New Post at RME: "Is Mathematics Teaching A Closed Book?"

[Michael Paul Goldenberg]

Please read the latest post at RationalMathEd.blogspot.com: "Is  
Mathematics Teaching A Closed Book?"

Excerpt:

"The nature of "Haim's Challenge" is his claim that "there are no open  
questions of mathematics pedagogy." He has his own reasons for making  
this claim that ostensibly have to do with the discourse on math-teach  
and his own agenda which primarily seems to be to assail public  
education at every turn, call for privatization, vouchers, dismantling  
public schools (if I understand him correctly, anyway), and blaming  
all our educational woes on unions, education professors, progressive  
educators, bad policy makers, etc., all of whom he lumps together  
alternately as "the Education Mafia," "Educational Mullahs," and  
similar witticisms. Our Haim is short on proof that any such entities  
exist, of course, but when you're arguing by name-calling, what need  
you evidence to offer?

Regardless, I am curious as to how readers of this blog feel about his  
fundamental claim, which he extends to the broader one that clearly he  
is right because "no one is interested in discussing math teaching; no  
active conversations are in evidence;" and so on.

Of course, I think this is arrant nonsense, but maybe I have a  
distorted understanding of what one is to make of the literature in  
mathematics education or the lists, web sites and blogs I frequent  
where concerns about pedgogy and the relationship between content and  
how to effectively teach it (be it to oneself, home schooled kids, or  
students in more traditional school settings) so as to improve  
learning are very much in evidence. Add to that active research  
programs by mathematics educators at universities and other  
institutions throughout the world and it's hard to imagine how anyone  
could offer a less accurate, more patently false claim.

But what do YOU think? Is mathematics teaching a closed book? Do we  
know all we need know about how to teach math? Is Haim/Ed right? And  
if not, what are some important open questions about mathematics  
pedagogy you are pursuing or would like to discuss with others?"

Read the entire post: <http://tinyurl.com/5x24hk>

[Richard Strausz]

Michael, I went to your blog and found the following reader's comment. I wanted to share it for two reasons:
1. I agree with both concerns.
2. I think others might find it interesting.
Richard
===================================
...when I think about whether there are open questions in math teaching, I think about my own teaching. I've been at it a long time; I try to get better every year...

Two of my big areas of concern are:
1. How can I teach a student at level N who hasn't mastered all the work of level N-1 (or maybe even N-2 for some things)? What can be done to help with the learning of the current content while remediating gaps in learning?
2. Is appropriate use of technology a tool which can help with the students in question #1, as well as increasing the learning of all students? What are some promising practices?

[Michael Paul Goldenberg]

The first question is nearly ubiquitous in mathematics education and  
of course there is no simple answer to it (not that we won't likely  
here some in this venue from non-teachers).

I suspect that we'll never see a time where teachers don't have to ask  
that question for the simple reason that there will never be a  
"system" that eliminates differential intellectual development,  
readiness, or individual variability in motivation for learning any  
given subject. It seems inevitable that kids will not be equally ready  
in all sorts of ways for any given mathematical topic at a given age/
grade level.

As long as we try to mass educate under the current model, with  
bizarre expectations that we can legislate kids to level N or  
intimidate schools, parents, teachers, or kids to some pull rabbits  
out of hats in an attempt to pretend that all kids are at level N at  
the appointed age/grade level, we'll miss the boat. A saner approach  
is to do the best we can to get kids ready for school and then teach  
them where they actually are, using differentiated instruction and  
methods to brings them as far as we can given where they start. That  
requires more flexible ideas about content and sequence, about  
instruction and tools, about the nature of classrooms, and about  
meaningful, useful assessment than we can reasonably expect to see  
given conservative/reactionary opposition to sensible approaches to  
public school in particular, and education in general.

Depending on how one views the first question,  answering the second  
seems more obvious. Appropriate use of technology will be a helpful  
approach for many kids. Some will find it less helpful. Lousy,  
deadheaded use of technology will prove no better and no worse than  
deadheaded use of most things, though it's always easier for  
troglodytes to point at ineffective and silly uses of innovation and  
shake their hairy fists at it than to recognize that there are no  
panaceas.

