NCTM's unconditional support might still be warranted if there was conclusive research supporting calculator use. However, a review of the literature published over the last two decades has revealed very little research. Furthermore, much of the research that does exist focuses as much on students attitudes toward mathematics as on actual achievement (Riley, 1996; Moore, 1982; Bartos, 1986, Liu, 1993). While attitudes are important, achievement differences, if they exist, should be the primary factor in deciding whether calculator use is appropriate.

When the research does address the effect of calculator use on achievement, the research has often been suspect and/or inconclusive. One study only lasted three weeks (Moore, 1982), which is not long enough to address concerns that calculator use might cause arithmetic skills to atrophy. Another was unable to establish a true control group (Riley, 1996). A third found no significant differences in addition, subtraction or division results, but did find that the control group (no calculators) did significantly better at multiplication (Bartos, 1986). Which is more important? The fact that there was no apparent difference for three skills, or that there was a difference for the fourth? Since the paper didn't even indicate how long the study lasted, it is difficult to draw conclusions.

So why isn't there more research? One obstacle to meaningful research may be that too many researchers believe that the issue has been decided and that resistance to the use of calculators is merely due to ignorance on the part of those urging caution. For example, a title such as "The Wasted Resource: Attitudinal Problems In Calculator Use In The Elementary Classroom" (Simon, 1990), demonstrates a clear bias toward more calculator use in spite of the lack of evidence. The bias is further demonstrated when the author cites the unconditional support of the NCTM for calculator use as apparently sufficient reason for using calculators, and proceeds to try and discover why elementary schoolteachers are resisting the NCTM' s recommendations.

One work, a meta-analysis by Ray Hembree while supervised by Donald Dessert, is cited so often that it appears that many people consider it the last word on the subject, thus alleviating the need for additional research. In it, Hembree concluded that "a use of calculators in concert with traditional mathematics instruction apparently improves the average student's basic skills with paper and pencil" for all grades except the fourth, but also concluded that sustained calculator use hindered the development of basic skills of average students in grade 4 (Hembree and Dessart, 1986). Many have chosen to ignore the latter result, and concentrate only on the former. However, further examination shows that there may be cause for concern about the effects of the calculator at early ages.

Although the research reviewed 79 studies, several studies involved more that one grade level. Hembree provided a table showing a distribution of grade levels for the 79 studies. It should be noted that the table, while both accurate and reasonable, might give a casual reader the false impression that there were actually 122 studies. This is not intended as criticism, but in the interest of clarity, this author will acknowledge that there are 122 grade level related divisions of data in the 79 studies. Of those, the meta-analysis included sixteen divisions of data involving students in the fourth grade, but only one for kindergarten students, two for first grade students, three for second grade students, and eight for third grade students. That leaves 93 divisions of data, over 76% of the total, spread out over the remaining eight grade levels (Hembree and Dessart, 1986). The data, therefore, may be more accurate for students at higher grade levels where basic skills have presumably already been learned. Also, among the early elementary levels, which represent the primary concern of this article, the fourth grade, where calculators proved harmful, was represented in more divisions of data than the other four grade levels (K-3) put together. It is reasonable to ask if there really was enough data to draw conclusions about the effect of calculators at the K-3 levels.

Another concern about the validity of the Hembree research stems from the questionable quality of the research used in the meta-analysis. Since a meta-analysis is a study of studies, its validity is dependent on the validity of the research that is analyzed. The lengths of the calculator treatment in the various studies lasted from one year to less than one class period, with a median of 30 school days (Hembree and Dessart, 1986). Even the longest study is not long enough to be meaningful in this author's opinion, and the median length is unacceptably short.

This author believes that a longitudinal study spanning several years with a genuine control group would be necessary to confirm or allay concerns about the possible deterioration of basic arithmetic skills. Admittedly, such a study would be nearly impossible to conduct. It is doubtful that one could obtain permission to use human subjects in this manner for that length of time. The budget for such research would probably be out of the reach of most researchers. And since an overwhelming majority of both teachers and parents oppose unfettered calculator use (Public Agenda, 1996), getting people to participate in the experimental group of such a longitudinal study might be difficult. Furthermore, if a control and an experimental group could be formed, keeping students in their assigned groups across several grade levels would be difficult, if not impossible.

Since little research exists, what does exist is suspect, and meaningful research may not be feasible, it is logical to examine arguments that do not rely on research. One concern among those who urge caution in the introduction of calculators, is that the ready availability of the calculator will lead to the impression that manual and mental arithmetic is no longer important. Jennifer Wand, an elementary teacher, seems to have adopted that very attitude. Wand argues that time spent teaching multiplication tables in the second grade is wasted since, in real life people would use calculators to perform computations (Wand, 1996). While Hembree's conclusions about calculator use at early ages may be open for question, at least he specified that the calculator be used alongside traditional instruction, rather than in place of it (Hembree, 1986). In fact, every research article discovered and cited here used the calculator in tandem with manual and mental arithmetic. The NCTM recognizes the need to develop paper and pencil skills, and puts great emphasis on the development of estimation skills, if for no other reason than to be able to tell if the figure in the calculator display is reasonable (NCTM, 1989). While the calculator itself will not make an error, the person entering the data can. Anyone who has used a calculator can tell stories of sticky keys, pushing the wrong buttons, and other simple mistakes that will lead to incorrect results. But the mental arithmetic skills that are needed for estimation, can only be developed by memorizing basic facts, such as multiplication tables, and by repeated exposure to arithmetic computation.

