Monday, October 4, 2010

No more p's and q's!

I decided sometime last week that this year I would bypass all the logical "Law of ..." names and just teach kids logical rules in the most straight-forward way possible: the way I picture all the facts in my head.

Being a super visual person, I know that when I look at a bunch of related facts, in my head I am not thinking about p's and q's. Instead, when I read something like,

In your class, there are 4 students who are born in August. Those who were born in August were all born on either a Saturday or a Sunday. Anyone who was born in July was born on a Friday.

I picture something in my head (or on my paper) that is broken down into symbols:
4 students --> August --> Sat or Sunday
July --> Friday
Then come the questions, which I then mentally add to this diagram, piecewise.

Question 1: If Amy was born in August, what conjecture can you draw about the day of week of her birthday?
Amy --> 4 students --> August --> Sat or Sunday
So, Amy was born on either a Saturday or a Sunday.
Question 2: If John was born in May, what conjecture can you draw about the day of week of his birthday?
That's still pretty easy:
John --> May
and May doesn't have any arrow leading us to any new conclusions. So, we don't have enough info to make a conjecture about which day John was born.
Question 3: If James was born on a Friday, what conjecture can you draw about the month of his birthday?
This question can be a bit tricky for kids, but it's actually pretty simple if you refer to the diagram. If you add James to the original diagram, most kids will automatically say that James should have an arrow pointing directly at Friday. Then, it's not a big stretch for them to see (diagramically or logically) that James wasn't necessarily born in July. "We can't go backwards in the diagram," is my reasoning, or "James could have been born on a Friday in another month, like March," is theirs. Either way, we're in agreement and everyone is happy.
Question 3B: Kid in class raises his hand and says, "But we know that James could not have been born in August." I was really happy when this kid said this. I said we'd come back to discussing his observation after going through Question 4.

Question 4: What conjecture can you draw about how many people from your class were born on either a Saturday or a Sunday?
At this point, we all agree when looking at the diagram we have on the board that we can't go backwards from Saturday / Sunday to being sure that we have exactly 4 students born on those days. (But, we do know there are at least 4 people born on Saturday/Sunday.)
We went back to Question 3B and discussed why Matias had drawn the conclusion that James couldn't have been born in August. I showed the kids how to reverse all the arrows while negating each statement, and explained using this specific example why that makes sense. ("If you weren't born on a Saturday or Sunday, you couldn't have been born in August. Because if you had been born in August, you'd have been born on a Saturday or Sunday, right? ...Similarly, if you weren't born in August, you couldn't have been one of those 4 students, and you're definitely not Amy.")

After that, they did some practice constructing arrow diagrams for other situations and questions, which they thought was really straight-forward (although they needed reminders to slow down and to use their diagrams as aid in answering the questions).

Tada! Not a very discovery-based lesson, no, and with no bells or whistles, but every kid got ALL the basic logical laws down pat in one day, no problem. I'm super happy with this outcome, because last year they could NOT figure out when to apply each logical rule, and I think this diagrammed way sort of combines all the individual rules into one, to be applied flexibly to each question. (And for the most part, it makes intuitive sense to be constructing a diagram like this, as opposed to turning simple situations into really scary p's and q's.)


  1. I like this. My students struggle with logic remembering the implications and arrows to and fro. They must have the p's and q's and truth tables however. Will you include them at all or what's your next step?

  2. Last year we didn't do truth tables at all. They needed to learn just basic relationships like:

    1.) If p --> q and p is true, then q is true.

    2.) If p --> q and p is false, then we can't make any conjecture about q.

    3.) If p --> q and q --> r, then p --> r

    4.) If p --> q and q is false, then p is false.

    5.) If p --> q, then to prove this statement false, we need to show one instance in which p --> ~q.

    6.) If p <--> q, then to prove this statement false, we need to show either p --> ~q or q --> ~p.

    I have no doubt that we can cover all of those rules this year, with this simple diagram method. (They kids already learned rules #1 through #4 from ONE day of working with the diagrams. We're going to cover the last two rules tomorrow, and then do some mixed practice.) We didn't do truth tables last year and so I'm not familiar with what additional facts/value they provide. Can you give me an example? :)

    But, in general, my approach is to hopefully NOT overwhelm 9th-graders with knowing every last rule/strategy. They need to have a solid grasp of those important ones, and to be able to apply those to analyzing situations and basic mathematical statements/definitions. I figured if down the road they need to learn something else more specific/advanced, their discrete math(??) teachers can build on what we have.

  3. But, anyway, I think if I HAD to introduce p's and q's as part of the curriculum, I'd do it after a couple of days of practicing basic (concrete) statements without using p's and q's. Introduce the rules last only as a "generalization" of the trends they already have observed. By that point, kids should be able to fill out missing parts of the rules themselves, if you leave out the most important parts of each rule.