Why Calculus is Hard

A blackboard with calculus work in progress

Calculus is widely regarded as a very hard math class, and with good reason. The concepts take you far beyond the comfortable realms of algebra and geometry that you’ve explored in previous courses. Calculus asks you to think in ways that are more abstract, requiring more imagination. You have to wrestle with new vocabulary, new symbols, and new processes. The problems are often longer and more involved, sometimes taking a full page or more of written work to complete.

Those are the obvious reasons why calculus is hard. But there’s more to it than that.

The Main Reason Calculus is Hard

In my experience as a tutor, the primary reason calculus is so difficult is that it is sitting atop an upside-down pyramid of previous math concepts. Calculus ties together everything you’ve learned (or were supposed to have learned) in algebra, geometry, and precalculus, as well as the more basic math from elementary and middle school.

Everyone accumulates knowledge gaps as they progress through math, and for many students, calculus is the course where these gaps finally come back to haunt them. At the same time that they’re trying to learn the new, weird concepts that make calculus what it is, they’re also struggling with all the old things they don’t know. This makes new concepts more confusing. It makes it hard to keep up with lectures. And it makes doing the problems very challenging.

Why do students have so many knowledge gaps?

First of all, the way our school system does math makes knowledge gaps inevitable. Let’s say the average student finishes a typical math class with an 80%. We say to them, “You got at B-. Not bad. Now move on to the next course.” So next year, they start out missing 20% of the information they’re supposed to know! We do this year after year, until their math knowledge pyramid looks like Swiss cheese.

But it’s worse than that, of course, because students routinely forget things that they used to know. The mastery path is a muddy slope, so even if you understand a concept at test time, you’ll quickly forget it without additional practice. This forgetting happens during the school year as well as over the summer.

And right now, in 2022, knowledge gaps are at an all-time high because students didn’t learn as much as they were supposed to during COVID. So if you’re going into calculus this fall, you can be sure that you have knowledge gaps that are going to make your life difficult.

Don’t be ashamed of your knowledge gaps.

You might be ashamed to admit that you don’t remember quadratics or trigonometry, but it’s okay. I’ve worked with calculus students who didn’t know how to do basic algebra. I’ve worked with calculus students who didn’t know how to add or multiply fractions. I’ve worked with calculus students who didn’t know their multiplication facts.

Having these gaps doesn’t make calculus impossible, but it does make it a great deal more difficult. So if you’re about to take calculus, or even if you’re in the middle of the course, consider hopping on Khan Academy and patching up some of the holes in your math knowledge. Yes, it’s extra work now, but it will make things easier for you in the future.

Don’t be ashamed of your knowledge gaps; deal with them. If you do, calculus won’t be quite so hard. You’ll avoid slipping into a downward spiral of math avoidance by transforming this daunting course into a manageable, even enjoyable, challenge.

How to Patch Holes in Your Math Knowledge: 3 Steps

Every math student collects knowledge gaps as they progress through school. Sometimes you don’t understand a concept when it gets taught, and the class moves on before you figure it out. Sometimes you get it, but since the mastery path is a muddy slope, you quickly forget. And some key pieces are, for whatever reason, never taught at all. This year, with limited schooling due to COVID, that final case is more common than ever.

And if you understand that math is an upside-down pyramid, then you know that knowledge gaps make a very unstable foundation for future learning. They increase the likelihood of getting confused, falling behind, and slipping into a downward spiral of math avoidance. So patching holes in your math knowledge is critical. But how do you do it?

What follows is a step-by-step process for identifying and filling knowledge gaps in mathematics.

1. Identify

Step one is to identify what you don’t know, back up to the earliest, easiest things that you don’t have mastered, and start from there. To do this, I recommend using Khan Academy’s course challenges. These are like miniature final exams for Khan’s math courses. There’s one for every grade. You can also take the unit tests that are at the end of each major subject within these math courses.

When you take a course challenge or a unit test, Khan keeps track of what you get right and what you get wrong. (You have to be logged in to your free account for this.) Then, when you scroll through the exercises of the course, each activity will have a little icon next to it that will have zero, one, two, or three purple bars filled in.

  • Three purple bars represents mastery: You’ve gotten it right several times and haven’t recently missed a question.
  • Two purple bars represents proficiency: You’re approaching mastery.
  • One purple bar represents familiarity: You’ve gotten a couple questions on this topic right, but you’ve also missed some questions, so this needs work.
  • Zero purple bars means you haven’t demonstrated any understanding of this topic.

Topics with zero or one purple bar are your knowledge gaps.

Let’s say you’re about to take high school geometry because you’ve just finished Algebra I. You might take the Algebra I course challenge. And if you have a pretty good memory of what you learned this year, you might miss only five or ten questions out of 30. If that’s the case, those five questions represent content areas for you to target. Go back into the course and work on them.

