I'm informally helping a few students with college Calc 1. This isn't the first time I've aided people with calculus, and so they've sought me for help, though I don't consider myself to be particularly good teacher or good at math.
In general they are good students, have good work ethic, follow what I tell them to do to succeed, but recently I ran into something of a road block with tangent lines and derivative trends in graphs.
They seem to at least be able to perform all other tasks, If I tell them they need to use X formula to solve Y problem, they can easily figure out the rest, and use that later in a different part of the problem, perform basic algebra or trig, and do the rest of the "calculus" part of the problem.
But when they seem super confused when I talk about tangent lines. First there seems to be an aversion to "precise" language. At first I tried to get them to describe to me what they thought a derivative physically meant, because I could tell they just weren't getting why parts of the graph meant positive or negative derivative, and were clearly not connecting rate of change with things like "a tub draining or filling with water".
I couldn't parse what they were trying to express. They claim that they just don't use the same "big words" I do. I tell them, the things they are saying are in-comprehensible, mathematics is in part communication, and being un able to communicate ideas isn't going to allow them to even let their professors know they know what they are talking about.
To be clear, I don't expect "big" words, but they weren't even getting close to anything related to "rate of change", it was like they were trying to find the "teachers favorite color" except I'm not their professor so that straight up doesn't work with me.
From that point, I decided okay, lets take step back. I'd previously explained in earlier sessions what the derivative is, and why the book was saying the things it does. But that hadn't stuck or they hadn't internalized it or something. So I went in to try to figure out what they weren't understanding. Eventually I got them to understand, at least I believe, the whole rate of change thing, and that this is a generic concept, it can apply to any quantity, any X, and they proved they understood by applying that knowledge to word problems using it.
So then I went forward and asked them what the significance of the tangent line was. They seemed to have no clue, and when I tried to get them to understand, they seemed to be waay too focused on the line, rather than what it represented. They seem to want to plug and play values with the line, then get confused when it doesn't work, I don't know how to explain it. It's like they don't seem to grasp the concept of something representing something else in the way it is framed with tangent lines. I ask them what is the relationship between the tangent line and the derivative, but they struggle, get it wrong, start guessing (note they've already been taught this). I'll end up telling them that the value of the slope of the tangent line is the value of the derivative at the point the on f(x) associated with the tangent line and they claim to understand that, only for them to clearly not be able to apply that knowledge later on.
There's something about it not being a straight "equation" that they can apply somewhere that makes it harder for them to use than if I just said "Here's an equation that approximates the derivative" or something.
Coming back to graphs, they would get the wrong answer and they would seem to mix up the derivative and properties of the actual f(x) line up, and I tried to show them how they could just visualize the derivative but because they struggled so much with tangent lines, they seemed to be unable to grasp visualizing what the derivative was (and maybe still haven't internalized what the concept was yet?)
They kept having homework questions over this, got frustrated that we were "spending so much time on this" (understandably, I was spending over an hour trying to get them to understand these concepts on questions that should have taken less than a minute), and I told them we could move on, but you needed to get help from someone who could figure out why they didn't understand these things.
One problem is their professor does not speak intelligible English, uses "reverse" classes (do 'homework' in class, watch lectures at home, and also do more homework) the lectures are recorded in poor quality. But the books material seemed to be similar to what I had when I was in school, and they get ample time with TAs who they seem to like. Their TA's have already aided them on similar problems is my understanding.
Come test time, they couldn't answer these types of questions correctly, and did poorly. I want to help, but I've told them for the time being they need to reach out to their TAs (Professor is useless, office hours are almost suspiciously poorly timed to avoid student needs). The next sections they seem to have little problem with, again, it's just plug and play equations for them, and now they are getting to more advanced derivative rules. I really don't understand how to get them to understand what should be much simpler things than the other problems they are able to solve.
What are some strategies, methods I can use to help these students understand tangent lines and trends of derivatives (and second derivatives) on graphs?