Exponential

A MathematicalFire

Exponential -

A MathematicalFire-Alarm

II

In today's world, a J-curve is the mathematical equivalent of afire-alarm going off in a burning building. The fact thatnential sequences, however, can so easily deceive us, combinedwith their power and counterintuitiveceedingly dangerous. Any numbers that mimic the fission eventsin a nuclear detonation should warrant our most serious attention.

When graphed, exponential sequences produce a characteristicJ-curve approximating the illusencounter a J-curve, we should react to the data that produced itjust as we would react to a fire alarm in a building.plosion, something powerfulgerous is underway, demanding thatand precautions be exercised without de

In our previous article ( Exponential 1)into $21,000,000 in thirty-oneduce a J-curve. If we graph the fission events that took place in thenear Alamogordo, New Mexico in 1945man population over the millennia between 8,000sults, like the one shown above,

The point should be unsettling, for there it is, set forth dramatically, mathematically, and inescapably: A graph of our own demographicsnuclear detonation. Thus, even though terms such as “a population explosion” or “abomb" seem frightening, they are actually a precise and mathematically accurate description of

Exponential -A Mathematical Fire

Because scientists work with exponential mathematics intheir fields of research and in their college curricula, theyare routinely cognizant of the late-phase characteristicsthat occur in exponential sequences. In contrast, however,the rest of us generally have little occasion to work withexponential progressions. As a result, many of earth's topscientists warn of a potentially catastrophic collision(their words) of humanity with earth's natural environment while the rest of us find it hard to believe that it isreally so.

If one is at work and hears a fire-alarm, the sound warnsthat potential danger exists and that precautionary measures are needed immediately. It signals that antion in ordinary conditions exists and attention to a potential crisis is urgently required. Anyone who ignoressuch an alarm does so at one's own peril.building, of course, occupants can all hear a fire alit sounds, so long as their hearing is npaired.

curve is the mathematical equivalent of aalarm going off in a burning building. The fact that expo-

nential sequences, however, can so easily deceive us, combinedwith their power and counterintuitive behavior, makes them ex-

Any numbers that mimic the fission eventsin a nuclear detonation should warrant our most serious atten-

When graphed, exponential sequences produce a characteristiccurve approximating the illustration depicted above. When we

curve, we should react to the data that produced itjust as we would react to a fire alarm in a building. Like an ex-

powerful, out of the ordinary, and highly dan-is underway, demanding that both emergency attentionautions be exercised without delay.

In our previous article ( Exponential 1) we saw that exponential mathematicsone days. If we graph the daily numbers in that progression,

If we graph the fission events that took place in the first test of a nuclear weaponnear Alamogordo, New Mexico in 1945, they too produce a J-curve. And if we graph earth's hu

millennia between 8,000 B.C. and the present day,, like the one shown above, is a J-curve.

The point should be unsettling, for there it is, set forth dramatically, mathematically, and inA graph of our own demographics over a span of ten millennia mimics the graph of

. Thus, even though terms such as “a population explosion” or “aing, they are actually a precise and mathematically accurate description of

IIA Mathematical Fire-Alarm

Because scientists work with exponential mathematics intheir fields of research and in their college curricula, they

phase characteristicsIn contrast, however,

the rest of us generally have little occasion to work withexponential progressions. As a result, many of earth's topscientists warn of a potentially catastrophic collision(their words) of humanity with earth's natural environ-

t while the rest of us find it hard to believe that it is

alarm, the sound warnsthat potential danger exists and that precautionary mea-sures are needed immediately. It signals that an interrup-

exists and attention to a po-tential crisis is urgently required. Anyone who ignores

n peril. In a burningcan all hear a fire alarm as

hearing is normal and unim-

we saw that exponential mathematics can change one centIf we graph the daily numbers in that progression, they pro-

first test of a nuclear weaponcurve. And if we graph earth's hu-

, the diagram that re-

The point should be unsettling, for there it is, set forth dramatically, mathematically, and in-nnia mimics the graph of a

