## Shape of development

A classic developmental question is whether change is continuous or discontinuous. Examinations of the shape of psychological growth has implications for diagnosis of stages in development. Relatively long plateaus have been considered to constitute stages and abrupt spurts to constitute transitions between stages.

New statistical techniques make these issues much easier to address. Functional Data Analysis, pioneered by Ramsay and Silverman, treats a growth curve as a function that can be analyzed as a function. After fitting the points in a curve with a smooth function, the derivatives of the function can be computed and plotted, thus revealing more clearly any spurts and plateaus in the data. More specifically, a B-spline basis function is applied between each of the adjacent pairs of data points in a growth curve. These polynomial functions are joined together end-to-end, in a way that matches not only the original function, but also the first four derivatives of the function, ensuring a continuous fitted curve. Then a roughness penalty is applied to smooth the fitted curve. Once a smooth function has been estimated for the data, the first and second derivatives can be computed and plotted to highlight the spurts and plateaus in the original function. Peaks in the first derivative (or velocity) can reveal spurts, whereas valleys in velocity indicate plateaus in the original function. Rapid descents in the second derivative (or acceleration) indicate velocity peaks and thus growth spurts.

Our results indicate that the overall shape of human growth, whether physical or psychological, consists of two plateaus joined by a spurt. If the data are sampled only crudely, as is common, then that may be all that is visible. More refined sampling of growth data reveals many such spurts and plateaus inside this overall shape, a sort of smooth staircase, where the steps are plateaus and the risers are spurts. The fact that this basic shape characterizes not only the overall development curve but also the fine details of the curve, if the data are sufficiently dense, is reminiscent of fractals.

The smooth-staircase shape of psychological development has often been noted, but interpretations of this shape have not been quite right. Some have held, for example, that this shape occurs only for optimal, feedback-corrected performance. However, with a sufficiently sensitive analysis of the higher-level derivatives, the smooth-staircase shape is apparent even with less than optimal performance. Some have argued that developmental change appears discontinuous when viewed from afar, but continuous when viewed from close up. Our results show the opposite - the closer one looks, the more discontinuity there is in developmental change. Our results also show that the oscillating-attention explanation of spurts and plateaus is contradicted by synchronies in the very psychological data that were used to formulate that explanation.

New statistical techniques make these issues much easier to address. Functional Data Analysis, pioneered by Ramsay and Silverman, treats a growth curve as a function that can be analyzed as a function. After fitting the points in a curve with a smooth function, the derivatives of the function can be computed and plotted, thus revealing more clearly any spurts and plateaus in the data. More specifically, a B-spline basis function is applied between each of the adjacent pairs of data points in a growth curve. These polynomial functions are joined together end-to-end, in a way that matches not only the original function, but also the first four derivatives of the function, ensuring a continuous fitted curve. Then a roughness penalty is applied to smooth the fitted curve. Once a smooth function has been estimated for the data, the first and second derivatives can be computed and plotted to highlight the spurts and plateaus in the original function. Peaks in the first derivative (or velocity) can reveal spurts, whereas valleys in velocity indicate plateaus in the original function. Rapid descents in the second derivative (or acceleration) indicate velocity peaks and thus growth spurts.

Our results indicate that the overall shape of human growth, whether physical or psychological, consists of two plateaus joined by a spurt. If the data are sampled only crudely, as is common, then that may be all that is visible. More refined sampling of growth data reveals many such spurts and plateaus inside this overall shape, a sort of smooth staircase, where the steps are plateaus and the risers are spurts. The fact that this basic shape characterizes not only the overall development curve but also the fine details of the curve, if the data are sufficiently dense, is reminiscent of fractals.

The smooth-staircase shape of psychological development has often been noted, but interpretations of this shape have not been quite right. Some have held, for example, that this shape occurs only for optimal, feedback-corrected performance. However, with a sufficiently sensitive analysis of the higher-level derivatives, the smooth-staircase shape is apparent even with less than optimal performance. Some have argued that developmental change appears discontinuous when viewed from afar, but continuous when viewed from close up. Our results show the opposite - the closer one looks, the more discontinuity there is in developmental change. Our results also show that the oscillating-attention explanation of spurts and plateaus is contradicted by synchronies in the very psychological data that were used to formulate that explanation.

- Dandurand, F., & Shultz, T. R. (2010). Automatic detection and quantification of growth spurts.
*Behavior Research Methods, 42*(3), 809-823. - Dandurand, F., & Shultz, T. R. (2011). A fresh look at vocabulary spurts. In C. Hoelscher, T. F. Shipley, & L. Carlson (Eds.),
*Proceedings of the 33rd Annual Conference of the Cognitive Science Society*(pp. 1134-1139). Boston, MA: Cognitive Science Society. pdf - Shultz, T. R. (2003).
*Computational developmental psychology*. Cambridge, MA: MIT Press.