Furthermore, doing paper and pencil arithmetic will reinforce and expand those mental skills. A personal example involves multiples of forty-five and twelve. During an earlier career selling home improvement materials, particularly floor tile which is sold in boxes of 45 tiles and ceramic tile which sold in boxes of 12 square feet, this author eventually memorized several multiples of forty-five and twelve, simply from having to multiply by those numbers so many times when making a sale. While knowing those multiples of forty-five or twelve by heart is no longer as useful as it was when selling tile, it certainly is not harmful, and may prove advantageous someday when checking computations on a calculator using values close to either number.

Another concern is that students will cease to believe that they are capable of doing arithmetic and accept it as something only done by a machine. This author can recall a student who insisted that she had "never done fractions, except on a calculator" and thought it unreasonable to expect students to be able to do that work by hand. Virtually any math teacher can recall instances of students accepting whatever is in the calculator display as correct, even though the number shown there is clearly not reasonable.

Nor is this concern limited to calculator skeptics. In an article on choosing the proper calculator for the classroom, Demana and Osborne note that young students trust calculators. They argue that classrooms should use scientific calculators, which will perform operations in the correct order, rather than basic four-function calculators, which perform operations in the order that they are entered (Demana and Osborne, 1988). For example, a scientific calculator will correctly evaluate 3 x 4 + 5 x 2 as 22, while most four-function calculators will display 34.  Demana and Osborne state that "trusting students may incorrectly conclude that operations should be performed in order from left to right" rather than the correct order which calls for performing multiplication and division operations before performing addition and subtraction (Demana and Osborne, 1988).

If calculator proponents are concerned that the wrong calculator may lead to incorrect learning, how will learning be improved by using calculators which will perform the operations in the correct order without any knowledge or effort on the part of the student being required? There are a statistically small number of students with learning disabilities that might prevent them from learning basic arithmetic concepts. Those students will eventually have to use a calculator because they have no choice, and the proper time to have them start will require discussion that is outside the scope of this article. But for the typical student, one who is capable of learning basic arithmetic facts and skills, this author believes that it would make more sense to simply not use the calculator at all until the student has demonstrated the ability to follow the proper order of operations without mechanical aids.

Another concern is that a lack of basic arithmetic knowledge can interfere with learning more advanced concepts. Arithmetic is, after all, a foundation of all mathematics and much of higher mathematics is simply an abstraction of arithmetic. It would be difficult, if not impossible, to teach higher mathematics without performing some arithmetic. A student who lacks basic arithmetic skills may have difficulty following the underlying process simply because they cannot follow the arithmetic. In an example taken from this author's exerience, imagine a student who witnesses a teacher converting the equation ³/₄ x + ¹/₂ y = 2  into the equivalent form 3x + 2y = 8 by multiplying the first equation by four. Students with weak basic arithmetic skills can, and do, get lost trying to understand how the teacher arrived at the values of 3, 2 and 8. They miss learning the purpose for performing such a transformation in the first place because they cannot get past the arithmetic.

Implicit in Wand's argument that the time spent teaching basic skills and facts is wasted, is the notion that the time could be better spent learning more advanced concepts. This argument has some validity. It is probably not necessary for today's students to spend hours performing operations on multi-digit numbers over and over in a futile attempt to attain a level of mastery that any first-grader with a calculator could surpass. But that isn't the same as saying that students shouldn't learn and practice manual and mental arithmetic. At the very least, there is a value in simply learning the various arithmetic algorithms. Calculators do break or get lost. Almost any walk of life involves something that requires following some series of steps to obtain the desired result; be it baking a cake, programming a VCR, or changing a tire. Simply learning to follow steps could be considered a sufficient reason to teach arithmetic algorithms to young children. The fact that it simultaneously teaches a valuable skill can, under those circumstances, be viewed as an added incentive.

A more pertinent reason for the continued teaching of manual arithmetic algorithms is that many of those algorithms have analogs in algebra. Students who can manually perform long division, are better prepared to learn how to divide polynomials since the algorithms are nearly identical. A student who only knows how to add fractions by punching buttons on a calculator will be at a significant disadvantage in an algebra class when asked to simplify expressions such as ^a/_b + ^c/_d since that student will not be familiar with the process of finding common denominators, or converting from one form of a fraction to another. Even a student with limited proficiency with fractions will have the edge over a student who only knows how to use the calculator.

However, the main rebuttal to the "time better spent" argument is that both can be done. While some drill and practice is still needed, the amount of time that was once devoted to it would be excessive today. Much of that time can be spent teaching more advanced concepts. Meanwhile, students can also get much needed arithmetic practice simply by doing the arithmetic without the calculator. Teaching arithmetic skills and higher concepts does not have to be an "either/or" proposition. We can do both.

This author agrees that students should be taught, not only how to use the calculator, but when it is appropriate to use the calculator. The disagreement is in when that training should begin. The NCTM national mathematics standards call for calculators to be available at all ages (NCTM, 1989) while the majority of parents and teachers, including this author, believe that calculators should not be introduced until children have learned their basic skills (Public Agenda, 1996).