Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
Technological innovation has been likened to a two-faced Janus. To take an extreme example, the taming of nuclear reactions gave humanity a cheap and abundant source of energy. It also created a means of previously unimaginable and total destruction. A less fateful example comes from the automotive industry. A car with manual transmission can be started (if parked downhill) even with a dead battery. To start the car, shift into the second gear, keep the clutch down, release the brakes, let the car accelerate, release the clutch - and voilą! the engine starts and off you go. No need to wait for a service truck. A nice feature especially valued by forgetful people known to leave their car with lights on. With the introduction of automatic transmission, this feature was lost. On the positive side, the automatic transmission gave more people the freedom of movement. (At the same time, it put more bad drivers behind the steering wheel, making roads more dangerous, which just serves to show how difficult it often is to separate the benefits from the drawbacks of a new development.)
Let's turn to the controversial question of using calculators in the classroom. The question is far from being settled, as becomes obvious on reading the latest issue (Volume 2, Issue 3) of the Mathematics Education Dialogues (an NCTM publication). The small volume is a collection of short articles putting forward arguments, both pro and con.
I'd like to make a twofold contribution to the discussion. First here is a calculator modeled after the simple calculator tool that comes standard with the Microsoft® Windows. (I expressly relieve Microsoft® from any responsibility for the performance of the model.) Later on, I shall share what I think is a very practical idea of a possible way to assess the success or failure of calculator use in the classroom.
First comes the calculator. This particular instance of the ubiquitous tool happens to malfunction(*). Please do not misconstrue my intention as an attempt to convey a sentiment to the effect that A good calculator is a broken calculator. Nothing would be farther from the truth. In fact, the opposite is true. Rather, the reader should see in it an expression of my belief that even a broken calculator may have its uses.
In the calculator below only a few buttons are functional at any time. Obviously, the calculator cannot serve its intended general purpose. Instead, the task is to somehow compute the integers from 1 through 15 with the limited means the calculator offers. The 15 numbers may be computed in any order. Those already computed will be listed in the space just above the calculator.
We learn in the Dialogues that the elementary mathematics curriculum may emphasize any or all of the paper-and-pencil, mental, or technology-based skills. The technological wonder of the broken calculator, while helping accustom children to new technology, fosters their mental skills, which I believe to be the most valuable of the three.
Why so? Simply because it's the handiest of all. Once you got it it's always there, it's always available. It is utterly impractical to be dependent on a technological gadget (paper-and-pencil or a more advanced calculator) to digest all the information that one comes across in one's daily life. Naturally, some amounts of information simply exceed the mental ability of an average student or a citizen. Somehow I can't worry about that. If need be, it's OK to seek friendly advice, have a peek into an encyclopedia, or make some other use of technology. But there is a minimal number of skills and basic knowledge that make a person numerate.
In his review of John Allan Paulos' Innumeracy (Vintage Books, 1990), Douglas Hofstadter wrote, "Innumeracy - the mathematical counterpart of illiteracy - is a disease that has ravaged our technological society." Please reread the previous sentence. Lamentable as the situation is, the terrible disease of innumeracy did not prevent our society from becoming and being technological. Put another way, in our society, technology and innumeracy live hand in hand.
To state the obvious, anyone who has reached the noble goal of mastering mathematics (with the use of technology or without), must have conquered along the way the common malady of innumeracy. The converse is not true. Elimination of innumeracy is therefore a more feasible goal, a necessary step on the way to achieving mathematical mastery.
My second contribution to the discussion on the use of technology in the classroom is that there should be one goal - to teach mathematics better - but two assessment practices. One is to assess mastery of the subject. The other is to assess the comfort level with the acquired mathematical tools. Let's call the latter numeracy assessment. The NCTM Standards recommend that Methods and tasks for assessing students' learning should be aligned with the curriculum. This pertains to the first kind of assessment. Numeracy assessment should be rather aligned with the age group and modest anticipation of students' knowledge. It should be carried out without any kind of technology.
For the sake of reference, here's one quote from Innumeracy (p 4):
and another from A.K.Dewdney's 200% of Nothing (John Wiley & Sons, 1993, pp 9-10):
As a suggestion for numeracy assessment, it is reasonable to expect that 4th graders be comfortable with the addition and multiplication tables. 8th graders must be able to handle the quotes from Paulos and Dewdney with confidence. Grade 12 students should be able to detect a typo in the following passage (G. Burrill, A Revolution in My High School Classroom, Mathematics Education Dialogues, 2 (3) p13):
(Numerate students should see at least 3 ways to correct the typo. This can be done by adding 1 symbol, or 2 symbols, or by lowering the center of gravity of the whole expression.)
Calculators came to the classroom in the early 1980. This is somewhat discouraging that almost two decades later there's still no consensus on their use. The reason, I believe, is in that the results of calculator usage are mixed at best. The 15-page Dialogues managed to present a spectrum of viewpoints, most of which lean on personal experience rather than a systematic study. In general, I responded better to those who tried to warn of possible pitfalls of calculator use, because what they said jibed with my personal experience. Numerate students I met and colleagues I worked with did not use calculators that much. Among students, calculator users were mostly innumerate. (Hence my suggestion to institute regular numeracy assessment tests.)
Not a single article mentioned my beloved abacus. And although I realized that it was all about calculators and not just any kind of calculating devices, I felt grouchy nonetheless. Wonderfully, one of the articles in the Dialogues (Randy Charles, p 11) supplied an explanation:
That's it. I felt grouchy because I first saw a calculator well past the graduate school.
The most congenial article Do We Need Calculators? by Kim MacKey quoted (p 3) Ralph Raimi: "Education is not imitation of life; it's an artificial process designed to put ideas into mind and not answers on paper." Hey, elementary school teachers, pay attention. The broken calculator may merit being looked into.
(*) I learned of the utility of broken calculators playing with a very entertaining piece of software, Cheops' Pyramid, that wraps a long list of mathematical activities in an adventure game format.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1996-2000 Alexander Bogomolny