Calculators: History and Usage
Adam Carberry
Abstract
Calculating devices have been around since the time of the Babylonians, but it
wasn’t until the invention of the first hand-held electronic calculator in the 1970’s that
they became highly influential within schools. Over the past 35 years, calculators have
evolved from simple four-function capabilities to the now ever popular graphing
calculator, which not only is capable of graphing functions but also possesses the ability
of programming. With the advent of new and exciting calculators, mathematics as a
subject has become ever changing. The greatest advantage of involving calculators into
classes is the ability to avoid countless paper and pencil calculations consequentially
allowing for more time to be spent on mathematical concepts. Calculators also allow for
some mathematics which is not possible with paper and pencil to be accomplished. Of
course with all new technologies there are critics. The common criticism of the
calculator is that its involvement means students are learning less and becoming lazy. In
this paper I discuss how usage of calculators in the classroom has changed from the past
to the present. The paper also takes a look into the future of calculators in schools.
Introduction
Mathematics exists all around us. Whether someone is simply counting how
much change they have in their pocket or determining the flight pattern of an airplane, it
is a necessity to understand and be capable of performing at the minimum simple math
related tasks. For most, these subjects can cause uneasiness and result in avoidance. Over
time, the subject of mathematics has been labeled as “difficult”, “boring” and “tedious”.
To combat these stereotypes, tools have been designed to assist individuals in their
calculations. One such device and the most common today, is the calculator. The idea of
the calculator is to help solve many problems, as well as making problems easier and
faster to solve. The development and integration of the calculator has changed the way
that teachers teach and the way that students learn (Waits & Demana, 2000). Such a
technology has changed mathematics by making some aspects less important (such as
solving by hand), some mathematics more important (such as non-linear mathematics),
and some new mathematics possible (such as fractal geometry which would be near
impossible without the aid of technological advances) (Pollak, 1986). As a result,
repetition can be reduced or eliminated completely combating the drudgery of applying
arithmetic and algebraic procedures. The overlying question however is to what extent
calculators should be used in the classroom?
In this paper I will focus on giving a brief history of the various calculating
devices, which led to the ultimate design of the calculator, how these devices have
affected classrooms past, present and future, and the concerns/myths that parents and
educators have about calculator usage in the classroom throughout.
History
The notion behind calculating devices was to create a supplementary tool that
someone can use to perform simple to complex mathematical problems. These devices
can be traced back all the way to the days of the Babylonians circa 300 B.C. A timeline
of these devices starts with the Babylonian Abacus used to simply perform addition and
subtraction. It wasn’t until the early 1600’s, with the invention of Napier ’s bones and a
number of Slide Rule variations that multiplication and division were incorporated.
Following these simple calculating devices was the invention of the first mechanical
calculator known as the Calculating Clock in 1623. This invention integrated gears and
was capable of performing addition, subtraction, multiplication and division. Between
this time and the mid-1950s, a number of mechanical calculators were produced. They
include the Pascaline, the Stepped Reckoner, the Arithometer, the Difference Engine, the
Odhner calculator, the Comptometer, the Millionaire calculator, the Dalton, the Curta
(King, 1997). Improvements issued by these numerous mechanical calculators involved
key pads, printing capabilities, advanced mathematical ability and size reduction. In
1961 the first ever electronic calculator was produced. This model was a desktop version
which was succeeded by the first hand-held electronic calculator in 1970 by a number of
companies. Since that time calculators have evolved from simple four-operation
calculators, to scientific calculators and on to graphing calculators. Technologies
integrated into such devices within the past 35 years have been battery power, solar
power, liquid crystal device (LCD) screens and programmability (King, 1997). These
devices are far from being a tapped out area for improvement. Advancement is occurring
everyday.
As with most technologies, this new invention was initially designed primarily for
use in businesses. Businesses are more subject to change and welcomed the calculator
with open arms. The device was seen as a tool that would ultimately result in less errors
and higher employee satisfaction. High levels of success led to the introduction of
calculators to the mathematics classrooms around the mid-1970s (Chase, 1980). Today
you would be hard fast to find a classroom that does not possess at least one calculator.
Calculators in the Classroom
Past
When calculators were first being introduced into classrooms, there were mixed
reviews as to whether or not these tools were conducive to the learning environment. The
traditional view has always been that students should be able to do all of their
mathematics without the use of a calculator. Their designed purpose was not to replace
pencil and paper calculations, but to support it. Although the design was clear, these
devices questioned the traditional view due to the possibility that the calculator could be
used completely instead of pencil and paper. This notion bred a number of concerns
including calculators becoming a crutch producing lazy students, a loss of stimulation
and challenge for students towards mathematics, basic workplace mathematical skills not
being learned, and a high dependence on the calculator. Those who were concerned saw
the calculator as simply a tool that provided the answer with no thought process what so
ever. Although these tools would eliminate the need for high pencil and paper repetition
(one of the main sources of distaste towards mathematics for students), the only skill
required to operate the calculator would be the ability to punch the correct keys. Those
with this mindset decided early on that calculators should be kept out of the schools.
