Improving Math Homework Practices: Community College Students' Reflections, Schemes and Mind Maps of Mathematics

Download Improving Math Homework Practices: Community College Students' Reflections and more Schemes and Mind Maps Mathematics in PDF only on Docsity! 22 JournAl of DeVelopmenTAl eDuCATIon Abstract: Homework is considered an important aspect of learning mathematics, but little research has considered how students utilize feedback as part of the homework process. This mixed methods, quasi-experimental study examines how community college students in a developmental intermediate algebra course participated in a feedback reflection activity throughout a semester and compares their outcomes with a class that did not engage in this activity. Although developmental math students are often positioned as deficient in skills and motivation, most students took this activity as an opportunity for self-assessment, documenting resources for success and critiquing their work for improvement. These students did not outperform peers on summative course assessments; however, there were differences in their growth as effective learners This paper reports on community college students’ reflections on mathematics homework feedback. Researchers have documented the positive impact of homework on achievement in college mathemat- ics and science courses (Sasser, 1981, Weems, 1998; Yalcin & Kaw, 2011); this is consistent with the well-documented positive impacts of homework in middle and high school (cf. Cooper 1989; Cooper, Lindsey, Nye, & Greathouse, 1998). Homework is especially beneficial when students receive feed back from teachers (Austin, 1976; Elawar & Corno, 1985; Paschal, Weinstein, & Walberg, 1984). Feedback on student work in general has been shown to have effects on achievement, though the effects are not always positive (Hattie & Timperley, 2007; Shute, 2008). Feedback is most effective when students receive “information feedback about a task and how to do it more effectively” (Hattie & Timperley, 2007, p. 84). Yet, as Sadler (1989) notes, this information is only considered to be feedback when it is used for improvement. Students can only use feedback for improvement if they come to understand how teachers judge their work, learn to judge themselves in the same ways, and use these standards as they are producing new work (Sadler, 1989). Thus students must have regular opportunities not only to review feedback, but also to engage in self-assessment, which is highly recommended for community college mathemat- ics students (AMATYC, 2006). Accordingly, the current study examines a homework reflection strategy. The strategy under study was developed through the first author’s work as a facilitator of and participant in professional development focused on effective teaching practices related to homework in developmental math courses. The strategy, which prompted students to complete reflection logs, was designed to address the concern that students were not consistently reflecting on and utilizing the feedback they received on assign- ments. Our primary goal was to understand the extent to which students were able to incorporate reflection into their repertoire of homework prac- tices. The study was conducted in two sections of a community college intermediate algebra course. Using a mixed-methods, quasi-experimental approach, we compared the performance of stu- dents who engaged in reflection with those who did not. Within the experimental section, we used qualitative analyses to describe how students did or did not incorporate reflection into their homework practices. The research questions addressed by this paper follow: 1. How did students utilize the logs as a structure for reflection? 2. How did students make meaning out of the reflection process? 3. Were there differences between the class that participated and the class that did not in terms of course performance and their growth as effective learners (i.e., their skills and habits related to student success)? The first two research questions led to qualita- tive analyses of students’ engagement; the third research question led to a quantitative comparison of student performance. The paper is organized as follows: First, we describe the theoretical perspective that guides the study, and then we outline the research methods, including the context of the two courses, data sources, and methods of analysis. Following, we mara landers professor of mathematics Co-lead of Developmental mathematics education program los medanos College 2700 east leland road pittsburg, CA 94565 mlanders@losmedanos.edu Daniel reinholz research Associate The Center for sTem learning university of Colorado, Boulder Campus Box 393 uCB Boulder, Co 80309 Students’ Reflections on Mathematics Homework Feedback By Mara Landers and Daniel Reinholz Students must have regular opportunities not only to review feedback, but also to engage in self-assessment. Volume 38, Issue 3 • sprIng 2015 23 1999). That is, they have a different role than other participants and they are held responsible for the artifacts produced in practice. Therefore, it is the student that we “zoom in” on (Lerman, 2001) as he or she participates in the practice of homework. Through participation in practice, individu- als negotiate the meaning of their work, which includes learning about the meaning of collective work for participants (Wenger, 1998). This process includes taking ownership of meanings: Some ideas become personally meaningful to