In geometry we often start with definitions and work from there by deductive logic. For example we may be given the definition of a rectangle as "a four-sided polygon with opposite sides equal and adjacent sides at right angles." From this definition we can then identify which geometric figures are rectangles and which are not. We can further formulate certain theorems about rectangles. We might prove the theorem that the diagonals of a rectangle are equal, or that the diagonals form right triangles, and so on. In doing this we would be applying logic to a complex web of definitions, postulates, theorems, and methods.

      Now compare this to the concept formation that a six-year old child would engage in when learning about rectangles. Consider this scenario: The child is working in a first grade workbook. The page he is working on has the instructions, "Color the circles red. Color the rectangles blue." The child asks his mother, "What are circles?" His mother points to several circles and says, "These round ones, they're the circles." Then she points to several rectangles and says, "These are rectangles. They're not round. They have corners." The child proceeds to correctly color the figures according to the instructions. A few pages over in the book, and perhaps a week later, the child learns to distinguish between rectangles and parallelograms by going through very similar steps. He may also learn some relations and facts about the various geometric figures.

      In both of these scenarios geometric concepts are developed. New knowledge is added to the old. Associations are made, and these associations result in new concepts as well as new information. However there is a world of difference between the two. The first example, that of a high school geometry class, illustrates definitional concept formation. The second example, of the six-year-old, illustrates operational concept formation. In definitional concept formation the definition is first presented and then it is applied. In the operational method of concept formation the learner learns what to do (color the rectangles . . .) and from there the concept develops. The operational method may or may not lead to a formal definition after some period of time.

      I think it would be accurate to say, at least in general terms, that definitional concept formation is very much a left brain phenomenon, while operational concept formation is much more a right brain phenomenon. Also, operational concept formation works with inductive logic, while definitional concept formation works with deductive logic.

      I don't think it would be accurate to say that one form of concept formation is superior to the other. Rather, the two different forms have different applications, and the two different forms do not at all have the same result. The six-year-old child's concept of a rectangle would not be satisfactory in a high school geometry class, and the geometric definition of a rectangle would be of little use to a six-year-old.

      In this example there is little danger of using the wrong method for a given context. No one would expect to teach a six-year-old about rectangles from a high school geometry book, and no math teacher would accept a six-year-old's concept of rectangle as sufficient in a geometry course. However there are times when the situation is not so clear-cut. I will next present a situation in which a combination of the definitional and operational forms of concept formation must be used.

      Consider this situation: A sixth grade English teacher wishes her class to learn the following vocabulary words: century, navigate, inauguration, sentiment, belligerent, and institution. Such a word list might arise naturally out of a story the class has been reading. I have put these words in approximate order of abstractness, as I hope will become evident as I discuss them. How might these words be taught to the class?

      In teaching the meaning of "century" it would seem obvious to simply tell the students that a century is a hundred years. This is a definitional approach. The concept is defined and the learners then apply this definition when needed. Immediately after giving the definition the teacher might ask the class, "What was the date exactly a century ago today?" The correct answer would normally be forthcoming from some, if not most, of the students in a typical sixth grade English class. In this example the definitional approach to the formation of the concept of "century" is effective and entirely appropriate.

      However the next word on the list, "navigate" is not quite so amenable to the definitional approach. The teacher may start with a definition, "To navigate means to find your way without getting lost." Then she would most likely give an example, "The Portuguese were able to navigate around Africa in the 1400's." This word is a little more abstract than the first word. An example is hardly needed for "century", but it is certainly needed for "navigate".

      The next word in the list "inauguration" might again be introduced with a definition, "An inauguration is a ceremony marking the beginning of something new." However this definition is not quite so clear cut as the definition of a "century" or even of "navigate". If the teacher now asks a student to use the word in a sentence she may not get a satisfactory response. When pressed, the average student might come up with something like, "The inauguration was good.", which is not sufficient to indicate either an understanding or a lack of understanding. Another student might come up with the sentence, "The music was the inauguration of the radio.", clearly showing the word is not fully understood.

