Jonathan Groves
Posts:
2,068
From:
Kaplan University, Argosy University, Florida Institute of Technology
Registered:
8/18/05


Re: The instructional role of mathematical lectures
Posted:
Apr 25, 2011 2:09 PM


Clyde,
What you say here agrees with one of the biggest problems I have seen with relying too much on lecture as a teaching tool. Of course, some deniers have implied that I'm railing against any use of lectures, regardless of the reasons, regardless of how long or short or frequent or infrequent they tend to be. Perhaps I don't always say that clearly enough, but even speaking against "overreliance on lecturing" can still gets responses from some who believe I'm calling for a total ban on lecturing. As Paul Lockhart says in his essay "A Mathematician's Lament" (speaking in a fictional dialogue with an imaginary person who represents many skeptics), "If I object to a pendulum being too far to one side, it doesn?t mean I want it to be all the way on the other side."
Is it possible for lectures to present some of the art behind mathematical learning? Perhaps so, but I don't think it is effective to rely just or mostly on lecture for doing so. However, a sad fact is that few classes taught primarily by lecture make even the slightest attempts to discuss the art behind mathematical learning. I find that students struggle a lot because they don't know much about learning mathematics or how to approach mathematics either specifically (in regards to specific mathematical ideas) or generally (how we approach mathematics regardless of what specific ideas we want to think about).
Dave Moursund hits on this key idea in his book "Computational Thinking and Math Maturity: Improving Math Education in K8 Schools" (Second Edition), which is available for free on the Internet. His way of saying this is that math education rarely addresses the issues of how to help students develop mathematical maturity. Without mathematical maturity, learning math becomes a mighty struggle, especially after about third to fourth grade or so when what is being taught doesn't line up with intuition as well as it once didor at least is not so obvious anymore how it does. Yes, the curriculum addressing the mathematical ideas is also to take a lot of this blame as well. But I wonder if the tangles in the curriculum getting there and lasting this long reflect the fact that fractions and other mathematical ideas discussed beyond second or third grade or so are not as intuitively obvious as earlier mathematical ideas.
"Not as intuitively obvious" can mean that the ideas can be made to appeal to intuition but how we can do so is not so obvious or that phrase can refer to ideas that appeal only to the intuition of highlevel mathematics students or mathematicians or other mathematical professionals (those with a lot of mathematical maturity). For instance, the notions of groups, rings, and fields appeal to my intuition, but most nonmathematicians would disagree. Likewise, they would disagree with me about the intuition behind the notion of what it means for a group to be generated by a nonempty set.
Without knowing much about the art of mathematical learning, students will find mathematics in bits and pieces that don't seem to connect well, and learning to become a fluent thinker in the subject will also become an essentially impossible goal to reach. Later mathematics becomes a Herculian struggle to learn. And if these students ever needed to consult a mathematical reference to recall something they need but don't remember (and how many of us find we need to do that?), they will probably have a lot of trouble making sense of whatever reference they consult, even if the reference discusses math they supposedly had "learned" before. They may even have a lot of trouble identifying what they need to look up!
In short, the students we are generally trying to reach do not just struggle with the mathematical ideas we are trying to teach but struggle with a bigger notion: how to learn and think about mathematics. Since that issue is rarely addressed, students who come into college who struggle with math tend to struggle with math almost as much when they leave college. Until these issues are addressed, these students will continue to find studying mathematics a Herculian effort.
As for forgetting what we have learned, I was quite surprised a few years ago to learn that some students forget so much that they end up not even remembering that one of their previous classes had discussed something. For instance, one professor had told me about a student in Calculus II who did not remember ever seeing polar coordinates before, but he did have this student in a precalculus course in which he did lecture on polar coordinates. He had told the student (almost certainly not an exact quote but a good paraphrase, at least according to what he told me), "I know you have seen this before because we discussed this in our precalculus course. I can show you the exam if you want me to." I had figured that students who forget would generally at least remember seeing something before if that mathematical idea is regarded as important by the teacher or the course (as opposed to a tidbit mentioned just in passing).
He had told me this story because I was mentioning to him as to why high school Algebra II classes don't seem to discuss logarithms anymore because, as a TA at the University of Kentucky, I had seen students react to logarithms as if they had never seen them before (as opposed to reactions that indicate they have seen them before but couldn't ever learn to make sense of them), and I know that the students had to have taken Algebra II to get into the University of Kentucky. He never did say whether logarithms are discussed in Algebra II classes in Kentucky, but he did say that students' reactions like that do not mean that none of their previous classes had discussed that idea. In other words, the students might have seen logarithms before but had forgotten about them so much that they don't remember ever having seen them before.
