November 17, 2006

Logic Lesson Two

Are you ready for your second grade logic lesson two? (Lesson one is here.)

Here we go:

Lesson Two

Here's a rule: Fish live in water.

Say the rule without looking.

The rule tells about fish. What does it tell about? Answer: Fish. (highlight to see)

You must pretend that you don't know about anything else.

Item 1: Fish live in water. A trout is a fish. So what else do you know about a trout? Answer: It lives in water.

Item 2: Fish live in water. A frog is not a fish. So what else do you know about a frog? Answer: Nothing.

Why don't you know anything else about a frog? Because the rule only tells you about fish.

Item 3: Fish live in water. A whale is not a fish. So what else do you know about a frog? Answer: Nothing.

Why don't you know anything else about a whale? Answer: Because the rule tells only about fish.

Item 4: Fish live in water. A turtle is not a fish. So what else do you know about a frog? Answer: Nothing.

Why don't you know anything else about a turtle? Answer: Because the rule tells only about fish.

End Lesson Two

Lesson two was probably unnecessary for most of you high performers. For you, today's lesson could have been presented as independent work. But, in a classroom setting some students may have been absent yesterday and need the material taught by the teacher while lower performers may not yet clear on the concept and need the extra instruction. In future lessons this skill will be presented as independent practice.

These two lessons have been field tested, so we know that it takes this amount of teacher led instruction is necessary for lower performers to understand the skill. No doubt, you higher performers are getting bored with the repetition of the presentation. The interesting thing to note is that the presentation is the same for both the lower and the higher performers. The only difference is the rate at which they learn and master the material. The lower performers need more modeling and leading by the teacher, but by the end of the lesson, they will have learned the same skill as their higher performing peers. They didn't need a more elaborate explanation, a connection to some real world phenomena or the use of manipulatives. They learned it like the higher performers, it just took them longer and it'll take them longer to master the material and retain it, but they are capable of learning it.

And, this is not an easy concept to understand. I took a course in deductive logic in undergrad, and the professor told us that the class would fall into two groups, those that understood it and got A's and B's and those that didn't and got D's and F's. There were no C students in logic. He was right.

Many students had trouble with introductory logic concepts as presented in this lesson, especially when presented at the brutally fast rate of a college course. Some students think that when you're told that p = q, that you can deduce that if r ≠ p, then r = not q. So, in today's lesson, some students might be under the false belief that if we know that we are told that fish (p) live in water (q) that it may be fair to conclude that frogs (r) don't live in water (not q) or that frogs do live in water ( r = q) is false. As we know, both of these conclusions are wrong; we don't know anything about frogs, it is possible that they don't live in water and it is possible that they do. This simple misunderstanding is at the root of the problem when the deductions get more complex and/or the terms are more abstract or beyond the familiarity of the student.

This is a good demonstration of the folly of not grouping students by ability. In a typical classroom, instruction is targeted to the average student. Some of the instruction would be cut out. Thus, higher ability students are still forced to endure repetition they don't need and could be proceeding at a much quicker pace. While, lower ability kids aren't getting enough instruction to understand and master the concepts and skills presented. It is a bit cruel that our least capable students are being challenged the most by being forced to learn with less instruction and practice that they need to learn. Due to the cumulative nature of learning, this deficit slowly snowballs to the point where the lower performers learn less and less each lesson and soon aren't learning anything at all. Failure does not breed motivation.

A commentor in the post on Lesson one wrote:
The logic lesson here is interesting from my perspective as a college student interested in education, but I suspect that my second or first grade self would have found it senseless, repetitive, and frustrating if it were not accompanied by some kind of comprehensible explanation.
Now bear in mind that I am dropping you in the middle of the sequence. Previous lessons taught the students about following rules of the type (p = q, if r = p then r = q ) i.e., Fish live in water. A trout is a fish. So what else do you know about a trout? A trout is a fish. In these current lessons, students are learning p= q, if r ≠ p, then we don't know anything further about r, which is an incremental addition to what the know. The lesson presumes that students know this prerequisite information, so only the incremental change is taught, it does this by inductive teaching. It presents the student with an example rule ("Big men are heavy. An elephant is not a big man. So what else do you know about an elephant? Nothing student who doesn't understand the rule would not be able to perform the disc. This rule tells about big men, not elephants. The only thing we know is what the rule tells us.") and requires the student to discriminate between similar examples (whales are not fish) and non-examples (Jack is a big man). This is why such a simple teaching technique is not teaching by rote. the student is required to immediately apply what he has just been taught to perform the discriminations presented in the second part of lesson one and in today's lesson. If the teaching was by rote, the student would have to be taught the rule in the second part of yesterdays lesson and today's lesson. Instead, the students are already generalizing from what has been taught about the rule presented in the first half of lesson one. This ability to generalize is what efficient learning is all about. Teaching by rote would have required three times the amount of teaching since three separate rules would have been required to be taught.