. Thus, even though terms such as “a population explosion” or “a populationing, they are actually a precise and mathematically accurate description of

human demographics over a period of tengraphics follow a pattern whose graph mimics a graph of the fissionshima.

repeatedly warned government officials of the vulneraa powerful hurricane should strike the city.ed on luck and inaction to save the city, everyone involved ended up with a disastrous outcome asofficialdom failed to prepare

At Hiroshima, of course, the utter disasterwas relatively localized and took place ina matter of moments. On the other hand,most of our own detonation has occurredprimarily over the past 180 years insteadof a few minutes – and insteadtating a city and its residents, we are obliterating the natural systems and biota ofour planet in a way that makeexplosion is global in its impacts

We can use the graph shown rightflect upon the early stages of the Hiroshima detonation. Notice the region where the graph is still flat and rising slowly from the xlike an airplane rising from a runwaystages of the detonation, forthey seem to be harmless or unimportantHiroshima detonation did no damage to the city, its people, or its environment.not occur until shortly afterwards as the number of fission eventsing phases of the progression.

phics over a period of ten thousand years. The numbers that depict our demographics follow a pattern whose graph mimics a graph of the fission events

Our Moment in History

Of course, there are differences. During itsdetonation, the Hiroshima weapon exploded overa matter of seconds and flattened a city, whilemost of our own detonation has occurpast two centuries and we are flattening the biotaand natural systems of our planet. Addressingclimate change, climatologist Steonce noted that "it is precisely because theresponsible scientific community cannot rule out...catastrophic outcomes at a high level of confidence that climate mitigation policies are seriously proposed." He then goes on to assess thestatus of climate change this way: "We could belucky and see a mild effect or unlucky and getcatastrophic outcomes" (Schneider, 2002).

Faculty at Louisiana State University and othersgovernment officials of the vulnerabilities that the city of New Orleans faced if

ricane should strike the city. When city and government officials, however, countto save the city, everyone involved ended up with a disastrous outcome as

officialdom failed to prepare for reality.

At Hiroshima, of course, the utter disasterwas relatively localized and took place ina matter of moments. On the other hand,most of our own detonation has occurredprimarily over the past 180 years instead

and instead of devas-tating a city and its residents, we are obli-terating the natural systems and biota of

in a way that make our ownexplosion is global in its impacts.

We can use the graph shown right to re-flect upon the early stages of the Hiro-

Notice the region where the graph is still flat and rising slowly from the xlike an airplane rising from a runway. The city of Hiroshima was not dam

for in this region of the graph the growing numbers are still so small thatthey seem to be harmless or unimportant. The point to be made is this: TheHiroshima detonation did no damage to the city, its people, or its environment.

fterwards as the number of fission events skyrocketed upwardthe progression.

sand years. The numbers that depict our demo-that destroyed Hiro-

Our Moment in History

Of course, there are differences. During itsroshima weapon exploded over

a matter of seconds and flattened a city, whileof our own detonation has occurred over the

past two centuries and we are flattening the biotaand natural systems of our planet. Addressingclimate change, climatologist Stephen Schneideronce noted that "it is precisely because theresponsible scientific community cannot rule out...catastrophic outcomes at a high level of con-fidence that climate mitigation policies are seri-

proposed." He then goes on to assess theatus of climate change this way: "We could be

lucky and see a mild effect or unlucky and getcatastrophic outcomes" (Schneider, 2002).

Faculty at Louisiana State University and othersthe city of New Orleans faced if

ials, however, count-to save the city, everyone involved ended up with a disastrous outcome as

Notice the region where the graph is still flat and rising slowly from the x-axise city of Hiroshima was not damaged by the earliest

ng numbers are still so small thatearly phases of the

Hiroshima detonation did no damage to the city, its people, or its environment. The disaster didskyrocketed upward in the clos-

nuclear

Graphs of nuclear detonations (and human population growth frompresent) produce J-curves similar to the graph shown right.of the growth occurs, and most of the damage is done,progression as the graph rockets upwards. One example of this sort of exponentialprogression might be 1...2...4...8...16...32...64...128ply repeatedly (in this case, by two).

