This post is preliminary thoughts in response to a Google Thread [Math Future].
“An adult switching to a new career and drawing on math remembered from high school to get up to speed. If they don’t remember the math from high school, or never had it, they learn it.”
Objective: Prepare the student to use “math”, that is, quantitative reasoning with appropriate tools, in order to:
* create and exploit quantitative models of systems encountered in everyday life and in jobs to which they aspire.
Some years ago I kibitzed on an “Algebra Reform” project in which a colleague participated. This outline is based on my memory of their approach, plus my own ideas as a student of Physics, Math, and Computer Science.
- Collect quantitative measurements of significant features of the system of interest
* explore opportunities for automated data collection
- Organize the measurements in a form useful to discover and exploit relationships among the data.
Tools: spreadsheets and spreadsheet table and charting features can be used
- Recognize common mathematical relationships among the data:
* constant values
* constant difference
* linear proportion
* inverse linear proportion
* quadratic and cubic proportion (length:area:volume)
* inverse square proportion (gravity, density of central field from center)
* exponential growth and decay (compound interest, unrestrained reproduction, half-life)
* cyclic or periodic variation
* normal probability distribution
- Express common relationships as mathematical formulae;
* recognize common functional relationships by the shape of a graph
* evaluate formula with mechanical assistance (calculators, spreadsheets, computers)
* parameterize formulas to approximate empirical data, using appropriate tools
- Exploit formulas as models of the system of interest
* estimate the solution by visually interpolating/extending a graph
(It is important to estimate BEFORE doing a machine calculation, to avoid excessive confidence in computers.)
* solve linear (and possibly quadratic) formula using algebraic methods
* solve more complex formulas using appropriate tools and algorithms:
e.g. Spreadsheet Solver feature
* linear interpolation of empirical data
* binary search for solutions
* apply “common sense” test to results of machine calculations (e.g., is the computer solution consistent with the initial estimate?)
- Exploit formulas as predictors of future behavior
- Understand and recognize instability of systems:
* sensitivity to small changes in parameters (chaos potential)
* behavior at extreme values: out of bounds, evolution in time, etc.
* potential for random variations
* sensitivity to “unlikely” and “black swan” situations.
I would strive to have actual career-relevant examples,
from as many fields as practical, for all functions.
I would place the emphasis on understanding the implications of formulas and values, and leave the actual computations to calculating machines.
That is, focus on the meaning of the mathematics, not the methods. Focus on the methods for using tools, not the methods the tools use.
* Many of the topics covered here could be explored with a computer or even tablet/phone spreadsheet program.
* Certain algorithms, such as graphing functions and binary search, could be explored with a simple programming language.
More advanced topics could be covered depending on the specific interests of the students. A whole course could be devoted to statistical methods for students anticipating entering a relevant field.