Where it goes wrong
Traditional teaching
Exponents arrive as notation drills — evaluate 2⁵, memorize the product and power rules — and logarithms arrive years later as their menacing inverse, defined by a formula and practiced as symbol-shuffling. The two ideas are taught so far apart that most students never notice they are the same idea read in opposite directions.
The reteach
The Alice Method
Start with a physical shock: fold a paper, double a rice grain, let an allowance compound. Repeated doubling breaks human intuition — we expect ramps and get rockets — and feeling that breakage is the point. Only then name it: an exponent counts the doublings. And a logarithm is nothing but the inverse question a child has already asked out loud: 'how many folds until it reaches the Moon?' Teach the pair together, as forward and backward readings of the same growth story.
Mental model
Counting doublings
An exponent counts repeated multiplications the way ordinary numbers count repeated additions. 2¹⁰ isn't 'a big number' — it's ten doublings. A logarithm reads the same story backward: log₂(1024) asks 'how many doublings got us here?' Forward is the exponent; backward is the log. One story, two directions.
Transfer
Where this shows up for the rest of their life
Exponential thinking is modern survival equipment: compound interest, inflation, viral spread, population growth, algorithmic complexity, Moore's law. The child who feels why doubling beats any ramp understands savings accounts, epidemics, and why 'just 3% a year' quietly doubles in 24 years. Logarithmic thinking runs the other way — decibels, earthquake scales, pH — whenever nature's range is too wide and we count doublings instead.
Watch for these
Common misconceptions
The misconception
“2⁵ means 2 × 5.”
The repair
The near-universal first error. Counting doublings fixes it physically: five folds of paper is visibly not ten sheets thick — it's thirty-two. Exponents count multiplications, not multiples.
The misconception
“Exponential growth is just 'fast growth.'”
The repair
It's growth whose speed itself keeps growing — proportional to the current amount. A steep straight line never catches a doubling curve. 'Fast' is a slope; 'exponential' is compounding.
The misconception
“Logarithms are an unrelated advanced topic.”
The repair
A log is only the inverse question about growth a child has already asked: 'how many times until…?' Taught as that question, logs arrive pre-understood; taught as a formula, they arrive as fog.
Seen, not said
The visual explanation
A fold counter climbing linearly — 1, 2, 3, 4 — while the thickness readout races through mugs, houses, Everest, space. The mismatch between the calm slider and the exploding number is the entire lesson, made visible.
Try it
Interactive example
Slide the folds. Watch a tenth of a millimeter pass the Moon in 42 steps — then ask the backward question.
Fold a piece of paper. Keep folding.
Each fold doubles the thickness — 0.1 mm to start. Intuition expects a ramp. Doubling builds a rocket.
4 folds
thickness = 0.1mm × 2^4
1.6 mm
that's taller than
…nothing yet
Now the inverse question: how many folds to reach the Moon? Not millions — 42. Asking “how many doublings until X?” is what a logarithm is. Exponents run the growth forward; logs read it backward.