Cognition, Logic, and Instruction

Martin Kozloff

October, 2003

This little paper is an example of epistemology (theory of knowledge). It will make you real smart. Your roommates, family, friends, and even passersby who hear you yacking to yourself will think you’re a genius, –definitely not worthless, sorry, unskilled.

Hang on! Here we go.

We usually focus on texts we are reading; words we are hearing, speaking, or writing; and activities we are engaged in.

Ms. Bronzedagger: “We’re working on geography–about the northern vs. southern hemisphere.”

But this is superficial, –on the surface. The northern and southern hemispheres are examples of something larger –namely, the concept of hemisphere. Does Ms. Bronzedagger know that? Does she teach that our planet is merely one example of things with hemispheres?

In fact, everything you USE when you teach is an example of something larger. For instance, when you show your students how to solve 10 equations having one unknown (x = 3 + 12;   6 x 5 = y) do not think you are teaching the strategy for solving those 10 particular equations. Know that you are teaching the general strategy for solving all equations of that TYPE. The 10 problems and their solutions are examples of the general type of problem and its solution. That (the general, –the idea) is what you really want students to learn. (Otherwise, how can they generalize to new examples?)

The classical role of teacher is to educate, —to lead students out of the cave of illusion (superficial knowledge of particular and changing things) to knowledge of what is general–universal, stable, and enduring. You teach about the Revolutionary War, Civil War, First and Second World Wars, and Cold War, –not so your students can think and talk about these particular wars only, but so they can think and talk about concepts, rules, and theories of war generally. Likewise, you teach students to sound out slip, slim, man, rim, lip, lamp, and ram, not so they can read these words only, but so they learn the general strategy for sounding out all words.

But we can’t teach the general itself. We can’t see the concept red, –only red things that are examples of the concept red. The concept is in the head.]

It is cognitive. It is an idea abstracted from (taken out of) the examples. We can’t teach or learn the sounding out strategy itself, –without words to sound out. We can’t see a rule relationship. We can only abstract a rule from examples of the rule relationship.

We are stuck in the here and now of particular, concrete events (objects, problems). These particular, concrete events (examples) are the only means to reveal–or to help students to grasp or to induce (discover, construct, say), –the general IDEAS that are behind, woven through, revealed, or embedded in the examples.

The question is How do you get students’ minds to move from concrete things (examples) to cognitive knowledge of general ideas revealed by or embedded in the things, –and how do you do this in the surest and fastest way? The answer is, Logically precise design of communication.

Believe me or believe me not, your true project as an educator is philosophy.

Fortunately, there are only four kinds of general ideas, or cognitive knowledge that can be revealed by examples. In fact, there are four and ONLY four kinds of cognitive knowledge that human beings can think and communicate.

These four forms of cognitive knowledge (ideas) are (a) verbal associations (simple facts, verbal chains, verbal discriminations), (b) concepts, (c) rule relationships (propositions), and d) cognitive strategies.

Each form of cognitive knowledge (see above) is a LOGICAL FORM, or has a LOGICAL STRUCTURE. And each form involves certain LOGICAL OPERATIONS, —mental steps, you might say. For example, you have to perform certain logical operations or mental steps in order to move from seeing examples to getting (grasping, understanding, knowing) and then using (applying, generalizing) a concept.

Following is the logical structure and the logical operations (mental steps) involved in getting and using (applying) the four forms of COGNITIVE knowledge, –i.e., these are the mental/logical steps in moving from examples (things) to ideas (cognitions), –from the superficial to knowledge.

The logical structure of a verbal association

The logical structure of a verbal association is THIS ONE THING GOES WITH THAT ONE THING. To “get” a verbal association means to get THAT THIS ONE THING goes with THAT ONE THING. Therefore, the most effective and fastest way to teach a verbal association is simply to say that THIS ONE THING goes with THAT ONE THING.

“That (point) is blue …. What is that?… blue …. Yes, point to blue.”

“The six New England states are Maine, New Hampshire, Vermont, Massachusetts, Rhode Island, Connecticut …. Your turn. Name the six New England states …. What are Maine, New Hampshire, Vermont, Massachusetts, Rhode Island, and Connecticut?… Where are they?” …

The logical structure of a concept

The logical structure of a concept is ALL OF THESE EXAMPLES HAVE SOMETHING(S) IN COMMON. THEY SHARE A SAMENESS in their features.

