Heuristic Strategies and Deductive Reasoning inProblem Solving

Seminar Report

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

by

Prajish PrasadRoll No : 154380001

under the guidance of

Prof. Sridhar Iyer

Inter-disciplinary Program in Educational TechnologyIndian Institute of Technology, Bombay

November 2015

Contents

1 Introduction to Mathematical Problem Solving 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Importance of Teaching-Learning of Problem Solving . . . . . 31.3 Organisation of Report . . . . . . . . . . . . . . . . . . . . . . 3

2 Use of Heuristics in Mathematical Problem Solving 42.1 Introduction to Heuristics . . . . . . . . . . . . . . . . . . . . 42.2 Teaching-Learning using Heuristics . . . . . . . . . . . . . . . 42.3 Limitations of Heuristics . . . . . . . . . . . . . . . . . . . . . 5

3 The WISE Methodology 73.1 Weaken-Identify-Solve-Extend . . . . . . . . . . . . . . . . . . 73.2 Extending WISE to other topics and problems . . . . . . . . 8

3.2.1 Common Math Puzzles . . . . . . . . . . . . . . . . . 83.2.2 Basic Permutations and Combinations . . . . . . . . . 93.2.3 Recursive Algorithms . . . . . . . . . . . . . . . . . . 9

3.3 Insights and Future Scope . . . . . . . . . . . . . . . . . . . . 10

4 Deductive Reasoning 114.1 Introduction to Deductive Reasoning . . . . . . . . . . . . . . 11

4.1.1 Definition and Examples . . . . . . . . . . . . . . . . . 114.1.2 Why is it Important to Improve Deductive Reasoning 12

4.2 Processes of Deductive Reasoning . . . . . . . . . . . . . . . . 124.2.1 Deduction as a Formal Syntactic Process based on Rules 134.2.2 Deduction as a Semantic Process based on Mental

Models . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Future Directions 17

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Chapter 1

Introduction toMathematical ProblemSolving

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 requir-ing 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 aparticular method to solve a problem. Learners can learn this method bysolving practice problems of the given topic, handed down to them by ex-perts, which they have to memorize. They eventually master the methodand can apply it to other problems.

The second definition views problem solving as an art, which requires acertain amount of creativity from the students and application of variousmethods in order to arrive at the solution. The main proponent of thisdefinition of problem solving was George Polya. He states that mathematicsinvolves guessing, intuition and discovery similar to the physical sciences.[8]

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1.2 Importance of Teaching-Learning of ProblemSolving

Over the years, there has been a change in how mathematics and prob-lem solving is perceived. Educators realise that for mathematics educationto fulfill its objectives, there has to be a shift from the first definition tothe 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 finalanswer. Students should be encouraged to explore patterns, and not justmemorize formulas. They should be encouraged to formulate conjectures,not just do exercises.

Schoenfeld reasons that this perspective of learning mathematics is em-powering. Mathematically powerful students are quantitatively literate. Theyare capable of interpreting the vast amounts of quantitative data they en-counter on a daily basis, and of making balanced judgments on the basis ofthose interpretations. They use mathematics in practical ways, from simpleapplications 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 perspec-tives for learning to think mathematically, dealing with novel problems andsituations. They are analytical, both in thinking issues through themselvesand in examining the arguments put forth by others.[12]

1.3 Organisation of Report

In this seminar report, two topics are explored, “Heuristics in Mathemat-ical Problem Solving” and “Deductive Reasoning”. Chapter 2 details theuse of heuristics in the process of problem solving and limitations of us-ing heuristics. Chapter 3 gives details of a methodology called “WISE”[7],which is a specific example of a heuristic operationalized for a variety of top-ics. Chapter 4 gives a brief introduction of deductive reasoning and theoriesfrom cognitive psychology which explain how we reason. We have outlinedour 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 inMathematical ProblemSolving

2.1 Introduction to Heuristics

As stated in the previous chapter, mathematical problem solving involvesguessing, intuition and discovery similar to the physical sciences. Heuris-tics aid in this process of guessing and intuition. According to Wikipedia,“Heuristic is any approach to problem solving, learning, or discovery thatemploys a practical method not guaranteed to be optimal or perfect, butsufficient for the immediate goals.”

