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The Four Forms of Knowledge and Formats for Teaching Each Kind

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
about the subject—is brown.

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
(predicate, tells more about Jefferson, a feature of Jefferson).”—teach a fact by 
a.  Stating the fact clearly, and

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,
furniture, red, color, male, female, fast, forest, trees. 

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. 
Therefore, to teach a concept you teach what features are common to examples in the class, and what features are not.

 

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:
“The higher the number of orders per day, the higher the price of gold per ounce.”  Or, “The price of gold varies directly with orders for gold.”

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:
a.  Deductive method.

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, 
2. Solving a math problem, 
3. Searching for words in a text (“ant” “said”) in order to answer a literal question (What did the ant say?”)

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.

Steps  Knowledge
elements

 1.  Put finger or
focus on spot to
the left of the first letter.

2.  Move finger or
direct focus to the
first letter.

3.  Say the sound
that goes with the
first letter.

4.  Move finger or
eyes to the second
letter.

5.  Say the sound
that goes with the
next letter.

Repeat.

 1a.  What a
letter is. 

1b. Where the
first letter is.

1c.  Where the
spot to the left
of the first letter
is.

2a.  How to move
finger or eyes to
the right, and to
stop at the first
letter.

3.  Know the sound
that goes with the
first letter (i.e., has
said the correct
sound reliably
before).

4a.  Where the
second letter is.

4b. .  How to move
finger or eyes to
the right, and to
stop at the first letter.

5a.  Know the
sound that goes
with the second
letter (i.e., has
said the correct
sound reliably before).

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…”
“Now you’re ready to learn!”

Frame the instruction.
1.  Tell what you’ll be working on.  
“Now we’ll…”

2.  Tell what they’ll do when you are
done (objective).  
“When we’re done, you will…”

Model.  Say the fact. 

Lead.  [If you think they need it.]

Test.  Students say the fact.

Gain attention.  “Boys and girls. 
Eyes on me.  Show me ready.”…                                                           

“Yes, I love the way…” 
“Now you’re ready to learn!”

Frame the instruction.
1.  Tell what you’ll be working on. 
“Now we’ll…”

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.
“Acometis an icy small solar system body 
(genus) that sometimes has a thin, fuzzy, temporary atmosphere and sometimes also a tail, and orbits the solar system from a few years to hundreds of thousands of years (difference from other solar system bodies).” http://en.wikipedia.org/wiki/Comet

(3) Now say/show several examples, and point out the features from the definition.
“This is a comet. Notice the….”

(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.  
(1) “New word. Orbit. 
What’s our new word?...
Spell orbit…
What’s our new word?”…

(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.  
 “What’s another way to say a path shaped like
a circle that one thing makes around another
thing.”… orbit…Yes, orbit IS a path shaped like
a circle that one thing makes around another
thing.”

(4)  Examples of new word and synonym.
“Listen, the car is making a path shaped like a circle around the house.  I’ll say that sentence with our new word.  The car is making an orbit around the house. ”

More examples.

(5) Test.  “Your turn to use our new word. 
Listen, the moon was making a path shaped
like a circle around the sun. Say that with our
new word.” The moon was making an orbit
around the sun. Verification. “Yes,…”

(6) Give examples of USING the new word
“Here’s an orbit. [point.] The earth (one thing) is making a path (this circular line) around another thing—the sun…Here’s the earth’s moon (one thing) making a path
(circular line) around another thing---the earth.” Etc.

(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
the idea of circular path.

Later, work on generalization items.

c.  Examples and nonexamples only---for sensory concepts.

 (1) “New word/concept---elliptical.  What’s our new word/concept?....
Spell elliptical. …  

What’s our new word/concept?”...

(2) Examples.  
“This is elliptical.” Ellip1 
“This is elliptical.” Ellip2
“This is elliptical.” Ellip3

“This is elliptical.”Ellip4

“This is NOT elliptical.” Ellip5

“This is not elliptical.” Ellip6

“This IS elliptical.” Ellip7

“This is not elliptical.”Ellip8

“This is elliptical.” Ellip9

(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?”
Ellip10    Ellip11   Ellip12                                                               etc.

