Republic of Mathematics blog

Should mathematics teachers tell the truth?

Posted by: Gary Ernest Davis on: April 22, 2011

Adults do not always tell the truth to children.

It’s common to let children believe there is a tooth fairy, that Father Christmas brings presents at Christmas time, or that a Golem defended the Jews in Prague.

Are there similar tales that mathematics teachers tell children?

Does mathematics have fibs, lies, and myths?

I will let you decide. But here, first, are some candidates for myths of mathematics:

(*) When trying to divide a circular region into 3 equal pieces, having made a radial cut from the center of the circle to the circumference, we know where to make the next cut.

Often we do not: we are simply guessing, or approximating by measurement, such as using a protractor marked in degrees. Of course, if we do know how to exactly construct an angle of 120 degrees then we can figure out how to make that next radial cut.

But we can push this a bit further: having a rectangular candy bar we know where to make the first cut to divide it into 3 equal pieces. Generally we do not, not unless we know the ancient Greek method for dividing a line into equal pieces using parallel lines. Be honest: do you know how to do this?

(*) We know how to add, subtract, multiply and divide decimal numbers.

No we do not – at least most of us do not. A procedure for doing arithmetic on decimal numbers was described in the paper: F. Faltin, N. Metropolis, B. Ross & G-C Rota, The real numbers as a wreath product, Advances in Mathematics 16 (3), 278-304 (1975).

This is not an easy paper and I would be tremendously surprised if many university or college instructors understood the ideas in it, let alone high school or elementary teachers. The problem, of course, is the infinite carrying we have to do when we try to do arithmetic operations on infinite decimal strings: without converting to fractions, how would you multiply 0.428571428571428571… by 0.384615384615384615… ?

(*) Decimal fractions are fractions written as decimals.

No they are not if we take the common school understanding of a decimal fraction as a finite decimal. With this understanding, decimal fractions are those fractions that, in lowest terms, have a denominator of the form 2^m\times 5^n.

(*) We know the numerical value of the area of a unit circle.

No we do not. We know a name for it: \pi, and we know the decimal expansion of \pi to any decimal places, but we do not, and probably never will, know the exact numerical value of \pi as a decimal number; that is, its exact position on a number line.

(*) Polynomial algorithms are fast.

No they are not. Try a \it{O}(n^2) algorithm on a large number of points, and see how slow it really is.

(*) We understand continuous functions.

No we do not. Continuous functions can exhibit hideous pathologies too horrible to even describe.

(*) Real numbers are well described as infinite decimals.

No they are not. We can only “describe” very limited classes of infinite decimals. Most behave more or less randomly.

(*) Randomness is a well-defined idea.

No it isn’t. We can test, through a bunch of sensible tests, if some sequence of numbers isn’t random, but we have no definition of “random”, despite its constant use in probability and statistics.

Then there are pedagogical myths:

(*) The idea of a variable doesn’t need explanation.

In fact, the foundations of mathematics do not contain “variables”. In that sense the idea of a variable is an extra mathematical concept imposed on the practice of mathematics. No “thing” can be variable, otherwise it would not be a thing. We say things like “the temperature is variable”. No it isn’t. The mercury in a thermometer varies, but we construct a “thing”, called “temperature” and then declare it varies. The most famous variable is “it”: it is raining, it is cold, it is hot, it is sunny, …

We often say “x is a variable”, but it isn’t: no matter how often we write it, it is still “x”.

Sometimes people say “x varies over the real numbers”. What does this mean? How is “x” varying?

Other people will say “x is a number”. Is it? If so, is it bigger than or less than 3?

Others will say “x is a container for a number”. If that’s the case then what is varying?

It’s a little confusing for a beginner. Variables are sophisticated ideas, not at all simple.

(*) Parallelism is an intuitive idea.

In fact much research shows that young children have great difficulty understanding parallelism. We might want to say the perpendicular distance between parallel lines remains constant, but it turns out that young children also have difficulty with perpendicularity.

(*) All students can see the lines in a grid.

In fact they cannot. Refer to this post for a fuller explanation.

(*) Students can see aspects of geometric figures if those features are pointed out to them.

Generally they cannot. Seeing is an active mental process, not just a mater of letting light fall on one’s eyes. Seeing geometric  properties and features is a process that requires active intelligence from a student, such as answering questions about a geometric figure when asked by others. Just “looking” is not enough to allow students to “see”.

There’s a lot more that could be included in a list of mathematical and pedagogical myths or fairy tales.

If you have any of your own, or are in violent agreement with mine, I would like to hear from you.

Shouldn’t we be doing our best to be truthful to students?









8 Responses to "Should mathematics teachers tell the truth?"

>In fact, the foundations of mathematics do not contain “variables”

You are mistaken somehow. Variables are a very fundamental part of the foundations of mathematics. Namely, first-order logic.

Thanks for this comment Xamuel.

I was expecting for someone to say this, and I did not want to deal with it at length in the post itself.

Of course in first-order logic we declare that certain symbols, or syntactical marks, are variables.

But from a syntactic point of view these signs do not vary in any sense: they always sit there as x1, x2, ….

The variation is in our minds, which is were the origin of variables lies.

We can claim to “substitute” something for a variable expression, but all we do syntactically is replace one sign with another.

I guess the best I can say is that, like Thurston, I believe mathematics is a branch of psychology, and that to really understand ideas like “variable” we have to look to human psychology, not to mathematics itself.

My point in the post is that this is not a simple or trivial task, and many students have trouble understanding what a variable is supposed to be.

Hmmm . . . But certain symbols (variables) can have something substituted for them, while others (constants, connectives, quantifiers) cannot. Yes, substitution is syntactic replacement, but variables also play an important part in proofs (which are syntactic processes).

Sure they can, Ben.

I can substitute 2/4 in the place of .5 for instance. And .5 is a constant.

but we do not, and probably never will, know the exact numerical value of pi as a decimal number;

pi doesn’t have a terminating decimal representation, so we definitely won’t ever, however there are a plethora of different representations of pi

Thanks Tony.

Someone commented to me that it wasn’t clear who was the intended audience for this post, and I should have made it clear that it was largely for elementary and secondary teachers of mathematics, some – perhaps many – of whom believe that we do know the exact decimal representation of pi.

In “(*) Decimal fractions are fractions written as decimals,” did you mean, Gary, that n=m in the denominator?

Oh – and on chatboard there was a discussion where a teacher was telling students that 7/8 is 0 – because they don’t understand decimals yet!

No – anything with a power of 2 in the denominator will terminate. Similar powers of 5, or multiples of powers of 2 and multiples of 5.

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