On Aug 19, 2008, at 5:39 PM, Richard Strausz wrote:

> Michael, I went to your blog and found the following reader's  
> comment. I wanted to share it for two reasons:
> 1. I agree with both concerns.
> 2. I think others might find it interesting.
> Richard
> ===================================
> ...when I think about whether there are open questions in math  
> teaching, I think about my own teaching. I've been at it a long  
> time; I try to get better every year...
>
> Two of my big areas of concern are:
> 1. How can I teach a student at level N who hasn't mastered all the  
> work of level N-1 (or maybe even N-2 for some things)? What can be  
> done to help with the learning of the current content while  
> remediating gaps in learning?
> 2. Is appropriate use of technology a tool which can help with the  
> students in question #1, as well as increasing the learning of all  
> students? What are some promising practices?
>

[Haim-5]

Richard Posted: Aug 19, 2008 5:39 PM

>Two of my big areas of concern are:
>1. How can I teach a student at level N who hasn't
>mastered all the work of level N-1 (or maybe even N-2
>for some things)? What can be done to help with the
>learning of the current content while remediating gaps
>in learning?
>2. Is appropriate use of technology a tool which can
>help with the students in question #1, as well as
>increasing the learning of all students? What are some
>promising practices?

 
Richard,

   Regarding question 1., my own hypothesis is:  nothing.  I.e., you cannot teach "current content" before remediating gaps in content that should have been learned previously.  What is your hypothesis?

   You have raised question 2. in this forum a number of times.  Can you share with us some of what you have done to explore this question.  Have you come to any conclusions?

Haim
Unashamedly White and Unapologetically Jewish

Michael, I agree with what you say. Some critics use such real-world observations as more ammunition for their crusade against public education.

However, in talking with math teachers in Catholic and Jewish high schools, I hear similar situations in their classrooms.

Richard

> The first question is nearly ubiquitous in
> mathematics education and  
> of course there is no simple answer to it (not that
> we won't likely  
> here some in this venue from non-teachers).
>
> I suspect that we'll never see a time where teachers
> don't have to ask  
> that question for the simple reason that there will
> never be a  
> "system" that eliminates differential intellectual
> development,  
> readiness, or individual variability in motivation
> for learning any  
> given subject. It seems inevitable that kids will not
> be equally ready  
> in all sorts of ways for any given mathematical topic
> at a given age/
> grade level.
>
> As long as we try to mass educate under the current
> model, with  
> bizarre expectations that we can legislate kids to
> level N or  
> intimidate schools, parents, teachers, or kids to
> some pull rabbits  
> out of hats in an attempt to pretend that all kids
> are at level N at  
> the appointed age/grade level, we'll miss the boat. A
> saner approach  
> is to do the best we can to get kids ready for school
> and then teach  
> them where they actually are, using differentiated
> instruction and  
> methods to brings them as far as we can given where
> they start. That  
> requires more flexible ideas about content and
> sequence, about  
> instruction and tools, about the nature of
> classrooms, and about  
> meaningful, useful assessment than we can reasonably
> expect to see  
> given conservative/reactionary opposition to sensible
> approaches to  
> public school in particular, and education in
> general.
>
> Depending on how one views the first question,
>  answering the second  
> seems more obvious. Appropriate use of technology
> will be a helpful  
> approach for many kids. Some will find it less
> helpful. Lousy,  
> deadheaded use of technology will prove no better and
> no worse than  
> deadheaded use of most things, though it's always
> easier for  
> troglodytes to point at ineffective and silly uses of
> innovation and  
> shake their hairy fists at it than to recognize that
> there are no  
> panaceas.
>
> On Aug 19, 2008, at 5:39 PM, Richard Strausz wrote:
>
> > Michael, I went to your blog and found the
> following reader's  
> > comment. I wanted to share it for two reasons:
> > 1. I agree with both concerns.
> > 2. I think others might find it interesting.
> > Richard
> > ===================================
> > ...when I think about whether there are open
> questions in math  
> > teaching, I think about my own teaching. I've been
> at it a long  
> > time; I try to get better every year...
> >
> > Two of my big areas of concern are:
> > 1. How can I teach a student at level N who hasn't
> mastered all the  
> > work of level N-1 (or maybe even N-2 for some
> things)? What can be  
> > done to help with the learning of the current
> content while  
> > remediating gaps in learning?
> > 2. Is appropriate use of technology a tool which
> can help with the  
> > students in question #1, as well as increasing the
> learning of all  
> > students? What are some promising practices?

