By A.K. Whitney, journalist. In 2009, Whitney went back to school to find out, once and for all, if journalists really are as bad at math as they fear they are; her blog about the experience, Mathochism, runs on Medium three days a week.
This fall, the Mathematical Association of America released a five-year study on college calculus that showed that, no matter how elite their learning institution may be, far too many students lose confidence in their math abilities after Calculus 1. As someone who recently spent a lot of time in calculus classrooms, I understand how that can happen.
Between 2012 and 2013, I enrolled in four different Calculus I courses. This may seem excessive even to the math-loving crowd reading this blog, but let me explain. Of the four, I dropped two, failed one and passed one. Of the four, two were in a community college classroom (the dropped and the failed), while two were Massive Open Online Courses, or MOOCs (one dropped, one passed, the latter with an 89.3 percent).
To be honest, I never set out to take this many calculus courses. Ideally, it would have been one and done. Some quick context: I am a print journalist with 20 years of experience in print and online. While always interested in science, I gave up on math at age 12. I spent the next 26 years as an avowed word person and math phobe, until leaving my full-time newsroom job to go freelance. Suddenly having so much time to think (the freelance career took a while to get going) made me question my youthful decision, and since I was already taking a computer class, I gave a remedial pre-algebra class a try. This turned into the Mathochism Project, where I was determined to revisit high school math as an adult, and write a blog about the joys and terrors of the experience.
To my surprise, there were mostly joys. From pre-algebra to pre-calculus, I did very well, and became delighted not just with math as a subject but also with my ability to understand it, getting mostly As and high Bs. I finished pre-calculus with a high B, and a strong level of confidence. Then the terror began, though I didn’t realize it at first.
If at first you don’t succeed…
In my first calculus course, the lectures were crystal clear. The homework, while not super easy, wasn’t hard. But the red flags were there: The instructor was not personable and seemed unwilling to answer questions. There was a lot of information, it came at breakneck speed, yet there was very little depth to it. Surely there was more to this, I thought as I went through limits, delta epsilon proofs and the squeeze theorem. Calculus can’t be that easy?
Then we had our first test, and all hell broke loose. Most questions barely resembled what I’d been working so hard on for weeks; others introduced new material, such as applying calculus to trigonometry. This was the homework if it had been on steroids, and it frustrated me. Why had we been wasting our time on simple limits involving rational equations, when the limits we were expected to do required much fancier algebraic footwork? Why not give us something meatier to practice on? True, they were limits I could have correctly computed, but they required deeper thinking and more time, and the sudden leap in expectations on the exam was unnerving, and threw me off my game. In previous courses, doing all the homework, attending all classes, studying and reviewing hard was enough. But apparently not now.
For the first time since I started Mathochism, I failed spectacularly, with a 33 percent. So did most of the rest of the class. No one got As. A select few got Bs. After telling us he didn’t think of us as test scores, the instructor announced that those who got those lower scores really should drop the class, since it would only get worse. This was disheartening, yet I decided to stay. I was no quitter.
But after a few more weeks, and another test (which while less difficult was still at a much higher level than the provided study material) I quit. I might have soldiered through, but the instructor lied about the test score thing. I discovered this when he refused to help me with a concept I was stuck on when I consulted him during office hours. As it turns out, my confusion was over something very simple, but I was 33 percent, and therefore not worth his time. He was not interested in further communication.
My calculus adventure could have ended there, but in spite of the awful experience, I found the subject fascinating and wanted to keep going. So I hit a dud instructor! I would try again!
Try, try again
My second instructor more personable, but she always seemed stressed because we had such a lot of material to get through and very little time to cover it. In previous courses, my teachers had made an effort to answer student questions and solve the occasional homework problem when most of the class was stuck on it, but that never happened now. Occasionally, we were asked to solve a problem during the lecture, but had less than a minute to do it, which was frustrating when the concept was particularly complicated, and I needed time to let it sink in.
Then we had our first test. I failed, though with a more respectable 56 percent. What did me in this time was root functions and absolute values. Again, not impossible, but the problem sets I’d been given focused heavily on quadratics and cubics. As before, I needed practice, but as before, barely had enough to practice on, with only two problems out of more than 60 tackling roots.
This time, though, since the instructor was encouraging and we had rapport, I stuck with it. In the next three months, I did everything I could to meet her expectations, though she was terrible at communicating them. I quickly figured out that she would test us on more obscure concepts, and did my best to practice those, supplementing problems since the book invariably failed to provide them. I bought or borrowed other calculus texts and consulted online and real life tutors. But it was never enough. The pace was simply too punishing, and I never caught up, and my confidence and energy were gone by the end of the semester. Ironically, I understood the material just fine, and she knew it.
