What is Knowledge? How Do You Get It?
And What Does THAT Tell You About How to Design Instruction?
Martin Kozloff
2014
What Teaching Is—Applied Logic
Teaching is in the branch of philosophy called epistemology.
λόγος
ἐπιστήμη [epistemeh]--knowledge.
[logos]--study of. What branch of epistemology? Applied logic. What is logic?
A branch of philosophy and mathematics that deals with the formal principles, methods and
criteria of validity of inference, reasoning and knowledge. Let’s look at those three words.
1. Inference. Generalizations that are the product of reasoning.
a. Inductive reasoning. Start with events--specifics. The teacher holds up objects that differ in size
and shape, but are the same in one way, and for each one she says, “This is kokkivo.” [Gr. v = nnn] Then
she juxtaposes objects that are the same in size and shape, but that differ in one way---color. She calls
one of them, “This is kokkivo,” and she calls the other one “This is not kokkivo.” At each step in the
communication, the learning mechanism performs a logical operation---a reasoning step.
The learning mechanism:
(1) describes each event; (2) compares events all called “kokkivo” to find sameness (“These look
the same in one way and are called kokkivo.”), (3) contrasts events to find difference (“This one
looks different in only one way from all of them that were called “kokkivo,” and the teacher called this
one ‘not kokkivo.’”), (4) draws a conclusion (generalization) about what the events reveal.
“Kokkivo must mean the color that all the ones called kokkivo had and the ones called not kokkivo
didn’t have.”
b. Deductive reasoning. Start with a general (a definition of a concept, a rule about how things are
connected, a routine for solving a kind of problem). Examine a new event and see that it has the
defining features of the concept, rule, or routine. Deduce how to treat the example (name it,
make a prediction about it, solve it).
“Here’s a new one. It has the same color as the ones called ‘kokkivo’---the very feature that the
ones called ‘not kokkivo’ didn’t have. We know that kokkivo is the name of a color. This new one
HAS that color. So, by deduction, this new one must be the color kokkivo.”
2. Reasoning. The sequence of logical operations (steps) in thinking from specifics to a generalization
(inductive) or from a general (e.g., rule) to a prediction about specifics. An inference (#1 above) is
valid if and only if the proper steps were followed, and the conclusion (inductive or deductive
inference) clearly follows the steps in the argument/reasoning.
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3. Knowledge is the conclusion, the inference, the generalization made on the basis of valid reasoning.
The person has learned a new concept by induction (kokkivo), has learned the color name of a new
event by deduction from the definition, or has performed a new routine (e.g., solved a new
problem based on a deduction from a general problem-solving routine).
“Every time she said ‘I’ll sound out this word,’ she said the first sound on the left; then the next
sound on the right; and then the last sound on the right; and she didn’t stop between sounds.”
ma, am, sa, sam, mam.
“Now she’s showing a new word. mas. I bet she does it the usual way.” [Or, “My turn to sound it
out? No problem. I’ll just use the routine that I have internalized.”]
Here’s the logic of generalization via deductive reasoning.
1. “So far, you sound out words by following these steps…..” [First premise. Rule.]
2. “Here’s a word.” [Fact relevant to the first premise.]
3. “So, I’ll sound out this word by following the steps.” [Conclusion deduced from new fact and
first premise.]
Well, teaching students to acquire new knowledge (by inductive reasoning) and to use or apply
knowledge (by deductive reasoning) is exactly what successful teachers do.
But it’s easy to confuse the learning mechanism. What if the teacher uses examples that don’t reveal
the defining features of a concept? What if she uses too few examples for the learning mechanism to
“get it” (to infer the proper generalization)? What if students don’t know what they are supposed to
pay attention to? What if students don’t have the pre-skills needed to do what the teacher does?
So, instruction has to be isomorphic with the logical operations of induction and deduction that the
learner is trying to perform. It has to support each operation/step.
What Logical Operations the Learning Mechanism
is Prepared to Do.
How Instructional Communication Should be
Isomorphic with the Operations of the Learning
Mechanism.
1a. Teach how to examine events. A pre-skill.
1. Examines events--which will after a process of
inductive reasoning be seen as examples of
something larger---a concept, rule, or routine.
1b. Engage students with the communications by
framing instruction: Topic, what they will do,
standards.
