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I've been a TA across every class in the calculus sequence, under the assignment of professors with different teaching styles and curricula. It's often clear to me ahead of time when a certain subject is going to be a lot easier or harder for students to digest in the allotted amount of time, and in those situations I feel a desire to assuage any frustrations students might have ahead of time. I think it's more common than not, for example, to preempt a discussion of epsilon-delta with disclaimers like "this is a definition with a lot of moving parts," "if this doesn't make sense right away, that's okay," "it's a concept most students struggle with," and "there's no way to get used to it but by grinding through a lot of practice problems."

If I don't offer a warning about a difficult subject, the students may become stressed out, ascribing their slower progress to something about themselves rather than about the material or the circumstance. And pragmatically speaking, it is in a student's best interest to have a keen sense of where they stand relative to their peers, so I sympathize with their concerns.

On the other hand, students stress out even when I give them warnings. I also worry if this doesn't prime them to give up or become frustrated with the material sooner than if I had said nothing. "Okay, no use trying to understand this lecture, I'll just wait for the practice problems."

Are there any good guidelines for when the warning is worth the effort?

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    $\begingroup$ Not a direct answer, but related: it's good (always, and particularly at times like this) to repeatedly reinforce the idea that mathematics is a skill that can be learned and improved with practice, like any other skill, and that making mistakes or being temporarily lost is a normal part of all learning. In other words, emphasizing a growth mindset, rather than a fixed mindset, helps students have resilience with difficult subjects. $\endgroup$ Oct 17, 2023 at 6:05
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    $\begingroup$ @GregMartin: Recall that the effects of "growth mindset" are a myth, i.e., fail replication studies -- like a lot of quick-fix pop-psychological fixes. matheducators.stackexchange.com/questions/24418/… $\endgroup$ Oct 17, 2023 at 15:51
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    $\begingroup$ @DanielR.Collins I always thought of the "mindsets" as a simple description of something very basic: having confidence that you can improve makes improvement more likely, and being convinced you can't do something is a good excuse to give up (whether it really makes it harder or not I don't know). I use the terms in that way but I don't try to legitimize it in any kind of scientific way. $\endgroup$
    – Thierry
    Oct 17, 2023 at 16:50
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    $\begingroup$ @DanielR.Collins I believe that research, but I also believe that (a) growth is math is in fact possible, and (b) a growth mindset improves student morale independent of grade outcomes. Also thank you for providing the link $\endgroup$ Oct 17, 2023 at 16:50
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    $\begingroup$ Worst case scenario: snopes.com/fact-check/science-textbook-gloomy-intro $\endgroup$ Oct 18, 2023 at 16:27

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I know this is generally accepted as normal behavior by teachers, but personally, if I ever find myself tempted to attach disclaimers like "don't worry if you feel overwhelmed by this", then I view it as an indication that I'm about to try to take too big a conceptual leap, and instead, I need to figure out a way to better scaffold the curriculum.

As a TA I guess you don't have control over how the teacher introduces the material, but in the off chance that you have enough bandwidth during TA sessions to re-teach from the ground up when needed, here's a rough outline of how I might scaffold $\epsilon$-$\delta$ limit proofs:

  1. Before I even mention the general definition of $\lim\limits_{x \to a} f(x) = L,$ I instead tell students that "the general definition is easiest to understand if you first understand some concrete examples."

  2. I state the definition in the case of $\lim\limits_{x \to 0} 2x = 0,$ $$\forall \epsilon > 0 \,\, \exists \delta > 0: |x| < \delta \Rightarrow |2x| < \epsilon,$$ and we play a game where I choose an $\epsilon$ and they win by stating a $\delta$ that makes $|x| < \delta \Rightarrow |2x| < \epsilon$ true.

  3. Once they find this game easy to win, I ask them to describe their "winning strategy" for general $\epsilon,$ i.e. create a formula for $\delta$ in terms of $\epsilon$ that will allow them to win regardless of what $\epsilon$ I choose.

  4. Then I say "you guys are too good at this, I'm tired of losing, let's make the game harder," and state the definition in a slightly more complicated case like $\lim\limits_{x \to 3} 2x = 6.$

  5. Once they find that game easy to win, we discuss the general definition of $\lim\limits_{x \to a} f(x) = L,$ and then we continue playing the game with progressively more complicated $f(x),$ discussing the different kinds of strategies that are necessary to win at each "higher level."

  6. And finally I say something like "like any game, you're going to need plenty of practice to become good at it," which conveys to students that they're going to need to grind practice problems, but doesn't prime them to give up or become frustrated.

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    $\begingroup$ I agree. In my experience, there is a significant fraction of students who will twist anything they can into "The teacher said this is hard, so I'm never going to get it." So you can't say anything even three steps away from "This is hard." $\endgroup$
    – user22788
    Oct 17, 2023 at 0:32
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I can remember two teachers warning us about the difficulty of a subject.

  • The first warning was by a teacher of measure theory and probabilities.

In the first half of the semester we'll study measure theory. There will be a midterm exam on measure theory. Then in the second half we'll study probabilities. The final exam will be entirely focused on probabilities. Every year I warn the students about the second half of the semester. Students seem to interiorise the concepts of measure theory perfectly, and then they understand nothing of its application to probabilities. In the past twenty years I have never managed to understand why. So I'm warning you. The second half of the semester is apparently more difficult than the first half, although I don't know why.

And his prediction turned true; most students struggled with the second half of the semester, and failed the final exam.

It wasn't a warning so much as an admission of failure on his part.

