Good geometry text with proofs
Heath
I'm looking for a good geometry text with proof techniques. I'd like
to learn the subject for my own enjoyment. I took the course in high
school, but I was a poor math student then (I earned D's all four
quarters), and my teacher was not very inspirational.
Since high school, I finished my B.S. in Chemistry (A's in Calculus,
Linear Algebra, and Differential Equations), and am finishing my M.S.
in Geochemistry, I then plan to enter a Ph.D. program.
I have several of the Dover Geometry books, but they focus on
non-Euclidian Geometry. I'd like to learn the ideas and proofs of the
Euclidian system. Are there inexpensive, but thorough books available
that would help me learn the proof techniques?
Thanks,
Heath
maky m.
> Hi,
> I'm looking for a good geometry text with proof techniques. I'd like
> to learn the subject for my own enjoyment.
recently i came across a high school/junior college level geometry
textbook.
elementary geometry by gustafson and frisk. imo, appropriate for a
self-learner: straightforward. adequately formal. adequate exercise
sets, both routine and proofs.
once you build good foundations, you may want to dive into any of the
following excellent geometry books
1. foundations of geometry by david hilbert
2. euclidean and non-euclidean geometries by marvin jay greenberg
3. Introduction to geometry by h. s. coxeter
> I took the course in high school, but I was a poor math student then
> (I earned D's all four quarters), and my teacher was not very
> inspirational.
one thing that we as students fail to understand is that 'inspiration'
is self induced. with few exceptions, ie k-12, it is almost irrational
to expect the educator to bear the burden of 'inspiring' a subject
such as geometry. of course, today's math "educators" and their bosses
will normally disgree with the notion that it is entirely the student
who should decide what inspires him. but then, it is necessary for
this group (math "educators" and their bosses) to practically reject
any notion of student responsibility. not doing so invalidates their
existence.
Walter Whiteley
do you want to learn 'proof'
OR
do you want to learn geometric reasoning?
Speaking as a geometer (who first trained in logic) I find
those are quite distinct. My own training (learning proofs)
left me without any real preparation in geometric reasoning
of the form needed for my current work over a range of geometric
applications ranging from structural engineering through robotics
and CAD to protein modeling. It also left me without a sense of
how truly enjoyable, and distinct, geometry could be.
Since about 1870's (Felix Klein), the practice of geometries has
centered on
transformational reasoning, and I find that essential for my work.
So a book like Hilbert's is a very interesting excursion into the
world of axiomatics, and mathematical logic, but has little to
do with the practice of geometric reasoning.
Not a text book - but a very interesting way to work with
transformations,
and get an exposure to the layers of geometric transformations
(translation, rotation, scaling or dilation, affine, projective) are the
three volumes Geometric Transformations by I. Yoglom.
These are problem books, with solutions, written for Russian high school
students (and difficult for most North American senior undergrads).
They have extensive solutions, and work well when combined with some
dynamic geometry program (such as Geometers Sketchpad, Cabri,
or Cinderella, or other shareware / freeware equivalents).
Yoglom's books (in translation) are inexpensive paperbacks, available
from the
MAA (Mathematical Association of America), online.
Coxeter's book is also very good, but I have not found people able to
work with it as their first book - rather as a second book pulling things
together and expanding connections.
Proof has its place in all fields of mathematics, including geometry.
However, it is not at the core of what mathematicians do day by day,
nor of what users of mathematics need as grounding for the use of
mathematics to make sense out of what we experience in the world.
I have found that most teachers (in fact most people with degrees in
Math in North America) have no real exposure to geometric reasoning.
It is hard to be inspired when one does not know the subject, or believe
the subject is important. Sadly, for several decades, that is the
experience
and the implicit message of most math programs, where geometry is not
taught, or if taught has become an exercise in logic with a teacher who
themselves is not inspired by geometric thinking but by the axiomatics.
Walter Whiteley
maky m.
> Without getting into a long debate, let me just raise the question:
>
> do you want to learn 'proof'
> OR
> do you want to learn geometric reasoning?
perhaps both. but really, what did you think when this student
acknowledged that his geometry background was un-"inspired" highschool
geometry? it seems to me that he wants to learn a bit of geometry with
proofs, or as you call it "geometric reasoning."
did you read the statement "once you build good foundations, you may
want to dive into any of the following excellent geometry books?"
> Proof has its place in all fields of mathematics, including geometry.
> However, it is not at the core of what mathematicians do day by day,
> nor of what users of mathematics need as grounding for the use of
> mathematics to make sense out of what we experience in the world.
