Introduction to Problem Solving in the Information Age




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Introduction to Problem Solving in the Information Age

Introduction to Problem Solving in the Information Age

David Moursund

College of Education, University of Oregon

Email: moursund@uoregon.edu

Website: http://uoregon.edu/~moursund/dave/index.htm

Website: http://i-a-e.org/

Brief Abstract of Book

“Never doubt that a small group of thoughtful committed citizens can change the world: indeed; it's the only thing that ever has.” (Margaret Mead)

In this book, the term problem solving includes posing and solving problems, posing and accomplishing tasks, posing and answering questions, and posing and making decisions. Problem solving is an integral component of every academic discipline.

The Information Age officially began in the United States in 1956 when the number of “white collar” workers first exceeded the number of “blue collar” workers.

The primary audience of this book is preservice and inservice teachers, and others who help students to learn. The goal of this book is to help K-12 students get better at problem solving in the various disciplines they study.

We get better at problem solving through informal and formal education, and through reflective practice. We also get better at problem solving through learning to make effective use of tools. In some sense, a tool incorporates the problem-solving insights of the inventor of the tool. When you learn to make effective use of a tool, you are building upon the knowledge and skills inherent to the tool.

This is true whether the tool is designed to aid your physical capabilities or your mental capabilities. A bicycle, motorbike, and a piece of factory machinery are aids to one’s physical capabilities. Reading, writing, and a library are aids to one’s mental capabilities.

This book explores how Information and Communication Technology (including calculators and computers) aid both your physical capabilities and your mental capabilities. By learning to make effective use of ICT tools, you can improve your problem-solving capabilities.

Copyright and Right to Copy

This book is Copyright © David Moursund 2007. However, it can be accessed free on the Web in both PDF and Microsoft Word formats and given to students and others for non-commercial purposes. This is done under the Creative Commons Attribution-NonCommercial 3.0 License. More detail is available at http://creativecommons.org/licenses/by-nc/3.0/.

About Dave Moursund, the Author

“The wisest mind has something yet to learn.” (George Santayana)

• Bachelor’s degree in mathematics, University of Oregon.

• Master’s degree and doctorate in mathematics (specializing in numerical analysis) from the University of Wisconsin-Madison.

• Instructor, Department of Mathematics, University of Wisconsin-Madison.

• Assistant Professor and then Associate Professor, Department of Mathematics and Computing Center (School of Engineering), Michigan State University.

• Associate Professor, Department of Mathematics and Computing Center, University of Oregon.

• Associate and then Full Professor, Department of Computer Science, University of Oregon.

• Served six years as the first Head of the Computer Science Department at the University of Oregon, 1969-1975.

• Full Professor in the College of Education at the University of Oregon for more than 20 years. Currently retired with rank Emeritus Professor.

• In 1974, started the publication that eventually became Learning and Leading with Technology, the flagship publication of the International Society for Technology in Education (ISTE).

• In 1979, founded the International Society for Technology in Education ). Headed this organization for 19 years.

• Author or co-author of about 50 books and several hundred articles in the field of computers in education.

• Presented about 200 workshops in the field of computers in education.

• Served as a major professor for about 50 doctoral students (six in math, the rest in education). Served on the doctoral committees of about 25 other students.

• Founding member of the Math Learning Center. Served on the MLC Board of Directors since its inception in 1976, and chaired the board for several years.

• For more information about Dave Moursund and for free online, no cost access to 20 of his books and a number of articles, go to http://uoregon.edu/~moursund/dave/.

Contents

"The strongest memory is not as strong as the weakest ink." (Confucius, 551-479 B.C.)

