Energy and Human Ambitions on a Finite Planet, 2021a

A.8 Equation Hunting 368
to perform in my head. Recognizing that 24 is divisible by 8, I am
strongly tempted to re-express the first number as 24 × 10 12 . See what I
did there? I multiplied the prefactor by 10, <strong>and</strong> decreased the exponent
by one accordingly to end up at the same place. Now I have 24 8 ×1012−7 ,
which reduces to 3 × 10 5 . All methods get the same answer, 23 which
turns into another lesson that math provides many paths to the same
answer, which can be used to check <strong>and</strong> reinforce.
23: The frolicking dolphin tries several <strong>and</strong>
revels in the reinforcement that comes from
consistency.
A.8 Equation Hunting
Students often form a counter-productive dependency on formulas.
Experts focus on learning the concept expressed by an equation, since an
equation is very much like a sentence that speaks some truth. 24 Once the
fundamental principle is mastered, the equation or formula is automatic,
<strong>and</strong> can be generated from a place of underst<strong>and</strong>ing—which is more
permanent than memorization.
24: . . . perhaps within some context or set
of assumptions
The practice is more common <strong>and</strong> natural than it might seem at first.
Let’s say a person has a take-home pay of $50,000 per year. Rent is
$2,000 per month, groceries <strong>and</strong> other bills come to $1,000 per month.
How much is left per month for discretionary spending? Where is the
formula for that problem? Of course, you wouldn’t bother hunting for
a formula in this case <strong>and</strong> would instead build your own math. You
essentially create your own formula on the fly. Whether you first divide
the annual figure by 12 <strong>and</strong> then subtract the monthly expenses, or
multiply monthly expenses by 12 before subtracting from the annual
amount <strong>and</strong> then dividing by 12, the result is the same: a little more than
$1,000 per month.
It is also clear in this context that it makes little sense to perform math
down to the penny, since the grocery <strong>and</strong> other expenses are not going
to be exactly the same each month. The lesson is that most people are
expert enough in managing money that they don’t scramble to find
printed formulas whenever they want to figure something out, <strong>and</strong> they
are also forgiving on precision because they know from context not to
take it all too literally.
This book tries to foster a more expert-like approach to the material.
For instance, Def. 5.3.1 (p. 71) introduces the concept of power without
explicitly saying P ΔE/Δt. It just says that power is how much
energy is expended in how much time. If a student internalizes that
idea, then why print a formula? By doing so, a student may bypass real
underst<strong>and</strong>ing 25 <strong>and</strong> rely on the formula as a crutch, never planting
the core idea firmly in the brain. Shortcuts can end up disadvantaging
students, as attractive as they may look in the moment. The student who
masters the concepts will be in a far better position to deploy them in a
wider variety of circumstances—including unfamiliar test questions.
25: . . . which in this case is not a heavy lift
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.