Exploring the Beauty of Calculus with Corequisite Models
As a teacher, I repeatedly observed the disconnect between what the algebra students learned in prerequisite courses and how they needed to apply it in calculus. I taught Calculus 1 about a dozen times at Clark College in Washington, using a variety of formats (traditional, flipped, and inquiry-based).
Every semester, I found myself telling students that algebra is the hard part of calculus. The idea that students failed calculus not because they didn’t understand the core concepts, but because they couldn’t jump through the algebra hoops, became an ever-increasing concern for me.
In 2018, I joined the Dana Center and immediately dove into the work on implementing corequisite supports for gateway college math courses (e.g., college algebra/precalculus, introductory statistics, and quantitative reasoning) in Texas and throughout the U.S.
For the past couple of years, I have been researching the emerging trend of corequisite supports for Calculus 1, which is traditionally considered the gateway math course for STEM-intending students. The more I learn about Calculus 1 corequisite models, the more I wish they had been an option at my institution when I was teaching calculus.
Corequisite courses are designed to allow students, who would otherwise enroll in a series of remedial courses, to enroll directly into their gateway math course with corequisite support. This allows students to earn college math credit in their first semester or year.
What’s the Point of Algebra?
As a student, I didn’t really see the point of algebra until I took calculus. This isn’t surprising, as most of the algebra included in the “standard” path—Algebra 1, Algebra 2, Precalculus—are geared to prepare students for calculus.
Looking back, with the perspective of both a student and a teacher, I can compare teaching algebra to teaching students how to do fundamental dance moves without their ever seeing or experiencing the whole dance with music.
When the modern calculus course was developed in the 1950s, students were typically required to simplify a lot of their work by hand. Due to limits in technology at the time and the lack of handheld calculators, it was easier to interpret answers after they were simplified to a specific form.
Colleagues and I often complained that the students clearly understood the calculus but just couldn’t do the algebra. Students didn’t apply algebraic procedures correctly or made basic errors when simplifying their answers, but they accurately performed the calculus processes, such as applying the chain rule to get a correct unsimplified answer.
However, we traditionally graded on both the initial calculus process and on the format of the final answer, so students lost points for errors not specifically related to the calculus procedures. It is no wonder students struggle to identify which strategy to use when algebra isn’t contextualized until a later course, such as Calculus 1.
When students learn the algebraic procedures in isolation, they don’t know how to combine the steps or “dance moves” they have learned until they finally hear the music.
An Emerging Trend: Calculus Corequisites
Having seen an increase in student success in gateway math courses when paired with corequisite support, I would love to see them broadly extended into the traditional, entry-level math class for STEM students—Calculus 1.
If students are able to enter directly into calculus and succeed with corequisite support, they can experience core courses earlier in their educational trajectory and gain confidence in their ability as scientists, engineers, or mathematicians.
Several institutions around the country have decided to implement corequisite support for their introductory calculus courses. Schools are finding that, given the opportunity, many students who have not necessarily learned all of the prerequisite content in a stand-alone format can succeed in calculus.
Through this approach, students can see the relevance and application of the algebra skills immediately, helping them make the explicit connections that lead to long-term resilient learning.
In addition to providing just-in-time instruction, corequisite models can promote student belonging and allow students to develop a peer group within the course, which is known to increase STEM persistence and retention.
For these reasons, Calculus 1 corequisites are a promising tool to remove calculus as a barrier for students who wish to pursue a STEM degree, and thus increase the diversity of the STEM field.
Calculus is a fascinating topic and one that I believe can be understood and appreciated by most people.
However, our habit of teaching the mechanical steps—the foot position, sequence, and routines—takes away from the beauty and power of calculus before it comes together in an elegant dance. Supporting algebra development with just-in-time supports allows students to see the role algebra plays in the bigger picture of calculus.
There is still a lot of work that needs to be done to determine best practices for calculus corequisites, but the successes of early adopters make a compelling case for ongoing investigation and implementation.
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