In our next article in this series,much of our schooling) produce in usmost everything in our world using a lineartine mathematics like the linear set depicted above

This is a problem because if wederstand and deal with problemserrors of catastrophic magnitude are virtually inevitable.

LinearProgression

Linear

If we examine a graph of humankind’sover the past ten millennia to 2011,of that history, our graph rose slowly from the xour most destructive impacts are occurring todaynumbers rocket upward along the y-axis

Our. moment in history is characterized by numbersare rocketing upward in the closing phases of annential progression, and we are all participants in the calamitous unfolding of events.

On the graph shown left, billions numbers 8, 9, 10, 11, 12, 13, 14, and 15 (plus800 million more after that) are based on the most recent U.N. “highworld population projections to 2100.

Linear versus Exponential

Our next two graphs (shown left and below)summarize several important differences between linear and exponential

This graph shown left results from agrows by repeated additions of like amountsof such a sequence might be 7....14....21....28....35....Notice that in this case, we added seven each time. In suchsequences, we can count on a steady and predictable progression in which tomorrow's conditions will be a continution of conditions that characterize yesterday and today.

Graphs of

detonations (and human population growth from 8,000 B.C. tosimilar to the graph shown right. In such sequences, most

and most of the damage is done, in the closing phases of theas the graph rockets upwards. One example of this sort of exponential

8...16...32...64...128 and 256, etc. in which we multi-

in this series, we will see that our intuitions (andour schooling) produce in us an inclination to interpret al-

using a linear mind-set based on rou-e linear set depicted above.

if we mtry tto use linear reasoning to un-derstand and deal with problems that are behaving exponentially,errors of catastrophic magnitude are virtually inevitable.

Progression

Linear

Exponential

examine a graph of humankind’s population growthwe see that for most

of that history, our graph rose slowly from the x-axis. Butare occurring today, as our

axis our graph.

characterized by numbers thatare rocketing upward in the closing phases of an expo-

, and we are all participants in the cal-

the graph shown left, billions numbers 8, 9, 10, 11, 12, 13, 14, and 15 (pluse based on the most recent U.N. “high-fertility”

inear versus Exponential

(shown left and below)summarize several important differences be-

exponential patterns.

results from a linear progression thatof like amounts. One example

7....14....21....28....35....42.... etc.this case, we added seven each time. In such

sequences, we can count on a steady and predictable pro-gression in which tomorrow's conditions will be a continua-tion of conditions that characterize yesterday and today.

Exponential

The perils inherent in exponential mathematics, however, are only apparent if our schooling en-sures a thorough understanding of the power and the deceptive nature of such progressions. Ifour teachers, textbooks, and curricula omit J-curves (and the behavior of the exponential pro-gressions that produce them) then we cannot hear the alarm they send and their dangers areinvisible.

Thousands of us will achieve success in life even if we are not experts at geometry and even if wenever hear of a fractal, a tessellation, a tangent, Andrew Jackson, or the value of pi. On the otherhand, if we do not understand the unique power and deceptive nature of J-curves and exponentialmathematics, then we imperil ourselves, our societies, and the survival of earth's biota and func-iioning systems.

Linear progressions are straight-forward and obvious and present themselves in a completelypredictable way so that we can count on relatively steady increments of change from one day tothe next. Exponential progressions, on the other hand, are extremely powerful and deceptive.And their numbers, so seemingly small and unimportant in the early phases of a progressionmultiply repeatedly in a catastrophic self-intensifying pattern that is typical of, among otherthings, both explosions and nuclear detonations.