      The use of examples would very much be needed here. The teacher might say, "The inauguration for a new president is on January 20." As another example she might say, "You can expect the inauguration of a new policy on gum chewing if we find any more gum on the floor." The use of examples moves the learner away from a definitional approach to a more operational approach. The essence of the operational approach is learning what to do. Thinking follows action. What the learner does, at this point is to follow the pattern given by the examples. If a president has an inauguration then perhaps a mayor has an inauguration. So the learner might offer the sentence, "The mayor's inauguration is on May 5th."

      The next word on the list, "sentiment" is yet a little harder to define. It is a word that I use frequently, but I do not find it easy to produce a definition. It is a word that I would not expect would be taught definitionally. It would be taught by example, which is an operational approach. The next word, "belligerent" is similarly hard to define. It might be given a synonym such as "warlike" but that hardly suffices to give the subtle meanings of the word. Examples would definitely be needed. The last word, "institution", is the most abstract of the list. I cannot imagine teaching it without a number of examples.

      I will expand a bit on this matter of learning vocabulary, as it was this that got me started thinking about operational and definitional modes. I observed over a period of some years that my children would often bring home vocabulary assignments from school. Sometimes the assignment would be to copy the definitions out of a dictionary or glossary. Educationally this is highly suspect. I often wondered if my children would tell me that's what the teacher said to do, when in fact the assignment was to define the word in their own terms with or without the help of a dictionary. It would be a poor teacher indeed who would assign a page of copying out of the dictionary to the whole class, but it would be a very common teacher who accepted such an effort from a slow-learning student who has a great deal of difficulty with language. Another assignment my children would often bring home would be to use a list of vocabulary words in sentences. This makes more sense educationally, but with the slow-learning kids in my family it still didn't seem to work very well. I devised a method that I used for some years with my own children. This method consists basically of filling in the blanks and is thus very much an operational approach. I would use a set of 3 x 5 cards. I will give an example of one such set. On the top card of the deck I would type:

      This vocabulary set contains only three words (counting adjective and adverb together), not the six words I discussed above. This method is geared for slow learners. and I have found three words to work best. Also I have presented them in a noun-verb-adjective format. I will have more to say about that shortly.

      The next card would have a sentence, such as:

      The learner's task, obviously, is to write down a word from the top card that will fit in the blank. Also, the correct form of the word must be chosen. Successive cards would be:

      I guess I ________ pretty well. I came clear across town and didn't get lost once.

      People have a lot of noble ________, until it's time to vote for higher taxes.

      "Love your enemy" is a very idealistic ________. But it's hard to put it into actual practice.

      Explorers began to ________ around the globe in the 1500's.

      The customer came in and _________ demanded his money back. He never did really explain what the problem was.

      We're__________ as best we can, but we lost our map so I'm not sure just where we are.

      And so on, for a total of about 12 to 20 sentences on as many cards. I would attempt to have cards requiring both singulars and plurals for the noun, and several tenses for the verb, and of course both adjective and adverb forms of the modifier. Over the years I have typed out about a hundred of these vocabulary sets.

      There are some words for which a synonym can be quickly found. I could use "cease" as a vocabulary word. However I can also simply tell the learner that "cease" means "stop". The definitional approach is appropriate here, and is more efficient than an operational approach. If a word can be defined so easily and explicitly, it hardly seems worth the effort of typing out a dozen fill-in-the-blank vocabulary cards. So I do not choose such words for this type of operational approach. Rather I choose words for which there is no easy synonym or definition.

      This approach to vocabulary learning is operational because the learner is faced with a problem of what to do - what word should be put in the blank - as opposed to what to think - how to put ideas together. It is not my intention that at any point the learner will write down, or even formulate in his mind, a definition for a given vocabulary word. Rather he will learn what word goes into what blank. I will give at least a rudimentary definition when asked, but more importantly, I will say, "Choose one of the words and put it in. Just guess if you need to. You'll miss some, but that's okay. We'll try them again another time, and then you'll get more of them. . . . . " The same set can be mixed up and used again and again, until the learner either gets the concepts, or gets sick of trying.

