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Student Attitudes and Achievement in Active Learning Calculus

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Title: Student Attitudes and Achievement in Active Learning Calculus
Language: English
Authors: Pablo A. Duran (ORCID 0000-0001-5685-9699), Adam J. Castillo (ORCID 0000-0002-8970-9176), Charity Watson (ORCID 0000-0003-0288-4180), Edgar Fuller (ORCID 0000-0003-4130-090X), Geoff Potvin (ORCID 0000-0002-8164-2309), Laird H. Kramer (ORCID 0000-0003-1035-8471)
Source: International Journal of Mathematical Education in Science and Technology. 2024 55(3):759-780.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 22
Publication Date: 2024
Sponsoring Agency: National Science Foundation (NSF)
Contract Number: 1832450
Document Type: Journal Articles
Reports - Research
Education Level: Higher Education
Postsecondary Education
Descriptors: Calculus, Student Attitudes, Academic Achievement, College Mathematics, Undergraduate Students
DOI: 10.1080/0020739X.2022.2150902
ISSN: 0020-739X
1464-5211
Abstract: The present paper explores the relationship between attitudes towards mathematics (ATM) and achievement in college calculus in active learning (AL) and lecture-based (LB) classrooms. Previous work on this relationship has mainly been limited to LB instruction, neglecting the impact of innovative approaches such as AL. Less attention has been paid to the roles played in this relationship by gender, year in college, and initial ATM. Results from a sample of 535 undergraduate students enrolled in 9 AL and 9 LB sections are presented. Data included ATMI surveys' responses, final grades, and demographics. Correlation and multiple regression analyses were conducted. The influence of instruction on students with low ATM was also examined. Gender and year in college were the main demographic variables considered. Achievement in AL was found to be less dependent on initial ATM in terms of correlation. AL showed higher gains in grades than LB, when controlling for ATM and demographic variables. Effect sizes of AL instruction on grades of students with low ATM were larger than those of students with higher ATM. Furthermore, AL courses had a large effect size (d = 0.81) on female students with lower ATM, confirming its role as a gender equalizer.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1413917
Database: ERIC
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  Value: <anid>AN0175640999;imt01mar.24;2024Feb27.06:47;v2.2.500</anid> <title id="AN0175640999-1">Student attitudes and achievement in active learning calculus </title> <p>The present paper explores the relationship between attitudes towards mathematics (ATM) and achievement in college calculus in active learning (AL) and lecture-based (LB) classrooms. Previous work on this relationship has mainly been limited to LB instruction, neglecting the impact of innovative approaches such as AL. Less attention has been paid to the roles played in this relationship by gender, year in college, and initial ATM. Results from a sample of 535 undergraduate students enrolled in 9 AL and 9 LB sections are presented. Data included ATMI surveys' responses, final grades, and demographics. Correlation and multiple regression analyses were conducted. The influence of instruction on students with low ATM was also examined. Gender and year in college were the main demographic variables considered. Achievement in AL was found to be less dependent on initial ATM in terms of correlation. AL showed higher gains in grades than LB, when controlling for ATM and demographic variables. Effect sizes of AL instruction on grades of students with low ATM were larger than those of students with higher ATM. Furthermore, AL courses had a large effect size (d = 0.81) on female students with lower ATM, confirming its role as a gender equalizer.</p> <p>Keywords: Active learning; gender equalizer; student achievement; college calculus; competitiveness; student attitudes; 97D40</p> <hd id="AN0175640999-2">1. Introduction</hd> <p>Although college calculus has continuously been confirmed as a vital component of STEM careers (Bressoud et al., [<reflink idref="bib11" id="ref1">11</reflink>]), its steady presence in the curriculum has been linked to large reform efforts (Bressoud, [<reflink idref="bib10" id="ref2">10</reflink>]). Some of the main drivers of these reform efforts include students' persistent low achievement (Bressoud et al., [<reflink idref="bib12" id="ref3">12</reflink>]), poor conceptual understanding of core ideas, and inability to transfer these ideas to other contexts in STEM (Hallett, [<reflink idref="bib42" id="ref4">42</reflink>]). These efforts have primarily focused on changes in instructional practices, content selection to highlight core ideas, and the emphasis on the application of core ideas to different disciplines in STEM (Carreon et al., [<reflink idref="bib16" id="ref5">16</reflink>]; Rasmussen et al., [<reflink idref="bib80" id="ref6">80</reflink>]).</p> <p>In terms of changes in instructional practices, special attention has been paid to classrooms that have implemented active learning (AL) approaches. These approaches have been characterized as a shift from teacher-centered, lecture-based classrooms to more student-centered classroom environments where students are actively engaged in meaningful learning activities and have an opportunity to reflect on their learning process (Bonwell & Eison, [<reflink idref="bib9" id="ref7">9</reflink>]). Examples of these approaches include flipped classrooms (DeLozier & Rhodes, [<reflink idref="bib27" id="ref8">27</reflink>]; Jungić et al., [<reflink idref="bib50" id="ref9">50</reflink>]), inquiry-based learning (Kogan & Laursen, [<reflink idref="bib53" id="ref10">53</reflink>]; Laursen et al., [<reflink idref="bib55" id="ref11">55</reflink>]), and courses that incorporate a larger component of group work supported by mathematical software (Armstrong & Hendrix, [<reflink idref="bib3" id="ref12">3</reflink>]).</p> <p>The effectiveness of active learning (AL) approaches started to appear in literature over three decades ago (Davidson, [<reflink idref="bib26" id="ref13">26</reflink>]). Since then, it has been extensively investigated in undergraduate STEM education (Freeman et al., [<reflink idref="bib35" id="ref14">35</reflink>]; Johnson & Johnson, [<reflink idref="bib47" id="ref15">47</reflink>]). For instance, Freeman et al. ([<reflink idref="bib35" id="ref16">35</reflink>]) in a meta-analysis of 225 studies, found that these approaches led to an increase in student performance on examinations and concept inventories by almost one-half standard deviation over traditional lecturing. Furthermore, students enrolled in courses based on these approaches were found to be 1.5 times less likely to fail the course than traditional lecturing courses. This success has also proven to be consistent across STEM areas, leading to high odds of student success in biology (Chambers, [<reflink idref="bib20" id="ref17">20</reflink>]), chemistry (Paulson, [<reflink idref="bib71" id="ref18">71</reflink>]), and computer sciences (Lasserre, [<reflink idref="bib54" id="ref19">54</reflink>]). In other STEM disciplines, such as modelling in physics, these odds have shown to be as high as 6.73 times greater than lecture-based instruction (Brewe et al., [<reflink idref="bib13" id="ref20">13</reflink>]). The effectiveness of AL approaches has motivated multiple institutional calls for its large-scale implementation (Cipra, [<reflink idref="bib24" id="ref21">24</reflink>]; Ganter et al., [<reflink idref="bib37" id="ref22">37</reflink>]; Olson & Riordan, [<reflink idref="bib68" id="ref23">68</reflink>]) increasing interest in a better understanding of its broader impact in college education, especially across STEM programs.</p> <p>In the specific case of college calculus, current studies on the effectiveness of AL, however, are still limited. These studies have focused on high-stakes assessments and concept inventories, neglecting other relevant outcomes such as students' attitudinal changes. Furthermore, meta-analysis studies on the impact of AL have included only a small sample of calculus courses with a wide variety of AL strategies, from courses with a high-lecture component and marginal emphasis on student collaboration (Lindaman, [<reflink idref="bib58" id="ref24">58</reflink>]; Maggelakis & Lutzer, [<reflink idref="bib61" id="ref25">61</reflink>]), to courses with minimal lecturing in classes and a large computer-based component (Armstrong & Hendrix, [<reflink idref="bib3" id="ref26">3</reflink>]). Further research on the effectiveness of AL on additional measures such as student attitudes in calculus is still needed.</p> <p>As an additional measure of the impact of AL on students' learning process, attitudes towards mathematics (ATM) in calculus has only recently been examined (Castillo et al., [<reflink idref="bib17" id="ref27">17</reflink>]). Developing a better understanding of this influence is important because Calculus has been found to produce sharp declines in ATM in college (Bressoud, [<reflink idref="bib10" id="ref28">10</reflink>]). These declines can have dire consequences in students' experiences, impacting students' persistence in STEM programs (Bressoud et al., [<reflink idref="bib11" id="ref29">11</reflink>]; Maltese & Tai, [<reflink idref="bib62" id="ref30">62</reflink>]), classroom equity (Ellis et al., [<reflink idref="bib30" id="ref31">30</reflink>]), and driving subsequent mathematics avoidance (Popham, [<reflink idref="bib75" id="ref32">75</reflink>]). Finding evidence of a direct link between AL and ATM positive gains would provide further support for the implementation of this and similar instructional strategies in Calculus, contributing to addressing these persistent issues in STEM education.