Going Over Homework in Mathematics Classrooms: An Unexamined Activity
by Samuel Otten, Beth A. Herbel-Eisenmann & Michelle Cirillo - August 22, 2012
Background: Homework is a key component of students’ school mathematics experiences, especially at the secondary level. Past studies have shown that because a substantial portion of class time is spent going over homework assignments, homework does not remain an at-home activity. Yet, little is known about what takes place during the classroom activity of going over homework.
Purpose: Drawing on systemic functional linguistics, this research note describes the discourse structures of homework review in eight secondary mathematics classrooms. Particular attention is given to the ways in which students participate in this discourse.
Participants: The 8 teachers involved in the study come from various districts in a Midwestern state. Their certifications range from elementary to middle school to secondary mathematics, and their years of teaching experience range from 2 to greater than 20. The students, in Grades 6–10, are racially diverse in some districts and socioeconomically diverse in all participating districts.
Research Design: This study is a qualitative discourse analysis of episodes of classroom interaction.
Data Collection and Analysis: Video recordings of lessons from the 8 participating teachers were made in 1-week segments at four different time points throughout an academic year, yielding 148 such recordings. These videos were transcribed, and all instances of homework review were identified for further analysis. Descriptive statistics (e.g., durations, turn lengths) were compiled, and themes in the structure of the discourse were identified.
Conclusions: Nearly 20% of class time was spent going over homework, with some variation between teachers in how they organized this activity. A commonality, however, was that the discourse in all eight classrooms tended to be structured in a problem-by-problem manner. Implications of this structure are discussed in light of curricular recommendations and past research, which suggests that it may also be beneficial to engage students in discourse that looks across sets of problems for regularities, connections, or key ideas.
The relationship between homework and student achievement has been a focus of research for several decades (see the following reviews: Cooper, Robinson, & Patall, 2006; Paschal, Weinstein, & Walberg, 1984). This research has typically been quantitative and has found positive relationships between achievement and homework, but some have warned that it would be a misinterpretation to endorse homework as a uniformly good thing (Corno, 1996), especially since Lange and Meaney (2011) revealed the potentially traumatizing nature of homework for students. Furthermore, as Trautwein and Koller (2003) pointed out, many of the past correlational studies lacked strong connections to theories of learning and failed to consider important mediating factors. Recent work has attempted to redress these overlooked issues by investigating, for example, the role of parents (Hyde, Else-Quest, Alibali, Knuth, & Romberg, 2006; Van Voorhis, 2011), the role of teachers (Epstein & Van Voorhis, 2001), and the relationship between homework and student emotions (Xu, 2011).
Another important factor to consider is the way in which homework is handled in the classroom, especially given the recent emphasis on quality instructional time (e.g., Sparks, 2011). In school mathematics specifically, going over homework constitutes a significant portion of class time; lessons often begin by reviewing the previous days homework assignment (Stigler & Hiebert, 1999). U.S. mathematics teachers report spending approximately 15% of class time reviewing homework (Grouws, Tarr, Sears, & Ross, 2010; Mullis et al., 2000). Past research (e.g., Good & Grouws, 1977) and more recent studies (e.g., de Jong, Westerhof, & Creemers, 2000) have handled homework review by measuring duration, but it is necessary to understand the quality of this use of classroom time in addition to the quantity. In this study, we examine homework review in secondary (Grades 610) mathematics lessons, measuring duration but also using theories from sociolinguistics to map the structure of the discourse and to explore the nature of students participation. Although this study focuses on the subject of mathematics, where homework is especially prevalent, we feel that such reflection on the use of instructional time to go over homework can be beneficial in all subject areas.
