Energy and Human Ambitions on a Finite Planet, 2021a

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An attempt is made to prevent students from equation-hunting, promoting instead development of a
core underst<strong>and</strong>ing <strong>and</strong> intuition. This can require an adjustment on the part of students, who often
treat equations as algorithmic tools to file away for use later when solving problems rather than as the
embodiment of concepts to be internalized. Students often want a clear recipe so that when presented
with a problem for homework, they can mimic a parallel example clearly laid out in the book. Doing
so may be convenient <strong>and</strong> time-efficient, but short-circuits actual learning—bypassing the neural
development that would accompany mastering the mental processes that are involved in solving a
problem. Only the student can form these neural connections, <strong>and</strong> only through some struggle <strong>and</strong>
effort. In this sense, learning is like climbing a hill: the only way to get to the top is by investing the
effort to gain elevation—no shortcuts can bypass the inevitable climb.
Problems in this book are formulated to emphasize underst<strong>and</strong>ing the underlying concepts, rather
than execution of a mathematical recipe. When students say they have math difficulties, it is usually
not a problem carrying out the operations (+, −, ×, ÷), but in formulating an approach. Therefore, the
main difficulty is a conceptual one, but blamed on math because casting a problem in a mathematical
framework forces a mastery of the conceptual underpinning: nowhere to hide. Given two numbers,
should one divide or multiply them to get the answer sought? Resolving such questions requires a
deeper underst<strong>and</strong>ing of the meaning behind the numbers in the problem (<strong>and</strong> associated units, often).
By focusing on what the numbers represent <strong>and</strong> how they relate to each other, problems aim to build
a more meaningful <strong>and</strong> permanent underst<strong>and</strong>ing of the content.
In soliciting feedback from students about problems, comments frequently pointed out that “Problem X
used exactly the same math approach as Problem Y, so was redundant.” This exposes a glaring
difference in how students <strong>and</strong> instructors might view a problem. To the student providing such
feedback, the problem seems to merely mirror an algorithm, devoid of contextual meaning. To the
instructor, it is a window into a richer world: insight <strong>and</strong> personal ownership of the material is at
stake. Problems are an opportunity to learn, as students are perhaps most actively engaged, mentally,
when attempting to solve them. Instructors are trying to recreate their own learning experiences for
students, through the imperfect mechanism of assigned work.
A similar revelation stems from comments that express the sentiment: “this problem has unnecessary
information that is not required to solve the problem.” Is the point to churn out a number, or to
embed the result into a deeper context (i.e., learn)? It’s a matter of context over algorithm. Context is
where the real learning happens. It’s where deep <strong>and</strong> lasting connections are made to the real world.
The point is not to exercise a student’s ability to perform mathematical operations, but to absorb a
greater insight into the issue through its quantitative analysis. Math is like the airplane that delivers a
skydiver to the jump. The jump/dive is the whole point, but the airplane is a necessary conveyance.
When it comes time to jump, clinging to the familiar safety of the plane won’t accomplish the goal.
A student who bypasses the context for just the math operation has not embraced the intended
experience <strong>and</strong> attendant mental growth.
The book’s format sometimes weaves math <strong>and</strong> numbers into the text, which is unfamiliar to some
students who are accustomed to clear delineation between math <strong>and</strong> text. Students are advised to
approach sections containing mathematical developments by treating equations as statements of truth
(within the appropriate context <strong>and</strong> assumptions) that help define <strong>and</strong> complete logical arguments.
Or, think of equations as short-h<strong>and</strong> sentences that encapsulate a concept. Experts work to underst<strong>and</strong>
the concepts, by reading rather than memorizing equations. What is the equation trying to say? What
truth does it impart? What relationships does it elucidate? Equations in the text are surrounded
by sentences to help bring the equations to life as guides to intuition. Students who just want a
step-by-step recipe to utilizing equations in an algorithmic autopilot mode are missing an opportunity
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.; Freely available at:
https://escholarship.org/uc/energy_ambitions.