Combative students in proofs classes

Question

When teaching my first discrete math class recently, I found a subset of about 5 out of 35 of my primarily computer science students who I struggled to reach. If these students simply struggled with the material this would be one thing, but they ended up disruptive to the rest of the classroom environment, so I would like suggestions on how to better handle this subset of students in the future.

I will try to characterize them through example behavior, though of course this characterization is bound to be inaccurate and no one student met all bullet points below. My question is, how to most effectively teach proof-based content to students who exhibit subsets of this behavior, or perhaps how to most effectively handle their impact on a classroom's attitude towards the class, and proof in general (since these students are often very vocal and self-assured, as they are perhaps accustomed to having better grades in computation-based courses). Some example behavior:

• Argues against the notion of doing problems on homework that are not exactly identical to the problems in class (this is common in lower classes, but it is a key feature of the students I am describing)
• Refuses to acknowledge differences in if-then statements in natural language versus propositional logic, even when those differences are explicitly addressed in one-on-one conversation (e.g. argues against the professor's statement that "if 3>5 then 3>4" is a true proposition in class and in office hours, and will argue that the instructor does not know anything rather than concede that they could be wrong.)
• Insists that the model proofs provided by the instructor contain superfluous information, which is in fact necessary to include at their level.
• Calls inductive proofs "circular" in complete confidence, not realizing that they do not understand induction.
• Makes negative comments about how nothing in the class makes sense during weekly group work, derailing their groups.

I thought that, for example, explicitly discussing the purpose of homework in the class, explicitly addressing the differences between natural language and propositional logic in class, providing 3 model proofs for the students to grade and rank before midterms with my own grade and rank provided later, and going over the logic of induction during every example inductive proof we did for a week would at least dissuade these students. But unfortunately my class had the most negative attitude I have ever experienced, and this was reflected in my student evaluations that term, despite always receiving high evaluations before.

Can these conflicts be avoided? Is this something that is bound to appear in any early-proofs course, and if so, how can these students be better reached -- by instruction or simple "classroom management" ?

Answer 1

There's probably no silver bullet.

But one tool I use is in these situations (e.g., I teach discrete mathematics etc. at a U.S. community college) is to very closely align with a good textbook. In fact, here's my personal note to myself in my checklist for preparing a new course:

The most important thing is to TEACH FROM A GOOD BOOK.

My motivation here is precisely that aligning with the book gives voice to a "second opinion" in the course, that I as the instructor have not gone off the rails or become delusional in what I'm presenting or asking students to do (in terms of exercises and test questions). If someone does push back, I can point to a particular section or wording in the book for backup. If they claim to know everything in the class, I ask if they can complete a certain exercise in the book.

I'll confess that I wish this made more difference than it does in practice. A vanishingly small number of my students are actually consulting the textbook at all; the vast majority accept being fully dependent on what I personally say. But philosophically it gives me confidence (which might help the OP), and in certain circumstances it's been helpful for me to be hyper-aware of the book sequencing so I could refer students there, or quote some definition or exercise as a backup.

Answer 2

You don't say in the question what kind of school this is. It must be a four-year school rather than a community college, but there is no indication of what its admissions standards are like. If this is a state school that's easy to get into, say Cal State Fullerton, and a lot of these students are math education students, then you're extremely lucky that only 10% of your students are behaving in this way. On the other hand, if this is Berkeley or the University of Chicago, then what you describe would seem very anomalous.

Your students are probably going through a very drastic transition right now. If your student population is anything like the one at the school that I recently retired from, then about half of them have probably passed their two years of lower-division math under COVID conditions where the tests were online and they cheated on them. They may have learned little or nothing. The test asks them to differentiate $\sin(x^2)$, so they use Wolfram Alpha to find out the answer. If the test asks them something that Wolfram Alpha can't answer, they get on their Discord server during the exam and see if anyone knows. Now, they're in a class where someone is actually reading their work carefully. So not only is it their first time actually having their learning evaluated, but they also completely lack the background knowledge needed in order to tackle the material in your class.

From your username and some comments above, we know that you're female. This doesn't help. Many students will discount your expertise.

