Note to Teachers | Explaining Logarithms

A math book by Dan Umbarger on solving logarithms
Explaining Math Logarithms
A Progression Of Ideas Illuminating an Important Mathematical Concept

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Note to Teachers

This text is not written for you. With the exception of parts of chapters 5, 6, and 7 and Appendix A, I assume that you already understand all the ideas presented. This is a book written for students who do not understand logarithms even if they can apply the rules and get correct answers. However, it would greatly gratify me if a teacher were to tell me that he or she enjoyed my organization and presentation.

I am a high school math teacher, not a mathematician. As such, I live and work in a world where sequence and progression of concepts leading to key ideas, along with pacing, “anticipatory sets,” evolution and organization of ideas, reinforcement, examples and counterexamples, patterns, visuals, repeated threading and spiraling of concepts, and, especially, repetition, repetition, and repetition are all more important than rigor. It has always seemed ironic that authors and teachers, so knowledgeable about mathematical sequences, could be so insensitive and clumsy about the sequencing of curriculum … how they could be so knowledgeable about continuity of functions but so discontinuous in their writing. There are plenty of materials available on teaching logarithms that are mathematically rigorous. I believe that “rigor before readiness” is counter-productive for all but the most gifted students.

As such, I present many, many examples to help the student to see patterns and only then do I present the abstraction which will allow for generalization to all cases. Induction is a powerful teaching tool. Because of economy imposed by the publisher or perhaps because the material is so “obvious” to the authors most textbooks present the abstraction (generalization) first with little attempt to develop the rationale behind it or to connect the material to previous material such as the Algebra I Laws of Exponents or the history of logarithms. Those texts then proceed hurriedly to applying the abstraction to specific situations. I believe that the best way to introduce a new idea is to somehow relate it to previous ideas the student has been using for some time. Using this approach, new concepts are an extension of previous ideas … a logical progression. Logarithms are a way to apply many of the laws of exponents taught in Algebra I. It is important that the students understand that!! I also believe in introducing an idea in one chapter and revisiting that idea repeatedly in different ways throughout the book.

The materials presented here are usually spread over two years of math instruction: precalculus and calculus. Doing so, however, separates ideas and examples that are helpful in the synthesis that leads to a deeper understanding of logarithms. For example, most high school text books seem to shy away from a meaningful discussion of why scientists and other professionals prefer to work with base e, the natural log, rather than the more intuitive common base, base 10. They do so because the pre-calculus student has not yet been exposed to the ideas that are necessary to justify the use of base e. If the goal is “rigor” then indeed many ideas associated with e must be postponed until calculus. But if your goal is to create familiarity with logarithms and appreciation of the number e, I do not believe that all that rigor is required. I have tried to bring all those ideas down to the pre-calculus level. I hope that I have done so.

My approach, however, has been done at the expense of rigor. If I get consigned to one of the levels of Dante’s Inferno because of my transgression it will be worth it if I am able to help young students past what, for me, was an unnecessarily difficult multiyear journey. When I did make an attempt at “rigor,” I chose the formal two column proof over the abbreviated paragraph proof.

I see three different audiences for this text: 1.) students who have never worked with logarithms before, 2.) those students in calculus or science who did not manage to master logarithms during their algebra/pre-calculus instruction, and 3.) summer reading for students preparing for calculus. The former students will need to receive instruction, but the second and third group of students, if sufficiently motivated, should be able to read these materials on their own with little or no help. There are questions at the end of each chapter to use to evaluate student understanding. Heavy emphasis is placed upon practicing estimation skills!!!


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Copyright © 2006-2013 by Dan Umbarger, Math Teacher Website: Last updated: 07/04/13.
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