Amy J. Hackenberg

Students’ Reasoning With Reversible Multiplicative Relationships

In an 8-month teaching experiment, I investigated how 4 sixth-grade students reasoned with reversible multiplicative relationships. One type of problem involved a known quantity that was a whole number multiple of an unknown quantity, and students were asked to determine the value of the unknown quantity. To solve these problems, students needed to produce a fraction of the known quantity that could be repeated some number of times to make the known, rather than repeat the known quantity to make the unknown quantity. This aspect of the problems involved reversibility because students who do not make a fraction of the known quantity tend to repeat the known quantity (Norton, 2008; Steffe, 2002). All four students constructed schemes to solve such problems and more complex versions where the relationship between known and unknown quantities was a fraction. Two students could not foresee the results of their schemes in thought—they had to carry out some activity, review its results, and then carry out more activity in order to solve the problems. The other two could foresee results of their schemes prior to implementing them; their schemes were anticipatory. One of these two also constructed reciprocal relationships, an advanced form of reversibility. The study shows that constructing anticipatory schemes requires coordinating three levels of units prior to activity, a particular whole number multiplicative concept. The study also reveals that even students with this multiplicative concept will be challenged to construct reciprocal relationships. Suggestions for further inquiry on student learning in this area, as well as implications for classroom practice and teacher preparation, are considered.

Continuing Research on Students’ Fraction Schemes

Directly or indirectly, The Fractions Project has launched several research programs in the area of students’ operational development. Research has not been restricted to fractions, but has branched out to proportional reasoning (e.g., Nabors 2003), multiplicative reasoning in general (e.g., Thompson and Saldanha 2003), and the development of early algebra concepts (e.g., Hackenberg accepted). This chapter summarizes current findings and future directions from the growing nexus of related articles and projects, which can be roughly divided into four categories. First, there is an abundance of research on students’ part-whole fraction schemes, much of which preceded The Fractions Project. The reorganization hypothesis contributes to such research by demonstrating how part-whole fraction schemes are based in part on students’ whole number concepts and operations.

Mathematical caring relations: A challenging case

Developed from Noddings’s (2002) care theory, von Glasersfeld’s (1995) constructivism, and Ryan and Frederick’s (1997) notion of subjective vitality, a mathematical caring relation (MCR) is a quality of interaction between a student and a mathematics teacher that conjoins affective and cognitive realms in the process of aiming for mathematical learning. In this paper I examine the challenge of establishing an MCR with one mathematically talented 11-year-old student, Deborah, during an 8-month constructivist teaching experiment with two pairs of 11-year-old students, in which I (the author) was the teacher. Two characteristics of Deborah contributed to this challenge: her strong mathematical reasoning and her self-concept as a top mathematical knower. Two of my characteristics also contributed to the challenge: my request that Deborah engage in activity that was foreign to her, such as developing imagery for quantitative situations, and my assumption that Deborah’s strong reasoning would allow her to operate in the situations I posed to her. The lack of trust she felt at times toward me and the lack of openness I felt at times toward her impeded our establishment of an MCR. Findings include a way to understand this dynamic and dissolve it to make way for more productive interaction.