The Inverser, the Contrapositive, and the Background Info
(February 19, 2013) The kids were pretty disappointed that today was our final session. C was even fake crying. Unfortunately, 4 of our 9 participants were absent, which totally changed the dynamic of the group. M wanted to solve the “All Puddings are Nice” riddle, but everyone was on edge. D had done an exciting magic trick before class started, L wanted that trick explained, and V wanted to review and clarify what we had done last week. In other words, each student had a different agenda from the start.
The group discussed and re-enacted last week’s roles, The Backwardser and the Converser. The converse was confusing; more was required than a simple algorithm. With Waggy the puppet playing the Converser role, the kids tried, unsuccessfully, to find a pattern in the effect of the converse on the truth value of the original statement. Then I introduced the puppet Baby Puppy (as named by the kids). Baby Puppy wore a baby bonnet. I asked the students to give Baby Puppy some true or false categorical or conditional statements.
“All puppies wear bonnets.”
“Puppets are pretend things.”
“If it is a puppy, then it is young.”
Baby Puppet stated the inverse of each statement, and the students easily grasped the concept. But then they began to argue over the truth value of the inverse. Several insisted, for instance, that the statement “If it is not a puppy, then it is not young,” is obviously true. Others didn’t agree with this at all. A few more examples led to the consensus, however, that the inverse is not necessarily true, but could be true in certain cases.
The debate got out of hand when I had the puppets interact with each other. Puppy the “Inverser” had responded to Waggy the Converser, who had first responded to a given statement. The result was the inverse of the converse: “If it is not young, then it is not a puppy.” Students argued vehemently over whether the inverse of the converse of a true statement must be true. They also wondered whether the inverse of the converse is the same as the converse of the inverse. Unfortunately, on this verge of mathematical insight, some students lost interest in the debate and engaged in paper-airplane making and chair tipping. The collective curiosity of the group vanished. So I moved the kids to the floor on the other side of the room.
On each of the boards I had written one line of our original Dodgson riddle:
All puddings are nice.
This dish is a pudding.
No nice things are wholesome.
I asked the students how Waggy and Baby Puppy would respond to these lines, and to each other. Unfortunately, since we were in the last 20 minutes of our last session, time was insufficient, kids were wild, and I was rushing. Or, maybe it’s more accurate to say that I was rushing, and therefore the kids were wild. I wanted to get back to this problem since it was our initial mystery and there was student desire to solve it. I realized at this point, however, that 6 weeks was just not enough time for this problem with this group. This topic is often taught in college, and these guys are in third grade.
I had come to class today solely prepared to use the puppets to tackle this problem. Since the puppet approach had been magical for the first 5 weeks, I thought of Bob Kaplan’s advice to avoid over-preparing as I prepared for class today. I came to class without my typical “Plan B” activities. So now I was faced with a choice: follow Bob’s advice for when things descend toward chaos (“function machines!”); or try to somehow refocus the kids on the problem.
I hurriedly put the puppet Waggy on one of my hands and Baby Puppy on my other, and told the kids that whoever can accurately predict Waggy’s response to the first line of the poem could hold Waggy. (Yes, I resorted to bribery.) Everyone got quiet and thought. Several students did come up with the converse to the first line. I then asked for Baby Puppy’s response to Waggy, in other words, the inverse of the converse, formally called the contrapositive. Everyone loved saying that word, but people got frustrated here because the second and third lines are hard to work with without rephrasing. We didn’t have time to adequately discuss rephrasing. I was asking too much of the kids in too little time, so we did not solve the riddle. I sent them home with the hint that arriving at the contrapositive for each line will lead to the solution.
Several of the students and parents asked me to email background information to solve the riddle at home. If you are working on this with your child (or if you are a child from the class), click here for a very nice explanation of the logic behind Dodgson’s riddles. In this article, G.N. Hile (University of Hawaii) solves them by rephrasing the statements as implications. This is not the only way to do it, though. Click here for an article on how to handle categorical statements by putting them into a standard categorical form. You might not need these articles, however. You might just need time, collaboration, and curiosity at this point.
Also as requested, here’s a list of terms we used during this session – a list that might best be called “Logic for Third Graders.”
Statement: a sentence which is true or false. In the game “Knights and Liars,” all characters spoke in statements.
Conditional Statement: a statement in the form “if A then B” (also called an implication – a term we did not use in class)
Categorical Statement: a statement relating two categories or sets of things, i.e. “A is a type of B” or “All A’s do B.” There are other standard forms of categorical statements, but we only used these.
True: right or correct (as defined by the students). Characters who always told the truth were called “Knights.” Those who always told lies were “Liars.” Those who did both were “Normals.”
Truth Value: the property of a statement that defines it as true or false
Negate: to state the opposite of a statement, often done with the insertion of the word “not.” Of course, you can rephrase. “Not” is not a required word. Characters who negated every statement were “Negators.”
Converse: a statement that reverses the position of the premise (“A”) and conclusion (“B”). We did not use the terms premise and conclusion in class, but discussed how the converse rearranges a conditional or categorical statement. Our characters who did this were called “Conversers.” At least one student lobbied to call them “Rearrangers,” but I thought that might be confusing.
Inverse: a statement that negates both A and B. Characters who did this, of course, were “Inversers.”
Contrapositive: the converse of the inverse (or, the inverse of the converse)
Logical Contradiction: the conjunction of a statement and its negation (The kids called characters who uttered logical contradictions “Blenders.”)
I know that some of you reading this know a lot more about logic than I do. I used more of Bob and Ellen Kaplan’s advice when I planned this Math Circle: choose a topic that you’re curious but know only a little bit about. I was learning some of this material along with the kids. So if I’ve unwittingly made any mistakes here, please let me know.
Also please let me know if you can explain the concept of the Pragmatic Truth as it applies to mathematics. During a tour of the Barnes Foundation recently, I learned that Dr. Barnes studied Pragmatism with John Dewey. This definition of truth is very different from the logical and scientific definitions I’m familiar with. I got to wondering whether anyone has applied Pragmatism to mathematics, so poked around online some more. Someone has, but don’t really understand what I’m reading. Please let me know if you can summarize or explain this in a paragraph or so. Thank you!
Finally, I promised some of you more history of Dodgson, so here you go. The kids had two questions about Dodgson’s life that I couldn’t answer, so I’ll throw them out to you:
- When he took Holy Orders at Christ Church College and gave up the right to marry, was he allowed to date?
- Why did some of his diaries go missing?
Thank you all for reading, advising, questioning, and participating.