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Unformatted text preview: Shulman Original Sources to Teach Mathematics in Social Context MATHALIVE!
USING ORIGINAL SOURCES
TO TEACH MATHEMATICS IN SOCIAL CONTEXT Bonnie Shulman ADDRESS: Department of Mathematics, Bates College, Lewiston ME
04240 USA. ABSTRACT: There is a growing trend in mathematics education to use
history and original sources to motivate students to learn content. An
added beneﬁt of using these sources is that it allows instructors to
integrate a humanistic understanding of mathematics into a. mathe
matics class by embedding the content within its motivating social
context. This article describes how a simple. question led to a fasci
nating quest through primary sources. Once collected, these sources
proved to be a treasure trove of material illuminating mathematics
and mathematicians in historical context. KEY WORDS: History of mathematics, original sources, logistic equation,
modelling, humanistic mathematics. INTRODUCTION There is a common perception of mathematics as a ﬁnished product invented
by dead geniuses. In an effort to dispel this notion and convey to my
students the excitement of mathematics as a. living, breathing, and growing
body of knowledge, created by human beings very much like ourselves, I have
turned to original sources. A careful reading of these works can also reveal
the social context within which mathematics is embedded. It is easier (and
less threatening) to discern the influence of biases and hidden assumptions
on the thinking of those from a previous era, than to question the underlying
paradigms of one’s own time. Once students are exposed to an historical
case study where it is clear how selfinterest and prevailing norms helped Film; March 1998 Volume VIII Number 1 determine what questions were asked, as well as how they Were answered,
they become more willing and able to evaluate critically and understand
the social context as a. component in the creation of all mathematics. THE PROBLEM How does one ﬁnd original sources to use in the classroom? Often this
happens serendipitousiy, as is illustrated by the foilowing story of my search
to ﬁnd out how the logistic equation got its name. The logistic equation beat 3!: 1+ce‘” is used to model natural systems, including human behavior, involving
growth with limited resources. This simple equation, along with the dif
ferential equation it solves, and its familiar Sshaped curve, is ubiquitous,
familiar to mathematicians, and natural and social scientists alike. Certain I was not the ﬁrst to wonder how this particular equation came
to be called the “logistic,” I began asking around. One colleague, in Bi—
ology, suggested that it had to do with the lodging of troops. Indeed,
Manfred Schroeder writes “The designation logistic . . . is derived from the
French Iogistique, referring to the lodgment of troops.” (emphasis in origi
nal) [9, p. 269]. Elsewhere, I read it simply meant “rational.”1 Both these
explanations seemed plausible, but not conclusive. Then I serendiptously
came across the following usage by Maria Agnesifi an Italian mathemati
cian from the eighteenth century, eldest of 21 children, who wrote one of
the ﬁrst calculus texts as a study guide for her yOunger brothers: “In these
cases, therefore, we are obliged to have recourse to other methods. There
are two of these which will assist us. One is, by means of a curve which
is called the Logarithmic Curve, or the Logistic.” (emphasis in original) [1,
p. 12]. Hmmmm, I thought, perhaps logistic refers to “loglike” rather than
lodgments or logic. I decided to try to trace back the usage through original
sources. In the process of digging, I unearthed a gold mine of texts that
went far beyond answering my question. First, I will address the question,
then I will suggest a few of the many ways these materials can be used
not only to teach the content (modelling, differential equations, etc.), but 1 “The equation for the Verhulst model, if graphically plotted, gives rise to an S—shaped
curve known as a logistic curve {which is from the Greek Iogistikoa, meaning rational)”
[16, p. 1231. 3A student of mine was writing a thesis on Agnesi, using some of her original work. Shulman Original Sources to Teach Mathematics in Social Context also to impress upon students how new mathematics arises out of social,
political, and economic, as well as intellectual needs. THE SEARCH The ﬁrst place I looked was in the oftcited classic paper by Pearl and Reed
[7], who are usually credited with being the ﬁrst to use the logistic equation
to describe the growth of the population of the United States. A thorough
reading revealed that although they do indeed introduce the equation as
an appropriate model, nowhere do the authors actually call it “the logistic
equation.” In the endnotes to the chapter on ecological models in Michael
Olinick’s excellent text on modelling [5], I found a. citation for G. Udny
Yule’s presidential address to the Royal Statistical Society in London {18]
which contained complete references and a critical history of the logistic
model. From Yule,3 I learned that PierreFrancois Verhulst (13041849),
a Belgian sociologist and mathematician, was actually the ﬁrst to propose
and publish a formula for the law of growth for a population conﬁned to a
speciﬁed area {17]. Yule states, “[p]robably owing to the fact that Verhulst
was greatly in advance of his time, and that the then existing data were
quite inadequate to form any effective test of his views, his memoirs fell into
oblivion.” [18, p. 4]. Apparently, some eighty years later Pearl and Reed
had arrived independently at the same result. Verhulst’s work did eventually
come to their attention, and in fact, Yule acknowledges that he is indebted
to Pearl’s book The Biology of Death [6] for the references to Verhulst.
