Calculators in the classroom
Calculators have become an integral part of educational settings, with a history that dates back to the invention of the abacus in the ninth century. Today, calculators range from simple arithmetic devices to advanced scientific and graphing calculators capable of handling complex mathematical operations. In primary classrooms, the use of calculators is a topic of debate; while some argue they may hinder the development of fundamental arithmetic skills, others believe they can enhance problem-solving abilities when used appropriately. Research indicates that calculators can support students in exploring number patterns and verifying their calculations, thereby fostering higher-order thinking and inquiry skills.
In secondary education, calculators are widely accepted, often being allowed in university entrance exams. The use of graphing calculators, in particular, facilitates independent learning and allows students to visualize mathematical concepts, such as quadratic functions, rapidly. However, concerns persist about over-reliance on calculators potentially undermining students' grasp of basic mathematical principles. Thus, a balanced approach is essential, emphasizing the importance of knowing when and how to use calculators to enrich mathematical understanding rather than replace it. Ultimately, the effectiveness of calculators in the classroom hinges on thoughtful implementation and guidance from educators.
Calculators in the classroom
Summary:Calculators can be used in classrooms to augment rather than replace learning mathematical calculations.
Calculators have a long history. They can be traced back to the ninth century when the original compact calculator, the abacus, was developed in China. Nowadays, calculators are small (often handheld), electronic, digital, and inexpensive devices to perform various operations of mathematics. There are many kinds of calculators. Simple calculators just perform the basic operations of arithmetic.
Advanced calculators include scientific calculators and graphing calculators. Scientific calculators can work on complex operations such as trigonometric, logarithmic, and statistical calculations; some are even able to perform computer algebra. Graphing calculators usually have similar abilities to scientific calculators; however, they can be used to graph functions defined on the real line or higher dimensional Euclidean space. Since the advent of handheld calculators in the 1970s, the daily lives of people have been changed profoundly. Almost each business office and every high school student has at least one calculator. People can solve tedious computations in a few seconds by calculator, which was impossible before 1970.
Calculators in Primary Mathematics Classrooms
The availability of inexpensive calculators in primary classrooms has been increasing rapidly. However, the debate on their effectiveness in student learning and their role in mathematics instruction continues. Whether calculators should be used in primary classroom remains a controversial issue. On one hand, some people worry that calculators may hinder students’ learning and obstruct the development of basic arithmetic operations such as addition, subtraction, multiplication, and division. On the other hand, research has shown that calculators can promote problem solving in students with a strong foundation in basic skills.
In primary classrooms, the use of calculators aims at facilitating the learning of mathematics rather than replacing mental arithmetic and written calculation. Pupils can use calculators to explore number patterns, construct concepts, and check different methods and results of problem solving. With the help of calculators, children can strengthen their abilities in mental arithmetic and estimation, as well as judge the sensibleness of the results of calculation. For instance, pupils may be asked to estimate the sum of 9+99+999 and explain how they get the answer. One method of estimating the sum is the calculation 10+100+1000-3.
After they have done the estimate mentally, they can check their estimation by calculator. Depending on their abilities, pupils may be asked to estimate the sum of more complicated operations such as 999+9999+99,999+999,999+9,999,999 and then check the answer by calculator. This kind of activity facilitates the development of inquiry mind and higher order thinking in children. When pupils are allowed to use calculators to check the answers they have come up with by themselves, they have immediate feedback, have more time for solving additional problems, and make fewer errors. Calculators help pupils concentrate on thinking rather than on computation.
Pupils cannot benefit much if they are requested to compute traditional calculations such as 2 + 7 or 3 × 5by using a calculator. However, when they are asked to explore what calculations would give an answer of 10 with the aid of calculator, the effect can be very positive. Pupils may find patterns such as 1+9, 2+8, 3+7,… ; 11-1, 12-2, 13-3,… ; and 10÷1, 20÷2, 30÷3,… ; and so on. Such open-ended tasks provide opportunities for children to explore basic arithmetic operations, natural numbers, fractions, and decimals. Through these exploratory activities, children can develop number sense and strengthen inquiry mind by making and testing conjectures. Calculators help children quickly detect and correct their misunderstandings experientially.
