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Educator's Voice

Volume 5, Issue 12
December 8, 2004

True Understanding

What do we mean when we say that we understand something? Not necessarily just knowing something, but truly understanding why it is the way it is. For example, when students recite multiplication tables, does this demonstrate knowledge or understanding? I think few would argue that performing a behavior, even if done with fluency (quickly with high accuracy), does demonstrate that the individual has knowledge, but does not necessarily show evidence of understanding. After all, I can read Spanish off of a printed page (without offending too many ears), but I cannot compose a sentence on my own or engage in conversation.

As teachers, we are continually faced with how to design lessons and assessments to determine whether our students have knowledge or understanding. There are many issues to consider about understanding--here, I offer just a few practical ideas.

Abstraction
One of the basic building blocks of knowledge (and, hence, understanding) is discrimination. Here, discrimination refers to learning to differentiate between two or more concepts or objects so as to identify not only what something is, but also what it is not. Abstraction is a particular type of discrimination that reveals understanding of a concept. When a student labels the number 4 as an "integer," we should naturally ask ourselves what it is about the number 4 that prompts that response. Ideally, we'd like the answer to be the fact that it is a natural, whole number. But the number 4, despite the fact that it appears to be pretty simple, is composed of multiple dimensions: in addition to the fact that it is a whole number, it is an even number, it is a prime number, it is made up of three straight lines, it is positive, it has a non-zero value, and it is a single digit. Believe it or not, even how the number is formatted in instructional materials (e.g., that two of the lines connect at the top, as well as font style, size, and color) is potentially important. The tricky thing is that without proper teaching methods, any of these dimensions can serve as the basis for the response "integer."

How do we determine that the critical feature of the number 4 produces the response "integer"? Essentially, we need to make sure that students can abstract this critical feature, that is, that they can identify this one feature regardless of the other features of the numeral 4. As long as the number is natural and whole, all other features are irrelevant. Complete teaching and testing of abstraction of the concept of "integer" would involve presenting examples in as many different forms as possible by varying all irrelevant features. Not doing so will produce faulty concepts. For example, not varying the positive/negative feature of an integer (i.e., presenting only positive numbers during the learning process) may lead students to label negative numbers as non-integers. When you learned the concept of "red," you were taught to label all sorts of red objects as red - the actual object itself was the irrelevant feature. If you had been taught to label only red trucks as red, you might have learned that "truck" was the defining feature of "red" and thus started labeling all trucks as red, blue, green, puce, or otherwise. (By the way, developing a list of defining and irrelevant features of a concept is a great class exercise. It forces students to think about what a concept is and what it is not. Try it with complex concepts, such as "human" or "anger.")

What does abstraction have to do with assessing true understanding? An abstraction is the defining feature of a concept. If students can discriminate this feature across a wide variety of novel examples of the concept, we can say with a good deal of confidence that they understand the concept. If all they can do is identify only examples they were taught, they have engaged in little more than rote learning.

Generalization
Few, if any, of us test students on exactly what they learned in class. Rather, good test construction involves creating questions that alter the presentation of the course material in some way, thus requiring students to respond to the material in a novel way. Furthermore, most, if not all, teachers would say that they want their students to be able to generalize what they learn in class to novel situations, particularly those in the world outside the classroom. Often, generalization is referred to as transfer. Even though it is based on what has been learned, true generalization occurs without explicit teaching. We exhibit generalization daily. My mother learned to drive on a tractor--her driving skills eventually generalized to an automobile. Students generalize their addition skills to new problems, their verb conjugation skills to verbs never seen before, and analytical skills to new word problems. For example, French regular verbs that end in -er have the following conjugation endings for each pronoun:

Pronoun	Ending
Je (I)	-e
Tu (you)	-es
Il/elle (s/he)	-e
Nous (we)	-ons
Vous (you)	-ez
Ils/elles (they)	-ent

After students learn to conjugate several regular verbs, such as penser (to think: je pense, tu penses, il/elle pense, nous pensons, vous pensez, and ils/elles pensent), generalization occurs when they are faced with novel -er verbs, such as jouer (to play) and can conjugate them without having been explicitly taught how. Overgeneralization, of course, can occur, as when they encounter an irregular -er verb (e.g.,) and conjugate it as if it were regular. Undergeneralization is the other potential problem: when students approach each new verb as if its conjugation follows unique rules.

