Professional Development

Rachel: Did anyone come up with a way that we could simplify rather than having to prove all of these implications? Is there a way that we could be, I guess, work smarter to just prove a simplified statement that actually demonstrates by proving that statement that every single one of these implications has been proven?

Student 1: Could you say something like P(n) implies P(n+1).

Rachel: P(n) implies P(n+1). So all of these implications, or all of the form P of one case implies P of the next case. So if I say one case is P(n), then the next case is talking about the n+1 case. So each of these I could restate as they follow the form P(n) implies P(n+1). Now we need to be a little bit careful with our use of n here. The reason why we need to be careful with our use of n is because we just said n was 500, and now we're using n as if it's variable.

Student 2: Should we just pick a different letter?

Rachel: Should we just pick a different letter? Sure. Yes.

Student 3: I would argue the only information that you can necessarily extrapolate is that you're trying to show the successor

Rachel: There you're showing the successor is true. If I wanted to rewrite this so that, think about the principle of universal generalization. When you're proving a "for all" statement, you say, introduce an arbitrary n.

Student 3: Oh yeah yeah

Rachel: I'm going to give you the argument that shows that it's true for that n. So here, I've introduced 500 as my n, and I want you to tell me how the argument's going to get me up to 500. So in that case, I'm being clear that I am still specifying a finite natural number, a specific natural number, and I'm just trying to give the argument that's going to help me climb my way up to showing that that number is true. So PJ said, could we just pick a different variable? Sure. The standard variable that you'll see is someone, we'll just call it a k. Here k is serving the role of an intermediate variable. This is our way of saying all of the intermediate cases that we need to prove in order to finally get to P(n) being true for the n that we've chosen. So these are our intermediate implications. The question is, how is this quantified? What's k? Where might, ya know, I want you to go back into your groups and I want you to figure out how you precisely quantify which k we must show this implication is true for if I want to make it all the way up to 500 being true.

Carmen : This is the part where I always mess up on. I'm not good at writing them out. So my thing is I think that with negation you're trying to find a way that the original statements, um like not necessarily the opposite, but just um like an alteration I guess to the original statement. So I don't think you're looking for a counterexample because that would just be proving that the original statement would be false. And So if negating it, you... it's still for every x and in the universe for x's in the universe or in the set. Yeah.

Andy: Um, if you're trying to prove, if you were trying to show that the original statement for all x, P(x) implies Q(x) is false, what would you do?

Carmen: P(x) implies Q(x)? So if I'm trying to show that that's false, I would just find one value in the set. Or if, are you saying x is a set for all?

Andy: So the implication I'm trying to falsify is for all x, P(x) implies Q(x). How would I show that's false?

Carmen: To show that that's false? I would just find one x where that is in P(x) but it's not in Q(x).

Andy: And that's not necessarily the same as the negation?

Carmen: I don't think so um because that's just finding a counterexample. Maybe it is. I think the negation is just kind of like doing the opposite of what it's asking for. It's not necessarily, it's not asking you to prove that something is false or to make it false, or it's like asking you to rewrite this statement that would - in a way that would make it false. Not asking you to prove that it's false, I guess. I dunno, I could be wrong, but I think...

The success of a task does not hinge on whether students generate correct solutions and proofs. Transition to Proof courses are playing fields where students have opportunities to connect their own initial ways of reasoning to the formal logic, language, and notation that is considered normative within the larger mathematical community. We want to help students think critically about their ideas, evaluate whether their ideas generalize, and begin to notice how subtleties of logic and language in mathematical statements impact their meaning. We consider task implementation to be successful when students have space to explore, challenge, and refine their own ways of reasoning to establish an intellectual need for making connections with normative mathematical logic. Following group work, students should be ready to talk about the key ideas -- they don't need to have solved the task.

We try not to interfere with student reasoning. Instead, we see the instructor in the role of eavesdropper -- listening in on group discussions so that they are better prepared to facilitate whole-class discussion following the task. It's important to give students ample freedom to explore their own ways of reasoning. Interjecting to push them toward more normative reasoning may reduce their intellectual need for the logical idea at hand. Sometimes, however, intervening is warranted. For example, it may be necessary to make a clarifying announcement to the class when instructors notice students have common confusions about what they have been tasked to do. We also discuss below circumstances when instructors probably should intervene to encourage equitable and inclusive group interactions.

When we ask students to share and challenge logical arguments, we need to pay careful attention to equity and inclusion during classroom interactions. For example, in inquiry-based classes, white men students often participate more frequently in discussions than women and students of color (Reinholz and Shah, 2018). In addition to assigning roles to group members to support equitable, productive collaboration on tasks, here are some simple strategies for creating a positive environment:

• Although active learning encourages students to think aloud, we shouldn't overlook the value of ``private think time.'' Many students benefit greatly from a few seconds to collect their thoughts or jot something down before group conversation. This allows students to reflect on their own thinking and share when they're ready, rather than just agreeing with what one or two more dominant, talkative students say. You might say, ``Take a minute to think to yourself about this before discussing in your groups.''
• Instructors should circulate around the room to listen in on group discussion. When one or more students are dominating the conversation, instructors should intervene to encourage plurality of contributions and ensure that other students are not being marginalized or left behind. For example, teachers might interject, ``Can you take a moment and make sure that all of your group members are on the same page?"
•  Jigsaw Tasks -- tasks that require students to take responsibility for different features of the problem and rely on each other -- are even better. By designing tasks that emphasize both the responsibility of the individual and the need for collaboration, we can mitigate the potential for low- or no-participation from less dominant students. For example, if students are tasked with reading and verifying a proof, you might assign conceptual questions to each student, making it their responsibility to lead a discussion on that topic during groupwork. e.g., How is the definition of a bijection being used in this proof? How can we tell what proof technique is being used here? Probing questions you would use to generate discussion often work well for these roles. Try to create questions that "jigsaw", building off each other so that students can't just work on their on questions independently.
• Because everyone brings their own ways of reasoning to the table, instructors should help students normalize the habit of measuring progress relative to where they started rather than relative to whether they have the solution. Attend to helping each individual student move from \textsl{where they are} toward even more powerful ways of reasoning. In particular, you may take time to celebrate how the hard work students have done to overcome a particular challenge from earlier in the course is paying off in your lesson.