He was an awesome teacher in my opinion, but this was one big failure in his teaching. So, if you find yourself warning your students about the difficulty of a subject: try to fix it. Warning the students that they'll fail is not a solution, it is not helpful, it is only an admission of failure on your part.

In the case of measure theory and probabilities, my understanding is that students were very good at learning all the very formal definitions and theorems. Students were very good at handling measurable functions $f : A \mapsto B$. But when we applied measure theory to probabilities, suddenly we were dealing with random variables, which are measurable functions $X : \Omega \mapsto B$ except space $\Omega$ "doesn't matter" or is left unspoken and undefined, so really $X$ feels like a function with a codomain but no domain, and students were confused because the very dry formalism of measure theory was now cohabitating with the very blurry vagueness of functions with undefined domains.

But the teacher never identified this precise difficulty; so year after year he warned the students about a difficulty without being able to make his class less difficult. The materials covered in the second half of the semester is in my opinion not harder than the material covered in the first half; but it does require a shift in philosophy because of these unspoken domains.

  • The second warning was by a teacher of the C programming language

So far we have only introduced variables. Now we will introduce pointers. This is a very hard topic and it makes language C much harder to learn than some other languages. It's okay if you don't get it at first.

Again, the prediction turns out to be true: many students start learning C, but never understand pointers and never become good C programmers, or at least it takes them an awful long time before they understand pointers.

Pointers are variables that hold the memory addresses of variables.

That's it. Pointers defined in one sentence, and it's actually super simple. So why are pointers so difficult? It's a self-fulfilling prophecy. Most teachers gloss over the topics of variables, which is reputed to be easy, and then dive into pointers and spend a lot of time on it, because it's reputed to be hard. But pointers are actually extremely simple, provided the student really understands variables. Trying to teach pointers on the shaky foundations of mistaught variables is what makes it hard.

Warning the students that "pointers are difficult, it's okay if you don't get it" is not helpful at all. It's laziness on the part of the teacher. Instead of warning the students of the difficulty, the teacher should identify where the difficulty stems from, and do a better job of explaining what a variable is.

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    $\begingroup$ Pointers are tricky before you have a memory model in your mind. And a memory model provides little-to-no benefit until that point. I think that's what leads teachers into the trap. (Personally, I like your explaination, and tend to take the time to explain memory models... but it always does feel clunky) $\endgroup$
    – Cort Ammon
    Oct 17, 2023 at 20:07
  • $\begingroup$ It's not as simple. Pointers are values, not variables. They need not point to variables, they need not be stored directly in variables. To add to the confusion, "arrays" decay to just pointers to the first element, so "an array is just a pointer". Pointers can be invalid addresses (NULL for example). Pointers have different lifetimes - some point to variables with static storage duration, some point to local variables (that only exist as long as the stackframe of the function), some are heap-allocated and need to be freed. $\endgroup$
    – Luatic
    Oct 18, 2023 at 18:31
  • $\begingroup$ C is also one of the topics where I think teachers should definitely warn students, especially if they're coming from scripting languages like Python where they have much less to think about (automatic memory management, dynamic types, proper error handling, powerful stdlib etc. all built-in). C is a language that isn't very forgiving when you make mistakes (this can be mitigated to some extent using the right compile flags etc.), and learners need to be aware of this upfront. A beginner will make mistakes all the time, and will need to have the patience to debug and learn from them. $\endgroup$
    – Luatic
    Oct 18, 2023 at 18:35
  • $\begingroup$ @Luatic Easy fix: Pointers are variables that hold memory addresses. What makes them difficult is 1. keeping track of all the indirect references to a memory location and making sure it's never used incorrectly nor just lost without clean up, and 2. understanding when C's compiler will (or won't) generate code to fetch the other location's value. The concept is extremely simple; it's the usage that's difficult because it involves managing so many easy to make mistakes if you're not really understanding what's happening or allow widespread access to the locations involved. $\endgroup$
    – jpmc26
    Oct 19, 2023 at 1:17
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    $\begingroup$ @Stef Pointers were tricky for me not because of the concept itself, but because of the conversion rules, i.e., in x=y; it puzzled me why it meant writing the content at the address associated with y to the address associated with the variable x. I would rather interpret it as replacing the address associated with x by that associated with y (which is what happens in some languages if you write A=B; for arrays). The usual meaning should then be coded as %x=%y; where % is the value operation or, even more logically, x<-%y; Pointer operations bring that potential confusion to its extreme. $\endgroup$
    – fedja
    Oct 19, 2023 at 4:56
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I delay that chapter for later in the course. I want them to understand derivatives (which have such great physical meanings) before they try to understand this very technical topic.

As a TA, you can't do that. But you could tell them that the reason for this weirdness will be clearer after they've done derivatives.

Also, I have a fun exercise that I use when teaching this. I use a (made-up) Cookie Crispness Index, where the x-axis is time in the oven, and the y-axis is crispness of your cookies. Before we start, I ask everyone their favorite kind of cookie. (Chocolate chip almost always wins.) Then I get everyone to say how crisp they like them, on a scale of 0 to 10. (0 is cookie dough, 10 is charred.) Then I'm hoping someone will be cute and give a decimal. (If no one does that, I ask someone to play the game with me, of they're the pickiest person in the world.) Then we try to find a just right amount of cooking time, but we're never perfect.

I draw a curve from (0 min,0 crispness) to (15 min, 10 crispness).

They want to be within .01 of the perfect crispness, and I show the epsilon band for that, and we figure out how close the time needs to be. But of course, then they want to be even pickier. Etc.

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