Why students should invest in becoming good at their tools

effectiveness.rb

# Why a student should invest the time to become effective with their tools

class Student
  attr_reader :learned, :effectiveness

  def initialize(effectiveness)
    @learned, @effectiveness = 1.0, effectiveness
  end

  def practice(topic_difficulty, time)
    # Current knowledge and effectiveness reduce the difficulty of topic
    relative_difficulty = topic_difficulty / @learned / effectiveness

    # The amount they learn is based on the time they put in, and how hard it is (see note at bottom)
    learned_during_practice = time / relative_difficulty
    
    @learned += learned_during_practice
  end
end

# These reflect the same student, but in the latter case, they have increased their effectiveness by practicing their tools.
# Their tools either hinder them or help them, thus affect their effectiveness.
ineffective = Student.new(0.1)
effective   = Student.new(0.2)

# 10 practice hours, on increasingly difficult material, we see there is already a significant gap
1.upto(10) { |n| ineffective.practice 20, 60 }
1.upto(10) { |n| effective.practice   20, 60 }
ineffective.learned # => 13.785849184900002
effective.learned   # => 109.9511627776

# Double the practice hours, and we see the much more important detail: the difference is not linear
1.upto(10) { |n| ineffective.practice 20, 60 }
1.upto(10) { |n| effective.practice   20, 60 }
ineffective.learned # => 190.04963774880807
effective.learned   # => 12089.258196146295


=begin
NOTE:
The math for the amount learned while practicing is oversimplified.

It's probably hyperbolic opening downward to reflect the Zone of Proximal Development (ZPD).
Something like y = d**2 - (x-d)**2

My model is good enough to express the general idea, but not good enough to reflect reality.
The other model should be able to express how differences in difficulty can lead to stagnation of all students:
  if it's too easy, no one is challenged so no one grows
  if it's too difficult, no one understands what they did, so no one grows
or ineffective students catching up
  target the ineffective students ZPD and they excel quickly, while effective students stagnate b/c the material is too easy
or effective students leaving them behind)
  target effective students ZPD and they excel, while ineffective studetns stagnate b/c the material is too hard
=end