Heuristic Strategies and Deductive Reasoning in
Problem Solving
Seminar Report
Submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
by
Prajish Prasad
Roll No : 154380001
under the guidance of
Prof. Sridhar Iyer
Inter-disciplinary Program in Educational Technology
Indian Institute of Technology, Bombay
November 2015
Contents
1 Introduction to Mathematical Problem Solving
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Importance of Teaching-Learning of Problem Solving . . . . .
1.3 Organisation of Report . . . . . . . . . . . . . . . . . . . . . .
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2 Use
2.1
2.2
2.3
4
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5
of Heuristics in Mathematical Problem Solving
Introduction to Heuristics . . . . . . . . . . . . . . . . . . . .
Teaching-Learning using Heuristics . . . . . . . . . . . . . . .
Limitations of Heuristics . . . . . . . . . . . . . . . . . . . . .
3 The WISE Methodology
3.1 Weaken-Identify-Solve-Extend . . . . . . . . . .
3.2 Extending WISE to other topics and problems
3.2.1 Common Math Puzzles . . . . . . . . .
3.2.2 Basic Permutations and Combinations .
3.2.3 Recursive Algorithms . . . . . . . . . .
3.3 Insights and Future Scope . . . . . . . . . . . .
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4 Deductive Reasoning
4.1 Introduction to Deductive Reasoning . . . . . . . . . . . . . .
4.1.1 Definition and Examples . . . . . . . . . . . . . . . . .
4.1.2 Why is it Important to Improve Deductive Reasoning
4.2 Processes of Deductive Reasoning . . . . . . . . . . . . . . . .
4.2.1 Deduction as a Formal Syntactic Process based on Rules
4.2.2 Deduction as a Semantic Process based on Mental
Models . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . .
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5 Future Directions
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1
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Chapter 1
Introduction to
Mathematical Problem
Solving
1.1
Introduction
In [12], Alan Schoenfeld refers to two definitions of the word “problem” Definition 1.1.1. In mathematics, anything required to be done, or requiring the doing of something.
Definition 1.1.2. A question... that is perplexing or difficult.
The first definition of problem solving seems to suggest that there is a
particular method to solve a problem. Learners can learn this method by
solving practice problems of the given topic, handed down to them by experts, which they have to memorize. They eventually master the method
and can apply it to other problems.
The second definition views problem solving as an art, which requires a
certain amount of creativity from the students and application of various
methods in order to arrive at the solution. The main proponent of this
definition of problem solving was George Polya. He states that mathematics
involves guessing, intuition and discovery similar to the physical sciences.
[8]
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1.2
Importance of Teaching-Learning of Problem
Solving
Over the years, there has been a change in how mathematics and problem solving is perceived. Educators realise that for mathematics education
to fulfill its objectives, there has to be a shift from the first definition to
the second. Therefore there needs to be a shift from content to processes.
The process of arriving at the solution is primary, as compared to the final
answer. Students should be encouraged to explore patterns, and not just
memorize formulas. They should be encouraged to formulate conjectures,
not just do exercises.
Schoenfeld reasons that this perspective of learning mathematics is empowering. Mathematically powerful students are quantitatively literate. They
are capable of interpreting the vast amounts of quantitative data they encounter on a daily basis, and of making balanced judgments on the basis of
those interpretations. They use mathematics in practical ways, from simple
applications such as using proportional reasoning for recipes or scale models,
to complex budget projections, statistical analyses, and computer modeling.
They are flexible thinkers with a broad repertoire of techniques and perspectives for learning to think mathematically, dealing with novel problems and
situations. They are analytical, both in thinking issues through themselves
and in examining the arguments put forth by others.[12]
1.3
Organisation of Report
In this seminar report, two topics are explored, “Heuristics in Mathematical Problem Solving” and “Deductive Reasoning”. Chapter 2 details the
use of heuristics in the process of problem solving and limitations of using heuristics. Chapter 3 gives details of a methodology called “WISE”[7],
which is a specific example of a heuristic operationalized for a variety of topics. Chapter 4 gives a brief introduction of deductive reasoning and theories
from cognitive psychology which explain how we reason. We have outlined
our proposed solution for teaching-learning of deductive reasoning. Finally,
Chapter 5 gives details of possible extensions of this seminar.