We view learning as inseparable from coming to actively participate in a discourse community (Lave & Wenger, 1991; Wenger, 1998). Thus, a theory of language becomes integral to any disciplined inquiry into students learning, and we use systemic functional linguistics (Halliday & Matthiessen, 2003) to guide our analysis and interpretations. In particular, we rely on Lemkes (1990) articulation of activity structures within classroom lessons. Because social activities have patterns of organization, one can view learning as the process of coming to function legitimately within these patterns. This is not to say that the patterns of interaction predetermine the activity, but if you can recognize that you are in the midst of a patterned, organized kind of social activity . . . you know the probabilities for what is likely to come next (Lemke, 1990, p. 4). An activity structure, then, is a particular, recognizable organization of interactions. Examples of activity structures within lessons include taking attendance, lecture, seatwork, groupwork, and going over homework (GOHW). From a discourse perspective, teachers and students behave differently depending on the activity structure they find themselves in.
Lemke (1990) articulated a set of activity structures in science classrooms, and there has been a call to do the same for mathematics classrooms (Herbst & Chazan, 2009). This work has recently begun with, for example, Mesiti and Clarke (2006) examining the first 10 minutes of mathematics lessons and Jablonka (2006) investigating the form and functions of student boardwork. The present study adds to this effort by cataloging the activity structure of GOHW in mathematics classrooms, guided by the following question: What is the nature of discourse during GOHW in secondary mathematics classrooms, and in what ways do students participate in that discourse?
Eight secondary (Grades 610) mathematics classrooms from a Midwestern state in the U.S are included in this study. The classrooms were selected based on their teachers involvement in a multiyear professional development (PD) project focusing on classroom discourse. The PD facilitators, both of whom are authors of this article, selected these 8 teachers to maximize variability along several dimensions, such as teaching experience, school setting, curricular materials, certification level, and gender (see Table 1). The data for this study came from the baseline year, prior to any PD interventions. All but one teacher (Denise) tended to give daily homework assignments as a part of their normal practice.
The eight classrooms were located in seven buildings in four school districts. The diversity of the student populations was evident both within and across classrooms. As Table 1 shows, two classrooms were located in a rural district, two in suburban districts, and four in an urban district. The English language learner percentages ranged from less than 4% to nearly 12% across the districts, and the percentage of students with an individualized education program ranged from approximately 10% to approximately 50% of students. Moreover, the students varied socioeconomically, with free and reduced lunch percentages in the schools ranging from 12% to over 65%. Selectivity also varied; one classroom was located in an International Baccalaureate World School.
Table 1. Participating Teachers and Their School Settings
During the 20052006 academic year, 4 weeks of video recordings were collected from a single class for each of the 8 teachers. These four rounds of data were collected during 1-week segments in September, November, January, and March, resulting in 148 classroom observations. For classes on a 4055-minute schedule, this meant five recorded class periods each week. However, two classes were on a block schedule and thus had fewer recorded class periods of longer duration.
External transcribers were hired to transcribe the classroom observations. Project personnel then coded the transcripts based on a modified version of Lemkes activity structure framework that was developed for these particular mathematics lessons (e.g., do now, seatwork, GOHW, classroom business). The transcripts of all GOHW segments constituted the primary data for this study, though we returned to classroom video as needed to help with interpretation of events. Teachers lesson reflections were also used as needed to elaborate classroom events. Using Lemkes (1990) categories, we considered collecting homework (i.e., when students merely handed in their assignments) as a separate activity structure from GOHW and therefore did not include it in this analysis.
The percentage of time spent in each activity structure was tabulated, and turn counts and turn lengths for each teacher and for their students in aggregate were compiled. The GOHW excerpts were analyzed for each teacher, focusing on the structure of the discourse. In particular, we identified discourse moves (made by teachers or students) that seemed to open up or close down the potential language choices that might follow. For example, a teacher saying, Lets look at number 7 made it unlikely (though not impossible) that a student would follow by commenting about a problem other than number 7. We recognize that such moves have significant interpersonal implications (i.e., the authoritative status of the teacher gives a degree of power to such a move), but the present analysis focused on the discourses textual features, that is, the organization and cohesiveness of the interactions. Next, we examined the ways in which teachers and students actually continued the discourse following the types of moves previously identified. For instance, what questions did students ask following a narrow prompt by the teacher (e.g., Which ones would you like to go over?)? What questions did students ask following a broader prompt (e.g., What questions do you have?)? After identifying these structural features of the discourse in individual teachers classrooms, the research team discussed patterns and distinctions across the teachers GOHW classroom activities.
