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"The Role of Calculators in Math Education Research" compiled by Heidi Pomerantz, Rice University under the direction of Bert Waits, Professor Emeritus - Dept of Mathematics Ohio University,.
This document outlines the benefits of calculator use in mathematics classrooms from Kindergarten through the University level. Calculators are tools for doing mathematical computations. This document describes how calculators, when used appropriately, can also be a tool for learning mathematics. Appropriate use of calculators is a way of increasing the amount and the quality of learning afforded students during the course of their mathematics education.
Since its invention over thirty years ago, the electronic calculator has evolved from a machine that could only perform simple four-function operations (addition, subtraction, multiplication, division) into one that can now also execute highly technical algebraic symbolic manipulations instantly and accurately. Each new generation of calculators builds on the previous one with heightened speed and more advanced capabilities. At the same time, the cost of a basic calculator has dropped so low that virtually every household in the United States can afford at least one.
Calculators allow students access to mathematical concepts and experiences from which they were previously limited with only paper and pencil. Because calculators make possible mathematical exploration, experimentation, and enhancement of learning mathematical concepts, the National Council of Teachers of Mathematics (NCTM) and various other organizations and individuals recommend that appropriate calculators be made available for use by students at every grade level from Kindergarten through college. Despite the extensive research documenting the benefits of calculator use, there are still many skeptics who worry that calculator use will impair students mathematical ability and result in increased mathematical illiteracy.
The reality, however, is that calculators are valuable educational tools that allow students to reach a higher level of mathematical power and understanding. By reducing the time that, in the past, was spent on learning and performing tedious paper-and-pencil arithmetic and algebraic algorithms, calculator use today allows students and teachers to spend more time developing mathematical understanding, reasoning, number sense, and applications. Four-function, scientific, and graphing calculators, as well as calculators with computer symbolic algebra manipulation capability provide new pedagogical enhancement opportunities. They afford students learning tools that complement but does not replace, mental and paper-and-pencil skills and they expand student’s ability to solve problems by providing multiple solution techniques.
Rote computations and tedious algebraic manipulations have historically turned many students away from mathematics. The subject of mathematics has traditionally been thought of as memorising formulas and substituting numbers in equations, drilling endlessly, and performing long, monotonous computations. The students who could perform these manipulations and computations quickly and accurately were considered to be mathematically inclined; those who were turned off by the mechanical operations were thought to be poor math students. Calculator technology allows students whom would ordinarily be frustrated or bored by these tedious manipulations to have access to the real mathematics itself, thus gaining a higher level of mathematical understanding, rather than giving up. The fact is, calculators are better tools to do some of the computations and manipulations that were once done with paper and pencil. In the past, paper and pencil were the only tools available. Appropriate use of technology and associated peagogy will get more students thinking and reasoning mathematically. Thus more people will develop useful mathematical understanding and mathematical power.
Calculators now come in a number of sizes and styles, and they cover a tremendous range of capabilities, functions, and prices. Despite the myths of harmful consequences resulting from their use, calculator is a pedagogical tool of great value. Teachers of different grade levels and the general public harbor varying preconceived beliefs as far as the use of calculators in the classroom is concerned. Fears regarding the ill effects of calculators use, however, are unfounded. Research has proven that calculators are beneficial to students at every level of education. ( Specific references are located in footnotes at the end of the last three modules) Calculators serve as an equaliser in mathematics education. Not only do they allow students who would ordinarily be turned off by traditional mathematics’ tedious computations and algorithms to experience true mathematics, but they also help students to more quickly and readily develop number sense, gain mathematical insight and reasoning skills, value mathematics, and cultivates mathematical understanding, while they enjoy what they are learning.
Dispelling the Myths
Greatly impeding the universal acceptance of calculators into classrooms are the myths that exist regarding calculator use. These myths only serve to slow the inevitable implementation of technology in classrooms and put students at a disadvantage in a world that is rapidly embracing technology. Evidence from research has proven calculators to be effective learning tools; yet, because of the circulation of misinformation with regard to their use, many people continue to believe they are harmful. It is important that these myths are addressed, so calculators can be appropriately incorporated into curricula from kindergarten through the university level.
