Global Change Seminar

Kurt Kreith

kkreith@ucdavis.edu

Teachers CollegeJune 25, 2012

NOTICES of the American Mathematical Society Mathematics of Planet Earth (http://mpe2013.org)

Jeffrey Sachs on Information Technology and Sustainable Development (www.tc.edu/news.htm?articleID=8475)

Royal Society report on People and the Planet (royalsociety.org/policy/projects/people-planet/report)

What are their implications for teacher preparation?

Today's gathering is consistent with some noteworthy events:

"Encourage mathematics teachers at all levels to communicate issues related to Planet Earth through their instruction and curriculum development"

More specifically, what is it that -

• Students should know about the world they are soon to inherit?

• Teachers should know in order to cultivate such understanding?

And given answers to these questions -

• How should policy makers and teacher educators act on them?

One Possible goal: Develop "The Mathematics of Global Change" as a blended online courses offered in 2013 by Teachers College/UC Davis.

In this context• What should students know about the world they are soon to inherit?

Students should be numerate about the world.

How large is the earth?How many of us share its resources?What would global equity look like?

Mathematics content standards are a sensitive matter.

For many people, thefuture of planet earth is

a matter of deeplyheld beliefs.

"Everyone is entitledto his own opinions butnot to his own facts." --Patrick Moynahan

• Students should be numerate about the world.• Students should be numerate about global change.

Aaron and Betty use Excel to keep weekly records of their savings. Aaron records the amount he saves in the course of each week, while Betty records the amount she has accumulated at the end of each week. How can they reconcile their records?

Aaron and Betty have just discovered the Fundamental Theorem of Calculus (albeit in discrete form). Given this form of numeracy they are able to deal with linear growth, exponential growth, and a host of mathematical concepts traditionally thought of as "calculus based."

• Students should be numerate about the world.• Students should be numerate about global change.

• Students should be cognizant of forms of global change that we encounter in every day life.

Why is it hotter in summer than winter?

What is the transit of Venus?

Are we really spinning at 750 mph?

A discussion of such phenomena can be a good lead-in to thinking about planet earth abstractly. It can also be used to convey an appreciation for the turmoil and controversy that accompanied arriving at an understanding of these forms of global change.

What is a lunar eclipse?

What is it that teachers should know in order to cultivate such student understanding?

Basic geometry and proportional reasoning are powerful tools for developing global numeracy:

• Eratosthenes at the well in Syene.

• In New York the north star is 40° above the horizon. In Toronto, 400 miles north of NYC, its inclination is 46°. How large is the earth?• A ship whose mast extends 52.8 feet above the water sails from Genoa at 3 mph. After three hours its flag disappears from view. How large is the earth?

The internet provides a wealth of reliable data regarding the earth, its people, and its resources.

.01 miles

What else might teachers know about global change?Each year the AMS sponsors an Einstein Lecture, and in 2010 Terrence Tao used this occasion to talk about "The Cosmic Distance Ladder." Step 1 was "Eratosthenes measures the size of Planet Earth." Soon after Step 3 the Copernican Revolution takes center stage.

Synodic vs. Sidereal views

Perceptions are deceptive

Revolutions are messy affairs"The Mathematics of Global Change" may be party to a similar drama that is yet to be played out.

Is there "a Copernican Metaphor"?

What about mathematical knowledge?Teachers should know about exponential growth, but they should also be prepared to "move beyond Malthus."

1. Since 1960, the growth of world population has been essentially linear.

by one million every five days!

2. Meaningful forms of demography require an ability to break populations into cohorts.

Age cohorts

Human population has increased by one billion every 13 years;

This observation is at the heart of the Royal Society study.

Economic cohorts

Many teachers include it now!

But isn't this topic too advanced for schools?

1, 1, 2, 3, 5, 8, 13, …Everybody loves to teach about the Fibonacci numbers.They have their origins in a problem about a fecund breed of rabbits in which pairs have a pair of babies each month.

That sounds a lot like exponential

growth at 100%

But only adult pairs have babies, and babies take a month to mature.

Fibonacci's population consists of two (age dependent) cohorts. Instead of growing at 100%, this population grows at about 61.8%.

Injecting delay reduces the exponential growth rate.

So eliminating delay should

increase the growth rate!

Fibonacci's problem also allows us to introduce "rules for change" based on matrix arithmetic.

u(n) =

€

a(n)

b(n)

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=1 1

1 0

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⎦ ⎥a(n -1)

b(n -1)

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⎤

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The 2x2 "transition matrix" embodies Fibonacci's problem.

all adults survive all babies mature

adult fertility is 100% babies can't have babies

Matrix equations of the form u(n) = Tu(n-1) provide a powerful tool for formulating and solving rules for change.(In demography, Fibonacci's rabbits lead naturally to an important tool called the Leslie matrix.)

