Designing Instruction Based on the Logic of Learning

Martin Kozloff

August, 2005

Examples and Non-examples ---à Learning Mechanism --à Inductive Generalizations

+ Performs a Set of a. Verbal associations

Assistance Logical Operations b. Concepts

c. Rule-relationships, or

propositions

d. Cognitive strategies

Here’s a big idea that connects learning and instruction

Ø Knowledge is general. If you show a child a pencil, the child does not have knowledge of pencils. The child merely sees something with color and shape held in your hand. What the child sees does not signify (point to) anything. It does not mean anything.

Ø When the child sees enough examples of things that you label “pencil,” of different colors, and held in different ways, the child will GET the IDEA of pencil—the CONCEPT pencil. In other words, the child INDUCES the GENERAL concept, pencil, from the examples. At this point, examples signify (point to) the whole set of pencils (the concept).

Ø There are only four kinds of knowledge--only four kinds of inductive generalizations from examples. They are verbal associations (this goes with that); concepts (these things have something in common); rule-relationships (things in one set are connected to things in another set (e.g., how temperature is connected to pressure); and cognitive strategies (sequences of steps for accomplishing tasks).

Ø The learning mechanism “goes” from the examples to the generalization by enacting a series of logical operations; for example, comparing and contrasting examples and identifying what is common to them.

Ø The subject matter does not matter. The learning mechanism “gets” a concept (for example) by using the same set of logical operations no matter whether the examples are in history (kinds of wars), biology (nucleus), chemistry (exothermic reaction), or reading (anything that looks like m says mmm.)

Ø To make learning (going from examples to the general) most effective, teach so that it is easy for the learning mechanism to perform the needed operations.

Ø Since there are only four kinds of knowledge, there are only four main instructional formats—one for each kind of knowledge.

Now let’s look at each of the above points in more detail.

1. Examples (of something – granite) and non-examples (not granite) are the only things that a teacher can show, demonstrate, or talk about. You can’t show the idea of granite. And you can’t show all of the examples of granite (to teach students what granite is), all long division problems, all poems, all metaphors, all connections between pressure and temperature. You can only show a sample of things that are, and things that are not examples of the idea—the general.

2. The learning mechanism (sense organs and brain and hands, etc.) somehow “gets” what is in the examples and non-examples. The learning mechanism gets the larger (general) thing that the examples are all about. For example, the learning mechanism gets what granite is (the features of rocks that define the concept granite); or it gets the connection between pressure and temperature (stated as a rule relationship. As pressure increases/decreases, temperature increases/decreases); or it gets the general strategy for solving long division problems (first do this; next do this; next do this…).

3. The teacher must select examples that clearly reveal the important features. For example, the teacher should select chunks of granite that clearly show the mica, quartz, and feldspar that define granite.

4. The teacher must also select non-examples that make it easy for students to compare and contrast the examples and non-examples so that the differences are obvious. To reveal the defining feature of democracy (effective representation, voting) the teacher would give an example of a democratic society and then an example of a society that is the same in most every way EXCEPT representation and voting. Then it’s easy for the learning mechanism to get the difference that MAKES the difference.

5. In addition to presenting examples and non-examples, the teacher must provide assistance to the learning mechanism.

a. Gaining student attention. “Boys and girls!”

b. Expectation. “New figure of speech. Simile.”

c. Preparation. “Here’s a simile. Listen. The dancer’s muscles are like ropes wrapped around her bones.”

d. Model.

Analysis (Taking apart). Directing attention to the important features (in examples) or to their absence (in non-examples).

example 1. [Point to letter m.] "This sound is mmm."

example 2. “Listen again. Notice when I say ‘like.’ I don’t say the dancer’s muscle ARE rope. I say they are LIKE rope. The dancer’s muscles are LIKE rope wrapped around her bones.”

e. Compare, contrast, identify. The teacher ensures that students see, hear, feel the essential features in examples and their absence in non-examples.

Example 1. [Point to letter m] "This sound is mmm. [Point to a different m] "This sound is mmm." [Point to f] "This is NOT mmm." [Point to m] "This IS mmm."

Example 2. “Listen to the difference. Simile. The dancer’s muscles are LIKE rope wrapped around her bones….. NOT simile. The dancers muscles ARE rope wrapped around her bones. What’s the difference between the simile and the not simile?”

Give an immediate acquisition test for each example.

Example 1. [Point to m] "What sound?"

"Example 2. [Read sentence.] "Simile or not simile?"

The teacher uses numerous examples and non-examples so that students see examples with a wide range of features. Otherwise, student might think (wrongly) that similes (for example) apply only to dance, or to rope. Likewise, the teacher uses many examples of long division to teach the general strategy for doing long division. For instance, if the examples all involve even numbers, students might get the wrong idea (generalization) that the strategy applies only to problems with even numbers.

f. Connect. The teacher helps students to make the inductive generalization from experience with particular examples and non-examples to knowledge of something larger (something general) that the examples represent.

Example 1. "So, what is common to all the examples of simile?"

Example 2. “So, what is common to all of the examples of granite? And what was not in any of the non-examples of granite?” Quartz, mica, feldspar.

