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Amy J. Hackenberg

In a small-scale, 8-month teaching experiment, the author aimed to establish and maintain mathematical caring relations (MCRs) (Hackenberg, 2005c) with 4 6th-grade students. From a teacher's perspective, establishing MCRs involves holding the work of orchestrating mathematical learning for students together with an orientation to monitor and respond to energetic fluctuations that may accompany student–teacher interactions. From a student's perspective, participating in an MCR involves some openness to the teacher's interventions in the student's mathematical activity and some willingness to pursue questions of interest. In this article, the author elucidates the nature of establishing MCRs with 2 of the 4 students in the study and examines what is mathematical about these caring relations. Analysis revealed that student–teacher interaction can be viewed as a linked chain of perturbations; in student–teacher interaction aimed toward the establishment of MCRs, the linked chain tends toward perturbations that are bearable (Tzur, 1995) for both students and teachers.

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Amy J. Hackenberg, Robin Jones and Rebecca Borowski

We tiered instruction for a class of seventh grade students during a proportional reasoning unit by providing the same problem with different numbers to different groups of students. We explain why we tiered, show students' work, describe what students learned, and give recommendations about differentiating instruction.

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Holly Garrett Anthony and Amy J. Hackenberg

An activity for “making quilts without sewing” that enables high school students to develop their understanding of planar symmetries and wallpaper patterns. This activity incorporates the culture and traditions of quilting into the study of geometry. From the same block, students can make quilts with different patterns by using various combinations of transformations, in addition to the possible combinations of transformations that might have been used to create a quilt or wallpaper pattern. Students may also reflect upon the cultural activity of quilt makers and the art of quilt making.

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Amy J. Hackenberg and Mi Yeon Lee

To understand relationships between students' fractional knowledge and algebraic reasoning in the domain of equation writing, an interview study was conducted with 12 secondary school students, 6 students operating with each of 2 different multiplicative concepts. These concepts are based on how students coordinate composite units. Students participated in two 45-minute interviews and completed a written fractions assessment. Students operating with the second multiplicative concept had not constructed fractional numbers, but students operating with the third multiplicative concept had; students operating with the second multiplicative concept represented multiplicatively related unknowns in qualitatively different ways than students operating with the third multiplicative concept. A facilitative link is proposed between the construction of fractional numbers and how students represent multiplicatively related unknowns.