By Lew Ludwig
I am always amazed by the authors of textbooks. They take years of expertise and distill it into something meaningful for the rest of us and our students. I know that for many of them, the process started in exactly the same place I find myself: not having the perfect textbook. So they develop their own notes, refine them, develop them further, refine again, until eventually they have a textbook they share with publishers and the rest of us. I am grateful for all their work.
But just as those authors couldn’t find the perfect textbook for themselves, I’ve yet to find one for my Linear Algebra and Differential Equations course. Over fifteen years of teaching it, I’ve used no fewer than six different texts. Ironically, the first text I tried and abandoned is the one I’m currently using. It turns out I wasn’t ready for it the first time through. I now appreciate its slim, essentials-focused approach, but that slimness means I need to fill in many gaps for myself and my students.
When the Dots Don't Connect
For years, my solution has been “skeleton notes” that accompany the textbook. The idea is sound: students read the text and fill in roughly 60–70% of the notes ahead of time—definitions, theorems, small examples—and then we use class to work through the remaining 30–40% together, focusing on the harder material. In theory, this makes a lot of sense. In practice, it has been a persistent struggle. Even for diligent students, the textbook isn’t always organized in the best pedagogical sequence.
Take the Wronskian. In my current text, it first appears in the linear algebra chapter as a small subsection on linear independence and basis. Since we start the course with linear algebra, my students haven’t thought about differential equations since their previous class, so they don’t see the need or purpose. The Wronskian resurfaces in the next chapter on differential equations, but there the focus shifts to when it works as an “if and only if” test, specifically for constant coefficient linear homogeneous equations. The concept is scattered, the motivation arrives too late, and every year I’d jot corrections in the margins that never got made before the next time I taught the course. You know how that goes.
Building Something Better, Together
This semester, I’m finally doing something about it, with AI as my co-author. Using a dedicated AI Project workspace like the one I described in last month’s column, I’ve shared my prior notes, old tests, and the relevant concepts from the textbook. Together, we’re rebuilding my course notes from the ground up, reorganizing them to read less like a textbook supplement and more like a coherent mini-textbook in their own right.
The Wronskian is a good example of what this looks like. Instead of students encountering it as a disconnected aside in the linear algebra chapter, the revised notes develop it as a natural extension of something they already know from calculus: the quotient rule. Students see that testing whether two functions are linearly independent is really asking whether their ratio is constant. The Wronskian for two functions is precisely the numerator of the quotient rule—something they’ve already worked with in calculus. From there, the notes show how scaling up from two functions to three moves the problem out of calculus and into linear algebra, and the importance of the zero vector, y=0. The concept that used to be fragmented across chapters now has a single, coherent home.
The process matters as much as the product. I bring the pedagogical judgment: what order concepts should appear, what students consistently struggle with, which connections the textbook misses. AI handles the drafting, helping me produce readable prose and embedded examples. Where prose isn't enough, I record short videos—5 to 10 minutes—drawing on 30 years of knowing exactly where students get lost. Judging by the view counts, they’re watching. Either that, or one very dedicated student is watching each video 50 times.
Better notes change what’s possible in class. Now, instead of students sorting through the textbook on their own, I give them the notes for the next class and ask them to read through beforehand. Class time shifts to what matters most: deliberate practice, where students try out new ideas, discuss with a neighbor, and get coaching from me rather than hearing everything for the first time from the front of the room. And I’m doing this with 61 students across three sections.
Want to try it? Here's a prompt that gets the collaboration off the ground.
Project Prompt: Rewriting Course Notes for Student Clarity
You are a pedagogical writing assistant helping me transform my existing course notes into clear, student-friendly materials. My project folder contains my source documents—lecture notes, past assessments, and reference materials. Use these as your primary source.
To guide our revisions, prioritize clarity over completeness, build from concrete examples to general principles, and use accessible language appropriate for undergraduate students. Match my existing formatting conventions. Ask before making structural changes that alter the mathematical flow or emphasis.
Before reviewing my materials, ask me three questions that will help you understand my students, my goals, and what ‘better’ looks like for these notes.
If you’ve got a set of course notes that never quite match your textbook, this is worth trying. You don’t need to rebuild an entire course at once. Pick one topic that’s always felt scattered or poorly sequenced, drop your materials into a Project workspace along with the starter prompt above, and see if “You + AI” can produce something your students would actually want to read.
An Update from the Shire
All of this—the rewritten notes, the AI collaboration, the human-in-the-lead approach—reminds me of a student I wrote about back in May 2024. He was my earliest proof that the cyborg model works.
In that two-part post, I shared the story of my student “John,” who learned to use generative AI as a genuine learning partner on his calculus assignments. John even presented this work with me at MathFest 2024.
Now a senior, John applied to several top master’s programs in AI: Columbia, Boston College, University of Chicago, Cornell, and Carnegie Mellon. To his excitement and mine, he got into every one. This fall, he’ll join the Cornell Business program in New York City.
I’m excited to see this young hobbit leave the Shire and take his skills to the big city. Perhaps I should leave a few blank pages at the end of this column for him to fill in his own chapter when he’s ready. After all, that’s what Frodo did for Sam.
AI Disclosure: This piece was written in partnership with Claude, which helped me organize the structure, edit for clarity, and identify gaps. The ideas and experiences are mine.
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.