Also, providing more of a comprehensive explanation would not necessarily improve learning or understanding for the student. A brief explanation was provided in the beginning of the lesson. If the student did not grasp the concept then, he would not have been able to perform the discriminations later. Providing more explanation might aid in more understanding, but it might also serve to decrease engagement as the teacher prattles on with an explanation that might be outside the students ability to understand since the bit of knowledge being taught has not yet been internalized. It is difficult to synthesize gibberish. A balance must be struck between student engagement and understanding. The theory behind this sort of instruction is to keep the student maximally engaged, to provide a brief an explanation as possible to perform the discriminations, and to get the student performing those discriminations as soon as possible.

If the lesson were senseless to the student, this implies he does not yet understand the material, which would not be the case if he can perform the requested discriminations, i.e., answer the questions asked. Students get frustrated when they can't perform the task assigned, which is again would only be the case if the student couldn't answer the questions correctly. I think what the commenter means was that the lesson was too repetitious and boring for her since she immediately understood the skill and was forced to endure the rest of the lesson. This is no doubt the case for many adults and high performer children (my son understood the skill by the end of one example and one non example and proceeded to finish the lesson on his own). This is where good teachers come in; good teachers know when to move on as well as when to go back and reteach. In the instant case, the teacher would know to teach the first half of lesson one to a group of high performers and assign the second half as independent work, making sure that all the students were answering the questions correctly. Lesson two would be independent practice.

In the next lesson we'll get to the drill and kill portion of the sequence. I hope you can bear the agony and not have your precious natural curiosity crushed by this mindless practice.

Go to Lesson Three


Anonymous said...

" ... and not have your precious natural curiosity crushed by this mindless practice."

It's interesting that hard work and practice are taken for granted (mostly) in high school and college. What is the magic age where this transformation takes place? Of course, the answer is that there is no magic age. Kids of all ages need to learn the value of hard work and mastery, whether they like it or not. Educational success that expects any sort of natural love of learning is going to fail.

Many in the educational world see interest in a topic as an indicator that they are ready for the material. No interest implies that they aren't developmentally ready yet. But how can they tell the difference between no interest, boredom, learning difficulty, bad teaching, bad curriculum, or do they just need a swift kick in the rear? Low expectations propagate problems, they don't fix them.

KDeRosa said...

I have a good quote from Bob Dixon on the interest vs. content debate.

Content, it seems to me, is independent of a child’s interests. Let’s say that I’m a young boy who is interested in sports. The nature of reading, rather than the boy’s interest, dictates about 90% or so of what has to be taught to that boy. Content, and the nature of content, doesn’t change according to the interests of children, nor according to any other characteristic of children. If we were trying to teach a gorilla to read, the nature of reading wouldn’t change. Obviously, when it comes to the nature of content, differences among learners don’t have much to do with anything.

Unknown said...

One thing that bothers me about this lesson is the use of "big men" in the first example (Lesson 1).

My sense is that students would be tripped up by Item 4, because, although they are told that Bob is not a big man, they would presume he is a man. In other words, they DO know something about Bob (or at least would presume to).

If we draw two circles of a Venn diagram--one containing "Big Things" and one containing "Men," their intersection could be labeled "Heavy." In other words, if it's both big and a man, then it's heavy. Students are told that Bob is not big, but may presume he is a man, in which case he clearly lies outside of "Heavy."

The logic is transparent enough, but I think the combination of the noun and the adjective in that first example could be productively reconsidered.

Students may read "big men" logically as "p and q" (big and man).

KDeRosa said...

I see your point.

Perhaps, this kind of tricky questions that is designed to make sure the kids are "pretend[ing] that you don't know about anything else."

Another thing is that in previous lessons the kids learned about diets and "getting heavy" and "growing big" by eating too much, so this concept of "heavy men" should be familiar to them.

These lessons are always field tested before publication, so I suspect this potential problem probably turned out to be not a problem in actual use. And, my son didn't have a problem. However, in other lessons he was confused by some wordings and/or examples and required clarification.