I'm male, but when I started teaching I was fairly young, and although for the most part that made it easier to connect with students, it did also tend to make some of them treat me with less respect. It helped somewhat when I started dressing a little more formally. On the first day of class, you might also want to give a brief professional bio. This could be stuff like "Gee whiz, I work on topology, and here's why that's super interesting -- here's a picture of a coffee cup turning into a doughnut!," but it can also be an occasion when you briefly mention your educational background.

As far as disruption of class, you have to strike a balance. If you shut them down completely, then it wrecks the atmosphere of intellectual inquiry, fails to model normal collegial behavior in academia, and risks worsening their lack of confidence in you, since authoritarianism is often a refuge of the incompetent. But if you let them revisit the same issues over and over, take too much time, or behave disrespectfully, then it diminishes the education that can happen in the room. If they're taking too much time on their bogus objection, say that: "Sorry, but we've spent quite a bit of time on your question, and now it's time to move on." You will probably see other students smiling and nodding when you say this. If they're revisiting something you've already addressed, say that: "This is the same issue you raised last Wednesday, and we discussed it then and I suggested that you take a look at section 16.7 in the book on induction. Unfortunately it's not appropriate to revisit this further at this class, but please do feel free to come by my office hours if you're having trouble with this topic."

Over the years I have visited many people's classes to evaluate their teaching. When there was inappropriate behavior by students, I almost never saw a response from the instructor that seemed firm enough to me. Most people hate conflict and public confrontation, so they err on the side of softness. A typical example would be that the instructor hands back a pile of homework papers, then puts the stack of unclaimed papers on the desk at the front of the room and begins class. Then a student comes in 15 minutes late, walks over to the stack of papers, and stands between the teacher and the rest of the class digging through the pile. Most instructors will simply ignore this, which IMO is wrong because it's extremely distracting and disrespectful. All it takes is: "Ah, John, what you're doing is inappropriate. You're blocking people's view of the board and it's very distracting. Please wait until after class to pick up your graded paper."

Answer 3

EDIT: I would like to clarify that my response below is not intended to be definitive. This is an extremely difficult problem to have. It is perhaps the most difficult problem one can have as a teacher: a complete breakdown of the trust in each other which is needed to make communication possible. The idea below is only a stab in the dark, which has many obvious potential drawbacks. Very open to hearing other approaches which are less problematic.

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Unfortunately, it seems that these students do not trust you as a subject matter expert. This is a difficult place to be. If they are convinced that your proofs are invalid, that your understanding of logic is faulty, and that your understanding of mathematical induction is circular, it will be very difficult to regain any sort of healthy classroom atmosphere.

Put yourself in their shoes: if you were taking a class and the instructor was clearly spouting nonsense (for instance, a biology class where the instructor clearly had a Lamarckian rather than Darwinian understanding of evolution), you too might act disruptively. I certainly would!

This is an extremely serious problem and it calls for serious solutions. Here are my suggestions:

1. Hold a meeting with the disruptive students and see if I am right: do they consider you to be mathematically or logically incompetent?
2. If so, you need to be prepared with a plan to address their concerns. Since the students themselves (clearly) do not have the logical prowess necessary to judge you, all you are left with is an appeal to authority. See if there are any instructors at your institution who they trust as mathematical authorities.
3. Invite these students, and their chosen authority, to a meeting together. Discuss these issues together (vacuous implications, mathematical induction, the standards you are holding them to in their logical arguments, etc.). When their authority figure confirms that they are wrong, and that you are right, this might have some impact. Hopefully, once they realize that they are wrong, they can begin really listening to your instruction and trying to learn from it.
4. Make it very clear to them that you will not tolerate disruptive conversations and behaviors during class. Make a clear distinction between genuine questions (in which both the asker and the receiver carry a curiosity about each other, and a humbleness in their responses), to questions which are designed only to inflame or to dismantle authority. Let the students know that the later sort of question will be identified as inappropriate and that you will ask a student who persists in these attacks to leave the classroom. Same comments apply to derailing groupwork: let them know that they are derailing their group, and that if they cannot stay on task they will be asked to leave.