Dave 2

Contents 3

Preface 6

Chapter 1: Introduction and Overview 8

Improving Education 8

Academic Disciplines 9

Subconscious Thinking and Problem Solving 10

Our Formal Educational System 11

What is Problem Solving? 12

Critical Thinking 12

Higher-Order Thinking 12

Some Important Aspects of Problem Solving 13

Chapter 2: What is a Formal Problem? 17

Definition of a Formal Problem 17

More Detail About the Four-Component Definition 18

Dealing With Problem Situations 19

Immediate Actions for Chapter 2 21

Chapter 3: Information and Problem Overload 23

Problem Overload 23

Who or What Should Do the Solving of a Problem? 25

A Story About Building on Work of Others 25

Should We Drop Some Content from the School Curriculum? 26

Summary 27

Chapter 4: Problem and Task Team 30

Introduction 30

Project-Based Learning 32

Problem-Based Learning 33

Problem Solving as a Process 33

Chapter 5: Transfer of Learning 36

We All Do Transfer of Our Learning 36

Transfer of Learning Theories 37

High-road and Low-Road Transfer of Learning 38

Final Remarks 40

Chapter 6: Expertise and Domain Specificity 42

Expertise and being an Expertise 42

Domain-Specific Knowledge in a Productivity Tool 43

Chapter 7: Some Problem-Solving Strategies 46

Introduction 46

Information Retrieval as a General-Purpose Strategy 47

Other General-Purpose Strategies 47

Still More Strategies 53

Chapter 8: Representations of a Problem 55

Introduction 55

Non-Mental Models 55

Special Vocabulary and Notation in Various Disciplines 58

Chapter 9: Representing & Solving Problems Using Computers 60

Computational Thinking 60

Computer Models 61

Spreadsheet Models 62

Some Applications of Computer Models 62

Artificial Intelligence (Machine Intelligence) 63

Highly Interactive intelligent CAL 64

Virtual Reality 65

Chapter 10: Computer Programming 68

Procedural Thinking 68

Types of Computer Programming 69

Using a General-purpose Programming Language 70

Testing and Debugging 71

Some History of Programming Languages 72

Data Structures and Control Structures 73

Object Oriented Programming 74

Transfer of Problem-Solving Skills From Programming 75

Final Remarks 75

Chapter 11: Summary of Important Ideas 77

Some Key Ideas 77

Appendix A: Brain/Mind Science and Problem Solving 79

Introduction 79

Brain/Mind Science 81

Gaining a High Level of Expertise 85

Nature and Nurture 86

Piagetís Developmental Theory 89

Augmentation of Brain/Mind 91

Concluding Remarks 92

References 94

Index 97



Preface

"What is written without effort is in general read without pleasure." (Samuel Johnson)

"In short, learning is the process by which novices become experts. " (John T. Bruer)

This short book provides an overview of a large and complex field—problem solving and roles of Information and Communication Technology (ICT) in problem solving. Quoting from the Wikipedia:

Problem solving forms part of thinking. Considered the most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills (Goldstein & Levin, 1987). It occurs if an organism or an artificial intelligence system does not know how to proceed from a given state to a desired goal state. It is part of the larger problem process that includes problem finding and problem shaping.

Reading, writing, and arithmetic are powerful aids to problem solving. These aids have been available for more than 5,000 years. They are so important that they are considered the basics of education.

We now have computer technology that builds upon and extends uses of reading, writing, and arithmetic. ICT tools have become routine aids to problem solving in every discipline. However, our educational system has been slow to thoroughly integrate ICT aids to problem solving into the everyday curriculum. While many people argue that ICT aids to problem solving should now be part of the basics (as are reading, writing, and arithmetic), we are a long way from achieving this in our schools.

The book has three main audiences:

1. It is intended for use in computer in education courses. Many computer in education courses focus mainly on the computer tools, rather than the deeper idea of computers as an aid to solving problems and accomplishing tasks that are part of “doing” or using a discipline. Use of this book in such courses will help redress this imbalance.

2. It is intended for use in non-ICT courses for preservice and inservice teachers. Thus, for example a person taking a Social Science, Science, or Math Methods course would benefit by reading it. Solving problems, accomplishing tasks, answering challenging questions, and making difficult decisions lie at the heart of every discipline. Thus, every discipline teaches problem solving, and every discipline benefits by having ICT available as an aid to representing and solving problems.

3. It is intended for use in workshops for inservice teachers, school administrators, and teachers’ aides. Here, the intent is to improve education by helping educators understanding the steadily increasing power of ICT to empower students in posing, representing, and solving complex problems in each discipline they are studying.

The single most important idea in problem solving is building on the previous work and learning done by ones self and others. ICT bring some new dimensions to these endeavors.

