I really never thought very much about my students’ mindsets. I knew that they might not “get” things immediately, but I always expected that they would consider my explanations with an open attitude. I guess that I subconsciously viewed students who raised questions or objections as being skeptics rather than deniers. For the most part, my subconscious assumption seems to have been correct. But on some rare occasions, the skepticism ran so deep that it really was denial.

I’m pretty sure that you have probably run into this one. When first confronted with the infinite repeating decimal 0.999 . . . , most students think that it must be less than 1.0. When you tell them that,

*if you are willing to accept that*0.999 . . .

*is actually a real number*,

*then is has to be exactly the same as*1.0, the skeptics among them are waiting anxiously to be convinced. They are willing to consider your arguments (and there are several convincing ones). The deniers, however, are absolutely certain that the numbers are different.

Their denial is almost surely mathematically based (usually because of the belief that there is an “infinitely small number” between 0.999 . . . and 1.0). Even though the denial may be very strong, and even though you may not be able to fully convince them, it is still possible to have a fruitful mathematical conversation about it. [As an aside, it is not far-fetched to talk about infinitesimals. Although they do not live within the real number system, there are number systems that do allow for them, and some mathematicians have used them quite routinely. For the most part, however, we can get along quite well without invoking the existence of infinitesimals.]

There is another type of denial that is somewhat deeper. I encountered it in a Calculus course when I showed how Willard Libby used a simple differential equation to help find the age of the remarkable prehistoric paintings found in the Lascaux caves in France.

I don’t want to give a Calculus lecture—suffice it to say that a Calculus student would not have difficulty with the math. The upshot of the lecture was that the solution of the differential equation using the C-14 measurements provided by Libby showed that the wood in the fire was 14,000 years old give or take several hundred years.

After going through what I thought was a very clear and convincing presentation, one student proclaimed that I must have made an error, because “

*How can this be possible, since the Earth itself is only 6000 years old?*”

This sort of denial is of a different order altogether. I don't believe that it is possible to have a mathematical conversation about it. As far as I know, there is no common ground between mathematics and young-earth creationism.

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Addendum April 8, 2015

Two posts about overcoming a fixed mindset that you might be interested in are by cheesemonkeysf and Heidi Siwak, found respectively at

cheesemonkey wonders

and

The Amaryllis

Web references:

A proof that 0.999 . . . = 1.0

Lascaux cave paintings

James Lunney

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Q: How many mathematicians does it take to screw in a lightbulb?

A: 0.999999 . . .

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