But if it doesn’t go so well, and you miss say, 15 or 20 out of the 30 questions, you probably need to back up further than Algebra I. Drop back to the Pre-algebra course challenge, or go back to the 5th grade course challenge. Feel it out. Maybe you need to go way back and relearn your math facts. Maybe your math comfort zone is so small that the right move is to start at the beginning and work your way up.

I’ve worked with Algebra II students who didn’t know their multiplication facts. I’ve worked with calculus students who needed to relearn fractions. And I myself had to relearn subtraction with borrowing at the age of 27. It’s all okay. There’s no shame in not knowing something. You just need to identify what’s missing from your math knowledge so you can go get it.

2. Understand

Once you’ve identified your knowledge gaps, it’s time to get to work. The first milestone on the mastery path is achieving understanding. For this, Khan has videos for every topic. Watch them, and pause often to take notes and try things out yourself. If Khan’s videos aren’t sufficient to “get it,” there are numerous articles and videos online that teach math, along with workbooks and textbooks. And, of course, you can ask other people for help.

Being confused is normal. Work through it by using resources and making written product.

Once you think you understand, try the exercises. If you get stuck, consult your notes or use the hints provided with each problem. The hints walk through the questions step-by-step. And don’t just look at the solutions – write them out.

3. Master

After you achieve understanding and do an initial round of practice, you need to follow up on that with spaced repetition in order to solidify both your memory and your comprehension of the topic. Revisit this topic several times in the days after you learn it and then for several weeks after that.

Start with massed practice. For this, Khan isn’t the best choice. Worksheets are a better option. Do a quick google search for the topic plus the word “worksheet,” and you’ll find plenty of choices. Include the words “answer key” if you’re not immediately finding worksheets that come with answer keys. For high school math, kutasoftware is great. Hit print, and get to work. But don’t necessarily do the entire worksheet in one sitting. It’s better to do a little bit each day than to do it all at once and then ignore the topic for a week.

Repeat this process for several topics within a unit on Khan, and then take the associated quizzes and the unit test. These present you with a variety of problems, forcing you to remember which methods apply to which situations. This is called “interleaving,” and it is critical for advancing as a math student. Just remember that you don’t want to go straight to interleaving; it is the combination of massed practice and interleaving that cultivates deep mastery.

Make Progress

Patching holes in your math knowledge is hard work. So if you engage this process, and you discover that you have many knowledge gaps, you might be disheartened. But please don’t see this as an all-or-nothing endeavor. Any holes you patch will make your life as a math student easier. Any progress you make here is worthwhile. Everything counts.

Mastery Learning: In and Out of the Classroom

Image from https://www.ted.com/talks/sal_khan_let_s_teach_for_mastery_not_test_scores?language=en

Last year, two different teachers of my students said that their classroom was using a “mastery learning” model. Maybe your child is in one of these so-called mastery learning classrooms right now. Or maybe you’ve seen Sal Khan’s TED Talk about mastery learning, and you’d like to either support your child with this or practice it yourself. In any case, we should talk about what mastery learning is and how it can be practiced, both in and out of the classroom.

Mastery learning is an academic model very different from the way traditional classrooms are run.

Your typical math program groups kids by age rather than by knowledge or skill-level. Almost the only time this rule gets broken is when we push kids ahead of their age group, but having students skip ahead in math is highly problematic. For the most part, students move through school at a uniform pace, and get variable outcomes – different letter grades or percentiles.1 The kids who struggle to keep up might be perfectly capable of understanding the math, but because the curriculum moves too quickly, they fall behind, get confused, and often fall into the downward spiral of math avoidance.

Khan explains that, after each test, unless you got 100%, you’ve demonstrated that you’re missing some of the knowledge, but, “Even though we’ve identified the gaps, the whole class will then move on to the next subject, probably a more advanced subject that’s going to build on those gaps. It might be logarithms or negative exponents. And that process continues, and you immediately start to realize how strange this is. I didn’t know 25 percent of the more foundational thing, and now I’m being pushed to the more advanced thing. And this will continue for months, years, all the way until at some point, I might be in an algebra class or trigonometry class and I hit a wall. And it’s not because algebra is fundamentally difficult or because the student isn’t bright. It’s because I’m seeing an equation and they’re dealing with exponents and that 30 percent that I didn’t know is showing up. And then I start to disengage.”1

Each semester, everyone gets a grade, and as long as that grade is a D or higher, they move on. For example, if you finish Algebra 2 with a C because your final grade was 75%, you’ll take PreCalculus next year. The trouble with this, of course, is that PreCalc is built on Algebra 2, and you’re missing one fourth of that knowledge, not to mention what you forget over the summer. Khan demonstrates the absurdity of this system by comparing it to home building:

“So we bring in the contractor and say, ‘We were told we have two weeks to build a foundation. Do what you can.’

So they do what they can. Maybe it rains. Maybe some of the supplies don’t show up. And two weeks later, the inspector comes, looks around, says, ‘OK, the concrete is still wet right over there, that part’s not quite up to code … I’ll give it an 80 percent.’

You say, ‘Great! That’s a C. Let’s build the first floor.’