Answer 4

Some helpful feelings of mine about teaching:

1. You can't force someone to learn. As an undergraduate professor, I am responsible for being a resource to provide my students with the information they need and some motivation and structure. However, my students are welcome to pass up those resources.
2. I am responsible for providing a classroom environment conducive to learning. If your troublesome students are calling out and disrupting class or disrupting groupwork, provide them with clear warnings and then ask them to leave the classroom if it continues. This is much easier said than done: it is very unpleasant to be confrontational.

Answer 5

The problem with induction is common (I'm sure you are aware of that). With an audience of CS majors I would try and utilize the connection between induction and recursion. Basically driving home the point that induction is the tool for proving that a function defined by recursion gives the predicted outcome.

I would go for the throat and use Ackermann's function. We define the function $f(x,y)$ with natural number inputs $x$ and $y$ by the following innocent looking rules:

• $f(0,y)=y+1$,
• $f(x+1,0)=f(x,1)$,
• $f(x+1,y+1)=f(x,f(x+1,y))$.

The students may have the desire to code this in whichever programming language they choose. But the last one is a real killer, and makes the most powerful computer scream for mercy.

A sequence of exercises around the theme could be:

1. Why/how do these relations define the function for all inputs $x,y\in\Bbb{N}$? (this is actually somewhat delicate, you may want to put it last)
2. Calculate (or run) a few test values $f(1,y)$ with $y=0,1,2,\ldots,5$. Form a conjecture as to what $f(1,y)$ is in simpler terms. Prove it by induction on $y$.
3. The same with $f(2,y)$. Calculate a few values, form a conjecture, prove it by induction.
4. The same with $f(3,y)$.
5. Then $f(4,2)$ (or make that $f(4,2021)$ is you feel particularly cruel).

Anyone who tries with a few lines of code alone will run into a limit on recursion depth. Guaranteed :-)

Anyway, using the connection that when proving anything about recursion induction is the only tool may make the penny drop.

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I only had to teach a related class to CS majors once. I gave them the answer for $f(1,y)$ and $f(2,y)$ and had them prove it for general $y$ by induction. $f(3,y)$ and $f(4,2)$ where extra assignments. I may have given the formula for $f(3,y)$ as well. It did generate a lot of discussion. Like one student asked whether I'm at all worried about recursion depth :-) Which is really the point for computer scientists. You can actually do $f(1,y)$ in class.

Anyway, it may be worth your while to use a (simpler) recursively defined function when explaining induction. Fibonacci may be enough. If Fibonacci is too much, start with integer multiplication $0*y=0$, $(x+1)*y=x*y+y$.

Answer 6

This is a tricky situation. Here are some strategies that I have followed in my Discrete Math for CS class, perhaps they will be of some use to you.

1. I make it clear that the purpose of the class is to understand how to read mathematical definitions and what constitutes a proof and that this understanding is to be inculcated in a hands-on manner, i.e., by reading and understanding definitions and learning how to write proofs of theorems based on those definitions. This goes some way in answering your first question: homework questions must be different because it is not the specific topic that is important but the process.

2. I begin with logic and present it as an abstraction, i.e., I actively discourage the students (who are at the sophomore level) from using their "real world" ideas while trying to understand logic. A good way to do this is to begin with the law of the excluded middle which doesn't hold in the real world. Essentially I present logic as a formal system with its own rules. Individual propositions like "3 > 4" are thought of as statements that bring their own truth value with them from some other place and, while we are studying logic, we do not dispute the truth value of a particular proposition, but just look at what truth values emerge when we put propositions together.

3. At the beginning of the semester I share with them guidelines on how to write a proof based on a discussion provided in Sec 2.7 of the text by Lehmann, Leighton and Meyer (MIT OCW Fall 2010). With this as a basis it can become clear what is essential in a proof and what is superfluous. Perhaps having such a benchmark will give you a stronger platform for arguing for the necessity of including extra material in proofs. However, my suggestion to you would be: don't include anything superfluous in a proof. You can always consider having an extra discussion before or after. You can also consider having versions of the proof from the baggiest to the leanest. In fact such a discussion might benefit the students immensely.

