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A long-ish wall of text, and I apologize.

Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A student piped up and said, "that's a rhombus." The professor stopped mid-stride, looked at him squarely, and said, "rhombus? That's a stupid word. What's a rhombus? I don't even think that's a word. The word I was thinking of was 'parallelogram'." This was shocking, because this was an American professor, at an American university, and in my American public education, I was taught what a rhombus was in the second or third grade.

Recently, however, I was thinking that maybe my professor wasn't wrong. Consider the naming system for quadrilaterals. The term "quadrilateral" makes some sense: "quad" from Latin for "four", and "lateral" meaning side. And then you get parallelogram, with "parallel" meaning "parallel" and "gram" from Greek meaning "drawn". But then a rectangle is a special case of a parallelogram where the angles are all right angles, which follows clearly enough, and a square is a special case of a rectangle, and important enough to merit its own term.

But then a quadrilateral with only two parallel sides is a trapezoid, which derives from Greek for "table shaped". And then a rhombus is the complement to the square in the special cases of parallelograms -- its angles are anything but right angles!

Confusing yet? We've got the following suffixes describing shapes: -lateral, -gram, -zoid.

We also have triangles, which makes sense because it's "three angles." Yet a "quadrangle" is a region in a university campus.

Increasing the number of sides in the shape, we go from "quadrilaterals" to "pentagons". Ok, now we've gone from the Latin prefix for "four" and a suffix meaning "side" to the Greek for "five" and a totally different suffix. Sometimes we describe the word using a root that means "drawn", and sometimes we describe it by the way that it looks.

And still "rhombus" fits in nowhere in this crazy, convoluted scheme!

To bring this all back to mathematics, and to ask my original question:

Individually, I can find the etymology of each of these terms. But why did the mathematics community adhere to these terms, particularly in elementary education? Did these terms get translated haphazardly from Elements? Is this one of those consequences of the somewhat insular nature of the mathematical community during the Renaissance era? The mathematics community has evolved to be fairly precise in its use of terminology. Why is the terminology surrounding elementary geometry so fragmented?

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Well, you could try to start speaking about "trigons" (triangles), "tetragons" (quadrilaterals), "equilateral tetragons" (rhombuses) etc., but I guess you'll only get blank stares (unless you're teaching, then it's your pupils who will get the blank stares later :-)) (Although "trigon" is already used implicitly in "trigonometry") –  celtschk Jul 24 '12 at 19:00
Well, that's the crux of my question: in many other branches of mathematics, there were competing terms to describe one thing (e.g. a "group" didn't always mean a group -- the nomenclature was standardized as abstract algebra matured as a field). "Equilateral parallel tetragon" is a perfectly suitable term for a rhombus, in my view. Corrections were made to the nomenclature standards for order of operations, arithmetic, and other elementary tools. Why not shapes? –  Arkamis Jul 24 '12 at 19:05
It has irritated me for years that a square is considered to be an example of a rhombus but is not considered to be an example of a trapezoid. –  MJD Jul 24 '12 at 19:21
Perhaps this site will be of interest: "Earliest Known Uses of Some of the Words of Mathematics" ( jeff560.tripod.com/mathword.html ) ... or Schwartzman's book "An Eymological Dictionary of Mathematical Terms Used In English" ( books.google.com/books/about/… ). –  Blue Jul 24 '12 at 19:33
If teachers did not have all these words (denominator, scalene, mantissa, distributive, etc.) to test students on, they would have to teach mathematics. –  André Nicolas Jul 24 '12 at 19:51
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2 Answers

up vote 6 down vote accepted

We also have, for example, add/sum/negation vs. multiply/product/reciprocal. As with natural language (be/is/was, go/went, speak/spoke), the oldest terms tend to be the most irregular, because they became established before the currently used structure emerged.

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In many cases it's not really that the oldest terms are irregular, as that they are survivors of earlier patterns that have been mostly lost as the language evolved. The commonest words tend to be the most resistant to change. –  Robert Israel Jul 24 '12 at 19:51
@Robert: Ah yes, this is true in most cases in natural language as well. "Historical linguists rarely use the category irregular verb. Since most irregularities can be explained historically, these verbs are only irregular when viewed synchronically, not when seen in their historical context." I've edited my answer to reflect this a little. –  Rahul Jul 24 '12 at 20:05
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I can't resist adding this, since it's been on my mind in recent months.

This vignette in the OP highlights part of a big problem with modern mathematics education: enshrinement of ideas and terminology as being written in stone. Another closely related thing is the unwillingness of textbook writers from departing from what has already been written. They apparently cannot dream of departing from the herd into more sensible pastures.

I acknowledge, of course, that it's necessary to select a vocabulary so that students can discuss things with you. For the sake of uniformity, teachers tend to stick with what they were taught, both because it is familiar and because other texts do it.

What this has come to in practice, though, is students being rigidly taught that "this is what you call it" along with the undertone that nothing else would be considered correct. As a result we have the patchwork of terminology to forcefeed children with.

I don't really think that this is a result of rigidity so much as it is an ignorance of what mathematics is about. The perception that mathematics is fixed or rigid causes teachers to treat it as such, whereas more mathematically experienced people recognize it is more like a canvas.

One annoying legacy of this insistance on sticking with tradition (that someone else mentioned in the comments) that (in the US anyway) we are stuck with thousands of primary school textbooks insisting that a trapezoid have exactly one pair of parallel sides. It has amazed me that there can be supporters of this position so entrenched, when their viewpoint flies in the face of the rest of the classification of quadrilaterals. (Another thing like this is insisting that kites have exactly two side-lengths, so that squares cannot be kites. This is more rare than the trapezoid thing, but it equally annoys me.)

This usually persists into the minds of secondary school teachers, and hence into their students' minds. Then, a student reaching college might discover that nobody in post-secondary education would entertain such a bad system, and they quickly have to relearn quadrilaterals according to the inclusive system. Or, more usually, the changes are just swept under the rug as stuff primary and secondary schools taught less than ideally.

Anyhow, it is difficult to see a good solution. Authors of textbooks are simply unaware of or unwilling to risk the transition to a 'new' system. Primary and secondary teachers apparently cannot (or will not) be persuaded en-masse that the 'new' system is more coherent and worth adopting.

But that's OK: it's not the worst problem education faces. Maybe overcoming these imperfections are important for developing mathematical minds.

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Great response! Part of learning higher mathematics is the periodic unlearning of things -- or rather, the re-learning of things once taken for granted. This problem is perhaps an extension of that. However, this impediment won't severely affect those with the academic prowess to retrospectively re-learn things once taught. I still think this hinders the development of those who might have potential otherwise. But, as you said, this is not the biggest problem that math education faces; it is more like a symptom. –  Arkamis Feb 11 '13 at 16:18
I love getting downvotes on answers like this because it amuses me to think such unreasonable people exist. (My theories on the downvote, in increasing order of probability: they think the US school system is perfect; they are a primary text author; they refuse to let rectangles be trapezoids; they are my angry petulant m.SE downvote stalker who enjoys doing such things.) –  rschwieb Feb 11 '13 at 21:46
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