For example, we now have the Web, the world’s largest library. It substantially helps in providing access to the previous work of others. Moreover, personal computers and connectivity allow a person to build their own virtual library, putting into it content that they find particularly useful in solving problems and accomplishing tasks that they encounter in their everyday lives.

We now have computers that can solve or help solve a wide variety of problems. In essence, we are automating many mental tasks that used to be done using simpler tools, such as pencil, paper, and calculators. More and more problems can be solved by just appropriately presenting the problem to a computer.

Computers have given us a new way of thinking about problems, called computational thinking. Computational thinking involves using one’s own thinking abilities, those of other people, and those of computer systems. Many complex problems are now being attacked by developing computer models and computer simulations of the problem situation. The world’s most powerful computers are being used to model weather and global climate (including global warming) and various aspects of proteins and genes in the study of genetics.

This book is divided into a number of short chapters. Each contains some suggestions for immediate actions (implementations) for teachers. Each chapter ends with a set of activities that are suitable for self-study, use in a workshop, or use in a course’s class meeting, and use in homework assignments.


Chapter 1

Introduction and Overview

"Learning without thinking is labor lost; thinking without learning is dangerous." (Chinese Proverb)

"Once you have learned how to ask relevant and appropriate questions, you have learned how to learn and no one can keep you from learning whatever you want or need to know." Neil Postman and Charles Weingartner. Teaching as a Subversive Activity.

This chapter contains some general background that lays foundations for the rest of the book. Keep in mind that this is a scholarly, “academic” book. This means that you need to read it while you are mentally alert and your mind is in gear. It means that after you read a paragraph, you should stop and reflect about what the paragraph means to you. Learning comes from the concentrated, alert mind reading process and from careful reflection about what you are reading. Learning comes from using what you are learning and from seeking possible uses of what you are learning.

Improving Education

This book focuses on improving education by helping preservice and inservice teachers, and their students, to get better at problem solving. Problem solving is certainly core to a good education. Thus, better curriculum, instruction, and assessment in this area can lead to improvements in our overall educational system.

However, there are some still larger issues than can be addressed our educational system. From time to time over the years, I have attempted to make a short list of Big Ideas that help me as I work to improve our educational system. Here is my current list:

1. Empower and enable the learner and those who directly help learners to learn.

2. Help students learn to sell-assess and to take a steadily increasing level of responsibility for their own learning.

3. Help students get better at asking researchable question and in learning to do the various types of research that are used to answer such questions. The Web plays a significant role in this endeavor, since a literature search is an important component of trying to answer a researchable question.

4. Help all people to get better at being both teachers of themselves and others, and learners. (We are all lifelong teachers and learners.)

5. All technology (not just computers) plays a role in the above list. Any particular technology is potentially more empowering and enabling for some than for others. Indeed, any particular technology has the potential to be disempowering and disenabling for some people.

Please work with me as I briefly explore the first item —about empowering and enabling—on the list from a combination of your point of view and from our educational system’s point of view. You, a learner, already know a great deal. You have good insights into what your want to learn, and why. You want your new learning to empower and enable you as you work toward achieving goals that you set for yourself.

Contrast this self-centered point of view with that of our educational system, your course instructors, book authors, and so on. The “establishment” feels that it know what is best for you. It develops requirements, courses, tests, and so on—hoops that you must jump through to achieve a degree, certification, a job, and so on. At the precollege level, these hoops include high stakes tests at the regional, state, and national levels. This assessment system has grown so powerful that it is shaping the lives of students and their teachers.

You, and all other students, are thus faced with the problem of dealing with these two different approaches to education. Some students find it easy to give in to the establishment. They develop an attitude summarized by the statement: “Just tell me what to do and I will do it.” Indeed, there is some evidence that this is a growing trend in our educational system. It is accompanied by increased efforts on the part of teachers to carefully specify what it is a student is to do and learn, and increased efforts on the part of students to challenge the teachers who do not provide detailed specifications of what they want students to learn and how they will assess this learning.

My personal opinion is that this is a poor approach to improving education. I much prefer to see an educational system that is more strongly student centered, that empowers and enables students from their personal points of view. I much prefer an educational system in which teachers are empowered and enabled in a quest to empower and enable individual students, working to achieve an appropriate balance between the needs and wants of students, and the needs and wants of the establishment.