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Chapter 2
Use of Heuristics in
Mathematical Problem
Solving
2.1
Introduction to Heuristics
As stated in the previous chapter, mathematical problem solving involves
guessing, intuition and discovery similar to the physical sciences. Heuristics aid in this process of guessing and intuition. According to Wikipedia,
“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.”
A comprehensive set of heuristics were first compiled and presented by
George Polya in his book “How to Solve it”[9] An example of a heuristic
is the analogous problem heuristic, which states “To solve a complicated
problem, it often helps to examine and solve a simpler analogous problem.
Then exploit your solution.” [9] Other examples of heuristics are 1. Draw a figure. Introduce suitable notation.
2. Solve a part of the problem
3. Look for a pattern
4. Consider special cases
2.2
Teaching-Learning using Heuristics
The use of heuristics is a useful tool in the process of mathematical problem solving. However, the question arises - “Does teaching heuristic strategies improve problem solving?” Schoenfeld conducted an experiment [11] in
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which two groups of students were given a problem solving training, in which
five heuristic strategies were taught. Each student worked on 20 problems,
then saw the solutions. They were given a list and explanation of the five
strategies used in the experiment and an “overlay” to each solution explaining how the strategy had been used. Figure 2.1 is an example of the solution
Figure 2.1: An example of the heuristic strategies solution shown to students
to a problem. The right-hand side is the solution seen by all students. The
left-hand side was seen only by the “heuristics” students.
Evaluation was done using post test. Students who were explicitly taught
heuristic strategies outscored the other group with a significent difference
in pretest to post test gains. Moreover, transcripts of the solutions show
that explicit use of the strategies accounted for differences between the two
groups.
2.3
Limitations of Heuristics
Although the experiment stated above shows postive results, Schoenfeld
is not quite optimistic. He states the following “But even if we succeed in teaching students to use a series of important
heuristic strategies, I see no guarantee that there will be clear signs of improvement in their general problem solving. Knowing how to use a strategy
isn’t enough: the student must think to use it when it’s appropriate.”[11]
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The set of heuristics can be considered as a set of keys. Only one of
the keys can unlock the problem. However, deciding which to use for a
particular problem is difficult. Polya’s book “How to Solve it”[9] has around
40 heuristics.
Even after one decides a particular heuristic strategy, the descriptive nature of the strategy makes it hard to directly apply it to the problem. For
example, the analogous problem strategy states “To solve a complicated problem, it often helps to examine and solve a
simpler analogous problem. Then exploit your solution” .
In order to use this heuristic, several other decisions have to be made.
1. Identifying that the particular problem indeed can use the ”analogous
problem” heuristic
2. Generate analogous problems
3. Choose the appropriate analogous problem
4. Solve the analogous problem
5. Extract important information from the problem i.e either the solution
or the method.
The next chapter uses a methodology called WISE, which operationalizes
Polya’s heuristic of solving easier problems first and can help alleviate some
of the limitations stated above.
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Chapter 3
The WISE Methodology
3.1
Weaken-Identify-Solve-Extend
The WISE methodology operationalizes Polya’s heuristic of solving easier
problems first. The four steps involved are as follows
1. Weaken Analyze the given problem P and try to figure out its instances, constraints and objectives. Instances and constraints in the problem are
easy to identify by looking at the nouns phrases and verb phrases in the
problem description, respectively. For each instance, we select a representation and list their properties.[7] After identifying the instances,
constraints and objectives, we try to weaken either the instance or
the objective. We can weaken the instance by considering extremal
instances. The objective can be weakened by relaxing one or more
constraints.
2. Identify Choose a candidate problem P 0 which is a problem obtained by weakening P .
3. Solve Try to solve P 0 . If you cannot solve P 0 , weaken the problem further.
If you can solve P 0 , try to find as many solutions as possible.
4. Extend Use insights gained when P 0 was solved and try to solve P . If P
still cannot be solved, add a previously removed constraint to P 0 and
repeat the Weaken, Identify and Solve steps.
