Handling Student Difficulties When Going Over Homework in Secondary Mathematics Classrooms Samuel Otten University of Missouri Beth Herbel-Eisenmann Michigan State University Michelle Cirillo University of Delaware
Going Over Homework in the U.S. 15-18% GOHW Grouws et al. (2010)Otten, Herbel-Eisenmann, &Cirillo (in press)
Discourse in GOHW • Structure of the discourse • Talk is typically organized around one problem at a time • Teacher presents answers or explanations problem-by-problem • Students ask “questions” about the homework by identifying a problem number • An alternative organization is to talk across problems by comparing/contrasting or attending to patterns in the homework problems Otten, Herbel-Eisenmann, & Cirillo (in press)
Discourse around Difficulties • Student Errors or Difficulties in the Discourse • Tarr et al. (2013) included the use of student misconceptions or mistakes as a learning site within their classroom environment protocol • Zahner et al. (2012) found a relationship between gain scores and incorrect responses being used as a launching point for discussion • These studies did not look at GOHW in particular
Discourse in GOHW • Guiding Question What was the nature of discourse aroundstudent errors or difficulties during GOHW?
Method • 8 math classrooms (grades 6 – 10) • Teachers varied in their background characteristics • 148 Video-recorded classroom observations • Recordings were parsed by activity structure—patterned social activities recognized by participants and shaping their interactions (Lemke, 1990) • Analysis focused on the instances of the GOHW activity structure, specifically, interactions where an error or difficulty was expressed. • Identified categories and examined the discourse practices within those categories
Results • Student errors or difficulties were relatively rare in the GOHW discourse. Talk tended to focus on correct explanations and correct answers to the HW items. • When errors or difficulties did arise, they played out in a variety of ways. • Was the source of the error the students’ present work or the teacher’s past experience? • Was the error addressed indirect or direct in the discourse?
Errors During GOHW • Most common occurrence in our data was for the errors to arise from present students’ work and to be handled indirectly.
Indirect Handling of Present Errors • Mr. M’s 8th Grade Class. • Students have written on the board answers to homework problems dealing with areas of compound shapes, and Mr. M is talking through each one when he notices an error. 10 7 70 9 14 81 70
Indirect Handling of Present Errors • Mr. M: “This [answer] seems off. Because … this has a distance of 9, and this distance is gonna be 7 going up and down there. Because the total length is 14, this length is 7 and so that means this [other length] has to be 7 because 7 plus 7 equals 14. That makes that 63. Add those together, I think you get 203 for your answer on that. If we used the same method that … Brandon showed yesterday, you people might have boxed it in and did an overall area of 14 times 19 and come up with 266. This [smaller] area is 63 and you subtract away 63 and you still get the same answer. You have a couple of different ways of doing it.”
Indirect Handling of Past Errors • Teachers did raise errors from past experience or knowledge of the content and students—for example, quickly mentioning a common error they had seen in the past. • Ms. H’s Advanced Algebra Class • Evaluating an expression containing x2 with x = 3 • Ms. H: “[Three squared] is six, right? That’s a popular mistake.” • Ms. P’s 8th Grade Class • Assignment in which Ms. P gave students problems and solutions, but some solutions were erroneous. Students evaluated the work and fixed errors. • GOHW discourse focused on students’ judgments and on the correct solutions rather than on having students articulate the mathematical thinking that might have underlay the errors.
Direct Handling of Errors • Occurred only a few times in 148 observations
Direct Handling of Errors • Ms. A’s Grade 7 Class • Problem: Find a fraction and whole number whose product was between one-half and one • She had solicited two correct answers when she asked for another response, calling on Sydney • Sydney: Um, I don’t know if this is right or not, but five-ninths of twenty-five. • Ms. A: [writes 5/9 × 25 on the board] That’s about what fraction, five-ninths? Close to what? • Sydney: Half? • Ms. A: So you want about half of twenty-five. • Sydney: Oh. • Ms. A: What’s wrong? • Sydney: It’d be twelve and a half. • Ms. A: It would be close to that, wouldn’t it? And that would not be between half and one.
Conclusion • Errors arising during GOHW were predominantly handled indirectly • i.e., providing a correct solution rather than investigating the reasoning or thinking behind an error • The teachers did not ignore student thinking in general. During other activity structures and even when handling correct responses from students in GOHW, they often made student thinking an explicit object of focus in the discourse—but not with errors or difficulties during GOHW. Why not?
Conclusion • Teachers may view direct handling of errors as too time consuming or better suited for other activity structures. • However, one could argue that handling the errors indirectly is missing opportunities for formative assessment as well as opportunities for student learning. • When mathematics learning is viewed as coming to participate in the disciplinary practices of mathematics, direct handling of errors may provide opportunities to explore reasoning and model perseverance through difficulty, which are hallmarks of mathematical practice.
Acknowledgments • This study was supported by the National Science Foundation (grant 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 and students for allowing us to work with them. firstname.lastname@example.org email@example.com firstname.lastname@example.org Twitter: @ottensam MathEdPodcast.com
References • Grouws, D. A., Tarr, J. E., Sears, R., & Ross, D. J. (2010). Mathematics teachers’ use of instructional time and relationships to textbook content organization and class period format. Paper presented at the Hawaii International Conference on Education, Honolulu, HI. • Lemke, J. L. (1990). Talking science: Language, learning, and values. Norwood, NJ: Greenwood Publishing. • Otten, S., Herbel-Eisenmann, B. A., & Cirillo, M. (in press). Going over homework in mathematics classrooms: An unexamined activity. Teachers College Record. • Tarr, J. E., Grouws, D. A., Chavez, O., & Soria, V. M. (2013). The effects of content organization and curriculum implementation on students' mathematics learning in second-year high school courses. Journal for Research in Mathematics Education, 44, 683-729. • Zahner, W., Velazquez, G., Moschkovich, J., Vahey, P., & Lara-Meloy, T. (2012). Mathematics teaching practices with technology that support conceptual understanding for Latino/a students. Journal of Mathematical Behavior, 31, 431-446.