We begin by presenting general features of GOHW, such as frequency and duration tabulations, before describing how aspects of the discourse shaped the GOHW activity in particular ways.
GENERAL CHARACTERISTICS OF GOING OVER HOMEWORK
Table 2 contains information about the frequency and duration of GOHW for each teacher. On average, teachers in this study spent 18.1% of their class time in GOHW, making it the most prevalent activity structure. By comparison, whole-class work only constituted an average of 14% of class time, and seatwork and groupwork were each approximately 10%. For individual teachers, GOHW ranged from 10% (in the urban Title I school) to 27% (in the urban gifted school) of class time. With the exception of Denise, GOHW occurred in the majority of observed lessons.
Table 2. Descriptive Statistics of the Prevalence of Going Over Homework
There were several different forms of GOHW, though individual teachers tended to have consistent patterns for the activity, and the majority of forms focused on answers to the homework problems. Three of the eight teachers usually presented answers to all homework problems by displaying them on the overhead projector, writing them on the front board, or reading them aloud. A fourth teacher did this on occasion. Students checked their work against these answers and sometimes asked their teacher to repeat or clarify an answer. Three teachers, instead of presenting the full set of answers, typically asked if the students had any questions on the homework, addressed whatever questions arose, and then asked for any additional questions, and they continued this cycle until no questions remained. Alternatively, two teachers tended to ask what questions the students had, collect them all on the board, and review them one at a time.
In every instance, the teacher remained the director of, and the most active participant in, GOHW. The teacher was usually the one to share answers or provide explanations, and he or she commonly used the initiate-respond-evaluate (IRE) interaction pattern (Mehan, 1979) during these explanations, which allows teachers to maintain control of the discourse. Amy, Cassie, and Elaine occasionally asked students to give explanations, but overall, student participation usually consisted of responding to teacher questions within IRE or asking the teacher closed questions (e.g., What was [the answer to] number 30? How many points were in total?).
STRUCTURING A PROBLEM-BY-PROBLEM DISCOURSE
As stated earlier, discourse within an activity structure such as GOHW is not predetermined by that structure, but the activity structure tends to demarcate a probabilistic range of interactions and language choices. Within the study classrooms, the discourse of GOHW was structured by individual problems from the homework assignment. We present evidence of this problem-by-problem structure, which existed in all eight classrooms, before sharing the few instances of discourse that deviated from such a structure.
As would be expected, the stating of problem numbers occurred in the discourse as a means of textual organization, tying subsequent utterances to a particular problem from the assignment. For instance, the following interaction took place in Gregs classroom:
Greg: Number 2 is 75 square inches. Number 3, how about put a question mark by the ones youre not quite sure of. Number 3, I had 1,400.
Student: What was number 2?
Greg: Number 2 was 75. Number 3 was one-four-zero-zero. Number 4 is three-point-six-eight. . .
Greg starts sentences with the problem number, allowing students to follow along as they check their answers. The students question also remains within this problem-by-problem organization. At other times, teachers stated problem numbers without including them in complete sentences (e.g., Number 7. Two triangles are similar because they have the same angle measures). If the problem numbers were removed, the remaining sentences would seem disjointed and difficult to follow.
This problem-by-problem organization, however, was not limited to the giving of answers but was also prevalent within the discourse of explanations. When teachers gave explanations during GOHW, they often asked students to identify particular problems that were troublesome. For example, Elaine once began GOHW with the question, Which ones does anybody want to talk about? This prompt promotes responses that involve single problems from the homework assignment. Indeed, in Elaines case, several students responded appropriately by calling out problem numbers, and Elaine proceeded to explain each one by one. Another example of how a teachers prompt shaped the types of questions students asked came from Howards classroom:
Howard: Okay, which ones do we need to talk about together from page 91? Which ones do we want to talk about?
