Abstract For many undergraduate students, Calculus I or II may be the first mathematics course they take at the university level. This course provides a foundation for students in solving mathematical problems analytically. Although many different assessment methodologies are applied by instructors, students generally appear to be unsuspecting of their performance. The purpose of this research is to quantify the relation between student self-assessment and their performance. This is achieved through weekly assessments in the form of quizzes. The results reflect an overall underestimation of performance. To promote a positive learning environment, explicit expectations, directions, and repetition was used to scaffold instruction. The impact of feedback is also under consideration.

Comparing Student Self-Assessments to Actual Performance in Calculus Mario Banuelos First Year PhD Graduate Student School of Natural Sciences University of California, Merced Introduction For a variety of majors in college, at least one course in Calculus is a requirement. For many undergraduates, Calculus I or II may be the first mathematics course they take at the university level. The ability for students to succeed in these courses will serve to be foundational for their problem solving skills, future courses, and their careers. At UC Merced, Calculus II serves as an introductory course to meet the program learning outcome of “solv[ing] mathematical problems using analytical methods.” Moreover, the class also aims to have students achieve specific course learning outcomes (CLOs). Integration methods and applications of integration and series are discussed throughout the semester. By having students assess their ability to perform these goals, they will be able to use problem solving in mathematics and in self-reflection. To further foster and encourage these outcomes, it is important to have students become more sophisticated learners [1]. To this end, I have implemented an empirical study surveying students self-evaluation on weekly quizzes. Students responded to questions regarding their preparedness as well as their quiz score prediction (See Appendix). The results of these surveys were utilized to inform instruction as well as provide self-feedback for students. Previous studies suggest students often portray overconfidence in their work. The deficits in metacognitive skill often leads to people overestimating their ability to complete a task [3]. This Kruger-Dunning effect, as it has become known, has been explored in a variety of fields since the study was first implemented [2]. Further understanding this concept in the context of a mathematics classroom is one main goal of this project. The other aim is to determine how much student confidence and preparedness plays a role in student performance on assessments. One study in mathematics suggests that students exhibiting higher confidence held a more “cohesive conception 1

of mathematics” [4]. Although it is not the main focus of this project, student comfortability and confidence with the material will be addressed in this study. Needs Assessment and Design of Survey Prior to implementing the self-assessment surveys accompanying weekly quizzes, a needsbased assessment was administered to students to gauge student opinion and readiness for the course. Not surprisingly, 94% of students expected to receive a grade of “A” or “B.” The figure below reflects these results of grade expectancy at the onset of the semester.

Figure 1: Needs assessments results for expected grade in course. Results reflect responses from two discussion sections. Moreover, a vast majority of the students identified the following four factors to attribute to success in the course: 2

• Attend discussion • Attend lecture • Completing homework • Study for quizzes and tests This is to suggest that students have at least a basic understanding of what is required to be successful in Calculus. Since a large population of students at UC Merced are first generation college students, I implemented different strategies to scaffold instruction. With this in mind, I implemented learner-centered teaching and assessment. Specifically, I focused on having students be actively involved via interpersonal cooperative groups [1]. Additionally, I developed mini-lectures for discussion sections through a backward-design process. Explicit instruction during the minilectures provided examples and repetition of the material for students. Having a clear goal for my students before creating the lesson assisted me in developing the appropriate assessments and examples to include for each discussion section. The design of the survey for this signature assignment focused on two key concepts: the comparison of student self-assessment and actual performance on quizzes, and self-reflection on what students did to prepare for the quiz. The surveys were administered to two discussion sections of Calculus II. For an example of the weekly surveys implemented, please refer to the Appendix. Mid-Semester Survey During the eighth week of instruction, a mid-semester survey was administered to students (See Appendix). This follow-up questionnaire asked students about the effectiveness of group work, mini-lectures, and self-reflection. The results reflected mixed feeling about asking students what they did to prepare for the quiz. However, 86% of students surveyed felt that the short lecture at the beginning of the discussion section to be helpful. The results of this survey helped inform my instruction and allowed me to focus on areas of improvement to reach both the CLOs and PLOs for the remainder of the semester. Mini-lectures were still incorporated into weekly discussion sections, but a greater emphasis was placed on group work and group learning. Moreover, the mid-semester survey also asked students about their learning and their grade expectancy. The latter did not change greatly and further informs that students want to succeed in these courses. As a TA, seeking more ways to motivate students is of utmost importance. When reflecting on material presented thus far, students identified “real world applications of integration,” “trigonometric substitutions”, “partial fraction decomposition,” and “integration by parts” as material learned thus far. These open-responses from students further supports meeting CLOs in the course.

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Weekly Survey Results The results of the self-reflection question were compiled and most students did not change their study practices. I believe this may be indicative of a more overarching theme in mathematics learning; students do not use systematic methods to effectively study and prepare for mathematical assessments. Specifically, first generation college students would greatly benefit from workshops on this topic. However, the department can take measures to counter this potential issue, so that all program learning outcomes are ultimately achieved. After analyzing student survey results as the semester progressed, a couple of major trends emerged. First, students, as a collective, became increasingly aware of their ability and comfortability with the material for the first three quizzes. In this time period, many students correctly identified how well they performed on the quiz. Secondly, those who did not predict correctly often underestimated their performance on the assessment. The latter trend does not agree with similar empirical studies [2]; however, the preparedness of the student population at UC Merced could be a cause for the discrepancy since first generation students comprise a sizable portion of the student body. The following figure illustrates the prediction differences observed from weekly survey responses. In the figure, positive differences reflect students’ underestimation of their performance while negative differences represent overestimation. Since quizzes are graded out of 10 points, classes generally underestimated their performances on these assessments.

