Everything that can be taught, learned, and stored (in language, music, painting, sculpture, and other representative media) is one or another of four forms of knowledge---
1. Concepts: the cognitive (thinking) organization of reality into categories, or kinds of things that share certain similarities. The class of housecats. The class of things that have it pretty easy, laying around and licking themselves. The class of felines. The class of mammals. The class of animals. The class of living beings.
2. Facts: statements (subject-predicate) about the features of individual things that are examples of concepts. Everything is an example of one, usually more than one, concept---at the very least, the concept of “thing.” “James Madison was the fourth president.” He was in the class of presidents as well as a member (example) of the class of things that wear pants, things that ride horses, things that write amendments, things that are male, etc.
3. Rules, or propositions. Statements about how categories/concepts are connected:
a. Categorical rules, or relationships:
All (things in the class of) dogs are (in the class of) canines.
Some (things in the class of) wars are (in the class of things that are) moral.
No (person in the class of) politician can be (in the class of person who is) trusted with your money.
b. Hypothetical or causal rules. The (more/less) X, the (more/less) Y. Whenever X occurs, Y
occurs. If and only if X occurs, does Y occur.
4. Cognitive routines. Sequences of steps that solve, explain, describe, sort, build, fix, make a model of, make a logical case for or against. Knowledge analysis reveals the steps of routines and the knowledge elements (concepts, rules) needed to perform each step. For instance, the second step in the routine for sounding out words might be saying the first sound----rrrr in ram. This step requires the knowledge elements of: (1) making the rrrr sound; (2) connecting a certain sound---rrrr--with the letter r.
These four forms of knowledge are our way of representing reality. Each form of knowledge is the result of logical operations of inductive and deductive inference, described by Aristotle, John Stuart Mill (A system of logic. 1843), Engelmann and Carnine (Theory of instruction, 1991), and Engelmann and Steeley (Inferred functions of performance and learning, 2003). The acquisition of knowledge (learning something new---concept, fact, rule, routine) involves the “learning mechanism” (sense organs and brain):
1. Examining events to identify features.
2. Comparing events to identity features that go together. For instance, a child’s brain says, “Teacher shows these objects, and for each one, she says, ‘This is red.’ The objects are different in lots of ways (size, shape), but are always the same color. So, I hypothesize that ‘red’ signifies the color.”
3. Contrasting events to find what’s different. For instance, the child’s brain now says,
“She held up one of the objects and said, ‘This is red.’ Then she held up an object just like it in size and shape, except for color, and said, ‘This is NOT red.’”
4. Drawing a conclusion from the evidence. The child’s brain says, “Okay, I get it. I am now prepared to draw the inductive inference (I have figured out, I have constructed knowledge) that ‘red signifies color.”)
Steps 1-4 are the steps in inductive reasoning.
The application or generalization of knowledge to new instances is also a logical process---deductive reasoning. It’s as if the learning mechanism:
1. States a rule. “The word ‘red’ signifies THAT color.”
2. Examines a fact. “This new object has THAT color.”
3. Draws a conclusion. “So, that new object must BE red.”
It makes sense to organize instruction (communication) so that words and demonstrations (examples) enable the learning mechanism easily to perform its built-in cognitive routines of inductive and deductive reasoning. It doesn’t make sense for students to struggle to figure out what the teacher is getting at. Lots of students fall by the wayside, which accounts for decades of low achievement in certain subjects. Of course, struggle is fine when the whole point IS for students to use certain logical tools to figure out what is going on---in inquiry projects. “Gee, what do you make of THIS?!”
It turns out that there is an effective format (communication) for teaching each form of knowledge. The following are not the only formats that could be used, but they are consistent with how the learning mechanism operates.
| Teaching Facts |
Teaching Concepts with verbal definition, synonym, or examples. |
Teaching Rules | Teaching Routines | ||||
| Definition | Definition | Definition | Definition | ||||
|
Facts tell features of a subject that is a single thing. “This table (singular subject) is brown (predicate). ”The subject part of the statement tells what the statement is about—this table. The predicate part of the statements tells more Since a fact is just a statement that connects a singular subject with a feature of the subject—“Thomas Jefferson (subject) wrote the first draft of the Declaration of Independence b. Having students repeat it. |
Concepts carve reality into chunks, or kinds of things. Concepts are classes of things grouped by certain ways they are the same. Dogs, tables, Everything (every instance) IN a concept or class is an example. Note: a word (“red,” “canine,” “republic”) is not the concept. A word is just a signal that points to examples of the concept. For instance, “red” points to (names) that object over in the corner. The concept is not the word; the concept is the class of examples that share certain features. |
Rules are statements of connections among concepts. Today, there were 1400 orders for gold in the word; and the price of gold was $1500 an ounce. That is two fact statements. 1. “The number of gold orders today (a singular subject) is 1400 (predicate, tells more about the subject).”2. “The price of an ounce of gold today (a singular subject) is $1500 (predicate, tells more about the subject). But what if we had daily facts on the number of orders and the price of gold for 200 days? We might find a connection between the class of things that are orders for gold (2 orders, 500 orders, 100 orders, 1400 orders, 3 orders) and the price of an ounce of gold ($1000, $900, $1500, $750).We might find—and state a rule: Since a rule states how change in one class of things (concept) is connected to change in another class of things (concept), teach rules with by: Telling the rule; showing and identifying examples and nonexamples; having students identify examples and nonexamples. Or b. Inductive method. Show examples and help students figure out and then state the connection. “As you go from one example to the next, is it the case that when one variable changes, the other variable changes?” Yes. “How?” |
A routine is anything performed as steps: 1. Sounding out words, 4. Finding and identifying figures of speech in a poem. 5. Stating 6 rules that constitute a theory. 6. Stating 8 facts that constitute a description. Routines are a sequence of steps that do something. But you must already know certain knowledge elements in order to perform the steps. Therefore, when teaching routines, do a knowledge analysis to determine the steps, and the knowledge elements needed for each step. Knowledge analysis of sounding out words.