Most readers of this book are preservice or inservice teachers and others involved in our overall educational system. I hope that each of you will think about the how you can empower and enable yourself through your studies, and how you can do the same thing for your current and future students.

Academic Disciplines

This book is intended for students who are studying many different disciplines. You might wonder why we have so many different academic disciplines. Indeed, you may wonder what distinguishes one discipline from another, or the extent to which the various disciplines one can study in precollege and higher education overlap each other.

Each academic discipline can be defined by a combination of:

• The types of problems, tasks, and activities it addresses.

• Its accumulated accomplishments, such as its results, achievements, products, performances, scope, power, uses, impacts on the societies of the world, and so on.

• Its history, culture, methods of communication, and language (including notation and special vocabulary).

• Its methods of teaching, learning, assessment, and thinking, and what it does to preserve and sustain its work and pass it on to future generations.

• Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.

• The knowledge and skills that separate and distinguish among (a) a novice, (b) a person who has a personally useful level of competence, (c) a reasonably competent person, (d) an expert, and (e) a world-class expert. Each discipline has its own ideas as to what constitutes a high level of expertise within the discipline and its subdisciplines.

Notice that this list emphasizes solving problems and accomplishing tasks. It emphasizes gaining an increasing level of expertise in “doing or accomplishing.”

One of the major issues in getting better at doing or accomplishing (solving problems, accomplishing tasks, etc.) is discipline-specificity versus discipline-independence. Does gaining an increased level of expertise in playing chess automatically make you better at solving math problems, or vice versa? Does learning how to solve chemistry problems make you better at solving the types of problems faced by a performing artist, or vice versa?

Clearly some cognitive tools cut across many disciplines. Reading and writing are useful in every academic discipline. Math is an old, broad, deep discipline in its own right, but also a component of many other disciplines. Math provides a language and a way of thinking that is useful in many other disciplines.

Information and Communication Technology is somewhat analogous to math in that it is both a broad and deep discipline in its own right, and it has become an important component of many other disciplines.

Thus, as you work to improve your knowledge, skills, and “doing” abilities in almost any discipline, you will find that you are continually drawing on your current levels of expertise in disciplines such as reading and writing, math, and ICT.

Subconscious Thinking and Problem Solving

Your brain and body are highly skilled in solving the problems of keeping you alive. Most of this goes on at a subconscious level. You do not have to consciously think about telling your heart to beat regularly, your lungs to keep breathing regularly and to oxygenate blood, or your digestive system to digest the food that you eat. You do not have to consciously tell your immune system do deal with infections.

You are also highly skilled at solving problems at a conscious level. When you carry on a conversation or read a book, you are solving complex communications problems. Interestingly, much of what you are doing as you carry on a conversation or read a book occurs at a subconscious level. For example, you think a thought and your brain somehow puts together the sequence of muscle, vocal cord, and other physical activities needed to deliver sentences about the idea. You look at small squiggly drawings on a page (that is, at letters, words, and sentences) and your brain turns these into silent sounds, pictures, and meaning in your head.

The conscious and subconscious processes of talking, listening, reading, and writing give us considerable insight into teaching and learning for problem, solving. You learned speaking and listening from our informal education system. You have an innate ability to learn oral communication—it is built into your genes. (Of course, some people’s genes don’t work correctly or physically injuries damage parts of their body that are necessary to oral speaking and listening.)

You probably learned reading and writing from our formal educational system. Reading and writing were developed just a little over 5,000 years ago. Learning to read and write is a major cognitive challenge for most people. It takes a relatively long period of instruction, learning, and practice to become skilled at reading and writing. Indeed, I find it somewhat surprising that with appropriate formal education, most people can learn to read and write relatively well.

Much of the initial learning of reading and writing is done at a conscious level. However, eventually much of what you do in reading and writing is carried out at a subconscious level. It is through considerable practice that the initial reading and writing learning becomes automatic at a subconscious level. That is, formal education and practice produce important results at both a conscious and subconscious level.

Your brain has considerable plasticity. When you are learning through our informal and formal education systems, your brain is changing. Your brain is forming new neural connection, strengthening some connections, and weakening others. If part of your brain is damaged, the overall brain plasticity helps to repurpose other parts of your brain to carry out tasks previously done by the damaged parts.