</p> <p>Although the existence of empirical evidence of the positive influence of AL models on ATM in the literature remains limited, at least three theoretical links strongly support it. First, when AL approaches situate cooperative learning as a classroom principle, learning goals are often shared by groups. According to Johnson et al. ([<reflink idref="bib48" id="ref33">48</reflink>], [<reflink idref="bib49" id="ref34">49</reflink>]), this process promotes positive interdependence and promotive interaction, inducing key psychological changes (such as substitutability, inducibility, and positive cathexis) hypothesized to expand self-interest to mutual interest (Johnson et al., [<reflink idref="bib48" id="ref35">48</reflink>]). This expansion strengthens interpersonal relationships and develops students' openness to being positively influenced by their peers, fostering attitudinal changes. Furthermore, scholars suggest that under such social influence, people tend to develop a more favourable self-concept by shifting their attitudes to align with positively valued groups (Pool et al., [<reflink idref="bib74" id="ref36">74</reflink>]; Prislin & Wood, [<reflink idref="bib77" id="ref37">77</reflink>]). Such attitudinal changes could include increases in motivation and enjoyment of mathematics.</p> <p>Second, the cooperative environment of AL approaches is also expected to promote further student engagement, increasing opportunities for students to develop a sense of being recognized, accepted, and encouraged by peers. This perception defines belongingness (Goodenow, [<reflink idref="bib40" id="ref38">40</reflink>]), germane to attitudinal changes as it has been linked to intrinsic motivation (Osterman, [<reflink idref="bib69" id="ref39">69</reflink>]) when present. More specifically, sense of belonging has been found to positively influence motivational beliefs in relation to that same course in college (Freeman et al., [<reflink idref="bib35" id="ref40">35</reflink>]). Additionally, the influence of belonging on students' motivation is particularly relevant for minoritized groups of students who have been found to start college with a much lower sense of belonging than their peers (Barbieri & Miller-Cotto, [<reflink idref="bib8" id="ref41">8</reflink>]).</p> <p>Last, the positive influence of AL approaches on ATM is also supported by social cognitive theory (Bandura, [<reflink idref="bib6" id="ref42">6</reflink>]) for two main reasons. On the one hand, as students collaboratively work in groups, they have more opportunities to witness peers successfully solving problems. These opportunities, in a phenomenon known as social modelling, are hypothesized to increase students' confidence in their own ability to solve problems (Bandura, [<reflink idref="bib7" id="ref43">7</reflink>]). On the other hand, AL approaches that incorporate culturally appropriate learning models lead to the creation of a safe, inclusive, and respectful learning environment (Ginsberg & Wlodkowski, [<reflink idref="bib38" id="ref44">38</reflink>]). In this environment, students would feel more comfortable giving formative feedback to peers. Since formative assessment enables students to gain a better sense of control over their progress (Chan & Lam, [<reflink idref="bib21" id="ref45">21</reflink>]), students would naturally increase their self-efficacy.</p> <p>The influence of ATM on achievement, on the other hand, has been widely ascertained in postsecondary settings, finding small-to-moderate effect sizes in calculus (House, [<reflink idref="bib46" id="ref46">46</reflink>]; Pyzdrowski et al., [<reflink idref="bib78" id="ref47">78</reflink>]; Sonnert et al., [<reflink idref="bib86" id="ref48">86</reflink>]). This effect size has been translated, when controlling for other relevant predictors of achievement such as mathematics preparation, to a 10% increase in final grade per 1-SD increase in ATM (Pyzdrowski et al., [<reflink idref="bib78" id="ref49">78</reflink>]). The nature of the close interaction between students' ATM and their achievement could be associated with the existence of a feedback loop between these factors (Ma & Kishor, [<reflink idref="bib60" id="ref50">60</reflink>]; Sonnert et al., [<reflink idref="bib86" id="ref51">86</reflink>]). As the semester progresses, students' higher achievement in preliminary assessment increases their ATM, and this renewed ATM, in turn, helps them to increase their future achievement in the class. Conversely, students' lower achievement in preliminary assessment decreases their ATM, and such change undermines their future achievement. This reinforcement cycle aligns with the influence that performance accomplishments can have over students' self-efficacy. As Bandura suggests, students' 'successes raise mastery expectations, repeated failures lower them.' (1977, p. 195)</p> <p>It is possible that the enhanced student engagement in AL classrooms could change the way in which this cycle evolves throughout the semester. The low-stake environment of AL classrooms and emphasis on student collaboration could contribute to a better distribution of skills, attitudes, and knowledge in the classroom (Slavin, [<reflink idref="bib85" id="ref52">85</reflink>]). Students with initially low ATM, often conditioned to lower achievement, might be positively influenced by students on the other extreme of the spectrum. Despite how plausible this mechanism seems to be, no studies were found that confirm its existence in AL classrooms in college calculus.</p> <p>Equally important in investigating the nature of the mechanism through which AL might change the relationship between initial attitudes and achievement, is the role demographic variables might be playing in this process. Clear interactions between ATM, achievement, and demographic variables, such as gender, year in college, career choice, race and ethnicity have been found. First, a recent study reported that women start and end the semester with significantly lower mathematical confidence than men, even when looking only at students with above-average mathematics preparation and skills (Ellis et al., [<reflink idref="bib30" id="ref53">30</reflink>]). Second, in terms of year in college, freshman students in college calculus have shown to have a more positive ATM than their sophomore counterparts (Sonnert et al., [<reflink idref="bib86" id="ref54">86</reflink>]). In terms of career choice, students who were interested in pursuing STEM careers had higher ATM than the rest of the students enrolled in the course (Sonnert et al., [<reflink idref="bib86" id="ref55">86</reflink>]). Last, literature on the influence of race and ethnicity on ATM is much more limited, but a previous study reported evidence that African American students showed higher values of ATM than their White counterparts, while the opposite relationship was true when examining their achievement (Stanic & Hart, [<reflink idref="bib88" id="ref56">88</reflink>]).</p> <p>Another critical aspect regarding the influence of AL in the classroom is its impact on students with particularly low attitudes. A regression model from a recent study suggests that students with initially low ATM (those with less than 2 standard deviations in mean ATM scores) are likely to earn low grades, fail or drop the class (Sonnert et al., [<reflink idref="bib86" id="ref57">86</reflink>]). There is evidence that these students benefit more from conventional good teaching with less group work than other instructional strategies better aligned with AL (Sonnert et al., [<reflink idref="bib87" id="ref58">87</reflink>]). However, a recent study found that certain AL approaches could lead to better outcomes in students' ATM (Castillo et al., [<reflink idref="bib17" id="ref59">17</reflink>]). This study, however, did not examine achievement, suggesting the need for further research.</p> <p>The main purpose of the present study is to examine the relationship between attitudes towards mathematics (ATM) and achievement in AL and LB calculus classrooms. The following five research questions guided this investigation: (<reflink idref="bib1" id="ref60">1</reflink>) To what extent are students' initial ATM associated with their achievement in both LB and AL classrooms? (<reflink idref="bib2" id="ref61">2</reflink>) How does this association vary by demographics? (<reflink idref="bib3" id="ref62">3</reflink>) To what extent do initial ATM predict achievement when controlling for instructional strategy and students' demographics including gender, year in college, and STEM choice? (<reflink idref="bib4" id="ref63">4</reflink>) To what extent does AL influence the achievement of students with initially low ATM differently, when compared to traditional classrooms? (<reflink idref="bib5" id="ref64">5</reflink>) How does this difference vary by demographics?</p> <hd id="AN0175640999-3">2. Related literature</hd> <p></p> <hd id="AN0175640999-4">2.1. Attitudes towards mathematics</hd> <p>Extensive research on student affect has been conducted in mathematics over the last decades (Reyes, [<reflink idref="bib81" id="ref65">81</reflink>]; Zan et al., [<reflink idref="bib97" id="ref66">97</reflink>]). The general idea of how student attitudes towards mathematics (ATM) are portrayed in these reviews is summarized by Reyes as 'students' feelings about mathematics, aspects of the classroom, or about themselves as learners of mathematics' (1984, p. 1).