1c. Teach what a fact, concept, rule, and routine
are. Teach the logical operations by which they
are learned so that the students can do the steps
with intention and in a planful way, and can talk
themselves through the reasoning.
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“Well, she called that one linear and this one not
linear. She treated them differently, so they must
BE different in some way. I’ll compare and
contrast them to find the difference in features
than made the difference in how she named
them.”
2. Identifies features of events.
2. Present events that are examples of a concept,
rule, or routine.
Use events whose features are at first easily
available, distinct. Teach students to say and list
the features.
3. Compares events to identify sameness of
features and sameness of treatment of each event
in the midst of differences.
3. Use a set of examples, one after another
(acquisition set) that differ in nondefining ways
but are the same in the defining features. Treat
each of these “same” examples the same way--name them, sort them, list them, solve them.
@$!*^ “This is flerm.”
Q#)!=+ “This is flerm.”
M!%>! “This is flerm.”
Teach students to note and to identify the
samenesses and the differences across events.
Teach students to note that events that have
sameness are treated the same. Have them say
how they are the same and how this sameness
leads to how they are treated.
[The enables the learning mechanism to use the
method of agreement.]
4. Contrasts events to identify differences in
features and associated differences in treatment.
4. Present events whose nondefining features are
the same but whose defining features are
different. Treat them differently---different name,
list, group, solution.
@$!*^ “This is flerm.”
@$ *^ “This is NOT flerm.”
Teach students to note and to identify the
samenesses and the differences in features across
events; to note differences in how they are
treated; and to identify the difference in features
that is associated with differences in how they are
treated.
This enables the learning mechanism to use the
method of difference.
Note: if you first use a set of examples (#3), and
then juxtapose examples and nonexamples (#4),
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this enables students to use the very powerful (in
terms of clarity and validity) joint method of
agreement and difference.
5. Makes a generalization.
5. Teach students to state a generalization that
connects the samenesses and their associated
treatment.
“! (how events called ‘flerm’ were the same) is
flerm (the common treatment of events with !).”
“Every rock that she called granite consisted solely
of mica, feldspar and quartz. No rock that did not
consist of all three—mica, feldspar, and quartz--was named granite. Therefore, granite (the
concept) is an igneous rock (genus) composed
solely of mica, feldspar, and quartz (difference
between granite, as an igneous rock, and other
igneous rocks).” (Concept as a generalization)
a. Teach students to use proper form for simple
declarative statements of fact knowledge.
“She performed steps 1-6 every with the 10
problems arranged like this---(a +/- b)(c +/- d). The
numbers were different, but she treated them (the
steps) the same way. When the numbers were
arranged differently—a (a +/- b)(c +/- d)---she did
NOT perform steps 1-6. So, steps 1-6 must be the
general way to solve that kind of problem.
(Routine as a generalization).
Subject is in the larger class of (genus) and it has
the defining features a, b, c (in contrast to other
classes in that larger genus= the difference).
“Planet 1 is 1,000,000 miles from its sun, and its
orbit is 4,000,000 miles.
Planet 2 is 10,000,000 miles from the sun, and its
orbit is 35,000,000 miles.
Planet 3 is 50,000,000 miles from the sun, and its
orbit is 160,000,000 miles.
Therefore, the greater the distance a planet is
from its sun (at least with these examples), the
greater is its orbit around the sun.” (Rule as a
generalization)
(2) Whenever X (happens, increases, decreases), Y
(happens, decreases, increases).
“James Madison (subject) was the fourth President
of the United States (predicate).
b. Teach students to use proper form for verbal
definitions of abstract concepts.
c. Teach students to state rules in proper form
that reflects the examples.
(1) All (no, some) things in the class of S are in the
larger class of P.”
(3) Whenever X (happens, increases, decreases), Y
(happens, increases, decreases), but only if Z
(happens, increases, decreases). –intervening
variable.
(4) X is a necessary condition for Y. [Y doesn’t
happen without X.]
(5) X is a sufficient condition for Y. [Whenever X
happens, Y happens.]
(6) When A, B, C, and D happen (in a sequence of
connections), W, X, and Y happen. State and
diagram.
d. Teach students to state the steps in a routine,
as well as the concepts and rules that govern each
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step. “Multiply the numbers in the ones column.
