Guided Notes: Unlocking Pedagogical Possibilities in Introductory College Mathematics Courses

Course(s): Precalculus (MTH 167, MTH 168), Calculus (MTH 181, MTH 182)

Department: Mathematics and Statistics

Institution: Cleveland State University

Instructor Name: Sara Froehlich, PhD

Number & Level of Students Enrolled: 35- 40 students per class section, primarily first and second year STEM majors

Digital Tools/Technologies Used: LaTeX, Digital Tablet, PDF annotation app such as GoodNotes

Author Bio: Dr. Sara Froehlich is an Assistant Professor of Practice in the Department of Mathematics and Statistics at Cleveland State University. She coordinates the precalculus to calculus sequence, which comprises approximately sixteen course sections each semester. She is also the Director of Operation STEM (OpSTEM), a student support and academic success program, and oversees the STEM Peer Teacher (SPT) supplemental instruction program at CSU. Having taught mathematics in various contexts for the past fourteen years, Dr. Froehlich is always seeking out innovative ways to improve her teaching and her students’ learning experiences.

Abstract: In this essay, I will reflect on my transition to using guided notes while teaching introductory college mathematics classes and demonstrate how this decision has sparked further growth in my personal pedagogical practices. Essential features of guided notes will be described, and I will provide examples of the guided notes I use in my classes. A discussion of recent research demonstrating the positive impacts of guided notes on student learning, and links to additional resources are also included.


First utilized as a means of improving the structure of my remote lectures in Spring 2020, I made the decision to continue using guided notes after returning to in-person instruction in Fall 2021. I have noticed several positive by-products of this choice in the two years since I have been using guided notes coupled with in-person instruction. These include an increased connection with my students, more effective pacing of my lectures, and the ability to incorporate a wider variety of teaching methods in my classes. Using guided notes has helped me to slowly move away from a strict “I lecture, you listen” mode of teaching to a more varied and interactive approach.

Creating a Framework for Growth

I had my first experience in an instructional role as a teaching assistant for introductory mathematics classes when I was still an undergraduate myself. There was no pedagogical training in place for teaching assistants, so I mimicked what I had seen my own professors do: “chalk and talk”. My teaching style remained largely unchanged for the next decade as I moved through various teaching assistant, and then part-time adjunct positions while a graduate student. When I accepted my first full-time faculty position, I began to reflect seriously about my role as a teacher and how I utilize time in the classroom. I knew that “active learning” was a critical component in creating a worthwhile learning experience for students. But in practice I was teaching tightly coordinated introductory level classes which had long lists of learning objectives and a common final exam, written by a committee of faculty, by which my students would ultimately be evaluated at the end of the semester. I struggled to envision what active learning could look like in this setting. So, I continued with “chalk and talk”, believing that my lectures formed a contract with my students: if the information was on the board, then I had taught it, and it was fair game to appear on any assessment.

In the spring semester of 2020, everyone’s learning and teaching experiences changed dramatically. During the initial switch to remote instruction, I wrote on my tablet as if it were a digital whiteboard while students followed along at home. At the end of each lecture, I would upload a PDF of the notes I had written. I found the pages of notes produced in this way to be uninspiring – they were not nearly as organized and insightful as they had been in my mind. I was left wondering if my whiteboard lectures in the classroom were also lacking in their effect.

Over the summer of 2020 I had time to properly prepare for what everyone correctly assumed would be another semester of online learning in the fall. I decided to use this time to develop a set of what I called skeleton notes: a document prepared in advance of each day’s lecture which included typed formulas, theorems, and problem statements, along with space in between these for myself and students to write comments, complete proofs, and solve problems together. My goal was to lay a framework for each day’s lecture, as well as to save time by having some parts of the lecture typed out ahead of time. I have since learned that such documents are alternatively referred to as gap notes or guided notes, and there is a fair bit of research available to support their use. Instructors can design their guided notes in a variety of ways. See for example a 2016 paper by Reynolds and Tackie which includes examples of skeleton notes taken from two undergraduate engineering courses. Below are sample pages of notes that I have used when teaching Calculus II. The handwritten parts were completed during the lecture.

I provide a blank version of the notes to students at the beginning of each week. In a typical three credit hour course, I will cover about 12 pages of such notes per week. At the end of the week, I post the completed note packet so that students can correct any mistakes or missing information in their own version of the notes. I have not found that posting my notes negatively affects attendance, perhaps because students must wait till the end of the week to receive them. When designing my guided notes, I include complete definitions and statements of theorems in boxes. I do not leave any ‘gaps’ in these boxes, because I want students to have this information stated precisely and accurately in their notes for future reference. I leave gaps as large as half a page – or sometimes an entire page for an elaborate problem – to solve illustrative examples. There are also gaps left for explanatory comments. In an introductory calculus class, there is limited time available to give formal proofs of theorems, so I will often provide a detailed outline of essential proofs with certain steps left as short exercises to be completed together as a class.