Example 3. “So, make a rule that states the connection between change in pressure and change in temperature.”

Give a delayed acquisition test using examples and nonexamples presented.

Example 1. [Point to each letter in an array of letters.] "What sound?"

Example2. [Read or have students read a set of passages.] "Is this a simile?" Or, which ones are simile?" Follow up with "How do you know?"

g. Confirm. The teacher tests/checks to see if students got it (made the proper inductive generalization from the examples presented) by presenting NEW examples. It’s one thing to get what connects the examples shown. But can students use what they GOT from the examples shown to respond to examples and non-examples they haven’t seen?

“Okay, here’s a new box of rocks. Make two piles. Granite and not granite. Be ready to tell why you grouped each one.”

“Listen. His voice sounded like a dying duck. Simile or not simile?”

h. Verify, firm, correct. When students get it right (in f or g) the teacher verifies this. “Yes, "His voice sounded LIKE a dying duck" IS a simile.”

When students are unsure, the teacher goes back to earlier examples and non-examples and tries again. Maybe some students were not focusing attention on the word “like” or did not notice that both yellow and white feldspar are flat and shiny.

When students continue to make errors, the teacher identifies the kind of error, and either corrects it and repeats the task, or gives extra instruction on the weak link. For example, a student says a rock is not granite because the student does not get it that feldspar is defined by its structure (flat planes, shiny) and not by its color.

“It IS granite. Look, here’s mica. Here’s quartz. And here’s feldspar. It’s yellow, but it’s still feldspar. It’s flat and shiny. What does feldspar look like?

Flat and shiny.

Right. Feldspar is flat and shiny. So, does this rock consist of mica, quartz, and feldspar?

Yes.

So is this rock granite?

Yes.

How do you know?

It consists of mica, quartz, and feldspar.”

Excellent!

h. When students have correctly made the inductive generalization from examples/non-examples to general knowledge (a concept, rule-relationship, strategy), the teacher works on various applications of the knowledge.

For example:

(1) Apply the general knowledge to new examples, or make up examples (e.g., similes). This is generalization.

(2) Increase speed. For example, speed at using the strategy for solving long division problems. Do more problems per minute, accurately. This is fluency.

(3) Retain knowledge over time, even though new things have been learned. “Okay, we’ve been working on double-digit multiplication. Let’s go back and review our long division.” This is retention.

(4) Independence. The teacher fades herself out as students work more on their own.

6. There are only four kinds of inductive generalizations that students can make. There are only four kinds of general knowledge. They are:

a. Verbal associations—basically, knowledge of names.

The capital of Massachusetts goes with Boston.

The first draft of the Declaration of Independence goes with Thomas Jefferson.

The phases of mitosis goes with interphase, prophase, metaphase, anaphase, telephase.

In other words one thing goes with another thing that has either one part or more than one part.

b. Concepts--knowledge of kinds.

Granite, igneous rocks, figures of speech, metaphors and similes and onomatopoeia, types of political systems, democracies and monarchies and aristocracies.

c. Rule relationships--knowledge of connections.
The higher the pressure, the higher the temperature.

No snakes have legs.

All mammals are warm blooded.

The lower the socioeconomic status the higher the rate of mental illness.

d. Cognitive strategies--knowledge of routines for getting something done: solving math problems, analyzing texts, writing papers, developing a diagram that represents how things are connected in a marine ecosystem

7. Now, the learning mechanism has to perform certain logical operations in order to get the verbal association, concept, rule relationship, or cognitive strategy revealed by the examples.

It is easier for the learning mechanism to GET what the examples reveal IF teachers design and deliver instruction in a way that enables the learning mechanism easily to perform the needed logical operations.

a. The logic 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, THIS ONE THING goes with THAT ONE THING.

“Thomas Jefferson wrote the first draft of the Declaration of Independence. Say that with me…Who wrote the first draft of the Declaration of Independence? … Now, tell me a fact about Thomas Jefferson?”

b. The logic of a concept is THESE EXAMPLES HAVE SOMETHING(S) IN COMMON. THEY SHARE A SAMENESS in their features. To get the sameness, the learning mechanism must

(1) Examine examples and non-examples.

(2) Compare and contrast their features.

(3) Identify the features common to the examples that are not in the non-examples.

So, to get a concept means to get the sameness common to the examples (particulars) but NOT found in the NON-EXAMPLES.

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

(1) Teach students how to examine examples and non-examples.

(2) Show examples one at a time, label them as examples, and point out the common features that make them “same.”

(3) Juxtapose examples and non-examples, and point out the absence in the non-examples of the features common to the examples.

(4) Acquisition test. Present the examples and non-examples just shown. Ask students to identify each one.

(5) Later, work on generalization/discrimination with new examples and non-examples.

c. The logic of a rule relationship, or proposition, is THIS SET of examples/things (CONCEPT) GOES WITH THAT SET of examples/things (CONCEPT) in a certain way.

The learning mechanism has to

(1) Examine examples of the relationship; e.g., social class/the corresponding rate of mental illness.

(2) Identify the value of each of the two variables in each example.

