Understanding Calculus and Higher Mathematics without Geometry?
Christopher M. Vanderwall-Brown
I'm an engineering student who decided to take calculus last Winter at my current institution, and for all intensive purposes I thought I was prepared. I had taken all necessary prerequisites (making up for an appalling high school education), including Algebra I, II, Intermediate Algebra, College Algebra and Analytic Trigonometry (Elementary Functions I & II).
I received an A in all 5. (I had great instructors) I was in my Jr. Year. Upon switching institutions I took calculus and was thrust into a new world. A wonderful world. However, my instructor, being an older gentleman, saw fit to assume all the students in class were well versed in geometry.
I sadly did not have the opportunity in high school, and seeing most colleges, especially community colleges these days, do not provide coursework in the subject, I was not. My previous instructor at my previous institution was great about realizing the students you get, especially those in a community college environment, may not have all the necessary prerequisites. As such, you cannot just assume they know everything.
This was a problem for me.
So I have a question for all the mathematicians out there.
Can you adequately learn and comprehend higher mathematics without understanding the fundamentals of geometry? And, I'd like to make the added suggestion, we need a comprehensive system for people who were not able to study it in HS. Perhaps a comprehensive compilation of videos, texts, exercises. Something that can help out students like myself who have to scrounge the entire internet for much of their summer trying to find what little is out there.
I have only the option of studying old text books. I admit, perhaps I am a spoiled fool, but I tend to learn a lot more from a lecture, simply because my mind is more audio/visual than just visual/linguistic. I need math humor and analogies to keep me alert and cognizant.
Currently I am making up for my high school deficiencies, ironically after I thought I had already done so, but this is life and it often throws you a curve ball.
I guess this is half a plea, and half a question to mathematicians out there to realize, many of your undergraduate students who enter, especially those who come from the community college section, can be deficient in geometry. Why? Because our college system does not consider it an important venture.
All I'm asking is if I can understand ideas like cords, lines of tangent and sine effectively without understanding the fundamentals of geometry, and if not, what can I do to foster my understanding.
I live in the Seattle Metro area, and there is not a single institution teaching the subject besides a high school. (I don't think the district would like a college student crashing a math class...I would be interesting though.)
I'm studying a few books, and in general they are low level texts on geometry. One more advanced college text went missing with my copy of the Elements after I moved, I think it's in a box somewhere...
Besides reading that copy of the Elements, and what few texts have been recommended out there, do you all have any suggestions?
I have acquired some videos on geometry from the teaching company on HS geometry, but besides that, is there anything else out there? Lectures on geometry that really give the subject a full and intuitive look. Something of classical rigour that we lack today. I really want to walk back into my calculus class when I transfer to Berkeley and be able to take whatever geometric problem they throw at me. I can't have these cases of needing to understand the relationships of geometric objects I've never looked into.
Advanced studies on trigonometry. All the fun stuff I wasn't able to cover because of time restraints in my Elementary Functions II class. (We were on a Quarter System)
Please, if anyone has any suggestions I'd truly appreciate it. Sorry if this post is so long, but I've been struggling with this for over three terms.
-Chris
Don Watland
Here are a few of my thoughts regarding your questions, coming from a
retired engineer. I went through high school in Washington, then graduated
from the U of W in Seattle in 1968 with BS in physics and math. I might add
that I have privately tutored high school students in Arizona for the past
10 years, so for all intents and purposes, I have been an instructor. While
I was in HS, I took every math class available, which included one year of
Plane Geometry which focused an entire year on using only formal proofs
(definitions, theorems, hypostulates, etc.), followed by an second year of
Analytical Geometry, which overlapped much of the same material, proving
concepts by use of graphs and equations. This was topped off with a half
year of trigonometry. Somewhere in all of that was the notion of both
rectangular, cylindrical, and spherical coordinates, along with
transformations from one to another. The basic notion of vectors was also
covered. However, when I entered the U of W, I had to take a placement test
in math, which resulted in my having to take a quarter of elementary
algebra, and a quarter of trigonometry again, before I could start the
freshment Calculus series. In hind sight, I'm glad I had to, even though at
the time it seemed like it was all going to be redundent. However, the
intensity and depth of those two classes was invaluable. That was 40 years
ago, and much has changed in the type of instruction given in high school
today. At the potential risk of offending some of today's instructors that
might read this, I feel that the primary concern in HS is to move the
students through the system, literally leaving "no students behind." Modern
text books have evolved into graphic wonders, suitable for anyone's
livingroom coffee table. Their physical size and weight is overwhelming, as
are the tremendous glossy pages abounding with full-sized photographs of
bridges, buildings, and doo-dads. Mixed in with these inspirational images
and marvelous graphics are some less appealing blocks of actual text and
meaningful definitions. However, many of the math classes in our local high
schools never use a text book, no matter how pretty they are. The teachers
hand out one or two zeroxed pages of homework problems that include small
blocks of space on the page for the actual work to be done. They solve a few
similar problems in class, and whatever notes the students might take become
the only reference material the students use. To my old-fashioned mind, this
is not only absurd, it's criminal. I might add that I live in one of the
most affluent neighborhoods in metro Phoenix. None of this has said anything
about what to do to better you situation, but it might serve to explain why
so many students rolling out of high school are in total shock when the
reality of their preparation sets in , should they enroll in a University,
but not necessarily a commuity college curriculum.
