Age, years figure 1-2 The growth of De Montbeillard's son 1759-1777: Distance. (Redrawn from Tanner JM. Growth at Adolescence, 2nd ed. Oxford: Blackwell Scientific Publications, 1962.)

weight, and head circumference on a sample of 31 children on daily, biweekly, and weekly measurements.6 The resulting data demonstrated that growth, in height at least, may not be a continuous phenomenon but actually occur in short bursts of activity (saltation) that punctuate periods of no growth (stasis) (see Chapter 12). However, the data we have for De Montbeillard's son was collected approximately every 6 months and thus, at best, can tell us only about half-yearly or yearly patterns of growth.

It is clear that the pattern of growth that results from these 6-monthly measurements is composed of several different curves. During "infancy," between birth and about 5 years old, there is a smooth curve that we can describe as a "decaying polynomial," because it gradually departs negatively from a straight line as time increases. During childhood, between 5 and about 10 years old, the pattern does not depart dramatically from a straight line. This pattern changes during adolescence, between about 10 and 18 years old, into an S-shaped, or sigmoid, curve, reaching an asymptote at about 19 years old.

The fact that the total distance curve may be represented by several mathematical functions allows us to apply mathematical "models" to the pattern of growth. These models are parametric functions that contain constants, "parameters." Once we have found an appropriate function that fits the raw data, we can analyze the parameters and, by so doing, learn a good deal about the process of human growth (see Chapter 3). For instance, in the simplest case of two variables, such as age (X) and height (Y), being linearly related between, say, 5 and 10 years of age (i.e., a constant unit increase in age is related to a constant unit increase in height), the mathematical function Y = a + bX describes their relationship. The parameter a represents the point at which the straight line passes through the Y-axis and is called the intercept; b represents the amount that X increases for each unit increase in Y and is called the regression coefficient. Fitting this function to data from different children and subsequent analysis of the parameters can tell us about the magnitude of the differences among the children and lead to further investigations of the causes of the differences. Such time series analysis is extremely useful within research on human growth because it allows us to reduce large amounts of data to only a few parameters. In the case of De Montbeillard's son, 37 height measurements were made at 37 different ages, yielding 74 data items for analysis. The fitting of an appropriate parametric function, such as the Preece-Baines function,7 which we discuss later (see Chapter 3), reduces these 74 items to just 5. Because of their ability to reduce data from many to only a few items, such parametric solutions are said to be parsimonious and are widely used in research into human growth.

the velocity growth curve and growth spurts

The pattern created by changing rates of growth is more clearly seen by actually visualizing the rate of change of size with time; that is, "growth velocity," or in this particular case, "height velocity." The term height velocity, coined by Tanner,8 was based on the writings of D'Arcy Wentworth Thompson (1860-1948). D'Arcy Thompson was a famous British natural scientist, who published a landmark biology text, Growth and Form, in 1917 with a second edition in 1942.9,10 In the later edition (p. 95), Thompson wrote that, while the distance curve "showed a continuous succession of varying magnitudes," the curve of the rate of change of height with time "shows a succession of varying velocities. The mathematicians call it a curve of first differences; we may call it a curve of the rate (or rates) of growth, or more simply a velocity curve." The velocity of growth experienced by De Montbeillard's son is displayed in Figure 1-3. The Y-axis records height gain in cm/yr-1; and the X-axis is the chronological age in years. We see that, following birth, two relatively distinct increases in growth rate occur at 6-8 years and again at 11-18 years. The first of these "growth spurts" is called the juvenile or mid-growth spurt (see Chapter 2) and the second is called the adolescent growth spurt (see Chapter 3).

There is, in fact, another growth spurt that we cannot see because it occurs prior to birth. Between 20 and 30 weeks of gestation, the rate at which the length of the fetus increases reaches a peak at approximately 120 cm/yr-1, but all we can observe postnatally is the slope of decreasing velocity lasting until about 4 years of age. Similarly, increase in weight also experiences a prenatal spurt but a little later, at 30-40 weeks of gestation. Of course, information on the growth of the fetus is difficult to obtain and relies largely on two sources of information: extrauterine anthropometric measurements of preterm infants and intrauterine ultrasound measurements of fetuses. Ultrasound assessments of crown-rump length indicate that

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