Figure 3.1 is a flowchart representing the WISE methodology.
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Figure 3.1: Flowchart representing the WISE methodology.
3.2
Extending WISE to other topics and problems
The WISE methodology has been used in [7] to solve problems related
to graph theory. We have applied WISE to other topics to investigate its
applicability to other domains and types of problems.
3.2.1
Common Math Puzzles
Example 3.2.1. 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?
Solution:
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We first try to weaken the instance for 5 light switches. At the first pass,
all the switches are ON. At the second pass, the 2nd and the 4th switches
are OFF. At the third pass, the 3rd switch is turned OFF. At the fourth
and fifth pass, the 4th is switched ON and the 5th switch is turned OFF
respectively. Hence, in the end the 4th light switch is turned ON, all the
others are OFF.
Can we gain certain insights from the weakened problem which will enable
us to solve the original problem?
We try to solve the problem by weakening the instance upto 10 numbers.
At the final pass, the 4th and 9th switches are ON. We notice that 4 and 9
and perfect squares, and try to come up with an explanation.
Each of the light switches changes its state on passes whose number is a
factor of the light switch’s number. For example, the 8th light will change
its state on the 1st , 2nd , 4th and 8th passes. Therefore, if the number of
factors are even, the switch will be OFF, otherwise the switch will be ON.
The number of factors are odd only for perfect squares. Hence the switches
will be ON for all perfect squares. Since there are 10 perfect squares between
1 and 100, 10 switches will be ON in the end.
3.2.2
Basic Permutations and Combinations
Example 3.2.2. How many words of length 8 can you form, where the first
letter is the same as the last letter?
Solution:
First weaken the instance to 2 letters and weaken the objective to any two
letters. A total of 262 words can be formed.
Now extend to 8 numbers with the above objective. A total of 268 words
can be formed.
We can now extend the objective. The first and the last letter can be chosen
in 26 ways, the remaining 6 letters in 266 ways.
Therefore, a total of 26 × 266 i.e 267 words can be formed.
3.2.3
Recursive Algorithms
A recursive algorithm is an algorithm which calls itself with ”smaller (or
simpler)” input values, and which obtains the result for the current input by
applying simple operations to the returned value for the smaller (or simpler)
input [1]. Consider the following example
Example 3.2.3. Write the recursive algorithm which will calculate the factorial of a given number
Solution:
Use WISE to weaken the instance to calculate the multiplication of 2 con-
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Problem Type
Permutations
and
Combinations
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?
How many words of length 8
can you form, where the first letter
is the same as the last letter?
Recursive
Algorithms
Write a recursive algorithm to
find the factorial of a given number
Math Puzzles
Insights
Good candidate problems
are those in which we can
weaken the instance
Good candidate problems to use
WISE since both instances and
objectives can be weakened
Good candidate problems
to use WISE since instances
can be weakened
Table 3.1: Insights gained from applying WISE
secutive numbers. The algorithm is as follows.
Data: Value of n
if n > 0 then
return n × n − 1
end
This insight will help in extending the solution for any given number.
The final algorithm is as follows:
Data: Value of n
if n == 1 then
return 1
end
return n × f actorial(n − 1)
3.3
Insights and Future Scope
Table 3.1 gives a summary of the insights gained from applying WISE to
problems of some topics. Certain type of problems like Permutations and
Combinations are ideal problems to apply WISE, since both objectives and
instances can be weakened. However, application of WISE to other classes
of problems is not straightforward.
Future scope of this exploration can involve teaching certain class of problems using the WISE methodology, and compare the effectiveness of WISE
with traditional methods of teaching the topic.
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Chapter 4
Deductive Reasoning
Reasoning is an integral and often unnoticed part of our lives. The ability
to make deductions is a central component of human thinking [10]. Special
training is not required by individuals to perform reasoning in their daily
tasks.
This chapter aims to address what is meant by deductive reasoning and
the mental process associated with it. Section 4.1 gives a brief introduction
and definition of deductive reasoning. Even though deductive reasoning
seems to occur so naturally, the underlying mental process of reasoning cannot be explained conclusively. Section 4.2 gives an account of two prominent
theories which explain how we reason. Finally, Section 4.3 outlines our proposed solution for teaching-learning of deductive reasoning.
