"Ignorance Isn't All That It's Cracked Up to Be!"

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

Ed Parker, James Madison University

In my last year of graduate school, I went to the national meetings to interview and spent my "spare" time going to talks. I heard one by Herbert Amann that had an idea in it that intrigued me and I followed up on it, and, after I had done enough background work to put the ideas firmly enough in mind to be able to articulate my conjectures, I went to talk to John Neuberger, my thesis advisor, about it. Since my take on the problem (it was in dynamical systems) was topological in nature, John suggested that I go talk to Bill Mahavier about it. When I got my audience with Dr. Mahavier, he listened carefully, thought about it a few minutes, got up from his desk, and walked over to a stack of off-prints stacked on the floor to about shoulder height (there was a time before the Internet when correspondence consisted of exchanges of letters on paper and off-prints of publications!). As he worked his way about a third of the way down from the top, his first remark was, "This is why old mathematicians know so much more than young mathematicians.". He pulled a paper from the stack, went back to his desk, peered at the paper for a few minutes, and then proceeded to remind me of some work I had done, gave me some definitions that he said should be accessible, and suggested a couple of problems that he indicated might be of some use if I could solve them. Mind you, he didn't show me the paper nor did he even give me the reference; I still don't know what it was!

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".

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.