An S-Curve

Some populations (but not all) exhibit a pattern ofpopulation growth called an S-CURVE, like the sig-moid-shaped curve shown here.

Notice that the initial stages of an S-curve beginwith an essentially exponential pattern of accelerat-ing growth during which births exceed deaths.

The arrow in the diagram, however, denotes a critical inflection point which marks the beginningof a “deceleration” phase, in which births still exceed deaths, but at an increasingly slower rate.This increasing deceleration in the rate of growth results from increasing environmental re-sistance and density-dependent feedbacks as limiting factors begin to affect the crowded pop-ulation more and more. Finally, notice that an idealized S-curve eventually flattens out, stabilizesand thereafter oscillates gently around an equilibrium in which births and deaths are essentiallyequal. .

The density-dependent feedbacks that cause growth to slow as crowding increases include nega-tive factors that tend to become worse as population densities increase. Thus, intensely crowdedpopulations face greater and more pronounced adversities such as: Accumulating wastes, mal-nutrition, limited resources, increased aggression, predation, competition, environmental destruc-tion, and exposure to epidemic disease. As crowding becomes more severe, such "feedbacks"commonly intensify, thereby causing rates of population growth to slow down more and moreuntil births and deaths finally begin to offset each other and the size of the population graduallyflattens out and becomes relatively stable for extended periods.

A retired physicist once told me, incorrectly, that humans do not follow a J-shaped curve. As acareer Ph.D., he was both highly intelligent and mathematically adept. He had also read enoughbiology to know that many animals with long lives and relatively small numbers of young(known as "k-strategists") commonly exhibit the stabilizing pattern of population growth epito-mized by an S-curve.

time

Our own data, however, between 8000already seen, there is no S-curve in that datathem yourself, generate the J

Graph above depicts human history to2011, and then shows U.N. mediumfertility population projections of 10billion by the end of this century.

Notice that BOTH of these graphs of human population growth arenumbers are rocketing upward along thetaken place in the last two hundred years, and with the bulk of that growth having ocpopulation milestone of two billioour numbers will have reachedby 2100. These graphs should be all the more sobering if one considers the effect that a Jon Hiroshima, Japan.

In nature, some sets of populations do exhibit ansee in the section that follows, climb rapidly,surge.

data, however, between 8000 B.C. and the present, do not support his claim. As we havecurve in that data -- our actual numbers, should you wish to graph

generate the J-curves shown below.

Graph above depicts human history to2011, and then shows U.N. medium-fertility population projections of 10billion by the end of this century.

This larger graph also depicts human history to2011, and then shows U.N. highlation projections to 15.8 billion

of these graphs of human population growth are. J-CURVES and that in both caseward along the explosive y-axis. Also notice that essentially all of our growth has

taken place in the last two hundred years, and with the bulk of that growth having ocstone of two billion in 1930. United Nations medium projections in May

our numbers will have reached NINE billion by 2043 and high-fertility projections send us toby 2100. These graphs should be all the more sobering if one considers the effect that a J

ome sets of populations do exhibit an S-curve, but many other populasee in the section that follows, climb rapidly, climb too far, and suddenly collapse

do not support his claim. As we haveour actual numbers, should you wish to graph

This larger graph also depicts human history to2011, and then shows U.N. high-fertility popu-

15.8 billion by 2100.

and that in both cases, ouraxis. Also notice that essentially all of our growth has

taken place in the last two hundred years, and with the bulk of that growth having occurred since ourn in 1930. United Nations medium projections in May 2011 estimate that

fertility projections send us to 15.8 billionby 2100. These graphs should be all the more sobering if one considers the effect that a J-curve once had

but many other populations, as we willcollapse as death rates

Climb-and-collapse disasters exhibiter, a population exceeds one or more critical environmentalThe exuberant growth then suddepicted in the graph below is after a quintessentialmented in a herd of reindeer on St. Paul Island, Alaska

Twenty-five reindeer were introduced to the 41wolves, predators, or competitors for the reindeer, so thattheir numbers peaked at more than 2000 individuals by 1938population peak). Following that peak, however, the ensuingwiped out more than 99% of the herd resulted in only eight surviving reindeer by the close of thestudy.

