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 ...
- 879
41
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 ...
- 7,195
39
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 -- ...
- 575
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 ...
- 510
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 ...
25
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 ...
- 5,719
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 ...
- 20.4k
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 ...
- 7,137
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 ...
- 879
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 ...
- 3,584
22
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 ...
- 4,858
21
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 ...
- 777
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 ...
- 8,846
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 ...
- 1,420
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 & [...
- 641
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....
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 ...
- 23.2k
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 ...
- 4,853
16
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 $am + bn$ ...
- 490
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 ...
- 7,137
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, ...
- 7,137
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 ...
- 5,719
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 ...
- 2,650
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 ...
- 10.7k
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 ...
- 11.5k
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 ...
- 8,883
13
votes
Vocabulary for giving just numbers, not a full answer
(From a comment)
On American math exams, I see the phrase "justify your answers".
- 7,195
13
votes
Use of language: "perfect square". is this useful or a hindrance?
Suppose the term "perfect square" was not in use.
Then when I asked you if
$x^2 + 6x +7$
was a square, you could conceivably say "sure, everything is a square:"
$\sqrt{x^2 + 6x +...
- 20.4k
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