students, and they may negotiate meanings, even constructing new ones. For example, students may come to see homework as “busy work” that should be done to satisfy teachers or, alternatively, as an opportunity to learn. When examining students’ perspectives on homework, researchers have given more attention to K-12 students (see Landers 2013b for a summary of research on homework meanings). However, like their younger counterparts, college students do not necessarily view homework as a positive experience. High school and college students have mixed feel- ings about homework, and they admit that they copy in order to get assignments done and to earn the best grades possible (Pope, 2001; Robinson & Kuin, 1999). In the current study we seek to under- stand a specific aspect of students’ participation in the practice of homework: how they incorporate reflection into their existing repertoire of practice and how they make meaning out of this work. present the results of the study, organized by the three research questions. The findings lead us to outline implications for future implementations of the reflection activity and implications for future research. Theoretical Perspective Although most homework research defines home- work as a task, we conceptualize homework as a social practice that students participate in with teachers and others in their lives (e.g., family members or peers). Following Wenger, we define practice as engagement in work or activities done in social contexts that “gives structure and meaning to what we do” (Wenger, 1998, p. 47). Researchers have long studied social practices as central to learning and development, examining how participation in a practice provides opportunities for learn- ing, problem solving, and cognitive and identity development (e.g., Cole, 1996; Lave & Wenger, 1991; Rogoff, 2003). For several decades the field of mathematics education has seen a “social turn” or “the emergence…of theories that see meaning, thinking, and reasoning as products of social activ- ity” (Lerman, 2000, p. 23). Often described as a sociocultural perspective, researchers in this tradi- tion attend to the “socially and culturally situated nature of mathematical activity” (Cobb, 1994, p.13). Researchers who take a sociocultural perspec- tive often argue that the unit of analysis should be a “community of practice,” like the world of the claims processors Wenger studied (Wenger, 1998), or the “figured world” of a school or classroom (Boaler & Greeno, 2000; Holland, Lachicotte, Skinner & Cain, 1998). That is, the unit of analysis is a social group or unit in which individuals engage in collective practice, occupy social positions, and negotiate the meanings of their experiences. However, the nature of homework as a practice leads us to a different analytical focus. Homework is different from other social practices in two ways. First, homework is done across contexts, specifi- cally in a cycle of contexts: Students and homework artifacts move in and out of school, daily, over time. In classrooms, teachers assign tasks, and students may work on assignments with classmates and teachers. The practice continues when students leave school. They work on assignments outside of school, either with peers, family members, or independently. They then return to class to review, finish, and turn in work. Figure 1 illustrates this homework cycle. In the context of the current study, students not only participate in the cycle as described, but they also complete in-depth assignments using the resources of a drop-in lab, where they complete tasks with classmates, student tutors, and math instructors. Second, although students do homework with other people, the students are positioned as the “owners” of homework (Varenne & McDermott, Method Setting and Participants The current study was conducted in two sections of intermediate algebra at a community college in Northern California. With over 12,000 stu- dents, this college is one of three in a district that serves a single county in the San Francisco Bay Area. The college’s student body is diverse along several dimensions, including age, race/ethnic- ity, and socioeconomic status. Although we did not formally collect demographic information in the sample populations, they were generally representative of the college as a whole, in which 18% of students identify as African American, 5% as Asian, 4% as Filipino, 36% as Hispanic, 7% as Multi-Ethnicity, and 27% White. In recent years the college has seen a growth in the number of students enrolling directly after high school and many of these students assess into developmental courses. As is the case at many of California’s com- munity colleges, the majority of students at the college must take developmental math and English courses before they can complete college-level courses needed to transfer to a four-year institu- tion. Approximately 70% of students at the college place into developmental math and 80% place into developmental English. The National Center for Developmental Education (NCDE) defines developmental education programs as supporting “traditional and nontraditional students who have been assessed as needing to develop their skills in order to be successful in college” (NCDE, 2015). Developmental math programs include courses from arithmetic to intermediate algebra, and progressing through this sequence has proven to be a challenge and even a barrier to educational advancement for an alarming number of students (Bonham & Boylan, 