If the product is 10 or more….”
What is Knowledge?
There are no sounds, no colors, no odors, no hot, no cold, no objects, no voices, no any THING outside
of our consciousness.
There is only energy in the form of waves and particles.
Pictures do not come into our eyes. Only photons emitted by atoms.
http://science.howstuffworks.com/x-ray1.htm
Sounds, melodies, and words do not come into our ears. Only sound waves.
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http://www.personal.psu.edu/afr3/blogs/SIOW/audio-sound-waves-img1%5B1%5D.jpg.
The scents of roses, fish heads, and baby breath does not come into our nose. Only volatile molecules
given off by objects.
http://blog.lib.umn.edu/efans/ygnews/olfaction.jpg
The human learning mechanism transforms incoming waves, particles, and molecules into a
representation of reality consisting of objects, persons, groupings, places, and events, all happening and
connected in time and space.
These representations of reality are what we call “knowledge.”
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Because human learning mechanisms operate much the same way, human beings experience a world in
common.
Human beings communicate our representations (knowledge) with language, music, dance, pictures,
and sculpture.
Here’s how it works.
Coming in From the Outside
The Learning Mechanism: Sense Organs and Brain
(1) Energy particles/waves,  (2) The neural (electro- 
and molecules, strike or
chemical) apparatus
enter sense organs:
of each sense organ
eyes, ears, skin, nose .
transforms incoming
energy particles/waves,
and molecules, into
sensations: color, sound,
touch, odor.
(3) Specialized centers in the brain transform
these sensations into the experience of things,
kinds of things, and connections among
things---or knowledge. In other words, the
learning mechanism transforms incoming
energy and molecules into a representation
of reality---a reality of individual things, classes
(kinds) of things, and connections among things.
http://www.mpoweruk.com/images/emspectrum.gif
Let’s take vision as an example.
Incoming photons (light) are transformed into the sensation of color by the neural apparatus of rods,
cones, and nerve endings at the back of the eye---retina.
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The retina is lined with many millions of photoreceptor cells that consist of two types: 7 million
cones provide color information and sharpness of images, and 120 million rods (Figure 3) are
extremely sensitive detectors of white light to provide night vision. …The outer segments (tops)
of the rods and cones contain a region filled with membrane-bound discs, which contain
proteins bound to the chromophore 11-cis-retinal. (A chromophore is a molecule that can
absorb light at a specific wavelength, and thus typically displays a characteristic color.) When
visible light hits the chromophore, the chromophore undergoes an isomerization, or change in
molecular arrangement, to all-trans-retinal…. The new form of retinal does not fit as well into
the protein, and so a series of conformational changes in the protein begins. As the protein
changes its conformation, it initiates a cascade of biochemical reactions that result in the
closing of Na+ channels in the cell membrane… When the Na+ channels are closed, however, a
large potential difference builds up across the plasma membrane (inside the cell becomes more
negative and outside the cell becomes more positive). This potential difference is passed along
to an adjoining nerve cell as an electrical impulse at the synaptic terminal, the place where
these two cells meet. The nerve cell carries this impulse to the brain, where the visual
information is interpreted. [As colors and shapes. mk]
http://www.chemistry.wustl.edu/~edudev/LabTutorials/Vision/Vision.html
Here’s the big question. How does a kaleidoscope of sensations---color, touch, odor, sound….
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Sensations
become (1) individual things and their features, (2) classes of things defined by common features
(concepts), and (3) connections among things (connections understood as systems, causal relationships,
and category relationships—such as all cats are in the class of felines)?
Knowledge of…..
Shape
Plane figure
Triangle
Equilateral
triangle
Red equilateral
triangle
Living being
Part of plant
Flower
Rose
Red rose
Beautiful red rose
that smells great-and not el stinko.
Living beings
Human beings
Mother and infant
Loving and protective
mother and comfy
infant
Systems of living beings
Ecosystems
Forests
Evergreen forest
Here’s the answer. Look at column (3) in the above diagram.
1. We are born with a “learning mechanism”—brain, eyes, ears, skin, and nose (Engelmann and
Carnine, 1991).
2. This learning mechanism is set to perform a series of logical steps (“operations”) with the sensations
of color, sound, odor, and touch made by our sense organs.
