All the material that is typically learned on the K-12 level is amenable of being placed into meaningful relationships (or connections) with pre-existing skills and knowledge of the student. Learning will be more difficult when:
1. the student does not possess the prerequisite skills and knowledge assumed by the new material; and/or
2. the teacher has not displayed the meaningful relationships inherent in the new material or has given an explanation that is difficult for the student to follow.
In these situations learning will require higher-levels of analytic ability. And, the students most likely to learn in these situations are the high-IQ students because these students are better able to solve problems they haven't seen before (or that contain untaught vocabulary).
For the purposes of this discussion I want to focus on the purposeful failure (often for pedagogical reasons) of the teacher to display the meaningful relationships inherent in the new material to novice K-12 students under the belief that the ensuing "struggle" will better learn the new material and the underlying meaningful relationship/connection. (I think it's generally recognized that providing confusing explanations by the teacher and failing to ascertain whether a student possesses the skill and knowledge prerequisites is an indication of poor teaching.)
Let's start with the kinds of information, skills, and knowledge that the K-12 learns and what neds to be taught and learned. I've adapted this explanation from Martin Kozloff's Making Sense of What You Read and Hear, and Making Sense When You Teach.
Regardless of the subject (math, history, science), there are only six kinds of information, skills, or knowledge that can be communicated to and learned by the K-12 student.
Each kind of knowledge represents a connection. To understand the knowledge is to understand the connection. To use the knowledge (to apply it to possible examples of it) is to apply the connection.
Ex: “The U.S. Constitution was written in Philadelphia.”
For purposes of instruction a fact is a true and verifiable statement that connects one specific thing (Constitution) and another specific thing (Philadelphia).
Teach the connection.
Ex. 1: “The elements of sugar are carbon, hydrogen, and oxygen.”
Ex. 2: “Here is a list of facts about the U.S. Constitution. Written in Philadelphia between May and September, 1787; the draft was sent to the various states for ratification; the Constitution plus the Bill or Rights is a compromise between advocates of strong central government (Federalists) and advocates of strong state governments with a limited central government (anti-federalists); the Constitution was finally ratified in 1789.
Like with facts, these statements connect one specific thing (elements of sugar, Constitution) and a list (of other specific things).
Teach the connection(s).
3. Sensory concepts
Exs: blue, on.
The specific things (examples) of the concepts differ in many ways (size, shape), but they are connected by a common feature, such as color or position.
All of the defining features of the concept are in any example. Therefore, the concept can be shown by one example. However, a range of examples is needed for the learner to see what the common feature is and to cover the range of variations (e.g., from light to dark red).
Teach the range of examples needed for the learner to determine the common feature and the range of variations.
4. Higher-order concepts.
Exs: Democracy, society, mammal.
The specific things (examples) of the concepts are connected by a common feature or features; e.g., making societal decisions through elected representatives (representative democracy).
The defining features of higher-order concepts, however, are spread out. Therefore, you can’t simply show examples to teach a higher-order concept. You have to give a definition (that states the common, defining features) and then give examples and nonexamples to substantiate the definition.
Teach the definition of the common features and then substantiate the definition through suitable examples and non-examples.
5. Rules or propositions
These are statements that connect not specific things but whole groups of things (concepts or categories).
- Categorical Propositions. Some rules or propositions state (assert, propose) how one kind of thing (concept or category) is part of or is not part of another kind of thing (concept of category). These are called categorical propositions. For example, all dogs (one kind of thing) are canines (another kind of thing). Or, No birds (one kind of thing) are reptiles (another kind of thing). or, Some bugs are delicious.
Teach the rule or proposition.
- Causal or hypothetical propositions. Other rules or propositions state, assert, or propose how one kind of thing (concept or category) changes with another kind of thing (concept or category). These are called causal or hypothetical propositions. You can tell that a statement asserts a causal or hypothetical proposition because it states (or suggests) something like “If…. If and only if.… Whenever…. The more… The less….one thing happens, then another thing (happens, comes into being, changes, increases, happens more often, decreases).
The “thing” (variable, condition, antecedent event) that is the alleged cause of something else can work (have an effect) in different ways. For example, the alleged cause might be considered a necessary condition for something else to happen or change. (“If X does not happen, then Y will not happen.” Or, “If and only if X happens will Y happen.”) Or, the alleged cause might be considered a sufficient condition for something else to happen. (“Whenever X happens, Y will happen.”)
For instance, Whenever temperature increases (one kind of thing), pressure increases (another kind of thing). [This proposition suggests that a rise in temperature is a sufficient condition (by itself) to cause an increase in pressure.] Or, If and only if there is sufficient oxygen, fuel, and heat (one category of thing) will there be ignition (another category of thing. [This proposition suggests that sufficient oxygen, fuel, and heat are a necessary condition for ignition.]
Note: When you have identified all of the necessary conditions, you now have a set of variables that are a sufficient condition. Think of a causal model of fire, a cold, and a revolution.
Teach the rule or proposition.
Routines are sequences of steps that usually must be done in a certain order. Solving math problems, sounding out words, and stating a theory or making a logical argument (each proposition in the theory or argument is like a step that leads to a conclusion).
Teach the routine.
NB: A routine is a connection of a number of events, such as steps in solving a problem or a listing of events leading up to a war. There are different arrangements of steps or events in routines. You want your students to see what these arrangements are.
- Sequence in one direction. A leads to B leads to C leads to D. Ex.: sounding out words, solving math problems.
- Sequence with feedback loops. A leads to B and the change in B produces a (reciprocal) change in A which produces more change in B until some limit is reached. Exs.: Outbreak of war, onset of illness, falling in love, divorce, getting porky and out of shape.
- Stages or phrases. A sequence of events or steps can be seen as a process divided into stages in a process.
Ex1: Load rifle: steps a—b—c--d; Fire rifle: steps e—f—g; Clear rifle: steps h—i; Clean rifle: steps j—k, etc.
Ex2: In history: If you examine enough (examples of) genocidal movements, you notice that one group has some features (e.g., property, social status) that produces envy in another group, or does something that threatens another group (e.g., resists power). This might be seen as the background (first) phase. Then (phase 2) the genocidal group demonizes the first group with racial slurs and propaganda. Then (phase 3) the genocidal group begins to mistreat the victim group; e.g., attacks, job loss, confiscating weapons, special (degrading) clothing. If (phase 4, escalation) the victim group fights back, this provokes worse treatment. If the victim group submits, it furthers the genocidal group’s perception of the victim as degraded. The genocidal group then (phase 5) creates an organization for killing or transporting. Then the killing begins (phase 6).
- Logical argument. A text might be arranged as a logical argument. There are two sorts of logical arguments:
a. Inductive. Facts are presented. Then the facts are shown to lead to a general idea, such as a conclusion. For example, examine five examples of genocide and INDUCE (figure out) the common phases and the activities in each phase.
b. Deductive. Or, text may be arranged so that it presents a deductive argument. It begins with a general idea, such as a rule--first premise.
“If X happens, then Y must happen.”
It then presents facts relevant to the first premise—evidence or second premise.
It then draws a conclusion.
“Therefore, Y must happen.”
In the next post we'll discuss how a student learns this knowledge.