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The disputes over the use of calculators in the mathematics classroom have raged for almost 30 years now. It's remarkable, really; there is no comparable dispute over the use of computers or the Internet, and in an earlier age there was no comparable dispute over the use of slide rules. What's different about calculators that we have such a problem with them?
Surely the answer involves our feelings about computation itself. For us, the ability to compute is one of the hallmarks of an educated person. If children use calculators, the argument goes, they won't learn to compute, and then they won't be educated people.
The ancient Greek philosophers would have laughed at this argument, because, for them, it was the ability to reason that was the hallmark of an educated person. Computation was something tradesmen did (generally using concrete manipulatives of some kind, the calculators of that time). The idea that an educated person can compute with numbers comes out of the Middle Ages, when Europeans had just learned from Arab mathematicians the algorithms for decimal arithmetic we still teach today.
What computational skills are important for an educated person in our culture? There is a name for this group of skills: numeracy. What does a numerate person know today?
First of all, a numerate person knows her number facts. No doubt about it, folks; if I don't know that 6 times 9 is 54, I am not a numerate person.
But there is more to "basic skills" in mathematics than knowledge of the multiplication and division tables. The numerate person has an agility in mental arithmetic that enables him to juggle figures very quickly with neither pencil nor calculator and say things like, "Well, 6% of $1543 is maybe $100 or a little less." This kind of skill doesn't come from doing pages of pencil and paper work. It comes from a lot of practice thinking about numbers, comparing them and combining them.
The numerate person has confidence in her ability to solve numerical problems, a confidence that comes from having solved many problems successfully. This skill must be based on knowledge of the fundamentals of the subject and also on knowledge of the tools used in the subject.
Finally, the numerate person has a healthy skepticism about numerical arguments. He knows the numbers presented, like the numbers we compute ourselves, are not always right. We must be ready to check and challenge, and to ask if they cannot be presented in a different way with perhaps a different conclusion.
The evidence from international testing is that we teach the number facts quite well. We do not teach mental arithmetic very well, and our students' problem-solving skills are clearly not as good as we want them to be. Our algorithmic instruction produces unskeptical students who often fall for the sucker answer or are derailed by a simple computational error.
Now it's time for the hard question. If we use calculators in the classroom more, will things get better or worse? Proponents and opponents of calculators should agree on one thing at least: It can go either way!
If we use calculators to excuse students from learning the fundamentals, then surely they will be less numerate adults. The opponents of calculators are undoubtedly correct about this.
If we do not demand that students become agile in mental arithmetic, then not only will they be less numerate, they will also be poor users of calculators. To use a calculator (or spreadsheet) effectively, a student must learn to actively monitor and control the calculation. This requires skill in mental arithmetic,
On the other hand, if we do not use calculators, then it will be very difficult for most students to develop the confidence they need as problem solvers. The hard fact (a fact we have never wanted to admit) is that most students will never become confident enough in their pencil-and-paper arithmetic skills to be really effective problem solvers. Furthermore, a large fraction of the problems arising in the real world are too difficult to be solved without the use of calculators or computers.
Finally, calculators are indispensable in checking and challenging numerical arguments. The numerate person needs to be able to say, "Wait, I have a bad feeling about this. Let me crunch a few numbers here and see if my suspicions are correct."
To produce numerate citizens, we must do two things simultaneously. We must be clear and rigorous in teaching the fundamentals. This doesn't mean everything in the traditional curriculum, but it does mean such things as the multiplication and division tables, the standard algorithms as applied to numbers of modest size, the use and comparison of fractions, the operations of percentage and rates, the basic geometric formulas, averages and other tools of data analysis.
At the same time, we must put tools in students' hands and teach them to use those tools. We believe that mathematics will be crucial for students' economic success in the new century. As adults, our students will use tools to apply mathematics, including calculators as well as spreadsheets and a variety of other computerized tools. It is our responsibility to teach our students to use those tools, not as crutches but as routine extensions of their own abilities to cover more complex situations than they could cover without them.
When the Adult Numeracy Network, a group of adult educators working to improve the mathematical skills of adults in the workplace, surveyed working adults to find out what they needed to advance themselves and their careers, the question of using calculators actually did not come up. The adults simply assumed that calculators would be available, because they are available in every workplace, and workers are expected to use them. On the other hand, it was very clear to working adults that they needed to improve their knowledge of the mathematical fundamentals, especially is such areas as the use of fractions and percentages.
This is where the debate should end. There is not a choice between the fundamentals and the calculators. We must have both the fundamentals and the calculators. The teacher who forbids calculators is cheating his students, and the teacher who fails to stress the fundamentals is cheating her students in the same measure.
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Originally posted November 3, 1997.
Copyright © 2001, Center for Mathematics and Science Education. Teachers have permission to duplicate this page for use in teaching their own classes. All other rights reserved. You are welcome to link to this page, but do not copy its contents.
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Center for Mathematics and Science Education http://www.unc.edu/depts/cmse/MSEdNC/calculators.html |