A comprehensive set of heuristics were first compiled and presented byGeorge Polya in his book “How to Solve it”[9] An example of a heuristicis the analogous problem heuristic, which states “To solve a complicatedproblem, 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 prob-lem solving. However, the question arises - “Does teaching heuristic strate-gies 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 whichfive heuristic strategies were taught. Each student worked on 20 problems,then saw the solutions. They were given a list and explanation of the fivestrategies used in the experiment and an “overlay” to each solution explain-ing 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. Theleft-hand side was seen only by the “heuristics” students.

Evaluation was done using post test. Students who were explicitly taughtheuristic strategies outscored the other group with a significent differencein pretest to post test gains. Moreover, transcripts of the solutions showthat explicit use of the strategies accounted for differences between the twogroups.

2.3 Limitations of Heuristics

Although the experiment stated above shows postive results, Schoenfeldis not quite optimistic. He states the following -

“But even if we succeed in teaching students to use a series of importantheuristic strategies, I see no guarantee that there will be clear signs of im-provement in their general problem solving. Knowing how to use a strategyisn’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 ofthe keys can unlock the problem. However, deciding which to use for aparticular problem is difficult. Polya’s book “How to Solve it”[9] has around40 heuristics.

Even after one decides a particular heuristic strategy, the descriptive na-ture of the strategy makes it hard to directly apply it to the problem. Forexample, the analogous problem strategy states -

“To solve a complicated problem, it often helps to examine and solve asimpler 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 ”analogousproblem” 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 solutionor the method.

The next chapter uses a methodology called WISE, which operationalizesPolya’s heuristic of solving easier problems first and can help alleviate someof 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 easierproblems first. The four steps involved are as follows

1. Weaken -Analyze the given problem P and try to figure out its instances, con-straints and objectives. Instances and constraints in the problem areeasy to identify by looking at the nouns phrases and verb phrases in theproblem description, respectively. For each instance, we select a rep-resentation and list their properties.[7] After identifying the instances,constraints and objectives, we try to weaken either the instance orthe objective. We can weaken the instance by considering extremalinstances. The objective can be weakened by relaxing one or moreconstraints.

2. Identify -Choose a candidate problem P ′ which is a problem obtained by weak-ening P .

3. Solve -Try to solve P ′. If you cannot solve P ′, weaken the problem further.If you can solve P ′, try to find as many solutions as possible.

4. Extend -Use insights gained when P ′ was solved and try to solve P . If Pstill cannot be solved, add a previously removed constraint to P ′ andrepeat 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 relatedto graph theory. We have applied WISE to other topics to investigate itsapplicability 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 turningevery other one off. Then, you walk by them changing every third one. Onyour 4th pass, you change every 4th one. You repeat this for 100 passes. Atthe 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 switchesare OFF. At the third pass, the 3rd switch is turned OFF. At the fourthand fifth pass, the 4th is switched ON and the 5th switch is turned OFFrespectively. Hence, in the end the 4th light switch is turned ON, all theothers are OFF.Can we gain certain insights from the weakened problem which will enableus 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 9and perfect squares, and try to come up with an explanation.Each of the light switches changes its state on passes whose number is afactor of the light switch’s number. For example, the 8th light will changeits state on the 1st, 2nd, 4th and 8th passes. Therefore, if the number offactors 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 switcheswill be ON for all perfect squares. Since there are 10 perfect squares between1 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 firstletter is the same as the last letter?

Solution:First weaken the instance to 2 letters and weaken the objective to any twoletters. A total of 262 words can be formed.Now extend to 8 numbers with the above objective. A total of 268 wordscan be formed.We can now extend the objective. The first and the last letter can be chosenin 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 (orsimpler)” input values, and which obtains the result for the current input byapplying 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 fac-torial of a given number

Solution:Use WISE to weaken the instance to calculate the multiplication of 2 con-

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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 themturning every other one off.Then, you walk by themchanging 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 problemsare those in which we canweaken the instance

PermutationsandCombinations

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

Good candidate problems to useWISE since both instances andobjectives can be weakened

RecursiveAlgorithms

Write a recursive algorithm tofind the factorial of a given number

Good candidate problemsto use WISE since instancescan be weakened

Table 3.1: Insights gained from applying WISE

secutive numbers. The algorithm is as follows.

