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Calculations About Calculators By Gerald W. Bracey Illustration © 1998 by Mario Noche

WE seem to be of two minds about the use of calculators in this country. Legislation in California would prohibit the use of calculators before the sixth grade. (Legislators do such a good job of micromanaging education these days that maybe we should try to interest them in health care.) On the other hand, the state of Virginia just purchased 200,000 graphing calculators to put every student on an equal footing in middle school and high school math classes.

The argument against calculators generally turns on the presumption that knowing how to do elementary calculations -- such as the multiplication tables -- is one of the "basics" and needs to be done by memory. I think one could argue equally persuasively that one must actually "know" more to use a calculator and determine whether the answer shown in the display is reasonable. That is, if a problem requires a student to divide 100 by 4 and the display reads .04, then the student needs to recognize that such an answer is not reasonable.

To me, the argument that students need to know how to calculate by rote is akin to Socrates' argument that people shouldn't be taught to write because it would destroy memory. Socrates, himself an illiterate, was right, of course. We cannot approximate today the feats of oral recitation found in preliterate cultures. But I think most people believe the tradeoff we made has been worthwhile. Indeed, the entire history of human civilization can be seen, in part, as an attempt to get stuff out of people's brains and into other, more reliable and more accessible locations. But that's another story.

In a recent issue of Educational Assessment (vol. 4, no. 3, 1997), David Meel of Bowling Green State University reviews the literature on the use of calculators during assessments. Students who use calculators, he finds, have better attitudes toward assessments and feel empowered. Using a calculator frees students from having to devote a lot of time and energy to actual calculation. Thus they can spend more time solving problems conceptually. The use of calculators also allows teachers to present the students with more real-world problems than would otherwise be possible.

Although he cites no specific study to back up his contention, Meel states, "With the aid of calculators, students can encounter complex mathematical topics at younger ages than previously thought possible. For example, the exploration of maxima and minima, which previously had been reserved for students studying algebra, can now be introduced to middle school students with graphing calculators during a study of functions."

Meel observes that test items can be calculator-active, -passive, or -neutral. In the first type of item, the calculator is required for speedy solution; in the second, it is impossible or inappropriate to use a calculator; in the third, it doesn't matter. Exposing children to a variety of items of all types can help teach them when the use of calculator technology is useful.

Meel spends some time discussing the kinds of foils (distractors) that would present problems for calculator use, such as those that require translation from one numerical representation system into another. He also observes that permitting calculators to be used for open-ended questions can be problematic for diagnostic analysis, because most computational work will reside in the calculator, not on the student's paper. Teachers must be able to determine for what purposes it is important for students to show their work.

Finally, not all calculators are created equal. The calculator that my real estate agent uses would give students an enormous advantage in finding answers that involve repayment times, loans, and interest rates. But the real estate agent's calculator would not be so cool for handling the derivatives and integrals of calculus. Virginia solved this problem by selecting the same calculators for everyone. But in other states students often have a wide range of calculators, especially if the students or their parents provide them.

Attrition from a School of Choice

WITH THE trend toward charter schools and other schools of choice, it becomes important to determine what attracts people to a program, why they stay with it, and why they leave. One of the few studies of this kind that I have seen appears in the spring 1997 issue of ERS Spectrum, a publication of Educational Research Service in Arlington, Virginia. The study was conducted by Kay Thomas of the Austin-based Texas Center for Educational Research, described in the credit line as an organization that "conducts and communicates nonpartisan research on educational issues."

Because the program studied is highly specific in terms of curriculum, some findings might apply only to that particular school, but some appear to have general application as well. The school is a middle school serving a largely Latino group in an inner city. It provides intensive study of foreign languages. Half of all qualified applicants are turned away. The article does not specify precisely what qualifies a student for the program; it states only that the school uses a combination of grades, standardized test scores, and recommendations from parents and teachers.

In looking at leavers versus stayers, Thomas found that stayers were far more likely to have been involved in the decision to attend. Ironically, parents who were concerned with their children's educational opportunities might have contributed to the students' leaving the school by not involving them in the choice. Said a principal, "Some of the attrition is due to the parent wanting this more than the child. The child gets into the program and finds he doesn't want to study a foreign language after all." A counselor felt that some children (and their parents) thought that they would be "privileged" and honored by the selection. "Then the kids get here, and [the foreign language is] just another hard subject," said the counselor.