Your brain has a considerable ability to learn. Learning and practicing what you have learned are natural and ongoing activities within your brain. That is, we are all life-long learners.

Our Formal Educational System

Our PreK-12 and higher formal education systems are designed to develop the capacity of your brain to deal with the problems that our society (that is, the establishment) feels a person might encounter in adulthood. As you progressed along this formal education trail, you gradually took more responsibility for yourself in deciding what courses to take and what general academic areas to pursue. You developed your knowledge and skills in knowing how to learn. You gradually gained increased expertise in being an independent, self-sufficient learner in the types of areas covered by formal education and other areas that interested you. You got better at solving the types of problems and accomplishing the types of tasks that you encountered at home, work, school, play, and in other parts of your everyday life.

It may feel strange to you to think about life from the point of view of getting better at solving problems and accomplishing tasks. However, that is one (useful) way to think about our informal and formal education systems. Thus, if you are going to spend your life increasing your capacity as a problem solver, likely you will find it worthwhile to gain efficiency in this endeavor. If you are a preservice or inservice teacher, then certainly you want to get better at solving the problem (accomplishing the task) of helping your students get better at problem solving. That is the purpose of this book.

This book gives a brief overview of the subject or discipline of problem solving and of roles of Information and Communication Technology (ICT) in problem solving. It is targeted specifically toward preservice and inservice teachers. The ideas from this book can be woven into instruction in almost any curriculum area.

What is Problem Solving?

In this book, I use the term problem solving to include all of the following activities:

• posing, recognizing, clarifying, and answering questions

• posing, recognizing, clarifying, and solving problems

• posing, recognizing, clarifying, and accomplishing tasks

• posing, recognizing, clarifying, and making decisions

• using higher-order, critical, and wise thinking to do all of the above.

This broad definition is intended to encompass the critical thinking and higher-order thinking activities in every discipline. An artist, mathematician, musician,scientist, and poet all do problem solving.

Critical Thinking

Problem solving and critical thinking are closely connected fields of study. Diane Halpern's area of specialization is critical thinking as a component of cognitive psychology. In her 2002 article “Why Wisdom?” she says:

The term critical thinking is the use of those cognitive skills or strategies that increases the probability of a desirable outcome. It is purposeful, reasoned, and goal directed. It is the kind of thinking involved in solving problems, formulating inferences, calculating likelihood, and making decisions. Critical thinkers use these skills appropriately, without prompting, and usually with conscious intent, in a variety of settings. That is, they are predisposed to think critically. When we think critically, we are evaluating the outcomes of our thought processes—how good a decision is or how well a problem is solved. Critical thinking also involves evaluating the thinking processes—the reasoning that went into the conclusion we have arrived at or the kinds of factors considered in making a decision. (Educational Psychologist. 36(4), 253-256)

Indiana University Purdue University Indianapolis provides a student-oriented definition of critical thinking (IUPUI, 2007).

[Critical thinking is] the ability of students to analyze information and ideas carefully and logically from multiple perspectives. This skill is demonstrated by the ability of students to:

• analyze complex issues and make informed decisions;

• synthesize information in order to arrive at reasoned conclusions;

• evaluate the logic, validity, and relevance of data;

• use knowledge and understanding in order to generate and explore new questions.

Higher-Order Thinking

The term “higher-order” thinking is often used in discussing critical thinking and problem solving. The work of Lauren Resnick is often quoted in discussing this issue (Resnick, 1987). She states that higher order thinking:

• Is nonalgorithmic—the path of action is not fully specified in advance;

• Is complex—with the total path not visible from any single vantage point;

• Often yields multiple solutions, each with costs and benefits;

• Involves nuanced judgment and interpretation;

• Involves the application of multiple criteria, which sometimes conflict with one another;

• Often involves uncertainty, because not everything that bears on the task is known;

• Involves self-regulation of the thinking process, rather than coaching at every step;

• Involves imposing meaning, finding structure in apparent disorder;

• Is effortful, with considerable mental work involved.

Probably you have heard about Benjamin Bloom's six-part taxonomy of cognitive learning. This was developed in 1956, and its focus was mainly on college education. However, it is applicable to education at all levels. . Quoting from Donald Clark (n.d.), the six levels are:

Knowledge: Recall data or information.

