Nobody can take a powerful idea and turn it into hash like he can, I said to myself about a colleague at work.

The problem seems to me to be that he doesn't have
a **deep** understanding of the concept he is working with.
He is intelligent enough to recognize that the idea is powerful
and to get excited from being in touch with a profound insight,
but in some way he is unable to internalize the concept sufficiently
so that it takes root and lives in his own intelligence.

That's a picturesque way of describing the matter,
but it doesn't quite define what I mean by a **deep** understanding.
In my own experience, as a former math major
and the holder of a license to teach mathematics,
the first illustration which comes to mind is quadratic equations.
Quadratic equations aren't so difficult to understand
at a more superficial level; most high school students master
the ability to work with them, to match the equations with graphs
and calculate the trajectories of imaginary baseballs thrown upward
in a rectilinearly uniform gravitational field in a vacuum.
Certainly I never had any particular difficulty with them.

When I became a student teacher and presented quadratic equations to high school students, I realized that my understanding of this class of equations was not very deep. I remember knowing that the constant factor of the squared term is a scaling factor, but what does the x-squared itself signify? And why are the quadratic curves the same as the conic sections? Unfortunately, I haven't yet gone back to understand them better.

Recently, I had a similar experience with matrix multiplication. In the last few decades there has never been much reason to work with matrices, and none at all to combine them. Something I read induced me to think about matrix multiplication and, having forgotten the technique, to look up the algorithm. It is simple enough, but what does it mean to multiply matrices? Having rediscovered the method, I realized that I had no understanding. No wonder I forgot how to do it! The operation had no deep significance for me; it had not, as I said before, taken root in my understanding.

The issue is not limited to mathematics. This week I'm working on a
sermon
about Jan Hus, the leading reformer of Christianity in the 1400s.
It occurred to me that the opponents of Hus read the scriptures
but never attained a **deep** understanding of anything they said.
Hus preached from the 34th chapter of Ezekiel, which is something
of a diatribe against the kings and priests of Israel.
The opponents heard the criticism,
and the implications of that criticism for the kings and priests
of 15th Century Europe.
They did not seem to understand the deeper promise of that chapter,
which is that God is personally interested in the welfare of the people
and will not leave peace, justice, and prosperity entirely to human leaders.

We could consider the Constitution of the United States.
Nearly every citizen knows that it is against the Constitution
for Congress to "establish" someone else's religion.
That is, the Government may not force Baptist Christians
to sacrifice chickens in a Santeria ceremony.
For many people, it is not as clear why Congress should not mandate
the beliefs and practices of their *own* religion.
Without a deep understanding of the reasons
behind the anti-establishment clause,
without a memory of the historical abuses of religion by government,
the idea of making the United States into a "Christian" nation
does not seem unreasonable at all, at least to Christians.
With that deeper understanding, however, it becomes clear
that identifying faith and secular power is a danger
to religion and to people of faith.
(It seems particularly odd to me that anyone who professes
to be any kind of Baptist would be willing to cede to any government
the right to define their religion.)

Perhaps advances in neuropsychology will eventually allow us
to distinguish **deep** understanding from operational competence
in a more rigorous way.
Until then, I will remain susceptible to the illusion
that I understand quadratic equations.