55 votes

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

Colloquially, there's a lot of conceptual overlap between all of these terms, but "sameness" is not a well-defined mathematical property. Congruent shapes need not be "the same" or ...
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38 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- ...
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  • 525
30 votes

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually "the same" or "equal". Usually when we say two things are "the ...
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27 votes
Accepted

What is the proper verb for "doing" an integral?

I'm no native English speaker, but you can tackle that question from the mathematical point of view as well. The best verb depends on how you view the nature of definite and indefinite integrals. ...
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  • 3,362
26 votes

Why don’t American school textbooks recognize negative numbers as whole numbers?

I’m more curious about incorrect things in them. Yet, this is the first thing I found. There's absolutely nothing "incorrect" about this. As Dave L Renfro noted in a comment: and whole ...
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24 votes
Accepted

Why are $m$ and $b$ used in the slope-intercept equation of a line?

This is also borderline not-an-answer, but it might be a nice broadening of your students' worldview to know that the "$m$" and the "$b$" are not universally accepted. Showing them this map (even ...
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  • 19.1k
24 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular ...
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  • 5,083
23 votes
Accepted

Should high school teachers say “real numbers” before teaching complex numbers?

Short Answer You should not avoid use of the term real numbers. This is a term-of-art in mathematics, and it is important for students to learn the correct jargon. However, this technical term ...
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22 votes

Is there a name for paths that follow gridlines?

Generally, this math falls under the scope of what is commonly called Taxicab Geometry. I would use taxicab path as a noun to describe the specific paths illustrated in the original question; whereas ...
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21 votes
Accepted

Metonymy in mathematics

Metonymy and its relatives, metaphor, polysemy, synecdoche occur all over the place in mathematical writing, and sometimes cause students problems and sometimes don't, because those thought processes ...
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21 votes
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Allowing nonstandard mathematical language and/or notation

I think, while teaching, the principal way to judge mathematical language is not whether it's standard, but whether it's effective communication. This difference applies principally to communication ...
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21 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for ...
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20 votes

Why don’t American school textbooks recognize negative numbers as whole numbers?

I don't think that "textbooks" decided this, usage did. The term "integer" covers positive and negative, so it would be redundant for whole numbers to refer to that category. And ...
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19 votes
Accepted

Does this property of subtraction and division have a name?

This is "left involution". ("left" because it doesn't work when you try it on the right.) \begin{align*} x \circ y &= z & \\ x \circ (x\circ y) &= x \circ z & [...
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18 votes
Accepted

Where does the word "roots" come from when talking about zeros

Doctor Peterson in The Math Forum refers to the following sources. From D. E. Smith, History Of Mathematics Vol II (1925), footnote page 393: The Arabs also used jidr (dyizr, root), whence the ...
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18 votes

What is an intercept?

This is a case where you might be looking for a distinction that's pretty subtle. By definition, the y-intercept occurs at x=0. In one notation, it's literally f(0), where the x is explicitly offered....
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18 votes
Accepted

Is there a more telling name for "Calculus 2"?

In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, "Calculus 1" typically refers to single variable differential calculus up to the fundamental ...
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  • 7,585
16 votes

Are the words "easy," "basic," "clearly," "obviously," etc., ever helpful?

I can think of a few instances where it might be useful: To situate the current piece of the concept among others coming up To call-back to something earlier in the course that really should be easy ...
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  • 1,950
16 votes

Difference in meaning of 'algebra'

Elementary algebra is the study of a few specific rings: integers, rational numbers, polynomials with rational coefficients, and rational functions. All the rules for "rearranging" equations are ...
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  • 4,765
16 votes
Accepted

What is a recommend way to describe a negative number with large absolute value?

It depends a little on context and how careful I am trying to be: If I were being very careful, I might call such a number "a negative number with a large absolute value" or "a negative number with ...
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15 votes

What is the proper verb for "doing" an integral?

I would avoid the verb solve as I reserve this for things like equations, inequalities and problems. An integral is equal to a number or a function, so verbs like find, evaluate etc are more ...
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  • 1,912
15 votes
Accepted

The word "and" rather than "or"

Express $\cos(x + \pi)$ in terms of $\sin x$ or $\cos x$ (possibly both). In my opinion, the "or" is logically more correct and the parenthetical recalls/stresses/clarifies this. In general, I am ...
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  • 7,582
15 votes

Are the words "easy," "basic," "clearly," "obviously," etc., ever helpful?

The main point here is that these words/expressions should not be used as substitute for an argument. They obviously have some negative effects: You evaluate your students by them and not in the ...
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14 votes

Why isn't the term *inequation* widely used in English?

You ask "Why isn't the term inequation widely used in English?" The answer, however tautological, is that the term isn't used in English because it isn't used in English. Perhaps more usefully, ...
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14 votes
Accepted

Should I describe the function $x \mapsto f(x_0) + f'(x_0)(x - x_0)$ as "linear" in a freshman calculus class?

In calculus one calls $x \to ax + b$ a linear function. In linear algebra one calls $x \to ax + b$ an affine transformation, and says that it is linear only if $b = 0$. The first order Taylor ...
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  • 5,083
14 votes

In teaching mathematics, should one always follow some international standards such as ISO 80000-2?

No. This standard may be useful for professionals in international settings. Most teaching happens in smaller, localized settings and things will differ from country to country (e.g. how large ...
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  • 2,182
14 votes

Vocabulary for giving just numbers, not a full answer

The problem with the word "solution" is that it could mean the final answer or it could mean how the final answer was obtained, so I suggest that you don't use it. How about the following? Clearly ...
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  • 10.2k
13 votes

What could be good non-mathematical analogies to explain the difference between the words theorem, proposition, lemma and corollaries?

Origami has things like lemmas, theorems and corollaries. Definition/axiom: The basic folds such as book, mountain, valley etc Lemma: a folding procedure that is used as part of another, such as the ...
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