"I'VE ALREADY DONE THIS":

This project will be presented as a poster at the 27th Annual Conference on Research in Undergraduate Mathematics Education (February 2025) in Alexandria, Virginia.

Click here for the conference proceedings article. (Link to be included once available)

For students pursuing STEM degrees in the U.S., Calculus I is a notorious obstacle that causes many students to leave STEM for other majors (Weston et. al, 2019). This is no different at our institution – historically, the DFWI rate (the percentage of students earning D, F, W, or I grades) in Calculus I ranges from 30-40%. In Fall 2023, one of us led 2 of 8 recitations for a large-lecture style (~250 students) Calculus I course whose goal was to decrease the DFWI rate significantly. However, interventions were limited due to the course’s coordinated setting; homework assignments were fixed across sections, and all students enrolled in Calculus I took the same final exam. One small intervention we could include was a series of short “How’d it go?” prompts (see Figure 1) after students completed problem sets, which we will discuss in more detail later. These were designed to develop students’ metacognitive skills over the semester. Students answered the prompts voluntarily and earned a small number of points for turning in a scan of their response, whether it was blank or not.   

At the end of the semester, the section with this intervention had a DFWI rate of ~12%, significantly lower than both the historical rates and the other sections of Calculus I in Fall 2023. While we were unable to conclude that the “How’d it go?” prompts were the sole reason for this success, many of the answers provided a glimpse into the students’ thoughts, beliefs, and feelings during the course. In this presentation, we will investigate how Calculus I students’ attitudes evolved throughout the course, as manifested through regularly assigned reflective prompts.  

This study adheres to the psychological theory of Meaning Systems, or beliefs people develop that organize their world and give their experiences meaning (Dweck, 2000). This theory holds that people’s self-theories, or beliefs about themselves, can lead them to think, feel, and act differently in identical situations (Dweck, 2000). One of the main ideas in Meaning System Theory is mindsets, or established attitudes held by someone. 

Dweck started looking at the idea of mindsets in the 1970s (Dweck & Yeager, 2019) and a lot of the work since then has been focused on students (Blackwell et al., 2007; Hong et al., 1999; Mueller & Dweck, 1998; Robins & Pals, 2002). Some work has been done with mindsets of undergraduate STEM students (Bosman & Fernhaber, 2018; Limeri et al, 2020; Santos et al., 2021), but undergraduate mathematics courses have been understudied. In undergraduate mathematics courses, there are only a few examinations of mindsets, including those from Campbell (2023) and Shively & Ryan (2013). However, these two studies have contrasting results. 

In this study, we focus on four established mindset pairings: growth/fixed, mastery-oriented/helpless, strategic/non-strategic, and entrepreneurial/non-entrepreneurial. 

• A student with a growth mindset believes that intelligence is malleable and knowledge is infinite, whereas a student with a fixed mindset believes that intelligence is unchanging and knowledge is finite (Dweck, 2016).  

• A student with a mastery-oriented mindset responds to failure hardily, remaining focused on achieving understanding in spite of failure, whereas a student with a helpless mindset believes that once failure occurs, nothing can be done to achieve success (Dweck, 2000). Often, helpless students will blame an innate lack of intelligence for their failure and demonstrate lower persistence (Dweck, 2000).  

• A student with a strategic mindset engages in metacognitive strategies while pursuing a goal. This involves identifying actions he or she can do to overcome an obstacle (Chen et. al, 2020). Conversely, students with a non-strategic mindset are unable to identify actionable ways to overcome struggle or failure. The inspiration for Chen et al. (2020) to identify the “strategic mindset” came from work looking at strategic resource use by undergraduate statistics students (Chen et al., 2017). So, it makes sense to consider strategic mindset in calculus. 

• A student with an entrepreneurial mindset possesses “the inclination to discover, evaluate, and exploit opportunities” (Bosman & Fernhaber, 2018). A student with a non-entrepreneurial mindset does not possess these traits. The majority of our students are engineering majors. So, we wanted to study this well-documented mindset that has appeared in previous studies of undergraduate engineering students (Bosman & Fernhaber, 2018).  

This study contributes to the existing literature by broadening the scope of mindset theory applications to tertiary mathematics education; we investigate the evolution of mindsets in university Calculus I students through responses to voluntary metacognitive assignments over a semester. To our knowledge, this study is unique because it was conducted in a calculus context, in an actual class setting, and relied on more open-ended responses than past studies.  

How do university calculus students’ mindsets differ depending on the setting of their first exposure to a calculus class (high school or college)?  

