Building Concept Maps - Clemson University



Building Concept Maps

Introduction

The note on “Rules for Reading” introduced the concept of concepts, no pun intended. The basic idea in that note was to explain to the students one possible approach to reading technical material. The bottom line is that the reading should propose concepts, events and propositions. It this were all there was, then the reading task would be fairly easy to complete.

Unfortunately, almost any interesting subject is very complicated and involved. Even at this stage of development of the natural sciences, mathematics, and computer science, new “knowledge” is being discovered every day. But what is knowledge? How does knowledge relate to what we need to do on a given project? What does knowledge have to do with learning? Studying?

We are going to follow the plan laid out in [1] and briefly touched on by “Rules for Reading.” A concept map is a graphical representation of concepts (the nodes) and the propositions that connect them (the weighted arcs). Concept maps are aids in the learning process. Concepts and propositions are always taken in a context. Related to, but separate from, the concept map is the knowledge vee.

Knowledge: The Basic Idea

Knowledge itself is a slippery subject, having been studied in Western philosophies since antiquity. For our purposes, we will divide knowledge into two distinct categories: knowing that and knowing how [2], sometimes called explicit knowledge and implicit knowledge, respectively. This relates computer science in the following way:

• Knowing that is the same as mathematics and logic. In these subjects, we are not so concerned with computing objects as classification and relationship.

• Knowing how is programming, or at least computation. Here the specific issue is how we can write programs that fulfill the problem statement. The two ideas are deeply related, for example, in computing theory.

Explicit knowledge is how knowledge in the natural sciences and mathematics is organized. Rarely, if ever, are the issues in the more formal aspects of explicit knowledge ever discussed in texts. In fact, these theories are usually called informal precisely because the epistemological and belief issues are not discussed. Concept maps are graphical representations of explicit knowledge issues.

Implicit knowledge is more difficult to study precisely because it is not written down. Implicit knowledge is knowledge internal to an individual. The knowing how of the painter or sculptor is the classical example of such knowledge. However, in mathematics, implicit knowledge is the basis of intuitionistic mathematics or constructive mathematics.

Explicit Knowledge

Explicit knowledge is knowledge that is written down so that all individuals can have access to the same ideas. There are three basic things to consider: vocabulary, logical coherence, and overarching beliefs.

Overarching Beliefs

Knowledge isn’t so much a thing as it is a system of interrelated ideas. Generally, we work within a framework of ideas, called a belief system or worldview, that we inherit from our environment. This environment is everyone you talk to and everything you read, touch, smell, etc.

Beliefs and worldviews are the guiding values under which philosophies are formed. A philosophy is a general belief system guiding inquiry. Often we are not consciously aware of these ideas. This paper presents a belief system — that technical subjects can be understood in a particular way. For most of us from the West, the philosophical system is the Cartesian system proposed by René Descartes and most often used in technical investigations.

Within the philosophical belief system we form epistemologies. An epistemology is a set of rules about nature of knowledge and the nature of knowing that guide the current inquiry. There is an epistemology for science that says that we know something when we have verified a theory with observations of real objects. In mathematics, one of several epistemologies is that we know something is we can prove a theorem about the subject starting from the axioms. There may be many epistemologies regarding the same subject. No wonder there is often confusion when “experts” proclaim the “truth” of completely contradictory ideas — Who do you believe?

A large portion of your education is aimed at getting you to believe the “standard epistemology” of the various subjects. Only recently has the business community seen that “thinking out of the box” (outside the standard system) has huge merit.

Another way of thinking about beliefs and epistemology is that they set the context of a discussion. Context provides the connection for the elements of a discussion, providing coherence between the parts of a discourse. Coherence means that the parts are consistent and “hang together.” Things that are connected in context are said to have correspondence. Things are in correspondence if they have congruity, harmony, or agreement. Correspondence and coherence are concepts to be tested to determine if a set of ideas “belong together.”

Logical Coherence

Logical theories (such as algebra and theoretical computer science) are composed of three basic types of statements: (1) formal structures composed from an abstract vocabulary; (2) principles that explain formal structures and (3) theories that provide justification. In scientific theories, for example, the theory provides an explanation of why the objects fitting the abstract vocabulary behave as they do.

1. Formal structures abstract a vocabulary. Vocabularies can be very specialized; indeed, in technical topics, they are almost guaranteed to be specialized. This means that the words in the vocabulary are often English words with very specialized meanings. The vocabulary provides the underlying semantics. Concepts can be objects or events (see below). While propositions can refer to “soft and squishy” ideas, it is more likely in technical areas for propositions to refer to theorems, axioms, postulates or specialized methodologies.

2. Principles are statements of relationships between concepts that explain events or how objects can be expected to appear or behave. Principles are generalized formal structures.

3. Theories are generalized principles that explain why events or objects exhibit the observed behavior. Theories represent specific issues about specific objects and events. The rules under which theories are produced come from epistemology. Theories always include an epistemological touch: how do I know the subject to be “true”?

