PEARSON MATHEMATICS WEBINAR April 7, 2011 Gary Rockswold, Ph.D
Emeritus Professor of Mathematics Minnesota State University,
Mankato
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In mathematics we spend a lot of time covering the details.
After all, putting the decimal point in the correct place can make
a big difference! Photo by Wonderlane
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Arithmetic, factoring, solving equations, Its tough to do it
all. Most of our time is spent studying the trees. Less time is
spent studying the forest. By Grant MacDonald
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In this presentation we will look at the forest. This view is
essentialparticularly when we are communicating with students and
public! Travis S. Flickr
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Math in the Natural Sciences
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Mathematics usage in the natural sciences is more well-known.
Many of these examples show how we are interpreting the world more
and more in terms of mathematics. By CERN Large Hadron Collider
(LHC)
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In one word: Higgs The Higgs particle is important because it
gives objects in the universe mass. There is no experimental
evidence for the Higgs particle.
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The symmetry in the equations suggests that it exists. In 1964
Peter Higgs proposed the Higgs particle in a paper that contained
only 79 lines and 5 equations.
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Scientists hope to create black holes in the LHC. Fortunately,
Stephen Hawking (1974) has shown mathematically that tiny black
holes will disintegrate in a fraction of a second. Scientists
really do trust mathematics!
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The Unreasonable Effectiveness of Mathematics in the Natural
Sciences by Eugene Wigner in Communications in Pure and Applied
Mathematics, February 1960
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Mathematics can lead us in a direction we would not take if we
only followed up physical ideas by themselves. --Paul Dirac
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Mathematics in the Social Sciences
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Next we will concentrate on the amazing impact that mathematics
is having in the social sciences. Social sciences often deal with
human behavior and the choices that people make. By Viola
Damiani
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We often think that behavior is unpredictable. But is behavior
outside the realm of math? Modeling the choices that people make is
one way to predict behavior. By MinivanNinja Flickr By OrphanJones
flickr
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Most of our students major in humanities or social sciences.
Math is meaningful and it is important to show how math relates to
their lives.
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Consider teaching the concept of intersection in a Venn diagram
in todays classroom. People wearing socks People wearing sandals MY
DAD!!!
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A Famous Example A Famous Example
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Mathematics can predict the next shot that an excellent pool
player will make with a high degree of accuracy. By misswired
flickr
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By sbma44 flickr Does this mean that a pool player is actually
using trigonometry, algebra, vectors, and properties of physics to
determine the next shot? Probably not But does this mean that our
mathematical model is incorrect and not worthwhile?
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Math is not modeling reality in that math is not modeling how
the pool player does it, but somehow mathematics is modeling the
pool players choices. Whatever is happening, there is an
equivalency: Pool players choices vs. Mathematical calculations By
misswired flickr By sbma44 flickr
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Modeling Non-Rational Choice
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Question: Is it better to give your significant other two $100
presents or is it better to give one $200 present? Next Question:
If a person wins $50 and then gets a $25 parking ticket, do they
feel happy because the net gain is $25? By puggles flickr By Bahi P
Flickr
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People weigh losses significantly more than they do gains.
(About double) People demonstrate loss-averse behavior. This
somewhat non-rational behavior is modeled by the following value
function. By Dale Moore
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Modeling: Losses and Gains versus Value
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Yes, people forget to send them in, but theres more. The small
rebate (gain) gives more value than does tacking on the additional
small loss to the already high price. Peoples value functions are
concave not linear!
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The Recent Debate in Economics
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How Did Economists Get It So Wrong? by Paul Krugman, Nobel
Laureate The New York Times 9/6/2009 By David_Shankbone flickr
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As I see it, the economics profession went astray because
economists, as a group, mistook beauty, clad in impressive-looking
mathematics, for truth. --Paul Krugman, Nobel Laureate
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I am in shocked disbelief because the whole intellectual
edifice has collapsed. -- Alan Greenspan, October 2008 By
trackrecord flickr
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Math continues to guide our thinking: Only after doing the
math, was I able to express my models in plain English. It really
took the equations to understand what was happening. -- Paul
Krugman, Nobel Laureate
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Social Networking
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Social networking is transforming our society. Social
networking is generational. Should this affect how we teach math?
By Emilie Ogez flickr
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However, Google reads our email and sends us advertisements
based on keywords in our private messages. Our privacy is a
currency. For example, Google charges nothing for gmail. By
Gubatron Flickr
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Three-quarters of Facebook users do not live in the U.S. The
privacy issues surrounding Facebook are global issues. In some
sense: Sharing data is the point of Facebook. By @superamit
flickr
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Using mathematics, it is possible to analyze a persons facebook
friends and determine with reasonable certainty things like
hobbies, interests, health issues, and even sexual orientation.
