Math Teacher explains "Why I Wrote This Math Book on Logarithms"
Why I Wrote This Math Book
In the fall of 2005 a former computer science student of
mine, at that time enrolled in a calculus class, asked me to tutor him in
logarithms. As I had taught him AP Computer
Science for two years and saw him pass both of his AP Computer Science tests
I knew that he was extremely bright. He could “sign” for the deaf (his
hearing is fine), play jazz piano, and had taught himself to program
firmware while doing a high school internship …(a fact that did not go
unnoticed by his company…they hired him to do paid work
during the summer.) This student had a highly qualified and highly
experienced Precalculus teacher. I was puzzled over his “knowledge gap”
regarding logarithms.
The following inequality generally holds true:
Bright student + Highly qualified and experienced teacher
≠ Missing skills.
It was a paradox…sort of like that barber who shaved every man in the
village who did not shave himself. I just could not figure it out. I
realized, in helping this student, that I could assist him in working all
his exercises in the two 10 minute sessions I spent with him but I could not
get at his fundamental problem: lack of internalization of fundamental
concepts relating to logarithms. My tutoring degenerated into “Here is a log
formula…use it.” I also realized that all my knowledge about logarithms was
disorganized and not suited to efficient communication. I wanted to share
with him ideas, concepts, and problem solving not just “plug and chug”
formulas. I wanted him to know “why” as well as “how.” I wanted him to see
the connection between some of the log formulas and the Algebra 1 formulas
he had been using for years.
I started thinking about how I had been introduced to logarithms back in the
1960’s and about the impact that calculators have had on the Precalculus
curriculum. (Thank you calculator engineers at Hewlett-Packard and TI!!!)
However, in freeing us from the tyranny and misery associated with log
tables those calculators have also made the curriculum much more compact and
abstract in areas that used to be pretty extensive and much more concrete.
Logarithms, for 350+ years, were used to “do” (actually approximate)
extensive arithmetic expressions involving multiplication, division, and
taking rational roots. I do not believe that students need to learn be
skillful with log tables before using calculators involving logarithms… but
I do believe that many of the skills and evolution of ideas that were taught
in the 60’s can still be helpful in the internalization and mastery and long
term retention of logarithmic concepts.
I found out in searching (unsuccessfully) for a publisher that commercial
publishers constrain how many pages are allowed to explain an idea. For the
most part publishers just told me that my book did not “fit into their
publishing plans”. One publisher did, however, share with me that the large
public schools had purchasing guidelines that precluded purchasing books
that were not an all inclusive curriculum…such as a Precalculus book. Any
such book would have innate limits on the amount of explanatory material for
any given topic. That is exactly what that same editor told me.
I also noticed in my research that my student’s knowledge gap on logarithms
was not unique. I found several references on the web that let me know that
he was not alone.
Check out the following website:
http://mathforum.org/library/drmath/view/55522.html
(website viable April, 2006)
The Math Forum, “Ask Dr. Math.”
“I have a bunch of rules for logs, properties and suchlike, but I find them
hard to remember without a proof. My Precalculus book has no proof of why
logs work or even what they are, nor does my calculus book. I understand
what logs are…but I don’t understand why they are what they are.”
This plea for help is from a calculus student but clearly he or she does not
even know enough about logarithms to articulate a clear question about what
he or she wishes to know. This student presumably got through his or her
unit tests in Precalculus but really did not understand logarithms.
Something is wrong here!
So 600+ hours and several thousand dollars later I have developed curricular
materials on logarithms that I would like to share with math students and
math educators alike. These materials show the evolution of logarithmic
ideas over 350 years. I do not propose to go back in time and require
students to learn about log tables!!!! I do believe that a quick review of
mathematics as it was practiced for hundreds of years would be helpful for
many students in understanding logarithms as they are still used today. As
teachers we should be emphasizing comprehension not “answer getting.” If the
students truly understand the material then the correct answers will be
there when the time comes. If you are an administrator who requires his math
teachers to teach to a test it might be good for you to re-read this last
sentence.
Although the material is copyrighted you have the permission of the author
to download, save, and print one copy for educational, nonprofit purposes.
If you think it worthy please consider a $5-6 donation (students, ages 1-18)
or $10-12 (adults) to a good cause…namely placating a wife who saw my
efforts and expenses as taking away time and money from our family. Single
and multiple bound copies are available by accessing the order form located
on this website.
I see three potential audiences for this material: 1.) students who have
never studied logarithms, 2.) students who have studied logarithms but who
did not master the concepts or have forgotten key ideas, or 3.) summer
school reading for students taking calculus in the fall. ( There are a few
sections loosely involving the idea of a limit. Those units were necessary
because of several extensive discussions of the constant “e” which grew out
of discussion of ln x vs. log x.) While the former group would probably
benefit from having an instructor the later two groups should be able to
read the material on their own.
It is my hope and would greatly please me if many, many students and math
teachers alike benefit from my labors. Drop me an email at
dan@mathlogarithms.com and
let me know of your impressions.
Dan Umbarger, 2006