</p> <p>Understanding the impact of these feelings in the learning process is important as they have been linked to student achievement (Evans, [<reflink idref="bib33" id="ref67">33</reflink>]; Yee, [<reflink idref="bib95" id="ref68">95</reflink>]), and have also proven to play a predominant role in shaping students' persistence in STEM programs (Bressoud et al., [<reflink idref="bib11" id="ref69">11</reflink>]; Maltese & Tai, [<reflink idref="bib62" id="ref70">62</reflink>]). Furthermore, students' ATM have been recently related to issues of equity. A study examining the effect of attitudes by gender, for instance, found that odds of a female student being discouraged from continuing in calculus is 1.5 times greater than that for a male student (Ellis et al., [<reflink idref="bib30" id="ref71">30</reflink>]).</p> <p>Due to the complexity of students' feelings, developing reliable instruments that measure students' ATM has proved to be a long process (Chamberlin, [<reflink idref="bib19" id="ref72">19</reflink>]; McLeod, [<reflink idref="bib63" id="ref73">63</reflink>]). This process started over six decades ago undergoing multiple iterations focused on capturing the most salient traits of student attitudes. Most recently, the attitudes toward mathematics inventory (ATMI) was developed to capture the most essential dimensions of student attitudes but with less items than previous instruments (Lim & Chapman, [<reflink idref="bib57" id="ref74">57</reflink>]; Tapia & Marsh, [<reflink idref="bib92" id="ref75">92</reflink>]). It is composed of 40 items and four factors: confidence, enjoyment, motivation, and value. Moreover, its psychometric properties have been confirmed in a college setting, yielding good model fit statistics and high Cronbach alpha coefficients (above 0.87) for each factor (Tapia & Marsh, [<reflink idref="bib91" id="ref76">91</reflink>]) in college settings.</p> <hd id="AN0175640999-5">2.2. Influence of active learning on student attitudes</hd> <p>Fewer studies have been conducted to understand the impact of AL approaches on students ATM than those examining students' achievement. Moreover, most of these studies have focused on pre-tertiary education. However, results in general suggest a positive impact on students' attitudes. In the US, findings from a meta-analysis of 65 studies on primary and secondary school (E. Savelsbergh et al., [<reflink idref="bib83" id="ref77">83</reflink>]) provide some evidence of this impact. According to this study, significant effects (0.35 < d < 0.4) were found in general attitude, general interest, and career interest in science. These results included a variety of innovative instructional strategies such as inquiry-based, computer-based, and collaborative learning strategies with no significant difference between approaches. Another interesting outcome of this study is that the effects of instruction were found to be weaker for older students.</p> <p>Outside of the US, the positive influence of AL on student attitudes has also been confirmed in secondary science education. For instance, in a study of over 12,000 students in the UK, student engagement in high school classrooms was linked to positive attitudes towards science, and particularly higher levels of student enjoyment and motivation (Hampden-Thompson & Bennett, [<reflink idref="bib43" id="ref78">43</reflink>]). Similarly, studies recently conducted with students in Turkey also found significant effect sizes on the impact of AL in students' attitudes (Akinoglu & Tandogan, [<reflink idref="bib1" id="ref79">1</reflink>]; Demirci, [<reflink idref="bib28" id="ref80">28</reflink>]).</p> <p>In college calculus, Alkhateeb ([<reflink idref="bib2" id="ref81">2</reflink>]) found that adding a hands-on, technology-based component to the course further enhanced students' attitudes and significantly redacted students' mathematics anxiety. Finally, a more recent study (Castillo et al., [<reflink idref="bib17" id="ref82">17</reflink>]) noted that incorporating AL strategies in a calculus course can improve student attitudes gains, when compared to a traditional lecture-based course. Moreover, this study found that AL had a particular positive impact on female students' self-confidence, acting as a gender equalizer.</p> <p>Although there still is a need for more systematic studies, experiences in multiple educational settings in pre-tertiary education and STEM disciplines in college points to a positive influence of AL approaches on student attitudes.</p> <p>With respect to how AL interacts with demographics variables including gender, year in college, and STEM intended choice a few, yet limited, studies suggest the existence of significant interactions. For example, Laursen et al. ([<reflink idref="bib55" id="ref83">55</reflink>]) conducted a quasi-experimental study of over 100 course sections at four academic institutions over a period of two years to understand differences between inquiry-based learning (IBL) courses and comparable non-IBL courses. Their results indicated not only an overall positive impact of IBL, but a significant decrease in attitudinal and achievement gaps. While female students in non-IBL courses still showed much lower cognitive and attitudinal gains than their male counterparts, in IBL courses, no significant differences in these gains were found.</p> <p>Research on the influence of AL on other demographics variables such as year in college, and STEM intended career choice is extremely limited. Only one study, by Fuselier and Jackson ([<reflink idref="bib36" id="ref84">36</reflink>]) was found regarding this issue. This study examined students' views on how collaborative science might change depending on their coursework, finding that the fewer science courses students take, the more collaborative they report science is for them. This progression in students' views of science could influence the impact of AL classrooms by year in college, indirectly suggesting that AL might be more effective in students in their first years of college. The present study is intended to contribute to the limited evidence found. Before presenting the main findings, details of the methodology used are included as follows.</p> <hd id="AN0175640999-6">3. Methods</hd> <p>The present study investigated the influence of AL on the relationship between students' initial attitudes towards mathematics and their achievement on an introductory calculus course at a large, urban, research intensive (R1) university. It initially included a randomized control trial experiment during the Spring 2019 and Fall 2019 terms to establish strong and reliable evidence. Instructors in the treatment group used an AL curriculum following the Modelling Practices in Calculus (MPC) model. Instructors in the control group, on the other hand, followed a lecture-based model.</p> <hd id="AN0175640999-7">3.1. Modelling practices in calculus</hd> <p>Faculty who followed the MPC model, participated in professional development that included a three-day workshop prior to teaching, and weekly planning meetings throughout the semester to support the model adoption. The MPC model integrates three core elements at its foundation: cooperative learning (Johnson et al., [<reflink idref="bib48" id="ref85">48</reflink>]), social metacognition (Chiu & Kuo, [<reflink idref="bib22" id="ref86">22</reflink>]), and a culturally appropriate learning environment (Ginsberg & Wlodkowski, [<reflink idref="bib38" id="ref87">38</reflink>]). First, in terms of cooperative learning, students work most of the class in groups on a set of notes and learning activities that develop student understanding of core calculus ideas. The notes introduce the key topics of the day with some examples and questions for the groups, and the learning activities contain a set of problems that lead students to reflect on and challenge their understanding of these topics. For example, after just the second class, students are led through the notes through group and whole-class discussions to have an intuitive understanding of limits using visual representations. With these notes, groups are then asked to develop an idea of how to compute limits without the need for a graphical or tabular representation. The associated learning activity presents a question involving a piecewise-function and asks to compute limits without any visual representations. By working cooperatively on the learning activity, students note how they use the domain of the piecewise function to 'visualize' what happens which helps them build an understanding of direct substitution, the first technique for computing limits presented.</p> <p>Second, the MPC model includes social metacognition as an essential element of the class. Opportunities for developing social metacognition are promoted on a typical day of class as students work together to write up and present ideas and solutions to problems they developed in their groups on whiteboards. Group members are asked to monitor each other's thinking and make suggestions to control their group problem solving.</p> <p>Last, MPC's culturally appropriate learning model allows students to try out their ideas in a low-stakes, safe environment, receive ongoing formative feedback from an instructional team, and participate in a community of learners. The instructor promotes a safe learning environment by messaging to students regularly that making mistakes and asking questions are acceptable and a natural part of mathematics. The low-stakes environment is also enhanced as Learning Assistants (LAs), or trained undergraduate classroom facilitators, are integrated into the classroom to support learning with groups and provide valuable information to instructors about student interactions (Otero et al., [<reflink idref="bib70" id="ref88">70</reflink>]). LAs are natural agents of this culturally appropriate model, as their demographics are that of the students, who provide insights and connections from the point of view of a former student in the course.