
Recursive 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 PROBLEMS
Problem 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
Solution sheet of “Heuristic” students
SOLVING
Experiment -[3]
○ Two groups of students - same problemsolving 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.
Figure taken from [3]
LIMITATIONS OF HEURISTICS
Too 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 HEURISTICS
Descriptive 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 problems
Choose the appropriate analogous problem
Solve the analogous problem
Extract important information from the problem i.e either the solution or the method.
DEDUCTIVE
REASONING
DEDUCTIVE REASONING
Definition, Motivation and
Examples of Deductive
Reasoning
Introduction
Literature survey of existing
strategies and solutions.
Proposed solution
Literature
Processes of
Deductive
Reasoning
Survey
Literature survey of how
reasoning is done by
individuals
Teaching-Learning of
Deductive Reasoning
DEDUCTIVE REASONING - INTRODUCTION
Definition, Motivation and
Examples of Deductive
Reasoning
Introduction
Literature
Processes of
Deductive
Reasoning
Survey
Teaching-Learning of
Deductive Reasoning
DEDUCTIVE REASONING - INTRODUCTION
Example I have to present my seminar at 9.30 am
It takes me half an hour to reach IIT
Therefore, I have to leave at 9 am
It takes me an hour to reach IIT if I leave between 8am and 10am
Therefore, 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 REASONING
Definition, Motivation and
Examples of Deductive
Reasoning
Introduction
Literature
Processes of
Deductive
Reasoning
Survey
Literature survey of how
reasoning is done by
individuals
Teaching-Learning of
Deductive Reasoning
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
A
Therefore 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 P3
Therefore, 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
If the ink cartridge is empty,
then the printer is not working
i
̴p
...
The ink cartridge is empty or
the printer is not working
i
̴p
i
̴p
The ink cartridge is empty, if
and only if the printer is not
working
i
̴p
...
If the ink cartridge is empty,
then the printer is not working
(P1)
i
The printer is working (P2)
̴p
p
...
Can we conclude that the ink cartridge is not empty - NO
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]
ILLUSIONS IN PROPOSITIONAL REASONING
If the ink cartridge is empty,
then the printer is not working
(P1)
i
̴p
...
Mental Model
If the ink cartridge is empty,
then the printer is not working
(P1)
i
̴p
̴i
̴p
̴i
p
Fully Explicit Model
If the ink cartridge is empty,
then the printer is not working
(P1)
i
̴p
̴i
̴p
̴i
p
The printer is working (P2)
Can we conclude that the ink cartridge is not empty -YES
p
EXPERIMENTAL RESULTS - I
If the ink cartridge is empty,
then the printer is not working
i
̴p
̴i
̴p
̴i
p
Two experiments
Score on modus ponens
was significantly higher
than modus tolens
Conclusion - Fallacies result due to construction of mental
models and not fully explicit mental models
EXPERIMENTAL RESULTS - II
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
Score on modus tollens was significantly higher in biconditional than modus tolens
with a conditional
Conclusion - Greater the number of models, greater is the
difficulty in performing deductions
DEDUCTIVE REASONING
Definition, Motivation and
Examples of Deductive
Reasoning
Introduction
Literature survey of existing
strategies and solutions.
Proposed solution
Literature
Processes of
Deductive
Reasoning
Survey
Literature survey of how
reasoning is done by
individuals
Teaching-Learning of
Deductive Reasoning
TEACHING-LEARNING OF DEDUCTIVE REASONING
Title
Pedagogy
Features
Methodology
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
Parameters to
measure
effectiveness
Results
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 REASONING
Title
Pedagogy
Features
Methodology
MIZAR-MSE, Proof checker WINKE
check
(1993)
assignments
[7]
Students given
assignments. Use
tool to complete
assignments
Syllog
[8]
4 day course on
logic. Fourth day
use of system
Interactive
Proofs,
Gamification
Parameters to
measure
effectiveness
Results
Scores in post test
Significant difference
between pre and post test
scores
PROPOSED SOLUTION
Mental 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 sensemaking 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 sensemaking 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