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Math 30: Calculus I

Material Preview, Active Recall, Self-Testing

Reflection on Instructional Change using Evidence-Based Lecture Alternatives:

This semester I am teaching Math 30: Calculus I, a prerequisite for many mathematics, engineering, computer science, and physics courses at Sacramento State. I chose to implement frequent "mini quizzes" in the form of daily entry tickets throughout the semester. My goal was to offer a low-stakes opportunity for self-assessment as a built-in part of our class time. Research suggests that frequent testing can significantly improve material retention, so I aimed to provide a formalized version of this learning strategy of this for my students.

To implement this change, I began several class sessions this semester with an entry ticket or warmup that took five to ten minutes total. This was done in three phases:

Phase One (Weeks 6-7): Four online multiple-choice entry tickets were administered through Gradescope. I alternated between awarding credit based on correctness and based on participation.
Phase Two (Weeks 8-12): I gave daily warm-up problems that did not require submission. Students worked individually or in small groups for the first five to ten minutes of class, after which we reviewed the solutions together.
Phase Three (Week 13): I administered daily free-response paper entry tickets, totaling four for the week. Each ticket included a question related to the previous class session's material. Students recorded an initial answer in the first five minutes of class, and submitted their final answer at the end of class. Credit towards the "Class Activities" grade (worth 10% of the overall course grade) was rewarded for participation, regardless of answer correctness.

Students responded positively to these changes overall, with a clear preference for the Phase Two warm-ups. During Phase Two, I administered an anonymous survey, to which 29 out of 30 students responded. Students were asked to rate, on a scale from 0 (not at all) to 5 (extremely), how helpful they found (i) the Phase One entry tickets and (ii) the Phase Two in-class group work. On average, students preferred the group work (mean rating: 3.86) over the entry tickets (mean rating: 3.36). A chart summarizing the survey responses is included below. Additionally, several students commented that they appreciated the opportunity to check their understanding independently before seeing the solution.

During Phase Three, I conducted another short survey, to which 19 out of 30 students responded. Students were asked to rate (on the same 0–5 scale) the extent to which the Phase Three entry tickets (i) helped their learning (mean: 3.05), (ii) reinforced their understanding (mean: 3.11), and (iii) supported their recall outside of class (mean: 2.79). A chart summarizing these responses is included below. I also asked a separate question about how many days they arrived to class on time during that week. In addition, students had the opportunity to submit open-ended comments. The feedback was mixed: several students praised the entry tickets, with one noting that they "help[ed them] remember the past lessons and regain a new understanding of the problems," while others indicated that they preferred the non-submission warm-ups from Phase Two. Only one student explicitly stated that they did not enjoy the Phase Three entry tickets.

See caption above image for details.

I believe this project contributed significantly to my students' overall success. The primary evidence for this is the students' own feedback: many found the activities helpful, especially the Phase Two warm-ups. Survey results from Phase Three also revealed that students who attended all four days of class that week rated the entry tickets significantly higher—by an average of 0.5 to 0.66 points across the categories—than the overall average. This suggests a correlation between attendance and the perceived benefit of the entry tickets. Although the surveys were anonymous and I cannot directly track academic outcomes, multiple students commented that the activities improved their learning and understanding, which makes me hopeful that their performance benefited as well. Additionally, I observed an increase in attendance during Phases One and Three when participation was tied to course credit. However, given the students' clear preference for non-submission warm-ups where they could keep their work in their notes, I plan to refine the structure in the future by finding a way to offer participation credit while still allowing students to retain their work for study purposes.

**Evidence of Instructional Change**
Phase 1: Gradescope Entry Ticket	Phase 2: Warm-Ups
An online entry ticket from Phase One. Q1 0.5 Points True or False: The derivative of sin(x) is -cos(x). (You may select "I don't know" for half-credit.) ○ True ○ False ○ I don't know Q2 0.5 Points Let $f (x) = e^{x} + 2 x$ . The slope of the tangent line of $f (x)$ at $x = 0$ is... (You may select "I don't know" for half-credit.) ○ 2 ● 3 ○ 0 ○ I don't know	A warm-up problem from Phase Two. Warm-Up Consider the function $y = g (x) = 4 \ln (x) - \cos (π x) - 3 x + 4 \sqrt{x}$ (a) Compute a formula for $y^{'} = g^{'} (x)$ . (b) Determine the slope of the tangent line to $y = g (x)$ at the value $x = 1$ . (c) Find an equation for the tangent line to $y = g (x)$ at $(1, g (1))$ . Write your result in point-slope form. (d) Using technology, sketch an accurate, labeled graph of $y = g (x)$ along with its tangent line at the point $(1, g (1))$ . (e) Using a calculator, compute $g (1.1)$ . Round to the nearest thousandth. (f) Let $y = a x + b$ be the tangent line equation from part (c). By hand, compute $t (1.1)$ . (g) By hand, compute $∣ g (1.1) - t (1.1) ∣$ and round to the nearest thousandth. (h) We say that $t (x)$ is a local linear approximation of $g (x)$ around $x = 1$ . Discuss with your group what this means.

**Changes in Course :**
Pre FLC Example:	Post FLC Example:
Fall 2024: I began class by working through this problem. Practice Find the area of the largest rectangle that can be inscribed in a right triangle with legs of length 4 cm and 6 cm, if two sides of the rectangle lie along the legs.	Spring 2025: I began class with this entry ticket. Math 30 (Loucks-Tavitas) Entry/Exit Ticket April 23, 2025 Name (print) ___________________________ Student ID # _______________ You are attempting to find the rectangle with the largest area that can be inscribed in a right triangle with base 6 cm and height 4 cm. Draw and label a picture of the situation, and identify the quantity to be optimized. Initial attempt (work out on your own in the first 5 minutes of class): Final answer (submit at end of class):

**Faculty Bio**
Faculty Image	About me:
	Dr. Jessie Loucks-Tavitas is an assistant professor of mathematics at Sacramento State University. She earned her bachelor's degree at Sac State in 2018 before completing her Ph.D. in June 2024 at the University of Washington in Seattle. Her research focuses on applications of algebraic geometry to computer vision. Outside of teaching and research, she enjoys running, fiber arts, and spending time with her husband, Ryan, and their two cats, Tim and Otto.

STEM4Equity: US Dept. of Education STEMHSI P031C210012
*Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Sacramento State.