Student A: 34 and 35.
Howard: 34 and 35. Other requests, other than 34 and 35?
Student B: 48.
Student C: 31.
By phrasing the question as which ones, Howard narrowed the students responses toward the single-problem variety, and students complied. In a later lesson, however, Howard used a more general prompt:
Howard: Okay, any questions on this stuff? Things that we want to talk about together on the Frieze patterns? Question?
Student: Um, number 18.
Howard: Number 18. Name and describe the transformation that maps A onto F. So we have these pointy, hat-shaped things. . .
In this interaction, Howard used a more open-ended prompt (any questions), but the student simply stated a problem number. This response is evidence that the norms of the discourse of GOHW in Howards classroom, as in other classrooms, was structured toward students identifying single problems that the teacher can then explain.
EXCEPTIONS TO A PROBLEM-BY-PROBLEM DISCOURSE
Although it was common in all eight classrooms for the discourse to focus on one problem at a time, there were isolated exceptions to this pattern. For example, Greg once concluded the GOHW activity by making a comment about the purpose of the assignment overall: The whole purpose of the assignment was to give you experience in identifying rational and irrational and the different forms that numbers may appear. Theres some nice computational practice along with it and evaluating expressions. Elaine also once began GOHW by making an observation about the problems as a set: Okay, now, in the problems that involve a circle, which is many of these, theres some variation in the accepted answer because it depends on what you use for pi and how you round it.
Another exception occurred in Cassies classroom when the topic was multiplying fractions. The discourse began in the typical individual-problem structure but moved into a discussion of how number 16 (i.e., find a fraction and a whole number whose product is a whole number) differed from the previous problems on the homework assignment (e.g., multiply two given fractions):
Cassie: 16. I think what [the textbook authors] are doing with you in 16 is theyre giving you a situation to let you practice multiplying fractions because thats what were supposed to be able to do by the end of the chapter. Notice all that we were doing here [in previous problems]? You get to practice, but 16 is making you go even further. What is 16 making you do? What, [Student 1]?
Student 1: You have to know the answer before you know the problem.
Cassie: Can you describe what you mean by that, You have to know the answer before you know the problem? What do you mean?
Student 1: They want you to give them a problem in which the answer is in between a certain number that they give.
Cassie: So what does that force you to have to do, you guys? Just think about the deciding that has to go on inside your mind to be able to do this. What do you have to do thats different than me just saying, OK, solve this problem? How is the deciding to do that problem different from the deciding that youve got going in sixteen? Go ahead.
Student 2: Um, its different because you are deciding in your head which one would equal that [answer] because youre already trying to equal something, instead of trying to figure out what it equals. You know it has to equal a certain answer, or else the problems wrong.
Cassie: Would you say its more complicated?
Student 2: Yes.
Cassie: What were you going to say, [Student 3]?
Student 3: You kind of have to do the problem in your head in order to make sure that its going to work out.
Cassie: That goes along with [Student 1]. Its like you gotta get the answer to know whether the problems going to fit. I said its more complicated a minute ago, but would you agree that theyre forcing you to think even more than if they just said, Solve this problem? Deeper thinking, so its kind of like theyre doing good things here. Theyre making you practice, but theyre also making you think more deeply.
In this example, Cassie commented on the purposes of homework problems and asked students to contrast problem 16 with other problems on the assignment. These moves opened up the discourse, allowing multiple students to look across problems rather than remaining focused only on individual problems. The interaction continued, however, by proceeding into a more standard discourse related solely to problem 16.
It should be noted that overall, these exceptions to the problem-by-problem structure of GOHW were relatively rare and consisted almost entirely of teacher talk. The example from Cassies classroom is the only exception that involved substantial student talk.