4. I am hopeful that if you present Induction after propositional logic and proofs by deduction, you can present it as iterative deduction. This presentation in LLM (see ref above) changed the way I approached Induction. In line with my comment #1 above, I find that students come in with certain wrongheaded ideas that need to be unlearned. LLM's text provides some examples of how proof techniques like Induction, when wrongly applied, lead to preposterous conclusions ("All horses are black"). Such examples can be used to try and convince the students to follow things in a more programmatic way so that they get things right at the sophomore level. As they proceed to later years the intuition will also come.

Your 5th point about negative comments is something I have not encountered, possibly because I teach in India where the students tend to be deferential to professors.

Answer 7

Take Them Seriously

Authority cannot be demanded or mandated; it must be earned. Frankly, I think you can and must earn it. You obviously take your job seriously, because you took the time to think carefully about these students, their disruptive behavior, take notes, and solicit advice. That's great! When a student challenges you in class, you should buy yourself time by saying: "That's a great point. I'm going to take a note and we will discuss that later." Then write down their objection and move on. Within the next day or so, bring up the objection in class, and delay the rest of your content to do so, so that the rest of the class can see that 1) you are taking all the students seriously, and 2) frivolous or unfounded interruptions cost the entire class.

Example Responses

Since you mentioned these are mainly CS students, I will provide responses tailored for that community.

• Argues against the notion of doing problems on homework that are not exactly identical to the problems in class

We're going to do a mock interview. You three have worked at Twitter for the last 5 years as back-end server engineers. I am the hiring manager at Google where you are looking for your next career transition.

Bobby, tell me a bit about the work you've done at Twitter.

Let him BS an answer for a few seconds to warm them up and build up their confidence. Nod and smile and give small affirmations ("Very good! Impressive!").

Ok, now why do you think you would be a good fit for Google?

More rambling BS...

Ok, that's all well and good, but based on your performance in MATH-243, I am led to believe that you only work on projects that are identical to work you've already done. You are aware that Google does not have any micro-blogging products, are you not? You see, we need engineers to design and build things that have never been done before. There are no examples for them to look at and study. They need to apply the principles they learned elsewhere, and generalize them to completely new areas. Which part of your MATH-243 history demonstrates this skill?

• Refuses to acknowledge differences in if-then statements in natural language versus propositional logic, even when those differences are explicitly addressed in one-on-one conversation (e.g. argues against the professor's statement that "if 3>5 then 3>4" is a true proposition in class and in office hours, and will argue that the instructor does not know anything rather than concede that they could be wrong.)

This one is actually very easy to address. You just need to learn to speak in their language. Ask them this:

Let's try a different proposition:

if rocket_launchers > rifles then rocket_launchers > pistols

Is this proposition true or false?

Any self-respecting gamer will instantly see that this is a trap. Because the natural-language version is true for some games, false for others, and both true and false for yet other games, depending on the map, the team, the enemies, etc. And they know that their friends in the same class will vehemently object to any possible answer they give with embarrassing counter-examples.

But let's not debate this amongst ourselves. Let's ask a computer to decide!

let a = 3

let b = 5

let c = b - 1

if a > b then a > c

Is this proposition true or false?

This instantly teleports the problem directly into the center of their world, and challenges them with an idea that they should have already encountered if they have taken any CS classes at all yet.

• Insists that the model proofs provided by the instructor contain superfluous information, which is in fact necessary to include at their level.

The basic theme here is that you have a core of students with logic bugs in their programming. Unfortunately, you have to put on your hacker hat and debug their wetware. This means coming up with a similar proof which fails without the "superfluous information". That can be quite challenging, but if demonstrated clearly and effectively in class, should also be a real eye-opener for everyone. And I guarantee you it would earn you a tremendous amount of authority, as well. It isn't easy to come up with these on the fly, so don't. Make sure you are well-prepared with whatever background research you need to do before responding to these challenges.

Most importantly, whenever possible, get the students themselves to lead the class down the path of the answer until they get stuck. They and the class should ideally "discover" the bug on their own. Try to avoid saying: "That doesn't work" and replace it with: "What happens if?" "What about this?" Whatever statements you want to put in the students heads will work best if the students themselves say it out loud and believe they are the originators of the statement. Try to work out the "failure modes" of the flawed proofs beforehand so you can readily point them out in class.