2011). Given these issues, the math department at this college created a developmental program We define practice as engagement in work or activities done in social contexts that “gives structure and meaning to what we do.” Figure 1. The homework cycle involves practices inside and outside of school. Volume 38, Issue 3 • sprIng 2015 27 work-habits question were coded for (a) whether or not the students reported changing work hab- its and (b) how they accounted for the changes. Responses to the question about homework not being graded were coded for (a) whether or not students indicated they would do the work and (b) categories of justifications. Quantitative analyses were used to compare the two classes on measures of course performance and to compare the class participants’ growth as effective learners, using the self-assessments given at the beginning and end of the semester. To compare performance, t-tests were used to determine differences in means on exams and course grades. To compare the classes for effective learning growth, chi square analy- sis was used to determine if the distributions of students in the three categories of self-assessment (excellent, satisfactory, and “not there yet”) shifted from the beginning of the semester to the end. The analysis was completed for each item on the self-assessment. Results With little prompting, the majority of the LS class (over 71%) participated in the reflections. Nevertheless, analyses revealed differences in how students engaged with the reflection log sheets and how they negotiated the meaning of the activity. These patterns within the LS class are summarized in response to the first and second research ques- tions, and then the differences between the LS and NS sections (Research Question 3) are addressed. Research Question 1 (reflection participation) LS students were categorized according to three different levels of participation, summarized in Table 3. The students in the nonparticipation group did not submit sufficient reflection materials for analysis, so analyses of reflection practices focused on the first two groups (frequent participation and decreased participation). Students from all groups completed surveys, thus analyses of students’ understandings of reflection come from all groups. Overall, students who participated in reflection received higher grades in the course (see Table 4). Comparative case analyses revealed dif- ferences in exactly how students engaged with reflection. The students with frequent participa- tion routinely answered all of the questions on the reflection sheets, except that they sometimes did not rate themselves for every skill listed. In contrast, the students with decreased participation often submitted incomplete or incorrect logs. For example, some students did not complete all the self-ratings in questions 3 and 4 of the logs, or they would circle/highlight a given skill both as achieved and as needing improvement. There were also differences in how students responded to question 5 on the reflection sheets, which asked them to “explain what you did that led you to be successful on this assignment, or what led you to not be as successful as you may have wanted.” The frequent participation students used this prompt to document their self-assessment as they reflected on their assignments. Six of these ten students routinely described their strengths and weaknesses, and three students focused almost exclusively on critiquing their work. The tenth student in this category was less specific in his responses, though he focused on the need to improve his communication and general study skills. To account for their success, these students discussed using their resources, such as getting help in the lab, or using their classwork. They also referenced specific math content and several of the learning outcomes for the course (e.g., problem solving, communication). They referenced these same topics when discussing areas for improve- ment, especially specific math content, the problem solving outcome, and the communication outcome. Further, students who frequently participated uti- lized the reflection logs to write about, and even negotiate, other aspects of their work, including critiquing the quality of help they received in the lab and questioning how their work had been graded. In contrast, only two students from the decreased participation group exhibited this same level of self-assessment. In accounting for their success, students in the decreased participation group rarely referenced specific math content or the learning outcomes, instead focusing primarily on the resources they used. In describing areas for improvement, their focus was also on resources, describing resources that they should have used. And these students did not use the reflection logs to discuss other aspects of their work. Table 5 shows examples of students’ responses to question 5. Research Question 2 (meanings of reflection) Regardless of their level of participation, survey responses indicated that all students understood the purpose of the reflection activity as self- assessment (see Table 6 for representative student responses). The coding of students’ responses to the question of why they engaged in the reflection activity yielded two main themes: self-assessment and teacher compliance. Of the 10 students who frequently participated, 6 gave self-assessment as Students who frequently participated utilized the reflection logs to write about, and even negotiate, other