As society has changed, post-secondary education has become available not only to the social elite, but to the masses [17, 36, 38], and increasing enrollments in undergraduate mathematics courses would suggest comparative increases in the numbers of students majoring in mathematics, but this is not the case. The book *A Challenge of Numbers* (1990) outlines historical trends and statistics in mathematics programs. The authors, Madison and Hart, describe the ‘boom’ in the numbers of mathematics bachelor degrees and mathematics enrollments after the launch of Sputnik in 1957 (pp. 2; see also [16]). Madison and Hart [24] note a “disparate trend” between undergraduate mathematics enrollments, which have been increasing steadily since the 1950s, and the number of mathematics majors, because most of the enrollment increase is happening at the remedial levels (pp. 36; see also [29]).

Data from the Office of Institutional Analysis (2013) show that undergraduate mathematics enrollments and number of graduates holding a degree in mathematics at our own large, North American Institution follows a similar disparate trend. Overall, undergraduate enrollment has increased nearly 500% from 1957 to 2013, and from 1980 to 2013 the number of total undergraduate student credit hours taught by this large, Canadian University increased by 42%. Meanwhile, the undergraduate student credit hours taught by the Department of Mathematics increased by over 120%, while the number of Science, Technology, Engineering and Mathematics (STEM) degrees increased by less than 30%, and the number of mathematics degrees remains extremely low, with a record high of 13 in 1985 and a low of 4 in 2013. The mathematics department, with one of the smallest budgets on campus, is now teaching over 5% of the total undergraduate credit hours at the university (with the Faculties of Science and Engineering teaching well over 25% of the total undergraduate credit hours), and mathematics undergraduate degrees account for less than 0.1% of all degrees conferred. This trend occurring at our university implies that the increase in credit hours taught by the mathematics department is attributed to the increase of students enrolling primarily in first-year math courses, including terminal courses, in order to meet the requirements of their degree.

A related trend, one better illustrating the limited persistence in undergraduate programs in which mathematics plays a significant part, is the decline of enrollment in mathematics courses above first year. In the 2014–2015 school year, the Department of Mathematics had 6340 students enrolled in first-year courses, 1859 in second year, 464 in third year, and just 32 in fourth, meaning that the department had ‘filtered’ out over 70% of its students each year. Although the numbers in first year and to some extent in second year represent service teaching to students in other programs, most students in first year at this university are registered as “University 1” students and have not yet selected a major. With so many students coming through our classes in their first year of post-secondary study, it seems that we are not doing a very good job of attracting them to continue with the study of mathematics. As one particular illustration, we can compare undergraduate students in a large, non-STEM faculty such as the Faculty of Arts to those in the Faculty of Science. Whereas Arts has 4570 students registered in first-year courses, 1803 students in second year, and 1789 students in third year, Science has 4270 in first year, 2050 in second year, and only 1303 in third year (most of the degrees conferred in these two faculties are 3-year degrees). The sharp decline in students registered in second-year courses versus third-year courses (and thus being able to graduate with a 3-year degree) in Science is not seen in Arts, where there is very nearly the same number of students registered in second-year courses as that in third-year courses.

Success in mathematics courses may not only determine retention in STEM majors, but in overall university success, as there seems to be a correlation between timely completion of a 4-year degree and success in introductory mathematics courses [32]. Thus, although university enrollments are increasing and mathematics departments are teaching more students than ever, persistence within mathematics-based programs is not reflecting these developments, and mathematics continues to be a barrier for many post-secondary students.

What is happening in our first-year mathematics courses that is so effectively closing the door to further participation in mathematics courses, STEM fields, and post-secondary education as a whole?

Currently, failure and withdrawal (FDVW) rates in undergraduate mathematics courses across North America and internationally are disappointingly high, ranging anywhere from 30% to as high as 60% [1, 11, 17, 19, 37], and there are very low grade-point averages for the students who do in fact manage to complete the courses [19]. This situation is reflected in the grade data collected from our own University. *A Challenge of Numbers* (1990) found that students with lower grade-point averages (GPAs) were less likely to continue with their freshmen choice of a mathematics major (pp. 38), and when the reality of the matter is that failure and withdrawal rates for first-year mathematics courses are near 50% [19], it becomes clear to see why so many students are being ‘filtered’ out of mathematics majors and in fact all disciplines that require higher level mathematics courses (NRC [29], pp. 21). Rates this high re-enforce the outdated notion of mathematics as a gatekeeper—which “refers to the exclusion of students from further involvement in school mathematics, in school, and beyond, based on their lack of success as mathematics learners” [25]. As Derek Holton points out, “if mathematics is seen as a major that is harder to complete it may also be avoided in favour of easier subjects that have similar financial rewards” ([16], pp. 9). In institutions that require their students to complete a math course for graduation, rates as high as these mean that mathematics courses could be filtering approximately half of the undergraduate population out of further mathematics study, STEM degrees, and even post-secondary study altogether.

The trends discussed above describe the historical lack of participation and persistence in undergraduate mathematics, which is what we attempted to address with our strategy to re-think the structure of one of our own first-year mathematics courses.

### The course

During the Fall and Winter terms of the 2014–2015 school year, the issues of high FDVW rates and low average GPAs were addressed by re-structuring the format of the terminal course *MATH 1010: Applied Finite Mathematics*. There were three sections during that school year, each containing 150, 150, and 240 students. Historically, this course has had extremely high FDVW rates, with the average over the last decade being nearly 49%. Even more concerning was the fact that this number has been increasing in a statistically significant way (Cox–Stuart test for Trend, *p* < 0.05), and grade-point average has also been quite low, hovering around a C+ over the last decade.

