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 MerriamWebster 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
3.
4.
5.
6.
7.
8.
9.
helpful to refine the focus question. If you substantially modify the focus
question, then do part 1 again. Continue ordering all concepts.
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.
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.
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.
Rework the structure, using new concepts and propositions as they arise in the
context of the focus question.
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.
You can put specific examples on the relationships to help clarify the relationship.
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.