Yule goes on to say that Verhnlst’s memoirs “are classics on the subject,
and as the originals are not very accessible, I have given some account of
them in Appendix I.” [18, p. 4]. Excitedly, I turned to the appendix where
Yule carefully explains Verhulst’s derivation of the formula, only to ﬁnd the
tantalizing hint: “Verhulst now christens the curve a ‘logistic,’” [18, p. 44]
with no further explanation as to why. So I continued my quest by searching
for the “not very accessible” original paper by Verhulst. Several months later, after consulting with our reference librarian (who
eagerly tool: up the challenge), I received a note informing me that the
document was available on microﬁlm through interlibrary loan. Verhulst
wrote in French, but with dictionary in hand and my rudimentary high
school background, I was able to comprehend the paper. On page 8, my efforts were rewarded, and I found what I was looking for. Here is the
evidence; judge for yourself. 30linick is also fully aware of Verhulst’s contributions and includes a biography of
Verhulst in his text. Palm March 1998 Volume VIII Number 1 THE QUESTION ANSWERED The logistic equation is usually written with population expressed as a func
tion of time (population is the dependent variable). This familiar form of the
equation involves an exponential function. Verhulst wrote the relationship
with time as the dependent variable.“ Since the logarithmic function is the
inverse of the exponential, his equation had t (time) equal to a (somewhat
complicated) logarithmic function of 1) (population): p(m  nb) t = (El/m) log lbim _ up) In the next sentence, Verhulst indeed “christens” the equation: “We give
the name logistic to the curve (see Figure 1) characterized by the preceding
equation.”5 {17, p. 8]. The diagram clinched it for me: there two curves
labelled “Logistique” and “Logarithmique” are draWn on the same axes,
and one can see that there is a region where they match almost exactly, and
then diverge. I concluded that Verhulst’s intention in naming the curve was indeed
to suggest this comparison, and that “logistic” was meant to convey the
curve’s “toglike” quality. As an unanticipated bonus, along with satisfying my curiosity, I now
had in my possession a rich collection of original sources (the papers of
Verhuist, Pearl and Reed, and Yule). I have used these materials to teach
content“ to students in calculus, differential equations and modelling, and
in a summer institute for K—12 teachers. I also assign readings from these
papers that lead naturally to discussions of mathematics in social context.
Here I give you a sample of some of the issues that arise. My goal is not to
provide speciﬁc lessons, but rather to suggest by example how to integrate
a. humanistic understanding of mathematics into a mathematics class by
embedding the content within its motivating social context. MATHEMATICS IN SOCIAL CONTEXT Pearl and Reed begin their paper by explaining why “mathematical expres
sions of population growth [that are] purely empirical. . . have a distinct and
considerable usefulness.” [7, p. 275]. The census is only taken every ten ‘The diii'erentiai equation model he derived and solved was dt = (Mdp)/(p(m — 1112)). 5“Nous donnerons le nom de logistique a la combe (voyez Ea ﬁgure) caractérisée par
l’équation précédente” [17, p. 8]. 6See [10]. Shulman Original Sources to Teach Mathematics in Social Context years and it is often necessary to get an accurate estimate of the popula
tion in inter—censal years (the years following the last census, as well as the
years lying between prior censuses). “For purposes of practical statistics
it is highly important to have these intercensal estimates of population as
accurate as possible, particularly for the use of the vital statistician, who
must have these ﬁgures for the calculation of annual death rates, birth rates,
and the like.” [7, p. 275]. We wilt return to the uses to which these ﬁgures
are put, as Well as the important social issues affecting even this seemingly
straightforward calculation of birth and death rates. fermn'tlmr‘v (l6 Figure 1. Verhulst's diagram after he christens the curve
logistic. [ 17, between pages 15 and 16.] The drawback of a purely empirical expression is that it does not give
a general law of population growth: I‘No process of empirically graduating
raw data with a curve can in and of itself demonstrate the fundamental law
which causes the occurring change.” [7, p. 275]. Dissatisﬁed with em’sting
empirical representations of population growth in the United States, even
when their accuracy is “entirely sufﬁcient for all practical statistical pur
poses” [7, p. 280}, Pearl and Reed ﬁnd it “worthwhile to attempt to develop
such a law {of population growth}, ﬁrst by formulating a hypothesis which
