Calculators in the Math Class
The Singapore government realizes that its future prosperity depends not on educating its people in the knowledge and skills for a particular kind of economy, but in developing its peoples capacity for learning and dealing with change so they can respond quickly and flexibly, adapting and retraining, as future economic opportunities or recessions arise (p. 20).
All children must be properly prepared for the knowledge society and its economy (p. 21).
Quotes from Andy Hargreaves’ Teaching in the Knowledge Society. The students names have been changed in this essay.
Taking these into account, coupled with my experience as a tutor for Jane and Jill, I want to focus on one rather small but significant fact in the way these two students have learned math. Jill has little experience with a calculator. She does a lot of the arithmetic she encounters by hand. Jane relies on a calculator so much that she was reluctant to do a relatively simple multiplication problem on paper.
As a future math teacher I have my concerns about the effect calculators have on the students ability to do math in their heads or on paper. I am afraid, and Jane is an example of this, that students will become so dependent on the calculators and machines to do the work that they won’t be able to form the most basic of mathematical concepts. Jane has struggled with fractions and percentages. According to the Oregon Department of Education standards she should have a solid understanding of fractions and percentages by the end of the eighth grade. In this small respect she is over three years behind. Why? She lets the calculator do the work. She doesn’t know how it works or why, she only knows how to plug the numbers in and press the enter button. She lacks the basic concepts. If she knows them, they are buried deeply under the crutch of the calculator. Jane is not the first student I have worked with who relies on the calculator so heavily.
The calculator as a crutch will not help any student survive in the knowledge society. Information is the bread and butter of the economy. Information is so prevalent that there are companies whose business is purely focused on information about information: ‹metadata.'
Writing about the dangers of calculators may seem like sour grapes on my part, after all I didn’t grow up with powerful calculators and I had to memorize addition and multiplication tables and use them throughout my academic and professional careers. Hargreaves points out that we will have to learn to teach using methods that we were not taught with (p. 24), but this is a skill that I don’t think we can give up. They are foundational in mathematics and need to be learned. I also realize that as a math teacher my job will be to teach students how to use calculators. I don’t have any objections to students using calculators when they have the conceptual framework of what they’re doing behind them, but the calculators allow students to remember only the processes, and those are quickly forgotten when they are not called upon. In fact my students don’t have a clear idea about how to translate the process from paper to the calculator, usually getting an answer of 27 instead of 15 when reducing the fraction (75/(3+2)). Building procedural knowledge on top of procedural knowledge doesn’t last, either, it seems.
I can forgive anybody who can’t do a problem like 647.25 times 13.62 in their head, in fact Miller’s number lets us (as educators) know that it is practically impossible. It also takes some time to do it on paper, but even the cheapest calculator will provide an answer in a fraction of a second. If the person who needs this information (8815.545) for whatever reason and doesn’t understand the concepts involved, they won’t even know what to plug into the calculator. Most of the features of the common graphing calculators we make every student purchase are ignored anyway. The manuals for the latest line of Texas-Instruments’(tm) are larger and heavier than the calculators themselves, but the calculators get used minimally.
Estimation is another problem exacerbated by calculator dependency. If someone is faced with needing to order 23 widgets and they cost $74.99 each, they should (according to Oregon State Benchmarks at least) be able to estimate a cost of $1,875, which is a high estimate but it is what the situation calls for. People who are calculator dependent can’t even reason this far because they’ve given the calculator all the authority over their own ability. Most advertisers who talk about price hope that their audience doesn’t have the number sense to know that at some rates and with some “low payment options” the price of the thing they are buying could double.
Any sufficiently advanced technology is magic. The knowledge society cannot afford people to rely on magic. As technology grows more and more complex, the more it is misunderstood by the general populace and there are fewer people who know what makes things tick. Hargreaves lists items that he believes teachers should avoid on pages 79 and 80 and these certainly have a negative effect on their students. I would add ‹overdependence on technology’ to that list and have teachers teach their students the same thing.
I am not a proponent of removing calculators from the mathematics curriculum. I am a proponent of keeping calculators in their proper place as tools for the student to use. They do not do much to help students learn the concepts of math, all they can do speed up grunt work. That is their biggest strength. Calculators and computers can give teachers and students more time to explore mathematical concepts. They make lousy substitutes for the real thing.