>
> I have found that most teachers (in fact most people with degrees in
> Math in North America) have no real exposure to geometric reasoning.
this may be true. what is the approximate percentage?
Walter Whiteley
>> Proof has its place in all fields of mathematics, including geometry.
>> However, it is not at the core of what mathematicians do day by day,
>> nor of what users of mathematics need as grounding for the use of
>> mathematics to make sense out of what we experience in the world.
>>
>> I have found that most teachers (in fact most people with degrees in
>> Math in North America) have no real exposure to geometric reasoning.
>
> this may be true. what is the approximate percentage?
There have been some sessions at MAA meetings on this.
Very few undergraduate programs have more than one discrete
geometry course (distinct from differential geometry, algebraic
geometry ... ).
I am not aware of any that require geometry as an upper level course,
where
most require analysis, algebra, .... .
When there are 'geometry' courses, they are often taught be people
in logic (I trained in logic at grad school) and taught as an exercise
in axiomatics. This does not do much for geometric reasoning (see
below).
I have been encouraging undergrads to consider graduate work
in geometry - but there are few graduate programs in North America
where there are more than one person doing discrete geometry.
U of T (where Coxeter taught for decades) made a decision that as
he retired, and then a number of other geometers retired / left that they
would hire NOBODY in this field.
I am not aware of any graduate program 'comprehensive exams'
which include geometry as a required part of the background of
future Ph.D students (that is of future university and college teachers).
Certainly I have not taken a formal geometry course since high school.
This type of standard is an implicit statement by the dominant culture
in math departments that geometry is not part of growing, research
mathematics. I believe they are wrong - but I think that is the
standard response.
I do not have statistics, but there has been an open discussion of this
over the last decade, with a number of public statements made by a
variety of people, without any evidence to the contrary. Many people
have agreed that this matches their experience and their observations.
I am not sure I have encountered contrary evidence about undergrad
programs (programs which prepare future teachers at all levels)
but it would be good to know of situations where it is better. Cornell
is an exception - having about 8 undergrad geometry courses
(and some good geometers). However, the statements I am backing
are strongly made by the people at Cornell, in these public sessions
at meetings of mathematicians.
When teachers see that 'geometry' is not central to undergrad programs,
there is some immediate impact. They may not take it themselves.
When they are teaching a course with geometry as one unit, and
they run out of time because there is too much material (which almost
always happens around here) they will leave geometry to the end
and then cut it short or drop it. They judge, correctly, that doing
geometry
will not help survive first year calculus.
I have been teaching a geometry course for people who are currently
teaching high school for some time. I have worked with people who
include
writers of curriculum for the largest province in Canada - Ontario. The
most
recent group included nobody with a background in geometry,
and it shows in many subtle ways. Just more stories perhaps, but
the they accumulate over time.
To be clear, I do not consider anything resembling two column proofs
to prepare one for geometric reasoning. It can be (if done well) a
nice
introduction to formal logic. I enjoyed it when I took such a course.
However, I did not learn geometry and missed something very important
which I had to learn, on my own, after graduate school.
Walter Whiteley
maky m.
> >> Proof has its place in all fields of mathematics, including geometry.
> >> However, it is not at the core of what mathematicians do day by day,
> >> nor of what users of mathematics need as grounding for the use of
> >> mathematics to make sense out of what we experience in the world.
> >>
> >> I have found that most teachers (in fact most people with degrees in
> >> Math in North America) have no real exposure to geometric reasoning.
> >
> > this may be true. what is the approximate percentage?
>
> There have been some sessions at MAA meetings on this.
can you be more specific?
let me try to understand your point a little better.
an introductory course in geometry would naturally have to expose
students to "two column proofs" and expect that they use such approach
to prove simple geometric statements. how do you arrive to the
conclusion that such process is, or implies, lack of "geometric
resoning?"
me
> an introductory course in geometry would naturally have to expose
> students to "two column proofs" and expect that they use such approach
> to prove simple geometric statements.
Please explain why this should be so. I don't believe it for a moment.
--Lou Talman
Department of Mathematical and Computer Sciences
Metropolitan State College of Denver
Art Mabbott
*I can remember spending three months doing two column proofs in the 70’s and 80’s. No longer...we missed so much of what geometry is all about doing that.
Art
Art Mabbott
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Walter Whiteley
some years back.
He had done a series of formal proofs, then gave an assignment
which included: Give a convincing argument that .... .
The assignments came back with many explanations, but no formal
proofs. When he returned the assignments and asked the class why?,
the response was: "You asked for a CONVINCING argument!"