Another way of "moving beyond Malthus" is to introduce logistic growth. Suppose you have money in a bank that pays an annual interest rate R. Letting u(n) denote your balance at the end of the n-th year, your balance will grow exponentially according to

u(n) - u(n-1) =

u(n) = (1+R)u(n-1)u(n) = u(0)(1+R)n

Pierre Francois Verhulst, a contemporary of Malthus, modified this rule for exponential growth by calling for an annual fee E based on the square of your balance.based on the square of your balance.

u(n) - u(n-1) = Ru(n-1) - Eu(n-1)2

u(n-1)R

What will happen to your money under this rule?

Can it be that the fee will equal interest?

Eu2 = Ru

If u > E/R, fee exceeds interest.

Eu = R u = E/R

Your balance will grow toward, but never exceed, E/R.

Students who are numerate about global change can discover this for themselves:

Logistic growth is a staple in mathematical ecology, but its significance may be overstated. The "sigmoid graph" we identify with logistic growth arises in many contexts.For example, given a nonrenewable resource whose annual production rate a(n) is bell shaped, its cumulative production A(n) will be sigmoid.

Aaron and Betty are now able to understand how, in 1956, M. King Hubbert is said to have arrived at his prediction of "peak oil" (aka Hubbert's Peak).

While logistic growth may be one of many different forms of sigmoid growth, it can be useful as a prototype. In a political context it may suggest a grim "no growth" future. But in population dynamics, it corresponds to a "soft landing" at the carrying capacity of a given environment.

What, on the other hand, can lead to overshoot?

A suggestive model for overshoot is reminiscent of Fibonacci. Lesser species may be forced to adapt to their environment in accordance with

u(n) - u(n-1) = Ru(n-1) - Eu(n-1)2- Eu(n-1-d)2

but clever species will find ways of dealing with the nonlinear environmental damping term.

Aaron and Betty are able to ponder such rules for change:

Fibonacci has enabled us to deal with age cohorts and delays. Both of these are important concepts that calculus intensive modelers may be tempted to overlook. But what about the economic cohorts that figure so heavily into the Royal Society study? At this point the situation becomes complicated!

globalizationeconomic

development migration

While admittedly part of a bigger picture, migration is a phenomenon that is interesting to single out. People move from where it is worse to where it is better - much like the flow of heat in a rod!

150200 80148191 94

••• •••

146.9etc.

Of course, such drudgery can be relegated to a spreadsheet. Consider seven people seated in a circle.un(t) - un(t-1) = .1[un-1(t-1) - un(t-1)] + [un+1(t-1) - un(t-1)]

un(t) - un(t-1) = .1[un-1(t-1) - 2un(t-1) + un+1(t-1)]

€

∂u∂t

= c∂2u∂x2

with c = .1

Consider seven people seated in a circle.

un(t) - un(t-1) = .1[un-1(t-1) - un(t-1)] + [un+1(t-1) - un(t-1)]

un(t) = .8un(t-1) + .1un-1(t-1) + un+1(t-1)

€

.8 .1 0 0 0 0 .1

.1 .8 .1 0 0 0 0

0 .1 .8 .1 0 0 0

0 0 .1 .8 .1 0 0

0 0 0 .1 .8 .1 0

0 0 0 0 .1 .8 .1

.1 0 0 0 0 .1 .8

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=

€

u1(t -1)

u2 (t -1)

u3(t -1)

u4 (t -1)

u5 (t -1)

u6 (t -1)

u7 (t -1)

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€

u1(t)

u2 (t)

u3(t)

u4 (t)

u5 (t)

u6 (t)

u7 (t)

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We are back to u(n) = Tu(n-1) where T is a stochastic matrix.

In a migration context, air travel eliminates the constraints of geometry and dimension.

=

More general models, allowing for growth within cohorts and varying degrees of "conductivity," will resemble a stochastic process.

Again, the transition matrix is (doubly) stochastic and diffusion resembles a Markov chain. €

1- 6c c c c c c c

c 1- 6c c c c c c

c c 1- 6c c c c c

c c c 1- 6c c c c

c c c c 1- 6c c c

c c c c c 1- 6c c

c c c c c c 1- 6c

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€

u1(t)

u2 (t)

u3(t)

u4 (t)

u5 (t)

u6 (t)

u7 (t)

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u1(t -1)

u2 (t -1)

u3(t -1)

u4 (t -1)

u5 (t -1)

u6 (t -1)

u7 (t -1)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

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Synodic View

Category Terrestrial Change

Great Book

Technology

Orthodoxy

Geocentrism

Celestial Change

Heliocentrism

De Revolutionibus…

Telescope

Religion

Sidereal View

Martyrs Giordano Bruno

Heroes

Is there "a Copernican metaphor?"

Computers

Limits to Growth

Context Determinism Free Will ?

The Mathematics of Global Change

Math 98 at UC Davis in Winter, 2013.