Comprehension: Understand the meaning, translation, interpolation, and interpretation of instructions and problems. State a problem in one's own words.

Application: Use a concept in a new situation or unprompted use of an abstraction. Applies what was learned in the classroom into novel situations in the work place.

Analysis: Separates material or concepts into component parts so that its organizational structure may be understood. Distinguishes between facts and inferences.

Synthesis: Builds a structure or pattern from diverse elements. Put parts together to form a whole, with emphasis on creating a new meaning or structure.

Evaluation: Make judgments about the value of ideas or materials.

Bloom's taxonomy is designed to help differentiate between the lowest order (knowledge; recall data and information) and the highest order (evaluation) of human cognitive activity. One way to think about the scale is that it starts at rote memory of data and information with little or no understanding, and it ends at the highest level of understanding and critical thinking.

Our educational system attempts to differentiate between lower-order cognitive (thinking) skills and higher-order cognitive skills. While there is no clear line of demarcation, in recent years our educational system has placed increased emphasis on the higher-order skills end of such a scale. In very brief summary, we want students to learn some facts (a lower-order skill), but we also want them to learn to think and solve problems using the facts (a higher-order skill).

Some Important Aspects of Problem Solving

Often the thinking and problem solving that we want students to do is to recognize, pose, clarify, and solve complex, challenging problems that they have not previously encountered. For example, consider the teaching of writing. You may consider good penmanship and correct spelling to be important, but most people would consider these lower-order goals. Learning to write in a manner that communicates effectively is a higher-order, critical thinking goal. In some sense, each writing task is a new problem to be solved.

Moreover, writing is a powerful aid to the brain. George Miller (1956) discusses the magic number 7  2. He and many others have observed that a typical person’s short term memory is limited to about 7  2 pieces or chunks of information. Thus, probably you can read a seven-digit phone number and remember it long enough to key it into a telephone pad. Your short-term memory is easily overwhelmed by a problem that contains a large number of components that need to be considered all at one time. Skill in using reading and writing extends the capabilities of your brain to deal with complex, multi-component problems. That is, reading and writing are brain tools that significantly increase your problem-solving abilities.

A few schools actually offer specific courses on problem solving. For the most part, however, students learn about problem solving through instruction in courses that have a strong focus on a specific content area such as art, history, reading, science, mathematics, music, and writing. Every teacher teaches problem solving within the specific subject matter areas of their curriculum. Some are much more explicit in this endeavor than others.

Many people have observed that the "every teacher teaches problem solving" is a haphazard approach, and that the result is that students do not get a coherent introduction to problem solving. When a student reaches a specified grade level, can the teacher assume that a student knows the meaning of the terms problem, problem posing, and problem solving? Can the teacher be assured that the student has learned certain fundamental ideas about posing, representing, and solving problems? Can the teacher be assured that the students know a variety of general-purpose strategies for attacking problems? In our school system at the current time, the answer to these questions is "no."

Thus, each teacher is left with the task of helping students to master the basics (fundamentals) of problems solving and then the new problem-solving topics that the teacher wants to cover.

This book covers the basics of problem solving. It is designed as a general aid to teachers who need to cover the basics with their students. Of course, the basics need to be interpreted and presented at a grade-appropriate level. This book does not try to do that. It is left to individual teachers to understand the basic ideas and then present them in a manner that is appropriate to their students.

This book places particular emphasis on several important problem-solving ideas:

1. Posing, recognizing, clarifying, representing, and solving problems are intrinsic to every academic discipline or domain. Indeed, each discipline is defined by the specific nature of the types of problems that it addresses and the methodologies that it uses in trying to solve the discipline’s problems.

2. Some traditional tools (for example, reading and writing) are useful in addressing the problems in all disciplines. Information and Communication Technology provides us with some new and powerful tools that are useful aids to problem solving in every discipline.

3. Much of the knowledge, techniques, and strategies for posing, recognizing, clarifying, representing, and solving problems in a specific domain requires a lot of knowledge of that domain and may be quite specific to that domain. However, there are also a number of aspects of posing, recognizing, clarifying, representing, and solving problems that cut across many or all domains, and so there can be considerable transfer of learning among domains. Transfer of learning is discussed in Chapter 5. Our educational system should help all students gain a significant level of expertise in using these broadly applicable approaches to problem solving. Learning to effectively do transfer of learning is one of the more important goals in education.