The effect of attitude on mathematics success is well documented, and studies have shown that attitude is an important factor in achieving success in mathematics [23, 31]. A majority of the students in MATH 1010 are from faculties outside of the Faculty of Science, and for all but a small few of them, this will be their last encounter with mathematics at the post-secondary level. For these students, it is especially important that the effect of this course on their attitude toward learning mathematics be positive.

Another issue surrounding attitude toward mathematics is that approximately 10% of students who complete this course continue on to earn degrees from the Faculty of Education specializing in early and middle years teaching. Several authors have noted that mathematical anxieties can be passed down from teacher to student [2, 18, 27]; thus, it is crucial that we take the opportunity presented to us in this course to reverse any negative attitudes and mathematical anxieties from the past, and structure the course in such a way that all students are provided the opportunity for success and a positive experience in mathematics.

There has been a multitude of investigation into instructional practices of mathematics courses at the undergraduate level. Some of these include the use of technology to promote discovery and the use of multiple representations [5, 9, 12, 28], the use of discourse and methods such as scientific debate in the classroom [14, 22], and the inclusion of real-world applications and project-oriented problem solving to motivate theory ([10], vii, pp. 10, pp. 20; [15], pp. 65), as well as classroom techniques for large classes [20]. Although these practices certainly played parts (in varying degrees) in our MATH 1010 classroom, we chose to focus on addressing the possible effects that an alternate course *structure* could have on student success, rather than focusing on the effect of specific teaching *strategies*.

Mastery-based learning is an educational philosophy in which students achieve a desired grade demonstrating mastery of given content before moving to new content in a course. The idea of mastery-based learning has been observed in classrooms since the 1920s where students were required to demonstrate sufficient skills in an area assessed by a formative test before moving to new content. If the student was unable to achieve mastery of the content, tutoring or extra instruction was provided in order to help the student reach mastery (Kuilk et al. 1990). In a particular mastery-based approach called Bloom’s Learning for Mastery (LFM), course content is teacher presented and divided into short units, and at the close of each unit, students complete a test in order to demonstrate mastery of the unit. Based on this model, Bloom [3] predicted that 90% of students would achieve grades in the top 10th percentile. Bloom [4] also states that weak students will not need more time to complete the tasks but generally only need more time to reach proficiency in beginning stages of a course [21]. In a paper by Kulik et al., [21] a meta-analysis was performed on mastery-based programs, and they concluded that in 12 of 14 studies that examined student attitude, students showed an increased positive attitude toward subject matter in mastery-based environments (1990). Other positive effects observed in mastery-based environments are that teachers feel more personal responsibility for students’ learning, hold high expectations for their students and have a more positive attitude toward teaching [39].

Examples of the implementation of mastery-based learning can be seen in schools around the world. In Dryersburg High School in Tennessee, mastery-based learning was incorporated into Algebra I with great success. Students seemed to retain what they had learned, gained self-confidence by achieving grades of A or B, and improved the quality of their work [39]. In Kelana Jaya, Selangor, Malaysia, mastery learning was introduced in a discrete mathematics course for 30 students majoring in mathematics under a Bachelor of Education program at the University Tun Abdul Razak. Students were taught short units and provided a formative assessment at the end of each unit. If the student achieved the desired grade, they were moved to the next unit, and students who did not achieve mastery were permitted to repeat a unit until mastery (a grade of 80% or higher) was achieved. Students who were repeating a unit worked simultaneously on the new unit(s) as well. At the end of the study, students commented that they felt they had understood the material, felt mastery learning encouraged them to study independently and that the repetition helped increase retention [34].

The literature on the effects of class size on various factors such as student achievement, engagement, and satisfaction in mathematics is mixed. There are some researchers who find no significant effects of class size on these variables [7], others who find that effects can be negated by the use of technology in the classroom [13], and some others who claim that the effects of small class size in mathematics are most significant for females [30]. Still others highlight the differences in classroom practices that arise from the use of large class sizes in undergraduate education. Cuseo [8] notes that in classes with large enrollments, “there is an increased reliance on the *lecture* method of instruction” (pp. 2), leading to less active student involvement in the learning process and reduced frequency of instructor interaction and feedback to students. In terms of class size in conjuncture with mastery learning, smaller class sizes have also been shown to aid in the reduction of extra time needed to reach mastery of the material [21].

The course structure that we used as a model for our own was first implemented by C. Card and D. Siewert of the Black Hills State University in response to low success rates in their Basic Algebra course (2013). In this course structure, students were provided an early “alert” mechanism, which allowed them a second chance to learn and master the course outcomes, while still covering the same course content as in previous terms. With this course structure, the developers saw an improvement in pass rates from approximately 50% to approximately 80% [6]. However, this structure was developed in order to address students’ lack of success in secondary mathematics courses. We wanted to see if a structure based on the model by C. Card and D. Siewert, with which they had much success, could offer the same opportunity for success in a large, tertiary level mathematics course by maximizing the effects of mastery learning in conjuncture with the positive effects of small class sizes. Although, due to resource limitations, it is in most cases not viable to simply reduce class sizes across the board, this course structure allows for targeted identification of students who would most benefit from a smaller class, and an opportunity for these students to benefit from the positive effects of a more active learning environment and more frequent instructor interaction in the classroom.