So, to get a concept (a general) means to get the sameness that is common to the examples (particulars) but that is NOT found in the NONEXAMPLES. The mental steps for moving from examples to the general idea (concept) are:

  • Identifying (locating, pointing out) and recognizing (“This …”) the features of examples (in all examples labeled “crystalline”) and the features of nonexamples (all called “not crystalline”). The features of each piece of rock might be color, size, hardness, and molecular structure.
  • Comparing and contrasting the identified features of labeled examples (“This IS crystalline”) to find the features that are common to them, –i.e., molecular structure is the same in all examples named crystalline, but other features (size, color) vary from sample to sample. Therefore, we hypothesize, it may be that molecular structure is the feature that makes the examples crystalline.
  • Comparing and contrasting the labeled examples and labeled nonexamples (“This is NOT crystalline”) to find the common features that are in the examples but are not in the nonexamples, –i.e., there is a different molecular structure in the nonexamples. Therefore, we conclude (induce, infer, generalize), that a certain molecular structure is what makes things crystalline vs. not crystalline. We now get the concept embedded in the examples.

    Therefore, the most effective and fastest way to teach a concept is to teach WHAT THE SAMENESS IS that is COMMON to the EXAMPLES but that is NOT found in the NONEXAMPLES. Specifically,

  1. Teach students how to examine examples and nonexamples.
  2. Show examples, label them as examples, and point out the common features that make them “same.” “See the planes in the structure.”
  3. Juxtapose examples and nonexamples; label the nonexamples and point out the absence of the features common to examples that make the nonexamples different from the examples. “This is NOT crystalline. No planes in the structure.”
  4. Acquisition test. “Is this (crystalline)? How do you know?”
  5. Later, work on generalization/discrimination with new examples and nonexamples.

The logical structure of a rule relationship

The logical structure of a rule relationship, or proposition, is THIS CLASS (group, set) of examples/things (CONCEPT) GOES WITH THAT CLASS (group, set) of examples/things (CONCEPT).

All beings are mortal. [All of the class of beings is INSIDE the class of mortals.]

No terrorist can be trusted in peace negotiations. [None in the class of terrorist are in the class of persons that can be trusted in peace negotiations.]

The more the enemy’s infrastructure is destroyed, the less the enemy resists after being defeated. [The class of events that involves destroying enemy infrastructure CAUSES the class of events that involves resistance.]

The mental steps (logical operations) for moving from examples of the rule to the general rule itself (the idea) woven through (common to) or revealed by the examples would be:

  • Identifying (pinpointing) and recognizing the value (amount) of one variable in an example and the corresponding value of the other variable in the example. “Total destruction of Carthage. [Goes with] No resistance.”
  • Comparing and contrasting the corresponding values across all the examples, –to see if there is a common way that they go together (co-vary).

    “Carthage and Rome: Total destruction goes with zero resistance.
    Sherman’s march: High destruction goes with low resistance.
    Iraq: Low destruction goes with high resistance.”

  • Stating a rule summarizing the common goestogetherness, or co-variation. “The greater the destruction of enemy infrastructure, the less the resistance after defeat.”

Therefore, the most effective and fastest way to teach a rule relationship is to teach THE RELATIONSHIP (GOESTOGETHERNESS) COMMON to the EXAMPLES. Here’s how.

  1. Teach students how to look at a range of examples. What are the features in each war?
  2. Show examples and point out the goestogetherness (co-variation) of the corresponding values of each variable in each relationship/example. Or, show examples and have students identify the goestogetherness. [It is better for you to do this first, –model, –and then have students do it with new examples.]
    “Rome totally destroyed Carthaginian infrastructure. Carthage then put up zero resistance.”
    “Sherman burned homes, materials, and fields. The Confederate soldiers put up little resistance after that.”
    “Grant killed soldiers, not cities. Even though Confederate soldiers died by the tens of thousands, the rest kept on fighting.”
    “The Marines beat the North Vietnamese and Viet Cong during the Tet Offensive, but they did not destroy Hanoi. The North Vietnamese and Viet Cong came right back.”
    “The U.S. and British forces made the Iraqi army quickly surrender or scatter, but did not destroy their weapons, food, fuel, homes, roads … Resistance began shortly after the defeat.”
  3. Have students say the relationship (goestogetherness) common to the examples. “The greater … the less … “
  4. Give new and/or hypothetical examples and/or have students invent them. Ask what will happen if (there is more or less destruction of infra-structure). Then ask, How do you know? (Students give rule.) And give new examples of more and less resistance after defeat, and ask why.

