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Topic: Mathematician
Replies: 28   Last Post: Oct 15, 2010 8:47 AM

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Jonathan Groves

Posts: 2,068
From: Kaplan University, Argosy University, Florida Institute of Technology
Registered: 8/18/05
Re: Mathematician
Posted: Oct 7, 2010 3:43 PM
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So any form of inquiry-based learning is not honest? Just because some
teachers botch it by either taking it way too far or using it
inappropriately or otherwise not using such learning techniques skillfully
proves that inquiry-based learning does not work, that such learning must
be fake? If we do try to teach inquiry-based learning without guiding
students or insisting that they discover ideas or mathematical or scientific
truths way beyond their level, then we do not go anywhere. And unfortunately
I have seen others distort what discovery learning means by appealing to
textbooks that take this approach or perhaps appealing to something else.
For an example of the former, see

If Paul Kirschner's idea of discovery learning that it be minimally guided
simply because of what he saw in a textbook (and that does appear to be the
case because he fails to mention any other sources of what he had learned
about discovery learning), then that is false reasoning because this is a
textbook for students in a math course, not a teacher's manual. A textbook
gives teachers and students ideas for discovery learning, but it is up to the
teacher to guide it because a textbook cannot replace a student's dialogue with
the teacher and peers. A textbook cannot give personal guidance to a student
while the student is working to discover something.

Though Paul Kirschner raises some valid issues and mentions other issues
that I believe are based on misunderstandings of discovery learning, I will
not go into that right now since my primary reason for citing that website
in this post is to give an example of one who gives a very narrow definition
of discovery learning that is almost certainly based solely on a students'

I will go into one valid idea he does raise: He is correct in that students
must have more than mere knowledge to discover something--that is,
students must have the ability to think. That also means that anything we
wish to have students try to discover does not require thinking beyond what
they are capable of thinking. There are ample opportunities for elementary
school students and students at any level to discover ideas that are well
within their reach. The traditional algorithms of arithmetic are not
within the reach of most students to discover, but they can discover
other meaningful ways to compute with large numbers. But discovery learning
even in these cases does not dictate that students must continue using
forever these methods they have discovered and never be given the
opportunity to learn the traditional algorithms. Many criticisms of
discovery learning I have seen do not say that explicitly, but there
is certainly a strong hint that those critics suggest just that.
Another problem with such criticisms is that they imply that any method
of computing that is not the traditional algorithm is fake mathematics.
But any method of computing that can be proven mathematically valid is
true mathematics. An algorithm is fake mathematics only when it does
not work.

As for thinking as opposed to just mere knowledge to discover something,
which is true, who says that discovery learning must be about discovering
something highly complex or something that few people, even ones talented
at math or science, can ever hope to discover without numerous hints?

Isn't it possible that students' attempts to discover something can help
teach them to think? I'm sure that is indeed the case since that works
for learning other kinds of problem solving. Few students who are good
at problem solving learned to be good at it immediately. Most become good
problem solvers by having worked at trying to solve a lot of problems.
And no doubt that even these efforts helped the ones who become good at
problem solving rather quickly to become even better problem solvers.
In short, the way to learn to become good at problem solving and to become
good at thinking is to keep working on problems and other activities that
require us to think and not just simply grind out answers mechanically
or recite them from rote. Doing lots of work without much thinking
makes the brain grow sluggish. Students who have done virtually nothing
beyond this kind of work fail to develop problem solving skills and thinking
skills simply because we must practice using those skills to develop them.
We cannot give them just mere knowledge and then hope that they eventually
learn to become good problem solvers and good thinkers someday.

Who says that if a student fails to discover something on his or her own that
the student must be forbidden to learn about it any other way or to be forbidden
to use it till he or she discovers it? Put it another way: Who says that all
students must be required to keep working to discover something, even if that
takes them the bulk of the school year? Who says that the teacher cannot move
on to something different until all students have discovered what the teacher
wanted them to discover? I hope no teacher takes discovery learning to this
extreme. If any do or have, their failure simply says that we cannot take
discovery learning to this extreme, not that discovery learning cannot work
at all. I wonder if any critics of discovery learning seem to be
concerned about this because I know I have picked up that concern from
somewhere. Whatever the case may actually be about this particular concern,
the teacher is certainly free to move on--and will probably be required to do
so--even if some students have not discovered what the teacher wanted the
students to discover. And we cannot conclude that these students' failed
attempts to discover that idea or truth did not teach them something
because even experts generally agree that their failed attempts have taught
them something, even when they had not succeeded at discovering what they
had hoped to discover.

