How Prevalent is Students' Proving Theorems in Plane Geometry???
Daniel A. Asimov
school (class of '64).
So it came as quite a surprise when a friend recently informed me that it is no
longer automatic that a high school course in Plane Geometry have a strong
emphasis on students' proving theorems from axioms.
My question to those of you who are high school math teachers is,
How prevalent in Plane Geometry courses these days is the inclusion of
a strong emphasis on students' proving theorems?
I'd like, if possible, a ball-park figure: Is it closer to 5%, or 25%, or 50%,
or 75% or 90% or (you name it) of all high school Plane Geometry courses that
strongly emphasize students' proving theorems?
Thanks,
Dan Asimov
Andrei TOOM
> My question to those of you who are high school math teachers is,
>
> How prevalent in Plane Geometry courses these days is the inclusion of
> a strong emphasis on students' proving theorems?
>
> I'd like, if possible, a ball-park figure: Is it closer to 5%, or 25%, or 50%,
> or 75% or 90% or (you name it) of all high school Plane Geometry courses that
> strongly emphasize students' proving theorems?
I an not a high-school teacher. I am a college teacher. Even the best of my
students have no idea not only of proving theorems, but also of basic facts
in geometry. Andrei Toom
Christopher D. Lusena
> there is less emphasis on proofs now. Most kids didn't understand them
> back then and still don't now...unless you get them ready for it (it being
> deductive reasoning...an alien form of thought to most people and only
> used professionally by the very highest level research folks).
Deductive reasoning only by professionals at research level? How
about undergrads learning mathematics though problem courses,
generally though to be the best way to _learn_ math at least
compared to standard Lectures. (theorem-proof, theorem-proof-machinery,
or worst of all machinery-machinery-machinery)
OR Detectives, mystery novel writers, anyone who hast to solve
a problem that they don't have a method for.
Deductive reasoning is a very useful skill, and should be
introduced much earlyer then it is now, like in elementary school. IMHO
I personally was teach some then and it made the next 5 years of math
and sciences a joke. Everything followed from one thing to
another, in logical steps. (I an supposely pretty good at such
things so feel free to not generalize this.)
>A develop-
> mental approach to proofs makes it much more comprehensible. Proofs are
> still important...I personally teach proofs primarily in the late 2nd
> semester of a yr course. -archie benton North Buncombe High
> NC
>
>
--Chris Lusena Trent University
CLu...@TrentU.Ca Peterborough, Ontario, Canada
Just a pain boring .SIG. #include<standard disclaimer>
Michael Keyton
approaches the task as providing an explanation that is acceptable to the
reader (listener) then every problem in mathematics from the earliest
stage should constitute "proof". If one means some formal, rigorous
approach that is grounded in abstractions, then one should incorporate
the elements gradually. I do not think that every topic that is presented
should me mastered. Students should encounter the elements of "formal
proof" gradually over several years. Geometry provides one of the best
vehicles for accelerating the process, but if the process is made into
the focal element then geometry suffers.
In the history of US education, mathematical topics have been pushed
lower and lower, but students are expected to master the material
nevertheless. When my father took geometry, he was 17; for me, I was 16;
for most of my students they are 13 and 14. Do I expect them to have the
same degree of philosophical development as a 17 year old? I hope not.
What then does one do with geometry at this level? Most of the schools in
our area have incorporated more and more courses called "informal
geometry"; since I am at a "College Prep. school", we do not include this
level, but we have slightly changed the geometry course so as to include
more exploration, more review of algebra, a little less formality.
Working in teacher workshops with other schools, however, I am appalled at
what is called a 2-column proof. Rarely, does a "proof" extend more than 4
or 5 steps; the rationale is that "the students can not comprehend beyond
that length". If the emphasis is placed upon explaining and outlining then
filling in details, I find students even at the age of 13/14 can produce
reasonable explanations (proofs). One of the extended problems is that
rarely are the students asked to struggle and "not be rewarded for their
attempts, which are treated as failures". Textbooks provide solutions and
hints for any problem of any difficulty, which confirms in most students
minds that all problems are easy and solvable in a short length of time.
From geometry, I want a student to know what the distinction between a
conjecture and a theorem is, to recognize a proof, to have some idea of
the philosophy of why there are postulates and how one differs from a
theorem, some competence with concepts of converse, contrapositive, proofs
by contradiction, definitions (how to break one down, how to analyze one
as to acceptability, how to use one in a problem); with a theorem, to be
able to identify the hypothesis and the conclusion and to know the
sequence they occur so as to use in a subsequent problem. When I see a
student during the next three years, I expect him to have these tools as
well as those algebraic skills "mastered in Alg. 1 and 2 so as to work
with higher levels of mathematics. We continue to work on "proof", how to
construct one, how to recognize one, how to refine one, and to realize
that the process is not always easy, can be exhilirating, and should be
utilized to separate mathematics from other investigative processes.
Here is where the proof of my thesis becomes validated, those students
who have not gone through a fairly rigorous study of geometry typically
flounder at the calculus level. Not because they can't think, but rather
because they do not know how to think effectively mathematically.
Michael Keyton
St. Mark's School of Texas
Dallas
North Buncombe High Schl
Sherlock Holmes certainly did.
Chris D. Lusena
> sounds like a lot of folks have deductive and inductive confused.
> Sherlock Holmes certainly did.
>
Actually I did not deductive reasoning is useful in real life.
Though at the more informal level than was one would do in a symbolic
logic course.
And learning how to reasoning, is probablely the most important life skill
that one can learn (IHMO). Reasoning in mathematics is much easyer to learn
because often we either know the answer already or we can easily varify
that our reasoning is correct. Thus one can look at Deductive Reasoning as
at less an important tool for learning life skills, and therefore
should be taught in high school, if not earlier. (again IHMO)
Thirdly and least important, high school math should be seen as
prepority to Post Secondary education, where if one is to
encounter at Real (read pure if you like) math courses one must
be able to understand and construct proofs in order to get very far.
You are doing students a disservice by sweeping proofs and thinking under the
carpet.
My apologies for ranting but to may of classmates can prove their
way out of a paper bag, and being a Third/fourth year math major I find this
a little frustrating.
Carl Lau
The reality of teaching is that the course is driven primarily by the
ability levely of the students. Unfortunately that means that in a large
amount of LAUSD senior highs, proofs are NOT emphasized. Textbooks for
the most parts still contain a significant amount of proofs [Serra, being
the exception], however, the classroom teacher must do a large amount of cut
and paste in order to make the course meaningful to the students. This
means a lot more informal and computational geometry.... YES, I would
love to teach a rigourous axiomatic type Geometry course like I did 15
years ago but if I attempted to do this today, 90% of the students would
get 0% out of the course. Many students simply are not capable of
comprehending proofs let alone appreciate the beauty of them. Even the
few that do comprehend them, have little appreciation or enjoyment of
them...To the typical student, proofs are incomprehensible and irrelevent...
As a teacher it would be nice if they could do them, but there are a lot
more important skills [finding area, pythagorean thm, computing
angles....]that take precedence.
Anyway I'm speaking from my own experience as a 20 yr math teacher at
Hollywood H.S. and of course there is a tremendous variance in abilities
and target populations withing the district; however I would say that at
least half the schools in the district are in the same boat...
Please don't ask why the students come into senior high schools with
fifth grade math ability and questionable motivation...that would take a
whole new newsnet to discuss
Anyway I hope this sort of answers your query.
Howard Lau
math teacher
Hollywood H.S.
[visiting on LAFN on may brother's acct]
--
Carl S. Lau <ag...@lafn.org>