Jonathan Groves
On 4/25/2011 at 12:52 pm, Clyde Greeno wrote:
> Alain: > Excellent point about instruction. Instruction in the > mathematical arts is > best not limited to the communications of > mathematical content. It best also > includes education in the arts of mathematical > learning ... including the > arts of digesting books, discussions, or any other > kind of mathematical > discourse. Curricular mathematics instruction has > long been woefully guilty > ... of expecting (or requiring, even demanding) that > students learn the > mathematics ... without ever educating them in how to > do so. > > Of course, lectures can do only whatever they can do > ... sometimes what > *only* lectures can do. The fact that lecturing is > not always the best tool > is no justification for denying its value. Just as > with your carpentry > toolbox ... you use each tool that you have, as best > you can. Unfortunately, > if a hammer is the only tool that one knows, the > tendency is to treat > everything else as if it were a nail. As yet, the > science of mathematics > instructology is too meager to provide educators with > rich collections of > strong tools that can be applied for alternative > purposes. > > Does anyone know of a serious attempt in that > direction? > > Curiously, > Clyde > >  > From: "Alain Schremmer" <schremmer.alain@gmail.com> > Sent: Monday, April 25, 2011 9:41 AM > To: "mathedcc list" <mathedcc@mathforum.org> > Subject: Re: The instructional role of mathematical > lectures > > > On Apr 25, 2011, at 9:02 AM, Ed Laughbaum wrote: > > > >> Clyde, > >> > >> Your post is quite interesting and I do not > disagree. I look at any form > >> of instruction as a process to help students > create a memory of the > >> mathematics being taught. That is, learning is > simply the process of > >> creating a memory. The brain does it automatically > and it is an ongoing > >> process. Forgetting is another normal process. In > teaching, the issue > >> becomes how to create a longterm memory of, for > example, how to solve > >> any equation. But longterm memory isn?t worth > much unless students > >> understand what it means to solve an equation. We > want students to learn > >> (create memories), understand what they learned, > keep the memory long > >> term, and be able to recall it in a month or five > years. The brain can > >> do this on a molecular/ cellular level in response > to external > >> events/actions. > >> > >> So what external events fosters the release of > dopamine? The release of > >> dopamine is a key to longterm memory creation. A > dry lecture is likely > >> not an external event that causes the release of > dopamine, which causes > >> the brain to give value to the math which > translates into longterm > >> memory creation through the hippocampus. But we > are left with > >> understanding the math in the memory and recall > of the memory as needed. > >> These primarily involve the proper use of the > external events of > >> meaning, visualizations and connections. > >> > >> Are these things possible through lecture? As you > indicate, educational > >> or psychological research suggests it is a > function of audience > >> knowledge. But it is also a function of the > inclusion of visualizations > >> of the math being taught, contextualizations that > model the math and > >> give meaning to the math being taught, > connections to the math being > >> taught, and even pattern building to a > generalization of the mathematics > >> being taught. Can you do these things in a > lecture? I would say probably > >> yes. > >> > >> I know there are implied nuances that I have > omitted in this post, but I > >> think the structure of direct instruction, or any > form must address > >> these basic brain function issues. > > > > > > > > I am struck by the fact that the focus above is > completely on > > "instruction" and that no mention is made by > Laughbaum of what the > > student has to DO to further her/his > understandingwhich, like most > > mathematicians, I believe to be "reading pencil in > hand". In what he > > wrote, the student seems to be no more than a > passive receiver of the > > teacher's ministrations. > > > > Moreover, I don't think that mathematics consists > only of such things as > > "how to solve any equation" which are only means > towards an end in the > > same way that the NavierStokes equations are only > a means towards > > understanding how fluids behaveeven though they > remain to be solved. > > > > My own focus is thus on helping the students do > whatever THEY have to do > > in order to acquire "a coherent view of > mathematics" and a "profound > > understanding of fundamental mathematics", namely, > for the most part, > > "read pencil in hand". And so I believe that > Einstein, although certainly > > not a mathematician, had it exactly right in terms > of mathematics when he > > said "I never teach my pupils; I only attempt to > provide the conditions > > in which they can learn." > > > > It is because of this focus that I think lecturing > is mostly a waste of > > timeonce the instructor has a text that fits > what s/he wants her/his > > students to understand and that: > >  does not assert anything without making a case > for the statement, > >  analyzes and develop ideas in at least some > detail, > >  preempts many likely misunderstandings, and > >  enables the students to formulate their own > questions with increasing > > precision. > > > > Regards > > schremmer > > > > > ****************************************************** > ********************** > > * To post to the list: email mathedcc@mathforum.org > * > > * To unsubscribe, email the message "unsubscribe > mathedcc" to > > majordomo@mathforum.org * > > * Archives at > http://mathforum.org/kb/forum.jspa?forumID=184 * > > > ****************************************************** > ********************** > > ****************************************************** > ********************** > * To post to the list: email mathedcc@mathforum.org * > * To unsubscribe, email the message "unsubscribe > mathedcc" to majordomo@mathforum.org * > * Archives at > http://mathforum.org/kb/forum.jspa?forumID=184 * > ****************************************************** > **********************

