Confession: I barely passed Multivariable Calculus in college. This came as an unpleasant surprise to me, as I had done well in high school math classes. How could this have happened? I was a good student: I spent hours reading and re-reading my lecture notes, practicing problems from the book, and always paid attention in class. Unfortunately, I was doomed from the start. Memorizing a bunch of procedures worked ok for me in high school, when the procedures weren’t particularly complex or long. This same strategy failed abysmally when I got to college and there were just too many things to memorize. Worse, I couldn’t improvise. If a problem on the test differed from something I’d practiced (even superficially!) I couldn’t solve it. Research shows that I was not alone. Many college students see mathematics as a series of disconnected steps you need to memorize (Stigler, Givvin, & Thompson, 2010). There is no connection among the steps, no connection among problems, no connection among the big ideas that organize the subject. To a distressing number of students, math has no meaning. This issue is hardly unique to mathematics; research shows that novices in many fields don’t see the connections among ideas in the subject and don’t understand how ideas are organized. In other words, they are missing the connections between ideas that make a domain “make sense” and give it meaning. Finding meaning in material is essential for long-term learning (Gray, Arnott-Hill, & Benson, 2021).
Fortunately for you, research in cognitive psychology has discovered several methods to help students find meaning, regardless of the subject matter. To get information from the page of a textbook into your long-term memory, you generally need to rehearse it in some way. One easy way to rehearse material is just straight repetition. For example, you can mindlessly repeat the definition of a derivative to yourself and practice a bunch of problems until you can reproduce the steps from memory. This works pretty well if you’re only going to be tested on the same exact problems you practiced, and if the test is in about 15 minutes. If you want your learning to be adaptable to different situations and longer lasting, you need to do something else. Elaborative rehearsal happens when you transform or expand the material you’re trying to commit to memory in some way. Instead of repeating the definition from your professor verbatim, you restate the definition of a derivative in your own words. When doing practice problems, you focus especially hard on word problems, which show the concept at work in a real-life situation. This gives the concept of a derivative meaning: you have transformed the idea into your own terms, and you can see it applied in a real-world situation. Elaborative rehearsal works. One study found that use of elaborative rehearsal was the biggest factor that differentiated high- and low-performing General Psychology students (Ratliff-Crain & Klopfleisch, 2005).
Another way to elaborate on course material is by connecting it to your prior knowledge. If a derivative is essentially a rate of change at a particular time, you might try to think about situations in which you you’ve worked with rates of change before (maybe when calculating the speed of a car traveling over some distance). To super-charge the beneficial effects of making connections to prior knowledge, utilize the self-reference effect and try to apply the material to yourself (e.g., Forsyth & Wibberly, 1993). Can you think of an experience in your life where you can apply the idea of a derivative? Maybe you run the 100m dash, and you want to know how fast you are going when you reach your top speed. A derivative would tell you your speed at the instant that you reach your top speed.
In addition to connecting material to yourself and prior knowledge, you should look for connections inherent in the material itself to organize it and help make meaning. Generally, domains like mathematics, physics, and American History are not a random disconnected set of isolated concepts. There are relationships among the concepts you study and there is some organization to the big ideas. Fortunately for students, this structure is something that many instructors try to help you see: take a look at the hierarchical organization of chapters, headings, and subheadings in a textbook or the “preview” slide in a lecture to get an idea of which topics are most important or which topics are similar and grouped together under the same subheading.
There are many different kinds of relationships you can look for to help you organize material into a meaningful, coherent whole. You can compare and contrast concepts, identifying their similarities and differences. How is a derivative, speed at a single instant, similar to average speed over an interval? How is it different? Which similarities are important to the definition of the concept? Which differences are irrelevant to how the concept operates? Answering these kinds of questions gives you an understanding of the organization of concepts and how they relate to one another, which gives them more meaning. In addition to comparing and contrasting, you can look for part-whole relationships, cause-effect relationships, and relationships that show a specific example of a general principle (see Gray et al., 2021 for further discussion of these relationships).
Now, there’s just a tiny bit of bad news: elaborative rehearsal is more difficult than repetition. But decades of research in cognitive psychology shows the payoff is worthwhile (Dunlosky et al., 2013). If you have to make your own meaning (i.e., do elaborative rehearsal), your memory for the concept is going to be better, and your understanding of the material will be more useful — you will be able to adapt it and apply it to different situations. I discussed making meaning with a concept from calculus, but these strategies are flexible and can be applied to any subject to improve your learning and understanding.
To close, I’ve been following my own advice: I’m currently in the process of re-learning calculus in a meaningful way. I’m connecting concepts together and learning how the concepts apply in a meaningful and fascinating domain (Mathematics for Life Sciences is an excellent course offered at UCLA, just fyi). And you know what? Yeah, it’s hard. Yeah, it takes a lot of time. But it is so fun. And that’s all because I’ve made the subject matter meaningful.
References:
Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4-58.
Forsyth, D. R., & Wibberly, K. H. (1993). The self-reference effect: Demonstrating schematic processing in the classroom. Teaching of Psychology, 20(4), 237-238.
Gray, K., Arnott-Hill, E., & Benson, O. (2021). Introduction to Psychology. COD Press.
Ratliff-Crain, J., & Klopfleisch, K. (2005). Studying for Introductory Psychology Exams: Lessons Learned from Successful and Unsuccessful Students. Paper presented at the Annual Conference of the Midwestern Psychological Association.
Stigler, J. W., Givvin, K. B., & Thompson, B. J. (2010). What community college developmental mathematics students understand about mathematics. MathAMATYC Educator, 1(3), 4-16.