The logical structure of a cognitive strategy

The logical structure of a cognitive strategy is a sequence of LOGICALLY ARRANGED, OR PROGRESSIVE STEPS (each next step depends, logically, on accomplishing the earlier ones; and each step, logically, makes the next step possible), GOVERNED BY RULES (“If the product is ten or more, write the … “), and USING CONCEPTS (product, tens, ones, column, times, carry) AND VERBAL ASSOCIATIONS (seven times five is 35).

So, to get a cognitive strategy is to get the logical arrangement of steps (and to see the necessary progression) and the concepts, rules, and verbal associations needed.

Therefore, the fastest and most effective way to teach a cognitive strategy is to teach the sequence of steps, the logical necessity of the progression, and all of the concepts, rules, and verbal associations needed. You may have to teach a strategy in chunks (forward chaining):

1,   12,   123,   1234,… Or     1,   2,  12,  3,  123,   4,   1234,…

Remarks

  1. Cognitive knowledge does not exist in the air. It is in thinking, speaking, or writing.
  2. Thinking (cognitive knowledge) is talking to yourself–communicating to yourself. You are INSTRUCTING yourself. “Seven times five is 35. (A simple fact) Write 5 (rule) and carry the 3 (rule) …”
  3. Speaking is cognitive knowledge that you are communicated to other persons. You are INSTRUCTING other persons. “Boys and girls, I look at the ones column first. (rule) I say the numbers to myself. (rule). The numbers are 7 and 5. (simple fact) Now I multiply 7 and 5. (rule) Seven times five is 35 (simple fact) …””Hey, I’m thirsty.” (simple fact)
  4. Besides physical routines, the ONLY things that you can learn, know, and communicate (think, speak, and write, –teach) are (a) verbal associations (simple facts, verbal chains, verbal discriminations), (b) concepts, (c) rule relationships (propositions), and (d) cognitive strategies. Therefore, every sentence that is thought, spoken, or written consists of verbal associations, concepts, rules, and/or steps in a cognitive strategy.
    “Yesterday was a great day.” (rule: categorical relationship. Yesterday is in the class of events that are great days.)
    “I stayed up too late. I’m really wasted today.” (rule: causal relationship. Staying up late caused being wasted.)
    “If you really love me, you will share your feelings.” (rule: causal relationship)”I think it’s raining
    ” (concept)
    “My name is Archilles” (verbal association)
    This is no accident. It’s murder!” (concept. Part of the conclusion of a long inductive strategy of homicide investigation in which examples/evidence point to murder.)”So, all the evidence clearly says, She is guilty.” (concluding rule of an argument, –a cognitive strategy)
    Three times four is twelve.” (verbal association)
    To be or not to be? That is the question.” (rule: categorical relationship. To be or not to be is inside the category of things that are questions.)
    Thus, conscience doth make cowards of us all.” (rule: causal relationship)
  5. Please note that solving a math problem, sounding out a word, analyzing a document, doing an experiment, and writing a paper involve the mental steps (thinking, talking to yourself) called cognitive strategies. Perhaps the largest cognitive strategies are deductive reasoning and inductive reasoning.
  6. Deductive reasoning (one thinking and communicating strategy) is reasoning that begins with a general rule (All beings are mortal); and then gives facts (Socrates is a being); and ends with a conclusion (Therefore, Socrates, –a being, –is mortal). But deductive reasoning can be a much longer chain of sentences. The Declaration of Independence is a long chain of deductive reasoning. Whole books, courses, and curricula may be organized as a deductive argument, –a logical progression of sentences leading to a conclusion. For example, we could begin with a general theory of conflict and then show how it applies to (explains) specific conflicts, –examples of the theory. Just as Socrates’ death is an example of the rule that all beings are mortal.
