One common problem that occurs to students during breaks is the retrogression of knowledge due to lack of practice. This problem occurs for students at all levels of education but is especially harmful to students who are taking sequential classes such as Calculus for Engineers I and Calculus for Engineers II where the retention of topics taught in Calculus for Engineers I are required for students to succeed. One solution to this problem is the Keep in School Shape (KiSS) program. The KiSS program is a very efficient and easily accessible program that allows students to stay warmed up and ready to go when they start a sequential course by having daily review material during academic breaks. During an academic break, students who are signed up for the KiSS program are sent a link through text message or email every day that allows them to access a multiple choice review problem. The review problem that they are given is a problem that presents material from the previous course that will be needed in the upcoming course. At the beginning of the review, students have the option to choose between a Level 1 or a Level 2 problem, where a Level 2 problem is related to its Level 1 counterpart but slightly more difficult. Before the students are permitted to solve the problem, they must first use a five point scale that indicates their confidence in their ability to solve the problem. After they complete either the Level 1 or Level 2 daily problem, those that got it wrong have the option to view a hint and try again or view a solution. The students that got the Level 1 daily problem right are also allowed to view the solution but will be permitted to go onto the next level right away whereas the students that got the Level 1 problem incorrect will need to try a similar problem before being able to move onto Level 2. For students who chose to do the Level 2 problem and were not very confident, they were given the option to solve a level 1 problem instead. Students who chose level 2 and got it wrong are given the options to view a hint and try again or simply view the solution before moving on to flashcard versions of the daily problems. Students who get the Level 2 problem correct are also given the option to continue practicing using the flashcards if they choose to. Once a week, there is also a trivia day where students have the choice to complete solely a mathematical trivia question or complete both the trivia question along with a daily review problem. This feature allows students to take a day off from doing mathematics if they choose, but still stay engaged by doing a related activity. Through this program, there is a lot to learn about whether doing Level 1 problems can help students improve their understanding of a concept enough to correctly solve a Level 2 problem. There are many factors to consider such as which question the student chose to answer first, student confidence, and student perseverance. Through the Summer Break 2023 KiSS program, there was data collected for every student answer for each day they accessed the daily KiSS activity. This thesis presents an analysis of the data showing how having two levels of problems is beneficial for students and the correlation between students’ results in Level 1 problems and Level 2 problems for students who chose to attempt both problems.

This study investigates the impact and experiences of students designated as English Language Learners (ELLs) as they engage with student-centered worked example videos (WEVs). Students from two southwestern high schools collaborated and provided their experiences as they watched WEVs and worked through four slope calculation problems. Although high school ELLs are placed in appropriate mathematics classes, the WEVs they engage with, by design, do not consider their diverse educational needs, one of which is the amount of cognitive load experienced when watching the videos. Through this Multi-Phase Mixed Methods study, I begin to understand inclusive design practices for WEVs, in which ELLs will not experience cognitive over-load, and as a result, will receive the needed remediation and/or instruction and develop concept proficiency through active learning as they engage with the videos. The research finds that specific design principles, closed captioning, conversational narration, and music, reduce cognitive load and provide ELLs a familiar and safe space from which to engage with mathematical content.

Learning loss occurs during academic breaks, and this can be detrimental to student success especially in sequential classes like Arizona State University’s Engineering Calculus sequence in which retention of the topics taught in a prior class is expected. The Keeping in School Shape Program (KiSS) is designed as a cost effective, efficient, and accessible way of addressing this problem. The KiSS program uses push technology to give students a way to regularly review material over academic breaks while also fostering a growth mindset.Every day, during an academic break, students are sent a link via text message or email to access a multiple-choice daily review problem which represents material from a previous course that is requisite for success in an upcoming course. Before solving the daily problem, students use a 5-point scale to indicate how confident they are that they can solve the problem. Students then complete the daily review problem and have a variety of resources to support them as they do so, as well as options after they complete it. Students are able to view a hint and try a problem again, view a solution, and attempt a challenge problem. On Tuesdays (aka 2’s-Days) students are given the opportunity to complete either an additional daily review problem or an additional challenge problem, and on Sundays (aka Trivia Days) students can decide between completing only a mathematics trivia question or trivia along with the daily review problem. There is much to be learned from each individual student who participates in the KiSS program. Three surveys were conducted during the Winter Break 2020 KiSS program that gave insight into students’ experience in the KiSS program along with their personal background and mindset regarding mathematics. Ten students responded to all three of these surveys. This thesis will present a case study for each of these ten students based on their data from program participation and survey responses. Conclusions will be drawn regarding ways in which the KiSS program is helping students and ways in which it can be improved to help students be better prepared for their upcoming studies.

