By Lew Ludwig

Last century, when I first taught calculus, there was a heavy emphasis on memorizing and then applying rules: the power rule, the product rule, the quotient rule, and the chain rule, to name a few. For students to perform well, they needed to memorize these rules and quickly apply them in a high-stakes timed test with a heavy dose of algebra. This played well for certain types of students (“the sprinters”) and not so well for others (“the long-distance runners”).

Also, during this time, graphing calculators were introduced. There was a considerable discussion of whether this was a useful exploratory tool for learning mathematics or a crutch that provided misunderstood shortcuts for students. Shouldn’t students commit to memory common graphs and their properties, just like they commit their multiplication facts to memory?

But where was the higher-level thinking here? Was memorizing and applying rules the type of “learning” I wanted for my students? Moreover, as I began to teach higher-level courses, the focus on rules and testing one’s memory shifted. I could no longer expect students to answer complex questions in an hour. Indeed, in some courses, one question could take that amount of time. So, in higher-level classes, I shifted to take-home tests. Tests that students could work on over 48 hours, allowing them time to think and ponder the more complex material – to engage in higher-level thinking.

In recent years, given the content challenge of higher-level courses coupled with the advent of search engines like Google, it has seemed as if the days of memorization have passed. Why memorize something when you can pull out your smartphone and have the rule in seconds? I thought students in my upper-level courses did not need to memorize as they could look things up over their 48-hour period.

During remote teaching, I finally caved and did away with timed tests. I developed a take-home assessment in which students create their own tests that I take. While this approach has numerous benefits, I found the students’ ability to respond to basic questions diminished. Their computations and applications slowed since they had not committed things to memory. Questions that the average student used to answer in seconds were now met with blank stares.

Looking at my higher-level courses, I've faced a similar issue. Students did not know the basic concepts or definitions because they did not commit them to memory. It was challenging to discuss the usefulness or applications of a vector space if students did not know what a vector space was without flipping through their notes. This reminds me of my college experience and my first open note test, which I did not prepare for. It went horribly. I spent most of the class flipping from page to page, hoping to find the answer or some understanding.

When it comes to higher-level thinking, memory still matters. Memory and thinking are more intertwined than we thought. As Dr. Michelle D. Miller, professor of psychological sciences at NAU notes, “when students have formed a more solid base of knowledge—such as through retrieval practice—they are more able to engage in processes like inference and extension, not less.”

If you currently use a take-home format, consider having an in-class version as well. While lower stakes, the in-class version can ensure students commit important things like definitions or examples to memory, ultimately improving their higher-level thinking.

I now include an in-class and a take-home test in my calc class. Like nearly anything, a balance is required. I cannot rely too heavily on one approach to meet my students’ diverse strengths and learning goals.

Upside

• Students are more likely to meet my learning goals with a more balanced approach.

• There is less emphasis on students performing quickly under pressure.

Downside

• With more types of assessments, I have more grading to do. However, I can grade more quickly as student understanding has increased.

Still learning from my misstakes mistakes.

Lew Ludwig is a professor of mathematics and the Director of the Center for Learning and Teaching at Denison University. An active member of the MAA, he recently served on the project team for the MAA Instructional Practices Guide and was the creator and senior editor of the MAA’s former Teaching Tidbits blog.