      One might certainly ask, "How do you know the concept is being formed? Maybe they are just memorizing answers." There are two lines of evidence that meaning is conveyed. First, the learner typically makes a jump from guessing to answering them all correctly. Memorizing would follow a more gradual path. Secondly, and more importantly, I can test for understanding by simply typing out a new card for an existing set. If answers were blindly memorized without formation of a concept, a new card would stymie the learner. I have done this plenty of times with success.

      I would argue that my vocabulary system is close to the way that vocabulary is learned by a young child in the home. Primitive vocabulary cannot be learned definitionally. How could you define the word "gone" to a two-year-old, or "turn" (as in "you have to wait your turn"), or "lunch", or a thousand other words that a young child learns? These early words are learned from context, which is to say they are learned operationally.

      Many vocabulary words are learned operationally even at higher levels of education. Consider again the vocabulary word "inauguration". As a vocabulary word, I have argued, "inauguration" must be taught at least partially by an operational approach. However it is likely that the word would first be introduced to most students in a social studies class when they study the election of a president. "Inauguration" would be presented as the name for the event in which the new president takes office. Perhaps emphasis would never be placed on the meaning of the term itself. Rather the students would be expected to learn facts. The teacher may tell the students that they are expected to remember that the presidential inauguration takes place on January 20 following the November election. She would require them to learn about the oath of office, about the transfer of power, about the customs of presidential inaugurations, and so on. It may never even occur to the teacher that "inauguration" could be a vocabulary word. However the meaning of the word is still learned. It is learned because the concept of inauguration is developed as the students learn about how we choose our president. It is learned operationally as a side effect of learning about presidential succession. This would be true of the vast majority of the words we learn at an early age, and of a great many words we learn when we are older.

      My main purpose in using these sets of vocabulary cards is to teach vocabulary. However there is a secondary purpose, and that is to teach parts of speech, at least nouns, verbs, adjectives, and adverbs, and this provides another opportunity to compare the definitional and operational approaches to concept formation. Parts of speech are normally introduced definitionally. The teacher tells the class "A noun is the name of a person, place, or thing." This is followed by extensive practice, exercises in which the students circle the nouns in sentence, and so on. One might suppose that the practice comes after the concept is learned, that the concept of "noun" is learned from the definition - a noun is a person, place, or thing. However I think it can be argued that the definition is insufficient to form the concept, that the concept is actually formed in the learner's mind operationally by doing all those exercises. Evidence for this is the weakness of any definition of "noun". I remember wondering, in fourth grade or whenever it was that we first were expected to learn about nouns, about that person-place-thing definition. "Hitting", I reasoned, is the name of an action. So in the sentence "I was hitting the ball", the word "hitting" must be a noun. And yet I knew it was not. And "green" must be a noun in the sentence "We lost the green ball" since "green" is a color, which qualifies as a thing in the "person, place, or thing" definition. And yet I also knew that "green" in that sentence was not a noun.

      A definition of "noun" may be a good place to start in order to learn the concept, but it certainly doesn't do the whole job. The concept is formed, to a large extent, operationally by doing exercises, making mistakes, and trying again.

      If the actual concept of "noun" is formed operationally then perhaps one should not try too hard to be definitional. Operationally one might approach nouns and verbs by saying "Nouns have singulars and plurals. Verbs have tenses." In a sense this is a definitional approach, if one defines nouns as "words that have singulars and plurals." But that is not really the primary definition of nouns, and even more importantly, it is operational in the sense that it gives the learner something to do, a test that he can apply to a word to determine if it might be a noun. My reasoning is that singular and plural is reasonably concrete, and even more important, already known and understood by students who are ready to start building the concept of "noun". For example, is "birthday" a noun? Using the "person, place, or thing" definition one might argue that "birthday" cannot be a noun, since it is neither a person, nor a place, nor a thing. However using the "singulars and plurals" definition it is a noun. We can say "Our class had three birthdays last week." It has a plural, so it must be a noun. And then one might reason that if it has a plural it must be a thing, and so it fits the first definition after all.

