Thinking Logarithm

ARTICLE:  Sharing teaching ideas -A simple way to teach Logarithms .

... Richard Hammack and David Lyons.  University of North Carolina, Chapel Hill, NC  27599-3250.


SOURCE:  Mathematics Teacher, May 1995, Vol. 88, No. 5, 374-375.



A review by Claude H. A. Simpson, Nova Southeastern University.


According to the Curriculum and Evaluation Standards for school math.

- In grades 9-12, the mathematics curriculum should include the continued study of functions so that all students can: model real-world phenomena with a variety of functions, ... and so that, in addition, college intending students can -understand operations on, and the general properties and behavior of, classes of functions (NCTM 1995).


The above quote has strong implications for the topic LOGARITHMS.  Logarithm of course is a very important component of algebra and calculus courses and there is always a search for appropriate teaching methods which would bring about an improved students' performance and better learning of the so called "tedious topic".

The  article  presents  a  technique  for  teaching  Logarithms  using  function
notation. It deals predominantly with the concept formation of Logarithms.  The core
theory is, its implication in mathematics education: mathematics instruction for secondary education and the use of a technique to improve students' performance.

The authors stated that they had tremendous success using the "a-box technique" to teach Logarithms. They claimed that many students had difficulties mastering the concept of Logarithms, more so than with other functions. A quick survey  of  the  method  is  as  follows:    Hammack  and  Lyons  showed  that  the conceptual way to understand the function y = log@x (meaning log base 'a' of 'x') is to view it as the inverse of  y = a*x where 'x' is the power to which 'a' is raised. The "a-box" as illustrated by the authors is simply a change of notation, that is, replacing log@ by a[], read "a-box".  The authors noted that the method used is to begin with examples rather than general definitions. Hence, the following would be written on the chalkboard: 2[] (8) =  ?  Students would repeat, "two-box of 8 equals blank".  The teacher would ask rhetorically, "What number goes in the box so that 2 raised to that power is 8?"
Teacher:  Since 2*3 = 8, ( meaning 2 raised to the power of 3 =8) we fill in the blank with a 3.  ... 2[] (8) = 3.

The authors emphasized that the teacher would begin with examples to establish a pattern.  Students would then be asked to supply answers to examples,

like:

2[] (16) = 4
2[] (2) = 1

3[] (27) = 3
3[] (9) = 2


Harder examples would later be introduced, like:


2[] (1/2) = ....-1 ?

Students would be given time to think through the answers. The teacher would then carefully explain how to use the appropriate properties of exponents.



2[] (-/2) = 1/2 'meaning two-box of the root of two equals half'

3[] (1/9) = -2  'meaning three-box of one-ninth equals minus two'



8[] (2) = 1/3  'meaning eight-box of two equals one-third'
2[] (1/16) = -4  'meaning two-box of one-sixteenth equals minus four'


Students would be asked to reason out examples like: 2[] (1) = 0; 3[] (1) = 0 before giving them the answers. Students would  observe that no matter what 'a' is, a[] (1) = 0, since a*0= 1.
Students would then be presented  with a special case:  2[] (-2) =        ?
Students attention would be drawn to the fact that since 2 raised to any exponent is positive and that by graphing y = 2*x . . . we see that 2[] (-2) cannot exist; then, any
a[] (-b) cannot exist.  Graphical representations of the functions y = 2*x and y = 2[]x proved that they are inverse relations. Thus, another way of writing 2[](x) is log to the base 2 of (x) and  in general, a[] = log@

The authors rightly stated that after some practice, "a-box" notation can be dropped  entirely.
The teaching method described by Hammack and Lyons is purely EXPOSITORY, utilizing examples and notations to account for the much "called for" hands-on activities.  The authors' attempts would all be in vain if the necessary prerequisites for learning Logarithms were not in place. As cited by Markovits and Sowder (1994), students should be quite competent to make sense of numerical situations. Thus enabling them to have an intuitive "feel" of numbers, using numbers flexibly and judging reasonableness of results. This implies that any attempt to teach Logarithms, students should show mastery in numeration such as: Prime Factors, Power Sequence, Transposition, Laws of Indices, Basic Functions and Graphs. To establish a strong conceptual foundation before the formal notation and language of Logarithm are presented, students in grades 9-12 should continue the informal investigation of functions and indices that they started in grades 5-8. Later concepts such as domain and range can be formalized.          ·

One of the strengths of the article is that it stresses recognition and application of the identity property of symbols/notations that can be used to help students overcome difficulties in learning Logarithm. I applaud the attempts by the authors to place students difficulties in contexts because I know how difficult it is to come up with good solutions to address the learning difficulties students have,especially with such an abstract topic as LOGARITHMS. Rahn and Berndes (1994) claimed that competency in Logarithms can be greatly improved by the teacher discussing activities to help students make visual generalization about power and exponential functions.

Among the strengths of this article, the "a-box" technique can be used to establish the basic property of logs.  For example a[](a*x) = x, similarly log to base of 10 of 10*x = x.


The authors however, rightly claimed that the box idea can be used for any inverse function.  For example, the concept of arcsin can be explained by using sin [].
Although the authors made a valiant attempt in utilizing a rather unique technique, there are still a few weaknesses in that method:  Wherefore y = log@x to be viewed as y = a*x could prove quite misleading to standard logarithmic transposition.  Note, if y = log@x then a*Y = x.
To substitute numbers for a, x and y
y = log@x  ... @ = 10, x = 100, y = 2  ie  2 = log base10 of 100          (true)
To write in index form          a*y= x

ie          10*2 = 100          (true)

From the "a-box" technique

y = a*x          ...if  a = 10, x = 100, y = 2
then          2 is not equal to 10*100

A second identifiable weakness of the article is the lack of appropriate statistical data to validate the authors' claim re-percentage of students in high school and college having difficulties mastering the concepts of logarithms.  Enrichment activities could be given.  As argued by Kullman (1992), indices logarithms may be examined through various examples, utilizing number tables and graphs to discover the functional relationship between two entities. Mayes (1994) claimed that software tools can be used  in the exploration  and  visualization  of  a relationship  between logarithmic and exponential functions.

In summary, the method may not be thorough enough to be accepted as the best strategy/technique but it certainly offered us as teachers with a technique that we can use in such a vital area of mathematics.


REFERENCES

Hammack, Richard., & Lyons, David. (1995, May). A simple way to teach Logarithms. Mathematics Teacher, Vol. 88, No. 5, 374-375.


Kullman, David E. (1992, March).  Pattern of Postage-Stamp Production.
Mathematics Teacher, Vol. 85, No. 3, 188-189.


Markovits, Zvia., & Sowder, Judith. (1994, January). Developing Number Sense: An Intervention Study in Grade 7. Journal For Research in Mathematics Education, Vol. 25, pt 1.


Mayes, Robert L. (1994, November). Discovering Relationships: Logarithmic and Exponential Function. School Science and Mathematics, Teaching Guide 052 Vol. 94, No. 7, 367-370.


National Council of Teachers of Mathematics (1995). Curriculum and Evaluation  Standards For School Mathematics.  Virginia: © 1989 by N C T M, Inc.


Rahn, James R. & Berndes, Barry A. (1994, March). Using Logarithms to explore Power and Exponential Functions. Mathematics Teacher, Vol. 87, No. 3, 161- 170.