Josef Stallings

Homeschooling 101: Math

INTRODUCTION

Welcome to the third installment in a series of primers for the new homeschooling family. My name is Josef Stallings, and my wife Anne and I have been homeschooling our 8 children for over 13 years. For more information on our background, please see the first primer in this series here.  

This third primer deals with the teaching of math in the homeschool setting.

WHAT IS MATH?
According to my go-to trusted source, Wikipedia (this article), Mathematics has no generally accepted definition. Here are some of the more prominent definitions:
 
  • Aristotle: The science of quantity
  • Auguste Compte: The science of indirect measurement
  • Benjamin Pierce: The science that draws necessary conclusions
  • Bertrand Russell: Symbolic logic
  • Ernst Snapper (intuitionism): A mental activity that consists in carrying out, one after the other, those mental constructions which are inductive and effective
  • Ernst Snapper (formalist): The manipulation of meaningless symbols of a first-order language according to explicit, syntactical rules
  • Walter Warwick Sawyer: The classification and study of all possible patterns
  • Stephen Wolfram: A broad-ranging field of study in which the properties and interactions of idealized objects are examined 
 
For your convenience, I’ve synthesized all of the above definitions into a single definition: 
 
Math is the mental activity and science of drawing necessary conclusions from the indirect measurement of quantities after carrying out a series of mental constructions and manipulations of meaningless symbols in order to classify and examine all possible patterns of idealized objects. – Josef Stallings 
  
 THE GOALS OF MATH INSTRUCTION
Given the above definition, the purpose of homeschool mathematical instruction is still not clear. In order to create an appropriate syllabus for your children, you need to focus on the teaching outcomes.
 
Again, I’ve turned to the experts to help to define these outcomes:
 
  • Bertrand Russel: [To] never know what we are talking about, nor whether what we are saying is true.
  • Charles Darwin: [To be a] blind man in a dark room looking for a black cat which isn’t there.
  • Henri Poincare: [To learn] the art of giving the same name to different things.
 
I’ve synthesized the above three outcomes by famous people into a single teaching outcome:
 
The purpose of teaching math is to instruct students such that they are blind to the world that surrounds them, they never know what they are talking about or whether it is true or false, and they make up names for things that have already been named.
 
This really sounds like where my kids come from before they start homeschooling.  
 
A PLAN FOR TEACHING MATH
After looking at the different fields of mathematics, Anne asked me to come up with a plan for teaching the kids math. We fudged on the older kids (we’re always learning, right?) but we wanted to get it right for the younger kids, especially Stevie, who is looking a bit confused these days.
 
After working through a dozen or so solutions, I decided to tackle the subjects in alphabetical order, since this seemed like as good of an approach as any. 
 
Here is how we plan to divide up the math teaching:
 
K: Arithmetic, Algebra (including order theory, general algebraic systems, number theory, field theory and polynomials, commutive rings)
 
1st: Analysis (rates of change, accumulated change, cryptography, abstract algebra)

2nd: Combinatorics (enumerative and extremal), and graph theory

3rd: Geometry and Topology (convex and discrete geometry, differential geometry, Algebraic geometry, general topology, algebraic 
topology, and manifolds)

4th: Probability and statistics (probability theory, stochastic processes, and statistics)

5th: Numerical analysis, computer algebra and machine learning

6th: Mechanics, mechanics of structures, mechanics of deformable solids

7th: Fluid mechanics, particle mechanics

8th: Operations research

9th – 12th: Review and applied studies
 
This is, of course, subject to Anne’s approval, and whether she knows this stuff; I surely don’t.
  
SUMMARY

I’ve got to say that math is not what it used to be. I remember spending the first 8 years of my education just memorizing times tables, doing a bit of geometry and trigonometry, and pre-algebra. We didn’t get to the hard stuff until high school. Oh well, times have changed, and kids are just smarter now than they used to be.

As always, I am looking forward to hearing your thoughts about this article, and our approach to homeschooling. Please leave any comments or suggestions that you have found useful in your own homeschooling experience. Thanx again to LTC Bob Smith for his patience in typing this for me, and working with the spell checker on the big words!

Our next article will cover teaching music appreciation in a homeschool setting.

Until next time,

Josef   

(Note from Anne: I don’t suppose you’d like to come home and teach the kids? Your deployment to Diego Garcia was VOLUNTARY!!! I’m thinking that it’s my turn for a six month vacation on some remote island. P.S. My hair turned grey)