• Calls inductive proofs "circular" in complete confidence, not realizing that they do not understand induction.

What is a "circular proof"?

Well, it's a thing where you assume the thing you are trying to prove!

Ok, let's try one out. Everyone come to the front and draw a number from this bowl. Now one of you will be the Prover, and another will be an Oracle. The rest will be the numbers. Now line up, in positions decided by the Oracle. Ok, the Prover will stand over here with his back to the rest of the class.

My theorem is this: the list is in ascending order from left to right iff, for every student, the student to their left has a smaller number. Now, Prover, you may ask the Oracle to interrogate any student and the one to their left to see which is larger. When you are done, you should announce whether the list is in ascending order or not.

This is not really an "inductive proof" in the conventional sense, but has the same elements as one, and should help programmer types visualize what is actually happening in the abstract mathematical space. The special condition where a student does not have a partner to their left is obviously the base case, and you should leave it to the class to discover it.

The Oracle can decide whether to put the students in order or not, and you should do both ways without announcing it to the Prover.

After the Prover announces their results, you can challenge them: "But you just assumed that to begin with, right? All the rest of the work was unnecessary, because this is circular, right?"

• Makes negative comments about how nothing in the class makes sense during weekly group work, derailing their groups.

This is more difficult, but indicates that some students may require tutoring. Ideally, your dept. has volunteer or paid tutors available to assist. If other students are succeeding in the class, you can simply point out that their sentiment is clearly not universal, but you would like to help them catch up, and here are some tutoring resources we can look at.

Answer 8

Their behaviour is classic Dunning-Kruger. The way to change their behaviour is to give them more knowledge about the subject (in this case e.g. mathematical induction).

I know this seems really difficult or perhaps even impossible: At the moment they seem to be working against you and seem to be actively refusing to acquire more skill.

Perhaps you could give them extra easy questions to work on to start with and gradually increase the level of complexity from there while making sure that you don't lose them on the way. That's how I would go about this. As long as they are having small experiences of success they have no reason to act up because it keeps them happy and motivated. You need to eliminate all sources of frustration for them.

It might also help to tell them, although in less direct words, that, just because they don't immediately understand something it does not mean that they are stupid. Maybe use 5 minutes to introduce them to the idea of mindsets and Carol Dweck.

Answer 9

You do not specify how many of these students have a background in cs, but you indicate some of them does. Computer science stress to a high degree concrete and clear logic used to solve concrete real world problems as effectively as possible. Therefore connecting whatever proof you are trying to teach to concrete, real world problems and demonstrating how you can use a specific type of proof to resolve such a problem would be helpful in making computer science students grasp that concept.

Proof based content is very close to the heart of cs. The subject matter will matter to those students at a very personal level.

Another alternative: Instead of trying to immediately teach a student that does not understand what you are talking about at a fundamental level, instead listen intently to him and try to pinpoint his underlying assumptions and why he reasons the way he does. It would be good to do this on a one to one basis initially, or at least in a small group. Once you understand the underlying assumptions and reasoning of the student you should have an easier time reaching not just that student, but most students with a similar psychology.

Another alternative: If you have the option to bring in a good guest lecturer this can be a good way to reach students you have trouble with. Sometimes you simply need a different approach to reach people. If the guest lecturer does manage to explain things to these combative students, you can learn from him.

Note that all of the above assume your combative students have a similar background or personality. Good luck.

Answer 10

Well, there are interesting philosophical questions that remain in logic. Regarding entailment, for example. Why not talk to them about it? eg. Relevance Logic. And induction too, might be seen as rather intemperate. Even, perhaps, as exhibiting a debauched lack of discipline. Maybe have a side talk about axioms vs. axiom schemes, and eg. Predicative Arithmetic. It's not like the topic of logic is closed. That might make them feel more comfortable about learning one particular model of rational, deductive thinking. Some of us instinctually rebel, when we're taught something that doesn't feel like it makes natural sense to us. Showing that it's one model, might help put a certain distance between them and the thing they are studying, and make it easier for them to swallow.