The impetus for this group of activity was born out of the frustration of teaching and reteaching logarithms to college calculus students who had experienced the type of instruction that I was providing at least once before in high school and once in either precalculus or college algebra. It seems that they did not recall many of the logarithm concepts that their instructors had tried so hard to teach them. After successive passes over logarithms, the meaning that some mathematics teachers hope to instill in their students is still missing. So how might we prepare students for our initial foray into the formal world of logarithms? Students seem to have very little experience with the ideas that might help prepare the way for formalized instruction in logarithms. The mathematical ideas that I believe build the foundation for logarithmic understanding are
1. multiplication,
2. integer exponents,
3. limiting process,
4. irrational numbers,
5. geometric and arithmetic sequences, and
6. the correspondence between geometric and arithmetic sequences.
The basis for this belief comes from at least three sources in the literature. First, Pirie and Kieren's work on understanding as a process (Pirie & Kieren, 1994). Which has lead me to believe that understanding develops through a process of mental/physical action and personal reflection on that action. This belief is portrayed in the activities that follow in which students are asked to act and then reflect on their actions. It is hoped that through this process students will grow from the primitive knowing stage through the image making and image having stages to the property noticing stage. According to Pirie and Kieren, students whose understanding is at the property noticing stage are ready for formalized school instruction.
The activities I selected to prepare the students for formalized logarithmic
instruction were developed based on my understanding of the genesis of logarithms
in a historical sense and an idea about what primitive knowing might be
necessary for such a genesis. In particular, I draw on accounts of Napier's
development of the idea of logarithms (Katz, 1995; Cajori, 1913; Ayoub,
1993; Lord Moulton, 1915) and Confrey's account of exponential understanding
(Confrey, 1991). From the historical development of logarithms I came to
believe that one possible route to logarithmic understanding might be modified
from Napier's path. That is by considering an arithmetic sequence { ...
,-2, -1, 0, 1, 2, ...} and a geometric sequence 
the student might
be able to determine a map from the geometric sequence to the arithmetic
sequence that could convert multiplication to addition. The sticking point
for many students, as it was for Napier himself ,would be the generalization
of the map to all real numbers. Being able to generalize the rule to the
real numbers necessitates at least a rudimentary understanding of the real
numbers and the limiting process that is part of the understanding of the
irrational numbers. I must say that the necessity of an understanding of
irrational and real numbers is a bit of speculation on my part. I feel that
this understanding is necessary, but I have no proof that the absence of
it impedes understanding. Based on Confrey's work I have conjectured that
student's have not developed an understanding of integer exponents and so
I have included a few activities that I hope will motivate that understanding.
I hope that these activities will provide both the teacher and the student will insight into their own understandings of multiplication, exponents, and logarithms.
Works Cited
Ayoub, R. (1993). What is a napierian logarithm?American Mathematical Monthly (April): pp. 351-364.
Cajori, F. (1913). History of the exponential and logarithmic concepts. American Mathematical Monthly 20 (1): pp. 5-14.
Confrey, J. (1991). The concept of exponential functions: a student's perspective. In L. P. Steffe (Ed.), Epistemological foundations of mathematicsl experience. pp. 125 - 159. New York: Springer-Verlag.
Katz, V. (1995). Napier's logarithms adapted for today's classroom. In F. Swetz, J. Fauvel, O. Bekken, B. Johansson, & V. Katz (Eds.), Learn from the maters. pp.49-57. Washington, DC: The Mathematical Association of America.
Moulton, Lord. (1915). The invention of logarithms, its genesis and growth. In C. Knott (Ed.), Napier tercentenaary memorial volume. pp. 1-32. London: longmans, Green and Company.
Pirie, S. & Kieren, T.E. (1994). Growth in mathematical understanding: how can we characterise it and how can we represent it? Educational Studies in Mathematics 26: pp.165-190.