"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.

Physicists Seek To Lose The Lecture As Teaching Tool

Note:  Thank you Matt Jones, CSU Dominguez Hills for the heads up.

Emily Hanford of NPR has a nice article about physicists, who are lecturing less and getting better results.  Be sure to also click on the link "Listen to the Story"for the radio broadcast.

http://www.npr.org/2012/01/01/144550920/physicists-seek-to-lose-the-lecture-as-teaching-tool

Physicists have collected data providing evidence that learning gains are better when students are deeply engaged in the subject and allowed to collaborate with peers.  Sandra Laursen, University of Colorado Boulder, also has data collected at the college level from mathematics courses.  Laursen's results are in alignment with this article.  Laursen's group has a number of findings.  Among the long list of findings, there is statistically significant evidence suggesting the more an instructor talks the less their students learn.  (Link to Colorado Study).

As Ed Parker says, "See What They Can Do."

See What They Can Do

My good friend and colleague, G. Edgar Parker, Professor of Mathematics, James Madison University, has written a wonderful story for The IBL Blog.  Thank you, Ed!


See What They Can Do

The bedrock for what became the basis for my teaching philosophy was forged when I was an undergraduate at Guilford College.  I was in college for all of the wrong reasons; I didn’t like school, but I knew my parents expected me to go to college and I was in no hurry to find gainful employment unless I could make a living playing baseball.  I was at Guilford for all the wrong reasons; I didn’t want to follow my older brother Elwood for another four years (the expectations associated with coming behind a straight-A student who was also a two-sport varsity athlete, all-conference in one sport and team captain in the other, even if they were imaginary, seemed real to me), but I went anyway because I had seen the baseball teams that my two older brothers played on there and knew that I could pitch better than anybody I had seen pitch in those six years (and besides, it had snowed on my campus visit to the school I visited that I liked the most and it was the week before the regular baseball season’s schedule back home was to start).

During the advising session before my first semester, fall of 1965, I jettisoned any ideas of following the subject that intrigued me most, Biology, because I found out that, at Guilford, you went to labs as a priority over practice and made up labs if you missed them for games.  My advisor, J. R. Boyd, placed me in two mathematics courses, Calculus I and Mr. Boyd’s Linear Point-Set Theory.  Interestingly enough, it was calculus that scared me.  I had actually heard of that and didn’t think I was prepared.  I should have guessed that Mr. Boyd was cut from different cloth; when I protested that I had only had four weeks of trigonometry in high school, he had “reassured” me by telling me, “Don’t worry, if you need to know more, you’ll learn it.”.

I was so naïve that I just assumed that the way point-set was being taught might be something that people did in college, so I just tried to solve the problems and hung on.  Since Mr. Boyd never said anything negative, for all I knew, I was doing okay.  I was getting some problems, or at least Mr. Boyd seemed satisfied with what he wrote on the board when I described my arguments.  Since I didn’t know the difference between a hard problem and an easy problem, I was trying them in the sequence they came in the notebook.  And then I ran into the problem that, looking back, could have been stated as “the continuum is not denumerable”, but was stated in a way that tempted a student to find a map from the natural numbers onto the numbers.  I thought I had an argument, but Hal Phillips, whom I considered to be the best student in the class, chose the problem when his turn came.  As Hal presented, I saw his ideas (which I had discovered myself) crash one by one and finally doom the argument on which I had worked so long and hard.  I was, at that time, still pretty meek in public and very reticent to call attention to myself, but the ardor of the moment overwhelmed me and I burst out in class in frustration (Mr. Boyd had made no suggestion as to how to remedy the now-evident flaws in the argument), “Mr. Boyd, why don’t you just show us how to do it?”.    Without missing a beat, he turned to me and said, “Mr. Parker, why should I limit you to what I know?”.  It was an opportunity to learn a lesson about teaching, but at the time, I was just relieved at the way Mr.
Boyd handled the situation and didn’t add to the embarrassment I had created by calling attention to myself.  Besides, I still wanted to pitch in the Big Leagues and education was just a diversion to enable me to continue to play the game I loved.  Education was not my intended vocation.

Fast forward to 1977.  I had finished my PhD in mathematics at Emory University, had courses given by Moore method from David Ford, Bill Mahavier, John Neuberger, Phil Tonne, and Mary Frances Neff, and chosen academe as my vocational home.  I recognized by now that Moore method is what allowed me to blossom creatively and provided me with the tools to learn other persons’ mathematics as well.  Still possessing, at that time, some sense of humility, I concluded that Moore method should provide the same growth potential for others it gave to me and decided to use it as the core for teaching the mathematics courses in the major at Pan American University that I gave.  Over the next seven years, for each such course, I painstakingly constructed a problem sequence with the self-assurance that, anybody who could do the problems in sequence would have to be able to see how to do the next one in the sequence because the connections were so obvious to me.  And each time, as often as not, the students leaped right past my lemmas to the important problems and found ways I hadn’t thought of to do the problems or ways that I considered “less natural”, or patiently proved my lemmas, but then showed me what the lemmas should have been by proving the theorem without using the lemmas.

Fast forward to the summer of 1988.  I had been at James Madison University since 1984 and I now understood how to follow the students’ ideas in the major courses, but still thought, for some reason, that non-majors had to be led to the fountain before they could drink.  That summer I was assigned yet another section of Mathematics 103: The Nature of Mathematics, the lowest numbered mathematics course in the catalogue and a course that I had taught every semester since it was introduced in 1985 (I had developed such a course for a less well-prepared clientele at Pan American, so, naturally, even though I had voted against our department offering such a course for General Studies at James Madison since I thought a school of our pretension should make calculus the core requirement, I was picked to give one of the initial offerings and was blessed with it on a continuing basis.) .  On a lark, I had the following conversation with myself: “The students are in this course because all they want is a mathematics credit.  So no one will fail the course.  But I will not tell them that and I will teach them as if they were majors.”  So I designed a course in which I “taught” them rigor by having them justify, on the basis of the field axioms applied to the words number, +, and $\times$, some of the fundamental techniques from high school algebra, and then put them to work on proving, as theorems, the field axioms stated for the ordered pairs of numbers with addition and multiplication defined so as to make the structure the complex numbers (without having i a part of the notation, I hoped nobody would recognize that they may have studied this algebra already and I was correct in this guess).  The only modification of Moore method I made was that the class was split into eleven groups, each with two or three students, and each group was responsible for proving a single theorem (chosen by lot), with all students responsible for certifying the correctness of the arguments and for reproducing and using what they certified to be correct.  The class got them all!  It was not that the students populating this class could not do the mathematics, it was my not believing they could do it that never gave them a chance to do it.  The summer of 1988 may have been an act of God.  I have used this problem set many times since, and no other class has gotten them all.  But every class has gotten some of them, and most classes have gotten the existence of reciprocals, the problem I consider the most difficult in the problem set.

Dr. Parker, why should you limit your students to the way you think about things, and why should you impose limitations on them?  


The longer I teach, the more I wonder why I didn’t ask myself this question earlier.

G. Edgar Parker, James Madison University