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Background

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.

Question

What are some strategies, methods I can use to help these students understand tangent lines and trends of derivatives (and second derivatives) on graphs?

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    $\begingroup$ Few students who take calculus understand the prerequisite material. That is a predictable result of imbecility on the part of mathematicians who think that teaching calculus to the masses who don't know the prerequisite material is a way to make the public aware of the nature of mathematics. That doesn't work. Every math professor has been reminded each day for 50 years that that doesn't work. $\endgroup$ Commented Oct 7, 2023 at 5:38
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    $\begingroup$ The above is not to be understood as suggesting that the students should go back and learn the prerequisites. For a few students that is the right thing; for most it is destructive. Math education for broad masses of people should acquaint them with what mathematics is rather than trying to prepare them in prerequisites for later courses that they will fail to understand even if they get perfect grades. $\endgroup$ Commented Oct 7, 2023 at 5:52
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    $\begingroup$ From your extremely long and confusing Q, I can see why your students might be confused too. I have no idea what you believe the "intent of tangent lines" is or what you believe your students believe that to be. $\endgroup$
    – user103496
    Commented Oct 8, 2023 at 4:20
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    $\begingroup$ My confusion with tangents was always that the intersection point between the tangent line and the graph was only 1 point (by definition). So how can a derivative be a slope at 1 point? You need 2 points to calculate the slope, yet the tangent is through 1 point. No matter how much you zoom in on a continous graph, it's only ever 1 point. As a visual learner this was completely unintuitive for me. In my visual mind i got around it by imagining the one point to be two that followed the slope of the graph in a more discrete plot way. I get it now, but i can relate to your students. $\endgroup$
    – stan
    Commented Oct 9, 2023 at 6:42
  • $\begingroup$ I like to use the seasonal length of days when discussing derivatives (and early October is a good time to do it). The change in the length of days is miniscule in late June, but it's quite large in late September/early October. Today's tangent is simply calculated from yesterday's value and tomorrow's value (for any value of today). It's also good that if you get a full year's data you have a clean sine wave. $\endgroup$
    – Flydog57
    Commented Oct 9, 2023 at 16:43

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Sadly, these students seem to think of math as a bunch of rules. You have done great work, and probably gotten them a little closer. But they are resisting the reasoning that can't easily be put into rule form.

One strategy I use is to have students imagine themselves in tiny cars, driving along the function. Their headlights and taillights make the tangent line. The slope of that line is also the car's (or function's) slope.

I also draw a long curve on the board that goes up and down a bunch, and ask students to tell whether the slope is positive, negative, zero, or undefined at lots of points on this curve. (We do this with concavity too.)

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When a concept just won't stick, no matter how many ways you explain it, that is usually a sign of a gap in the student's background knowledge. So I'd recommend searching for the gap.

Of course, you've already tried this to some extent, and I think reinforcing derivatives and rates of change was a good first step.

Next, you might try reviewing lines/linear functions. I know it sounds basic, but can these students look at a graph of a linear function and understand how the slope relates to a rate of change? And can they apply this to simple modeling questions?

It's possible they "know" this stuff, but it's been too long and/or they just aren't connecting it to the new material they're learning.

After refreshing linear functions, you could return to rates of change, and try to hammer home the idea of an instantaneous rate of change and how it relates to the rate of change (slope) of a linear function. This is basically the difference between a car traveling at constant speed vs. variable speed. If they can understand this, then tangent lines are the graphical version of the same concept.

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I taught physics for 25 years at a community college, and the situation you're describing is one that I'm very familiar with. (I also got the oportunity to teach first-semester calculus a couple of times, which was super fun!) Students would show up in the first semester of a calculus-based physics class, and the first week or two would be velocity and acceleration. You would think this would be easy for them, since it's exactly the material they've covered in calculus, in the application that is probably the most heavily emphasized in calculus.

This problem may be exacerbated when there are other issues present, e.g., the student is not very intelligent, or they had inadequate preparation for calculus. However, I'm convinced that what you're describing also occurs among the best students. Part of this is just from having discussed with a couple of very intelligent people, who are good at math, what they got out of their college calculus class. (One of these people is a lawyer and one is a French professor.) It was clear from talking to both of these people that they had absolutely no idea what calculus was about at the conceptual level. When I said to them that calculus was the study of rates of change and of how change accumulates, their reaction was like "Oh, really? That's what it was?"

You discuss their difficulty in understanding the concept as "generic" and relating it to standard examples like water flowing out of a bathtub. I don't think their difficulties with this are mysterious at all. Calculus classes and textbooks simply don't do applications. Look at the homework problems: dozens and dozens of problems, one after another, that are pure symbol manipulation.

