Learning to reason in an informal math after-school program

Abstract

This research was conducted during an after-school partnership between a University and school district in an economically depressed, urban area. The school population consists of 99% African American and Latino students. During an the informal after-school math program, a group of 24 6th-grade students from a low socioeconomic community worked collaboratively on open-ended problems involving fractions. The students, in their problem solving discussions, coconstructed arguments and provided justifications for their solutions. In the process, they questioned, corrected, and built on each other's ideas. This paper describes the types of student reasoning that emerged in the process of justifying solutions to the problems posed. It illustrates how the students' arguments developed over time. The findings of this study indicate that, within an environment that invites exploration and collaboration, students can be engaged in defending their reasoning in both their small groups and within the larger community. In the process of justifying, they naturally build arguments that take the form of proof.

The co-construction of arguments by middle-school students

“…These features have been documented by Henningsen and Stein (1997) to include opportunities for multiple representations, multiple solutions, and mathematical discussion. According to Mueller and Maher (2008), open-ended challenging tasks that allow for exploration, multiple representations and multiple strategies contribute to the co-construction of ideas and thus to the development of multiple forms of reasoning. The tasks should be presented to students as strands of related problems that can be revisited over time and in different contexts, thus enhancing students' opportunities to overcome obstacles in problem solving.…”

“…When given the opportunity to reason about mathematical knowledge in a supportive environment, students as young as elementary school can create conjectures, reflect upon and evaluate them, and try to convince others to accept these justifications (Maher & Davis, 1995;Yackel & Hanna, 2003). Given contexts that allow students to share their ideas with others, and participate in mathematical discussions, debates, and reasoning about mathematics, students are able to offer convincing arguments that indicate various forms of reasoning in the development of solutions to problems (Maher, 2005;Maher & Davis, 1995;Maher & Martino, 2000;Martino & Maher, 1999;Mueller & Maher, 2008).…”

“…Maher (1998) described specific classroom conditions that form a culture in which students are able to support and represent their ideas and discuss the ideas of others while being supported by adults. These conditions include, but are not limited to: (1) providing a safe, supportive environment for collaborative learning; (2) providing students with open-ended tasks and sufficient time to investigate, discuss, and revisit tasks; (3) encouraging students to build representations; (4) having tools available so that models can be built; (5) inviting students to explain and justify their reasoning; (6) emphasizing sense making; (7) carefully planning teacher interventions; and (8) promoting mathematical discourse (Maher, 2005;Maher & Martino, 2000;Martino & Maher, 1999;Mueller & Maher, 2008, 2009). These ideas are expanded upon in the theoretical framework.…”

Teachers’ perceptions of students’ mathematical work while making conjectures: an examination of teacher discussions of an animated geometry classroom scenario

“…Research on student thinking shows that learners are capable of making reasoned and insightful conjectures (for examples, see Balacheff, 1988;Boero et al, 1999;Ellis, 2007a;2007b;Herbst, 2006;Lee and Sriraman, 2011;Mariotti, 2006;Martin and Pirie, 2003;Mueller and Maher, 2009;Yerushalmy, 1987Yerushalmy, , 1993. This body of research provides a template for the types of conjecturing activities that students could engage in during instruction.…”

Some aspects of executive functions as predictors of understanding textual mathematical tasks in students with mild intellectual disability

“…Tekst zadatka opisuje realnu situaciju koja treba da se reši primenom matematičkog postupka koji obuhvata definisanje poznatih, odnosno nepoznatih vrednosti. Matematički zadaci predstavljeni u numeričkom formatu su jednostavniji s obzirom na to da podrazumevaju samo izračunavanje vrednosti postavljenog izraza i stoga ih učenici bolje savladavaju u odnosu na tekstualno predstavljene zadatke (Carpenter, Cobitt, Kepner, Linquist & Reys, 1980;Mueller & Maher, 2009). …”