Calculators in the Classroom

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[edit] Descriptons, definitons, and History

The first four-function and scientific handheld calculators appeared in the early 1970s. Their appearance gave rise to simplistic yes-no articles in newspapers and magazines. Yet those opinions had little effect because cost, fragility, and short battery life limited calculator use.

By the early 1980s, those deterrents lost force because of the appearance of solar-powered, hard-case, four-function and scientific calculators costing less than $10 and $15, respectively. And so the new generation of calculators began to be used. In 1985, the first user-friendly calculators appeared that could graph functions. Like their simpler counterparts, these calculators were too expensive to be widely adopted when they first appeared, but today many high schools require them for all or virtually all their students. The use of these calculators in secondary school has not generated as much controversy as the use of simpler calculators in elementary school, and they are required on many college-entrance tests. More recently, user-friendly calculators have appeared that can solve literal algebraic equations, manipulate algebraic expressions, differentiate and integrate, and solve systems of equations. 4

[edit] Application in classrooms

Calculating, is what calculators are good at, which is why much more time could be spent on the other parts of the problem-solving process. The other parts are unavoidable when solving problems. Since these parts often are neglected in school-mathematics, I would say that students, who don’t have problem solving as a part of their own family tradition, actually are discriminated against this in school.1

An article in "Education World" lists several advantages to calculator use in the classroom. Calculators allow students to spend more time using critical thinking to solve problems, since they are not spending as much time on the tedious calculations that the calculators can perform for them. Calculators allow students to study concepts that would have been beyond their scope of understanding if the calculations would have to be performed manually. The calculators bring about a spark of interest to otherwise uninterested or bored students. They also simplify the tasks at hand allowing for more time to explore different methods to solving problems. Calculators also bring about more confidence in students. 13

[edit] CALCULATORS CAN CHANGE THE WAY WE TEACH

Bert Waits reports ten fundamental activities done with "hand-held visualization technology” in the classroom work of students in the Calculator and Computer Pre-calculus project (a project involving more than 1,000 schools in the USA). These activities are:

1) Approach problems numerically.
2) Use analytic algebraic manipulations to solve equations and inequalities and then support using
visual methods.
3) Use visual methods to solve equations and inequalities and then confirm using analytic
algebraic methods.
4) Model, simulate and solve problem situations.
5) Use computer generated visual scenarios to illustrate mathematical concepts.
6) Use visual methods to solve equations and inequalities which can not be solved or are
impractical using analytic algebraic methods.
7) Conduct mathematical experiments, and make and test conjectures.
8) Study and classify the behavior of different classes of functions.
9) Foreshadow concepts of calculus.
10) Investigate and explore various connections among different representations of a problem
situation.2

"According to the National Council of Teachers of Mathematics (NCTM), the use of calculators along with traditional paper-and-pencil instruction enhances the learning of basic skills." In fact, the NCTM supports "the integration of the calculator into the school mathematics program at all grade levels in class work, homework, and evaluation." (Roberts, 1991, p.51) In other words, the use of calculators should not eliminate the teaching of the basic algorithmic skills and processes of mathematics. It should be properly integrated to reinforce the basic concepts that are being taught and to aid in the application of these math processes to real-world situations. The key words here are "proper integration". This implies teacher-supervised activities relating to the mathematical concepts being learned.3 How could calculator use benefit students? When students do not have to worry about computation mistakes, they can focus on reasoning and problem solving. Teachers can help students see patterns, check estimates against reality, and solve complex problems, like those encountered in daily life, through the structured use of calculators. Children introduced to the calculator when they are young will find it easy and effective to use. Calculators should be used in the classroom for many reasons:

  • Calculators help students at all levels learn mathematically complicated material.
  • Even young children can use calculators to focus on the ideas behind computation rather than on the act of calculating.
  • Rather than hampering mathematical ability, calculator use can actually improve student achievement in mathematics, according to research.
  • Both the SAT and ACT now allow students to use calculators during testing, as do many state-level exams. Students who have not been comfortable with calculators from a young age may be at a disadvantage on these tests. In the recent Third International Mathematics and Science Study (TIMSS) fourth and eight graders who used calculators almost every day performed at higher levels than did those who never used one or only used it once or twice a month. 9

With guided calculator use even young children can begin to see differences and relationships among arithmetic procedures. The difference between memorizing 5 x 5 = 25 and hitting 5+ on the keypad five times is that the latter activity can lead to discussions about connections between addition and multiplication and make the underlying patterns of multiplication apparent. With teacher guidance young children can become aware of larger numbers or even negative numbers at an earlier age than they have in the past.