Another cause of the tangent-line issues is that students are never (or almost never) assigned any exercises involving sketching without a formula. It's not hard to make up such exercises. You can give a verbal description (the car speeds up, then slows down) and ask for sketches of x, x', and x'', stacked vertically. You can give a graph and ask where the derivative is positive, zero, etc. You can give a graph and ask for a sketch of the derivative. These tasks just don't show up in a calculus class. (For one thing, they can't be graded by the textbook publisher's homework grading software.)

The whole thing is not particularly easy conceptually. It doesn't come naturally to the human mind. Look at Star Trek, where shutting off the Enterprise's engines should cause =0, but the scriptwriters all believe it causes =0. When it comes to money, people often ignore these distinctions as if they don't exist or are too hard. For example, people talk about the national debt's ratio to GDP, as if both numbers had units of dollars and the ratio were unitless. Actually GDP has units of dollars per year.

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I can't figure out from what you say what definition of derivative you are using, or what you have said the connection is between derivatives, tangents, and rates of changes. So I can't tell what might be missing, or what the alternatives might be. But maybe if I described where I would start, that might help?

We start off with a concrete equation, like $y=x^2$. We pick a few values of $x$, calculate $y$ for each, plot them on a graph, and then draw a smooth curve through them. (Get the class to use their calculators, and shout out the numbers.)

Now pick a value of $x$ like $x=1$ and ask how does the curve behave 'close' to this point. Let's try $x={1,1.1,1.2,1.3}$, and we get $y={1,1.21,1.44,1.69}$. For every $0.1$ we increase $x$, $y$ goes up by a bit more than twice as much as that. Let's try looking closer. We take $x={1,1.01,1.02,1.03}$, and we get $y={1,1.0201,1.0404,1.0609}$, and can say that for every $0.01$ we increase $x$, $y$ goes up by very close to twice as much. If we plot them on a graph, the line looks straight. Now look even closer, $x={1,1.001,1.002,1.003}$, and so on. If we go from $x$ to $x+\Delta x$, then $y$ goes to a value close to $y+2\,\Delta x$, and the smaller the step $\Delta x$, the closer the approximation.

Next try the same technique on other points on the same curve. Like $x={2,2.01,2.02,2.03}$, then $x={3,3.01,3.02,3.03}$, and so on. It's not hard to spot the pattern in the multipliers - they go $2, 4, 6,\ldots$, as $x$ goes $1, 2, 3,\ldots$ . The multiplier is $2x$, and when $x$ is increased by a small step $\Delta x$, then $y$ increases by $2x\,\Delta x$.

This 'multiplier' is what we call the derivative. And this gives us a 'pocket calculator' way to find the derivative at a point $x=x_0$. If we calculate $y$ at the given point, and one just a millionth of a unit to the right, we see that $y$ increases by so many millionths, which tells us the derivative. (This method doesn't always work, if the function has sharp corners or tight wiggles, or involves very big numbers. But for all the examples you will meet in an introductory course, it will.)

If we plot the points on a graph, we get the chord from $(x,y)$ to $(x+\Delta x,y+\Delta y)$, and if we extend this straight line segment beyond the curve in both directions, we find the two crossing points get closer and closer together, and the line gets closer and closer to being the tangent, where the line touches the curve at a single point. (I have to say, that's probably more useful to 17th century mathematicians who grew up with Euclid's geometry and have already internalised the properties of tangents. To a more modern audience, I might say it's the straight line 'along' the curve at the point.)

Next we pick some other simple polynomial curves, and calculate derivatives at points along the curve with the 'millionth' method. Plot them out, note the patterns. The derivative is positive when the curve is sloping up, the derivative is negative when the curve is sloping down, and the derivative is zero when the curve is horizontal, at a minimum, maximum, or inflection point. We put in a few more complicated curves like $y=1/x$ and $y=\sin x$ to make clear that the definition applies to any smooth function, not just polynomials.

If they have access to computers, this is a great point to automate the process - to plot a graph of a function, and then plot its derivative using the 'millionth' method. Then if they want to check something, experiment, build intuition, they can do so without hours of labour with a calculator.

At this point, we can move on to the abstraction and introduce the algebra and notation for differentiating $x^n$, $f(x)+g(x)$, $kf(x)$for constant $k$, and build up to simple polynomials, and the formal idea of 'limits' and so on. They have something concrete to cling to, with the 'millionth' method, numerous simple examples, and mental pictures of curves and slopes.