Data Collection 

Permission to conduct this study was obtained from our Institutional Review Board. The participants in this study were 99 students enrolled in the same section of Calculus I at a large R1 public university in the Southeastern U.S. during the Fall 2023 semester. The course was conducted in a 2-2 lecture-recitation format. Throughout the semester, students were provided printed packets that contained all materials necessary for each two-day lesson: lecture notes outlines, pre-recitation prep problems, recitation problems, and (in some cases) extra practice problems. The whole semester included 21 packets (called lessons) divided into four units based on Stewart’s Calculus: Early Transcendentals (2016). 

• Unit 1 included limits and conceptual derivative topics. 

• Unit 2 included derivative computations and applications such as L’Hospital’s Rule and Mean Value Theorem. 

• Unit 3 included applications of derivatives such as extrema/optimization, concavity, curve sketching, and implicit differentiation/related rates. 

• Unit 4 included integration topics such as Riemann Sums, limit definition of the integral, Fundamental Theorem of Calculus, evaluating integrals, and substitution. 

Data for this study was collected from the reflective “How’d it go?” prompts included in pre-recitation prep assignments from 12 randomly chosen lessons (3 from each unit).  

Data Analysis 

To analyze the data, we used a priori coding (Saldaña, 2013) with codes defined by the described mindsets: 

• (a) growth, (b) fixed, 

• (c) mastery-oriented, (d) helpless, 

• (e) strategic, (f) nonstrategic, 

• (g) curious, and (h) not curious. 

Note that instead of entrepreneurial and non-entrepreneurial, we only coded for curiosity or explicit non-curiosity because the evaluative and exploitative components of an entrepreneurial mindset are also qualities of a strategic mindset. 

In one session, we coded one of the twelve lessons together. The codebook had already been established, and the definitions of each code were finalized during this session. After this, we each coded the rest of the data individually and met twice to resolve any differences. The source of our coding differences arose from ambiguity in the codebook, which we resolved by refining the codebook. Potential power dynamics were mitigated by both coders being senior Ph.D. students at the same institution. 

We disaggregated the data according to the students’ prior exposure to high school calculus (H-CALC), n = 65, or not (C-CALC), n = 34. Our final results visualizations included relative code frequencies due to having different sized groups. We focused on high school calculus experience because, during initial coding, we noticed a large number of responses that referenced assignments being “easy” because of having “done this already [in high school]”. The College Board’s AP® Calculus AB was the most common high school calculus class taken by our students. 

Mastery-Oriented & Helpless Mindsets 

H-CALC students saw an overall increase in mastery-oriented mindset and helpless mindset from the beginning to the end of the semester, whereas C-CALC students saw an overall decrease in both mastery-oriented and helpless mindsets (see Figures 4 & 5). Despite an overall decrease, C-CALC students achieved maximal mastery-oriented mindset during Unit 2. However, for H-CALC students, maximal mastery-oriented mindset occurred in Unit 3. Helpless mindset peaked for C-CALC students during Unit 3, whereas H-CALC students had relatively constant helpless mindset the entire semester (see Figures 4 & 5). Overall, C-CALC students had equal or greater levels of mastery-oriented mindset than H-CALC students (see Figures 2 & 3). 

Growth & Fixed Mindsets 

H-CALC students were more than twice as likely to have a fixed mindset than C-CALC students, whereas C-CALC students were as much as seven times more likely to have a growth mindset than H-CALC students (0.281 vs. 0.039, resp., in Unit 2). Specifically, H-CALC students saw a large increase in fixed mindset from Unit 1 to Unit 2 (0.230 to 0.392). During this time, C-CALC students saw an analogous increase in growth mindset (0.143 to 0.281). Following the peak in Unit 2, H-CALC students saw a steep decline in fixed mindset to Unit 3 (0.392 to 0.102) then stayed relatively constant through Unit 4 (see Figure 4). 

Strategic & Non-strategic Mindsets 

C-CALC students exhibited higher initial levels of strategic mindset than H-CALC students. C-CALC and H-CALC students had similar initial levels of non-strategic mindset. However, non-strategic mindset in C-CALC students increased overall by Unit 4, while H-CALC students saw an overall decrease (See Figures 4 & 5).  

Curious & Not Curious Mindsets 

Both H-CALC and C-CALC students had little change in their not-curious mindsets over the semester. However, H-CALC students were more explicitly not curious during every unit (see Figure 2). For both C-CALC and H-CALC students, levels of curious mindset decreased over the course of the semester (see Figures 4 & 5), with H-CALC students exhibiting more curiosity overall (see Figure 2). 

Students with High School Calculus Experience (H-CALC) 

H-CALC students were more likely to have both not curious and fixed mindsets. This could be because H-CALC students have a perceived well-defined notion of “calculus” since they have taken a calculus class before, but the College Board’s AP® Calculus AB curriculum (Mui & Tully, 2020) has different content than our university's Calculus I. The misalignment in ours and the students’ definitions of calculus may also account for the steep increase in fixed mindset in the first half of the course; this is when students’ existing ideas were most resistant to change.  

