Little White Lies

A little inaccuracy sometimes saves a ton of explanation.

—H. H. Munro

I have a confession to make: I lie to my students. Not only that, I do it all of the time. More scandalously, I’m not the only teacher that lies to their students. We might not be surprised to learn this of bad teachers, but even the best teachers often lie. In fact, the best teachers tell the most outrageous lies! Teachers just can’t help themselves—they are compelled to lie to their students.

This may seem like a bold assertion which casts doubt upon the ethics and morals of educators, but the reality is that it is necessary to lie to students. By telling lies, teachers can eliminate extraneous information, or create a simpler model of a complex idea. Little white lies here and there can save an instructor hours of explanation, and ultimately give students a far better understanding of the concepts under discussion. Nowhere does this seem to be more true than in the study of mathematics.

The world of mathematics is full of complex and abstract ideas. Consider even the most basic of all mathematical concepts: a number. What is a number? Do numbers really exist, or are they just an abstract concept? Can we see a number, or hear it, or hold on to it? These are difficult questions. Great minds, including Plato, Descartes, and Kant, struggled with them. Mathematicians have spent a great deal of time creating highly technical answers for them. A deep understanding of what a number is requires not just deep mathematical study, but a fairly broad understanding of philosophy, history, and neurology.

Of course, when numbers are first taught, we don’t expect our students to have knowledge in any of those fields. In fact, when students first learn about numbers, it is probably from their parents while they are still in diapers. In order to give children a solid foundation, we have to lie to them about something as fundamental as the nature of numbers. If we didn’t, it would be almost impossible for them to ever make any progress.

In mathematics, the most common lies are those that are told in order to simplify a problem. Instead of presenting students with a complete explanation of a phenomenon or process, we omit details or alter the model in order to make the object under study more easily comprehensible.

For example, consider the real numbers. To get a little technical for a moment, the real numbers are a field. This means that we can add and multiply real numbers, that every real numbers has both an additive and multiplicative inverse (think negative numbers and reciprocals), and that both addition and multiplication obey certain rules like associativity and commutativity. If that doesn’t make sense to you, don’t worry about it—the important point is that numbers and basic operations like addition are actually rather complicated.

When we first introduce students to the field of real numbers, we ignore most of them. We don’t talk about negative numbers, or irrational numbers, or rational numbers. In fact, we limit the discussion to natural numbers (i.e. the counting numbers 1, 2, 3, &c.). This is a lie of omission.

Using the natural numbers, we teach students how to count objects. Once again, we are omitting important ideas. For instance, natural numbers can be used for more than just counting things. Of course, such a discussion would require abstract thinking skills that most kindergartners don’t have, so we elide it altogether. Once again, we are lying by omission.

Once students get to be expert counters, we introduce addition. We tell students that addition is used to put two groups of objects together. It is a way of counting different groups, and getting a total count for the number of objects contained in both groups. This is more or less consistent with the underlying theory of addition, but is, once again, a simplification. Another lie.

Then comes the first outright fib: we teach students to subtract. In axiomatic set theory and abstract algebra (the areas of mathematics that give rise to the elementary manipulations taught in elementary school), there really isn’t a subtraction operation. What we think of as subtraction is just another form of addition—specifically, addition of the additive inverse. We teach students to perform an operation that, in a highly technical sense, doesn’t exist! There’s an awfully big lie.

Of course, the other problem with subtraction is that it is introduced before students are told about the existence of negative numbers. This is why subtraction isn’t initially taught as addition of the inverse, as negative numbers are the additive inverses of positive numbers. This also means that taking a larger number from a smaller number doesn’t make sense. We lie again, and tell students that it is impossible to subtract a larger number from a smaller number.

Obviously, mathematics education is full of lies. By the time students are in the first grade, we have already told them an uncountable number of little lies about numbers and operations. But is this a bad thing?

Of course not!

In telling these lies, we have given students a simplified model to work with. It is like putting a child on a tricycle, then a bicycle with training wheels, then finally a bicycle without training wheels. Just as it would very difficult for a child to simply get on a bicycle and ride, it would be quite difficult for a student to learn mathematics from the ground up in all of its gory detail. It is not only common for teachers to lie to students, it is a good and necessary thing!

And here is the real moral of the story: teachers have to lie to their students. It is an unavoidable fact of reality. The one of the differences between teachers is that the good ones know that they are lying. Not only that, they know why they are lying, and they are ready, willing, and able to provide a more truthful or detailed explanation if the students ask for it and are ready for it.

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