3. This series of logical steps (“operations”) transforms sensations (colors, odors, touch, and sounds)
into things, kinds of things, and connections among kinds of things.
4. Connections are in space (near, far, on, next to), in time (first, next, before, after), in togetherness
(red is a kind of color; poison is not a kind of food), in causation (When X happens, Y will happen.),
and in a series of steps (routine) that gets something done (“If you do 1, 2, and 3, you produce
effect 4.”).
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5. We use language (as well as painting, music, dance, and sculpture) to represent WHAT the learning
mechanism has accomplished when it transforms sensations into things, kinds of things, and
connections—that is, transforms sensations into a world that seems to be really there---solid.
6. These representations are called “knowledge.”
7. There are different forms or kinds of knowledge---that is, representations: facts (knowledge of
features of individual things), concepts (knowledge of features of kinds of things), rule-relationships
and routines (knowledge of connections). Like this.
a. Knowledge of features of individual things: fact knowledge. Statements of subject (individual
thing) and predicate (feature). “This table is brown, has four legs, and is made of pine.”
b. Knowledge of features of classes of things: concept knowledge. Statements of definitions.
“Monarchy (class of things) is a political system that involves (1) rule by one person (2) usually on
the basis of hereditary ascension (the two features of all things that are political systems grouped
together and called ‘monarchy because they share those two features).”
c. Knowledge of connections (three kinds) among classes of things:
(1) Category-rule knowledge. Statements of category propositions. “This class of things is inside
that class of things.” “All mushrooms are fungi.” Or “All things in the class of mushrooms are
also in the larger class of things that are fungi.”
(2) Causal (hypothetical or functional) rule knowledge. Statements of causal propositions.
“When this happens, then that happens.” Or, “When the temperature of water decreases to
32 degrees Fahrenheit, the molecules crystallize.” Or more precisely, “If and only if the
temperature of water decreases to 32 degrees Fahrenheit, do the molecules crystallize.”
d. Routine knowledge. “When you do steps 1, 2, 3, and 4, you end up with a product of some
kind.” “To sound out a word, say the first letter-sound slowly, and don’t stop; then say the
next letter-sound slowly; and don’t stop; and then say the last letter-sound slowly.”
And that’s all the kinds of knowledge there are.
Therefore, everything you know or can know or can communicate boils down to these kinds of
knowledge: facts, concepts (sensory and abstract), rules (categorical and causal), and routines.
Therefore, our reality IS of (and is ONLY of) facts, concepts (sensory and abstract), rules (categorical
and causal), and routines.
Let’s take a look at each one.
8. Fact knowledge is what we say, dance, write, paint, and chisel about the features of individual
things. Fact knowledge is best communicated with simple declarative statements of subject and
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predicate.
“This meat chunk (individual thing) is greasy (a feature of this meat chunk).” A fact statement
about ONE meat chunk.
“Today (individual thing: subject of a fact statement) is Thursday (a feature of this day:
predicate of a fact statement).
9. Concept knowledge is what we say, dance, write, paint, and chisel about features of kinds
(categories, classes) of things. Imagine that the learning mechanism sees how the following individual
things are the same in important ways (shape of body, teeth, claws, how eyes operate, how they hunt),
despite many differences (size, fur color, where they live), and so it puts these individual things in the
same place---a circle.
The class (of things that are) tiger felines.
The class (of things that are) leopard felines
The class (of things that are) felines.
The class (of things that are) lion felines.
The class (of things that are) house cat felines.
The class (of things that are) cougar felines.
The class (of things that are) cheetah
felines.
And the learning mechanism represents this ACT of categorizing all these beings on the basis of their
identified samenesses with a definition that tells the common features---the basis for the grouping.
Felines are a class of mammals (genus: the larger class in which felines and other mammals
exist) that have four legs, retractable claws, eyes with elliptical pupils and a large number of
rods that permit accurate vision in low light, long canine teeth, that hunt, and that make “the
motorcycle” with their hind leg if you tickle their tum-tum (difference between felines and other
animals in the class of mammals).
10. Category rule knowledge (knowledge of ONE kind of connection) is what we say, dance, write,
paint, and chisel about how things are connected in togetherness. One class of things is all part of,
partly part of, or not at all part of other classes of things. Here are examples. Notice kinds of
togetherness---all of one class is (a) INSIDE, (b) partly inside and partly outside, or (c) all outside another
class/category.
