# Topics for All Students

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### 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
- computing GPA, grades and midterm grades
- average paradoxes

- 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

- Linear Growth

- 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