In secondary school, calculators can help students develop their understanding of algebra and other advanced mathematics. Students will have an easier time learning advanced mathematical procedures if the foundation for complex calculator use is laid in the elementary grades.

Time for exploration is needed for effective calculator use. If students use calculators to figure out the relationship between the circumferences and diameters of many different round objects, they can get beyond problems of correct division and watch the concept of pi emerge. Doing such work adequately, however, requires that the teacher make the time to allow students to work with their own material until the concept is discovered and internalized. Once such time is set aside the calculator will repay with more time available for searching, developing hypotheses, and testing them. Without anxiety over basic mathematical processes, children will be able to concentrate on the applications and meanings of the world of numbers. 9

[edit] Evidence of Effectiveness

What does research show about classroom calculator use?

Researchers have studied classroom calculator use for several decades and in many countries. Research in this area began to take off in the late 1970s. In a 1986 study in the Journal for Research in Mathematics Education, Hembree and Dessart analyzed 79 studies of calculator use and found the following:

  • Children who use calculators on tests have higher scores in both basic computation skills and problem solving.
  • Students who use calculators within a mix of instructional styles do not lose their paper and pencil skills.
  • Calculator use in the classroom improves the paper and pencil skills of students regardless of their ability levels.
  • Those who use calculators in class have better attitudes toward mathematics than children who do not use them. 9

Hembree and Dessart (1992) reported the findings of a meta-analysis of the effects of pre-college calculator use. This research analyzed results from eighty-eight studies focused on students’ achievement and attitude. Each study involved one group of students using calculators and another group having no access to calculators. From their analysis, Hembree and Dessart concluded that the calculator did not hinder students’ acquisition of conceptual knowledge and that it significantly improved their attitude and self-concept concerning mathematics.

Smith (1997) conducted a meta-analysis that extended the results of Hembree and Dessart. Smith analyzed twenty-four research studies conducted from 1984 through 1995, asking questions about attitude and achievement as a result of student use of calculators. As in the Hembree and Dessart study, test results of students using calculators were compared to those of students not using calculators. Smith’s study showed that the calculator had a positive effect on increasing conceptual knowledge. This effect was evident through all grades and statistically significant for students in third grade, seventh through tenth grades, and twelfth grade. Smith also found that calculator usage had a positive effect on students in both problem solving and computation. Smith concluded that the calculator improved mathematical computation and did not hinder the development of pencil-and-paper skills.

A recent large study examined effects over a longer term. The purpose of the project, Calculators in Primary Mathematics, funded by the Australian Research Council, Deakin University, and the University of Melbourne, was to have primary and elementary school students explore and develop number sense using calculators before standard algorithms were taught (Groves and Stacey 1998). It involved one thousand students and eighty teachers over a three-year period. The performance of students in the project was compared with that of a control group for the same schools using a written test, a calculator test, and an interview. The results showed that the project students performed better overall on a wide range of items including place value, decimals, negative numbers, and mental computation. No detrimental effects of calculator use were observed. Until curricular innovations such as that tried in Australia are implemented, we believe that students and teachers should distinguish among three tools of computation: mental arithmetic, pencil and paper, and calculators. For example, we would chastise any student who reaches for the calculator to find 3 x 4; we would suggest pencil and paper for calculating 27 x 340; and we would insist on using the calculator for 2.7568 x 345.8972 after the student estimates mentally an answer of 900 (3 x 300).

In conclusion, we recommend that schools strongly encourage the use of calculators in all aspects of mathematical instruction including the development of mathematical concepts and the acquisition of computational skills. We believe that calculator education is an obligation of schools to our society where calculators are in common, daily use. 6

Emphasis is placed on the role of technology and the appropriate concepts and skills related to its use. Changes in technology and the broadening of the areas in which mathematics is applied have resulted in growth and changes in the discipline of mathematics itself. The new technology not only has made calculations and graphing easier but has also changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them.