On the other hand, many saw this tool as an excellent supplement for a difficult
subject. If used properly, calculators allow students the ability to check their answer
quickly and avoid tedious pencil and paper calculations. With such a device, more time
could be spent on the method of problem solving and less on computations (Dick, 1992;
McClendon, 1992; Runde,1997; Siskind, 1995; Vazquez, 2003; Whiles, 2002;
Wilkins,1995). The idea is that if the true nature of mathematics is understood, then the
use of technology in the learning of mathematics will be seen as natural enhancements
and extensions (Waits & Demana, 2000). This is in no way stating that calculators
should completely replace pencil and paper computations. The calculator should simply
enhance and expand their learning. Those who took this vantage point decided to
integrate calculators into the classroom keeping in mind the necessity for students to
understand the mathematics behind the calculator.
Integration of such a tool does not come without some difficulty. Change within
a classroom is an extremely difficult proposition no matter what the technology. Like
many other ideas, when the calculator was first introduced, if the teacher did not have
experience managing the technology, chances were that the calculators were not used. In
a number of schools where calculators were first introduced, teachers were not given
proper training resulting in calculators either collecting dust in a closet or being stolen.
Calculators were also very expensive 35 years ago, not allowing every individual student
to own one. Similar to the computer, the inability to use these devices outside the
classroom made it difficult for students to learn its uses and feel comfortable. It wasn’t
until the integration of in-service teacher training along with a high abundance of
students financially being able to own calculators that this technology really began to
take flight.
Present
The use of today’s calculators in mathematics can be classified under seven
categories: 1) laboratory, 2) tool, 3) teaching aid, 4) curriculum influence, 5) cheating
device, 6) nuisance and 7) status symbol (Kissane, 1995).
Calculators as a laboratory refers to the opportunities they afford towards
exploration in mathematical ideas. Similar to scientists in a laboratory, anyone doing
mathematics on a calculator is learning and exploring a number of powerful ideas related
to mathematics. The variety of calculators available today allow for a broad range of
ideas to be explored.
As a tool, the calculator is compared to the instruments one would use for any
vocation. Like a hammer to a carpenter, a calculator supplies to anyone performing
mathematics a tool to perform a particular task. This same broad category involves
knowing when to use the calculator and when not to.
Calculators can also be used as a teaching aid. Often calculators are simply
thought of as devices to help students learn as opposed to devices to help teachers to
teach. The calculator supplies a platform for both teaching and learning to be performed
bearing in mind that the calculator is a supplement only.
As stated previously, calculators can influence the mathematics curriculum.
Because calculators afford the ability to perform tedious tasks quickly as well as some
impossible pencil and paper tasks, the calculator can dictate what teachers are capable of
teaching. Based on the involvement of calculators in the classroom over the past 35
years, curriculum has changed allowing for not only less time to be spent on
computations but also new and interesting topics to be explored.
Every category of calculators is not always a positive. With new technologies
comes new ways of using them as a cheating device. An example would include using
the programmable graphing calculators to insert common equations. It is one thing to use
this technology properly as it was designed, but misuse can lead to more harm than good.
Calculators to some teachers and students may come across as a nuisance as well.
To some students as well as teachers, the addition of having to know and operate a
calculator adds an unwanted extra burden to what is to be learned in an already crowded
curriculum. Although in time it may become an advantage, initially it may not appear so.
Finally, calculators can be seen as a status symbol where individuals will always
want to have the latest and greatest. It is inevitable that as soon as someone becomes
comfortable with a new technology, something more complex and more involved will be
introduced. In order to take full advantage of a new technology involved with a
calculator, one must keep on top of the new evolutions.
In general today, the calculator with all its uses and classifications has been highly
integrated into every mathematics class from kindergarten through to the collegiate level.
