Publication 992

Hackenberg A. J. (2013) The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior 33: 1. Fulltext at
To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students, with each of three different multiplicative concepts, participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed. Relevance: In this paper the author uses second-order models of students’ multiplicative concepts and fractional knowledge built from radical constructivism to explore relationships between students’ fractional knowledge and algebraic reasoning. The paper is therefore one contribution to the construction of second-order models of students’ algebraic reasoning, which is sorely needed by the field of mathematics education, particularly for students who struggle to learn algebra.


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