# Topics for All Students

### Ruminations on how to teach mathematics for the liberal arts

Mathematics for Discovery (Understanding)

'The purpose of computing is insight, not numbers,' R.W.Hamming.

• Topics
• Linear Growth
• slope and y-intercept
• regression(below)
• proportionality
• cost of car ownership: including fuel// fuel increase more important at lower mpg// using 1/mpg=gpm
• Exponential Growth
• doubling times and contrast to linear growth
• caused by compounding percentage increase
• Interest, loans and mortgages, adjustable rates, credit cards
• population growth and (un)sustainability
• (un)sustainability of exponential growth (use of resources,etc.)
• Logistic Growth
• population growth, predicting the ultimate carrying capacity of planet earth (do we want to get there?)
• peak oil predictions
• growth of alternative energies?
• chaos (see below)
• Statistics
• measuring the worlds inequality - black vs white in US(prisons, health, income), world income levels
• curve fitting and least squares
• hypothesis testing and estimation??
• Anonymous Surveys
• Dynamical Systems - models of change ??
• population growth
• resource usage modeling
• predator prey - ecological modeling
• the game of life (complicated behavior from simple rules.)
• probability
• expected value
• surveying (see just above)
• Monty Hall problem and its application
• weighted averages
• Collecting Data Wisely
• Math in a democracy
• Voting, fairness and how methods influence results
• allocation of resources or representatives
• fair division among belligerent parties
• What's the big deal about Calculus?
• Ideas from science that impact our view of life
• Euclidean geometry as a model for modern science
• information theory, entropy and energy, 2nd law of thermodynamics(all tends to chaos - life is a local opposition to chaos)
• quantum theory - nothing can be measured exactly, life/learning is a series of approximations to truth
• relativity and non-Euclidean geometry - there is no ultimate truth (Euclidean geometry is not truth, there is no fixed standard of measurement(no Ether, no absolute length or time( which came first?)).
• 4th and higher dimensions - thinking about alternate realities
• black holes - a mathematical singularity takes its place in the universe
• A historical vignette - (how truth changes through the ages)
• Egyptian surveying and the beginnings of geometry
• Euclidean Geometry,axiomatic systems and logical deduction
• Greek geometry as a basis for modern mathematics and science (Newtonian physics)
• non-euclidean geometry and the loss of certainty
• Reimannian geometry (maybe not)
• Einstein used Riemanian geometry to discribe the world
• Einstein's 'mistaken' expanding universe - later evidence that the universe expands
• Discovery of black holes as mathematical singularities and recent evidence of black holes in abundance

• Other Reflections
• Teach it sideways (Let the students learn the main topics by trying to accomplish something else. What?)
• Have a consistent theme (math and democracy, sustainabilty, no absolute truth - just sure logical deduction from hypotheses).
• Must take care of home before we take care of world?
• All I learned in kindergarten: share everything - how do we put into practice?
• problem-solving in everyday life
• Mathematics and Mathematicians as part of humanity's battle against ignorance and chaos. Genius overcoming ignorance and the price paid. Culture as an accumulation of genius over the ages - schools as a repository of the knowledge. A cultural and humanistic endeavor of which we too are a part.
• Real education is the residue when all the facts have been forgotten. What is it our students will remember in 20 yeas?
• Technology is the knack of so arranging the world that we do not experience it -- Max Frisch