Another factor should also be quite disquietingulation, the combined bodies of all of the reindeer physicallypercent of the total island area that appeared to remain theoreticallyonly did a population of mammals undergo climbin seemingly “vast open-space”be ALMOST ENTIRELY EMPTY

Climb and Collapse

disasters exhibit exponential growth during their initial phasesone or more critical environmental thresholds, tipping points, or

suddenly ends, followed by a quick and massive diedepicted in the graph below is after a quintessential climb-and-collapse population study docu

eer on St. Paul Island, Alaska (after Scheffer, 1951)

five reindeer were introduced to the 41-square-mile island in 1911. The island had nowolves, predators, or competitors for the reindeer, so that their numbers grew exponentially untiltheir numbers peaked at more than 2000 individuals by 1938 (notice the J-curve that leads to their

Following that peak, however, the ensuing precipitous die-of the herd resulted in only eight surviving reindeer by the close of the

ctor should also be quite disquieting (and instructive), for at the time of their peak popbodies of all of the reindeer physically-occupied less than

of the total island area that appeared to remain theoretically-available. In other words, notonly did a population of mammals undergo climb-and-collapse, but their 99% die

space” conditions in a surrounding environment that visually appeared toEMPTY.

ing their initial phases, until, howev-thresholds, tipping points, or limits.ck and massive die-off. The data set

e population study docu-(after Scheffer, 1951).

mile island in 1911. The island had notheir numbers grew exponentially until

curve that leads to their-off and collapse that

of the herd resulted in only eight surviving reindeer by the close of the

(and instructive), for at the time of their peak pop-less than 2/1000ths of one

available. In other words, notbut their 99% die-off took place

nt that visually appeared to

In a follow-up to Scheffer’s classic study,tous climb-and-collapse outcome in a secondKlein's data is depicted in the graph belowreindeer died as the collapse occurredalso occurred in an environment that was

The Open-space Delusion

We have just seen two classical examples of population climb-and-collapse calamities and 99% dieseparate and independent studies of realmalian populations. And subsequentalyses have shown that both of these collapsesdependently, began when the combined bodies of allmembers of each herd physically.2/1000ths of 1%. of seemingly “vast openappeared to remain theoreticallyonment that visually appeared to remainly empty (illustration shown right depicts 2/10001%) (Note that even an intelligent species might find itdifficult to believe that danger is imminent when suchenormous quantities of open-

Remarkably, exactly the sametions also characterize deadly marias dinoflagellate red-tides.sions of dinoflagellates such asample, all one million one-celled individuals in a oneliter sample, combined, physically2/1000ths of 1% of the one-liter sample in which theyreside. Since the dinoflagellates induce environmentalcalamity by their release of wastes into their surroundings, we might, perhaps, find such information instructive since our own species exhibits a remarkably similar pattern of behavior.

194429

Reindeer

up to Scheffer’s classic study, D. R. Klein (1968) documented ancollapse outcome in a second reindeer study on St. Matthew Island, Alaska.

depicted in the graph below. Notice that in this study also, more than 99% of thereindeer died as the collapse occurred. (And subsequent analysis has shown that this collapsealso occurred in an environment that was 99.998% empty.)