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a. All (things in the class/concept of) maple trees are also in the larger class/concept of (things that
are) deciduous trees. Or, All maple trees are deciduous.
The class/concept of (all things that are) deciduous trees.
The class/concept of maple trees.
The class/concept of beech trees.
The class/concept of willow trees.
The class/concept of oak .trees,
Here’s another example of category rule knowledge of togetherness among classes/concepts.
All forests are ecosystems, or All things that are in the class/concept of forests are also into the
larger class/concept of things that are ecosystems.
And,
All (things that are in the class of) ecosystems are also (in the larger class of things that are)
systems of living beings. Or, all ecosystems are systems of living beings.
The class/concept of systems of living beings.
The class/concept of plant ecosystems: forests,
deserts, marshes, terrariums.
The class/concept of human ecosystems (civilizations,
societies, communities, families).
The class/concept of animal ecosystems (predators and prey).
b. No (things in the class/concept) of apples are also in the class/concept of (things that are)
vegetables. Or, No apples are vegetables.
The class/ concept of (all things that are) vegetables.
The class of (all things that are) apples.
c. Some (things in the class/concept of) mushrooms are also in the class/concept of (things that are)
poisonous.
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The class of mimes
The class of (all things that are) poisonous.
The class of things that are mushrooms.
The class of things that are Woody Allen.
The class of things that are drain cleaner.
The class of things that are arsenic.
Okay, so we looked at fact knowledge, concept knowledge, and category rule knowledge. Now….
11. Causal rule knowledge (knowledge of another kind of connection) is what we say, dance, write,
paint, and chisel about how things are connected through time----When X happens (earlier), Y happens
(later).
We aren’t usually sure that one earlier thing MAKES another later thing happen. One thing---a
change in X---is simply followed by a change in Y. An earlier KIND of thing may not MAKE another
later kind of thing happen; it just PREDICTS that a later thing will happen. So, another way to name
this kind of before-after CONNECTION is hypothetical or functional rule knowledge.
For instance, we can say that
In general, whenever you see lightning, you’ll soon hear thunder.
In other words, the learning mechanism has noticed that a whole bunch of lightning and thunder
facts SHOW the same connection: first lightning and then thunder. You can represent this
knowledge with a rule sentence (as above), or with hand gestures and noises (Kablooie!), or with a
picture [ZZZZZZ boom].
Another example of knowledge of a causal, hypothetical, or functional connection in time is….
In general, if and only if the temperature of steel increases to 2700 degrees Fahrenheit, will the steel
melt.
The learning mechanism has noticed a bunch of cases (facts) about steel and its temperature. It
found that when the temperature of the steel rose to 2700 degrees, the steel always melted, and
when the temperature was less than 2700 degrees, the steel never melted. And so the learning
mechanism SUMMARIZES or makes a GENERALIZATION from these individual facts, that, in general,
if and only if (and whenever) the temperature of steel increases to 2700 degrees Fahrenheit, will the
steel melt.
This learning mechanism is one smart guy---or gal.
“You bet! Without me, Pilgrim, working like a dog behind the scenes to transform mere incoming
energy, and then raw sensations, into things, kinds of things, and connections among kinds of things,
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you would not have a world.”
Albert E. Neuman
http://www.byrnerobotics.com/forum/uploads/
JohnByrne2/2011-02-07_181719_alfred_e_neuman.jpg#
alfred%20e%20neuman%20einstein%20298x414
Mona E. Neumann
http://www.who2.com/sites/ default/files/ imagecache/blog-full/
imagecache/blog-full/blog/inline/5/ mad_apes_backcover.jpg
12. Routine knowledge (knowledge of another kind of connection) is what we say, dance, write, paint,
and chisel about how things are connected by a series of steps that gets a job done. The series of steps
is a routine. Examples.
a. Get fuel. Add oxygen. Heat the fuel.  You get ignition.
b. ram Say rrr, say aaa, say mmm  You sounded out a word.
c. (a +/- b)( c +/-d) = what? Multiply the first, then the outside, then the inside, and then the last
numbers, and you get the answer.
