56 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 ...
Nuclear Hoagie's user avatar
42 votes
Accepted

Is there a canonical name for a polynomial-like expression allowing for negative powers?

There is also the term "Laurent series", where we allow an infinite number of terms... $$ \dots +5x^{-3} + 2x^{-2} + 3x^{-1} + 2 + 4x - 7x^2+\dots $$ So perhaps yours is a "finite ...
Gerald Edgar's user avatar
  • 7,369
41 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 -- ...
Flydog57's user avatar
  • 595
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 ...
Rivers McForge's user avatar
26 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 ...
Chris Cunningham's user avatar
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 ...
R.. GitHub STOP HELPING ICE's user avatar
26 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 ...
Dan Fox's user avatar
  • 5,728
24 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 ...
Xander Henderson's user avatar
  • 7,520
24 votes
Accepted

How can I validate the existence of percentages above 100?

The formula to express a value $X$ as a percentage of another value $Y$ is simply $Percentage= (X/Y)×100$. It is simply a ratio of numbers (a multiplicative factor), times 100. There is absolutely ...
Nuclear Hoagie's user avatar
23 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 ...
David E Speyer's user avatar
22 votes
Accepted

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 ...
Milo Brandt's user avatar
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 ...
Andrew Sanfratello's user avatar
20 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 ...
user52817's user avatar
  • 10.3k
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 ...
Acccumulation's user avatar
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 & [...
Eric Towers's user avatar
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....
JTP - Apologise to Monica's user avatar
17 votes

Examples of Mathematical Slang

One of the most colorful names I have heard is the Chicken Mc Nugget theorem: for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ ...
Federico Poloni's user avatar
17 votes

Is there a canonical name for a polynomial-like expression allowing for negative powers?

You can call it a "polynomial in $x$ and $x^{-1}$" or a "polynomial function of $x$ and $x^{-1}$". The idea is that you are taking a two variable polynomial $p(x,y)$ and then ...
Steven Gubkin's user avatar
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 ...
Xander Henderson's user avatar
  • 7,520
15 votes
Accepted

What's it called when multiple concepts are combined into a single problem?

The answer I am about to give is a little tangential, but I think that it will help to answer the question "How do I communicate to students that the problem I am setting for them is going to ...
Xander Henderson's user avatar
  • 7,520
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, ...
Xander Henderson's user avatar
  • 7,520
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 ...
Dan Fox's user avatar
  • 5,728
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 ...
Jasper's user avatar
  • 2,699
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 ...
JRN's user avatar
  • 10.8k
14 votes

The term "unique" for functions and operations

I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way. Students have trouble with the notion of a function because it's hard. The way ...
Henry Towsner's user avatar
14 votes

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

Educator here who has worked with many students in the aforementioned age range (9-13) on triangles and squares. In my experience, it has never come up that a confusion between the boundary and the ...
Rivers McForge's user avatar
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 ...
DavidButlerUofA's user avatar
13 votes

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

The standard that you link to (ISO 80000-2:2009) seems to be not available for free. That is, in order for me to follow the standard, I have to be able to read it, and in order for me to read it, I ...
JRN's user avatar
  • 10.8k

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