- verbal associations - facts and lists: (this one thing goes with that one thing);
- concepts - sensory and higher-order: (all these things have some features in common);
- rule relationships: (this set of things goes with that set of things); and
- cognitive routines: (to read all of these words, or to solve all of these math problems, or to write these kinds of essays, do steps 1, 2, 3, 4, 5 and 6)
In this post we'll discuss how these connections are learned and, thus, how knowledge is acquired by a learner.
Generally, facts and lists are learned differently than concepts, rules, and routines.
1. Facts and Lists
Facts and Lists (statements that connect specific things) are learned by simply memorizing the connection. The memorization could be long term or short term depending on how long the knowledge is needed by the learner. If the knowledge is important, it is often useful for the knowledge to be practiced (rehearsed) so that it retained in long term memory. Otherwise, the knowledge will have to be reactivated every time it is needed (this is where Google helps tremendously).
2. Concepts, Rules or Propositions, and Routines
In the case of concepts, rules or propositions, and routines, however, the student has to figure out the general idea that is revealed by the examples. The student gets from the examples (specifics) to the general idea embedded in the examples by inductive reasoning. In other words, the learner performs a sequence of logical operations, beginning with examples and ending with a general idea. That is, the learner:
(a) observes examples and nonexamples (examples of concepts, or rules/propositions, or routines);
(b) performs a series (a routine) of logical operations on what it observes; and
(c) arrives at (induces, figures out, discovers, “gets”) the general idea (the concept, rule, or routine) revealed by the examples and nonexamples.
The teacher's job is to induce this learning by helping the student to “get”(see, grasp, figure out, learn) that the examples reveal a connection—a general idea (verbal association, concept, rule-relationship, cognitive routine). The teacher uses examples (and nonexamples) so that students learn (grasp, figure out, induce) the general idea (the knowledge) that connects the examples.
There are a few ways this can be done.
a. Method of agreement.. The student identifies what is common to the examples and not found in nonexamples. What is common is the general idea; e.g., redness. This is because the examples differ in nonessential features but agree in the essential feature. If they are all “treated” the same way (named, solved), it must be because of the way in which they agree. So, the teacher should present a range of examples; help the student to compare and contrast the examples; and to identify the sameness—which is the general idea (concept, rule, routine).
b. Method of difference. The student identifies what is different between the examples and nonexamples. What is different must be what makes the difference. This feature that is different (present in examples, but missing in nonexamples) must be the general idea. Examples and nonexamples are the same in nonessential features but differ in the essential feature. If they are treated differently, the difference (feature) must be what makes the difference. This difference is the general idea. So, the teacher should juxtapose examples and nonexamples; help the student to compare and contrast and to identify the difference that makes the difference.
c. Method of concomitant variation. The student identifies how one kind of thing changes along with (in the same or different direction as) another kind of thing. If one thing changes, and everything else stays the same, and then another thing changes in a regular way, then it is reasonable to infer that they go together—they are in a functional or causal relationship. This form of inductive reasoning is used especially in discovering rule relationships.
Let's now take a look at how these play out in learning concepts, rules, and routines.
A concept is a set of events or things that have one or more common features. The common features are the concept. The word (“red”) is not the concept. It is merely a name for the concept. “Red” signifies or points to the feature—the redness. So, to teach a concept, the teacher should teach the student what are the common features of the examples and the name of the features (“red”) so that the student can communicate the concept to other persons.
What makes a concept a basic (or sensory) concept, as opposed to a higher-order concept? The “stuff” of basic concepts is right in front of the learners eyes, ears, or skin.
- Red: Not the word “red”; the color your eye perceives that is called “red.”
- On: Not the word “on”; the way things are arranged that is called “on.”
- Hot: Not the word “hot”; the way your skin feels when you touch “hot.”