Immediate Actions for Chapter 1

Chapter 1 suggests that every teacher teaches problem solving, lower-order skills, higher-order skills, and critical thinking. Every teacher teaches for transfer of learning. Talk to your fellow teachers about this set of ideas. Look for the nature and extent of agreement and disagreement among teachers of a variety of disciplines and grade levels. Engage your students in the same conversation. By carrying on such conversations with your fellow teachers and students, you will increase your understanding of problem solving, lower-order skills, higher-order skills, critical thinking, and transfer of learning.

Activities for Chapter 1

1. Select a discipline that is a standard part of the PreK-12 curriculum. List several important lower-order skills within the discipline. List several important higher-order skills. Compare and contrast lower-order and higher-order skills within the discipline. Keep in mind that there is no fine dividing line between lower-order and higher-order skills. However, try to select examples in which you feel there is a clear distinction.

2. Repeat Activity 1, but with a different discipline. Then: A) Compare and contrast the lower-order skills within the two disciplines; and B) Compare and contrast the higher-order skills within the two disciplines. Keep in mind that every discipline has lower-order and higher-order skills. This idea parallels the idea that every discipline can be defined by the types of problems that it addresses and the types of methodologies that it uses to represent and solve problems.

3. Select two different broad discipline areas such as social studies and science. Compare and contrast your problem-solving skills in these two areas. To do this, you might want to name some typical problems that each area addresses. Then analyze your current level of skill in addressing these problems. Pay particular attention to the differences that you find between your level and type of expertise in the two areas. This type of self-analysis is an important aspect of getting better at problem solving.

4. List several relatively challenging problems that you have solved during the past few days. Your problems should come from a variety of settings, such as home, work, play, school, and so on. Think about what you learned by solving these problems. That is, do metacognition, and be reflective. Metacognition and reflectiveness are key aspects of getting better at problem solving.

5. Select a specific grade level and/or subject area that you teach or would like to teach. Analyze the content of Chapter 1 of this book from the point of view of applicability to students at that grade level and/or in that content area.

6. Drawing upon the full range of your current ICT knowledge and skills, analyze roles of ICT within the ideas discussed in Chapter 1. Identify your current strengths and weaknesses in ICT from this point of view. One of the topics you might want to address is the issue of memorization versus learning to “look it up.” ICT has made it much easier to search for and retrieve needed information.

Chapter 2:

What is a Formal Problem?

"All progress is precarious, and the solution of one problem brings us face to face with another problem." (Martin Luther King Jr.)

See how a smart squirrel deals with a difficult problem. View the short video: http://www.youtube.com/watch?v=nWU0bfo-bSY.

Each of us has our own concepts as to what constitutes a problem. People wanting to share their thoughts and do research on the topic need to agree on a definition. This chapter presents both a general overview and a formal definition of problem.

Definition of a Formal Problem

Problem solving consists of moving from a given initial situation to a desired goal situation. That is, problem solving is the process of designing and carrying out a set of steps to reach a goal. Figure 2.1 graphically represents the concept of problem solving. Usually the term problem is used to refer to a situation where it is not immediately obvious how to reach the goal. The exact same situation can be a problem for one person and not a problem (perhaps just a simple activity or routine exercise) for another person.



Figure 2.1. Problem-solving—how to achieve the final goal?

There is a substantial amount of research literature as well as many practitioner books on problem solving. A recent Google search on the quoted expression “problem solving” produced over 9 million hits.

Here is a formal definition of the term problem. You (personally) have a problem if the following four conditions are satisfied:

1. You have a clearly defined given initial situation.

2. You have a clearly defined goal (a desired end situation). Some writers talk about having multiple goals in a problem. However, such a multiple goal situation can be broken down into a number of single goal problems.

3. You have a clearly defined set of resources that may be applicable in helping you move from the given initial situation to the desired goal situation. There may be specified limitations on resources, such as rules, regulations, and guidelines for what you are allowed to do in attempting to solve a particular problem.

4. You have some ownership—you are committed to using some of your own resources, such as your knowledge, skills, time, money, and so on to achieve the desired final goal.

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