Difference between pages "Lehrer, Strom, & Confrey (2002)" and "Confrey & Smith (1994)"

Latest revision as of 05:31, 23 May 2015

Exponential Functions, Rates of Change, and the Multiplicative Unit

Authors: Jere Confrey and Erick Smith
Journal: Educational Studies in Mathematics
Year: 1994
Source: http://link.springer.com/article/10.1007/BF01273661

Abstract

Conventional treatments of functions start by building a rule of correspondence between x-values and y-values, typically by creating an equation of the form y=f(x). We call this a correspondence approach to functions. However, in our work with students we have found that a covariational approach is often more powerful, where students working in a problem situation first fill down a table column with x-values, typically by adding 1, then fill down a y-column through an operation they construct within the problem context. Such an approach has the benefit of emphasizing rate-of-change. It also raises the question of what it is that we want to cal 'rate' across different functional situations. We make two initial conjectures, first that a rate can be initially understood as a unit per unit comparison and second that a unit is the invariant relationship between a successor and its predecessor. Based on these conjectures we describe a variety of multiplicative units, then propose three ways of understanding rate of change in relation to exponential functions. Finally we argue that rate is different than ratio and that an integrated understanding of rate is built from multiple concepts.

Corrolary

APA: Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2-3), 135–164. http://doi.org/10.1007/BF01273661
BibTeX

@article{Confrey1994,
author = {Confrey, Jere and Smith, Erick},
doi = {10.1007/BF01273661},
journal = {Educational Studies in Mathematics},
number = {2-3},
pages = {135--164},
title = {{Exponential functions, rates of change, and the multiplicative unit}},
url = {http://link.springer.com/article/10.1007/BF01273661},
volume = {26},
year = {1994}
}

@@ Line 1: / Line 1: @@
-{{Title|Grounding Metaphors and Inscriptional Resonance: Children's Emerging Understanding of Mathematical Similarity}}
+{{Title|Exponential Functions, Rates of Change, and the Multiplicative Unit}}
 __NOTOC__
-* Authors: [[Richard Lehrer]], [[Dolores Strom]], and [[Jere Confrey]]
+* Authors: [[Jere Confrey]] and [[Erick Smith]]
-* Journal: [[Cognition and Instruction]]
+* Journal: [[Educational Studies in Mathematics]]
-* Year: 2002
+* Year: 1994
-* Source: http://www.tandfonline.com/doi/abs/10.1207/S1532690XCI2003\_3
+* Source: http://link.springer.com/article/10.1007/BF01273661
 ==Abstract==
-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.
+Conventional treatments of functions start by building a rule of correspondence between ''x''-values and ''y''-values, typically by creating an equation of the form ''y=f(x)''. We call this a ''correspondence'' approach to functions. However, in our work with students we have found that a ''covariational'' approach is often more powerful, where students working in a problem situation first fill down a table column with ''x''-values, typically by adding 1, then fill down a ''y''-column through an operation they construct within the problem context. Such an approach has the benefit of emphasizing rate-of-change. It also raises the question of what it is that we want to cal 'rate' across different functional situations. We make two initial conjectures, first that a rate can be initially understood as a ''unit per unit'' comparison and second that a unit is the ''invariant relationship between a successor and its predecessor''. Based on these conjectures we describe a variety of multiplicative units, then propose three ways of understanding rate of change in relation to exponential functions. Finally we argue that rate is different than ratio and that an integrated understanding of rate is built from multiple concepts.
 ==Corrolary==
 ;APA
-: Lehrer, R., Strom, D., & Confrey, J. (2002). Grounding metaphors and inscriptional resonance: Children’s emerging understanding of mathematical similarity. ''Cognition and Instruction'', 20(3), 359–398. http://doi.org/10.1207/S1532690XCI2003_3
+: Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. ''Educational Studies in Mathematics'', 26(2-3), 135–164. http://doi.org/10.1007/BF01273661
 ;BibTeX
 <pre>
-@article{Lehrer2002,
+@article{Confrey1994,
-author = {Lehrer, Richard and Strom, Dolores and Confrey, Jere},
+author = {Confrey, Jere and Smith, Erick},
-doi = {10.1207/S1532690XCI2003\_3},
+doi = {10.1007/BF01273661},
-journal = {Cognition and Instruction},
+journal = {Educational Studies in Mathematics},
-number = {3},
+number = {2-3},
-pages = {359--398},
+pages = {135--164},
-title = {{Grounding metaphors and inscriptional resonance: Children's emerging understanding of mathematical similarity}},
+title = {{Exponential functions, rates of change, and the multiplicative unit}},
-url = {http://www.tandfonline.com/doi/abs/10.1207/S1532690XCI2003\_3},
+url = {http://link.springer.com/article/10.1007/BF01273661},
-volume = {20},
+volume = {26},
-year = {2002}
+year = {1994}
 }
 </pre>
 [[Category:Journal Articles]]
-[[Category:Cognition and Instruction]]
+[[Category:Educational Studies in Mathematics]]
-[[Category:2002]]
+[[Category:1994]]
-[[Category:Similarity]]
+[[Category:Covariation]]
-[[Category:Design Research]]
+[[Category:Exponential Functions]]
+[[Category:Rate]]
+[[Category:Ratio]]

Difference between pages "Lehrer, Strom, & Confrey (2002)" and "Confrey & Smith (1994)"

Latest revision as of 05:31, 23 May 2015

Abstract

Corrolary

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