In this study, we examined the discourse of GOHW in eight secondary mathematics classrooms. Like prior research, we found that a sizeable percentage of class time (nearly 20%) was spent within this activity structure. Although there was variation in the ways teachers organized GOHW, a commonality in the discourse was a problem-by-problem structure. If learning is viewed as coming to participate in specialized patterns of interaction, the data in this study demonstrate that teachers and students successfully developed and maintained patterns during GOHW, and these patterns were fairly similar across diverse school settings. One might ask, however, to what extent this structure aligns with the disciplinary practices of mathematics. Standards documents in mathematics (National Council of Teachers of Mathematics, 2000; National Governors Association & the Council of Chief State School Officers, 2010) emphasize, for example, connecting ideas, recognizing common structures, and expressing regularity in reasoning as hallmarks of mathematical practice. Such processes would seem to depend on a discourse that looks beyond a single problem at a time. Although it is possible that teachers use other activity structures to cultivate discourse that focuses on connections and ideas over time, one could argue that GOHW has rich potential for such discourse because students come to GOHW having already spent time working individually with the mathematical ideas. Also, GOHW, whether intended or not, may serve as a model for students of how prior work can be reflected on in the mathematics community. From this perspective, an emphasis on answers and individual problems may not be ideal because it reduces attention to overarching mathematical structures, big ideas, and mathematical connections and is indeed a different discourse than that used within the discipline of mathematics.
Furthermore, opening up the discourse of GOHW to big ideas and connections between ideas may constitute a quality use of that instructional time. Jitendra and colleagues (2009) found that middle school students randomly assigned to receive schema-based instruction that focused on structures across problems outperformed students in control classes on mathematical problem-solving tasks. The discrepancy between their schema-based instruction and the discourse structures found in the present study suggests that alternative patterns for GOHW should be investigated. Such investigations would seem likely to uncover relationships to student learning, even though past studies (e.g., de Jong et al., 2000) did not. It is plausible that such past work failed to uncover links between GOHW and student learning because it focused only on the amount of time spent in the activity, whereas our analysis suggests that the nature of the activity is important to consider.
This is not to say, however, that a problem-by-problem discourse structure during GOHW is inherently unconstructive. There may be times when students would benefit a great deal from focusing their attention on a single problem, or there may be other times when a teacher needs to move quickly through the answers of an assignment because there are more pressing academic matters to attend to that day. Moreover, it is unlikely that all homework assignments lend themselves equally to attention across problems or on overarching mathematical ideas, so there may need to be versatility in the discourse of GOHW. If, however, teachers and students are unconsciously operating within a problem-by-problem discourse structure and are unaware of its limiting nature on their interactions, they may focus on individual problems regardless of whether it is most beneficial at that particular point in time.
As a final point, our findings raise interesting issues with regard to participation during GOHW. Prior to this study, one might have hypothesized that students contribute a substantial amount of talk during GOHW because they have presumably completed the assignment and thus have a substantial amount of relevant experience to share in the discourse. Furthermore, by having students talk about their work, teachers may be able to assess students in ways not possible solely from written work while simultaneously using GOHW to increase student authority over the mathematical ideas being developed. We found, however, that students spoke infrequently and with short turns during GOHW. Students typically stated only problem numbers before the teacher provided an explanation or used IRE to lead the class through the solution. In other words, the discourse structures of GOHW provided more opportunities for the teacher to talk about how he or she would solve the problem than for the students to talk about how they actually tried to solve it.
Overall, this study suggests that homework does not necessarily remain at home but plays an important role in the classroom. In what ways can homework review constitute a quality use of instructional time? When might it be appropriate to focus on answers, on single problems, on sets of problems, or on broader ideas embedded in the homework assignment? These questions are relevant not only in mathematics but also in all subject areas as we strive to make all classroom activities maximally beneficial to students learning.
This study was supported by the National Science Foundation (Grant No. 0347906, Herbel-Eisenmann, PI). Any opinions, findings, and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of NSF. We thank the teachers for their graciousness in allowing us to work in their classrooms.
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