Developing middle school students' understanding of recursive and explicit reasoning

Current curricular recommendations (Kaput, 1995; National Council of Teachers of Mathematics, 2000) support the introduction of algebraic concepts in the elementary and middle school grades. Kaput (October, 1995) has suggested that generalizing numeric situations offers a natural connection between algebra and the K–8 curriculum (Kaput, October 1995). Hence, this study examined the impact of instruction on developing students' understandings of generalization. In particular, it focused on developing students' understandings of the strengths, weaknesses, and interrelationship between recursive and explicit reasoning when generalizing numeric situations.; A class of 25 students participated in a 10-day, whole-class constructivist teaching experiment (Cobb, 2000). Contextual problem situations were provided during the teaching experiment to provoke students to examine the strengths, weaknesses, and interrelationships between recursive and explicit reasoning. Quantitative and qualitative data were gathered regarding students' use of recursive and explicit reasoning. In addition, all students completed pre, post, and retention assessments that provided information about their ability to construct generalizations on three problem situations.; The results showed a significant increase in the students' abilities to develop and describe generalizations for numeric situations following the teaching experiment, and indicated that students could recognize the strengths and weaknesses of recursive and explicit reasoning. However, students exhibited difficulty recognizing general relationships while computing particular values of the dependent variable, and showed a lack of understanding of what particular components of their generalizations represented. These elements appeared to contribute to students' inability to construct generalizations. In addition, the students' limited understandings of the meaning of mathematical operations and their focus on finding procedures over developing understanding hindered their ability to generalize. Also, one student demonstrated a generalization strategy that had not been documented in prior research studies, the rate-adjust...