I cannot say for certain why you yourself believe that discovery learning is
fake learning, but I do know what some other critics have said either
explicitly or at least very strongly seem to be suggesting about it.
But I do remember one post of yours where you did attack an extreme form
of discovery learning, so I have some idea of what to think:

No one I know who supports discovery learning is requiring a "full-blown"
discovery learning environment. And no one I know who supports
discovery learning is requiring that the student discover "new and exciting"
mathematics if "new and exciting" is in reference to the mathematical
community at large. Though R. Moore did not teach K-12, he certainly did
not require that his students discover mathematics that is new to everyone
(except his own Ph.D. students). And those who support discovery learning
do not oppose students learning (in fact, they support it) some things from
books and teachers rather than via discovery. We know that mathematicians
learn some things via discovery but also learn some things from books and
articles. As I mentioned above, there are ample opportunities for students
at any level to discover some ideas that are new to them. Not everything
in mathematics that is new to the student requires a PhD education in
mathematics. If it did, then even elementary school mathematics ideas
would be inaccessible to those without a PhD education in mathematics.

I am not defending Professor Stage's work because I'm not familiar enough
with her work or her ideas on discovery learning. Even if her ideas do
not work, I would refute those ideas based on what she actually had said
and done and on actual data--if possible, not on some narrow or pre-conceived
vision of what discovery learning is.

Finally, as for attempts to get ALL students to learn math has failed because
that does not expose students to real math, see the following recent post
by Ralph Raimi in Math-Learn:

He believes that remedial courses in math are reduced to show and tell and
memorizing rules and algorithms simply because the ones who make decisions
about these courses (falsely) believe that communicating actual
mathematical ideas and reasoning to students might interfere with their
ability to do well on their versions of departmental exams.

I do not see much real math in what traditional American math education
offers because there is little emphasis on how to think about math beyond
procedures and facts and formulas. Perhaps an emphasis on reasoning and
understanding might be too hard for students? Sounds like this vision of
math education encourages students to take shortcuts in their learning
that simply do not work for the vast majority of them. To expose students
to real math, we need to expose them to genuine mathematical thought.
Yes, knowledge is vital; without it, we go nowhere. But exposing students
to mere knowledge without encouraging understanding and mathematical
thought and reasoning does not expose them to real mathematics.

Jonathan Groves

On 10/7/2010 at 11:46 am, Wayne Bishop wrote:

> I should have pointed out in my response to Michael
> that Dr. Stage
> has done more to destroy solid science education in
> California than
> any other single person and her efforts toward
> mathematics education
> are entirely consistent with that achievement. In no
> small part
> because of her "inquiry-based" science instruction
> instead of honest
> science instruction managed to keep his foot in the
> door in the late
> 90s and early 2000's as part of official state
> mandate in spite of
> its demonstrated failure and complete removal
> (officially, that is)
> of that crap from mathematics. As Haim just pointed
> out, this vain
> effort to successfully educate ALL (caps original)
> students in
> science (as in mathematics) only ensures that those
> students with
> real aptitude and interest are denied the opportunity
> for an honest
> exposure to the real stuff in their school
> experience. If they
> succeed, and some do, is in spite of their approved
> school
> experience. Behind the scenes by good teachers is
> one way), simple
> personal dedication is another, but it is almost
> certain science and
> mathematics education death to high potential
> students from low
> socioeconomic communities with a low priority on
> upward mobility
> through genuine education (i.e., more than just
> failing to drop out,
> the standard industry goal).
> Wayne
> At 11:41 PM 10/6/2010, GS Chandy wrote:

> >Michael Dougherty posted Oct 6, 2010 9:15 PM:
> > > > If only Wayne Bishop would occasionally take
> his
> > > own
> > > > advice to others and would search through
> Google,
> > > > he would easily find a biographical sketch
> about
> > > > Elisabeth Stage and what she may have taught at
> > > > the graduate level:
> > > >

> > >
> > >
> > > > ch.html
> > >
> > > Oh, I think Wayne has a good gripe here.
> > >

> > > > Quote
> > > > > (Dr Stage), a mathematician by training,
> thinks:
> > >
> > > The biography says nothing about being a
> > > "mathematician" at all.
> > >
> > > "Dr. Stage holds an Ed.D. in Science Education