[Michael Paul Goldenberg]

I'm not surprised in the least. Didn't mean to suggest there was  
something unique going on in secular schools. I suspect we'd hear  
similar issues and concerns from private, non-sectarian classrooms.
On Aug 20, 2008, at 2:12 PM, ArPeEs@... wrote:

> Michael, I agree with what you say. Some critics use such real-world  
> observations as more ammunition for their crusade against public  
> education.
>
> However, in talking with math teachers in Catholic and Jewish high  
> schools, I hear similar situations in their classrooms.
>
> Richard

[Wayne Bishop]

Richard,

You are right about nonpublic schools but you seem to have
misinterpreted the real problem with your identification of this
problem being "more ammunition for their crusade against public
education". It's ammunition for "their" crusade against colleges of
education, the problem so well identified by Reid Lyon.  The aversion
against standards-based education (prior to the collegiate level
where, at least in mathematics, standards tend to be used and
accepted) is a direct consequence of the teaching of schools/colleges
of education.  For inexplicable reasons, almost religious in
appearance, the industry prefers to place students by age-level even
when it is obvious that students are being placed into situations
where they are doomed to fail if honest performance standards are maintained.

Wayne

At 11:12 AM 8/20/2008, ArPeEs@... wrote:

>Michael, I agree with what you say. Some critics use such real-world
>observations as more ammunition for their crusade against public education.
>
>However, in talking with math teachers in Catholic and Jewish high
>schools, I hear similar situations in their classrooms.
>
>Richard
>
> > The first question is nearly ubiquitous in
> > mathematics education and
> > of course there is no simple answer to it (not that
> > we won't likely
> > here some in this venue from non-teachers).
> >
> > I suspect that we'll never see a time where teachers
> > don't have to ask
> > that question for the simple reason that there will
> > never be a
> > "system" that eliminates differential intellectual
> > development,
> > readiness, or individual variability in motivation
> > for learning any
> > given subject. It seems inevitable that kids will not
> > be equally ready
> > in all sorts of ways for any given mathematical topic
> > at a given age/
> > grade level.
> >
> > As long as we try to mass educate under the current
> > model, with
> > bizarre expectations that we can legislate kids to
> > level N or
> > intimidate schools, parents, teachers, or kids to
> > some pull rabbits
> > out of hats in an attempt to pretend that all kids
> > are at level N at
> > the appointed age/grade level, we'll miss the boat. A
> > saner approach
> > is to do the best we can to get kids ready for school
> > and then teach
> > them where they actually are, using differentiated
> > instruction and
> > methods to brings them as far as we can given where
> > they start. That
> > requires more flexible ideas about content and
> > sequence, about
> > instruction and tools, about the nature of
> > classrooms, and about
> > meaningful, useful assessment than we can reasonably
> > expect to see
> > given conservative/reactionary opposition to sensible
> > approaches to
> > public school in particular, and education in
> > general.
> >
> > Depending on how one views the first question,
> >  answering the second
> > seems more obvious. Appropriate use of technology
> > will be a helpful
> > approach for many kids. Some will find it less
> > helpful. Lousy,
> > deadheaded use of technology will prove no better and
> > no worse than
> > deadheaded use of most things, though it's always
> > easier for
> > troglodytes to point at ineffective and silly uses of
> > innovation and
> > shake their hairy fists at it than to recognize that
> > there are no
> > panaceas.
> >
> > On Aug 19, 2008, at 5:39 PM, Richard Strausz wrote:
> >
> > > Michael, I went to your blog and found the
> > following reader's
> > > comment. I wanted to share it for two reasons:
> > > 1. I agree with both concerns.
> > > 2. I think others might find it interesting.
> > > Richard
> > > ===================================
> > > ...when I think about whether there are open
> > questions in math
> > > teaching, I think about my own teaching. I've been
> > at it a long
> > > time; I try to get better every year...
> > >
> > > Two of my big areas of concern are:
> > > 1. How can I teach a student at level N who hasn't
> > mastered all the
> > > work of level N-1 (or maybe even N-2 for some
> > things)? What can be
> > > done to help with the learning of the current
> > content while
> > > remediating gaps in learning?
> > > 2. Is appropriate use of technology a tool which
> > can help with the
> > > students in question #1, as well as increasing the
> > learning of all
> > > students? What are some promising practices?
>
>No virus found in this incoming message.
>Checked by AVG - http://www.avg.com
>Version: 8.0.138 / Virus Database: 270.6.6/1623 - Release Date:
>8/20/2008 8:12 AM

[Richard Strausz]

> Richard,
>
> You are right about nonpublic schools but you seem to
> have
> misinterpreted the real problem with your
> identification of this
> problem being "more ammunition for their crusade
> against public
> education".