Supplemental Instruction (SI) is a well-known non-remedial academic support program for high-risk courses, not students. The program offers regular SI study sessions open to all students, and attendance is voluntary. SI has been shown to decrease the number of failure and withdrawals and to increase students' grades by one half to a full letter grade. A recent study has shown that SI bridges the gap in achievement between students who enrolled in college with low prior achievement and students with high prior achievement. Another study has shown that students of different academic abilities can all benefit from SI. It is well established in the education literature that perceived self-efficacy is an accurate predictor of academic performance and retention in college. Many of these studies have defined perceived self-efficacy as the belief in one's ability to excel in a task or achieve a goal. While many studies tackle the effectiveness of SI and its benefits to SI attendees and SI leaders, no prior study has examined the role of SI in fostering self-efficacy. The purpose of the study is to examine the relationship between perceived self-efficacy, attendance to SI sessions and academic performance. The hypothesis of this study is that attendance to SI sessions will increase students' self-efficacy. The SI participants in the high attendance group had marginally significant changes in self-efficacy while the SI participants in the low attendance group and the non-SI participants did not have significant changes in self-efficacy. The results from this study showed promising outlook that SI might be fostering self-efficacy and enhancing students' academic achievement. Additional studies are required to provide deeper insights into the role of supplemental instruction in fostering self-efficacy.

Past research has shown that students have difficulty developing a robust conception of function. However, little prior research has been performed dealing with student knowledge of function composition, a potentially powerful mathematical concept. This dissertation reports the results of an investigation into student understanding and use of function composition, set against the backdrop of a precalculus class that emphasized quantification and covariational reasoning. The data were collected using task-based, semi-structured clinical interviews with individual students outside the classroom. Findings from this study revealed that factors such as the student's quantitative reasoning, covariational reasoning, problem solving behaviors, and view of function influence how a student understands and uses function composition. The results of the study characterize some of the subtle ways in which these factors impact students' ability to understand and use function composition to solve problems. Findings also revealed that other factors such as a students' persistence, disposition towards "meaning making" for the purpose of conceptualizing quantitative relationships, familiarity with the context of a problem, procedural fluency, and student knowledge of rules of "order of operations" impact a students' progress in advancing her/his solution approach.

This study empirically evaluated the effectiveness of the instructional design, learning tools, and role of the teacher in three versions of a semester-long, high-school remedial Algebra I course to determine what impact self-regulated learning skills and learning pattern training have on students' self-regulation, math achievement, and motivation. The 1st version was a business-as-usual traditional classroom teaching mathematics with direct instruction. The 2rd version of the course provided students with self-paced, individualized Algebra instruction with a web-based, intelligent tutor. The 3rd version of the course coupled self-paced, individualized instruction on the web-based, intelligent Algebra tutor coupled with a series of e-learning modules on self-regulated learning knowledge and skills that were distributed throughout the semester. A quasi-experimental, mixed methods evaluation design was used by assigning pre-registered, high-school remedial Algebra I class periods made up of an approximately equal number of students to one of the three study conditions or course versions: (a) the control course design, (b) web-based, intelligent tutor only course design, and (c) web-based, intelligent tutor + SRL e-learning modules course design. While no statistically significant differences on SRL skills, math achievement or motivation were found between the three conditions, effect-size estimates provide suggestive evidence that using the SRL e-learning modules based on ARCS motivation model (Keller, 2010) and Let Me Learn learning pattern instruction (Dawkins, Kottkamp, & Johnston, 2010) may help students regulate their learning and improve their study skills while using a web-based, intelligent Algebra tutor as evidenced by positive impacts on math achievement, motivation, and self-regulated learning skills. The study also explored predictive analyses using multiple regression and found that predictive models based on independent variables aligned to student demographics, learning mastery skills, and ARCS motivational factors are helpful in defining how to further refine course design and design learning evaluations that measure achievement, motivation, and self-regulated learning in web-based learning environments, including intelligent tutoring systems.

This dissertation describes an investigation of four students' ways of thinking about functions of two variables and rate of change of those two-variable functions. Most secondary, introductory algebra, pre-calculus, and first and second semester calculus courses do not require students to think about functions of more than one variable. Yet vector calculus, calculus on manifolds, linear algebra, and differential equations all rest upon the idea of functions of two (or more) variables. This dissertation contributes to understanding productive ways of thinking that can support students in thinking about functions of two or more variables as they describe complex systems with multiple variables interacting. This dissertation focuses on modeling the way of thinking of four students who participated in a specific instructional sequence designed to explore the limits of their ways of thinking and in turn, develop a robust model that could explain, describe, and predict students' actions relative to specific tasks. The data was collected using a teaching experiment methodology, and the tasks within the teaching experiment leveraged quantitative reasoning and covariation as foundations of students developing a coherent understanding of two-variable functions and their rates of change. The findings of this study indicated that I could characterize students' ways of thinking about two-variable functions by focusing on their use of novice and/or expert shape thinking, and the students' ways of thinking about rate of change by focusing on their quantitative reasoning. The findings suggested that quantitative and covariational reasoning were foundational to a student's ability to generalize their understanding of a single-variable function to two or more variables, and their conception of rate of change to rate of change at a point in space. These results created a need to better understand how experts in the field, such as mathematicians and mathematics educators, thinking about multivariable functions and their rates of change.