The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.     

Deaf clients' perceptions of counseling expertise as a function of counselors' signing skill, gender, and therapy type

The study examined deaf clients' perceptions of counseling expertise as a function of several counseling variables: counselor's signing skill, gender, and therapy type. Twenty undergraduate students at a special college for the deaf who were enrolled in either counseling courses or psychology courses viewed 4 video clips reflecting a mock counseling session. After viewing the clips, the participants were asked to complete the Counselor Rating Form--Short Form (Corrigan & Schmidt, 1983). There was no statistically significant main effect for the sex of the counselor, the match between the counselor's gender and the client's gender, or the mode of communication. While the study demonstrates the utility of the method for future research with deaf clients, further modifications of the method are necessary.     

Learned helplessness and level of underachievement among deaf adolescents

The purpose of this study was to investigate the effects of a helplessness experience on the subsequent performance of deaf adolescents functioning at mildly underachieving or severely underachieving levels. Sixty deaf adolescents meeting the experimental subject criteria were separated into mildly underachieving or severely underachieving groups of 30 each on the basis of the discrepancy between their individual IQs and 1976 achievement scores. Group members were then randomly assigned to one of three pretreatment situations: a solvable Block Design test; an unsolvable Block Design test; or no pretreatment. A 2 × 3 multivariate analysis of variance posttest only group design then examined the solvable posttest performance of all 60 subjects. Results indicated significantly high posttest error scores among severely underachieving deaf adolescents experiencing an unsolvable pretreatment situation. Recommendations are offered for the practical applications of this research in the classroom.