This information is used to target advertisements toward a subset
of consumers. If businesses analyze social networks they can know
exactly where to insert their campaign so that it fuels itself and
reaches people who care. More effective than 30-second TV
commercials
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The Mathematics of Social Networks
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Max Levchin (CEO of Slide) is the number one creator of
applications for Facebook. He stated the following about his
success. Our competitive advantage is actually our math skills,
which is probably not something you would expect from a media
company. --Max Levchin, CEO of Slide
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People are vertices tied by friendship, interest, membership,
or beliefs. Person 1 and 2 are friends with each other. Person 3
likes everyone, but no one likes Person 3. 2 1 3 5 4
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A square adjacency matrix can be used to summarize this simple
social network: 2 3 1 5 4
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Matrices are often square, but not necessarily symmetric. If a
ij = a ji, then friendships are mutual. By looking at each row and
column, people with larger social networks can be identified as
having many 1s. The sums of columns and rows have meaning.
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The computation A can be used to determine people who have a
social distance of 2. 2 1 3 4 5
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The following graph shows a social network. People with
centrality become apparent, as do loners.
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One concept can be used to simplify a vast array of complicated
problems. Problems solved by adjacency matrices: Transportation
problems/shortest paths Internet pages and links Social networks By
HH- Michael flickr By ButchLebo Flickr
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VIDEO: From Vermont Public Television: Features Peter Dodds,
University of Vermont Mathematics, Data, and Social Networks
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Using Mathematics to Keep Secrets Safe
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Cryptography as been around for a long time as a way to hide
information. Using math, groups of people now compute virtually
anything from their collective data without revealing any
individual data.
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155 210184549 lb ++= Suppose that three people have individual
weights of 155 lb., 210 lb., and 184 lb. They want to calculate the
groups total weight without revealing their individual
weights.
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Each person privately selects three numbers between 0 and 1000,
whose sum equals their weight modulo 1000. For example, the 155 lb.
person could use That is, 1155 mod 1000 = 155 155 255 563 337 1155
+ + =
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First person (155 lb): 255 + 563 + 337 = 1155 Second person
(210 lb): 437 + 493 + 280 = 1210 Third person (184 lb): 323 + 601 +
260 = 1184 155 255 337 493563 280 184210 437 601 260 323
115512101184
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Each person distributes two of their numbers to each other and
keeps one private. 155 255 337 493563 280 184210 437 601 260 323
115512101184 437 260 337 255 493 601 323 563 280
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Each one adds their three new numbers and simplifies it modulo
1000. 34 mod 1000 155 255 337 493563 280 184210 437 601 260 323
115512101184 437 260 337 255 493 601 323 563 280 Add the 3 new
numbers 103413491166 === 349 mod 1000 166 mod 1000
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Now they share and add these three numbers. This sum gives
their combined weight. 155 255 337 493563 280 184210 437 601 260
323 115512101184 437 260 337 255 493 601 323 563 280 103413491166
=== 349 mod 1000166 mod 100034 mod 1000 34 + 349 + 166 = 549 +
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Other Applications in Social Sciences
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A group of Canadian mathematicians are going to Africa to model
the spread of infectious diseases, such as AIDS. Understanding
social networks is essential. By max_thinks_sees flickr
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About 15 major league teams, including the Yankees, have a
fulltime person whose only job is to do statistical crunching.
Locating team deficiencies is essential to winning. By Ohad*
flickr
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In the digital world, style is mathematical. By analyzing
vertical, horizontal, and diagonal brush strokes from a digital
print, they have been able to identify fakes of Van Gogh. By Jankie
flickr
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How do you gather food and stay safe from predators? Do you
spend time being innovative or copying others? Two graduate
students showed mathematically that Conquering by Copying is
superior to innovation with regard to survival. Not what we want to
teach to our students! By Myrrien flickr
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Some Conclusions
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Social scientists used to rely on intuition, but human
intuition can go wrong. It is difficult to keep track of every
factor in the interaction of millions of human beings. -- Viridiana
Rios
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New applications of math are constantly appearing because
peoples behavior is always changing. Any time there is a large
group of people behaving in a similar manner, there is mathematics.
Ive been involved professionally with mathematics for almost 40
years and . Math is more than a tool. It greatly influences how we
discover/create our realities.
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The Essence of Mathematics
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What Is The Essence of Mathematics?
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There is simplicity in mathematics: a + b = c By sogni_hal
flickr
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Most mathematics can be discovered by using only paper and
pencil. By quacktaculous flickr
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Yet amazingly, mathematics describes the complex world we live
in. By ecstaticist flickr
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The essence of mathematics is to.
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Solve many complicated problems with one simple, but profound
concept.
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Social Networking By MikeBaird Flickr
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Transportation Networks By Paraflyer flickr
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By adrenalin Flickr The Internet
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Dr. Flemming Flickr Fluid Flowing
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Forces By Scott Butner Flickr
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Electricity [Lightning? Laser Light show?] Electricity By
EpicFireworks Flickr
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F (x) Understanding the infinite By cindy47452 flickr
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Motion By Devils Rancher Flickr
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The Universe [NASA photo] Our Universe
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Change in Society
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The ESSENCE of Mathematics is
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To make complicated things simple.
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To ignore either the abstract beauty or profound applicability
of mathematics is like
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seeing a rose but never smelling one.
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Thank you for attending!
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Slides and a narration of this presentation can be found at:
www.garyrockswold.net