Some (including some frequent posters here) note these problems and assume that things are rosier in private schools. That was my point.

> It's ammunition for "their" crusade against colleges of
> education, the problem so well identified by Reid
> Lyon.  The aversion
> against standards-based education

I presume you are not including NCTM standards here.

>(prior to the collegiate level
> where, at least in mathematics, standards tend to be
> used and
> accepted) is a direct consequence of the teaching of
> schools/colleges
> of education.  For inexplicable reasons, almost
> religious in
> appearance, the industry prefers to place students by
> age-level even
> when it is obvious that students are being placed
> into situations
> where they are doomed to fail if honest performance
> standards are maintained.
>
> Wayne

As I made my original post here, I was writing of the experience of teaching a student in Algebra II who had passed Algebra I, but had some deficiencies that I want to bring up as I teach the Alg II curriculum. I think you were restricting your comments to students allowed in Alg II after failing in Alg I.

Richard

[Haim-5]

Richard Posted: Aug 21, 2008 10:16 AM

>Some (including some frequent posters here) note these
>problems and assume that things are rosier in private
>schools. That was my point.
 
Richard,

   I do not know whom you have in mind, but as one of the more frequent posters to this forum I would like the record to show that I am not such a one.  Quite to the contrary.

   I am well aware that many private schools are far more fuzzy than most public schools.  Furthermore, many public schools are chartered by people who think the public schools are not fuzzy enough.  I believe this is one important reason why the evidence on charter schools is ambiguous.

   The important difference, from my point of view, is that people in private and charter schools are there by their own personal choice.  Indeed, I just posted an anecdote about personal friends of mine who were committed to fuzzy education.  They invested THEIR OWN son and THEIR OWN money into it.  I have no objection to this.

   I vehemently object when government agencies foist fuzzy education upon the unsuspecting, the unwilling, and the helpless.

Haim
Unashamedly White and Unapologetically Jewish

[Wayne Bishop]

At 07:16 AM 8/21/2008, Richard Strausz wrote:

> > "more ammunition for their crusade against public education".
>
>Some (including some frequent posters here) note these problems and
>assume that things are rosier in private schools. That was my point.

More must be known about a school, public or private, to know whether
or not things are rosier in general or about unqualified social
promotion in particular.

> > The aversion against standards-based education
>
>I presume you are not including NCTM standards here.

You can extrapolate far beyond "here".  There are no standards in the
NCTM document abbreviated by that name and I have said so many
times.  I do believe in standards-based education but not in
deliberately misnamed bait-and-switch.

>As I made my original post here, I was writing of the experience of
>teaching a student in Algebra II who had passed Algebra I, but had
>some deficiencies that I want to bring up as I teach the Alg II
>curriculum. I think you were restricting your comments to students
>allowed in Alg II after failing in Alg I

In a no-fail world, "passed Algebra I" can be a meaningless
criterion.  Your statement had much more meaning, "How can I teach a
student at level N who hasn't mastered all the work of level N-1 (or
maybe even N-2 for some things)? What can be done to help with the
learning of the current content while remediating gaps in learning?"

If that was meant to imply an honest pass in Algebra I but still in
need of a bit of review, my apologies.  I read N-2 as implying some
pre-algebra problems; what did you have in mind?

Wayne

[GS Chandy]

Haim The Challenger's Assertion:
There are no important open questions in math pedagogy.

GSC's Hypothesis:

It is impossible for a person with a closed mind to see an open question anywhere, in any field.

- -- GSC

[Michael Paul Goldenberg]

For those of us who actually teach K-12, the issue that is far more  
onerous is not what book was used, what teaching methods were most  
prominent (well, that's a concern if one is using eclectic  
instructional approaches with students who've been taught in totally  
lecture-driven, teacher-centered classrooms in which student passivity  
is so ingrained that ANY requirement that they become actively engaged  
in mathematical thinking and mathematical discourse is useless until  
the teacher shows students HOW to become active in mathematics and  
convinces them that it is necessary and productive to do so), or even  
to some extent whether the previous class covers every topic that s/he  
would have. Rather, it is the passing of students from previous  
classes with grades of D and, generally, C. Frankly, I'd say that in  
my experience, unless we're talking about an honors course of some  
sort or an extraordinary school, anyone coming into Algebra II with a  
grade less than a solid B in Algebra I is going to have more  
mathematical holes in his/her head than it takes to fill the Albert  
Hall or are found in the typical argument from members of  
Mathematically Correct and NYC-HOLD about any aspect of mathematics  
education. And that's a lot of holes, let me tell you.