aspects of their work. Table 3 Categories of Reflection Participation Category N Description Frequent participation 10 These students completed most or all of the logs consistently throughout the semester. These students completed 14a -20 logs each (70% or higher). Decreased participation 10 These students started out completing logs, then decreased in frequency, or stopped turning them in. These students completed 6-13 logs each (30-65%). Nonparticipation 8 These students completed 0, 1, or 2b logs each (0-10%). Note. a Included in this group is one student who initially did not complete logs, but then began to do so midway through the semester. She completed a total of 14 logs. b There were no students who completed 3 – 5 logs. Table 4 Reflection Participation and Final Course Grades Grade Received Category A B C D F Frequent participation 3 4 3 Decreased participation 3 6 1 Nonparticipation 1 4 2 1 28 JournAl of DeVelopmenTAl eDuCATIon their reason, although one of these students also indicated teacher compliance. Two of the ten stu- dents explained that they participated for teacher compliance because they already reflected on their own. The last two indicated teacher compliance as well as other reasons: giving the teacher informa- tion and earning better scores. (This last reason is related to self-improvement, though not clearly linked to self-assessment.) Of the 10 students with decreased participation, 8 took the survey, and 4 of these students gave self-assessment as their reason for participation. Two students indicated self-assessment and teacher compliance, and two did not give responses that explained their rea- sons. Thus, most of the students in the frequent participation group and half of the students in the decreased participation group were classified as taking ownership of the value of reflection for self-assessment, in alignment with how the teacher introduced the practice. Additionally, these students negotiated the idea of participation for teacher compliance. For example, one student who completed all of the logs described his participa- tion: “I have done all the reflections. I did them because they were assigned as homework, but I did not get much out of them. I just wanted to make sure they were turned in. When I got my work returned, that is when I would learn from my mistakes, not by looking at the reflection sheet and trying to put it into words.” Students with decreased participation or no participation provided little information about why they stopped or did not participate, even though the survey directly prompted them to discuss this issue. Only three students responded: one explained that he forgot to do the reflections because he was focused on getting assignments done; one explained that she did not do reflections because she did not do the assignments in the first place; the third student described how his motiva- tion for school in general had decreased. The end-of-semester survey also asked stu- dents in the LS class how likely they were to reflect in this same way on their own in future classes. In the frequent participation group, 9 of 10 students indicated they would continue reflecting; however, four of these stated that they would not reflect at the same level of written work and detail. As one student explained: “most likely [I won’t] go over it as I did this semester, but if I actually have to do it, I would. But I don’t think I would go over it like I did this semester unless I had to, so it was a good thing to Table 5 Sample Responses to Reflection Sheet Question 5 Student Category Sample Responses to Question 5 (accounting for success or lack) Frequent participation Using my previous work as an example helped me with this assignment. I could have done better if I was done with [the skills software] for the exponent properties. Frequent participation Overall I think I understood the problems really well, which led to some success. I do tend to rush through things even when I try not to so I leave things out or inadvertently write the wrong thing down. For #8 I did the equation on my calculator and explained the answer but forgot to put in the process for my solution. Decreased participation I tried doing this assignment by myself, and I got a decent grade, but I should get help next time. I try to do each problem as if it was the first question. I take my time, and then go over it. Decreased participation I did not explain the meaning of things. I went to the math lab and worked on my assignment with two other people in my class but I still didn’t understand some things. Table 6 Examples of Students’ Descriptions of the Purpose of the Reflection Activity Student’s Participation Sample Responses to Survey Question Frequent participation To understand what helped you be successful on each assignment, and what you were not successful on. Areas where you could improve. Decreased participation Maybe for us to see what we need to improve on when we get graded work back and to do better on the next assignment. Nonparticipation I think we have been asked to do this because it gives us an opportunity to evaluate ourselves and to see a visual of our performance in the class. have to do.” The students in the frequent participa- tion therefore took ownership of the reflection by critiquing the way the reflection was structured. In contrast, students in the decreased participation and nonparticipation groups gave mixed responses about future reflection: Some indicated they were not likely to