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Grounding Metaphors and Inscriptional Resonance: Children's Emerging Understanding of Mathematical Similarity

The goal of this classroom study of third-grade students was to support and document the emergence of multiple senses of mathematical similarity. Beginning with grounding metaphors of scale, magnification, and classification, the classroom teacher helped students redescribe their perceptions of these everyday experiences with the mathematics of similarity. Each sense of similarity was mediated by a distinct form of mathematical inscription: a ratio, an algebraic "rule," and a line in a Cartesian system. These forms of inscription were tools for externalizing and structuring children's perceptions of magnification and scale ("growing") and of classification ("same shape"). Children's interpretations of the mathematical meanings of the notational systems employed were supported by successive forms of signification, which Peirce (1898/1992) described as iconic, indexical, and symbolic. We tracked student sense making through 2 sequences of lessons, first involving 2- and then 3-dimensional forms. The shift in dimension supported students' integration of multiple senses of similarity as ratio and as scale. The process of integration was assisted by resonances among diverse forms of inscription. Students' explorations of similarity served later in the year as a resource for modeling nature (e.g., by conducting explorations of density and growth). This shift toward modeling introduced a troublesome, yet ultimately rewarding, epistemological dissonance between mathematics and science. Postinstructional interviews suggested that most children came to appreciate the mathematical generalizations afforded by the algebraic and graphical forms of notation used to inscribe similar forms. A follow-up design experiment conducted in the fifth grade included some revisions to instruction that proved fruitful. Our concluding comments about design experiments and developmental corridors are motivated by a need to rethink these ideas in light of the contingent and historical nature of student thinking as it unfolded in these classrooms.