In learning, we always go from the concrete, to the pictorial, to the abstract. Mathematicians have a bad habit of starting with the abstract. It works, in terms of being able to give correct answers, but it quickly becomes a bit of a 'Chinese Room', where symbols are manipulated without understanding, to give the illusion of intelligence. I think you probably did a bit better trying to start with the pictorial. But when that's not working, I find you have to go all the way back to the concrete.

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First you need to think about what you are doing and what the priorities are or should be. You are an unofficial resource and not plugged in with the prof, imperfect as he may be. Physician, first do no harm...

Many students don't even master procedures. So it's actually half a loaf to be getting kids who are at least doing that.

The way to teach concepts is very similar to how to teach procedures. Imitation and practice. Lots of imitation and lots of practice. And weaker kids will need more. There is no royal road, no magic lock and key explanation.

Still, I would be very concerned about you derailing things and going from half a loaf to zero, if you actually implement what I just said...large amounts of practice on concepts. Especially if the REAL teacher is going to test them mostly on procedures.

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I used to do a lot of math tutoring, and I fear change in the curriculum prerequisite to calculus may be the problem. Way back when, a lot of time went into generic curve-sketching. And students worked more abstractly. But with the advent and use of graphing calculators, students today want to input a function then use the calculator to calculate and draw the graph. A level of abstraction has been lost, I fear.

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Time permitting, I would want to quiz them on whether they intuitively understand slope and tangent lines in the first place:

  1. See if they can answer speed-drills on graphing basic lines. This is one of 5 skills that I personally identify as "automatic" necessities for college algebra, and have a timed quiz at the link below. I'd want to see that they can answer this correctly within the timer; if not, they should practice a few minutes every day until they do so.

Automatic-Algebra: Graphing Lines

  1. Then, I would want to give them some random curvy graphs, name a point, and ask them to draw a tangent line there and estimate its slope. If they can do that, then the explanation "derivatives give you that slope" should be coherent.
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  • $\begingroup$ Timers make me nervous. I actually got one wrong each of the 2 times I tried this just now. Otherwise, I like this answer. $\endgroup$
    – Sue VanHattum
    Commented Oct 8, 2023 at 0:41
  • $\begingroup$ @SueVanHattum: I tend to hear that when people try it for the first time. I'd be curious how much quicker it becomes with a little practice, treating it like a video game, or multiplication tables? I set the timers to around triple what it initially took me; that particular quiz now takes me around 5 seconds (out of the 30) on average. $\endgroup$ Commented Oct 9, 2023 at 1:41
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    $\begingroup$ I see the value of this sort of thing, but I also know that some students are harmed by timed tests. I just did it 3 more times. The first time, I got one wrong (thought I saw a line where the axis was). The next 2 I got all right. If students can do it as many times as they want, perhaps it will do no harm. (And perhaps I should have considered this sort of thing. But I have now retired. Ha!) $\endgroup$
    – Sue VanHattum
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    $\begingroup$ @SueVanHattum: Glad you looked at that once more. Like the main page says, the intent there isn't as a test, but as daily practice over weeks of time. $\endgroup$ Commented Oct 9, 2023 at 5:07
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Might or might not work, but try a graph of how far a car has traveled with respect to time, some journey they have actually traveled, then ask them what the speedo would have been reading at a few points on the graph. Something they should be able to relate to.

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Here's my personal experience, as a former A(ish) grade student, now in his late 30's, doing math in my spare time:

There was a big disconnect inside my head between the abstract symbols I was easily manipulating (functions & equations, basically symbols and numbers), and the physicality of a function's graph - I applied stuff like f'(x)=0 to find min&max, the dips and peaks of a graph, but just as formulas and algorithms 'given' to us, without a mental picture of rates of change and infinitesimals.

Much, much later, I had trouble grasping why, for parametric curves, the calculated df/dg 'happened' to exactly be the 'physical' tangent to the sketched graph. What helped me bridge these two universes are the 'rise-over-run' mnemonic and understanding an x-y graph as an actual, precise, 2D 'physical' map, and not just some 'yet another' simplified representation of a function*.

*In class, a graph was most of the times a 'rough sketch' of a function, some squiggle meant to convey a rough outline of what the function was actually doing 'internally', in it's own universe of symbols and numbers, where it was precise - and without giving it much thought, that's how I internalised it, as an imprecise tool for showing someone else a glimpse into what your function does, not as an accurate 'image' of the function - precise graphing tools like desmos where I can "see" that the graph matches the grid and changes in x & y are coordinates I can draw and see them moving precisely according to equations helped me overcome this 'rough sketch' perception.

