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


  • 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