Students’ median grades on MATH 51: “Linear Algebra, Multivariable Calculus, and Modern Applications” exams rose at least 15 percent between spring and fall 2018 after a new textbook and syllabus were introduced with the aim of making the course slower-paced, more engaging and more applicable.
In spring 2018, the mean on the first midterm was 77 percent; in fall, the mean was 92 percent. On the second midterm, the mean was 68 percent in spring and 87 percent in fall.
MATH 51 is a core requirement for many majors at Stanford. Around 300 students take the course each quarter, the majority of whom are freshmen. Though the course has gradually changed since its founding in 1996, the fall 2018 redesign was one of the most drastic.
Students have claimed that there is a disconnect between the material taught and the material tested, that there is too much material being crammed into too little time and that students must teach themselves the material. The math department responded to such concerns in a June letter to the editor, in which they discussed the anticipated course overall.
During the past four years, professors from the mathematics department have been working on a new textbook to accompany MATH 51. Previously, the course used two different textbooks, but there were issues with adapting the textbooks to the material.
According to professor Brian Conrad, the Director of Undergraduate Studies in Mathematics, MATH 51 required two textbooks in the past because its curriculum, which integrates multivariable calculus and linear algebra, is unique to Stanford.
“There’s pretty much no other pure institution … which teaches a kind of general audience multivariable calculus that incorporates linear algebra right from the start,” he said. “And that’s one of the reasons there’s no off-the-shelf book that we could really use.”
According to mathematics professor Rafe Mazzeo, one of the faculty involved in the creation of the new textbook and syllabus, updating the course material involved communication with people across 15 different departments to determine what was most important for students to learn.
The math department decided not to reveal the textbook’s writers so that the book can be seen as a departmental effort with room for potential future revisions.
One faculty member took a quarter off from teaching MATH 51 to write a skeletal version of the text. Then the department worked together to fill in the chapters, finally finishing the full text and exercises in summer 2018.
“We have no intention of publishing this [textbook],” Mazzeo said. “We want it to be freely accessible to students. And we’re going to keep it easily updatable, because things change.”
One major change in the new textbook is the inclusion of more references to the material’s real-world applications.
“One goal of the wide array of real-world contexts discussed in the new book is to convey to students how n-dimensional considerations with n much bigger than three are extremely relevant to many contemporary applications,” Conrad said. “This is something that was difficult to convey in the earlier incarnation of the course because the texts that were used never addressed real-world applications of higher-dimensional linear algebra.”
Conrad noted that the new textbook includes information about how topics connect to various fields in order to encourage students to understand the importance of what they learn in the course. By writing their own book, the faculty could emphasize how different concepts interact with each other.
“I think the book is easier for students to access, it’s an easier read, it’s easier for them to find information and people were very positively responsive to that,” Mazzeo said.
A student who took MATH 51 in spring 2018 and retook it in autumn said the new textbook was “much more clear” because it was “written much more specifically for the course.”
“It was much easier to follow than the Colley and Levandoski books that they used last time,” said the student, who requested to not be named in this article.
In addition to replacing the course textbooks, the math department changed MATH 51’s pacing. Topics such as Eigenvalues and Eigenvectors, row reduction and determinants were either moved to Math 52 or 53 or removed entirely.
So far, the overall student response to MATH 51’s restructuring has been positive. According to course evaluations, 70 percent of students rated the new textbook as either “extremely useful” or “very useful.”
“I think the course was actually very well organized, and it felt like there was a lot of correspondence between what we learned in lecture, and what we saw in homework and what was in the textbook,” said Ophir Horowitz ’22, who took the course in fall 2018. “In general, the textbook was very coherent. It was easy to follow and I think just the fact that because they wrote it, it went in order.”
A more gradual change in MATH 51 over the past several years was the faculty’s application of active learning techniques. This involved the incorporation of iClickers in lecture and pre-readings that have students read the textbook before lecture and ask questions about potentially confusing topics.
Additionally, MATH 51 discussion sections have shifted from a lecture-based structure to one in which students receive worksheets and go over problems together.
“We tried very hard to find a model that incorporates features of both [traditional lecture and flipped-classroom class structure] that would allow us to proceed and keep students engaged,” Mazzeo said.
More changes to come
The math department looks to continue improving MATH 51 as it receives feedback from students and continues to adjust the new textbook. In addition, Conrad and Mazzeo said they aim to eventually write textbooks for MATH 52 and 53.
Inspired by mathematics education professor Jo Boaler’s course EDUC 105: “How To Learn Mathematics,” ASSU Senator Melissa Loupeda ’21 put forth a proposal to slow down the course by splitting its material between two quarters.
Loupeda said taking a challenging course like MATH 51 can decrease students’ confidence, especially if they come from a high school without a strong math background. She emphasized the importance of self-confidence in effective learning.
“If your first experience with math in particular is really negative, then that carries on,” said Loupeda. “Because that class is the basis for your next stats class, next ECON 50 … And if you feel not confident about the material because it was too fast, you’re going to carry that with you.”
While the project may take time, Loupeda’s overall goal is to improve accessibility and confidence, and her project has already been endorsed by the First Generation and Low-Income Partnership (FLIP) as well as the Stanford Students of Color Coalition (SOCC).
Loupeda’s other ideas include creating an experience similar to the Stanford Summer Engineering Academy (SSEA) but for mathematics, creating a preparatory course for MATH 51 or creating an online course to further supplement students. However, this idea doesn’t just apply to MATH 51; Loupeda is working with the Office of the Vice Provost for Teaching and Learning and the Office of Institutional Research to improve accessibility for STEM courses in general, which may possibly lead to slowed-down versions of other courses in the future.
Contact Jodie Bhattacharya at jodieab ‘at’ stanford.edu and Michelle Leung at mleung2 ‘at’ stanford.edu.