Construction of algebraic reasoning and mathematical caring relations
Hackenberg, Amy Jeanne
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The purpose of this study was to understand how sixth graders construct algebraicreasoning in interaction with a teacher-researcher who endeavors to enact mathematical caringrelations (MCR) with them. Reasoning quantitativelyÑthe purposeful functioning of schemes inquantitative contexts in order to model relationships between known and unknown quantities anddetermine unknownsÑwas taken as a foundation for studentsÕ work toward constructions andsolutions of basic linear equations (of the form ax = b). From a teacherÕs perspective, MCR wasformulated as an orientation to balance stimulation and depletion, or increases and decreases inlevels of energy and feelings of well-being, in student-teacher interactions aimed towardmathematical learning. From a studentÕs perspective, participating in MCR was articulated asbeing open to the teacherÕs interventions in oneÕs mathematical activity and pursuing oneÕsquestions of interest.As teacher-researcher I taught two pairs of sixth graders at a rural middle school inGeorgia in a constructivist teaching experiment from October 2003 to May 2004. All teachingepisodes were videotaped. Teaching practices included posing problems that involvedmultiplicative relationships between fractional quantities, adapting problem situations toharmonize with and challenge studentsÕ ways of operating, and tracking studentsÕ affectiveresponses to and engagement with this interactive activity. In a retrospective analysis ofvideotapes, I constructed second-order models that accounted for changes students made in theirmathematical ways of operating and in their affective responses to mathematical interaction.MCR was beneficial in facilitating studentsÕ engagement in mathematical activity andinfluential on the construction of the studentsÕ, as well as the teacherÕs, self-concepts in relationto doing and communicating about mathematics. StudentsÕ multiplicative structures weresignificant resources in their construction of schemes for making improper fractions and forsolving problems that could be solved with basic linear equations. The students had difficultyconstructing fractions as multiplicative operations on quantities. These results have implicationsfor how teachers and researchers conceive of student-teacher mathematical interaction and of theconstitution of algebra learning for middle school students.