# What holds your students back in Calculus?

I teach Precalculus to high school kids, and I know a lot of you all teach Calculus.

What are some issues that your students have in Calculus classes that you wish had been addressed in a Precalculus class?

e.g. "Man, my kids never know how to factor, and that really holds them back."

e.g. "Informal experience with finding speed graphs from distance graphs would be huge."

e.g. "Nothing! My calc students are perfectly well prepared. Keep it up, precalc teachers!"

e.g. "Downvote. Not specific enough."

• They manly make silly mistakes like $(x+y)^2=x^2+y^2$ or $\ln (x+y)=\ln x+\ln y$ or even $\cos(x+y)=\cos x+\cos y$... – حكيم الفيلسوف الضائع Mar 16 '14 at 14:33
• Should we have a big-list tag? On the one hand, this question would be perfect for it; on the other, I don't know that we should in general encourage lots of big-list type questions. – user37 Mar 16 '14 at 15:02
• Here's a pretty comprehensive list: math.vanderbilt.edu/~schectex/commerrs – Mark Meckes Mar 16 '14 at 15:20
• Logarithms...... – Brian Rushton Mar 16 '14 at 23:30
• The mistakes being made seem to come from a poor foundation in Algebra more than anything else. Understanding what you can and cannot do with an equation is the most common problem I've seen when tutoring students on the college level. – David G Mar 17 '14 at 4:39

They are not usually well-prepared, but factoring is not a big issue. I would like students to be able to:

Make meaning from graphs. [I want to get to introduce the connection between speed graphs and distance graphs, though, so I'd be happier if you focused on other graphs.]

See what is algebraically sensible, and what's not. e.g. you can't cancel from one term in a fraction that has more than one term on top or bottom ((x+h)/h does not = x), square root of (x^2 + y^2) does not equal x+y, ...

Understand the basic area and volume "formulas", instead of trying to memorize them. Then maybe they'll have a chance at solving optimization problems. [I asked the post office problem on the last test. (The post office will not allow packages to exceed 108 inches in combined length (longest dimension) plus girth (measure around the shortest way). Find the package, with square ends, of maximum volume.) Although I explained repeatedly what length plus girth meant, they bombed it. To me, that means they can't deal with thinking about something even a tiny bit new.]

This list is starting to take on a theme. It's not the topics you teach, so much as how well you help them learn to think mathematically. Except trig. Is trig part of your pre-calculus course? They need to be able to understand the trig functions. My students suffer more from their trig weaknesses in calc II than in calc I. I've told my pre-calc students that the main theme of the course is understanding how equations and graphs go together.

• Oops! I just finished a unit in Precalc on the connection between speed and distance graphs. I tend to think that spreading ideas over multiple years is typically good for students, though, so I don't entirely regret the unit. – Michael Pershan Mar 16 '14 at 15:50
• Oh, don't regret it at all. I just want to hog the fun! I'm sure you're right that seeing an idea again and again is helpful. I do sometimes have students coming in and assuming "I know that." when they don't really. But that's not the teacher's fault. – Sue VanHattum Mar 16 '14 at 17:09
• (x+h)/h =/= x : A fellow Math Ed student that I am in several classes with still doesn't fully understand this concept. She also doesn't understand that x/(x+h) =/= x/x + x/h and I'm not sure why. – David G Mar 17 '14 at 4:31
• @Skytso: I think a lot of problems with fractional expressions come down to students not having fully internalized that / means division. – Mark Meckes Mar 17 '14 at 8:25
• @David G: For the Math Ed student (who may want to consider changing to something-else Ed), you can try getting rid of the letters and just look at simple numerical examples. This doesn't get at the concepts, but it's a useful trouble-shooting strategy students should be encouraged to develop. As for the concepts, $(a+b)/c$ has to do with breaking up something into $c$ many pieces, and looking at $a$ many of those pieces and another $b$ many of those pieces, while $a/(b+c)$ has to do with breaking up something into $b+c$ pieces and looking at $a$ many of them, which doesn't give an addition. – Dave L Renfro Apr 8 '14 at 17:51

Here's one I haven't seen mentioned so far: students coming into calculus are very uncomfortable with the idea that there might be more than one way to do something, and especially that they might have to make decisions, especially decisions that depend on context rather than just a rule.

Two concrete examples of this phenomenon.

1) Simplifying. I see students routinely oversimplifying in the middle of problems---half-way through, they'll do a bunch of work to put an expression into some canonical form, even if it makes the next step in the problem much harder. They're happy to learn "don't simplify at all ever", or even "simplify in these problems but not these", but it's a real struggle to get them to ask, "Can I simplify here, and if so, will it help or hurt what I do next?"

2) Lines. My students believe that the only way to write the equation of a line is in slope-intercept form. Present a line in another form and they're not even sure it is a line; ask them for a line in point-slope form and they'll typically force it into slope-intercept anyway. (For specific reasons about material, this is a particular obstacle: working with the linear approximation to a function lends itself very naturally to dealing with the point-slope form.) And they're very uncomfortable with the idea that there are several ways to write lines, and that one should choose which way based on the application you have in mind.