However, by Unit 3, students were forced to reevaluate their preconceived notions of calculus because this unit had the largest divergence from the AP® curriculum. This was evident in the steep decline of H-CALC students’ fixed mindset.  

Students without High School Calculus Experience (C-CALC) 

C-CALC students were more likely to have growth and mastery-oriented mindsets. Since these students were entering the class with either no calculus experience or calculus experience at our university from a previous semester, they did not have the misaligned preconceived notion of “calculus” that the H-CALC students had. Unlike the H-CALC students, who felt they had already mastered much of the content, the C-CALC students approached the course with the knowledge that they had much to learn. This mindset pairing is not surprising, given that general psychology studies have demonstrated a correlation between growth and mastery-oriented mindsets (Robins & Pals, 2002), as seen in the overlap of these two plots in Figure 4. 

Despite these high levels of growth and mastery-oriented mindsets, C-CALC students also exhibited an increase in non-strategic mindset and a fluctuating helpless mindset. While this is seemingly contradictory to the two positively connotated mindsets, it could be due to these students being novice calculus learners coupled with an increasing difficulty in material as the semester progressed. In particular, we’ve found that students find Units 1 and 3 more difficult because of the more conceptual and applied nature of these units. Units 2 and 4 however, are much more computational, which students typically find more familiar. These heuristics correspond precisely to the trends in helplessness for the C-CALC students.  

Calculus Students in General 

Both the H-CALC and C-CALC students exhibited high levels of strategic mindset compared to most other mindsets we studied. We believe this was due to a couple of factors. First, the course was framed on syllabus day as preparation for the high-stakes, comprehensive uniform final exam. Since the course coordinator wrote this exam, we stressed to students the need to focus on learning rather than setting performance-oriented goals for exams. Furthermore, there was an emphasis on strategic learning through recitation activities (for example, constructing a flowchart describing limit evaluation strategies), in-class practice problems (called “Get Your Reps In”), and frequent low-stakes quizzes (twice per week) that emphasized vocabulary. 

Our study showed that, in the context of learning calculus, student mindsets are not constant. They fluctuate quite often, and are influenced by various factors in the course including prior experience, perceived difficulty of content, and instructor mindset, among others. In particular, previous experience with (high school) calculus creates a finite view of the scope of calculus, defined by the curriculum students are initially exposed to, which takes time to amend with different perspectives on what “calculus” is. First impressions are hard to change. 

On the other hand, students without these preconceived notions are invited to form their own meanings as they experience calculus either for the first time, or in accordance with the same instantiation of calculus they have previously experienced.  

This study was limited by the wording of the “How’d it go?” prompts, which were initially intended to improve student metacognition rather than capture student mindset and focused solely on the assignment at hand rather than capturing students’ broader academic context at the time. Furthermore, the way we chose to present the data, randomly sampled lessons aggregated into units, may have affected the results.  

Our study contributed to existing literature by broadening the applications of mindset theory to tertiary mathematics education, and specifically in a Calculus I context. Previous studies in this area looked at engineering students’ mindsets in a general mathematics context (Campbell, 2023), and the other looked at mindsets in a college algebra setting (Shively & Ryan, 2013).  

Our results were more similar to Campbell (2023) possibly because our calculus students are mostly engineering majors. However, the contrast between ours and existing results may have arisen because of methodological differences. Previous studies used surveys with specific prompts designed to measure growth and fixed mindsets (Limeri et al., 2020; Shively & Ryan, 2013). However, our “How’d it go?” prompts were more-open ended and voluntary, which not only gave students the choice to disclose or not disclose information about particular mindsets, but also gave us a broader insight into which mindsets organically evolve in calculus students.   

Our study also agreed more with Campbell (2023) methodologically and in the strong appearance of strategic mindset, which we believe was encouraged by instructor mindset. Likewise, Campbell (2023) noted that having a “teaching approach that promoted growth mindsets by encouraging improvement through effort” could be part of the reason for more growth mindsets in students, indicating that STEM instructor mindset has a correlation to student mindset. This claim is supported further by other work (Fuesting et al., 2019; LaCosse et al., 2021; Muenks et al., 2020). 

In the future, it would be interesting to replicate this study using “How’d it go?” prompts designed to illicit responses about mindset and the broader context of students’ real-time college experience to see if other classes students may be taking impact their mindset in calculus. 

Through this study, we have demonstrated that regular metacognitive reflections, even brief ones, offer valuable and authentic insight into students’ complex psychological processes during the mathematical learning process. Not only that, but engaging students in regular metacognitive reflection coupled with a strategic framing of calculus may significantly increase course success rates in a highly coordinated, exam-heavy grading system. 

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