The obvious solution is to stop making grades the joke they are and  
instead go to minimum exit/entrance exams with multiple sorts of  
assessment employed to determine whether students are ready to move  
on. I suggest both exit and entrance exams so that there is a double  
check in place: one on leaving level N-1 and one on beginning level N.  
And also so that those who marginally fail to gain exit at level N-1  
can get another shot at the beginning of the following year. Should  
they have gotten themselves up to speed by then, they should not be  
denied a chance to prove themselves ready to proceed.

Naturally, I'm not calling for a nationally-normed testing instrument  
here, especially if it's strictly some vapid multiple-choice test that  
values only one narrow set of skills. But within each state's  
standards, if there is a real attempt at meaningful formative  
assessment that leads directly to re-teaching areas of weakness before  
allowing students to take a crack at more challenging mathematics that  
builds directly on previous knowledge, then such a system would  
potentially reduce the number of students who are simply pushed  
further down the rabbit hole with nothing to hold on to.

For such a system to work, however, it can't be linked to a bunch of  
politicized punishments intended to help ideologues and politicians  
use teachers, schools, and kids as footballs for making propaganda.

Minimally, that would mean that people like Wayne Bishop would need to  
be keep as far as possible from influencing the assessment process.  
Instead, knowledgeable assessment experts who actually understand and  
stick to psychometric principles must have major input into each state  
and district's development of assessment tools. Close attention to  
some of the issues about cut scores, the theory of performance  
standards, and related issue raised by Alan Tucker at <http://www.ams.sunysb.edu/~tucker/StandardsProb.pdf 
 > and <http://www.ams.sunysb.edu/~tucker/MathAPaper.pdf> must be  
paid. A balanced approach to assessment, regardless of the religious  
objections of the nay-sayers, must be used, and any structure that has  
been proven to promote wide-scale cheating due to absurd political  
pressure on educators, parents, and students, must be closely examined  
and, if found incapable of improvement, abandoned.

Of course, the mileage of the usual suspects may vary and no doubt  
already does.

Expensive? You bet. But then, finding out what's really going on in a  
complex world is always dearer than looking for surface data that  
tells you what you already believed to be the case beforehand. And  
crafting real solutions rather than destroying public education so  
that the rich can get richer is always unpopular. . . among the  
powerful and their lackeys and mouthpieces.

On Aug 22, 2008, at 12:17 AM, Wayne Bishop wrote:

> At 07:16 AM 8/21/2008, Richard Strausz wrote:
>
>> > "more ammunition for their crusade against public education".
>>
>> Some (including some frequent posters here) note these problems and  
>> assume that things are rosier in private schools. That was my point.
>
> More must be known about a school, public or private, to know  
> whether or not things are rosier in general or about unqualified  
> social promotion in particular.
>
>> > The aversion against standards-based education
>>
>> I presume you are not including NCTM standards here.
>
> You can extrapolate far beyond "here".  There are no standards in  
> the NCTM document abbreviated by that name and I have said so many  
> times.  I do believe in standards-based education but not in  
> deliberately misnamed bait-and-switch.
>
>> As I made my original post here, I was writing of the experience of  
>> teaching a student in Algebra II who had passed Algebra I, but had  
>> some deficiencies that I want to bring up as I teach the Alg II  
>> curriculum. I think you were restricting your comments to students  
>> allowed in Alg II after failing in Alg I
>
> In a no-fail world, "passed Algebra I" can be a meaningless  
> criterion.  Your statement had much more meaning, "How can I teach a  
> student at level N who hasn't mastered all the work of level N-1 (or  
> maybe even N-2 for some things)? What can be done to help with the  
> learning of the current content while remediating gaps in learning?"
>
> If that was meant to imply an honest pass in Algebra I but still in  
> need of a bit of review, my apologies.  I read N-2 as implying some  
> pre-algebra problems; what did you have in mind?
>
> Wayne

[Haim-5]

GSC Posted: Aug 22, 2008 10:09 AM

>Haim The Challenger's Assertion:
>There are no important open questions in math pedagogy.
>
>GSC's Hypothesis:
>
>It is impossible for a person with a closed mind to see
>an open question anywhere, in any field.