4.1
4.1.1
Introduction to Deductive Reasoning
Definition and Examples
A simple example of reasoning is as follows I have to be present in office at 9.30 am.
It takes me half an hour to reach office.
Therefore, I have to leave at 9 am.
But it takes me an hour to reach office if I leave between 8am and 10am.
Therefore, I have to leave at 8.30am
[13] defines deductive reasoning as follows
Definition 4.1.1. “Deductive reasoning is the process of reasoning from
one or more statements (premises) to reach a logically certain conclusion ”
In the example, we see that the conclusion “Therefore, I have to leave
at 8:30am” can be logically deduced from the premises stated above. In the
process of deductive reasoning, the premises are assumed to be true.
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[5] cites three major domains of deduction
1. Relational inferences based on the logical properties of such 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?
2. Propositional inferences based on negation and on such connectives as if, or, and and. Example If the ink cartridge is empty then the printer wont work.
The ink cartridge is empty.
So, the printer wont work.
3. Syllogisms based on pairs of premises that each contain a single quantifier, such as all or some.
Example All artists are bakers.
Some bakers are chemists.
Therefore, some artists are chemists.
4.1.2 Why is it Important to Improve Deductive Reasoning
Deductive reasoning is an important skill needed in a variety of
contexts. The ability to reason well is essential in analyzing a problem
and deriving a solution to it. Reasoning well enables us to detect
fallacies and inconsistencies in arguments and ideas of others as well
as our own. Most of the aptitude exams for graduate education contain
sections which test logical and analytical reasoning.
4.2
Processes of Deductive Reasoning
Although reasoning is an essential skill and used ubiquitously, the process
of how the mind does deductive reasoning is not well understood even today.
This section outlines the two main schools of thought about the process of
deductive reasoning. Deduction is controversial, and there has been extensive debates between these schools. Some have concluded that the process
of deduction relies on a mixture of both these processes.[4]
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4.2.1
Deduction as a Formal Syntactic Process based on Rules
According to this theory, reasoners extract the logical forms of the premises
and use rules to derive conclusions. There are rules for sentential connectives such as “if” and “or”, and for quantifiers such as “all” and “some”.
Using the method of Natural Deduction, we can eliminate axioms or introduce sentential connectives by making assumptions or suppositions, until
we arrive at a conclusion. This theory was championed by many psychologists, such as Jean Piaget[3] who believe that the process of applying these
rules occur naturally and are embedded in the mind right from childhood.
[10] has implemented this theory as a computer program called PSYCOP.
Consider the following example of natural deduction
1. If the ink cartridge is empty the printer wont work. (Premise 1)
2. The printer is working (Premise 2)
3. Can we conclude that the ink cartridge is not empty?
4. The ink cartridge is empty (Supposition)
5. The printer wont work (Premise 3 - Modus ponens on Premise 1 and
Supposition)
6. Contradiction between Premise 2 and Premise 3
7. Therefore, our supposition is wrong. Hence the ink cartridge
is not empty
4.2.2
Deduction as a Semantic Process based on Mental Models
The theory of mental models accordingly postulates that reasoning is
based not on syntactic derivations from logical forms but on manipulations
of mental models representing situations.[6] Each model represents a possibility, and it’s structure and content represent different ways in which the
possibility might occur. Consider the following example “The ink cartridge is empty and the printer is not working”
Based on the mental model’s theory, a user constructs a model in their
brain, corresponding to the semantic meaning of the sentence. The mental
model of the above example is
∼p
i
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(4.1)
where i denotes the mental model of the statement, “The ink cartridge is
empty” and p denotes that “the printer is working” The ∼ symbol denotes
the negation of the premise. Thus mental models can contain abstract elements, such as negation, that cannot be visualized.[6]
The mental models of other sentential connectives are as follows
1. “The ink cartridge is empty or the printer is not working”
i
∼p
∼p
i
(4.2)
2. “If the ink cartridge is empty, then the printer is not working”
i
∼p
· · · (4.3)
3. “The ink cartridge is empty, if and only if the printer is not working”
i
∼p
(4.4)
The mental models of the conditional, conjunction and the biconditional
are the same in the figures above. This is due to what [6] calls as the “Principle of Truth” which states that “Individuals tend to minimise the load on
working memory by representing explicitly only what is true, and not what
is false.” In the mental models of the conditional and the biconditional,
models which represent the antecedant as true is only mentioned, hence the
similarity in the models of conditionals and biconditionals. This incomplete
information represented in the mental model accounts for difficulty in accounting for the validity of certain proofs as the one which we had seen
earlier.