</p> <hd id="AN0175640999-8">3.2. Participants</hd> <p>The sample in this study consisted of a total of 553 students enrolled in Calculus I at a large, urban, research-intensive institution in the US. In the Spring 2019 semester, a total of 168 of these students were randomly assigned to three control and three treatment sections. In the subsequent semester, the number of sections increased due to semester enrolment trends and the gradual AL curriculum implementation design. In the Fall 2019 semester, the total of students participating in this study then expanded to a total of 385 students randomly assigned to six control and six treatment sections.</p> <p>Additionally, students' demographics data was reported by students to the university and collected at the time of course enrolment. A breakdown of the number of students by treatment group and the demographics of all participating students, can be seen in Table 1.</p> <p>Table 1. Demographics by treatment group.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td /><td>Treatment</td><td>Control</td><td /><td>Treatment</td><td>Control</td></tr><tr><td>(<italic>N</italic> = 286)</td><td>(<italic>N</italic> = 267)</td><td>(<italic>N</italic> = 286)</td><td>(<italic>N</italic> = 267)</td></tr></thead><tbody><tr><td><italic>Gender</italic></td><td char="(" /><td char="(" /><td><italic>STEM choice</italic></td></tr><tr><td> Female</td><td char="(">147 (51.4)</td><td char="(">117 (43.8)</td><td>Non-STEM</td><td char="(">50 (17.5)</td><td char="(">45 (16.9)</td></tr><tr><td> Male</td><td char="(">119 (41.6)</td><td char="(">132 (49.4)</td><td>STEM</td><td char="(">216 (75.5)</td><td char="(">204 (76.4)</td></tr><tr><td> Missing/NA</td><td char="(">20 (7.0)</td><td char="(">18 (6.7)</td><td>Missing/NA</td><td char="(">20 (7.0)</td><td char="(">18 (6.7)</td></tr><tr><td><italic>Precalculus proficiency</italic></td><td><italic>Race/Ethnicity</italic></td></tr><tr><td> High</td><td char="(">75 (26.2)</td><td char="(">78 (29.2)</td><td>African Am.</td><td char="(">15 (5.2)</td><td char="(">16 (6.0)</td></tr><tr><td> Low</td><td char="(">186 (65.0)</td><td char="(">164 (61.4)</td><td>Asian/Pac. I.</td><td char="(">12 (4.2)</td><td char="(">8 (3.0)</td></tr><tr><td> Missing/NA</td><td char="(">25 (8.7)</td><td char="(">25 (9.4)</td><td>Hispanic</td><td char="(">185 (64.7)</td><td char="(">199 (74.5)</td></tr><tr><td><italic>Class Standing (Year in College)</italic></td><td>White</td><td char="(">32 (11.2)</td><td char="(">15 (5.6)</td></tr><tr><td> Freshman</td><td char="(">93 (32.5)</td><td char="(">79 (29.6)</td><td>Others</td><td char="(">22 (7.7)</td><td char="(">11 (4.1)</td></tr><tr><td> Sophomore</td><td char="(">86 (30.1)</td><td char="(">93 (34.8)</td><td>Missing/NA</td><td char="(">20 (7.0)</td><td char="(">18 (6.7)</td></tr><tr><td> Junior</td><td char="(">60 (21.0)</td><td char="(">49 (18.4)</td><td /><td char="(" /><td char="(" /></tr><tr><td> Senior</td><td char="(">27 (9.4)</td><td char="(">27 (10.1)</td><td /><td char="(" /><td char="(" /></tr><tr><td> Others/NA</td><td char="(">20 (7.0)</td><td char="(">19 (7.1)</td><td /><td char="(" /><td char="(" /></tr></tbody></table> </ephtml> </p> <p>1 Note<emph>.</emph> In parentheses: percentages of students in each category in both semesters.</p> <hd id="AN0175640999-9">3.3. Measures and procedure</hd> <p></p> <hd id="AN0175640999-10">3.3.1. Measures</hd> <p>Student attitudes were measured using the Attitudes towards Mathematics Inventory (ATMI) developed by Tapia and Marsh ([<reflink idref="bib92" id="ref89">92</reflink>]). This survey is composed of 40 items measuring the four subscales described in the previous section of this study: enjoyment (10 items), motivation (5 items), self-confidence (15 items), and value (10 items). Eleven items of this survey were reversed-coded later on for data analysis. Additionally, at the end of the semester, students' final grades were collected and converted to a 100-point scale (A + = 98, A = 94.5, A– = 92, B + = 88, B = 84.5, B– = 81, C + = 78, C = 74.5, C– = 71, D + = 68, D = 64.5, D– = 61, F = 40). This scale was previously used in a study (Sonnert et al., [<reflink idref="bib87" id="ref90">87</reflink>]) on calculus performance that included student attitudes as a predictor, yet not accounting for innovative instructional strategies as active learning.</p> <hd id="AN0175640999-11">3.3.2. Procedures</hd> <p>In order to obtain the final sample in this study, students enrolled in multiple, 80-seat (twice the normal size) sections of introductory calculus, chosen to fit their schedules as they normally would. Instructor names were invisible to students throughout this enrolment process. Two days prior to the beginning of each term, each of these 80-seat sections were then split into two 40-seat sections by assigning each student at random to one of either a treatment (MPC) or control (non-MPC lecture-based traditional instruction) section. After this random assignment was completed, students were still allowed to change sections prior to the enrolment deadline.</p> <p>In the Spring semester, a total of 261 students were randomly assigned to ten sections within the study, with 130 students of these in five treatment sections and 131 students in five control sections. In the Fall 2019 Semester, a total of 533 students were randomly assigned to 16 sections within the study, with 271 students in eight treatment sections and 258 students in eight control sections. Since only sections with matching schedules (same day/time teaching) were included in this study, the final sample included three sections per treatment in the Spring 2019, and six sections per treatment in the Fall 2019 semester. A group of students, no larger than 21% of the sample, were not part of the original random assignment. This group included students from other sections who decided to enrol in RCT sections after the split and prior to the enrolment deadline. However, these students enrolled evenly in control and treatment sections. The number of students per treatment per semester in the final sample can be seen in Table 1. Students were asked to complete the ATMI and the PCA survey, at the beginning (first week of classes) and end of the semester (last two weeks of classes). Surveys were administered by the instructors, following a protocol that involved ensuring students their participation was not going to influence their grade in any way.</p> <hd id="AN0175640999-12">3.3.3. Missingness</hd> <p>The overall unweighted unit response rate for both semesters was 82.7% for pre-surveys, 66.8% for post surveys, and 64.8% for students submitting both surveys. A breakdown of survey response rates by semester is presented in Table 2.</p> <p>Table 2. Initial enrolment and survey-response rates by treatment section.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td /><td>Control</td><td>Treatment</td></tr></thead><tbody><tr><td>Enrollment<xref ref-type="table-fn" rid="tfn2">a</xref></td><td>335</td><td>342</td></tr><tr><td>Pre-survey<xref ref-type="table-fn" rid="tfn3">b</xref></td><td>248 (74.0)</td><td>263 (76.9)</td></tr><tr><td>Post-survey<xref ref-type="table-fn" rid="tfn3">b</xref></td><td>178 (53.1)</td><td>235 (68.7)</td></tr><tr><td>Both surveys<xref ref-type="table-fn" rid="tfn3">b</xref></td><td>156 (46.6)</td><td>204 (59.6)</td></tr></tbody></table> </ephtml> </p> <ulist> <item>2 End of the semester enrolment count.</item> <item>3 Number of students answering at least one item of the respective survey (%).</item> </ulist> <p>A total of 82 (8.8%) out of 924 pre- and post-surveys collected were partially completed (at least one item completed). Excluding blank surveys, item non-response rate for both treatment and control groups for each item was less than 3.4% for both pre- and post-surveys. When including blank pre-ATMI surveys later paired with partially completed post-surveys, on the other hand, the percentage of missing values across the 40 pre-survey items of collected surveys ranged between 9% to 14%. However, this percentage was much higher for the post-survey items, ranging between 18% to 37%. This imbalance was mainly explained by high rates of student attrition, especially in the control section.</p> <p>Due to differences in attrition rates between control and treatment sections, missing data was considered missing not completely at random (Rubin, [<reflink idref="bib82" id="ref91">82</reflink>]). Potential loss of statistical power and biased estimates due to this missingness were addressed using a multiple imputation (MI) algorithm (Kang, [<reflink idref="bib51" id="ref92">51</reflink>]). The extensively validated expectation-maximization with bootstrapping (EMB) MI algorithm AMELIA II was used to impute unit and item non-response (Honaker et al., [<reflink idref="bib45" id="ref93">45</reflink>]). MI was carried out considering the pre–post design as a time series, using the pre-PCA survey results as a covariate. Since the percentage of missing data was less than 30%, using over 30 iterations was considered appropriate (White et al., [<reflink idref="bib94" id="ref94">94</reflink>]). Although assumptions of normality were violated for each survey item, given the sample size in this study and since these deviations were slight, the EMB algorithm was assumed to be robust against these violations (Demirtas et al., [<reflink idref="bib29" id="ref95">29</reflink>]).