“I know you understand calculus,” she told me. “You’re just not good at taking my tests.”
Unfortunately, since those tests were the entire grade, I failed the class. And my confidence was shredded.
Whither confidence in college calculus?
Or was it? What was really shredded was my confidence that I could have a good calculus experience at that college, which is a shame, since I deliberately took all my courses there in hopes that their system wouldn’t fail me.
But it did. The system failed me in that, in spite of doing well in the courses leading up to calculus, they were clearly not preparing me for the rigors of that course. Had I known I needed to seek out supplemental problems to train on, since the book wasn’t enough, I would have done that from the beginning. Had I known that the professors would be so busy getting through material that they barely had time (or in the first case, inclination) to go into any depth, or tell students that depth was even expected – well, I would probably just have stopped at pre-calculus. And then I would have bought myself a bunch of calculus textbooks, hired a tutor and homeschooled.
But my calculus story has a happy ending. After considering an extension course at a nearby university, I turned to massive open online courses, or MOOCs. They didn’t have the one-on-one element, true, but they were free, and Coursera was offering one by UPenn professor Robert Ghrist that was getting raves. Those raves were deserved. Unfortunately, I had to drop the course, because it was more advanced than Calculus 1.
I got my second chance in fall 2013, when Ohio State University offered “Mooculus,” again through Coursera. Taught by professors Bart Snapp and Jim Fowler, this was my dream experience. They were both great, personable lecturers, always available for questions on the online forums (and if not them, multiple TAs were available), and I loved their attitude that calculus was not impossible to learn, even if you stumbled at first.
Although it was two weeks shorter than the community college courses, this MOOC packed in way more material, including log and inverse trig calculus, and handy techniques like L’Hopital’s Rule. And yet, I never felt rushed, probably because I wasn’t spending time commuting to and from school or sitting in class waiting for lectures to begin. I could have this class any time I wanted, and even repeat video lectures over and over again when I missed something.
But best of all, they understood how important it was to have enough problems to practice on! They offered those problems in two formats. The easier ones were interactive, and offered step-by-step solutions if you got stuck. The software acted a lot like a video game; if you showed that you really understood a concept, it leveled you up and asked you tougher questions. If you were having trouble, it gave you as many problems as you needed to get it right before allowing you to the next level.
The hardest questions were in the course’s pdf textbook. Like any textbook, they first explained the concepts you needed, then gave you problems applying those concepts. But unlike in my community college text, most of these problems were at a high level, and once you solved them (no step-by-step solution available here, though you did get an answer, like \(\sin x\) or 2), you really felt a sense of accomplishment and that you had gone deeper.
Once you had tackled the problems in a particular section, it was time to take a quiz. The quizzes were all graded, as were the midterm and final. You could take a quiz any time of day or night you wanted (though by a certain deadline). They were also untimed. What really made the difference was that, if you had done both the interactive and pdf problems, the exams contained problems that reflected your prior work, even if they were more challenging.
A year earlier, I had finished Calculus 1 at a community college depressed and exhausted over having failed in spite of having understood the material. I finished Mooculus online exhilarated and exhausted, with my grade finally resembling the ones I had gotten in previous courses. And I had learned amazing things, like \(e^x\) is its own derivative.
Where Do We Go From Here?
In October, the Mathematical Association of America released its study. Financed in part by the National Science Foundation, it surveyed 213 colleges and universities, 502 instructors and more than 14,000 students. Not only did students report less confidence after Calc 1, they also reported lower levels of enjoyment, worries about readiness for future courses and about their ability to understand future material. Women were way more affected than men.
“What can be done?” the study authors asked, adding that such attitudes did not bode well for getting more people, particularly women, into number-reliant STEM careers.
To which I answer, even as I concede I am not an aspiring scientist or engineer (at least not this year): Take a cue from the guys at OSU.
Yes, it is possible to teach calculus effectively! No, you don’t have to offer untimed quizzes, and I understand why that is not doable. But interactive homework is doable. So are more challenging problem sets, a more vigorous curriculum that includes logs, and most of all, professors who don’t refuse to help students who struggle, whether it is by disdaining them or by not communicating expectations effectively because they are so stressed out by the pace of the course.
It is possible to get through a calculus course and still feel confident in one’s abilities. I have the faith, and lived experience, to say we should try. Don’t we owe our students that?