Social Class Upper Middle Lower

Chicago 1/1000 4/1000 16/1000 population

Detroit 2/1000 6/1000 20/1000

Jersey City 5/1000 10/1000 35/1000

Hickory, NC 25/1000 39/1000 78/1000

(3) Compare each example to determine if the values of the two variables are in the same direction across the examples.

“As social class goes higher, what happens to the rate of mental illness?”

(4) Find words that summarize the relationships across the examples.

“The lower the social class, the ______________ the rate of mental illness.”

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) Pre-teach students how to examine examples and to identify sameness and difference in the features.

(2) Either state the rule relationship (and have students repeat and write it) OR tell them that they are to compare and contrast examples and discover (induce) and state the relationship.

(3) Show examples and point out the goestogetherness (co-variation) among certain variables—that is, point out the relationship across the examples. 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.] For example,

“Boys and girls, here’s a rule. The more the enemy’s infrastructure (of water, waste, power, communication, roads, food) is destroyed, the less resistance the enemy will mount after the defeat of its military.”

“Here is an example of the rule. In the Third Punic War (149-146 BC), Rome totally destroyed Carthaginian infrastructure. Down to the ground. Even salted the fields. Carthage then put up zero resistance.”

“Here’s another example. In the War Between the States, Union General William Tecumseh Sherman burned southern cities, homes, materials, and fields. The Confederate soldiers put up little resistance after that.”

“Here’s a contrasting example. Union General Grant killed soldiers, but unlike Sherman, he did not destroy cities. Even though Confederate soldiers died by tens of thousands, the rest kept fighting.”

“A more recent example. United States Marines beat the North Vietnamese and Viet Cong during the Tet Offensive, but they did not destroy Hanoi. [They were not allowed to.] The North Vietnamese and Viet Cong came right back fighting.”

“Finally, the U.S. and British forces so overwhelmed Hussein’s Iraqi army that his armies quickly surrendered or scattered. However, allied forces but did not destroy Iraqi weapons, food, fuel, homes, and roads. Resistance in Baghdad, Falluja, Najaf and other cities began shortly after the defeat.”

(4) Have students identify and state the relationship (goestogetherness) common to the examples. “The greater….the less…”

(5) 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. Again, students apply the rule.

d. The logic 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 steps 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 that is the same across examples (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 should teach most of the concepts, rules, and verbal associations beforehand, so that during instruction on the strategy, you are mostly teaching the steps.

You may have to teach a strategy in chunks (forward chaining). 1; 12; 123; 1234.... Or 1; 2; 12; 3; 123; 4; 1234… It is also important to describe what you are doing as you demonstrate the routine so that students learn to verbalize the concepts, rules, and steps (conspicuous or explicit strategy instruction).

There is a general format for teaching anything. Basically, model, lead, test/check. There are four variations on this general format for teaching each of the four forms of knowledge.

Get attention. "Boys and girls, listen."

Frame.

The teacher states the learning task at hand. "New sound." "New figure of speech. "New law of physics." "New kind of math problem. Equation with TWO unknowns."

Model.

The teacher provides information (e.g., reveals the logical structure of a verbal association, concept, rule relationship, or cognitive strategy, or shows how to apply this knowledge) verbally or through demonstration. If needed, the teacher repeats the model to make sure all students heard or saw it.

"This sound is mmm."

Lead.

The teacher and students say the information or perform the routine together—several times if needed to ensure that all students do it correctly; that is, are firm.

"Say it with me." mmm.

Test/check.

Students perform the task independently, several times if needed to do it correctly. This is a test or check of whether the students have gotten it. It tells the teacher whether she communicated clearly, the students’ pre-skills were firm before this task, and the students were properly attending and trying.

"Your turn. What sound?" mmm

Verification.

The teacher provides specific praise—stating what the students learned.

"Yes, mmm."

Here's an example.

Attention.

Boys and girls. Listen.

Frame.

Now we're going to read words the slow way.

Model.

Everyone, listen. [Gets attention first.] I’ll show you how to read this word (Points to word on board) the slow way. [Preparation by stating the task. Uses a visual prompt to focus attention and a verbal prompt to distinguish between reading the slow way and the fast way.] Get ready. [Gives think time to orient to the model.] sssnnnaaaap. [The model. Moves her finger under the word and sounds it out.]

Lead.

Everyone [attention], read this word [points] with me, the slow [prompt] way. Get ready. [Think time]… (Signals students to respond by moving her finger under the word.) sssnnnaaaap. Again. Get ready… sssnnnaaaap. [Repeats another time to ensure that all students respond promptly, together, and correctly.]

Immediate Acquisition Test/ Check.

Your turn to read this word (points) the slow way. Get ready… (Signals students to respond by moving her finger under the word.) sssnnnaaaap.

Verification.

Yes, sssnnnaaaap. I love the way you said each sound without stopping.

More Examples. Juxtapose with Nonexamples.

And this is mmm (different color). And this is mmm (different size). This is NOT mmm. (same color and size but different letter)

Delayed Acquisition Test/Check

"Let's read ALL our sound. When I touch under a sound, you say the sound. Don’t let me fool you. Get ready. What sound…?"