1. Calculators are a crutch….. they do the work for the student
2. Because calculators do all of the work for the student, he/she will not be stimulated or challenged enough.
3. " If I didn’t need to use technology to learn math, then neither does my child"
4. Calculators prevent students from effectively learning basic mathematics
5. People will become so dependent on calculators that they will be rendered helpless without one.
Myth # 1:
Calculators are a crutch : They are used because students are too lazy to compute the answers on their own; they do the work for the student. There is almost no mathematical thinking involved in doing rote computations. A real comprehension of mathematics comes as a result of understanding what the question is asking, knowing how to set up the problem, deciding which operations are appropriate and determining whether or not the answer obtained makes sense. Calculators are simply tool students use to help solve problems. Since they eliminate tedious computations and algebraic manipulations that discourage many student, calculators allow more students to solve problems and appreciate the power and value of mathematics in the world today. When used appropriately, calculators enhance learning and thinking, they do not replace it.
Myth #2 :
Because calculators do all of the work for the student, he/she will not be stimulated or challenged enough. Calculators do only the low-level tasks of computation - they do not "think" calculators can speed up the learning process. Students understanding the appropriate use of calculators experience more time to explore challenging and interesting mathematics. Calculators permit students to work enough problems to discover and observe patterns in mathematics, which were seldom seen when computations were done by tedious paper and pencil methods. Students will also be able to focus on useful, practical applications for the theories and concepts they learn in class. In the past, many students in mathematics were doing little more than memorizing rules and formulas; they were doing little thinking, problems solving, or reasoning. With appropriate use of calculators, many more students will have the opportunity to get past the mechanics of computation and manipulation and learn about the true meaning and value of mathematics.
Myth # 3:
"If I didn’t need to use technology to learn math, then neither does my child. After all, I turned out just fine." Because the calculator technology that exists today was not present a generation ago, all computations had to be worked out with pencil and paper in a series of long, tedious steps. The world has rapidly moved in the direction of technology, however, and technology has rendered obsolete many of the techniques and methods that were used previously. Because of technology, more students are now able to explore territories in mathematics that are still uncharted, and they are able to do ‘real" mathematics and understand its meaning and value. Often the parents who argue against the use of technology in mathematics classes simply have a fear of the unknown. They remember mathematics as consisting of drills, algorithms, and paper-and-pencil manipulations. However, calculators have eliminated the need for great skill in paper-and-pencil arithmetic computation and algebraic manipulation - items that used to be the core of a "proper" mathematics education. Computations that took several minutes and many sheets of notebook paper before can now be executed with the touch of a button. Since technology is being implemented in classrooms all over the world, all students must begin to understand technology and its appropriate use now, in order to learn the new technology core skills that will be necessary in the future.
The use of calculators prevents students from effectively learning the basic mathematics they will need when they enter the workforce. Calculators facilitate the mathematics learning process by eliminating tedious and needless paper-and-pencil calculations. However, they also familiarize students with technology, increasing their comfort level with technology and giving them a competitive advantage over those who have never been exposed to technology. Moreover, this understanding of the benefits and limitations of technology, as well as a general knowledge of how it operates will increase openness and willingness to use new forms of technology. Employers want employees who can think, work cooperatively, solve problems using the most effective methods ( technology when appropriate), and communicate " solutions" effectively.
Myth # 5
People will become so dependent on calculators that they n when no calculator is available ?). It is very important that mental calculations as well as estimation and some paper-and-pencil skills continue to be taught in schools, when those are the most appropriate methods for solving problems. Such skills are necessary in the mathematical learning process. These skills will also come in handy when a calculator is not available and when it is necessary to determine the appropriateness of a calculator result. The fact is, calculators are more efficient and accurate at performing many computwill be rendered helpless without one. (e.g. What if the battery dies or the student has to perform a computatioations, and they are inexpensive and portable enough to keep in one’s purse, pocket, car or office.
Despite all of their benefits and capabilities, calculators will never be able to replace the human mind when it comes to knowing how to read and understand a problem situation, writing an appropriate equation for the problem, choosing which operations to use to solve the problem, correctly interpreting the solution displayed on the calculator, and determining the appropriateness of the answer. Calculators are only as effective as the information students enter into them. Calculators, in conjunction with mental, paper-and-pencil and estimation skills when appropriate, comprise the tools to help students work through the computations and manipulations necessary for solving problems. Calculators are like computer word processors to English students. Computer word processors do not "create" essays but they do considerably facilitate the creation of an essay. Calculators do not "understand" mathematics but they do considerably facilitate the understanding of mathematics. Despite all of their capabilities, however, they will never replace the important, complex thought processes of which only humans are capable.
5.0 Graphic Calculators : Issues Affecting Secondary School Teachers and University Professors
When students work with graphing calculators, they have the potential to work much more intelligently than they could if they were not using this valuable resource; they form an "intelligent partnership" with the graphing calculator (Jones, 1996). Research has shown that graphing calculators can improve classroom dynamics, boost students’ confidence levels, and promote the understanding of mathematical concepts and functions, and advance problem-solving ability. That is not to say that mental arithmetic and paper-and-pencil and estimation skills are no longer valuable - they are. But there must be a balance. It important to use the tool that is most appropriate for the job. When they are used appropriately, graphing calculators do not pose a threat to students’ ability to perform algebraic manipulations or procedure.