We represent routine knowledge with pictures that show “first do this and then do that” and with
spoken and written lists of steps.
How Do Human Beings Acquire Knowledge? Inductive Reasoning
How do human beings DO this learning? How does the learning mechanism find things, kinds of things,
and interconnections among kinds of things IN the kaleidoscope of sensations?
It performs a set of logical operations---called inductive reasoning---ON the sensations, until what
emerges is generalizations.
“All these lines and colors are (to be called) triangles.”
“All these trees are to be understood as a forest system.”
“The fact that the prices of all these goods increased when persons wanted to buy more and more of
these goods, is to be seen as a causal relationship, not just a bunch of coincidental happenings.”
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Interestingly, the methods of reasoning that scientists use to draw conclusions from data, are exactly
the methods of reasoning that everyone uses to draw conclusions from data. The methods are called
agreement, difference, joint method of agreement and difference, concomitant variation, and residues.
These methods were described by John Stuart Mill in A system of logic (1843). All quotations are from
Mill.
Method of agreement
"If two or more instances of the phenomenon under investigation have only one circumstance in
common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given
phenomenon."
Symbolically, the method of agreement can be represented as:
When events A B C D go with or are followed by events L M N O, and
When events A E F G go with or are followed by events L S T U, and
When events A Q V E go with or are followed by events L H I K….
The learning mechanism concludes that A and L are connected. For example, “A must be the definition
of concept L (concept).” or “A must be the cause of L (rule).” The instances agree in one way. This
method of inductive reasoning is the method of agreement.
Note: the more examples, the more confidently the learning mechanism can draw the conclusion. Why?
Because 10 examples with A and W always together is pretty unlikely unless they are connected.
Design instruction that enables students to use the method of agreement to draw a conclusion from
examples.
1. Make sure examples are DIFFERENT in irrelevant features, but are the SAME (they agree) in the
relevant feature.
For example, show 4 (or more) examples of linear functions. Each example differs in the angle (A, B, C,
D) and the scale of values (10’s, 100’s, 1000’s) on the X and Y axes (E, F, G, H), but each line is straight
(L).
This (A, E, L) is linear.
This (B, F, L) is linear.
This (C, G, L) is linear.
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This (D, H, L) is linear.
The learning mechanism says, “Therefore, L is what makes it linear; A-H are irrelevant features.”
More examples of how the teacher presents examples as above, and the learning mechanism uses the
method of agreement (comparing examples to identity how certain features that are the same go with
the same treatment.
Teacher: “This (red circle) is called roja.” “This (red square) is called roja.” “This (red apple) is called
roja.”
Roja is L, and the color red IN the examples is A.
Teacher: “This chunk of granite consists of mica, feldspar, and quartz. And THIS chunk of granite
consists of mica, feldspar, and quartz. And notice that THIS chunk of granite also consists of mica,
feldspar, and quartz. So, what do you think is the definition of the concept, granite?”
Method of difference
“If an instance in which the phenomenon under investigation occurs, and an instance in which it does
not occur, have every circumstance in common save one, that one occurring only in the former; the
circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part
of the cause, of the phenomenon.”
When events A B C D go with or are followed by events L M N O, and
When events B C D go with or are followed by events M N O…
The instances DISAGREE in one way. When A is there, L is there. When A isn’t there, L isn’t there.
Therefore A is the cause, or the effect, or a part of the cause of L (why events are named L or why events
are solved with L).
This is the design of an experiment. The experimental and control groups are the same on the variables
B, C, and D (for example, age, sex, pre-test scores) but differ in A (the intervention to increase scores).
And the group with A (the experimental group) ends up with higher post test scores---W. So, A (the
intervention) must be the feature of the group that makes the difference in achievement.
Design instruction that enables students to use the method of difference to draw a conclusion from
examples.
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Present events in this way (above and below) to show students the difference that makes the difference
in whether something IS an example of a concept, rule, or routine.
“This (red circle) is called roja. This (blue circle) is called not roja.”
Or, “This is granite. It consists of mica, feldspar, and quartz. But THIS is NOT granite. It consists only of
quartz. So, why is this one NOT granite?”
Or, “This graph shows the rule that when demand increases, price increases. This other graph does NOT
show the rule that when demand increases, price increases.”