Figure 2: Predicted differences for Class 1 reflect an overall underestimation of their quiz scores.

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Figure 3: Predicted differences for Class 2 does not necessarily follow one pattern; however, students generally gauged their performance within one letter grade.

In regards to a discrepancy in subsequent quizzes, this may be an area to increase focus. Thus, topics that may be of more difficulty for students in general may come to light. The results of this study would then be used to inform instructors and the department in order to better meet both course and program learning outcomes. Ultimately, this assessment project has provided insight into the metacognitive state of student learning and preparedness, while also supplying knowledge on topics that should be covered more in depth. Future Directions As a campus who serves a large portion of first generation college students, UC Merced can make further analysis on student self-assessment to better serve their students. Additional similar studies would further clarify whether or not students generally underestimate their mathematical performance on this campus. Measures can also be taken to promote better study practices for students who may not be familiar with a variety of study techniques. By doing so, students will be better equipped to meet expectations from not only mathematics courses but in other subjects as well. Finally, teaching assistants and professors can then address discrepancies in performance assessment by further motivating students in determining what they value from a particular course. As a continuing teaching assistant, I plan to consider the demographics of students in my courses closely and continue providing explicit instructions via mini-lectures at the beginning of discussion sections. Comments on end-of-semester evaluation survey results reflected positive response to the brief lectures. Moreover, exploring the relation between feedback and performance is also under consideration for future work.

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References [1] Huba, M.E.; Freed, J.E., Learner-Centered Assessment on College Campuses: Shifting the Focus from Teaching to Learning. Pearson. 2000, 32-63. [2] Karatjas, A.G., Comparing College Students’ Self-Assessment of Knowledge in Organic Chemistry to Their Actual Performance. Journal of Chemical Education. 2013, 90, 10961099. [3] Kruger, J.; Dunning, J., Unskilled and Unaware of It; How Difficulties in Recognizing One’s Own Incompetence Lead to Inflated Self-Assessments. Journal of Personality and Social Psychology, 1999, Vol. 77, No. 6, 1121-1134. [4] Tariq,V.N; Durrani, N., Factors influencing undergraduates’ self-evaluation of numerical competence. International Journal of Mathematical Education in Science and Technology, 2012, Vol. 43, No. 6, 337-356.

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Appendix Needs-Assessment Survey 1. What math classes have you previously taken? (Mark all that apply)

Math 05

Math 11

Math 12

Math 21

Math 22

AP Calculus

Other: 2. Which of the following do you think is important to be successful in this course? (Mark all that apply)

Attend discussion section

Attend lecture

Read the textbook

Complete all assigned homework

Study for quizzes and tests

Other: 3. What grade do you expect in this class? (Please respond with a letter answer)

4. What would you like to get out of this class?

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5. Complete the following questions to the best of your abilities: a)

b)

Rπ 0

sin(x) + 1 dx

x 4 d t dx c

R

dt,

(c is a constant)

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Self-Assessment Survey Example Math 22

Quiz 7

March 31, 2014 Section:

Name:

Problem 1 (10 points): Sketch the curve and find the area it encloses. r = 2 sin(θ)

What have you done to prepare for this quiz and what is your comfort level with the material?

Quiz Grade Prediction I predict to receive a grade of

9 − 10

8

7

6

5 or lower











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Mid-Semester Survey Thank you for taking the time to complete this anonymous survey. Please answer the following questions honestly. I will use these responses to inform my instruction, be a more helpful TA, and to ultimately improve the discussion section. Instructor and Course Questions 1. The TA’s explanations and comments are clear and help me in my understanding of the material. Agree

Somewhat Agree

Neutral

Somewhat Disagree

Disagree

2. The TA allows sufficient time for questions and comments. Agree

Somewhat Agree

Neutral

Somewhat Disagree

Disagree

3. I find the lecture review during discussion section helpful. Agree

Somewhat Agree

Neutral

Somewhat Disagree

Disagree

Somewhat Disagree

Disagree

4. I find group work to be effective for my learning. Agree

Somewhat Agree

Neutral

5. I find the feedback received on my work valuable to my learning. Agree

Somewhat Agree

Neutral

Somewhat Disagree

Disagree

6. I find the survey at the end class helps me reflect on my preparation for the quiz. Agree

Somewhat Agree

Neutral

10

Somewhat Disagree

Disagree

Self-Assessment Questions 7. How many hours per week, outside of lecture and discussion section, do you spend on this class? 1-2

2-4

4-6

6-8

more than 8

8. What grade do you expect in this class (Please respond with a letter answer)? A

B

C

D

F

9. What are three important things you have learned so far?

10. What comments or suggestions do you have to improve the discussion section?

Thank you once again for completing this survey.

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