In basic concepts, such as red, on, and s says /sss/, the “stuff” that defines the concept is right there to be seen, heard, or felt. However, the stuff that defines higher-order concepts is not right in front of the learner's senses. The learner can’t see in one place all the stuff that defines democracy, or justice, or furniture, or symmetry. The stuff that defines democracy (elections, for example) is spread over time and place and groups. So, to teach a higher-order concept the teacher should first give a verbal definition that draws a big circle around all the stuff in the concept. And then give examples and nonexamples so that student sees the actual stuff that defines the concept. Here's a simple example of teaching the higher-order concept "bird":
Rules are statements that connect not one thing and another thing (e.g., name and date), but connect whole sets of things (concepts):
- When demand increases, price increases.
- All dogs are canines.
- Rules can be shown on diagrams; e.g., graphs and models of interconnections.
1. Deductive method: from general (rule) to specific (examples). In the deductive method the teacher teaches the rule statement first. Then examples and nonexamples are presented, as with concepts. Then the teacher tests all examples and nonexamples to see if the student has learned the rule.
Teacher: “Is this (verbal description of graph) an example of the demand-price rule?”
Teacher: “How do you know?”
Student states rule.
Then the teacher generalizes to/tests new examples and nonexamples. Here's another example of using the deductive method.
“The question is, Is there a connection between how steep an inclined plane is and how long it takes a ball to roll down it?”
The teacher then tells the student the rule-relationship (the steeper the inclined plane, the less time it takes the ball to roll down the inclined plane) and then show examples using inclined planes of different angles. These examples would confirm the rule.
2. Inductive method: from specific (examples) to general (rule). In the deductive method the teacher presents a range of examples first (e.g., different price-demand curves): cars, oil, movies. Then the teacher shows students how to compare the examples and to identify the sameness—the relationship.
One variable goes up and the other variable goes up. “Demand varies directly with price.” Then the teacher present nonexamples, and show (in relation to the rule) how they are nonexamples. “Demand is increasing, but price stays the same. That does not fit the rule. Then the teacher gives new examples and nonexamples, and has the student say if they are or are not examples, and how he knows.
Here's an example of using the inductive method on the inclined plane ball-rolling example above.
The teacher has the student do an experiment by rolling balls down inclined planes of different angles, measuring how long it takes each ball to roll down, and then has the student draw a conclusion.
This way requires more skills. (In the deductive method, the student merely compares examples with the rule. “Yup, the ball takes less time when the angle is steeper.”) For example, the student has to change the angles, measure the times, write the measurements, compare and contrast the instances, and figure out the connection. This means the teacher would have to teach these pre-skills before students do the experiment.
A Routine is a sequence of steps for getting something done. Solving math problems, sounding out words, writing essays, brushing your teeth, brushing someone else’s teeth. Routines are taught in the same way that lists are taught. The teachers models each step (or a few steps), then adds a few more steps (and then models the whole sequence so far, then adds a few more steps; etc., until the routine is complete
Applying Acquired Knowledge
That's how students acquire knowledge, but we also we also want them to be able to apply and extend the knowledge they've acquired to new examples. In general, we apply or generalize knowledge through deductive reasoning. That is, the learner:
(1) has/knows/can say a general idea (concept, rule/proposition, routine);
(2) uses the general idea (definition of a concept, or statement of a rule, or features of the things handled by the routine; e.g., math problems, words) to examine a possible new example using the information in (2);
(3) “decides” whether the new thing fits (is an example of) the definition, rule, or routine (“Can you solve this with FOIL?”); and
(4) “treats” the example accordingly--names it (concept), explains it (with the rule), solves it (with the routine).
Here's an example: If a general rule or proposition (learned either via inductive reasoning or is being told) is that when demand increases, price increases, and if you notice that the demand for oil is increasing, the learner will deduce (predict) that the price of oil will increase.
It is a simple deductive syllogism. When demand increases, price increases. The demand for commodities are increasing. oil is a kind of commodity. Therefore, the price of oil will increase. “How do you know?” “Because when demand increases, price increases, and an increase in the demand for oil is an example of an increase in demand.”
Now we know about the forms of knowledge and how that knowledge is acquired and applied. Next we'll learn how knowledge is retained past the acquisition stage. And, then, we can get to the good stuff -- the differences between novices and experts and why the "struggling" of an novice is not the same as the "struggling" of an expert.
*So far these first two posts have been adaptive from the works of Martin Kozloff, Professor of Education. See Kozloff's site for much more detail on these topics.