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Maybe a physical demonstration would help, e.g. using a thin wooden disk (or dinner plate) and a ruler. With the ruler in the plane of the disk, touch it to various points on the disk noting (1) how the ruler touches the disk at a single point, and (2) how the "slope" of the ruler changes as you move the point of contact around the disk. Maybe try different shapes, some with corners. What happens at the corners and on the sides of, say, a triangle?

Or some animation. From Wikipedia: Tangent Line Animation

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I think @user22788 has made a good point about prior concepts, but there is a related issue, motivation. Why are the students doing calculus in the first place? I assume that they don't aspire to be mathematicians or physicists, because I wouldn't expect such people to struggle with calculus. Maybe the examples, e.g. cars driving up the graph of f(x), have no relation to anything that interests them. Can you find examples that are actually in line with their interests? If so, try to pick up @user22788 's other point, starting with simple linear examples, where you really don't need calculus, then introduce the tangent as "the linear function when you don't have a linear function".

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The attitude of the students is that this is a hoop they need to jump through. They aren't motivated to understand the thing your trying to teach them. They just want to jump through the hoop. I had this experience with many kids I tried to tutor. They don't really care to understand the thing at all. This is true for the whole society. If they could dispense completely with learning it they would.

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  • $\begingroup$ This may be true, but how does it provide a solution? $\endgroup$
    – breversa
    Commented Oct 9, 2023 at 13:38
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Perhaps this might help:

The derivative is (essentialy) the best linear approximation of a function (at a given point).

I never quite understood expressions such as "rate of change" or "slope of a line", even though I never had any trouble with the mechanics of derivatives. Much later I was told the quote above, and everything was clear.

Suppose you have a function $f(x)$ (with appropriate hypothesis), and you want to approximate $f$ at point $x_0$. Then to approximate by a constant $b$ the best choice is the value of $f$, i.e $b = f(x_0)$. To approximate by a linear function, i.e. some $g(x) = a \cdot (x - x_0) + b$, the best choice is $a = f'(x_0), b = f(x_0)$. The derivative is not the linear approximation, but the coefficient of the parameter; you already have the constant approximation $f(x_0)$ so the derivative $f'(x_0)$ is all the extra info you need.

("Best" means that the appropriately defined error goes to 0 when approaching $x_0$. You might be able to motivate the error formula, or you may get by on intuition.)

Applications:

  • Is the function increasing at $x_0$? Look at the linear approximation: when is it increasing? When $a > 0$ [plus technicalities]. :-)
  • What about a (local) maximum? (Skipping details) either $a < 0$ at the left edge, $a = 0$ inside, or $a > 0$ at the right edge.
  • Instantaneous speed, and many other Physics-motivated examples.
  • Solving equations! (approximately). You want $x$ such that $f(x)=0$ (e.g. to compute $\sqrt{u}$ solve $f(x) = x^2-u$). Then whenever you have a good approximation $x_k$, solve instead $g(x) = f'(x_k) \cdot (x - x_k) + f(x_k) = 0$ instead. You get $x = x_k - f(x_k)/f'(x_k)$, and that is a step in Newton-Raphson method. For $\sqrt{u}$ it'd be $x_{k+1} = x_k - (x_k^2-u)/(2 \cdot x_k)$.
  • Extends to Taylor expansion formula.
  • Generalises to diferentiation in multiple variables.
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Why not talk in terms of time series, say the annual population of some city over time. Show them a graph with some points lying on or near some curve in a neighborhood left of a point where the derivative is, say 1,000 (and the second derivative is fairly small, so the trend is fairly obvious), and ask them to predict what the population will be next year. When they find themselves confidently offering guesses, ask them how they arrived at them. Then do the same where the derivative is 2,000.

This kind of experience extrapolating should provide an entry point to discuss “where curves seem to be heading“ at various points.

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Easy. The issue is it's not a math problem. But one rooted in behavioral psychology and misappropriated expectations for consistent results. Expecting an exacting mathematical Datum consistency, where the measure doesn't apply in teaching math subjects.

Hint: Everyone's born as uniquely different as night and day. And it's obviously a mistake to expect a categorically (satisfactorily) consistent means and result thereof. Ya' can't force feed, what won't work anyway.

Remember now!

New Math Tom Lehrer "So very simple, that only a child can do it."

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    $\begingroup$ I don’t know what to make out of this. You say "easy", give a "hint", advise to "remember now!" and provide a book reference (I think?)… but what about giving an actual answer to the question? $\endgroup$
    – breversa
    Commented Oct 9, 2023 at 13:37

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