rigorously meets the logical requirements, and then by seeing whether in
fact the hypothesis ﬁts the known facts.” [7, p. 281]. So they list a set of
conditions that should be fulﬁlled by any equation meant to describe the Pﬁimu} March 1998 Voiume VIII Number 1 growth of population in an area of ﬁxed limits. These are the explicit un
derlying assumptions of the logistic model. They are reasonable enough:
we assume a the population is aiways increasing, 0 when resources are plentiful the rate of increase is itself increasing,
and o as resources become more scarce, the rate of increase begins to de
crease, until the population approaches the carrying capacity of the
restricted area. One equation which fuiﬁlls these requirements is beat y=1+ce°£ which is equivalent to the equation Verhulst named the logistic. Pearl and
Reed then demonstrate that this equation gives as good a ﬁt to the observa tions as the “logarithmic paraboia,”7 the curve which “describes the changes
which have occurred in the population of the United States in respect of
its gross magnitude, with a higher degree of accuracy than any empirical
formula hitherto applied to the purpose.” [7, p. 280]. This result is signiﬁcant because “[a iogistic] curve which on a priori
grounds meets the conditions which must be satisﬁed by a true law of pop
ulation growth, actually describes with a substantiai degree of accuracy
what is now known of the population history of this country.” [7, p. 284}. In this success lies a danger. It is a very human tendency to conﬁate
our symbols with the things they represent; to confuse our models, which
are merely our best attempts to make sense of the Welter of phenomena we
observe, with “laws” of nature — rules that phenomena must follow. The
statisticians of the time were aware of these pitfalls. In response to a paper
delivered by Dr. Stevenson, in the meeting following Yule’s Presidential ad
dress, a Professor A. L. Bowley said: “I regret that so much prominence has
been given to the iogistic equation. It certainly has the merit, and the dan
ger, of mathematical neatness. . . .” [13, p. 76]. He warns that although “the
iogistic equation is well adapted to represent rather roughly the recorded
changes of popuiation in selected countries,” he “fear[s] that, after the com
siderable advertisement it has received, many erroneous conclusions will be
based on it.” [13, p. 80]. We tend to enshrine the Iogistic model as the
logistic law of population growth. This is not something we do consciously,
but the way the subject is traditionally taught often leaves students with 7A quadratic equation with a logarithmic term added on, y = a + be: + c172 + dlogz. 6 Shulman Original Sources to Teach Mathematics in Social Context this idea. When the subject matter is presented in the context in which it
was created, students are properly warned of this danger. POPULATION OF UNITED STATES gqaoqooa mmmmmwmo‘wwmwmlmma YEAR Figure 2. ”Diagram showing observed and cal
culated populations (from logarithmic parabola)
from 1790 to 1920."[?. p. 289] Other mathematical models, based on different but equally reasonable
assumptions, also resulted in curves that ﬁt the data. Dr. Brownlee, another
discussant of Dr. Stevenson’s paper, “had been accustomed to graduate the
populations in the country from 1801 onwards to a curve of a different form y = A3"(ge)c: a curve which developed from the idea that a sudden impulse to increase
of population took place for some reason or other and then slackened in
a geometrical progression.” [13, p. 82]. Why was the logistic model the
most popular mathematical model? And if other expressions, determined
solely by curve—ﬁtting techniques (and not based on mathematical models
of assumptions about how populations should behave) yield' curves that ?¥im March 1998 Volume VIII Number 1 empirically ﬁt the data equally well or better,8 why Were statisticians so
anxious to embrace a. mathematical model? THE REST OF THE STORY There are good mathematical reasons that explain each of these questions,
but they do not provide the whole story. Part of the context in which
mathematics takes place includes social relations and societal values. In
the early part of the twentieth century, Statistics, as a discipline, was still
struggling to attain the status of a “science.” In our society, Mathematics,
the “queen of the sciences” confers respectability and higher status to any
discipline that can frame its arguments in the language of equations and
formulae. The presidential address by Yule set the Society ﬁrmly on the
path of greater mathematization. This made many members uneasy. “Some
apprehension has been expressed lest the highly technical trend taken by