do so (2 in each group), whereas others said they would (4 in each group), though it was ambiguous if they were referring to the reflection log process or to reflection in general. In contradiction to their actual practices, some students with decreased or no participation indicated that they found the reflections to be useful. This indicated that they did not fully take ownership over the activity (cf. Landers, 2013a). Two of these cases are summarized in Table 7 (p. 29), and this issue is examined further in the discussion section. Research Question 3 (comparison of sections) We conducted t-tests to assess whether there were differences in the mean level of performance between the LS and NS groups. (Scores of 0 were not included. These scores represent students who stopped attending class near the end of the semester but did not drop officially. They did not take the final exam.) Comparisons of the two classes revealed no statistically significant dif- ferences in final exam scores, t(48) = 2.011, p = 0.387 (two-tailed) or final course grades, t(52) = 1.421, p = 0.161 (two-tailed). However, the end-of- semester surveys and self-assessments revealed differences in students’ perspectives on homework and self-assessment of several effective learning characteristics. In both the beginning and end- of-semester surveys, students rated themselves as “excellent,” “satisfactory,” or “not there yet” on a set of 16 effective learning characteristics. The surveys were analyzed using chi-squared to compare the LS and NS sections, specifically to document shifts in each class of the distributions of ratings. Relevant findings are summarized in Table 8 (p. 30), rows 3 – 5. For instance, LS students indicated that they were more likely to revise their work and saw homework as a learning opportunity, not just a means of attaining a grade. There were also differences in the sections in terms of their perspectives on homework (see Table 8, row 1, p. 30); however, there were no clear differences in the students’ perspectives on how their work habits had changed during the course of the semester. In fact, in both classes, 75% of the students who completed the end-of-semester survey indicated that they had improved their study habits. Further, qualitative analysis revealed that both groups of students reported the same types of improvements: spending more time on work, being more focused, improving time management, and more frequently asking for help. Volume 38, Issue 3 • sprIng 2015 29 Table 7 Homework Nonparticipation and Examples of Reflection Perspectives Reflection Participation Description of Homework Practices Response to Survey Question Nonparticipation Completed all assignments but routinely missed deadlines. Only completed two reflections. I feel like the reflections was my favorite assignment because I was able to write out how I was feeling at the moment and it really helped me become a better student. Decreased participation Completed most, but not all, course assignments. Completed seven reflections, but filled out the form incorrectly. Only reflected on assignments on which he was highly successful. They do help, I know what my strengths and weaknesses by doing the reflection sheets therefore I know where I need to get help. Discussion Although most of the students in the LS class understood the purpose of reflecting, the students who regularly participated took ownership of the value of reflecting in ways that their classmates did not. These students aligned their claims about the value of the reflection with their actions, but as described previously, they also showed more understanding of the artifact itself and how to complete it. They went beyond the skills listed on each log, regularly describing their specific strengths and weaknesses related to the course content and learning outcomes. Several of these students also used the artifact as a means to negoti- ate other aspects of the classes’ collective work (cf. Wenger, 1998), such as how their assignments were graded and the quality of help they received in the math lab. Although it is not evident why this is so, it is perhaps the case that these students have felt more empowered to negotiate, or even to critique aspects of their experience, than their classmates. Our concern was to understand students’ par- ticipation in practice and their meaning making; yet the findings also point to issues of identity, a concept deeply connected to social practice and meaning making (Wenger, 1998). Specifically, at issue is how students are positioned and how they position themselves in practice (Holland et al., 1998). Through this lens, it becomes clear that the reflection activity positioned those who participated as “successful” and those who did not participate as “not successful.” This is also true for the practice of homework in general, and most other school-related practices. In this context, the students who already included reflection in their repertoire of practice, and those who were willing and able to expand their repertoire to include reflection, were positioned as successful and compliant with the teacher’s expectations. This positioning may have contributed to their willingness to negotiate. Other students were positioned as less suc- cessful. This may explain survey responses from some of the students in the decreased participa- tion and nonparticipation groups indicating that the reflections were very useful, even though they did not actually engage in the reflection activity