Handheld technology & mathematics: Towards the intelligent partnership.
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| 3 | 4 | 6 | ||
| 1 | 4 | 0 | 4 | |
| 9 | 3 | 6 | ||
| 7 | 0 | 2 | ||
| 8 | 0 | 9 | 6 | 4 |
Pencil and paper, as a recording device, and the long multiplication algorithm, as a device to reduce long multiplication to a sequence of single digit multiplications and additions, are both technologies which, in the pre-calculator era, were required by most of us to multiply multi-digit numbers accurately and reliably. If the numbers to be multiplied involved decimals, for example, 2.34 0.0346 the long multiplication algorithm became much more difficult to perform and it was usual to resort to another technology, four figure logarithm tables, to help with the task. Tables of logarithms enabled complex products to be transformed into less complex sums which could be systematically carried out with the aid of pencil and paper as shown in figure 1.
| Nž | Log |
| 2.34 | 0.3692 |
| 0.0346 |  |
| 0.08096 |  |
Up until the 1970's, this was the only computational technology routinely available to students and its key place in certain areas of the mathematics curriculum is clearly illustrated by the worked example of an application of the sine rule based on a solution given in a 1960's mathematics textbook shown in figure 2 (Rose, 1964).
ABC completely when c = 1916 ft., b = 1748 ft. and C = 59
. [This triangle is drawn to scale in Fig. 137
 |
| To find B - |  |
| and hence - |  |
Taking logs throughout -
 |  |
 | |
 | |
 |  |
| Then - A |  |
 |
Consequently, in that era, a considerable amount of time and energy was spent in the earlier years of secondary school level mathematics courses teaching students to become skilled at carrying out computations with logarithms. Unfortunately, because these skills were not generally used in practice until much later in schooling, and because of the time and energy needed to teach students these skills, in the minds of many teachers they came to be viewed as a significant part of the mathematics of the era, rather than as a skill made necessary because mathematics was essentially a pencil and paper based activity.
Once the electronic calculator became common place in the classroom, the need for tables of logarithm for computations became unnecessary. Yet, at a workshop conducted for teachers on the use of the first electronic scientific calculators in the early seventies (Barling, 1995), one of the reasons given to teachers for introducing the calculator into their classroom was that it would obviate the need for students to have logarithm tables. It was stated that the calculator could be used to generate the logarithms, do the additions and then take the antilogarithms to obtain the required answer!
Why do such things happen? In part, it is due to the general lack of recognition that mathematics, like all human intellectual activity, is always shaped by the available technology, but that, with time, the technologies "become so deeply a part of our consciousness that we do not notice them" (Pea, 1993, p. 53). As a result, the technology effectively becomes "invisible", while the activities it generates can come to be seen as mathematical activities in their own right, for example, carrying out calculations using logarithms. Hence, when a new technology such as the electronic calculator is introduced, it is common for it to be promoted as a means of "enhancing" the teaching of such activities, even though the technology itself has been designed to obviate the need for such calculations. The irony of using a technology such as a calculator to help do computations with logarithms should not be lost on anybody. Yet today, with graphics calculators having sophisticated numeric integration capabilities, we face a similar situation and similar responses.
For example, let us say that we wish to calculate the length of the arc of the curve 
between 
and 
. The solution reproduced in figure 3 is based on a solution given in a typical calculus text (Grossman, 1977).
SOLUTION. Here
so that
Let 
so that
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In looking at this solution we see that it bears an uncanny similarity to the 1960's textbook solution to the sine rule problem. First some theoretical knowledge is used to set up the solution to the problem with pencil and paper. In the case of the sine rule application this results in a complex arithmetic expression which is then evaluated with the aid of log tables. In the arc length problem the solution is set up in the form of a definite integral which is then evaluated using an appropriate substitution and some algebraic manipulation to enable the original integral to be transformed into standard form. The results of the manipulation are recorded with pencil and paper and presumably a table of standard integrals is used in the end to help evaluate the resulting integrals. From figure 3 we can see that the arc length is
 |
For most of us who learned calculus as a pencil and paper based activity, it would be hard to accept that the steps involved in evaluating the definite integral in the arc length problem are not worthwhile mathematics, yet, if the true purpose of the activity was to evaluate the arc length, then the process as a whole may have no more intellectual value to the majority of students than the mastering of the skills needed to carry out complex arithmetic computations with tables of logarithms. Just as the electronic calculator was designed to avoid the need for human beings to carry out complex arithmetic computations by hand, a graphics calculator with numerical integration capabilities is designed to avoid the need for human beings to, amongst other things, evaluate complex definite integrations. This is challenging to those of us for whom the only technology supporting our calculus activities was pencil and paper and possibly tables of standard formulae. We had to master integration methods to solve more advanced problems, just as students in the past had to master computations with logarithms to solve more advanced mathematical problems. Thus we see that the available technology is a prime determinant of what mathematics we do in the classroom and how we do it, both now and in the past. So what is different now?
There is a PDF file available for this paper.
Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom.
Mail comments to Pedro Gómez: pgomez@uniandes.edu.co
between 
and 
, based on a solution given in a typical current day calculus text (Grossman, 1977)