Here’s a test question to see if you have calculator addiction: What is 180 – 40?
Did you feel an urge to reach for your calculator to compute that? If so, please continue reading this post.
I was actually working with a student who had exactly that computation to perform. The problem was to find the measures of the base angles of an isosceles triangle with an apex angle measuring 40 degrees. This student always reached for her calculator to perform any computation, and she did the same here. However, she made a calculator typo and punched in “180 – 50 =”. She proceeded to get the wrong answer of 65 degrees for the measure of each base angle.
After having observed this student reach for her calculator for every single computation, I asked her if she really needed her calculator. She said, “yes.” I asked her to try to do the problem without her calculator. She protested, “I can’t.” I insisted that she give it a try. We spent about 3 minutes going back and forth like this. Finally, she gave in and tried without the calculator.
I was expecting to see her substitute scratch paper for her calculator and set up the subtraction problem 180 – 40. But to my surprise, she figured out the answer in her head and said, “180 minus 40 is . . . 140?” and then proceeded to get the correct answer of 70 degrees for each base angle.
“You’re better without your calculator!” I announced.
“What do you mean?” she countered.
“Well, you got the wrong answer when you used the calculator.”
Okay, that last bit might have been a tad facetious, but, in this case, I think it is true that this student would fare better if she left her calculator at home. And, there are important reasons for this student to try to avoid her calculator beyond the simple fact that she was faster and more accurate without it.
What is education all about, after all? I like to make education about improving one’s ability to think and learn. Mathematics is a fabulous subject by which to achieve those goals. Errors in one’s thinking reveal themselves in contradictions and the subject offers material to think and wonder about that provides a super-astronomical spectrum of difficulty levels.
To get better at thinking, a good strategy is to find things to think about that are challenging, but not impossibly so. Also, higher-level thinking ability requires competence at lower levels of thinking ability. So to reach a higher-level of thinking ability where one is not yet capable, it’s a very good idea to work with and master concepts and problems that require a lower level of thinking ability. Furthermore, abstractions in mathematics will not be meaningful without knowledge and understanding of the concrete examples which inspire the abstraction. And so many abstract concepts trace their roots to properties of numbers.
Therefore, playing with numbers sets the stage for many more advanced concepts and avoiding this number play will make it harder, perhaps even impossible, to achieve a sophisticated understanding of mathematics.
The human mind has this remarkable ability to notice things and make itself more efficient at doing things if given the chance. Reaching for that calculator on every computation is depriving the mind from having the chance to notice and learn useful ideas that may seem very tiny, but form the building blocks of ever grander concepts and abstractions.
Every arithmetic computation can be viewed as a tedious chore…or it can be viewed as an opportunity to notice something, reaffirm something, test something, refine something, or practice something.
For example, let’s take a look at that computation the student confronted: 180 – 40. Noticing that one can ignore the 0’s and compute 18 – 4 to get 14, then tack on a 0 to get 140 is something that is worth noticing. It may seem awfully small, but these awfully small observations are like little grains of sand that fill in the holes of our understanding. If one does a lot of computations, perhaps this observation will first be made empirically. Soon, a point in time will come where one wonders why it works, or if it really always works that way. This wondering will make the distributive law a friend. But the key point is that this kind of observation cannot be made if one always reaches for the calculator.
I’d like to give 2 real-life examples to show how calculator addiction can be holding you back.
Example 1. The problem: Concentric circles of radii 1, 2, 3, 4, and 5 comprise a target for a dart game. What fraction of the dartboard is contained within the 3rd ring?
I watched as a student reached for a calculator and punched in the following:
ENTER STORE A ENTER.
ENTER STORE B ENTER.
A / B ENTER ToFRAC ENTER.
If this student uses this calculator approach to solve such problems for the rest of his schooling years, he’ll miss out on seeing live examples of the power of simplification, for a student more adept at math might write
and see that the ‘s cancel. Entering 3.14 into the calculator is thus proven to be a total waste of time. It’s just more opportunity to make a mistake. However, to be able to notice the cancellation, one must think about the computation before jumping for that calculator. There is also the matter of using an approximate value for , which calculator addicts seem to do as a matter of habit, so much so, that I have met some who seem to believe that . Even more, computations like this one are instances of the general geometric concept that the ratio of the areas of similar figures is equal to the square of the similarity factor. By actually doing the computation without a calculator, the student is creating a bed of experience onto which the more abstract geometric concept can rest soundly.
Example 2. This example comes from an exercise that a student was given as part of a lesson on arithmetic progressions: Compute 1.2 + 1.6 + 2.0 + 2.4 + 2.8.
Again, I watched a student punch into a calculator:
1.2 + 1.6 + 2.0 + 2.4 + 2.8 ENTER,
and move on never noticing, or needing to notice, that the numbers were in arithmetic progression, thereby defeating the point of the exercise in the given context.
If this student were forced to do this computation without a calculator, probably the student would evaluate the expression left to right, adding the numbers one by one in that order. That would no doubt feel tedious. But if done properly, a series of such “tedious” questions could be posed that would induce the student to prefer to seek a pattern, and when they see it, the point of the lesson will really sink in, along with possibly a deeper understanding of the notion that the sum of a list of numbers does not depend on the order in which they are added up. A really strong experiential understanding of that principle could translate into a profound psychological difference in reaction upon hearing the commutative law stated for the first time. To someone who always reaches for that calculator, the reaction might be a sense of unfamiliarity and discomfort with the abstraction (“there’s nothing to put into a calculator…”). But to someone who has knowingly or not used the law to aid in computation, the reaction might be one of familiarity and even the feeling that the statement is “obvious.” And these kinds of psychological differences can mean the difference between going much further in one’s study of mathematics and quitting it as soon as possible.
The calculator can be a very useful tool, to be sure. But if you are using your calculator to avoid thinking about things, hide your calculator and remove its batteries because while your calculator may be saving you time for the moment, it’s costing you huge amounts of time in your future and it is preventing you from playing with things that are just what you need to play with to prepare yourself for more advanced mathematics.