Data: Value of nif n > 0 then

return n× n− 1end

This insight will help in extending the solution for any given number.The final algorithm is as follows:

Data: Value of nif n == 1 then

return 1endreturn n× factorial(n− 1)

3.3 Insights and Future Scope

Table 3.1 gives a summary of the insights gained from applying WISE toproblems of some topics. Certain type of problems like Permutations andCombinations are ideal problems to apply WISE, since both objectives andinstances can be weakened. However, application of WISE to other classesof problems is not straightforward.

Future scope of this exploration can involve teaching certain class of prob-lems using the WISE methodology, and compare the effectiveness of WISEwith 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 abilityto make deductions is a central component of human thinking [10]. Specialtraining is not required by individuals to perform reasoning in their dailytasks.

This chapter aims to address what is meant by deductive reasoning andthe mental process associated with it. Section 4.1 gives a brief introductionand definition of deductive reasoning. Even though deductive reasoningseems to occur so naturally, the underlying mental process of reasoning can-not be explained conclusively. Section 4.2 gives an account of two prominenttheories which explain how we reason. Finally, Section 4.3 outlines our pro-posed solution for teaching-learning of deductive reasoning.

4.1 Introduction to Deductive Reasoning

4.1.1 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 fromone or more statements (premises) to reach a logically certain conclusion ”

In the example, we see that the conclusion “Therefore, I have to leaveat 8:30am” can be logically deduced from the premises stated above. In theprocess 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 rela-tions 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 connec-tives 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 quan-tifier, 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 Rea-soning

Deductive reasoning is an important skill needed in a variety ofcontexts. The ability to reason well is essential in analyzing a problemand deriving a solution to it. Reasoning well enables us to detectfallacies and inconsistencies in arguments and ideas of others as wellas our own. Most of the aptitude exams for graduate education containsections which test logical and analytical reasoning.

4.2 Processes of Deductive Reasoning

Although reasoning is an essential skill and used ubiquitously, the processof how the mind does deductive reasoning is not well understood even today.This section outlines the two main schools of thought about the process ofdeductive reasoning. Deduction is controversial, and there has been exten-sive debates between these schools. Some have concluded that the processof 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 premisesand use rules to derive conclusions. There are rules for sentential connec-tives such as “if” and “or”, and for quantifiers such as “all” and “some”.Using the method of Natural Deduction, we can eliminate axioms or in-troduce sentential connectives by making assumptions or suppositions, untilwe arrive at a conclusion. This theory was championed by many psycholo-gists, such as Jean Piaget[3] who believe that the process of applying theserules 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 andSupposition)

6. Contradiction between Premise 2 and Premise 3

7. Therefore, our supposition is wrong. Hence the ink cartridgeis not empty

4.2.2 Deduction as a Semantic Process based on Mental Mod-els

The theory of mental models accordingly postulates that reasoning isbased not on syntactic derivations from logical forms but on manipulationsof mental models representing situations.[6] Each model represents a possi-bility, and it’s structure and content represent different ways in which thepossibility 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 theirbrain, corresponding to the semantic meaning of the sentence. The mentalmodel of the above example is

i ∼ p (4.1)

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where i denotes the mental model of the statement, “The ink cartridge isempty” and p denotes that “the printer is working” The ∼ symbol denotesthe negation of the premise. Thus mental models can contain abstract ele-ments, 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

i ∼ p

(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 biconditionalare the same in the figures above. This is due to what [6] calls as the “Prin-ciple of Truth” which states that “Individuals tend to minimise the load onworking memory by representing explicitly only what is true, and not whatis false.” In the mental models of the conditional and the biconditional,models which represent the antecedant as true is only mentioned, hence thesimilarity in the models of conditionals and biconditionals. This incompleteinformation represented in the mental model accounts for difficulty in ac-counting for the validity of certain proofs as the one which we had seenearlier.

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 con-dition 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 othercases. For example, conjunctions are easier than conditionals, which in turnare easier than disjunctions. Likewise, exclusive disjunctions (two mentalmodels) are easier than inclusive disjunctions (three mental models)[6]

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In fully explicit models, false affirmatives are represented by true nega-tions, and false negatives are represented by true affirmatives. [6]. For ex-ample, 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 bedrawn

1. Fallacies result due to construction of mental models and not fullyexplicit mental models.

2. Greater the number of models, greater is the difficulty in performingdeductions.

These insights from cognitive psychology theory can prove helpful whenwe want to design learning interventions for teaching deductive reasoning.Sufficient experiments confirming the mental model theory gives us confi-dence to use these conclusions for our interventions in the future.