• On the subject of your (2): my students are also uncomfortable with the idea that (a) linear functions can be represented by (b) many different equations relating input and output and that all of these graph (c) the same line. These concepts are all blurred together. I try to make these distinctions by using the analogy that (a) love is often represented by (b) giving jewelry and often results in (c) having the same address, but that one should be very careful to distinguish these three things. – Shay Apr 3 '14 at 18:05

My impression (as a former private tutor/current university employee and from my mind as a former student) is that mainly equation solving is a big problem for poor (sometimes also for advanced) students.

This is somehow some task one has to perform to do subtask in calculus (e.g., a new thing is to calculate a minimum of a function. The students learn that his is done by taking the derivative (=new thing in calculus!) and solve $f'(x)=0$.) Since f can now be any function, the students must have in mind how to solve a various number of equations, but normally they learn at some point how to solve quadratic equations, (forget it maybe), then learn how to solve exponential equations, (forget it maybe), learn how to solve trigonometric equations, etc. -- At that point, everything you want to do is limited how good students can solve equations and if you are unlucky, you have to spent more time on teaching/reminding how to solve equations than on the actual topic. - If one is aware of this, the pre-calculus teacher as well as the calculus teacher can do something to solve this issue by giving some homework in repeating that things.

A very similar analysis goes with the issue of manipulating terms.

When I teach calculus, there are two preparedness issues that frequently surface:

1. Understanding functions: I don't mind if my students can't write down the definition of a function (though ideally they could), but if I say something like "let $f$ and $g$ be functions," they shouldn't be horrifically confused. Similarly, if I ask very basic questions about functions (e.g. if $f$ is a function from the reals to the reals, and $x$ is a number, what type of thing is $f(x)$?), I expect they will have at least a nose for the answer.
2. An openness to being confused. Many students in my calculus classes expect to never struggle, never not know precisely how every step of a problem will be done in advance, etc. I try to emphasize to them that non-confusing things aren't worth spending time learning, but most of them have so internalized the inference chain "confusion leads to getting things wrong leads to getting bad grades leads to living under a bridge'' that they run from confusion at all costs, even when that means never engaging with confusing material that I've promised will be on exams.
• I love "confusion leads to getting things wrong leads to getting bad grades leads to living under a bridge'' and will share your thoughts with my students. – Sue VanHattum Aug 15 '19 at 3:19

As coordinator of a Maths Learning Centre, where many many students have actually seen some calculus at school but still struggle with calculus at University, these are my top few:

The notation f(x)

Quite a few students think that $f(x)$ means $f \times (x)$. This is particularly true with trigonometric functions, where many of them actually say "sine times ex" when they read $\sin(x)$ aloud. I teach them it's pronounced "sine of ex", but for some that doesn't help because they were taught at school that "of" always means "times" (as in "10% of 15"). Constantly reinforcing that $f(x)$ is the object produced when the action $f$ is performed upon the object $x$ -- or "the result of f acted upon x" -- would make life better for many of them.

Even if students do realise that $f(x)$ is not about multiplication, many of them do not know at a deep level that $f(x)$ is actually a number. They see it as a call to action, rather than the name of the result of this call to action. While it may be possible to calculate the value of $f(3)$ and it might come out to the answer $4$, say, you don't necessarily need to calculate it, and you can use the symbol "$f(3)$" every time you mean "$4$" if you want. Basically, $f(3)$ is literally able to stand in the place of whatever value it is.

This may seem like a small issue, but it is a problem when students attempt to differentiate $\ln(2)$ to give $\frac12$ (for example, given $y = e^{\ln(2)x}$, they write $\frac{dy}{dx} = e^{\ln(2)x}(\frac12 x + \ln(2))$ ). I keep reinforcing to them that $\ln(2)$ is just a number. It's somewhere between 0 and 1, but I don't have to calculate it to know it's a number.

The fundamental meaning of an x-y equation

I really wish students knew that the fundamental meaning of an x-y equation is to tell if a point is part of set in 2D space. That is, when we see something like $y = 3x+1$, it is telling us a rule for how x and y must be related in order for the point $(x,y)$ to be on a line. If we sub in the point and it works, then it's on the line, and if it doesn't work, then it's not on the line. The question of "Is the point (1,4) on the line with equation $y = 3x+1$?" stumps many of my students, even if they can tell me what the slope and intercepts are without even thinking.

Another point I'd like them to realise is that any equation such that all the points on the line satisfy it and all the points not on the line don't satisfy it, is a possible equation of the line. For example, $y - 3x = 1$ and $x = \frac{y-1}{3}$ or even $(y - 3x +1)^2 = 0$ are perfectly reasonable equations of the same line.

Basically my point is that I would like students to know that an equation does not have to be explicitly tied to a function, and even if it is, it can still be used to tell if points are part of a set.

The connection between algebra and geometry

The flip-side of the above point is that equations do represent geometrical objects, and you can glean information from the equation to tell you about the geometry of the object. I guess what I'd like them to realise is that writing an equation in a certain way will tell you certain information about the object, and especially that different ways of writing the equation will tell you different information. These ideas are often known separately for different equations (eg quadratic equations where completeing the square tells you the vertex but factorising tells you the x-intercepts), but I'd love them to have the general concept reinforced for everything so that they realise it's actually one idea worked out in different situations.