Same thing. We have two weeks, do what you can, inspector shows up, it’s a 75 percent. Great, that’s a D-plus. Second floor, third floor, and all of a sudden, while you’re building the third floor, the whole structure collapses. And if your reaction is the reaction you typically have in education, or that a lot of folks have, you might say, maybe we had a bad contractor, or maybe we needed better inspection or more frequent inspection. But what was really broken was the process. We were artificially constraining how long we had to something, pretty much ensuring a variable outcome, and we took the trouble of inspecting and identifying those gaps, but then we built right on top of it.”1

So that’s the traditional model. Mastery learning, by contrast, would have students move forward at their own pace, only advancing after they’ve demonstrated true mastery of the content they’re working on now.1 In my view, a mastery learning curriculum would also utilize the science of learning and build in both massed practice and interleaving, employ spaced repetition, and have students produce lots of written product. Students could receive on-demand, individualized instruction as well as on-demand, individualized homework and tests.1 Khan points out that this would have been impossible or too expensive in the past, but with online resources, such as Khan Academy (the platform he created), this level of individualization is actually possible.

Still, even with great tools like Khan Academy, switching our schools over to a true mastery learning program would be very challenging. Students like to be with kids their own age, it’s expensive and time-consuming to shift curriculum in a major way, most teachers aren’t trained to run classrooms in this way, and it’s not even clear exactly how a mastery learning classroom would be run because this idea hasn’t been widely tested. I’m also hesitant to set up any academic system that relies so heavily on computer-based learning because I’ve seen so many issues with that model. Plus, a true mastery learning program would start when kids first enter the school system and follow them throughout their school career. It would we difficult to implement this mid-way through a student’s career because they’ve already accumulated many years’ worth of knowledge gaps.

With all that in mind, let’s return to those two math teachers who claimed to be running mastery learning classrooms and take a look at what they were doing.

The one thing they had in common was that homework was assigned but not graded; the only grades came from tests. From there, they diverged.

In one, each test was worth 100 points, and everyone would complete a mandatory retake of that test one week later, regardless of their score on the first test. Whatever score was higher would be counted. After that, no retakes. And in between the test and the retake? The class moved on to new content.

In the other, tests would target specific math skills, and students would be graded on each math skill individually, receiving one of four possible scores: 1, 2, 3, or 4. Only the top two scores, 3 and 4, were considered passing. In order to pass the class, students needed to earn a 3 or 4 on a majority of the topics tested. They could retest individual skills at any time by special request. They could raise a low score to a passing score by doing well on a later exam that included the old content. This seems closer to the mark, but the scoring confused students and parents alike, and I felt it was too blunt to be very meaningful.

Neither of the students I worked with from these classes was consistently learning what the teachers were teaching. They often didn’t fully understand the material, and they weren’t receiving training at school on how to use resources, such as the textbook, to figure it out. One of the teachers was available for one-on-one support, and the other was not. Neither classroom allowed students to progress at their own pace. Neither made sure you actually arrived at “I get it!” on the mastery path.

Both teachers assigned homework on a regular basis, but both assigned very short problem sets that didn’t force students to do enough practice to move along the mastery path. The problem sets were always of one type – whatever the day’s topic was – and almost never included mixed review of old topics. So students were missing out on both massed practice and interleaving.

Both classrooms acted as though succeeding with a math topic one time was good enough for mastery – once-mastered, always-mastered. But of course that’s not how memory works. If you don’t revisit old content, you’re liable to forget it. Use it or lose it. Neither seemed to have much spaced repetition built into the structure of their teaching, homework, or tests.

So neither of these teachers was really facilitating mastery learning. And, of course, traditional classrooms aren’t necessarily doing better. But, in all these cases, it’s not the teachers’ fault. Classrooms are inherently limiting. They have to teach to the average student. They have limited time and resources. They have state-mandated curriculum to get through. They have to stay on good terms with parents, students, and administrators – three groups who often have competing interests.

And even if you did manage to set up an ideal, truly mastery-driven classroom, it would be an island of excellence in a sea of mediocrity. The students who entered that classroom would carry with them all the knowledge gaps in their upside-down knowledge pyramids acquired over years of non-mastery-based learning. To truly pursue mastery, some students would need to back up very far in order to expand their math comfort zone. You might have to spend an entire year just patching all those holes and rebuilding a sense of self-efficacy. 

Our classrooms can do a better job of facilitating mastery learning, and I applaud teachers for trying. But we shouldn’t expect them to transform into Sal Khan’s ideal any time soon. So, in the end, it’s up to students (and their parents) to facilitate masterly learning at home. For students, this means doing more than what is asked and using effective study techniques. It means not just practicing until you get it right, but practicing until you hardly ever get it wrong. For parents, this means providing resources, such as tutoring, textbooks, and a home study space. Parents can also lead by example by modeling their own pursuit of mastery in whatever they’re working on.

1 Khan, Sal. “Let’s teach for mastery – not test scores.” TED Talks Live. November 2015.