|
||||
|
Format or Procedure |
Format or Procedure |
Format or Procedure |
Format or Procedure |
||||
|
Gain attention. “Boys and girls. Eyes on me. Show me ready.”… “Yes, I love the way…” Frame the instruction. 2. Tell what they’ll do when you are Model. Say the fact. Lead. [If you think they need it.] Test. Students say the fact. |
Gain attention. “Boys and girls. “Yes, I love the way…” Frame the instruction. 2. Tell what they’ll do when you are done (objective). “When we’re done, you will…” Model. Three ways:a. Verbal definition. (1) “New word/concept---comet. What’s our new word/concept?.... Spell comet. … What’s our new word/concept?”... (2) Tell the definition. (3) Now say/show several examples, and point out the features from the definition. (4) Show nonexamples and point out absence of features that the examples had. Then do a few more examples for contrast. “This is NOT a comet. Notice it does NOT have…” (5) Test all examples and nonexamples (delayed acquisition test). “Is this a comet… How do you know?” [Student use definition.] Later, work on generalization items. b. Synonym. (2) Tell the new word and its synonym. “Listen, an orbit is a path shaped like a circle that one thing makes around another thing. Again, an orbit is…..” (3) Test. (4) Examples of new word and synonym. More examples. (5) Test. “Your turn to use our new word. (6) Give examples of USING the new word. (7) Give nonexamples. “This is an asteroid (one thing). It is going straight down to another thing—the earth. It is NOT making a circular path around the earth. So, it does not have an orbit.” (8) Test all. Use the examples and nonexamples. “Is this object making an orbit?....How do you know?” Students use Later, work on generalization items. c. Examples and nonexamples only---for sensory concepts. (1) “New word/concept---elliptical. What’s our new word/concept?.... What’s our new word/concept?”... (2) Examples. “This is elliptical.” “This is NOT elliptical.” “This is not elliptical.” “This IS elliptical.” “This is not elliptical.” “This is elliptical.” (3) Acquisition Test. Show all examples and nonexamples, and ask, “Is this elliptical?” Correct errors or verify. Now give a generalization test. “Is this elliptical?”   etc. |
Gain attention. “Boys and girls. Eyes on me. Show me ready.”… “Yes, I love the way…” Frame the instruction. 2. Tell what they’ll do when you are Model. Two ways: (1) Model. “Here’s a rule. The farther a solar system object is from the sun, the larger is its orbit.” [repeat?] (2) Test. “What’s our rule?” correct errors and verify. (3) Examples. “Look here (point). Mercury is the closest planet to the sun. Earth is farther from the sun than [More examples.] (4) Test. “Jupiter is farther from the sun than Earth. Which planet has a larger orbit?.... How do you know? [students use the rule.] Correct errors and verify. Later, work on generalization; e.g., the orbits of moons around planets. b. Inductive. “Boys and girls. Here are facts. Distance Orbit Merc 37 m 58 m Show solar system and give facts for each planet. Tell the rule each time.
Have students summarize the facts by saying the rule. (2) Test. Now give new examples—for instance, “Moon A is 2 million miles from Jupiter, and its orbit is 5 million miles. Moon B is 4 million miles from Jupiter. Whose orbit is larger? Moon 1 and Moon 2… How do you know?” [students use the rule.] More examples. |
Sounding out words Pre-teaching. 1. When I touch under these letters, 2. “Remember how you move your finger under the letters…. Like this… 3. “Let’s practice saying words Gain attention. “Boys and girls. Eyes on me. Show me ready.”… Frame the instruction. 2. Tell what they’ll do when you are done (objective). “When we’re done, you’ll read all these words.” Model. “Listen, aaaammmm." Lead. “You sound it out with me. Put your finger on the first ball….That’s it…Now, we’ll move our finger to each letter as WE say it…Get ready…aaaammmm" “Yes, aaammmm. Test. “Yes, aaammm. You sounded it out!” Listing the stages by which solar Pre-teaching. Frame the instruction. 2. Tell what they’ll do when you are done (objective). “When we’re done, you will explain how solar systems form by saying all the stages in order.” Model. “Stage one….” [name it and say a few things that happen.] Test. “What is the first stage…. Tell what happens.” Correct errors and verify. Model. “Stage two…. “ Test. “What is stage two…. Tell what happens.” Correct errors and verify. Model. “Stage three….” Test. “What is stage three…. Tell what happens.” Correct errors and verify. Model. “Stage four…” Test. “What is stage four…. Tell what happens.” Correct errors and verify. Integrate. “Stage one…” Correct errors and verify. Do all the stages. Now all together. When you teach routines, use several First format. Teacher does the steps and tells what she is doing. Student watches. Student joins in (e.g., tracks under letters, writes numerals in multiplication problem). Second format. Teacher tells students that they will do the routine (or a step in the routine). Teacher tells students WHAT they will do. (It is what students watched the teacher do in format Repeat with more steps. Repeat with more examples until students do all the steps with this sort of prompting (“Here’s what you’ll do…”) Third format. Teacher tells students that they will do the routine (or a step in the routine). Teacher has STUDENTS tell what they will do (which they did in format 2). Repeat with more steps. Repeat with more examples until students Fourth format. Teacher tells students that they will do the routine (or a step in a routine). Teacher reminds students of rules---“Make sure to….” Repeat with more steps. Repeat with more examples. Fifth format. By this format, students should be largely independent. They may need some error correction or part firming. |