[edit] Critics and their rationale

David Gelernter, professor of computer science at Yale University, believes calculators should be totally eliminated from the classroom. He feels that allowing children to use calculators produces adults who can't do basic arithmetic, doomed to wander through life in a numeric haze. In 1997, California legislation would prohibit the use of calculators in schools prior to the sixth grade.3

One of the problems experienced with children who have used calculators from an early age without proper integration is that they have not had the opportunity to develop good "number sense". This is sometimes referred to as a "feel for numbers". The importance of estimating an answer before or after calculating is not understood. Why estimate, when the calculator will give you the exact answer? After all, a calculator is never wrong. That is virtually true, but the operator of the calculator is capable of error. 3

Beyond the anecdotal, there is also the input from the Third International Mathematics and Science Study (TIMSS). For the 8th grade assessment, the majority (>50%) of the students from three of the five nations with top scores (Belgium, Korea, and Japan) never or rarely (once or twice a month) used calculators in mathematics classes. In contrast, the majority of students (>65%) from 10 of 11 nations, including the United States, with scores below the international mean, used calculators almost every day or several times a week in mathematics classes (Beaton, Mullis, Martin, Gonzalez, Kelly, and Smith 1996) While such data do not prove that calculator usage is damaging to the development of mathematical skills, it would be folly to ignore this. Taken from Mathematics Framework for California Public Schools K–12 (draft, 1998) 8

From the TIMSS results it is clear that mathematical competence at the grades K–6 level does not require calculators. Two of the highest-achieving countries at the fourth-grade and eighth-grade levels, Singapore and Japan, use calculators sparingly in elementary schools. 5

"Some of us who were very early to use technology to alleviate drudgery, to visualize graphs and surfaces, to conduct helpful experiments, etc., are now alarmed at its use as a substitute for thinking. It even seems to deter problem solvers from producing general mathematical proofs by holding their focus to computing a few numerical examples.” (John Duncan, mathematics professor, University of Arkansas, American Mathematical Monthly 102, p. 194)

The worst effects of calculator usage are the following:

1. The loss of experience in simplifying and the consequent loss of the student’s (and the teacher’s or examiner’s) expectation that expressions should have any meaning.
2. The destruction within half a generation of a hard-won, effective algebraic symbolism, developed and proved over centuries, capable of being manipulated as a "calculus” for exact numerical and symbolic calculations, and its replacement by slavish verbatim copies of what appears in calculator displays.
3. The collapse within ten years of arithmetical fluency within the very best students, with the resulting loss of meaning for symbolic generalizations of numerical expressions.
4. The loss of all attempts to teach pupils to present solutions in forms that others can make sense of, and the decline into mere personal jottings en route to an answer, which is related—I suspect—to the next effect.
(Anthony Gardiner, mathematician University of Birmingham, personal e-mail)

[edit] Considerations due to diversity and gender

Lest you are about to jump up and toss calculators into the classroom, let me caution you that this is where things get tricky. What if calculator use does not promote a "girl friendly” learning environment? According to the AAUW again, "Girls have developed an appreciably different relationship to technology than boys … and technology may exacerbate rather than diminish inequities by gender as it becomes more integral to the K–12 classroom.” More boys have and use technological tools and toys. Boys more often perceive themselves as going into careers, such as engineering, that require technology like calculators and computers. They can envision a payoff for learning to use these tools. We need to be sensitive to past and present inequities when structuring opportunities for all children. 7

[edit] Signed "life experiences", testimonies and stories

The calculator, especially the high end graphing calculators, can have a profound impact on learning and teaching. (to be continued Brad Frey - Johnsburg High School)

As an assistant math teacher for gifted children I learned first hand how a calculator can be a learning tool. At first I felt that the students should have to do their calculations on paper as many of us did when we were in school. What surprised me was how the students used the calculators. They didn't just enter the problem to be solved and hit enter - they chose to program the calculator. This is where I became convinced that using calculators in the classroom was a positive for these students. ~RSmall


Until recently I never allowed a calculator in my class. But now with the state testing situation, it has become a must. I guess we need to look at it realistically, in life most everyone uses a calculator. By not teaching the students to use a calculator, we would be denying teaching our students a life skill. Whether is balancing a checkbook, or using it on the job, calculators play an important role in our lives. We need to teach our students the right way to utilize the calculator. – Dale Donner

I think that calculators shouldn't really be used in the educaitonal setting because we are having our students rely on technology instead of the students using their brain. I have a student who currently counts on her finger when adding if she doesn't have a calcluator around to do basic arithmatic. She will use a calculator to do simple addition and subtraction. I just think that by allowing students to use technology we are cheating them because they aren't challenging themselves at all. --Brian Bucciarelli