I, personally, am a book freak. A have shelves of books of reference
material that I have purchased through eBay, old bookstores, and via a
website specializing on searching for used books: www.abebooks.com. Most of
these are older, to very old, books. As far as the math books are concerned,
they are filled with math....no colored pictures, but rather definitions,
theories, proofs, derivations, and many, many problems. Some of my favorite
books are dated in the late 1800's through the early 1900's. These contain
of wealy of knowledge, at a fundamental level, that does not appear in
modern texts. In your quest for basic geometry, find a good, used bookstore.
Downtown Seattle, or the University district should have several. Find
several geometry books published in the 1950-1970 era. If they have big
covers, and glossy pages with colored pictures, leave them behind. Find some
that are loaded with proofs dealing with lines, angles, triangles, and
circles. Get several. Don't worry about pencil marks or high-lighted
definitions. They are usually cheap, anywhere from $1-$5. Few people buy
them anymore. Reading seems to have gone out of fashion, especially of
non-fiction. These books are everywhere. Look on eBay. A majority of people
who have been through a university probably have a box or two of old text
books stacked in a back corner of their garage, or basement. If buying old
books doesn't appeal, take a trip to the public library downtown and you'll
find more old books than you can imagine, sitting on shelves waiting for the
likes of you to check them out for free (got to get a library card, another
invaluable privelege).
Then, read. Read several. Read a lot. Find a quiet place, take a pad of
paper and lots of pencils, and work problems...many problems. Learning
math is not a lot different than learning to play an instrument, or learning
a foreign language. If you do not practice, you do not learn (unless you
have a special abililty that most of us don't have). If you are lucky enough
to be attending a decent school, go to the math department and ask about the
availability of any teaching assistents, or volunteer grad students, or
study groups. I know the U of W has a study area set up where various tables
are designated to different topics. Studens can drop to study, and asks
questions to other peers. I doubt that anyone there asks for IDs, so if you
are not a student there, but live in the area, drop in and look for a seat.
If anyone asks, tell them an alumni told you to go!
Good Luck,
D. Watland
"Christopher M. Vanderwall-Brown" <christop...@hotmail.com> wrote in
message
news:11365212.1215594859...@nitrogen.mathforum.org...
triclino
EUCLID
THE THIRTEEN BOOKS OF THE ELEMENTS
Translated with introduction and commentary by
SIR THOMAS L.HEATH
THOSE BOOKS CONTAIN EVERYTHING THERE IS TO KNOW ABOUT GEOMETRY
Christopher M. Vanderwall-Brown
My biggest problem has been discussion. I've found a great number of books, many of which are excellent subjects on the topic, but none have answers to key questions asked. I want an explanation of thinking questions. I don't have all the answers and perhaps instead of a direct answer I need a hint or a new way of looking at it. Something that will enable my brain to see what is as yet unseen.
A previous poster (or one of our members who emailed me directly. Thank you in either case.) mentioned that I should look to finding a group of students and possibly teachers who have the necessary skills to answer my questions. For example, spending time at the UW and looking for a tutoring table, or study table for advanced math. I still need to build up the courage to try. What's the worst thing they can do, ask me to leave? Charge me with trespassing on public property? If it's just students I shouldn't have a problem.
In any regards, Heath made an excellent translation and I will study it. I was thinking at one point and perhaps still of attending St. Johns College as a way to shore up my deficiencies. Then again, I am not sure that would solve everything. It might give me the means to ask real questions though. Something I still struggle with.
Best Regards,
-Christopher M. Vanderwall-Brown
John B. Tant
Geometry, as (should be) taught in HS (HSG), is a course about deductive reasoning. It merely USES lines, circles, polygons etc as teaching tools. That said ...
Your typical Calculus course doesn't use a lot of the specific factoids from HSG. However, the ability to work a problem backwards (like I do when I'm trying to plan for a vacation or solve a puzzle) is invaluable to the student of any of the post-Trig math courses.
Trig, however, is one of those MUST have pre-requisites for Calc. If you don't know it cold going into Calc, you will by the time you finish it. You don't need it for "calculus" but you can't work the problems without knowing it.