Student Conjectures without Oracular Professor

One idea for mid-level college math classes (i.e. after calculus, before highly proof based courses) which I've seen people speak very positively of is to have a portion of the course consist of students producing conjectures on their own by coming up with examples and counterexamples until they have a viable conjecture.

This sounds like a great fit with, for instance, a discrete math course, where engaging with lots of examples is important to grasping the topic.

However every time I've tried this, I find myself serving as an oracle, because there's no "stopping point" if they get the answer right: if there's a counterexample, they might be able to find it if they looked harder, but once they have a right conjecture, the only tool they have for verifying it is asking me whether or not it's right. So every time they have a new conjecture, they ask me if it's right. (And, at its worst, we alternate between me giving a counterexample and them producing a new conjecture taking into account that one additional counterexample and then asking again.)

One thing I like about the concept is that it starts to move away from "professor as arbitrary decider of right and wrong" which many of my students have internalized. But when the definition of correct conjecture is "professor says you can stop now", it ends up reinforcing that notion instead.

My question is how to avoid this: for students who can't be asked to prove their conjectures, how do they know when to stop looking for counterexamples?

• I would think that a discrete math class would be a wonderful place to have them do "informal" proofs. i.e. When they get to a correct conjecture, ask them: "Why do you think that?"
Feb 28, 2017 at 15:07
• My answer: That's what a proof is for. Feb 28, 2017 at 15:13
• @Adam: And then what? In particular, how do you avoid being the arbiter of correct explanations: either you say "okay, that's good enough", or "no, your explanation overlooked a case, here's a counterexample"? (And how does one respond to incorrect explanations of correct conjectures? In other words, what's the difference between a good conjecture and a good explanation?) Feb 28, 2017 at 15:50
• This is why such courses require very careful planning so that students can, in fact, verify their conjectures (if that is in fact your goal). Feb 28, 2017 at 16:24
• This is hardly a useful solution, but if you can find a problem on which students can make conjectures, and you yourself have no idea what might be the correct answer, you can turn a class into interactive work on an open problem. The hard part is identifying such a problem. Mar 1, 2017 at 15:22