  7. Inductive reasoning (another thinking and communicating strategy) is reasoning that begins with facts and gradually builds (induces, discovers, figures out, constructs) general rules that ACCOUNT for the facts. For example, when you show kids examples of red things and not red things, and call them red and not red, the kids figure out (induce) the CONCEPT redness. Their mind (thinking, speaking) goes from specifics (examples) to a general idea.Likewise, you could examine many different examples of social conflicts and gradually help students to develop (induce) a general theory of conflict that accounts for (describes, explains) the examples.
  8. Rule relationships (propositions) are one of the four forms of cognitive knowledge. Propositions are asserted in sentences. There are two kinds of relationships asserted by rule relationships or propositions.Categorical propositions, or rules. These propositions assert that one class of things (concept) is inside, partly inside, or outside another class of things (concept). All mammals are warm blooded.Some bacteria are helpful.No music by Madonna is worth listening to. [You may have to think about a sentence to see that it asserts one or more of the above categorical propositions. “Love hurts” implies “All love hurts.”]Causal propositions, or rules. These propositions assert that one class of things (concept) causes, fosters change in, yields, is followed by another class of things (concept). If the rate of reinforcement increases (or decreases), then the rate of the reinforced behavior increases (or decreases). [This asserts a direct relationship.]If the rate of punishment increases, then the rate of the punished behavior decreases. [This asserts an indirect, or inverse relationship.]If and only if the material is hot enough will the material burn. [necessary condition]Whenever a nation is attacked, social cohesion increases. [sufficient condition]After you add numbers in the ones column, add numbers in the tens column.
  9. So, one of your jobs as a teacher is to do a knowledge analysis of the objectives (analyze a poem, calculate the slope of a straight line, state the main rule in each of the first Ten Amendments to the U.S. Constitution, conjugate a new verb). Ask, What are the forms of knowledge contained in DOING each objective? What verbal associations, concepts, rules, and steps in a cognitive strategy must students know/do?Your second big job is to figure out how to teach (communicate) these in (a) a logically clear way (i.e., precise wording and a range of examples/nonexamples that unambiguously [“It can only mean this …”] and quickly reveal the associations, relationships, ordering, features); and (b) in a logically progressive (deductive or inductive) sequence. See numbers 5, 11, and 12, above.
  10. And now THE FINALE …. If all cognitive knowledge is (nothing but) thinking and speaking and writing verbal associations (This goes with that.), concepts (This IS a that.), rules (These go with those in this way.), and cognitive strategies (e.g., explanations, arguments, problem solving), –often in the form of SENTENCES. (Even doing math problems is a series of sentences.)
  11. And if getting each form of cognitive knowledge involves certain LOGICAL OPERATIONS or mental steps, –moving from examples to the general (e.g., concept or rule).Then (besides physical routines) ALL teaching boils down to TEACHING STUDENTS TO THINK AND SPEAK AND WRITE IN LOGICALLY CLEAR SENTENCES ARRANGED IN A LOGICAL SEQUENCE.If YOU speak and write (model, ask questions–teach) in a logical fashion, then your STUDENTS will quickly learn the logical structures in and the mental steps needed for getting verbal associations, concepts, rules, and cognitive strategies you are trying to teach (the objectives).But if you don’t, they won’t. Their thinking, speaking, and writing will be ILLOGICAL, –invalid, incomplete, false, inept, ignorant.
    They will not have knowledge. They will have illusion.

Application of the four forms of COGNITIVE knowledge using deductice and inductive reasonining tell what it means to teach in a logical fashion.

 

DO IT! DO IT! DO IT! DO IT! DO IT! DO IT!

You DO NOT want MY spirit haunting YOU. I will come back, –hot from Hell.

A Wrath, a Demon …
Unfightable, unwarrable on, unholy,
A bold, black Ruin …
I will suck your life’s blood dry,
then hale you below
To pay the painful penalty …
For mighty Hades is strict
In calling men to account under the earth.
His mind keeps records,
Nothing escapes his control.

[Aeschylus]