As Klein noted in his paperthe experiment resulted in “ly complete annihilation of theherd.” In our graph offer’s results (previous page)that the same thing happened. Andsince a graph ofcurve is, if anything,treme than the graph of either reindeer herd, perhaps wearize ourselves withties, limiting factors,delayed feedbacksnamics.

space Delusion

We have just seen two classical examples of popula-calamities and 99% die-offs in

separate and independent studies of real-world mam-malian populations. And subsequent mathematical an-

both of these collapses, in-began when the combined bodies of all

physically-occupied less thanof seemingly “vast open-spaces” that

appeared to remain theoretically-available in an envir-visually appeared to remain almost entire-

ly empty (illustration shown right depicts 2/1000ths of1%) (Note that even an intelligent species might find it

that danger is imminent when such-space appear to remain.)

same 2/1000ths of 1% condi-tions also characterize deadly marine outbreaks known

. During population explo-sions of dinoflagellates such as Karenia brevis, for ex-

celled individuals in a one-liter sample, combined, physically-occupy less than

liter sample in which theySince the dinoflagellates induce environmental

calamity by their release of wastes into their surround-ings, we might, perhaps, find such information instruc-tive since our own species exhibits a remarkably simil-

196442

Reindeer

1963 - 6000

Reindeer

time

D. R. Klein (1968) documented an even more precipi-on St. Matthew Island, Alaska.

, more than 99% of theAnd subsequent analysis has shown that this collapse

As Klein noted in his paper (1968),the experiment resulted in “the near-

mplete annihilation of thegraph of V.B. Schef-

fer’s results (previous page), we sawthat the same thing happened. Andsince a graph of our own populationcurve is, if anything, even more ex-

than the graph of either rein-deer herd, perhaps we should famili-

ourselves with carrying capaci-ties, limiting factors, overshoot, and

backs in population dy-

(Unfortunately our own species does not confine itself toand biological wastes into our surroundings. Instead, weway that is utterly unprecedented in the history ofand ever-increasing avalanche ofbarked upon a worldwide trajectory that is not onlbut is multiple orders of magnitude

No other animals do this, and no other animals in the entire history of the earmented their biological wastes in the way that we donot an incidental or minor footnote to the biology of our speciesical eradication of ever-expandingknown to exist anywhere in the unicharacteristics.”

(Unfortunately our own species does not confine itself to releasing only our cellular, metabolic,and biological wastes into our surroundings. Instead, we supplement our biological wastes,way that is utterly unprecedented in the history of life on earth, with a daily,

avalanche of billions of tons societal and industrial wastesbarked upon a worldwide trajectory that is not only worse than that of an outbreak of red

multiple orders of magnitude worse, at that.)

No other animals do this, and no other animals in the entire history of the earmented their biological wastes in the way that we do. “And our exceptionality in this respect isnot an incidental or minor footnote to the biology of our species – but, along with our sheer

expanding portions of the only planetary life-support machinery so faranywhere in the universe, it is one of our most pronounced and all

(What Every Citizen Should Know About Our Planet, 2010)

Thus, population explosions of mindlessdinoflagellates induce calamity uponand the environment in which they livethe reindeer herds experienced 99% diemathematical assessment of each example shows thatin each case, the organisms making up the crisisulation physically-occupied less thanof the environments in which they livewords, in environmental conditions that visually appeared to be

e.almost entirely empty

Not to worry? Because, after all, we area herd of reindeer or a population of mindless onecelled dinoflagellates,

aren’t we?

(1) Unless, of course, the subject of the exam is exponential mathematics, demographics,tween a million and a billion, biospherics, wholetems ecology, the open-space delusion, and 2/1000thsof 1% … and …

(2) Unless we employ our technologies and ingenuityin ways that make us more efficient at exploititroying, eradicating and damaging earth’smachinery more completely, efficiently, and rapidlythan those systems can repair, susthemselves - (in other words, faster, more efficiently,and more completely than ever -earth movers, and bulldozers clearthat was once logged by hand, for examplesatellite tracking, sonar, and fishing technologies thatpermit a fishing industry to catch fish faster than thefish themselves are at reproducing and maturing).

