I remember struggling a lot with Algebra I in eighth grade. I’d been challenged in classes before, but somehow algebra was different; I felt like I just couldn’t really get into the swing of things. I would do really well on one test, then get a terrible grade on the next. I’d make silly arithmetic mistakes or misread part of a question all the time. To this day, my score on the systems of linear equations exam is still the worst grade I’ve ever gotten. I was inconsistent and often only had a surface-level understanding of the material.
Looking back on it now, it’s almost funny how hard I once found Algebra I to be. I’ve since taken countless classes that use basic algebra: physics, statistics, calculus, multivariable calculus, and linear algebra. And I haven’t struggled in any of these nearly as much as I did in my first introductory course to algebra.
You might say that part of why I struggled with algebra was just because I wasn’t yet mature enough to understand the subject. I’m not sure that’s true, though. I took Honors Physics in ninth grade—only one year later—and excelled in this notoriously challenging and math-heavy course. I was also capable of mature critical thinking in my other courses in eighth grade; I just couldn’t quite grasp math. It would also be easy to blame my struggle with the material on the teacher or the curriculum or the pitfalls of Common Core, but ultimately, I don’t think it was any of these factors, either.
I struggled a lot with Algebra I because learning, unlike the equations on the first test I ever bombed, is non-linear.
This phrase—that “learning is non-linear”—actually has a kind of funny backstory to it. Last fall, I took an Introduction to Probability course called Stat 110 that is known for being really challenging. A couple days before the final exam, another student in the course posted a meme in “Harvard Memes for Elitist 1% Tweens,” a group on Facebook where students post funny content about Harvard College. The meme was a parody of a quote from The Fault in Our Stars by John Green. It had a picture of the two main characters from the movie. Over the photo, instead of the quote “I fell in love the way you fall asleep: slowly, then all at once,” the student had written, “I learned Stat 110 the way you fall asleep: slowly, then all at once.” It was a reference to how many students felt like they were cramming to study a lot of rigorous material in a short period of time right before the final exam. The best part wasn’t the meme, though. It was that the professor actually commented on the post himself! He wrote, “Expectation is linear; learning is non-linear. Good luck everyone!” The first half of his message is a reference to a concept in statistics that the expected value of a probabilistic distribution (i.e., the average value) has a set of mathematical properties that make it linear. The second half is much more profound.
What Professor Blitzstein meant by the term non-linear here is that learning doesn’t always follow the timeline that you and your course syllabus want it to. Classes move through material at a pace necessary to cover a certain amount of critical information, but they certainly don’t move at a pace that is natural or even feasible for everyone to build strong mastery of the material. And when learning something as new and fundamental as algebra or introductory probability, there’s going to be a lot of variation in your performance as you get used to thinking in a different way. There were some concepts in Algebra I that I just needed to hear and write out hundreds or thousands of times rather than the 20 or 30 times that I could manage throughout my first year of learning the subject. As I took more classes that used algebra, and I began to see how the math worked in new contexts like physics or geometry, I got more comfortable with the material. My strength in algebra today is the product of years of building on concepts here and there, having random moments of breakthrough, sometimes months after being tested on a topic. It wasn’t a pretty, step-by-step process, and I wasn’t done learning when I was done with the class.
A visual representation of how learning is non-linear.
I’ve been struggling a bit with Organic Chemistry this semester and it’s reminded me a lot of my experience with algebra seven years ago. In Orgo, I’m capable of understanding the material and when I see how to perform a certain mechanism, it makes sense to me. But I’m still not at the place where I can comfortably intuit on my own how and why certain reactions work the way they do. As a result, I haven’t been doing as well on the exams as I’ve wanted to.
It sucks taking an exam and feeling like you still haven’t mastered the material on it. It sucks even more finishing an entire course and still feeling shaky on the content. But sometimes, no matter how hard you work or how much you try, that’s just how learning is. I didn’t do very well on my Orgo exam yesterday and I’m not sure I’ll really be able to do that much better on the final, either. I don’t mean this in a defeatist way; I still plan to study really hard and review a different unit each day up until the final exam. But I think Orgo is just one of those things that I personally need to see a lot more of—and in a lot more contexts and examples—before I can truly be comfortable with it. The college semester timeline doesn’t always allow for that. Even if I don’t grasp all of organic chemistry by the end of this semester, I’m confident that I will understand it over time. Maybe that time will be by the end of second semester Orgo, or maybe it will be when I study for the MCAT. Whenever it is, it will come naturally as long as I continue to put effort toward understanding the material.
If you’re like me and currently struggling with a course, take solace in the fact that the end of the semester is not the end of your academic growth. A bad grade at the end of the term is not a reflection on your ability to understand course material, but rather a reflection on where you’re at in a continuous process of learning and improving. As long as you’re trying your best—taking advantage of support from the teaching staff, reviewing lecture material, and doing as much practice and review as you can reasonably fit without sacrificing sleep and mental health—learning will happen in its own seemingly random, non-linear way.
Sending encouragement to everyone bogged down by final exams and assignments this week. We got this!!
On Ariel's Mind 