TITLE: The Tension Among Motivation, Real Problems, and Hard Work
AUTHOR: Eugene Wallingford
DATE: April 17, 2018 4:25 PM
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In today's edition of "Sounds great, but...", I point you
to
Learn calculus like Huygens,
a blog entry that relates some interesting history about
the relationship between Gottfried Leibniz and Christiaan
Huygens, "the greatest mathematician in the generation
before Newton and Leibniz". It's a cool story: a genius
of the old generation learns a paradigm-shifting new
discipline via correspondence with a genius of the new
generation who is in the process of mapping the discipline.
The "Sounds great, but..." part comes near the end of the
article, when the author extrapolates from Huygens's
attitude to what is wrong with math education these days.
It seems that Huygens wanted to see connections between
the crazy new operations he was learning, including second
derivatives, and the real world. Without seeing these
connections, he wasn't as motivated to put in the effort
to learn them.
The author then asserts:
*
The point is not that mathematics needs to be applied. It
is that it needs to be motivated. We don't study nature
because we refuse to admit value in abstract mathematics.
We study nature because she has repeatedly proven herself
to have excellent mathematical taste, which is more than
can be said for the run-of-the-mill mathematicians who
have to invent technical pseudo-problems because they can't
solve any real ones.
*

Yikes, that got ugly fast. And it gets uglier, with the
author eventually worrying that we alienate present-day
Huygenses with a mass of boring problems that are
disconnected from reality.
I actually love the heart of that paragraph: *We don't
study nature because we refuse to admit value in abstract
mathematics. We study nature because she has repeatedly
proven herself to have excellent mathematical taste...*.
This is a reasonable claim, and almost poetic. But the
idea that pseudo-problems invented by run-of-the-mill
mathematicians are the reason students today aren't
motivated to learn calculus or other advanced mathematics
seems like a massive overreach.
I'm sympathetic to the author's position. I watched my
daughters slog through AP Calculus, solving many abstract
problems and many applied problems that had only a thin
veneer of reality wrapped around them. As someone who
enjoyed puzzles for puzzles' sake, I had enjoyed all of
my calculus courses, but it seemed as if my daughters and
many of their classmates never felt the sort of motivation
that Huygens craved and Leibniz delivered.
I also see many computer science students slog through
courses in which they learn to program, apply computational
theory to problems, and study the intricate workings of
software and hardware systems. Abstract problems are a
fine way to learn how to program, but they don't always
motivate students to put in a lot of work on challenging
material. However, real problems can be too unruly for
many settings, though, so simplified, abstract problems
are common.
But it's not quite as easy to fix this problem by saying
"learn calculus like Huygens: solve real problems!".
There are a number of impediments to this being a
straightforward solution in practice.
One is the need for domain knowledge. Few, if any, of the
students sitting in today's calculus classes have much in
common with
Huygens,
a brilliant natural scientist and inventor who had spent
his life investigating hard problems. He brought a wealth
of knowledge to his study of mathematics. I'm guessing
that Leibniz didn't have to search long to find applications
with which Huygens was already familiar and whose solutions
he cared about.
Maybe in the old days all math students were learning a lot
of science at the same time as they learned math, but that
is not always so now. In order to motivate students with
real problems, you need real problems from many domains, in
hopes of hitting all students' backgrounds and interests.
Even then, you may not cover them all. And, even if you
do, you need lots of problems for them to practice on.
I think about these problems every day from the perspective
of a computer science prof, and I think there are a lot of
parallels between motivating math students and motivating
CS students. How do I give my students problems from
domains they both know something about and are curious
enough to learn more about? How do I do that in a room
with thirty-five students with as many different backgrounds?
How do I do that in the amount of time I have to develop and
extend my course?
Switching to a computer science perspective brings to mind
a second impediment to the "solve real problems" mantra.
CS education research
offers some evidence
that using context-laden problems, even from familiar
contexts, can make it more difficult for students to solve
programming problems. The authors of the linked paper say:
*
Our results suggest that any advantage conveyed by a
familiar context is dominated by other factors, such as
the complexity of terminology used in the description, the
length of the problem description, and the availability of
examples. This suggests that educators should focus on
simplicity of language and the development of examples,
rather than seeking contexts that may aid in understanding
problems.
*

Using familiar problems to learn new techniques may help
motivate students initially, but that may come at other
costs. Complexity and confusion can be demotivating.
So, "learn calculus like Huygens" sounds great, but it's
not quite so easy to implement in practice. After many
years designing and teaching courses, I have a lot of
sympathy for the writers of calculus and intro programming
textbooks. I also don't think it gets much easier as
students advance through the curriculum. Some students are
motivated no matter what the instructor does; others need
help. The tension between motivation and the hard work
needed to master new techniques is always there. Claims
that the tension is easy to resolve are usually too glib
to be helpful.
The Huygens-Leibniz tale really is a cool story, though.
You might enjoy it.
(The image above is a sketch of Christiaan Huygens's first
pendulum clock, from 1657. Source:
Wikipedia.)
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