Many formal proofs are almost completely detached from meaning,
conviction, understanding. In fact, formalists pride themselves on this
detachment from meaning. However, we do want students to work with
meaning, with conviction, with understanding. Some preliminary studies
of brain images of people working with even simple reasoning indicate
that
distinct parts of the brain do 'the same' reasoning when presented
formally
and when presented with meaningful terms. It is good if students can
connect,
but for geometry, I much prefer to engage with the meaning.
I have read many proofs which are not 'convincing'. In fact, I remember
working with peano's axioms for the natural numbers, and writing a proof
in which every sentence between the assumption and the conclusion was
less obvious that the conclusion! It was a game a could play, but it
did not
convince me of much besides the fact that these axioms could, under
duress, be made to work.
I have also written proofs, for publication, which did not generate
'conviction' in themselves, for me. The conviction proceeded them, or
sometimes
was generated by other explorations, including model building.
I have often wondered why (other than history of math) we choose geometry
as the place to introduce standard forms for proofs, rather than some
piece of algebra. Perhaps, algebra is 'too important' to be sacrificed
to teaching logic, but geometry is 'not too important'?
I do, of course, want lots of 'reasoning' and 'presentation of arguments'
with assumptions, explorations of when and how they are used.
I am not familiar with any structured form that does not distort things
so
much as to be seriously distracting and leading to narrow horizons and
possibilities. For example, most 'structured forms' have little room
for
reasoning with transformations. In fact, the structured forms become
an obstacle to working with these ideas which are at the core of
geometry.
Michael de Villiers has written about the many forms and uses of 'proofs'
and reasoning in geometry. One source is his book on Rethinking Proof
with Geometers SketchPad (a new edition has appeared for GSP 4).
He continues to write on this, and I recommend his explorations,
although I would probably be even more radical than he is on some
of the problems of too narrow a vision of reasoning in geometry.
Walter Whiteley
maky m.
> mman...@my-deja.com (maky m.) wrote:
>
> > an introductory course in geometry would naturally have to expose
> > students to "two column proofs" and expect that they use such approach
> > to prove simple geometric statements.
>
> Please explain why this should be so. I don't believe it for a moment.
what do you suggest? geometer sketchpad and cute pictures alone?
maky m.
> There should be at some form of formal proof/logical thinking learning
> in an introductory course. Not necessarily two-column proofs as we used to
> do years ago*. But some type of structured proof writing - two-col,
> paragraph, flowchart, or whatever - is necessary for almost all kids.
good. did you notice the quotations around
"two-column proofs?"
what you said above is what i had in mind.
> --
Art Mabbott
> m...@talmanl1.mscd.edu (me) wrote in message
> news:<200304181800...@talmanl1.mscd.edu>...
>> mman...@my-deja.com (maky m.) wrote:
>>
>>> an introductory course in geometry would naturally have to expose
>>> students to "two column proofs" and expect that they use such approach
>>> to prove simple geometric statements.
>>
>> Please explain why this should be so. I don't believe it for a moment.
>
> what do you suggest? geometer sketchpad and cute pictures alone?
Sketchpad would be a good start...to get to the conjecture. But then they
need to answer the questions...Why? How do you know? What convinces you
that it will always be true? And that is where some formalization comes
in...
>
>> --Lou Talman
>> Department of Mathematical and Computer Sciences
>> Metropolitan State College of Denver
>>
>> http://clem.mscd.edu/~talmanl
>
>
Art
maky m.
> On 4/21/03 2:38 PM, "maky m." <mman...@my-deja.com> wrote:
>
> > m...@talmanl1.mscd.edu (me) wrote in message
> > news:<200304181800...@talmanl1.mscd.edu>...
> >> mman...@my-deja.com (maky m.) wrote:
> >>
> >>> an introductory course in geometry would naturally have to expose
> >>> students to "two column proofs" and expect that they use such approach
> >>> to prove simple geometric statements.
> >>
> >> Please explain why this should be so. I don't believe it for a moment.
> >
> > what do you suggest? geometer sketchpad and cute pictures alone?
>
> Sketchpad would be a good start...to get to the conjecture. But then they
> need to answer the questions...Why? How do you know? What convinces you
> that it will always be true? And that is where some formalization comes
> in...
you have just admitted that
"an introductory course in geometry would naturally have to expose
students to "two column proofs" and expect that they use such approach
to prove simple geometric statements."
the methods might vary, but the expected outcome is the same.
thank you.