      Now try to fit either definition to the word "tact", as in the sentence, "It takes a lot of tact to get along with Mr. Jones." It might be argued that "tact" cannot be a noun on the basis of either definition. It is certainly not a thing in the sense of being tangible, and it doesn't seem like it has a plural, so can it be a noun? Perhaps we could define "noun" as "any word that can be the object of a preposition". But that hardly seems sensible. We expect kids to understand nouns and verbs pretty well before introducing prepositions.

      Definitions fail. At least I can not come up with a fool-proof definition of "noun" or "verb". We learn the concepts operationally, by doing exercises, not definitionally. A definition is at best simply a good place to start.

      Thus my purpose in using the noun-verb-adjective format in the vocabulary cards is simply to give more exposure to these concepts. By giving the singular and plural of the noun, and the principle parts of the verbs, I can emphasize those properties of nouns and verbs, and thereby help to develop the concepts, operationally of course.

      Now I would like to turn to arithmetic for further examples of operational versus definitional approaches. Recently I took a computer science class. One day the professor was explaining something, I don't remember just what, and he commented, "We're using this relation:" which he wrote on the board

           A/B = C + D/B       or       A = CB + D

"which any second grader knows." That phrase "any second grader knows" got me to thinking. Of course the professor was not suggesting that algebraic expression should be used with second graders. He is talking about the number relations, and he is quite right. Second graders do know that

This fits the what the professor put on the board. A, in the algebraic expression, is 7 in the division, B = 3, C = 2, and D = 1. When a second grader (more likely third grader I think) divides 7 by three he is indeed using the number relations expressed by A/B = C + D/B. For the average third grader these number relations are quite clear (except for the "D/B" part, which can only come with fractions). They can be easily shown on fingers or by drawing circles on paper. However it does not at all follow that the average third grader could derive any understanding or benefit from the algebraic expression, and that is where operational versus definitional approaches comes in. A definitional approach to building the concept of division would require starting with a definition of division expressed in something like this algebraic form.

      The important point is that the concept of division is learned operationally in the second, third, or fourth grade, not definitionally. In learning the concept of division, the child learns what to do. He learns a series of thought processes that lead to the desired result, which is getting the right answer. This is not to suggest that learning division is a mindless behavioristic process. While learning what to do, the child engages in some advanced cognitive processes, some of which could certainly be called definitional. However the overall approach is operational. By being told what to do, and how to do it, and why it works that way, the learner forms concepts.

      One might ask, in a given situation, whether one should use an operational or a definitional approach. It is a very logical question, but I would immediately argue that it is the wrong question. Operational versus definitional is a continuum, not a polarity. A better question would be, "Is the instruction given by this particular teacher to this particular class on this particular day leaning too heavily on either the operational or the definitional extreme?" That is a question that can only be answered by professional judgment, one situation at a time.

      I think it would be accurate to say that as a general rule early learning is very operational, while advanced learning is, or can be, very definitional. But the important thing is that the teacher should be aware of this type of thing and make adjustments when needed. A third grade teacher might wonder why her explanation of whole number division elicits a lot of blank stares from her students. A shift to a more operational presentation might prove beneficial. Or a college teacher might do the very same thing. Or either the third grade teacher or the college teacher might find him or herself neglecting to explain the subject to the depth that the students need, indicating a shift toward the definitional end of the continuum is needed.

      These shifts are made everyday by teachers of all kinds at all levels. Making such adjustments is usually a natural thing to do. Teachers don't need labels like "operational" or "definitional" in order to use their common sense. However there are times and situations in which such adjustments ought to be made but are not. In these times an awareness of the two perspectives can be helpful. I will try to describe several of these situations. I will start with what I will call the "definitional fallacy", which I think often tends to lock a teacher on the definitional end of the continuum. The definitional fallacy can be stated:

      "If I can say it, they can learn it."