Gain attention. “Boys and girls. Eyes on me. Show me ready.”…                                                         

“Yes, I love the way…” 
“Now you’re ready to learn!”

Frame the instruction.
1.  Tell what you’ll be working on. 
“Now we’ll…”

2.  Tell what they’ll do when you are
done (objective).  “When we’re done,
you will…”

Model.  Two ways:
(1) Deductive (rule then --> examples)
and 
(2) Inductive (examples then figure
out --> rule)
a.  Deductive.
You tell the rule about how things are connected, and then give examples of it.  (deductive)

(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.
Its orbit is the smallest.”  [with older kids use numbers, in miles.] 

Earth is farther from the sun than
Mercury (point and/or tell distance) and Earth’s orbit is larger than the orbit of Mercury.”

[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.  
(1) Give examples.

“Boys and girls. Here are facts.
               

Distance           Orbit
from sun 
m = millions of miles

Merc    37 m        58 m 
Venus  65 m        108 m 
Earth   93 m        150 m 
Mars    140 m      228 m 
Jupi     484 m      778 m 
Satur   884m       1427 m 
Uran    1786 m    2870 m 
Nept    2790 m    4497 m 
Pluto    3627 m    4914 m

Show solar system and give facts for each planet. Tell the rule each time. 


“Earth is farther from the sun than Venus, and the orbit of the Earth is
greater than the orbit of Venus.”

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.  Teach all of the pre-skill elements identified by knowledge analysis—early and continually. Review and firm these at the start of the new lesson.

1.  When I touch under these letters,
you say the sound.”

2.  “Remember how you move your finger under the letters…. Like this…
Now you do it.”

3.  “Let’s practice saying words
slowly… Say mmmaaaa…. Now say
aaaamm.”

Gain attention.  “Boys and girls. Eyes on me.  Show me ready.”…                   
“Yes, I love the way…”
“Now you’re ready to learn!”

Frame the instruction.
1. Tell what you’ll be working on.
“Now we’ll learn how to read words.”

2.  Tell what they’ll do when you are done (objective). “When we’re done, you’ll read all these words.”

Model.
“I’ll show you how to sound out this word. I’ll say it ssslloowwllyy. When I touch under the letters I’ll say the sounds. I don’t stop between sounds.
Put your finger on the first ball…
That’s it…Get ready to move your finger to each letter as I say it.

“Listen,  aaaammmm."
"Again, aaaammm."

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.
“Your turn to sound out this word. Put your finger on the first ball…That’s it…Now, move your finger to each letter as YOU say it…Get ready…aaaammmm

“Yes, aaammm. You sounded it out!”

Listing the stages by which solar
systems form

Pre-teaching.
1.  Teach all of the pre-skill elements identified by knowledge analysis—early and continually. Review and firm these at the start of the new lesson. For example, vocabulary/concepts that are used in defining each stage.

Frame the instruction.
1.  Tell what you’ll be working on.
“Now we’ll explain how solar systems form. ”

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.  
“Boys and girls. Here are the stages by which solar systems form. 
[Students might read from text.]

“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…. “
[name it and say a few things that happen.]

Test.  “What is stage two…. Tell what happens.”  Correct errors and verify.

Model. “Stage three….” 
[name it and say a few things that happen.]

Test.  “What is stage three…. Tell what happens.”  Correct errors and verify.

Model.  “Stage four…” 
[name it and say a few things that happen.]

Test.  “What is stage four…. Tell what happens.”  Correct errors and verify.

Integrate.
“Now let’s do all the stages that explain how solar systems form…”
[Prompt students to read from their notes or from text.”

“Stage one…”  Correct errors and verify.

Do all the stages.

Now all together.
“Now YOU say the stages all together.” 
Correct errors and verify.

When you teach routines, use several
formats, from mostly teacher to all students.

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). 
Repeat with more examples until students do their part.

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 
1).  Teacher has students REPEAT what they will do.

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
do all the steps with this sort of prompt.

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.

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