In this
post, I have invited my good friend and colleague, Ed Parker, to be a guest blogger. Ed sometimes mentions, “Ignorance isn’t all
that it’s cracked up to be,” when we have work talked. Typically this quote comes up when we are working in workshops, and the discussion is about prior knowledge and content coverage. Ed attributes John Neuberger for the quote, which he discusses in detail below. (John is Ed's thesis advisor.)
Ignorance
in teaching comes up in several ways, and I'll comment on one way specifically here, and then turn it over to Ed. One way
“ignorance” comes up is in the difference between (a) teaching a topic that students have never seen
before vs. (b) teaching a topic students have seen in a prior course. Let’s take Topology vs. Euclidean
Geometry, and focus on Euclidean Geometry first. College students have
been exposed to Euclidean Geometry in K-12, and often their incomplete
knowledge of Euclidean Geometry gets in the way of new learning, because they
have memorized some things correctly and others incorrectly in a way that is
disconnected from the axioms and first principles. When we ask students to prove a fact in
Euclidean Geometry, they often don’t know where to start, because it is hard to
distinguish between what is allowed as a fact or axiom. Many of the statements are “obvious,” and
thus the teacher has to navigate around teaching and learning obstacles through all this, which is no easy task!
On the
other hand, Topology is a topic students have not usually seen before. Hence starting with some assumptions and
definitions provides a clean slate.
Students are on a level playing field, and the focus is on taking the
definitions, understanding them deeply, and then moving to proving theorems. Students can do better sometimes when they
are completely ignorant, because they can then do math in a way that is exactly
the way mathematicians do math. They have to work with definitions and grapple with mathematical definitions, logic, and all that.
The
fact that Euclidean Geometry can be more difficult to teach than Topology lies
to a significant degree in the fact that something learned poorly has serious unintended consequences.
Students come in with prior incomplete knowledge, expectations, and perhaps unhelpful habits of mind. The same can be said of
subjects like Calculus, where college students are likely to have seen many of
the ideas at least once before. Thus, in
such courses IBL instructors have a more complex set of challenges to know
about, learn about, and address. All of this underscores the weight and significance of what students walk into a course with. We'll dedicate future blog posts to share some strategies for dealing with these kinds of issues.
Now back to "Ignorance Isn't All That It's Cracked Up to Be!"... Take it away, Ed!
-- By G.
Edgar Parker, Professor Emeritus, James Madison University
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| Ed Parker, James Madison University |
But he then offered me some professional advice. He told me,
"Ed (I had already defended, so I was no longer Mr. Parker), to my mind,
when to go to the literature is the hardest decision a researcher has to make.
If you take on a problem and the first thing you do is go find out what
everyone else knows about it, your thinking is almost certain to be channeled
by their approaches to the problem. The only way you'll be able to improve the
results that way is to be smarter than they are since they probably know the
literature well. (Not so subtle between-the-lines message: and you're not
smarter than they are.) On the other hand, if you can understand the problem
and go to work on it, normal minds running free can get unique perspectives
that sometimes allow us (sic) to see things that others don't. But suppose that
you really want to solve the problem, but after three months you aren't getting
anywhere. Is it time to go to the literature? And, if you choose to, do you
stay there until you are well-informed, or just until you have a new idea to
work with?" And he left it at that.
I don't know whether this advice was intended to have global applicability or
whether it was intended just for me. My history as a student was that I was a
painfully slow learner of other people's mathematics and couldn't tell the
"hard" problems from the "easy" problems so I had about the
same success with both and usually my proofs, I found when I later became
familiar with the literature, were consistently not made along the lines of
thinking that produced the proofs that made the books. Trevor Evans, from whom
I took my algebra courses at Emory, used to say after I would present,
"Yes, Mr. Parker, I suppose you are correct, but (as he would take the
chalk away from me) WHY DIDN'T YOU THINK OF THIS?", and he would proceed
to show the class a "reasonable" proof. Most of my work in the
literature my first decade out of graduate school was spent teaching myself
enough to be able to teach my courses or going to the literature to make sure
that what I wanted to write up hadn't been done.
Fast forward about 25 years. I was talking to Dr. Neuberger about a problem on power series connected to wherever Jim Sochacki and I were at the time in our exploration of polynomial projection and in passing, made the comment, "You know me, I didn't know hardly any of this stuff..." and kept on describing the idea. When I came up for air, John said, "You know, ignorance is highly overrated." I don't think he was being critical; the message I got was, you could be using the same brain cells that you are using to figure out stuff that other people don't know that you now have to expend to figure out stuff that many undergraduates know.
I'm guessing that the tension between Mahavier's advice and Neuberger's admonition has something to do, by analogy, with "the coverage issue". Hopefully, we put together our course notes well enough so that our students not only experience the growth of solving their own problems, but also come away with a corpus of "facts" that at least prepares them to be a few hours reading away from being literate about any one thing that they might "need to know". I continue to believe that if one is forced to choose between students "doing" and "being shown so as to achieve 'literacy'", one chooses "doing". I believe that the power associated with "learning to learn" is far greater than any level of mastery of someone else's bag of techniques. On the other hand, I'd like to be a whole lot better at communicating that, "now that you have done this, don't stop", is what we're after rather than "you are now certified to repeat some pre-packaged dose of curriculum neatly packaged and ingested".
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| Ed Presenting at the Legacy of RLM Conference, June 2013 |
Nevertheless,
we should, by intention, be purposeful in trying to make the “curriculum”
demarcated by our course notes as informative of the “curriculum” as dictated
by the contents of a standard text in the subject as is feasible. In my own case, I have spent considerable
time trying to optimize the fit between the mathematics my students will try to
make in my courses with the curricular expectation I perceive that my teaching
colleagues have. I would like what may
appear as ignorance to be a foundation for “I’ve seen that idea before.”. If our students learn to learn, our
colleagues will be happy with them, regardless of whether the students can
knee-jerk responses.