Or, “We read these REGULAR words (sad, ten) by sounding them out. We say the first sound, then the
next sound, and then the last sound. sssaaad, teeehhhnn. And that is how we SAY the word---sad, ten.
But we do NOT SAY these IRREGULAR WORDS (said, the) the way we sound them out. We sound them
out---sahiiid, t/heeh---and then we SAY them THIS way---sehd, the.”
This technique is called “juxtaposing examples and nonexamples” to reveal difference.
Make sure examples are the SAME in irrelevant features, but are DIFFERENT (they DISAGREE) in the
relevant feature. For instance,
Rome, 1st century, before 31 BC. Spoke Latin. Agricultural society. Representatives of the people--senate, tribunes. “This is a republic.”
Rome, 31 BC. Spoke Latin. Agricultural society. One ruler---emperor. “This is not a republic.”
The learning mechanism says, “Well, the one difference is representative (dispersed) rule vs.
concentrated rule in one person, AND the first is called republic and the second is called not republic. So
the concept---republic---must be defined by representative (dispersed) rule.”
Another----juxtapose a plot in which the line is straight (“This is linear.”) and a plot that is the SAME in
the values of variables on the axes and in the direction of the line (up, for example), but the line is
CURVED.
“This is linear.”
“This is not linear.”
The learning mechanism says, “Linear CAN’T be the way the axes are labeled (by 10s), because they are
the same, but one plot is called linear and the other not linear. So linear must be the ONE difference in
the plots that goes along with calling one linear and the other not linear. It was called linear when the
line was straight and not linear when it was curved (not straight). So, linear must mean straight.”
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Joint method of agreement and difference
"If two or more instances in which the phenomenon occurs have only one circumstance in common,
while two or more instances in which it does not occur have nothing in common save the absence of
that circumstance: the circumstance in which alone the two sets of instances differ, is the effect, or
cause, or a necessary part of the cause, of the phenomenon."
This method combines the methods of agreement and difference.
Symbolically, the Joint method of agreement and difference looks like this.
First set of juxtaposed examples that agree in one way.
A B C D occur together with x y z
E F G D occur together with s n z
Another set of juxtaposed examples that agree in one way.
L M N D occur together with w t z
I O H D occur together with s v z
So, by the method of agreement, I hypothesize that D goes with---is the definition of, is the way
to solve---z.]
Now a set of juxtaposed examples that DIFFER in one way.
A E D occur together with x w z
A E occur together with x w
Another set of juxtaposed examples that DIFFER in one way.
F G D occur together with t u z
F G occur together with t u
The learning mechanism uses the method of agreement to conclude that D goes with z. Then it
CONFIRMS that conclusion by applying the method of difference to the second two examples, to
conclude that D goes with z]
Design instruction that enables students to use the joint method of agreement and difference to draw a
conclusion from examples. For instance….
Method of agreement portion.
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Show examples of republics (Rome, Switzerland, United States), and label them as such. Examples differ
in size of country, time in history, and language, but are the same in a government in which
representatives are elected and make decisions for the whole.
The learning mechanism concludes that the ways the examples differ can’t be what defines republic. It
MUST be the way the examples are the same---representation.
Method of difference portion.
Now JUXTAPOSE an example of republic (Rome as a republic---“This is a republic.”) with Rome ruled by
an emperor---“This is not a republic.”
The learning mechanism concludes that republic must be the feature whose presence or absence makes
the difference in how the example is treated (named). “Republic must be the feature of
representation.”
Method of Residues
"Deduct from any phenomenon such part as is known by previous inductions to be the effect of certain
antecedents, and the residue of the phenomenon is the effect of the remaining antecedents."
If we know that certain factors go with one another, and we have matched them, and we find that one
“effect” remains, then some other one factor that remains must be the cause (or the effect) of the other
one that remains.
“It can’t be Professor Plum with a lead pipe. He was sick in bed, and the lead pipe was locked away.”
“It can’t be Colonel Mustard. He was dead drunk.”
“It can’t be Miss Primrose. She was busy dusting the ceiling.”
“The only remaining possibility is Miss Scarlet with a candlestick. Yup. It’s always the red heads!”
Symbolically, the Method of Residue can be represented as:
When A B C D happen, F G H I happen.