Several benefits of my summertime effort to develop complete sets of guided notes for Calculus I and Calculus II were immediately apparent in the fall of 2020. Perhaps most impactful: my lectures no longer felt rushed. I had time to cover the necessary material comfortably, without scrambling to transform a completely blank page into a detailed exposition of essential content within a one-hour class period. A second effect was that typing my notes out ahead of time forced me to think very deliberately about the flow of my lecture. Producing a typed document gave the notes the weight they deserved as the students’ permanent record of testable material. Although I had always considered myself to be an effective and organized lecturer, I believe that using guided notes has helped me improve in this regard.

The Argument for Guided Notes

Much of the research on the effectiveness of guided notes, particularly their use in undergraduate mathematics classes, is relatively recent. A small qualitative study by Iannone and Miller showed that the use of guided notes effected students’ note-taking behavior, with some students moving beyond simply creating an exact copy of what the instructor had written on the board. These students also included verbal comments made by the instructor, and sometimes even noted connections between different parts of the lecture. Achieving this type of real-time metacognition is difficult when students are tasked with copying down every word written on the board.

A study by Krapf and Pfefferkorn explored how a group of students in a mathematics education course perceived the guided notes used in their class. Among the positive effects most often noted by students were increased motivation to attend lecture and pay attention (when compared to being provided a fully completed set of notes), enhanced structure and organization of notes (when compared to not being provided any pre-made notes at all), as well as comments appreciating that writing less allows one to think through concepts during lecture, and noting the difficulty of thinking and writing at the same time. Some of the negative perceptions students expressed, such as being forced to attend class and take notes since full notes were not being provided, are positive effects from an instructor’s point of view. Other negative comments were related to the design of the guided notes used in the study – for example that there was still too much writing expected of the student.

The Ohio State University provides the following website with advice for instructors wishing to incorporate guided notes in their lectures, as well references to other relevant studies: Guided Notes: Improving the Effectiveness of Your Lectures.

Positive Spillover Effects

I was hired as a faculty member in the Department of Mathematics and Statistics at Cleveland State University in the summer of 2021, so at the same time I returned to in-person instruction, I was also starting a new job. I had to decide, then, whether to continue using guided notes to support my lectures in-person. It felt risky to overhaul my entire modus operandi in the classroom at a time of major professional and personal transition, but I had become used to the comfortable pace that the guided notes allowed me to maintain. I decided to revamp the notes I had created the previous summer and have used guided notes in every math class I have taught since 2020.

The positive results I have seen using guided notes since returning to in-person instruction have gone beyond what I anticipated. When I lectured by writing on a blank whiteboard or chalkboard, my end of semester course evaluations would often include comments that I faced the board too often, or that I blocked the board while I was writing. Now, by writing on my tablet with its screen displayed on a projector screen behind me, I spend the entire class period facing my students. I can easily scan their faces for comprehension or confusion at any point. This has increased my ability to connect with and understand my students and their learning needs.

In a recent mid-semester feedback form given in my Calculus II class, I asked the students whether the pace of the class was too fast, about right, or too slow. I was happy to see that 75% of the students said the pace was about right. For a class where we often cover an entire section of the textbook in each 65-minute-long class meeting, this was very encouraging feedback to receive. A too-fast lecture pace is a common concern in introductory mathematics classes which can be mitigated by using guided notes. My end of semester course evaluation comments also frequently refer to how organized and clear the class notes are, leading me to believe that the provision of guided notes is appreciated by my students.

Finally, by including some of the lecture content in pre-typed note packets, I have more freedom to experiment with how I use class time. That elusive gold-standard of effective pedagogy, active learning, now feels more available to me. There is time to pose questions mid-lecture with sufficient wait-time for students to provide meaningful responses. I often employ Socratic questioning to move my lecture forward when introducing a new concept and the resulting back and forth dialogue with the class has produced many satisfying moments of revelation for students. Pausing lecture for a few minutes to give the students a chance to work out the next example in the notes on their own before reconvening the class allows me time to walk around the room and correct misconceptions immediately in the moment. If students are struggling with a particular topic, I have also incorporated group or two-stage quizzes 5 to clarify and reinforce those topics. These are some of the ways in which guided notes have allowed me to move beyond “chalk and talk” and truly grow as a teacher.


Larwin, Karen H. and David A. Larwin. “The Impact of Guided Notes on Post-Secondary Student Achievement: A Meta-Analysis.” The International Journal of Teaching and Learning in Higher Education 25 (2013): 47-58.

Reynolds, S. M., & Tackie, R. N. (2016, June), A Novel Approach to Skeleton-Note Instruction in Large Engineering Courses: Unified and Concise Handouts that are Fun and Colorful. Paper presented at 2016 ASEE Annual Conference & Exposition, New Orleans, Louisiana. 10.18260/p.26383

Iannone, Paola, and Dominic Miller. “Guided Notes for University Mathematics and Their Impact on Students’ Note-Taking Behaviour.” Educational Studies in Mathematics, vol. 101, no. 3, 2019, pp. 387-404. JSTOR,

Krapf, R., Pfefferkorn, L. How Does the Provision of Guided Notes Affect Student Learning in Undergraduate Mathematics? Int. J. Res. Undergrad. Math. Ed. 8, 642–670 (2022).

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