Stayers were also more likely to have friends in the program. The role of social support is well known and might have been even more important in this case. The program was for middle-schoolers who tend to be trying out a lot of tentative roles and who need support.

Stayers performed better than leavers -- a not very surprising finding. Unfortunately, the article doesn't say explicitly whether those who left the program had exhibited lower achievement earlier. However, it implies that these differences occurred after the selection.

Phonics and Whole Language: In Sequence?

SOME people make whole language sound like the Devil's own invention. Phonics, on the other hand, has taken on a curious political direction: extreme right. Why people of a conservative bent should have latched onto this particular pedagogical approach remains an enduring mystery to me, and I simply note here that some odd things get said in behalf of both schools of thought. For example, Robert Sweet, head of the National Right to Read Foundation (funded by the Hooked on Phonics folks), declared flatly in conversation, "We teach them phonics. Then they can read anything." An amazing statement, that -- especially when held up against this passage:

The batters were merciless against the bowlers. The bowlers placed their man in slips and covers, but to no avail. The batsman hit one four after another with an occasional six. Not once did a ball look like it would hit the stumps or be caught.

While this passage bends American minds, many English and Australian children would be able to tell you precisely what it means, the citation being an account of an Australian cricket match.

Into the phonics/whole-language minefield stroll another pair of researchers: Colin Sacks of Diablo Valley College, Pleasant Hill, California, and John Mergendoller of the Buck Institute for Education, Novato, California. Their research appears in the winter 1997 issue of the American Educational Research Journal.

At least these researchers seem to have a handle on the complexities of the issues. For instance, they state early on that it is difficult to tie the phrase "whole language" to a specific set of classroom practices: "Some proponents conceive of whole language as a philosophy rather than an explicitly defined instructional methodology." Moreover, many teachers are eclectic in their use of strategies for teaching reading, so it is difficult to classify them as being in one camp or the other. Furthermore, teachers who say that they subscribe to one approach or the other often can't be distinguished by any of their teaching behaviors. Finally, few studies have used standardized measures or large numbers of students.

Using kindergartners and their teachers, Sacks and Mergendoller administered the Theoretical Orientation to Reading Profile to the teachers to identify them as phonics oriented or whole-language oriented. The children were given the Test of Early Reading Ability-2 to get a measure of their reading skills. This test measures a child's alphabet knowledge, ability to construct meaning from text, and understanding of arbitrary conventions used in reading and writing. The students were tested at the beginning of the school year and again at the end and were classified as high scorers or low scorers on the basis of the fall test scores. Although the achievement groups were defined only by their being above or below the 50th percentile, the differences between the two groups seem large: the low scorers at the end of the year were about where the high scorers were at the beginning.

The two approaches did not produce differences in how the classrooms were grouped, but children in the phonics classrooms spent more time completing worksheets. The two groups also did not differ in the amount of teacher talk when the class as a whole was addressed, but in small groups the whole-language teachers talked more. Even so, phonics teachers spent more time presenting information and giving directions. Whole-language teachers spent more time in literature and writing instruction.

The major finding concerning the two approaches to teaching children to read was that the whole-language approach benefited children whose initial reading scores were low. Children with high initial reading scores benefited more from a phonics approach. Does this mean that the two strategies should be used on different "types" of children? Sacks and Mergendoller don't think so. They argue for a stage model of learning to read.

There is an initial phase in which children develop skills imperative for later reading. These emergent literacy skills include learning the conventions of print, the register of the written language, and general expectations about the nature of reading. Later stages allow for the mastery, practice, and, finally, automatic execution of specific skills. . . . It may be that emergent literacy opportunities of the whole language-oriented classrooms, which were available less frequently in phonics-oriented classrooms, were especially beneficial to lower scoring students.

It might also be that for less-ready students an approach that emphasizes real literature is a lot more fun than an approach that emphasizes decoding skills. One other finding that would support this interpretation is that the lower-scoring students were more engaged in their reading activities in whole-language classrooms than in phonics-oriented classrooms.

It would have been interesting had the researchers continued their study into first grade, where reading would be emphasized more and treated more formally than in kindergarten. Studying first-grade classes might offer sharper differentiation between phonics-based and whole-language programs.

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Last updated 23 February 1998
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