1. If the ink cartridge is empty the printer wont work. (Premise 1)
2. The printer is working (Premise 2)
3. Can we conclude that the ink cartridge is not empty?
The mental model of “if” does not have a model which represents the condition where the printer is working(p) and the ink cartridge is not empty.(∼
i). Hence arriving at a conclusion in such cases is more difficult than other
cases. For example, conjunctions are easier than conditionals, which in turn
are easier than disjunctions. Likewise, exclusive disjunctions (two mental
models) are easier than inclusive disjunctions (three mental models)[6]
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In fully explicit models, false affirmatives are represented by true negations, and false negatives are represented by true affirmatives. [6]. For example, the corresponding fully explicit model of the conditional is as follows -
i
∼p
∼i
∼p
∼i
p
(4.5)
Based on experiments conducted in [6] the following conclusions can be
drawn
1. Fallacies result due to construction of mental models and not fully
explicit mental models.
2. Greater the number of models, greater is the difficulty in performing
deductions.
These insights from cognitive psychology theory can prove helpful when
we want to design learning interventions for teaching deductive reasoning.
Sufficient experiments confirming the mental model theory gives us confidence to use these conclusions for our interventions in the future.
4.3
Proposed Solution
The mental model theory states that reasoning is based on manipulations of mental models representing situations. These mental models are
constructed in the brain during reasoning. Our hypothesis is that explicit
construction of such models using a technology enhanced learning(TEL) environment will improve deductive reasoning skills. Our aim is to provide a
TEL environment which will allow learners to manipulate explicit models
while reasoning to arrive at a conclusion.
The TEL environment which we have chosen is Scratch. Scratch is a
programming language and an online community where children can program and share interactive media such as stories, games, and animation with
people from all over the world. As children create with Scratch, they learn
to think creatively, work collaboratively, and reason systematically. Scratch
is designed and maintained by the Lifelong Kindergarten group at the MIT
Media Lab. [2]
The advantage of using Scratch over other conventional programming
languages is that it allows us to create objects and models quickly and
easily. Learners can explicitly create and manipulate mental models using
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the Scratch programming language. The program can be executed and
learners can check if their reasoning leads them to the desired conclusion.
Hence it can provide a mental trace of the reasoning process.
We intend to provide this intervention in two stages.
1. Stage 1 - A set of premises are displayed to the user in Scratch, along
with explicit models of these premises. A set of conclusions are also
provided to the user. The user has to decide the right conclusion which
follows from these premises. Based on the response of the user, the
model changes and the user receives prompts and hints to arrive at
the solution.
2. Stage 2 - A set of premises are displayed to the user in Scratch. The
user has to construct models of the premises by programming the
model in Scratch. The conclusion is derived by writing a program in
Scratch and observing the output.
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Chapter 5
Future Directions
Two topics have been explored in this seminar - “Heuristics in Mathematical Problem Solving” and “Deductive Reasoning”. In the future, I plan
to work on the latter topic. Based on feedback from the presentation, I
plan to do a more extensive literature survey of mental models, especially
its use in other areas like science inquiry learning. I also intend to do a thorough survey of other teaching-learning interventions which teach deductive
reasoning.
I also intend to finalize on the domain and topic through which I will teach
deductive reasoning. Characteristics of the learner also has to be identified,
such as age of the learner etc. As of now, I am thinking of high school
students who are learning the basics of logic.
The use of Scratch as the technology intervention has to be explored
further. I intend to explore features of Scratch which I can use to teach
deductive reasoning. As a first step, I intend to code certain examples in
Scratch, conduct a pilot experiment and do certain preliminary evaluations.
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