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
Mathematicsexists 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 Babylonianscirca 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 inthe 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 calculatoras 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 nuisanceas 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 hasskyrocketed 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 commonplace 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 examplewriting 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 thatcomputational 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.