Multidirectionality
If A = B, B = ? Most, if not all, of us would reply "A." This is the relational property of symmetry. If A = B and B = C, then A = ? Again, most of us would answer "C," applying the property of transitivity. Symmetry is a bidirectional relation--each object/concept can be substituted for the other. Transitive relations are multidirectional: At least two symmetrical relations make up the transitive relation. As a result, each object/concept can be substituted for the other two (or more). The neat thing about such relational learning is that one need learn just one direction and the other directions "emerge" without direct training. This emergence happens exponentially, so that for every one new relation learned, an increasing number of new relations emerge.

Relational learning is best illustrated by taking an example from our childhood--say, learning the word "cat." With the help of others (mostly teachers and parents), we learn the relation between an actual cat and the spoken word "cat" and between an actual cat and the written word cat. Relational learning paradigms predict that, by virtue of this instruction, the relation between "cat" and cat would emerge without explicit teaching (i.e., children should be able to match these two words). The diagram below illustrates the learning of taught relations (solid lines) and the emergence of novel relations (dashed lines).

We can add to this set by teaching the relation between cat and chat, the French word for cat. By teaching just this one relation, notice how many new relations emerge (hence, the exponential growth of relational sets):

How does multidirectionality represent true understanding? If students demonstrate the emergent relations (the dashed lines), we can say that they truly understand the meaning of the word "cat." If the dashed relations do not emerge after learning the basic relations, all they have accomplished is rote learning. Demonstrating emergent relations requires higher cognitive processes.

Multiple Response Forms
Just as the same behavior can occur in novel environments (generalization), novel forms of a behavior can emerge in the same environment. To illustrate, consider a student learning French. At first, grammar is perfect, nothing like the fluent speech or slang that native French speakers tend toward. For example, in saying "I don't know," the learner will pronounce every phoneme of "Je ne sais pas" carefully and deliberately. After a while, this response may take on new forms that more closely approximate the speech of a native Francophone: "Je sais pas" or "J'sais pas." The same response is occurring in the same context; we call this response generalization, and its occurrence indicates that the learner truly understands the meaning of this phrase. That is, the ne is not the defining feature of this phrase, nor is the articulation of all phonemes necessary.

Final Thoughts
In no way have I intended to exhaust the discussion on true understanding, but I hope these points have provided you with food for thought and ideas for creating lessons, assignments, and assessments for your students.

       --Jennifer O'Donnell, Ph.D.

TIP

Course Homepage Tools: Ensure Instructor and Student Success

There are three useful tools available on the Course Homepage that will help ensure your success as an instructor and also help ensure student success in your course.

Announcements, posted at the top of the Course Homepage, help ensure that learners see important information relevant to their success in the course. Announcements help achieve redundancy in presenting information to students. It is recommended that a Welcome announcement always be posted for a course. In addition, you can post announcements any time you need to provide additional information to students or to reinforce information provided at an earlier time. For instance, post an announcement reminding students of a special course event, such as a guest speaker in a Chat. The judicial use of announcements will increase your effectiveness in managing your online course, help ensure student success, and reduce your course management workload.

What's New, located at the bottom of the Course Homepage, lets you monitor student activity in your course by seeing when individual students contributed to a select area of the course. Therefore, you don't have to search through every unit or content item to see which students have contributed to various course areas. For example, if you wanted to see if there were any new postings to the Unit 1 Threaded Discussion, you could sort your What's New list by content item and see that the threaded discussion for Unit 1 had 5 new responses and 3 new postings. Similarly, if you wanted to see how student Jill Hill was participating in the course, you could sort by name and see that she had posted 2 responses in the threaded discussion and a site to the Webliography. This is a great course management tool. If you haven't used it in the past, try it the next course you teach. You'll find it very useful.

Course Checklist, located beneath What's New, is more helpful for students than for instructors. From the student view, the Course Checklist section displays a list of all course assignments and their associated due dates, as well as a check box for each item so students can monitor their progress in regard to assigned course work. Ideally, this course "task list" will help students be aware of assignment due dates and prompt them to complete assignments on time. Students can sort their list by Unit or by Due Date. Remember, too, that (except for exams) students manually check off the items in their Course work checklist. As a result, a student may check off an item that is not necessarily complete. Remember, this is a useful management tool for students only if they are aware of it and if you, as the instructor, set it up through the use of the Scheduler. It's a good idea to mention this tool in the Syllabus and in an Announcement on the Course Homepage.

       --Kenneth Switzer, Ph.D.