Preservice elementary teachers' development of rational number understanding through the social perspective and the relationship among social and individual environments

A classroom teaching experiment was conducted in a semester-long undergraduate mathematics content course for elementary education majors. Preservice elementary teachers’ development of rational number understanding was documented through the social and psychological perspectives. In addition, social and sociomathematical norms were documented as part of the classroom structure. A hypothetical learning trajectory and instructional sequence were created from a combination of previous research with children and adults. Transcripts from each class session were analyzed to determine the social and sociomathematical norms as well as the classroom mathematical practices. The social norms established included (a) explaining and justifying solutions and solution processes, (b) making sense of others’ explanations and justifications, (c) questioning others when misunderstandings occur, and (d) helping others. The sociomathematical norms established included determining what constitutes (a) an acceptable solution and (b) a different solution. The classroom mathematical practices established included ideas related to (a) defining fractions, (b) defining the whole, (c) partitioning, (d) unitizing, (e) finding equivalent fractions, (f) comparing and ordering fractions, (g) adding and subtracting fractions, and (h) multiplying fractions. The analysis of individual students’ contributions included analyzing the transcripts to determine the ways in which individuals participated in the establishment of the practices. Individuals contributed to the practices by (a) introducing ideas and (b) sustaining ideas. The transcripts and student work samples were analyzed to determine the ways in which the social classroom environment impacted student learning. Student learning was affected when (a) ideas were rejected and (b) ideas were accepted. As a result of the data analysis, the hypothetical learning trajectory was refined to include four phases of learning instead of five. In addition, the instructional sequence was refined to include more focus on ratios. Two activities, the number line and between activities, were suggested to be deleted because they did not contribute to students’ development