It is evident from a yearly survey conducted by the National Assessment of Educational
Progress (NAEP) that calculator use has been on the rise. Statistics have shown that over
a 6 year span between the years of 1986 and 1992, calculator availability and usage has
skyrocketed from 21 percent to 81 percent. Today that percentage is inching towards 100
percent throughout all grade levels (Dossey & Mullis, 1997). Today’s calculators are
owned by almost everyone due to their ease of use, their size and portability, and their
relatively inexpensive price tag. Even the most simplistic calculator is capable of
performing addition, subtraction, multiplication, division, square roots, and percentages
of both positive and negative numbers. Use within the classroom depends on the level of
schooling and the individual teacher. Within the elementary levels, students tend to use
four-function calculators which assist them in learning how to perform addition,
subtraction, multiplication and division. Throughout middle school, high school and
college, graphing calculators have become common place to assist students in
understanding several fields of mathematics such as graphs, numbers and algebra. It has
been indicated that graphing calculators can help in developing deeper conceptual
understanding due to the graphing calculator’s innate ability to draw pictures leading to a
better understanding about graphs and functions (Hollar, 1997; Kinney, 1997; Browning,
1989; Slavit, 1994). Graphing calculators have made it possible to perform some
complex calculations that the four-function calculator is incapable of. These include for
example writing programs, logarithmic calculations, performing conversions, setting up
matrices, performing trigonometric calculations, integrations and exponentials. These
calculators also contain databases of known number constants such as п or the gas
constant. Reality is that these calculators have made it possible to perform entire
computations without ever having to write anything until you get the answer.
As previously stated, calculator use in classrooms is entirely dependant on the
teacher and/or the school system. Calculators can go from being used everyday to solve
problems, to being used solely to check pencil and paper solutions, to not being used at
all. The majority of students report that calculators are being used on a daily basis as
well as on mathematics tests. Although research has indicated that calculators do not
degrade basic skills and that they have a positive effect on students’ problem-solving
abilities and attitudes toward mathematics (Hembree & Dessart, 1992) , there are still
some examples of teachers who will completely not allow the use of calculators in their
class. These teachers feel strongly about the underlying concern that once students
realize that they can use a calculator to do the tedious computations they have been doing
with pencil and paper, they will begin to resort solely on the calculator.
No matter what vantage point a school system or teacher takes, there is no
denying the fact that calculators are everywhere and are difficult to avoid. Although
before these technologies were available, it was necessary for students to spend time
mastering and becoming proficient in the use of paper and pencil computational and
manipulative techniques, today there is an easier way. There is the calculator that can
allow for much of that computational time to be spent on developing deeper conceptual
understanding and valuable critical-thinking and problem-solving skills.
Future
Everyday advancements of the calculator are produced making the calculator out
to be more of a miniature hand-held computer than just a calculating device.
Advancements encompass not only computational abilities but also power capabilities.
The newest advancement involves the invention of the flash ROM. This new technology
for the calculator allows for a greater source of programmable memory as well as the
ability to install and run software applications (Waits & Demana, 2000). Mathematical
concept advancements are happening everyday. It will only be a matter of time before
calculators can perform any and every computation desired from addition to Fourier
Transforms.
As calculators become more advanced and complex, educators are going to have
to decide what role they will play. They will have to decide whether it is beneficial to
allow for their students to use such technology when learning mathematics. Extensive
research has been conducted showing that calculators could in fact enhance students of
all ages’ learning of mathematics while saving them time and energy as well. This may
be true but teachers still need to be mindful of how their students are learning. No school
should immediately jump to the conclusion that mathematics classes should be taught
throughout solely with calculators. Waits and Demana (2000) stress that there is a
definite need to maintain a delicate balance between traditional and technological
methods in order to fully develop the students’ understanding of mathematical concepts.
This will become more and more difficult as technologies become more and more
advanced. Teachers must remain conscience that whatever happens in the future of
mathematics education, students will without a doubt encounter calculators in their adult
life and they will need to be adequately equipped to use them effectively. They will also
need to be capable of living without one.
Conclusions
It has been shown that calculators are a useful tool for mathematical calculations
but that they should also not be abused. While calculators are a helpful tool, they do not
show the students the mathematics behind how they reach their answer. Without this
knowledge, mathematics becomes a subject of pushing buttons instead of problem
solving. To take this a step further, students will also not have the understanding of
whether their answer makes sense or not. In the real world, calculators are not always
handy when they are needed, and therefore students and adults alike must learn to
problem solve without them. One additional note is that calculator use is not fool proof.
Although most of the time the answer will be correct, due to the possibility of entering
information inaccurately, the calculator may not always be right. People need to be able
to recognize when an answer does not make complete sense.
It is apparent that in order for someone to be solid in mathematics, both methods
(paper and pencil and the calculator) are needed to teach students, but not one more than
the other. In fact, as mentioned previously, the use of them together will in all likelihood
produce the greatest results. I believe that using the methods together in the future is the
best course of action for both teachers and students. This way a student understands
mathematics and can use technology to expand their mathematics instead of replacing it.
Calculators can be seen as a tremendous technological advancement but must be
mediated.
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