Knowing what the graphs of various functions look like

My students self-identify this one as their own major problem. They just don't have an instinct for what the graph of a function looks like without having to get a computer/graphics calculator draw it for them. If you ask them "what does the graph of a cubic look like?", they will reach for their graphics calculator, when all you wanted them to do was draw a "swoosh".

I would love it if they simply "knew" what various graphs looked like. A bit of a stimulus-response thing where if they see either of the forms of the equation of a line, they'll think "line", or if they see $y = \text{quadratic in }x$ they'll automatically think of an up-curve or a down-curve.

I realise this may be at odds with the previous point, but I think you can achieve both. Plus it really is an issue for them when their university lecturers use these functions as their basic examples, and the students simply can't imagine them!

Skill at algebraic manipulation

Finally, this really is a prerequisite that gets in the way for a lot of people. It's not as bad a problem as you might think, because often learning algebra "in passing" while doing calculus is actually more successful than the students' previous algebra instruction. However, it would be nice if they knew how to expand/factorise/manipulate fractions.

This last one of manipulating fractions particularly comes up when they are doing differentiation by first principles with limits, and when they are using the quotient rule, and many students show that they simply do not know how to add fractions, even of the number variety. (Though there are some students who can do the algebraic ones and not the number ones.)

There are three main areas where students entering my calculus have trouble: keep in mind I teach a more formal calculus than most US schools these days. I do spend about 30% of the lecture time on proofs and I start with the axioms of the real numbers and some discussion of basic topology on the real line. I teach primarily from my notes and I'm not sure how much longer I'll be able to teach as I am given the invariable decline of the students with time. As it currently stands, I have been able to attract some excellent students with our calculus. They like the challenge.

1. trigonometry: many of them have been taught with calculators. Others not at all. Generally speaking there is a lot of ignorance about things they ought not need think hard about: $\sin(0)=0$, $\cos(0)=1$, graphs of sine, cosine, tangent. I just gave a physics (calculus-based, so these kids have already been through calculus) where no one could solve $0.8\sin \theta + \cos \theta = 1.2$. More generally, the problem is the bar is set too low in the previous courses. I suspect they are taught to solve standard problems rather than being taught how to think. Have they ever encountered a problem on an exam which was not in homework? Is that even on the table? Were they expected to take notes or just to fill in some template? Was homework done at home or in class? These are all factors.

2. set notation, reading definitions, original thinking. I've touched on this in 1. already, but it deserves it's own point. When I teach calculus I make a point of giving definitions. The student is expected to read it and use it. I see students totally shut-down when I ask a really simple question which requires nothing more than reading the set-builder notation and literally some number-line thinking. They go hunting for a problem like that, so they can follow the guide. This is a problem. The books we use teach them problem solving strategies as opposed to helping them think for themselves. I realize babies need milk, but at some point you gotta start to chew. Sooner would be better for them.

3. False confidence. Some students have taken a highshcool calculus before where it was mostly symbol-pushing. Well, I still make them push symbols around, but there is much more analysis involved. Can they actually write mathematics? As in , use sentences to explain what they are doing. Often, I spend a lot of ink on the first test just trying to communicate to them that they must connect the equations properly. To summarize, if getting the answer was sufficient in their previous course then it's an uphill battle when they get to my class and I tell them that the answer is the whole argument.

• The formatting in this answer makes it difficult to get through all the content. Also, consider streamlining your points. They seem to run together and lack specificity. – David G Mar 17 '14 at 4:37
• @Skytso I added bold. In so far a specifics go, the point of my answer is that it's not about a particular missing point. It's a family of missed concepts which plagues them. For some it's factoring, for some it's basic trig. for many it's an inability to graph, you pick a topic they don't really know it. (speaking statistically of course). Many of my colleagues, especially the more experienced ones, would like to funnel all the incoming students through a algebra course just to address the trouble. – James S. Cook Mar 17 '14 at 5:16
• @James: I don't have anything in particular to add, I just wanted to say hi. I don't know if you remember me, but I was an undergrad at NCSU while you were a grad student. – Jason DeVito Mar 18 '14 at 12:56
• How am I supposed to be able to solve $0.8 \sin \theta + \cos \theta = 1.2$ by hand? I can turn it into $\sqrt{1.64} \sin(\theta + \tan^{-1}(1.25)) = 1.2$ and go from there but that isn't much of an answer, and many trig courses wouldn't cover that; I don't think it's unreasonable that your students didn't know that trick. What were you looking for? – David E Speyer Mar 26 '14 at 19:47
• @DavidSpeyer Since $\cos^2 \theta + \sin^2 \theta = 1$ it follows that you can eliminate either sine or cosine to give an algebraic equation which when squared gives you a quadratic which when solved reveals the two solutions for $\theta$. They do have a scientific calculator for the numerics here. – James S. Cook Mar 27 '14 at 2:46