Howard Tanner wrote (in part):


> Sadly, I am one of those who should never be let near
> a geometry class,
> because I don't think that teaching about ordered,
> labelled triangles is
> the best way to approach this in lower secondary at
> least.



I would ignore that comment.  The one who made is clearly the kind
of mathematician who enjoys insulting others who don't agree with
him.  I have met some others on other discussion lists in Math Forum;
math-teach, for example, includes several mathematicians like that.
I had mentioned some details about this in an earlier message in
mathedu; this message is available at

http://mathforum.org/kb/message.jspa?messageID=6874856&tstart=0.

One of them enjoys attacking those who support at least some ideas
of reform mathematics education and attacks those who do not agree
with current standardized test policies in K-12.  Then I had mentioned
the problem with many students doing well on such tests and even classroom
exams who lack a decent understanding of mathematics because these tests
don't test their reasoning, only their ability to regurgitate stuff out of
a standard textbook or to make routine calculations or solve problems 
similiar to problems they've seen before (many of these are similiar to 
textbook problems).  There are many who can pick up a textbook on algebra
or calculus and work any standard exercise within the book and yet lack
a deep understanding of algebra or calculus.  Standardized tests and classroom
exams are the kinds of exams that label these kinds of students as talented
or competent in mathematics.  This mathematician says that he and the university
would be thrilled to see such students because their chances of graduating
with their degree of choice is high.  I wonder if he really means this; I doubt
he does and is saying it just to find an excuse for attacking me because he 
can't find an argument to refute what I've said.

The problem has become serious enough in recent years that another professor had 
told me that he's interviewed people with PhD's in mathematics for 
positions at his college who lack a basic understanding of elementary mathematics!
I'm not impressed to see students doing well on exams or in math courses who 
haven't learned anything; thus, I will not make any conclusions whatsoever about a
person's level of ability in mathematics if all I see are his grades and test
scores because massive test score inflation and grade inflation in recent years
allows students to make A's and B's and high test scores without learning anything.
Yet this mathematician I had mentioned tries to attack me by denying that this is
a problem; I'm sure he's well aware of the problem and most likely denies it because
he can't find an argument to refute my claims.



> At times on this list, I do have a sense of being
> lectured to by people
> who regard themselves as the cleverest kids in the
> room, whose word
> should be accepted on pain of ridicule. I am not
> picking out any one
> person as the biggest culprit in saying this. More
> than one person has
> behaved this way and as an educator, I don't find it
> helpful.



I agree, especially if there is no evidence that suggests that one approach
works well and that the other or others are damaging to students' confidence
or ability to understand and use mathematics.  I see no evidence that
teaching triangles with the ordinary definition is in itself damaging to their 
confidence or ability to understand and use mathematics.  In fact, I think
what causes the most damage to students' confidence and understanding of
mathematics is teaching rotely, relying on one approach only and refusing
to consider others, not giving students the opportunity to think about and
enjoy mathematics and be creative with it, telling or suggesting students
that they cannot learn to think for themselves--that they need a teacher who
will tell them everything up front because they will get it wrong otherwise
(yes, we can do things that suggest that attitude without ever saying it
explicitly), reducing mathematics to procedures and memorization and not
explaining the reasoning for why the mathematics they are learning works the
way it does, etc.  Such teaching fails to help students learn to see what
mathematics really is and may possibly give them the idea that only certain
people can learn mathematics and the rest of them are too dumb to learn it.         



> Teaching children about proof is not easy. Rigour is
> necessary, but we
> all need to start somewhere. Before demanding that we
> all start with the
> most refined definitions, we might remember Lakatos.
> 
> It is easy to criticise practice that is not rigorous
> in a way which is
> not helpful. I am currently concerned about a
> training video from
> "Teachers TV" in the UK that shows a teacher trying
> to move from "good"
> to "outstanding" (as graded by inspectors). 
> 
> The video shows different ways of "proving" that the
> internal angles of
> a triangle add to 180 degrees. Most of the examples
> offered showed but
> did not prove and no distinction was made between
> them. I don't want to
> attack the teacher or trainer involved, but I want to
> make sure that the
> concept of proof is taught and learned effectively in
> our schools from
> an early age.



Speaking of rigor in mathematics, these comments suggest to me that
the role of rigor in pre-college mathematics in the UK is completely
different from the role of rigor in pre-college mathematics in the 
United States.  Rigor is being gradually sucked out of mathematics
in the U.S. schools; more and more high school geometry classes are
de-emphasizing proofs.  The numbers of proofs in textbooks are
shrinking over the years.  For example, I had recently seen some
commentary about a pre-calculus book that is over 1000 pages long
and contains only six proofs in the whole book.  Calculus books
in the 1960s and 1970s had contained far more epsilon-delta proofs
than today's calculus books.  The explanation of such proofs in
calculus books is gradually being de-emphasized; in fact, some (many?)
calculus books don't contain these proofs or the formal definition of
limit.  One professor I had talked to had told me that the latest editions
of James Stewart's "Calculus" book contain fewer proofs than the older 
editions.

I haven't seen many arguments in U.S. mathematics education about whether
this approach or that approach is rigorous enough but instead whether
it gives students the proper skills they need (with more emphasis on
computational skills than problem solving skills).  Even if the word
"rigor" arises, it is often used to mean that the mathematics taught is
correct, not that the proofs and reasoning taught in the classroom
(very little of that is taught anyway) are rigorous.  I doubt this
argument we have seen about which definition of "triangle" should be
taught to high school geometry students that we have seen here on
MATHEDU would turn in heated debates in the U.S.  Even if the debate
were to become heated, I doubt the argument would be about which
approach is more rigorous and helps students better learn to understand 
and write proofs but which allows them to make better grades on tests.

I don't see much interest among U.S. math educators about trying to
bring rigor and proof back into the K-12 math classrooms.  Those who do
have interest in this are mostly mathematicians, not math educators
who are not mathematicians.  But they argue frequently about how
this should be done.  



> The Euclidean proof is within the capabilities of
> many 11 year old
> pupils, but it is not examined, so is often not
> taught. I don't want to
> go back to learning proofs of by heart without
> developing personal
> meaning. But I do want pupils to be challenged to
> think rigorously. It
> would be more productive to consider how we might
> achieve this.
> 
> Howard Tanner



Add to the problem that some mathematicians here in the U.S.
do not think most 11-year-olds are capable of understanding these
proofs; heck, many of them think most students are incapable of 
learning these proofs (or any rigorous proofs) at any age.



Jonathan Groves