</p> <hd id="AN0175640999-13">3.4. Data analysis</hd> <p>Four different strategies were used to analyze the data: a correlation analysis, a least square dummy variable (LSDV) fixed effects model, an optimal cutoff analysis, and an analysis of mean differences for students identified with low and high initial ATM. First, a correlational analysis was carried out to understand the strength of association between ATM and achievement overall and by demographics. Student achievement and ATMI scores were both considered continuous interval variables. Pearson product moment correlation coefficient preferred, since sample sizes were considered large enough to assume normal distribution (Bujang & Baharum, [<reflink idref="bib14" id="ref96">14</reflink>]). Since missingness was addressed using MI, correlation coefficients were pooled using Fisher's Z transformation to normalize data before using Rubin's rules (Enders, [<reflink idref="bib31" id="ref97">31</reflink>], p. 220).</p> <p>Second, it was expected that instructors' differences might have led to slight variations in MPC implementation. The lack of direct measures of fidelity of implementation is a limitation of this study, however, as previously indicated, weekly supporting meetings with instructors, and professional development workshops contributed to control for this issue. Additionally, a least square dummy variable (LSDV) fixed effects model, preferred over mixed-effect models (McNeish & Kelley, [<reflink idref="bib64" id="ref98">64</reflink>]; McNeish & Stapleton, [<reflink idref="bib65" id="ref99">65</reflink>]), accounted for differences in sections.</p> <p>The response variable was students' final grades (100-point scale), the only continuous predictor was students' initial ATMI scores, the remaining predictors were all categorical: treatment (0:Non-MPC sections; 1: MPC sections), gender (0:male students, 1:female students), year in college (0: freshmen, 1: sophomore, 3: others), and STEMdeclared (0:non-STEM and 1:STEM). The section variable included to account for differences in instructors had 12 levels in the MPC sections and 12 levels in the non-MPC sections.</p> <p>Following findings reported in the literature review (Sonnert et al., [<reflink idref="bib87" id="ref100">87</reflink>]), interactions between treatment and initial ATM were added to the model. Additional interactions were also included between treatment and each of the following variables: gender, year in college, and STEMdeclared. Although evidence of these additional interactions was not found, each variable was expected to have large main effects, thus likely to interact with treatment. Additionally, their inclusion was expected to expand the understanding of the relationship among these variables (Harrell, [<reflink idref="bib44" id="ref101">44</reflink>]). In terms of variable selection, since this study was exploratory, and there were a priori reasons to assume the initial variables were all relevant, the full model was preferred over stepwise methods. Furthermore, stepwise methods were also avoided to minimize bias in parameter estimation, and error inflation (Harrell, [<reflink idref="bib44" id="ref102">44</reflink>]).</p> <p>Third, from previous studies the effect of ATM on achievement was expected to be polarized (Sonnert et al., [<reflink idref="bib87" id="ref103">87</reflink>]). An optimal cutpoint analysis using the Youden index metric (Yin & Tian, [<reflink idref="bib96" id="ref104">96</reflink>]) was conducted on the control section as baseline to identify groups of students with low and high attitudes. This cutoff optimized the likeliness of misclassifying students' success based on their initial ATM. Students, with initial ATMI scores lower than this cutoff, were considered more likely to fail the class, based on the data collected from the non-MPC sections. Results of this analysis were also compared to the cutoff score analysis based on equal-frequency discretization to check for consistency. The optimal cutpoint analysis was conducted using the cutpointr R package (Thiele & Hirschfeld, [<reflink idref="bib93" id="ref105">93</reflink>]).</p> <p>Last, after identifying groups with low and high initial attitudes based on the previously determined optimal cutoff score, the ATMI scale was considered a continuous interval variable, given the Likert scale nature of the items and large sample size of this study (Carifio & Perla, [<reflink idref="bib15" id="ref106">15</reflink>]). This assumption followed previous studies (Asante, [<reflink idref="bib4" id="ref107">4</reflink>]; Karjanto, [<reflink idref="bib52" id="ref108">52</reflink>]; Primi et al., [<reflink idref="bib76" id="ref109">76</reflink>]) and was supported by two main theoretical positions. On one hand, the ATMI scale was considered to have no true zero. Each item in the ATMI survey was scored with a minimum of 1 (strongly disagree) and a maximum of 5 (strongly agree). A score of 3, corresponding to the 'Neutral' response, was also considered a student attitude and not the lack of it. When adding each item's score, the overall ATMI scale ranged accordingly between 40 and 200.</p> <p>On the other hand, the assumption of equal distance between points was deemed to be reasonable given the type of Likert scale involved (Strongly Disagree, Disagree, Neutral, Agree, and Strongly Agree). Furthermore, confirmatory analyses on the ATMI survey (Lim & Chapman, [<reflink idref="bib57" id="ref110">57</reflink>]; Ngurah & Lynch, [<reflink idref="bib66" id="ref111">66</reflink>]) suggest that the contribution of each item to each scale (overall, motivation, enjoyment, self-confidence, and value scale) is fairly homogeneous. Since Likert items on each scale in this study were not examined individually, but as summated scales, this homogeneity prevented certain items from over or under representation of the scale. Therefore, 1-point differences in scores were assumed to be similar.</p> <p>Limitations associated with the midpoint choice of neutral response (Chyung et al., [<reflink idref="bib23" id="ref112">23</reflink>]) and the presence of small differences in loadings from item to item in confirmatory analyses (León-Mantero et al., [<reflink idref="bib56" id="ref113">56</reflink>]) were assumed to be controlled by the robustness of the summated scales, given the sample size, the number of points in each item, and the number of items in each scale in this study (Carifio & Perla, [<reflink idref="bib15" id="ref114">15</reflink>]; Pell, [<reflink idref="bib73" id="ref115">73</reflink>]). Finally, multiple ANCOVAS were then conducted separately in two groups: low and high initial ATM. Results were used to compare students' achievement between MPC and Non-MPC sections, while controlling for initial ATMI scores, and semester of enrolment. Since the nature of this study was exploratory, corrections for multiplicity were not conducted to prevent error type II inflation (Streiner, [<reflink idref="bib89" id="ref116">89</reflink>]). All statistical analysis, unless otherwise indicated, were conducted in R, using the stats package (v4.0.2; R Core Team, [<reflink idref="bib79" id="ref117">79</reflink>]).</p> <hd id="AN0175640999-14">4. Results</hd> <p></p> <hd id="AN0175640999-15">4.1. Correlation analysis</hd> <p>Pearson product-moment correlation coefficients were computed to measure the strength of relationship between student attitudes towards mathematics, as measured by the ATMI scores, at the beginning of the semester and their course achievement, as measured by their total grade at the end of the semester. Correlation coefficients were pooled from the multiple imputed datasets using Fisher's Z transformation and Rubin's rules. Results of this analysis are presented in Table 3.</p> <p>Table 3. Pearson's pooled correlation coefficients of ATMI pre-scores and final grade by demographics.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Group</td><td>df</td><td>MPC<xref ref-type="table-fn" rid="tfn4">a</xref></td><td>df</td><td>Non-MPC<xref ref-type="table-fn" rid="tfn4">a</xref></td></tr></thead><tbody><tr><td><italic>Overall</italic></td><td>265</td><td char=".">0.193</td><td>248</td><td char=".">0.238</td></tr><tr><td><italic>Gender</italic></td></tr><tr><td> Female</td><td>146</td><td char=".">0.135</td><td>116</td><td char=".">0.252</td></tr><tr><td> Male</td><td>118</td><td char=".">0.274</td><td>131</td><td char=".">0.260</td></tr><tr><td> 1–2 Years</td><td>162</td><td char=".">0.118</td><td>150</td><td char=".">0.224</td></tr><tr><td> > 2 Years</td><td>79</td><td char=".">0.340</td><td>65</td><td char=".">0.247</td></tr><tr><td><italic>STEM declared</italic></td></tr><tr><td> STEM</td><td>215</td><td char=".">0.188</td><td>203</td><td char=".">0.270</td></tr><tr><td> Non-STEM</td><td>49</td><td char=".">0.244</td><td>44</td><td char=".">0.130</td></tr></tbody></table> </ephtml> </p> <ulist> <item>4 Pearson's correlations coefficients pooled using Fisher's Z transformation and Rubin's rules. df = N−1. All coefficients were statistically significant (<emph>p</emph> <.001).</item> <item>5 Based on a cutoff of 60 credits.