Vocabulary

There are usually two types of vocabularies in an area: concrete and abstract.

1. Concrete objects are things that one can observe, touch, feel, etc. The interaction between objects is called an event.

2. The first step in abstraction is to convert — generalize — objects and events. Concepts are generalized objects. Propositions are generalized events and relationships among concepts.

In order to form concepts, we need examples. How we talk about ideas is crucial. Linguistically, anything that is a noun is an object. Events are logical conditions. For example “on” is an event concept related to a light switch.

When you read a textbook, the authors have already digested the objects and events for you. This may be good or bad, depending on circumstances. For example, early computer science texts were written with a specific manufacturer in mind and abstraction was very difficult. Therefore, when you read, the problem is often the other direction: the author gives you a concept and you must determine what objects or events fulfill the definition. Here some examples of such instances.

1. Beauty. Everyone has his or her own idea, as in “Beauty is in the eye of the beholder.” Name twenty “beautiful” things that are not people.

2. Computer. Just what is a computer? The engineers have a concept, which looks like any computer you’re likely to see in a store. But they’re all so different. Theoretical computer scientists have concept called the Turing machine. No one has built a Turing machine because it can have an infinite tape (another concept: infinite tape). Are the engineers and theoretician talking about the same thing?

3. Program. What exactly is a program? Are the circuits in the chip a program? For programmers, a program is something they write in a specific programming language — but there a huge number of programming languages. Can you write the same algorithm (another concept) in any language?

Therefore, the first issue in any reading assignment is to identify the concepts the authors intended for you to grasp. For the record, the definition of concept given in the Merriam-Webster dictionary is “2. An abstract or generic idea generalized from particular instances.”

There can be other concepts, the implied ones. When you read a book meant for a senior level course, the authors assume you have mastered some concepts from earlier course. For example, a course in programming languages assumes that you can program in at least one “modern” programming language.” So, in your reading, the author is trying to make contact with those previous concepts. For the sake of clarity, I’ll call those earlier concepts grounded or inherited.

Developing Vocabulary Every Time You Read

Whenever you read, you are required to develop a list of the new and grounded concepts in the reading assignment. You will also be required to order them from the most general to the most specific. You must also establish the context under which these concepts are defined.

After you have an initial list you will be making a list of propositions. 99 times out of 100 you will be adding concepts after you start identifying propositions. This is perfectly normal and natural. If you already knew everything you wouldn’t be reading.

Linguistically, propositions are verbs. In the traditional Aristotelian ideas of concepts, the verb was some form of “to be.” In fact, in many ways, this relationship is still the most important. In object-oriented methodology the idea subclass-ISA-superclass. This is not broad enough for modern ideas, so a better way to think of it is the standard declarative paradigm subject-verb-direct object.

Making Concept Maps

We now have enough background to understand concept maps. In terms every computer scientists should understand, a concept map is a directed, weighted graph with the node set being concepts and the weights being propositions.

It is necessary to set a context in order to successfully develop a concept map. Writing down one or more focus questions sets the context. Focus questions are questions about problems, issues, or knowledge domains that the concept map is to address.

1. Identify a focus question that addresses a problem, issues, or knowledge domain to be addressed. The question and the area establish the context. Try to identify 10 to 20 concepts that are pertinent to the question and within the established context. You can use note cards, Post-it ™ notes, scratch paper … whatever. You need to be flexible.

2. Rank orders the listed concepts from the most inclusive and broadest to the most special and narrowest. If it is difficult to find the most inclusive, it is often helpful to refine the focus question. If you substantially modify the focus question, then do part 1 again. Continue ordering all concepts.

3. There should be a most one inclusive concept — however, you could have as many as three. If there are more than three or so, then reconsider the focus question again, repeating 1 if necessary. Put the most inclusive at the top of the page. The narrower the focus, the smaller the map.

4. Next, select, up to a maximum of four or so, sub concepts that seem to be immediately implied by the higher concept(s). If there is more than about four, then reconsider the concepts, looking for a more inclusive concept that these sub concepts fit in.

5. Now, connect the higher concept(s) to the lower ones with directed arcs with the head at the more specific concept. Label these arcs with words that indicate the relationship between the two concepts so that the whole thing reads sensibly. That is, the whole must be a valid, sound proposition or relationship between the higher and sub-concepts. This connection creates meaning. When you hierarchically link a large number of relationships you start to see the meaning for the given context.

6. Rework the structure, using new concepts and propositions as they arise in the context of the focus question.

7. Now, take in the whole picture. Look for cross-links between “distant” portions of the map. You should get a bit of the famous Aha feeling.

8. You can put specific examples on the relationships to help clarify the relationship.

9. Any change to either the focus question or the context will result in a changed map.

Bibliography

[1] Joseph D. Novak. Learning, Creating, and Using Knowledge: Concept Maps as Facilitative Tools in Schools and Corporations. Mahwah, NJ: Lawrence Erlbaum Associates, Publishers. 1998.

[2] Ryle. Concept of Mind.

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