4.3 Proposed Solution

The mental model theory states that reasoning is based on manipula-tions of mental models representing situations. These mental models areconstructed in the brain during reasoning. Our hypothesis is that explicitconstruction of such models using a technology enhanced learning(TEL) en-vironment will improve deductive reasoning skills. Our aim is to provide aTEL environment which will allow learners to manipulate explicit modelswhile reasoning to arrive at a conclusion.

The TEL environment which we have chosen is Scratch. Scratch is aprogramming language and an online community where children can pro-gram and share interactive media such as stories, games, and animation withpeople from all over the world. As children create with Scratch, they learnto think creatively, work collaboratively, and reason systematically. Scratchis designed and maintained by the Lifelong Kindergarten group at the MITMedia Lab. [2]

The advantage of using Scratch over other conventional programminglanguages is that it allows us to create objects and models quickly andeasily. Learners can explicitly create and manipulate mental models using

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the Scratch programming language. The program can be executed andlearners 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, alongwith explicit models of these premises. A set of conclusions are alsoprovided to the user. The user has to decide the right conclusion whichfollows from these premises. Based on the response of the user, themodel changes and the user receives prompts and hints to arrive atthe solution.

2. Stage 2 - A set of premises are displayed to the user in Scratch. Theuser has to construct models of the premises by programming themodel in Scratch. The conclusion is derived by writing a program inScratch and observing the output.

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Chapter 5

Future Directions

Two topics have been explored in this seminar - “Heuristics in Mathe-matical Problem Solving” and “Deductive Reasoning”. In the future, I planto work on the latter topic. Based on feedback from the presentation, Iplan to do a more extensive literature survey of mental models, especiallyits use in other areas like science inquiry learning. I also intend to do a thor-ough survey of other teaching-learning interventions which teach deductivereasoning.

I also intend to finalize on the domain and topic through which I will teachdeductive 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 schoolstudents who are learning the basics of logic.

The use of Scratch as the technology intervention has to be exploredfurther. I intend to explore features of Scratch which I can use to teachdeductive reasoning. As a first step, I intend to code certain examples inScratch, conduct a pilot experiment and do certain preliminary evaluations.

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Bibliography

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cs381content/recursive_alg/rec_alg.html. [Online; accessed30-Nov-2015].

[2] Scratch - Imagine, Program, Share. https://scratch.mit.edu/. [On-line; accessed 30-Nov-2015].

[3] Evert Willem Beth and Jean Piaget. Mathematical epistemology andpsychology, volume 12. Springer Science & Business Media, 2013.

[4] Rachel Joffe Falmagne and Joanna Gonsalves. Deductive inference.Annual review of psychology, 46(1):525–559, 1995.

[5] Philip N Johnson-Laird. Mental models, deductive reasoning, and thebrain. The cognitive neurosciences, pages 999–1008, 1995.

[6] Philip N Johnson-Laird. Deductive reasoning. Annual review of psy-chology, 50(1):109–135, 1999.

[7] Jagadish M. A Problem-Solving Methodology Based on ExtremalityPrinciple and its Application to CS Education. PhD thesis, IIT Bom-bay, 2015.

[8] George Polya. Patterns of Plausible Inference: Volume II of Mathe-matics and Plausible Reasoning. Princeton University Press, 1994.

[9] George Polya. How to Solve It:A New Aspect of Mathematical Method.Princeton University Press, 2014.

[10] Lance J Rips. The psychology of proof, 1994.

[11] A. H. Schoenfeld. Teaching problem-solving skills. The American Math-ematical Monthly, 87:794–805, 1980.

[12] A. H. Schoenfeld. Learning to think mathematically: Problem solv-ing, metacognition, and sense-making in mathematics. In D. Grouws(Ed.), Handbook for Research on Mathematics Teaching and LearningMacMillan, 1992.

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[13] Robert J. Sternberg. Handbook of Human Intelligence. CambridgeUniversity Press, 1982.

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