</item> </ulist> <p>As shown in Table 3, correlation between student attitudes and achievement in both MPC and Non-MPC sections was found to be small-to-moderate (Cohen, [<reflink idref="bib25" id="ref118">25</reflink>]). Students' attitudes in MPC sections were slightly less correlated to achievement. When considering specific groups by gender, year in college, and STEM declared. When examining gender, attitudes were significantly less correlated to achievement for MPC female students than in non-MPC sections. On the other hand, correlations for male students were similar in both MPC and Non-MPC sections.</p> <p>Regarding year in college, attitudes were correlated to achievement in a similar way for freshman and sophomore students in both types of instruction with significantly less correlation in MPC students. In senior students, on the other hand, this situation was reversed. Finally, correlation between students' attitudes and achievement was mixed for STEM and Non-STEM students. Correlations were significantly lower for MPC sections in STEM students, but higher in non-STEM students. Although most correlations were found to be small-to-moderate, differences in the strength of association by treatment confirmed the need to further investigate these relationships using a regression analysis.</p> <hd id="AN0175640999-16">4.2. Multiple regression analysis</hd> <p>A least square dummy variable (LSDV) fixed effects model was used to predict students' final grade based on students' ATMI scores at the beginning of the semester, type of instruction (MPC or Traditional), gender (female or male) and year of college (freshman, sophomore, junior or senior). The Section variable was considered a fixed effect to control for between-variability due to differences between instructors or groups of students who might attend different class days. Following a recent study (Sonnert et al., [<reflink idref="bib86" id="ref119">86</reflink>]) absence of non-linearity in the relationship between initial ATM and Grade was assumed. Basic descriptive statistics and regression coefficients of this model are shown in Table 4.</p> <p>Table 4. Descriptive statistics and regression coefficients of LDSV model.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td /><td>Estimate<xref ref-type="table-fn" rid="tfn6">a</xref></td><td>std.error</td><td><italic>t</italic></td><td>conf.low</td><td>conf.high</td></tr></thead><tbody><tr><td>(Intercept)</td><td char=".">78.24</td><td char=".">3.22***</td><td char=".">24.33</td><td char=".">71.92</td><td char=".">84.56</td></tr><tr><td>ATMI</td><td char=".">3.23</td><td char=".">1.02**</td><td char=".">3.18</td><td char=".">1.23</td><td char=".">5.23</td></tr><tr><td>ATMI:MPC</td><td char=".">−0.52</td><td char=".">1.33</td><td char=".">−0.39</td><td char=".">−3.13</td><td char=".">2.09</td></tr><tr><td>ClassOthers</td><td char=".">−4.60</td><td char=".">2.35.</td><td char=".">−1.96</td><td char=".">−9.22</td><td char=".">0.01</td></tr><tr><td>ClassSophomore</td><td char=".">−4.89</td><td char=".">2.16*</td><td char=".">−2.26</td><td char=".">−9.13</td><td char=".">−0.64</td></tr><tr><td>GenderFemale</td><td char=".">2.74</td><td char=".">1.83</td><td char=".">1.50</td><td char=".">−0.85</td><td char=".">6.33</td></tr><tr><td>STEMSTEM</td><td char=".">4.54</td><td char=".">2.36.</td><td char=".">1.93</td><td char=".">−0.09</td><td char=".">9.17</td></tr><tr><td>MPC</td><td char=".">9.82</td><td char=".">5.69.</td><td char=".">1.73</td><td char=".">−1.36</td><td char=".">21.00</td></tr><tr><td>MPC:ClassOthers</td><td char=".">−1.22</td><td char=".">3.19</td><td char=".">−0.38</td><td char=".">−7.49</td><td char=".">5.06</td></tr><tr><td>MPC:ClassSophomore</td><td char=".">0.22</td><td char=".">3.00</td><td char=".">0.07</td><td char=".">−5.68</td><td char=".">6.12</td></tr><tr><td>MPC:GenderFemale</td><td char=".">−0.89</td><td char=".">2.49</td><td char=".">−0.36</td><td char=".">−5.79</td><td char=".">4.00</td></tr><tr><td>MPC:STEMdeclaredSTEM</td><td char=".">−3.58</td><td char=".">3.28</td><td char=".">−1.09</td><td char=".">−10.03</td><td char=".">2.87</td></tr></tbody></table> </ephtml> </p> <p>6 ATMI scores are normalized by z-scores. Signif. Codes: <emph>p</emph> <.1, *<emph>p</emph> <.05, **<emph>p</emph> <.01, ***<emph>p</emph> <.001</p> <p>For the initial model, about 11% of the variance in final grades were accounted for by the model (adjusted r<sups>2</sups> = 0.107). Such a low percentage was partially explained by excluding other relevant predictors of achievement such as mathematics preparation or students' sense of belonging. ATMI initial scores were found to have a significant effect on Grade (<emph>p</emph> < 0.05), leading to an increase of 1-SD in ATMI initial scores per 3.23-point mean increase (SE = 1.02) in final grade in non-MPC sections. This increase in grade was found to be similar to the 3.15-point increase reported in a previous nationwide study (Sonnert et al., [<reflink idref="bib86" id="ref120">86</reflink>]).</p> <p>Since the year in college factor had four different levels, an omnibus F test was conducted, finding a significant effect (F = 8.7265; <emph>p</emph> < 0.0002). None of the interactions included in the model (MPC interaction with gender, year in college, and STEMdeclared) were significant.</p> <p>It is worth noting that the increase in grade in the model was found to be not significant, when controlling for all variables. Given the large standard error of the predictor, this result does not imply that AL had no impact on students' grade, but that follow-up studies are needed to better understand this impact. The model also failed to reflect the differences in gender and intended career choice previously reported in the correlation analysis.</p> <p>When visually inspecting residual plots, the existence of a heavy-tailed distribution for the dependent variable was identified. Although estimation of model coefficients was robust against deviations from normality when sample size is reasonably large (Pek et al., [<reflink idref="bib72" id="ref121">72</reflink>]), this distribution could imply that data points in the tails were excessively penalized. The extreme value analysis reported in the following section was expected to contribute to measuring the impact of this penalization.</p> <hd id="AN0175640999-17">4.3. Achievement in lower attitudes toward mathematics inventory scores</hd> <p></p> <hd id="AN0175640999-18">4.3.1. Optimal cut-off score</hd> <p>In order to identify groups of students with extreme values of initial ATM, an optimal cut point analysis using the Youden index as a metric was conducted on the control section as a baseline to identify students likely to pass the course as predicted by their initial ATMI score. A maximum value of 0.25 for the Youden index was achieved by setting the cut-off score for a passing grade in non-MPC sections at a raw ATMI initial score of 131 (sensitivity 69.6%, specificity 55.8%), corresponding to a mean score of 3.28 and a z-score of −0.28. Low correlation between ATMI scores and achievement led to a relatively low (AUC 0.62) accuracy of the cutoff in discriminating between students who pass or do not pass the class. However, the cutoff captured a high proportion of students with lower grades in both sections. This cutoff also seems to appropriately differentiate students earning higher grades (A or B), implying an extremely low probability of earning these grades with an initial ATMI score lower than this cutoff. Furthermore, equal-frequency discretization analysis led to similar results.</p> <p>It is important to notice that this cutoff score should not be used to primarily predict students' performance. The sensitivity of the cutoff score clearly discourages such interpretation, as it estimates that this score would misclassify 30% of students with low ATMI scores as failing the class in the control sample. The cutoff score in this study was used only as an optimal score given the control sample data to identify groups of students with extreme values of ATM.</p> <hd id="AN0175640999-19">4.3.2. Differences by gender and year of college for students with low attitudes towards math...</hd> <p>After identifying groups of students with lower initial ATM in the control sections, multiple two-tailed unpaired t-tests were conducted to investigate differences in final grade between MPC and Non-MPC sections. Using the previously found cut-off score (ATMI < 131), results were summarized in Table 5.</p> <p>Table 5. Final grade means, standard errors and effect sizes between MPC and Non-MPC sections by demographics for students under the optimal cutoff score (ATMI < 131).