releasing only our cellular, metabolic,our biological wastes, in a

, ongoing, worldwide,societal and industrial wastes, so that we are em-

than that of an outbreak of red-tide,

No other animals do this, and no other animals in the entire history of the earth have ever supple-And our exceptionality in this respect is

along with our sheer phys-support machinery so far

it is one of our most pronounced and all-encompassing

ld Know About Our Planet, 2010)

population explosions of mindless one-celledupon both themselves

and the environment in which they live, while both ofthe reindeer herds experienced 99% die-offs, while amathematical assessment of each example shows that

, the organisms making up the crisis pop-less than .2/1000ths of 1%

of the environments in which they lived – in otherwords, in environmental conditions that visually ap-

empty.

we are smarter thana herd of reindeer or a population of mindless one-

aren’t we?

Unless, of course, the subject of the exam is expo-mathematics, demographics, the difference be-

tween a million and a billion, biospherics, whole-sys-space delusion, and 2/1000ths

Unless we employ our technologies and ingenuityin ways that make us more efficient at exploiting, des-

ng earth’s life-supportmachinery more completely, efficiently, and rapidlythan those systems can repair, sustain, and perpetuate

in other words, faster, more efficiently,think of chain saws,

earth movers, and bulldozers clear-cutting a forestthat was once logged by hand, for example – or ofsatellite tracking, sonar, and fishing technologies thatpermit a fishing industry to catch fish faster than the

e at reproducing and maturing).

Our own numbers from 8000comfortable S-curve that causes some to suppose that all is wellalthough there is some minor retreat from ourresulting from stringent one-world nations of western Europe),.227,000. additional people to world population every day, almostmonth, and one billion additional

An actual graph of our demographics over twarranted, for there is no S-curve there

If we set aside our denials, our hopes and our wishes and simply graphographic history and then examine the resee shows that we are living in the closingalong its y-axis over the course of the past 180 years

Perhaps you recall a popular reggae tuneadmonishes us "don't worryof population growth, for some reason, almost neverappears on television. And, over a span of three decades, it has rarely, if ever, been brought up in a presidential news conference or in annumessages. Occasionally, of course, we may see a newspaper article or a book that addresses population.fortunately, however, many of these seemadvance special economic interests and to omit thenumbers and principles that we are addressing here.

Numbers making up an exponential sequencefire-alarm going off in a burning buildingarm can only be heard if some teacher, somewhere,teaches us the dangerous, powerful and deceptivehavior of exponential number sequences.new homework set, we are wiring our students’ brainsso completely with 1930s arithmetic that we leave themill-equipped to deal with, or even to perceive, theworld" mathematics of this 21

The Decades Just Ahead

As young people today live their lives in this 21ry, there are fundamental mathematical skills that each of them needs to master. For those whowill become scientists and engineers, quadraticfessional necessities. But what about the real world mathematics needed byzen? Civilization and our planet will all benefit if school districts, teachers, and publishers ensurethat “real-world” demographic, numeric, and exponential topics are mastered in every math, science, and social studies classroom.

Don't Worry - Be Happy

Our own numbers from 8000 B.C. to 2050 A.D., when graphed, emphaticallycurve that causes some to suppose that all is well – at least not so far. And

minor retreat from our percentage rates of three decades ago (largely-child policies in China and a general stabilization in the

world nations of western Europe), on a worldwide basis, we are still adding approximatelyadditional people to world population every day, almost seven million additional every

dditional every twelve to fifteen years.

n actual graph of our demographics over the past 10,000 years shows optimism to becurve there – at least not in the data that brings us to the present day

If we set aside our denials, our hopes and our wishes and simply graph the entirety ofthen examine the resulting graph as objectively as possible, what we actually

we are living in the closing stages of a J-curve that has been skyrocketing upwardover the course of the past 180 years

in a worldwide trajectory that is shockingly similarthe fission events inside the detonation that devastated Hiroshima

Perhaps you recall a popular reggae tune that cheerfullyadmonishes us "don't worry – be happy." The subjectof population growth, for some reason, almost never

vision. And, over a span of three dec-ades, it has rarely, if ever, been brought up in a presi-

or in annual state of the unionOccasionally, of course, we may see a news-

book that addresses population. Un-fortunately, however, many of these seem intended to

interests and to omit the rawthat we are addressing here.