      An example of this would be a professor in an advanced math or science course struggling to find just the right string of words to define an idea or concept. Sometimes his facial expression will indicate just when he feels he has succeeded. The fallacy, of course, is in thinking that the professor's success in expressing an idea guarantees the students' success in understanding the idea. Very often t hat is not the case. An accurate verbal expression of a concept or idea may be a long way from producing an understanding in the student's mind. The students will spend considerable time after class trying to understand their notes and the text book, and considerable time doing assignments. The understanding, in many cases, comes from this more than from lecture. This may be known by the professor on one level, but I suspect that level is often very superficial. The "assumption of fluency", which I maintain is the number one mistake made by teachers of any level or subject, leads the professor to vastly overestimate the amount of his lecture that the students are understanding and the amount of that understanding that they will retain.

      It is not generally true that a successful expression of an idea is a sufficient cause for the understanding of that idea. This is obvious at the lower levels of education, but can be still very true at higher levels. Good teachers at any level take this into account. They give examples, they assign homework and grade it. They elicit feedback from the students. When I'm sitting in a class and the professor is struggling to find just the right string of words to concisely, yet completely and rigorously, encapsulate a concept, idea, or fact, I often wish he would just quit and put an example on the board.

      To be sure, there are occasions when the rule, "If I can say it, you can understand it." can be used. When writing for an educated, intelligent, and motivated audience it may be quite true. Consider this example. Article five of the United States Constitution consists of a single sentence:

      "The Congress, whenever two thirds of both houses shall deem it necessary, shall propose amendments to this Constitution, or, on the application of the legislatures of two thirds of the several States, shall call a convention for proposing amendments, which, in either case shall be valid to all intents and purposes, as part of this Constitution, when ratified by the legislatures of three fourths of the several States, or by conventions in three fourths thereof, as the one or the other mode of ratification may be proposed by the Congress; provided that no amendments which may be made prior to the year one thousand eight hundred and eight shall in any manner affect the first and fourth clauses in the ninth section of the first article, and that no State, without its consent, shall be deprived of its equal suffrage in the Senate."

      That's quite a sentence. Apparently it is logically valid. I have never heard it charged that the Constitution is ambiguous or contradictory about the process of amendment. We know exactly what the process is. Logically the sentence is just fine. Pedagogically it stinks. There is no "definitional fallacy" here, to be sure. The writers of the Constitution were quite correct in thinking that if they could express it we could understand it. But it does not follow that a teacher can make the same assumption. Constitutional scholars have unlimited time to analyze this sentence, but students do not have unlimited time to decipher their teacher's pronouncements. And more importantly, analyzing and applying definitions is not always the best way to learn a concept. Concise definition certainly has its place in good teaching. The definitional fallacy consists of thinking that it is the essence of good teaching. It is not.

      One might question if there is an "operational fallacy" corresponding to the definitional fallacy. I believe one candidate for this would be the "discovery method" that was a much-touted part of "modern mathematics" of the sixties. By this method students were supposed to be led through a series of steps so that they would "discover" some concept or principle on their own. There was even purported to be evidence to the effect that expression of the principle or concept was detrimental, that retention is better if the students hold off on verbally expressing what they have discovered. There is some merit in this discovery approach, or course, but the merit is simply the merit of the operational approach. Too much emphasis on the operational at the expense of the definitional is an imbalance that should be corrected. The "discovery method" failed to be the magical idea that some people thought would revolutionize the teaching of math.

      The "Socratic method" is a similar example. In this method the teacher asks questions which presumably lead the learners to figure things out for themselves. It is not purely an operational approach, but it seems to me that it is much more operational than definitional. It may end up with a definition, but it does not start out with a definition, and on that account can be considered operational. Again, this method has no magic. It may be valuable at times, but as a general rule it overemphasis the operational end of the continuum.

      I have claimed to be talking about "concept formation" but my examples have obviously slipped over into the learning of principles and facts as well. It is certainly true that concept formation is a different phenomenon than the formation of relations between concepts. However I think many of the issues I raised about concept formation will also apply to relating concepts. I will discuss these other types of learnings in greater detail in other contexts.