</p> <p> <ephtml> <table><thead valign="bottom"><tr><td /><td>MPC</td><td>Non-MPC</td><td>Mean Differences</td></tr><tr><td>n<sub>1</sub></td><td>M</td><td>SD</td><td>n<sub>2</sub></td><td>M</td><td>SD</td><td><italic>t</italic><xref ref-type="table-fn" rid="tfn7">a</xref></td><td>Δ</td><td>SE</td><td>LCI</td><td>UCI</td><td>d</td></tr></thead><tbody><tr><td><italic>Low ATM</italic></td></tr><tr><td> Female</td><td>45</td><td char=".">80.72</td><td char=".">12.45</td><td>46</td><td char=".">73.71</td><td char=".">12.17</td><td char=".">2.74*</td><td char=".">7.01</td><td char=".">2.56</td><td char=".">1.91</td><td char=".">12.10</td><td char=".">0.81</td></tr><tr><td> Male</td><td>35</td><td char=".">76.82</td><td char=".">14.34</td><td>30</td><td char=".">71.64</td><td char=".">18.45</td><td char=".">1.22</td><td char=".">5.18</td><td char=".">4.24</td><td char=".">−3.34</td><td char=".">13.70</td><td char=".">0.45</td></tr><tr><td> Freshm/Soph</td><td>45</td><td char=".">80.81</td><td char=".">14.12</td><td>44</td><td char=".">70.68</td><td char=".">15.94</td><td char=".">3.2*</td><td char=".">10.13</td><td char=".">3.17</td><td char=".">3.82</td><td char=".">16.44</td><td char=".">0.96</td></tr><tr><td> Jun/Senior</td><td>35</td><td char=".">76.36</td><td char=".">12.39</td><td>32</td><td char=".">75.94</td><td char=".">12.85</td><td char=".">0.13</td><td char=".">0.41</td><td char=".">3.16</td><td char=".">−5.92</td><td char=".">6.75</td><td char=".">0.05</td></tr><tr><td><italic>High ATM</italic></td></tr><tr><td> Female</td><td>88</td><td char=".">83.01</td><td char=".">13.87</td><td>59</td><td char=".">82.04</td><td char=".">11.98</td><td char=".">0.45</td><td char=".">0.97</td><td char=".">2.16</td><td char=".">−3.30</td><td char=".">5.25</td><td char=".">0.12</td></tr><tr><td> Male</td><td>75</td><td char=".">83.99</td><td char=".">9.25</td><td>82</td><td char=".">79.26</td><td char=".">14.95</td><td char=".">2.41*</td><td char=".">4.73</td><td char=".">1.96</td><td char=".">0.85</td><td char=".">8.61</td><td char=".">0.53</td></tr><tr><td> Freshm/Soph</td><td>118</td><td char=".">83.91</td><td char=".">12.34</td><td>107</td><td char=".">81.96</td><td char=".">12.69</td><td char=".">1.16</td><td char=".">1.95</td><td char=".">1.67</td><td char=".">−1.35</td><td char=".">5.25</td><td char=".">0.23</td></tr><tr><td> Jun/Senior</td><td>45</td><td char=".">82.40</td><td char=".">11.22</td><td>34</td><td char=".">75.74</td><td char=".">15.44</td><td char=".">2.12.</td><td char=".">6.65</td><td char=".">3.14</td><td char=".">0.36</td><td char=".">12.95</td><td char=".">0.73</td></tr></tbody></table> </ephtml> </p> <p>7 Signif. Codes: <emph>p</emph> <.1, *<emph>p</emph> <.05, **<emph>p</emph> <.01, ***<emph>p</emph> <.001</p> <p>As can clearly be seen in Table 5, AL did not negatively impact any of the groups considered in the low or high ATM categories. Furthermore, in the low ATM category the effect of AL on achievement was particularly larger for two groups: female students, with a medium-to-large effect size (Cohen's d = 0.81); and students in their first two years of college with a large effect size (Cohen's d = 0.96). The only group where the effect of AL on achievement was not significant in this category was students in their last years of college, with a negligible effect size (Cohen's d = 0.05).</p> <p>It is also worth noting that the positive impact of AL in the achievement of students with high ATM was particularly higher for male students, with a medium effect size (Cohen's d = 0.53); and students in their last years of college, with a medium-to-large effect size (Cohen's d = 0.73). Some of the underlying reasons that could explain these trends are discussed in the following section.</p> <hd id="AN0175640999-20">5. Discussion</hd> <p>The main objective of this study was to investigate the relationship between attitudes towards mathematics (ATM) and achievement in two different college calculus settings: active learning (AL) and lecture-based (LB) classrooms. Previous work on this relationship has mainly been limited to LB instruction, and paid little attention to the roles played by gender, year in college, and low initial ATM. This study was intended to contribute to a better understanding of these issues through the results from an initial RCT design with a sample of 535 students enrolled in control and treatment sections during the fall and spring semester of 2019. Treatment sections adopted the Modelling Practices in Calculus (MPC) approach which incorporates AL strategies and enhances learning facilitation by Learning Assistants (Otero et al., [<reflink idref="bib70" id="ref122">70</reflink>]), and control sections were predominantly lecture-based (LB) classrooms. Data collected from this implementation included a measure of students' ATM, using the ATMI survey (Tapia, [<reflink idref="bib90" id="ref123">90</reflink>]; Tapia & Marsh, [<reflink idref="bib91" id="ref124">91</reflink>]), final grades, and certain demographics. After a multiple imputation algorithm was used to address data missingness issues, the analysis of this data consisted of a correlational analysis, a fixed-effect model, and a mean differences analysis of students with low ATM, using a referential cutoff score found by metric optimization. A brief discussion of this analysis is presented below.</p> <hd id="AN0175640999-21">5.1. Impact of initial attitudes towards mathematics on Achievement in active learning classr...</hd> <p>As explained at the beginning of this study, AL approaches have proven to positively impact both student achievement (Freeman et al., [<reflink idref="bib35" id="ref125">35</reflink>]) and ATM (Castillo et al., [<reflink idref="bib17" id="ref126">17</reflink>]). Given the existence of these links, it was reasonable to expect that AL would have a moderation effect on students' initial ATM and achievement. The collaborative engagement students can experience in AL classrooms with peers and instructors was expected to lead to increased positive interdependence (Johnson et al., [<reflink idref="bib48" id="ref127">48</reflink>]), sense of belonging (Goodenow, [<reflink idref="bib40" id="ref128">40</reflink>]), and social modelling (Bandura, [<reflink idref="bib7" id="ref129">7</reflink>]), providing students with a better support system to overcome the detrimental impact of low ATM on their learning process, and thus reinforce their achievement.</p> <p>The small positive correlations found between students' initial ATM and their achievement in MPC and Non-MPC classrooms were within the range of values previously reported (House, [<reflink idref="bib46" id="ref130">46</reflink>]). A significant difference between both classroom settings was found, indicating a lower correlation in AL classrooms. The direction of this correlation suggests the existence of a supportive role, in which AL helps students with lower ATM to earn higher grades than students in LB classrooms. Lastly, when considering specific groups of students, the correlation analysis suggested that AL's supporting role is particularly enhanced for female students, and students in their first two years of college.</p> <p>The regression model confirmed the supportive role of AL on students learning processes including a significant increase in final grade, when controlling for ATM, gender, year in college, and STEM intent. Although we expected to find significant interactions between the control variables, none were identified. These results do not necessarily contradict the correlational analysis findings, as the lack of significance of a t-test associated with a regression coefficient cannot be directly interpreted (Harrell, [<reflink idref="bib44" id="ref131">44</reflink>]). We believe that two main shortcomings related to the regression model might have hindered its ability to accurately capture these interactions.</p> <p>On the one hand, the available data did not include important predictors of achievement including variables associated with mathematics preparation. This restriction could explain the model's low goodness-of-fit (adjusted r<sups>2</sups> = 0.117) that prevented the model from accurately reflecting relationships between its variables (Fan & Huang, [<reflink idref="bib34" id="ref132">34</reflink>]). On the other hand, the existence of a heavy-tailed response variable distribution could have led to underrepresenting the influence of AL for low values of ATM (Catoni, [<reflink idref="bib18" id="ref133">18</reflink>]). An extension of our study to a larger sample and richer dataset that includes additional predictors of achievement will contribute to clarifying this issue. While addressing the limitations associated with the regression model, results from our analysis of students with low ATM not only helped to confirm some of the trends identified in the correlation analysis, but also provided further insight into the role that students' demographics might be playing.</p> <hd id="AN0175640999-22">5.2. Active learning as leverage to low initial attitudes towards mathematics</hd> <p>The mean difference analysis confirmed expectations of a supportive role of AL in students with low ATM (Sonnert at el., 2015). In terms of demographic variables, as shown in Table 5, the most compelling findings were the large effect sizes identified in gender and year in college for students with low initial ATM. Our analysis showed that well-designed AL strategies such as MPC can have a particularly large effect (d = 0.81) on female students with low initial ATM. This effect size seems to initially contradict a recent study by Sonnert at el. (2020) who found that ATM's influence on achievement was similar for both male and female students. However, the regression model used by these authors was not focused on active learning classrooms nor included interactions specifically measured on students with extreme values of ATM, as we did in our study.</p> <p>The large effect size on MPC female students with low ATM found in our study translates to an increase of over half a letter on final grade when compared to traditional lectures. Such a substantial increase is of particular significance since several studies have shown that female students enter STEM careers in college with lower ATM than their male counterparts (Good et al., [<reflink idref="bib39" id="ref134">39</reflink>]; Saxe et al., [<reflink idref="bib84" id="ref135">84</reflink>]). Furthermore, in terms of persistence, female students' odds of being discouraged from continuing in calculus have been estimated to be 1.5 times greater than that of their male counterparts (Ellis et al., [<reflink idref="bib30" id="ref136">30</reflink>]).