Numbers making up an exponential sequence are like aalarm going off in a burning building – but this al-

arm can only be heard if some teacher, somewhere,es us the dangerous, powerful and deceptive be-

f exponential number sequences. With eachnew homework set, we are wiring our students’ brainsso completely with 1930s arithmetic that we leave them

equipped to deal with, or even to perceive, the "real-21st century.

The Decades Just Ahead

live their lives in this 21st centu-there are fundamental mathematical skills that each of them needs to master. For those who

will become scientists and engineers, quadratic equations and polynomial expansions will be professional necessities. But what about the real world mathematics needed by

zation and our planet will all benefit if school districts, teachers, and publishers ensureworld” demographic, numeric, and exponential topics are mastered in every math, sci

ence, and social studies classroom.

, when graphed, emphatically do not generate theat least not so far. And

rates of three decades ago (largelya general stabilization in the richer first-

adding approximatelyseven million additional every

he past 10,000 years shows optimism to be most un-that brings us to the present day.

the entirety of our dem-sulting graph as objectively as possible, what we actually

e that has been skyrocketing upward

in a worldwide trajectory that is shockingly similar to a graph offission events inside the detonation that devastated Hiroshima

there are fundamental mathematical skills that each of them needs to master. For those whoequations and polynomial expansions will be pro-

fessional necessities. But what about the real world mathematics needed by each and every citi-zation and our planet will all benefit if school districts, teachers, and publishers ensure

world” demographic, numeric, and exponential topics are mastered in every math, sci-

At this precarious and decisive moment in history, we must ensure that tin the world’s classrooms, the openthe first questions on our standardized examsunderstandings that we need to navigate and mitigate the crises that lie ahead. After all, how caone lead wisely if one does not know:

The powerful and deceptive behavior of J

The difference between a million and a

That we currently add more thanthree days?

That we are addingyears?

And that current rates of population growth require us to complete more than 32,000additional classrooms every four dayscentury just ahead?

A continuationthe greatest single

Sources and Cited References…pending…

Anson, 2009.Anson, 1996.Campbell, et al., 1999Cohen, 1995Cohen and Tilman, 1996.Dobson, et al., 1997Duggins, 1980;Estes and Palmisano, 1974Mader, 1996Mill, J.S., 1848.Pimm, 2001.Prescott, et al., 1999.Raven, et al., 1986Wilson, 2002.

At this precarious and decisive moment in history, we must ensure that the opening presentationssrooms, the opening pages of math, science, and social science

the first questions on our standardized exams address the real-world mathematicsunderstandings that we need to navigate and mitigate the crises that lie ahead. After all, how caone lead wisely if one does not know:

The powerful and deceptive behavior of J-curves and exponential mathematics

The difference between a million and a billion?

That we currently add more than 600,000 additional people to world population every

That we are adding one billion additional persons to our planet every twelve to fifteen

And that current rates of population growth require us to complete more than 32,000additional classrooms every four days repeatedly and endlessly throughout the half

A continuation of today’s demographic tidal wave may constitthe greatest single risk that our species has ever undertaken

What Every Citizen Should Know About Our Planet

Copyright 2009. Randolph Femmer.All rights reserved.

ources and Cited References

he opening presentationsing pages of math, science, and social science textbooks, and

world mathematics and real-worldunderstandings that we need to navigate and mitigate the crises that lie ahead. After all, how can

curves and exponential mathematics?

ple to world population every

additional persons to our planet every twelve to fifteen

And that current rates of population growth require us to complete more than 32,000throughout the half-

may constituterisk that our species has ever undertaken.

Excerpted fromWhat Every Citizen Should Know About Our Planet

Used with permission.