GS,

   It should make no difference to you what I do or do not see.  If you have an open question in path pedagogy, that you care to explore, please get on with it.

   In fact, only recently you presented this forum with a list of questions.  Although I did not send you an engraved invitation, I did ask you to choose one of your own questions and to invite others to begin exploring it with you.

   Let's put aside the question of why you have not long since done so.  Please begin now.

Haim
Unashamedly White and Unapologetically Jewish

[Louis Talman]

On Aug 22, 2008, at 8:09 AM, GS Chandy wrote:

> GSC's Hypothesis:
>
> It is impossible for a person with a closed mind to see an open  
> question anywhere, in any field.

I think I said something substantially equivalent to this a few weeks  
ago...  :-)


- --Lou Talman
   Department of Mathematical & Computer Sciences
   Metropolitan State College of Denver

   <http://clem.mscd.edu/%7Etalmanl>

[GS Chandy]

> GSC Posted: Aug 22, 2008 10:09 AM
>
> >Haim The Challenger's Assertion:
> >There are no important open questions in math
> pedagogy.
> >
> >GSC's Hypothesis:
> >
> >It is impossible for a person with a closed mind to
> see
> >an open question anywhere, in any field.
>
> GS,
>
> It should make no difference to you what I do or
> or do not see.  If you have an open question in path
> pedagogy, that you care to explore, please get on
> with it.
>
> In fact, only recently you presented this forum
> rum with a list of questions.  Although I did not
> send you an engraved invitation, I did ask you to
> choose one of your own questions and to invite others
> to begin exploring it with you.
>
> Let's put aside the question of why you have not
> not long since done so.  Please begin now.
>
> Haim
> Unashamedly White and Unapologetically Jewish

> Haim The Challenger posted: Aug 22, 2008 8:50 PM  
> It should make no difference to you what I do or
> or do not see.  If you have an open question in path
> pedagogy, that you care to explore, please get on
> with it.
>
> In fact, only recently you presented this forum
> rum with a list of questions.  Although I did not
> send you an engraved invitation, I did ask you to
> choose one of your own questions and to invite others
> to begin exploring it with you.
> Let's put aside the question of why you have not
> not long since done so.  Please begin now.
>
> Haim
> Unashamedly White and Unapologetically Jewish

In case Haim The Challenger had not realised it yet, there are others at this forum besides Haim The Challenger.  Some of those others have indeed been invited and they are indeed exploring this issue with me.

As has been explained earlier (several times over) such exploration cannot be done at this forum because it lacks facilities to explore issues using prose + structural graphics.  This means, in other words (in order to make matters as abundantly clear to Haim The Challenger who evidently requires everything repeated to him many, many times over before it penetrates that Challenging Brain of His) that we have to move to other forums to perform such explorations.  Therefore, Haim The Challenger, we have indeed been exploring the issues at other forums.

I do trust this is, now at least, clear at last to you, Haim The Challenger ("Unashamedly White and Unapologetically Jewish").  If absolutely essential, I am sure it could be arranged that this message is repeated 10 or 100 or a 1000 times over - but I would prefer not to burden others at this forum with such repetitiveness.  In fact, I shall endeavour to ensure that the assertions in this message are not repeated even once at any thread in all of Math-Teach.

- -- GSC

[Dave L. Renfro]

Michael Paul Goldenberg wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6345559

> Rather, it is the passing of students from previous
> classes with grades of D and, generally, C. Frankly,
> I'd say that in my experience, unless we're talking
> about an honors course of some sort or an extraordinary
> school, anyone coming into Algebra II with a grade
> less than a solid B in Algebra I is going to have
> more mathematical holes in his/her head than it takes

This issue came up in the ap-calclus discussion group
(archived at Math Forum) this past May/June. Interestingly,
someone who I thought had a fair amount of teaching
experience (but perhaps it was all in the same school
or same type of school) held an opinion opposite of
what you said. On the other hand, most of the other
participants in the group felt closer to what you said
(myself included, although I don't know if I posted
any comments at the time).

Here are three of his posts, followed by a Math Forum
search that pulls up most of the discussion that took
place:

http://mathforum.org/kb/thread.jspa?messageID=6206969
http://mathforum.org/kb/thread.jspa?messageID=6234979
http://mathforum.org/kb/thread.jspa?messageID=6234988

http://tinyurl.com/5lq43j [53 posts; May 22 to June 14]

Dave L. Renfro