Quantifying exponential growth: Three conceptual shifts in coordinating multiplicative and additive growth

Alagic, 2006, Teachers explore linear and exponential growth: Spreadsheets as cognitive tools, Journal of Technology and Teacher Education, 14, 633 Carlson, 2002, Applying covariational reasoning while modeling dynamic events: A framework and a study, Journal for Research in Mathematics Education, 33, 352, 10.2307/4149958 Castillo-Garsow, 2012, Continuous quantitative reasoning, Vol. 2 Castillo-Garsow, 2013, The role of multiple modeling perspectives in students’ learning of exponential growth, Mathematical Biosciences and Engineering, 10, 1437, 10.3934/mbe.2013.10.1437 Castillo-Garsow, 2013, Chunky and smooth images of change, For the Learning of Mathematics, 33, 31 Cobb, 1983, The constructivist researcher as teacher and model builder, Journal for Research in Mathematics Education, 28, 258, 10.2307/749781 Confrey, 1994, Exponential functions, rates of change, and the multiplicative unit, Educational Studies in Mathematics, 26, 135, 10.1007/BF01273661 Confrey, 1995, Splitting, covariation, and their role in the development of exponential functions, Journal for Research in Mathematics Education, 26, 66, 10.2307/749228 Davis, 2009, Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge, Journal of Mathematics Teacher Education, 12, 365, 10.1007/s10857-009-9115-2 Dubinsky, 1991, Reflective abstraction in advanced mathematical thinking, 95 Ellis, 2007, Connections between generalizing and justifying: Students’ reasoning with linear relationships, Journal for Research in Mathematics Education, 38, 194 Ellis, 2013, Correspondence and covariation: Quantities changing together, 119 Ellis, 2013, An exponential growth learning trajectory, Vol. 2, 273 Farenga, 2005, Algebraic thinking: Part II. The use of functions in scientific inquiry, Science Scope, 29, 62 Goldin, 1991, Towards a conceptual–representational analysis of the exponential function, Vol. 2, 64 Gravemeijer, 1994 Harel, 2007, The DNR system as a conceptual framework for curriculum development and instruction Lappan, 2006, Vol. 2 Leinhardt, 1990, Functions, graphs, and graphing: Tasks, learning, and teaching, Review of Educational Research, 60, 1, 10.3102/00346543060001001 National Governors Association Center/Council of Chief State School Officers, 2010 Presmeg, 2005, An investigation of a pre-service teacher's use of representations in solving algebraic problems involving exponential relationships, 105 Saldanha, 1998, Re-thinking covariation from a quantitative perspective: Simultaneous continuous variation, vol. 1, 298 Simon, 2011, Studying mathematics conceptual learning: Student learning through their mathematical activity, 31 Simon, 2010, A developing approach to studying students’ learning through their mathematical activity, Cognition and Instruction, 28, 70, 10.1080/07370000903430566 Smith, 2003, Stasis and change: Integrating patterns, functions, and algebra throughout the K-12 curriculum, 136 Smith, 1994, Multiplicative structures and the development of logarithms: What was lost by the invention of function, 333 Steffe, 2000, Teaching experiment methodology: Underlying principles and essential elements, 267 Thompson, 1994, The development of the concept of speed and its relationship to concepts of rate, 179 Thompson, 2002, Didactic objects and didactic models in radical constructivism Thompson, 2008, Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education, Vol. 1, 45 Thompson, 2011, Quantitative reasoning and mathematical modeling Thompson, P.W., & Carlson, M.P. (in press). Variation, covariation and functions: Foundational ways of mathematical thinking. To appear in J. Cai (Ed.), Third handbook of research in mathematics education. Reston, VA: National Council of Teachers of Mathematics. Thompson, 1992, Images of rate Weber, 2002, Students’ understanding of exponential and logarithmic functions, 1