There are many good calculator activities that will enrich the learning experience for pupils. Different activities may be suitable for different classrooms; however, the focus should be on the mathematics rather than the calculator.
Calculators in Secondary Mathematics Classrooms
The use of scientific and graphing calculators in secondary school causes much less controversy than the use of simpler calculators in elementary schools. In fact, many countries allow their secondary students to bring in approved calculators for their university-entrance examinations.
Over the past 10 years, many innovative methods of teaching secondary mathematics have been developed with the advancement of handheld calculators and the needs of society. Many of the ideas require only basic calculators, but scientific and graphing calculators open up more possibilities—particularly for the learning of complicated functions, shapes, and graphs.
A graphing calculator typically refers to a class of handheld calculators that are capable of plotting graphs, solving systems of equations, and performing numerous other tasks with variables. For instance, graphing calculators allow students to explore the effect of varying the coefficients in the quadratic equation y=ax2+bx+c by plotting graphs for different set of values of a, b and c in seconds. Plotting quadratic graphs by paper and pencil would consume a lot of time and effort, which would slow down the learning pace and reduce learning interest in the topic.
Graphing calculators save students from laborious work and provide opportunities to facilitate independent learning and give scope for open-ended exploration. If students go further in their investigation, they may generalize the conditions under which only one solution is obtained for the quadratic equation 0=ax2+bx+c.
The power of calculators advances rapidly. Some people worry about the use of symbolic calculators that can perform symbolic computations. They have argued that the use of symbolic calculators can cause core mathematical skills to wither, or that such use can prevent understanding of advanced concepts. It is not unusual that students use a symbolic calculator to find
without realizing the mathematical principle or skills involved.
Concerns on Usage
From time to time, calculators are accused of making children lazy and replacing the need for them to use or remember number facts. They provide a means for getting answers without understanding mathematical processes. Some people worry that the extensive use of calculators in mathematics instruction interferes with students’ mastery of basic mathematical skills and the understanding they need for more advanced mathematics.
In reality, the calculator is a tool that, if used in the right way, can support and encourage children’s mathematical thinking. It is not calculators themselves that matter but when and how they are used that is important. To avoid overemphasis on the use of calculators, students should be guided to recognize the functions and limitations of calculators, so as to strengthen their abilities in exploring and solving mathematical problems. For instance, in a classroom activity, all pupils are given the same set of calculation questions, such as789+0, 25×4, 17×8, and 299×10. Pupils work in pairs; one is requested to find the answers by mental computation while the other uses a calculator.
At the end, they have to record the time needed and the number of correct answers. Pupils have to discuss and identify which calculations can be easily done mentally and which cannot. This activity can facilitate pupils’ communication in mathematics and understanding that mental calculation sometimes is more powerful than the calculator.
Research also suggests that inadequate guidance in the use of calculating tools can restrict the kind of mathematical thinking that students engage in. Therefore, it is important that schools implement a balanced program that develops students’ understanding of the appropriate use of the calculator.
Bibliography
Baxter Hastings, Nancy, ed. Workshop Precalculus: Discovery With Graphing Calculators. Emeryville, CA: Key College Publishing, 2002.
Guin, D., K. Ruthven, and L. Trouche, eds. Didactical Challenge of Symbolic Calculators: Turning a Computational Device Into a Mathematical Instrument. New York: Springer Science+Business Media, 2005.
Rising, Gerald R. Inside Your Calculator: From Simple Programs to Significant Insights. Hoboken, NJ: Wiley Interscience, 2007.
Sparrow, Len and Paul Swan. Learning Math With Calculators: Activities for Grades 3–8. Sausalito, CA: Math Solutions Publications, 2001.