> and an
> > > M.Ed., both from Harvard University, and an A.B.
> in
> > > Chemistry from Smith College."
> > >
> > > We do have to jealously protect the title
> > > "mathematician." Judging just from that

> biography,
> > > she was someone interested in Chemistry/Education
> who
> > > found herself in a position where she "directed
> the
> > > Mathematics Professional Development Institutes
> under
> > > the Office of the President of the University of
> > > California." As someone who was told how to

> teach
> > > math by an Ed.D. in charge of Academic Affairs at
> a
> > > previous job, I can tell you I've seen how such
> folks
> > > can manipulate into such positions where they do
> not
> > > have the credentials to be much more than grant
> > > writing assistants, yet this lady is called a
> > > "mathematician by training"? What am I missing?
> > >
> > > - --Mike D.
> > >

> >I take your point - but I still would like to claim
> that Dr Bishop's
> >"peeve" was rather petty. Perhaps it was the
> writers of that blurb
> >who put up the claim that "Dr Stage is a
> "mathematician by
> >training", possibly incorrectly. My underlying idea
> is always the following:
> >
> >In order to move forward in promoting math and

> thereby to enable
> >people appreciate the power and beauty of math, we
> do need the
> >support of all who are, as claimed about Dr Stage:
> > > .... guided by a vision of
> > > high quality mathematics and science education

> for
> > > all students.
> > >

> >This is very valuable (of course, if genuine), and I
> would not wish
> >to lose sight of that vision.
> >
> >It should have been easy enough for Dr Bishop to

> have discovered a
> >little more about Dr Stage, and whether her vision
> is based on a
> >real willingness alongside some ability to help
> actually promote
> >math. If that exists genuinely, a great deal could
> perhaps be achieved.
> >
> >Currently, a great many students are (I claim) being

> needlessly
> >turned off math by 'ineffective systems'. These
> ineffective systems include:
> >
> >- -- lack of understanding why students get turned

> off math;
> >- -- huge confusions about the right way to present
> math to students
> >and about the right way to elicit their interest;
> >- -- and so on and so forth.
> >
> >We've seen plenty of discussion right here at this

> forum on these
> >points - without ever coming to a workable
> conclusion on any of them.
> >
> >I claim that a great deal of progress can easily be

> made on such
> >issues simply by enabling the crucial faculty that I
> call the
> >learner's "question-asking frame of mind". I have
> previously posted
> >a sizable amount of documentation on simple tools
> and techniques
> >through which this faculty can be stimulated.
> >
> >(Much of this must have been before you started

> contributing to this
> >forum, I believe. Right now I am unable to put up
> that
> >documentation as I don't now have ready access to
> it: some general
> >documentation about the processes as a whole - not
> specifically
> >related to 'math-learning issues' - is available at:
> >
> /02%20-%20OPMS%20-%20Background%20Info/
> >
> >Right at this forum I have posted several models

> relating to math
> >learning issues being debated here - but I have no
> way of readily
> >locating those messages: perhaps the Moderator might
> on request be
> >able to point to some of my messages where such
> documentation is available?)
> >
> >With reference to your valid grouse about:

> > > As someone who was told how to teach
> > > math by an Ed.D. in charge of Academic Affairs at

> a
> > > previous job, I can tell you I've seen how such
> folks
> > > can manipulate into such positions where they do
> not
> > > have the credentials ...
> > >

> >I note that the tools I recommend would *definitely*
> prevent such
> >abuse of position - if abuse it was and that the
> Ed.D. was not
> >actually raising a valid "learning issue". I must
> observe that many
> >excellent mathematicians do not teach effectively -
> some
> >understanding about the processes involved in
> "understanding" &
> >"learning" is required for effective teaching [of
> any
> >subject]. Mathematicians may not be adequately
> aware of the
> >learning difficulties that many math learners
> confront as they
> >themselves may never have faced such difficulties.
> (That said, I
> >fully accept that those who wish to teach math MUST
> know *enough*
> >math to teach it).
> >
> >I observe that Louis Talman has rightly pointed out

> (though
> >implicitly) that Dr Bishop's gripe seems to have
> been put up merely
> >for the sake of putting up a gripe, nothing more. I
> would believe
> >that the real issue lies in the possibility of
> realizing the vision
> >as expressed for Dr Stage: "high quality mathematics
> and science
> >education for all students". In the words of that
> wonderful song:
> >"Accentuate the positive" - but to do that always
> though properly
> >understanding all the real difficulties we confront.
> >
> >GSC

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