Although I do not teach math, I have heard from math teachers about the students' reliance on calculators. Keep in mind, these are middle school students that are being referred to. One of the most disturbing facts is that students are using them to do basic mathematical operations such as addition, subtraction, and multiplication. When I was in school, using a calculator for such purposes would've been unheard of. We only used the calculator for complex problems and graphing purposes. We were expected to have those basic operations memorized. If students are allowed to use calculators for everything I think it is doing a disservice to our youth. I feel that calculators should be used when performing complex operations and when they are necessary, not just for calculating the simplest of problems. ~K. Kleckauskas

I understand the cognitive need for students to understand the basics of computation and the historical drive to keep that as a mainstay of K-12 math instruction. However, given the ever-growing breadth of information that students need to be exposed to prior to graduation as well as the increased demands of mandates, not to mention the increasing reliance on technology society-wide, I cannot help but to endorse the use of calculators in the classroom. Not only the use of the calculator but instruction on how to program for and complete complex functions. -Steve Svendsen

While some students do become too reliant on the use of calculators, their use should not be prohibited. Calculators, if used correctly, can enable a student that struggles with basic facts to learn more advanced skill without the worry of making a simple math mistake. In an upper level mathematics class, calculators can improve the comprehension of a student by making what may have been a long, tedious problem more concise. (My stats students would attest to this last fact.) I am not advocating that students should be allowed to use them in all cases. For example, in my stats classes the students are taught to work the problem by hand one day. The next class day, they are taught how to repeat the process using a calculator. In this way, they learn to appreciate what the calculator is doing for them. We have also been practicing programming skills - which require logical mathematics steps - in an effort to further apply our basic math knowledge. - M Foshee

[edit] References and other links of interest

1. Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom, Per Broman, Chapter 3 http://ued.uniandes.edu.co/servidor/em/recinf/tg18/Base/WWWfiles-1.html

2. Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom, Silva, J. C., Chapter 4 http://ued.uniandes.edu.co/servidor/em/recinf/tg18/Base/WWWfiles-1.html

3. Nancy Ayers, http://www.math.wichita.edu/history/topics/calculators.html#calc

4. Zalman Usiskin, Editor, Mathematics Education Dialogues, May/June 1999, Volume 2 Issue 3, A publication of the National Council of Teachers of Mathematics, http://www.nctm.org/dialogues/1999-05.pdf#search='calculators%20in%20the%20classroom'

5. Kim Mackey, "Do we Need Calculators”, Mathematics Education Dialogues, May/June 1999, Volume 2 Issue 3, A publication of the National Council of Teachers of Mathematics, http://www.nctm.org/dialogues/1999-05.pdf#search='calculators%20in%20the%20classroom'

6. Donald J. Dessart, professor of mathematics and mathematics education University of Tennessee—Knoxville (UTK), Charleen M. DeRidder, adjunct professor of mathematics education at UTK and supervisor of mathematics for grades K–12 in the Knox County Schools of Tennessee, and Aimee J. Ellington, instructor of mathematics at the University of Tennessee at Chattanooga. The Research Backs Calculators, Mathematics Education Dialogues, May/June 1999, Volume 2 Issue 3, A publication of the National Council of Teachers of Mathematics, http://www.nctm.org/dialogues/1999-05.pdf#search='calculators%20in%20the%20classroom'

7. Charlene Morrow, co-director of the Summer Math program and faculty member in the Psychology and Education Department Mount Holyoke College, South Hadley, MA. "Ensuring That All Children Are Powerful Technology Users” Mathematics Education Dialogues, May/June 1999, Volume 2 Issue 3, A publication of the National Council of Teachers of Mathematics, http://www.nctm.org/dialogues/1999-05.pdf#search='calculators%20in%20the%20classroom'

8. Johnny Lott, Editorial Panel, "A Calculator Tour Around Canada and the United States”, Mathematics Education Dialogues, May/June 1999, Volume 2 Issue 3, A publication of the National Council of Teachers of Mathematics, http://www.nctm.org/dialogues/1999-05.pdf#search='calculators%20in%20the%20classroom'

9. Quick Takes March 1998:a publication of the Eisenhower Southwest Consortium for the Improvement of Mathematics and Science Teaching, http://www.sedl.org/scimath/quicktakes/qt9803.html

10. Graphs for Calculus

11. Graphing Calculators for Quadratic Equation

12. Graph Paper

13 Starr, Linda. "Education World", 2002. http://www.education-world.com/a_curr/curr072.shtml

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