Graphing calculators help students visualize problems, discover mathematical theorems on their own, instantly check the validity of their answers, test out their own hypotheses, and explore different ways of solving problems. Graphing calculators allow topics to be discovered by students on their own, even before the teacher formally introduces them. They facilitate an active approach to learning, converting a classroom from a place where students sit back passively listening to the instructor, to one where students work with their classmates and produce their own ideas and solutions. Graphing calculators improve communication among students, and they allow students a faster, better way to produce graphs; this is a much more efficient, accurate method than drawing graphs by paper and pencil alone.
Computer graphing (e.g. using graphing calculators) provides an important new teaching and learning paradigm : Graphs can now be used to study math. In the past, students studied advanced mathematics (calculus) to learn how to draw graphs accurately. Now computer-generated graphs can be used to study important mathematical concepts.
Inexpensive graphing calculators make the power of computer visualization a reality for all secondary and college students.
Calculators offer students a method of performing computations and algebraic manipulations that is more efficient and precise than paper-and-pencil methods alone. Graphing calculators can store, manipulate and display data in many different ways. They can be thought of as well-programmed miniature special purpose computers that are very portable, much more affordable, and easier to use than desktop computers. They are inexpensive enough so that classrooms may have a set for students to use at any time, and many students will be able to afford their own.
They cost about as much as a pair of trendy new athletic shoes, and a class set can be purchased for as much as it would cost to buy just one or two computers. Each generation of graphing calculators is becoming more powerful and useful; yet they are not getting more difficult to use.
Graphing calculator technology is changing the types of problems that are important in mathematics classrooms. When algebraic manipulations were the most important concept to be learned, they naturally comprised the focus of most teaching. Now, However, arithmetic calculations and algebraic manipulations can be performed correctly in a matter of seconds using technology. Students can avoid time-consuming, tedious procedures and concentrate on understanding concepts, developing higher order thinking skills, and learning relevant applications. These powerful calculators provide students with another resource - in addition to paper-and-pencil, mental and estimation skills - to assist them in executing the procedures necessary to understand and apply mathematics.
Calculators allow for the cultivation of analytical adeptness and proficiency in complex thought processes, rather than just the development of mechanical, computational skills. Problems representing "real-life" situations with complicated numbers can also be addressed. And students will not have to worry about whether they are going to make a mistake in addition or multiplication and not come up with the correct answer, even though they worked the rest of the problem correctly. Students still need to be taught to develop good mental estimation skills, so they will know if the calculator solution makes sense in the problem situation. Use of calculators allows students to focus on the steps involved in problem solving. This is just as important, if not more so, than the actual answer.
Students value what is tested. Therefore, since learning how to use technology appropriately is so very important today, calculators should also be permitted on examinations. If students were allowed to use calculators in class and for homework, it would not be appropriate to deprive them of this valuable resource on tests. That is not to say that calculators must be allowed on every portion of every examination. For example, the College Board’s Advanced Placement Calculus exam has two parts : in one part-graphing calculator are required, while in the other portion, calculators are forbidden.
The use of graphing calculators in the classroom extends problem-solving and mathematical understanding by making both practical and possible an important learning theory called multiple linked representations. Mathematical concepts can now be experienced through, and problems solved by, numerical, graphical, and symbolic representations. Graphing calculators allow students to move easily between these representations. When students are able to choose between several methods of solving a problem, it is more likely that they will remember how to solve it and be able to solve a similar type of problem the next time they see one. Moreover, seeing the graphs helps reinforce abstract concepts and allows the problem to become more tangible. Students can also use graphical methods to confirm an answer obtained through algebraic methods or to solve problems that would be too difficult to solve using algebraic methods. Research has also shown that access to graphical representations has improved females’ ability to visualize functions and graphs.
When students are able to spend more time concentrating on understanding, setting up, and choosing the appropriate operations and equations for a problem, they tend to see mathematics as more useful and less tedious than those who are forced to perform all algebraic manipulations by paper and pencil alone. They are more likely to stay with the problem, and their confidence about their mathematical abilities is boosted.
Research not only proves that the use of calculators results in more positive feelings and better attitudes about mathematics for both students and teachers but it also confirms that calculators improve performance in a variety of areas including problem solving (Dunham, 1995)