We know that G is caused by B.
When know that H is caused by C.
When know that I is caused by D.
So, F must be caused by A.
Present a situation in which several variables could have caused some other event. See if you can
eliminate these causes until only one is left.
“So, what was the cause of the beginning of the decline of the Roman Empire? Various explanations
have been given. (1) Overwhelmed by barbarians. (2) Weakened by the pacific religion of Christianity.
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(3) Overextended the empire and exhausted resources. (4) Replacement of republican government by
monarchy (emperors). (5) Loss of martial spirit and patriotism. (6) Slavery. Let’s look at each of these
to see when they might have affected the strength of the empire and how much of an effect they could
have had on its decline… Well, it couldn’t be the barbarians; they invaded later….”
I have just given you the intellectual equipment to design instruction that will teach anything and
everything---because everything COMES as an example of one of the above arrangements. Either…
a. All events are alike (agree) in some way.
b. There are some differences among events that make a difference in how they are to be treated
(named, solved).
c. Two things change together.
d. It’s possible to eliminate everything but one variable, so that one must be the important one.
Method of concomitant variation
"Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular
manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of
causation."
Symbolically, the method of concomitant variation can be represented as (with ± representing an
increase or decrease):
When A B C D happen, F G H I happen
When A (+/-) B C D happen, F (+/-) G H I happen [+/- means increases or decreases]
[Let’s say A and F change in the same direction.]
Or A increases and F always decreases. [They change in the opposite direction.]
Since the only things that change together are A and F, the learning mechanism reasons that A and F
must be connected in some way (perhaps a change in A causes a change in F). This is called the method
of concomitant (both together) variation.
Therefore A and x are causally connected.
Examples.
The more a government uses threats and punishment [jail, fines, SWAT teams breaking into homes
without a search warrant, confiscation of property, violating the Constitution, recording internet and cell
phone activity, police road block identification checks) to force compliance from citizens, the more the
government comes to be seen as illegitimate by citizens, who form opposition groups and resist the
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government. [Peter Blau. 1965. Exchange and Power in Social Life. Also, the historical basis for the
American Revolution.]
The less you exercise and the more carbs you eat, the more weight you will gain, the more likely you are
to get diabetes, and the lower is your life expectancy. So, eat nothing and run all day.
Notice how these variables vary sometimes in the same and sometimes opposite directions.
Present examples in which everything else (as in B C D and G H I) above, is the same, but only two things
vary (A and F).
Summary of the Learning Process
From Specific
Events
Students Learn
General Ideas: Four
Forms of Cognitive
Knowledge
How?
Teacher Presents ---->
Examples and Treats
Them the Same Way
(e.g., names, solves,
analyzes them).
+
Teacher presents
Nonexamples and
Treats Them a
Different Way
(e.g., names, solves,
analyzes them).
+
Teacher provides
Assistance such as
Gaining attention,
Review, Framing the
Task, Modeling the
Information, Leading
Students Through the
Information, Testing/
Checking to Ensure
Learning, Correcting
Errors, Outcome
Assessment.
The Learning Mechanism --->
Performs a set of Logical
Operations***. It:
a. Examines examples;
observes their features
b. Compares and contrasts
examples; identifies
features that are the same
c. Contrasts examples (that
share some of the same
features and are treated
the same way) with
nonexamples (that don’t
have those features and are
treated differently).
d. Identifies the differences
(in the features) between
examples and nonexamples,
and how they are treated.
e. Makes a generalization:
“This thing goes with that thing.”
“All things that have features
A, B, and C are called
republic.” [concept]
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Makes generalizations
a. Simple fact
X goes with Y
(Name---event)
(2) verbal chain, or list
X goes with Y1-Y5
(mitosis ---list of
phases)
b. Concept
(1) sensory, or basic
(all defining features
can be seen, heard,
felt) red, on, faster
(2) higher-order
(defining features
are spread out and
must be synthesized)
sandstone, justice
c. Rule-relationship, or
proposition
(relationship between
sets of things/concepts)
“When things that are X
happen, then things that
are Y happen.”
[rule-relationship]
“To accomplish Z, do steps
1, 2, 3, 4, and 5.”
[cognitive routine]
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d. Cognitive routine
(sequence of steps
for accomplishing
task)