statistical investigation during the past thirty years may not have kept some
from joining the Society, in consideration of their want of acquaintance
with the higher mathematics.” {18, p. 60]. Nevertheless, a mathematical
model is to be preferred over a purely empirical one in part because it is
mathematical. And why did the logistic win out? Partly because its champion was
President Yule. Professor EdgeWorth says, “Altogether, on the question
which I have raised concerning the credentials of the logistic law, I am dis
posed to defer much to what I should call Mr. Yule’s ‘authority.’ . . . He is
conversant with the canons of evidence which satisfy a society dealing with
sciences more rigid than statistics.” [18, p. 59]. Scholars in science stud
ies maintain that when several competing theories or models em'st, other
(social) factors, along with mathematical and scientiﬁc plansibiiity and rea
sonableness, help determine which theory or model prevails.9 I believe we
have here some evidence in support of this controversial claim. OTHER INTERESTING DISCOVERIES Let us now look more closely at some of the other assumptions hidden
within this modelling process. We can ask, does the census really include
everyone? Our logistic model may he a. very good fit to only certain segments 8“The most that can be asserted is that [the logistic} equation gives nearly or quite
as good a ﬁt to the observations as dines the logarithmic parabola.” {7", p. 284]. 9“{W]hether an account is powerful or not depends not simply on the force of the
argument but also on the positioning and resources of its advocatea.. ..[W]e need to
ask why particular readings hold such sway and others disappear without a. trace.” [4, pp. 3523]. Shulman Original Sources to Teach Mathematics in Social Context of the population. There are many illuminating examples throughout Yule’s
address (as Well as in Verhuist’s paper) of assumptions made about (and
encoded into the models of) populations. Two examples from Yule are
whether to take into account (or not) catastrophes like World War I and
the inﬂuenza epidemic of 191819; and whether or not to include Alsace
Lorraine in the population of France. We read in Verhuis‘tm that “We
haven’t taken account of immigration, because we have observed that it has
been oﬂ‘set by the impediments that slavery brings to the growth in the
number of blacks in the South.” 11 [17, p. 5]. As many feminists. have pointed out, categories create our reality.12 In
particular, census categories determine what is visible — what is counted as
work, who is deﬁned as a. worker, how (or even if) class is measured, and so
on. The census feeds almost every other deﬁning and decision—making pro
cess in the country.13 In fact, the United States census purposefully started
out to include only a'percentage of slaves as property and to exclude Native
Americans.“ Again, the logistic model may not ﬁt the actual population. Pearl and Reed explore some of the “consequences which ﬂow from [the
logistic] equation...” and make the claim that “the United States has al
ready passed its period of most rapid population growth.” Although they
caution that “prophecy is a dangerous pastime,” they go on to say that
according to their model, “the maximum population which the continental
United States, as now really limited, will ever have will be roughly twice
the present population” (i.e., it will level off at just under 200 million). [7,
pp. 2846]. To support this claim, Pearl and Reed use mathematical concepts no
more sophisticated than ratio and proportion. These few pages provide
a. fascinating case study for students to examine and critically evaluate
quantitative arguments made by “experts” from a previous era (and they
can check the validity of the predictions). To encourage them to apply the
same skills to arguments made by the pundits of their own time, I turn
to the pages of local newspapers and national magazines for examples in
current events. The key point to emphasize and discuss is that important 1“Verhulst uses popuiation data from the United States, 17901840, as an example of
a pepulation growing in geometric progression (exponential growth). 11“Nous n’avons pas tenu compte des immigrationa, parce que nous les regardons
comma largernent compensées par les obstacles que l‘esclavage apporte a la multiplication
des noirs, dans les Etats du Sud.” [17, p. 5!. 12For instance, see Gloria Steinem in [12, pp. 2256}.
13For more information about eariy U. S. censuses and their social inaccuracy, see [3}. 1“See [12] and references therein. FREE!!!) March 1998 Volume VIII Number 1 polic...
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 Logistic function, original sources, President Yule

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