</p> <p>The impact of AL on female students whose initial attitudes might otherwise prompt them to fail the class, could contribute to create more equitable Calculus classrooms in college. Results from a recent study (Laursen et al., [<reflink idref="bib55" id="ref137">55</reflink>]) that compared inquiry-based learning (IBL) college mathematics to lecture-based classrooms are consistent with the supporting role we found AL's classrooms are playing in female students. In this study, differences between female and male students found in mastery gain in lecture-based classrooms vanished in IBL classrooms.</p> <p>The underlying mechanism that explains the influence of AL on female students has not been fully explained. However, it is possible that the enhanced collaboration in AL classrooms provides female students with support they normally do not encounter in more lecture-based classrooms. This explanation is aligned with several studies that have shown that women, in general, respond more favourably to collaborative than competitive environments (Ash et al., [<reflink idref="bib5" id="ref138">5</reflink>]; Niederle & Vesterlund, [<reflink idref="bib67" id="ref139">67</reflink>]). Studies that support this explanation in calculus are still needed, but studies on college physics confirm that the collaborative components of AL courses might be the main factor responsible for diminishing gaps in conceptual understanding (Lorenzo et al., [<reflink idref="bib59" id="ref140">59</reflink>]), and some attitudinal variables such as students' self-efficacy (Espinosa et al., [<reflink idref="bib32" id="ref141">32</reflink>]).</p> <p>Regarding the achievement of students with low ATM by year in college, as shown in Table 5, AL was clearly more beneficial to those in their first or second year of college. This result is aligned with findings by Savelsbergh et al. ([<reflink idref="bib83" id="ref142">83</reflink>]) in which the effect of innovative instruction was found to be weaker for older students. Laursen et al. ([<reflink idref="bib55" id="ref143">55</reflink>]), also found similar results, where first-year students had greater gains in social and cognitive measures in IBL college mathematics than last-year students. Reported gains by Laursen et al. included mathematical thinking, persistence in solving problems, and collaboration. Although the reasons why these gains were observed were also not fully explained in their study, we suspect that since AL classrooms are still underrepresented in the STEM curriculum, junior and senior students might have a more extended experience with lecture-based courses in STEM, and that this experience makes their transition to AL more difficult. It is also possible that students' different views by year in college regarding collaboration and competition in science (Fuselier & Jackson, [<reflink idref="bib36" id="ref144">36</reflink>]) might be interacting with the benefits of AL classrooms.</p> <hd id="AN0175640999-23">5.3. Limitations and further research</hd> <p>The main limitations of this study were related to its sample size and the existence of unaccounted variables. In the first place, a larger sample size would allow for higher statistical power in each bivariate analysis conducted for each demographic variable and alleviate issues related to deviation from normality of the ATM scale. Having a larger sample would also allow the use of other regression models that account for extreme values such as spline regression (Harrell, [<reflink idref="bib44" id="ref145">44</reflink>]). Similarly, including more participants in the study could have helped to address a limitation related to natural constraints of students' enrolment in college calculus. Giving students the freedom to switch sections after the ATMI survey was administered resulted in a small group of students who were not part of the original random assignment. Although the fact that these students enrolled evenly in AL and traditional calculus sections led to an overall balanced sample, a proximity score matching algorithm (PSM) (Granger et al., [<reflink idref="bib41" id="ref146">41</reflink>]) could have contributed to better control for these changes. The reduced number of observations after PSM would however prevent us from exploring issues related to students' demographics with appropriate power. Expanding our study to additional semesters and other institutions would also help to address this limitation.</p> <p>The authors also recognize the need to incorporate additional variables in a subsequent model. Some of the important unaccounted variables in this study included student mathematics preparation, instructors' characteristics, and measures of student collaboration. First, measuring students' mathematics preparation would have helped to provide a more robust regression model to capture the main interactions investigated. Second, this study assumed that the instructors' weekly professional development and planning meetings in the AL sections were effective in ensuring a high level of fidelity of implementation, yet no associated measures to confirm this assumption were included in the study. To partially address this issue, the effect of the instructor on the interactions was examined through a fixed-effect model with a dummy variable on instructor and its analysis yielded consistent results. Last, other variables such as students' course engagement or their level of collaboration throughout the semester were not available. Qualitative studies that explore students' classroom interactions in groups are also needed to avoid overlooking these and other relevant variables in this process.</p> <hd id="AN0175640999-24">Acknowledgements</hd> <p>We thank the instructors, students, and Learning Assistants involved in this study and their support during administration and collection of data. 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  Group: Date
  Data: 2024
– Name: SourceSuprt
  Label: Sponsoring Agency
  Group: SrcSuprt
  Data: National Science Foundation (NSF)
– Name: NumberContract
  Label: Contract Number
  Group: NumCntrct
  Data: 1832450
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Calculus%22">Calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Attitudes%22">Student Attitudes</searchLink><br /><searchLink fieldCode="DE" term="%22Academic+Achievement%22">Academic Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22College+Mathematics%22">College Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Undergraduate+Students%22">Undergraduate Students</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/0020739X.2022.2150902
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0020-739X<br />1464-5211
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The present paper explores the relationship between attitudes towards mathematics (ATM) and achievement in college calculus in active learning (AL) and lecture-based (LB) classrooms. Previous work on this relationship has mainly been limited to LB instruction, neglecting the impact of innovative approaches such as AL. Less attention has been paid to the roles played in this relationship by gender, year in college, and initial ATM. Results from a sample of 535 undergraduate students enrolled in 9 AL and 9 LB sections are presented. Data included ATMI surveys' responses, final grades, and demographics. Correlation and multiple regression analyses were conducted. The influence of instruction on students with low ATM was also examined. Gender and year in college were the main demographic variables considered. Achievement in AL was found to be less dependent on initial ATM in terms of correlation. AL showed higher gains in grades than LB, when controlling for ATM and demographic variables. Effect sizes of AL instruction on grades of students with low ATM were larger than those of students with higher ATM. Furthermore, AL courses had a large effect size (d = 0.81) on female students with lower ATM, confirming its role as a gender equalizer.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2024
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1413917
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1413917
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/0020739X.2022.2150902
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 22
        StartPage: 759
    Subjects:
      – SubjectFull: Calculus
        Type: general
      – SubjectFull: Student Attitudes
        Type: general
      – SubjectFull: Academic Achievement
        Type: general
      – SubjectFull: College Mathematics
        Type: general
      – SubjectFull: Undergraduate Students
        Type: general
    Titles:
      – TitleFull: Student Attitudes and Achievement in Active Learning Calculus
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Pablo A. Duran
      – PersonEntity:
          Name:
            NameFull: Adam J. Castillo
      – PersonEntity:
          Name:
            NameFull: Charity Watson
      – PersonEntity:
          Name:
            NameFull: Edgar Fuller
      – PersonEntity:
          Name:
            NameFull: Geoff Potvin
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          Name:
            NameFull: Laird H. Kramer
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          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2024
          Identifiers:
            – Type: issn-print
              Value: 0020-739X
            – Type: issn-electronic
              Value: 1464-5211
          Numbering:
            – Type: volume
              Value: 55
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: International Journal of Mathematical Education in Science and Technology
              Type: main
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