54

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 "equal" in all respects - they can be rotated differently, or be in different positions, or be different colors, or have different names, or differ ...


38

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 -- much, much older) think that a triangle is a triangle. It's a polygon formed by three non-colinear points. A triangle has many ways you can think about it. ...


30

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 same", we mean that they are not just indistinguishable, but that they are literally the same exact thing. "Equality" means two numbers are the ...


27

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. Operators/Functionals Indefinite integrals are operators mapping functions to a set of functions or function (as representative of the equivalence class). ...


26

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 number is rarely used as a precise designation outside of school mathematics there is no agreed-upon rigorous definition of the term, and in fact it's largely ...


24

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 though I do not know its original source, so it may not be accurate in its details) could help their thinking out a bit: Substantial edit: I now no longer believe ...


24

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 the word sphere is particularly confusing in this context. (Mathematicians use sphere to mean the the two-dimensional surface; colloquial speech and some ...


23

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 should be introduced as such—emphasize that "real" in mathematics does not mean the same thing that "real" means in everyday vernacular English. ...


22

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 taxicab geometry would be a term I'd use for the subset of mathematics covering these types of scenarios.


21

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 are basic to our understanding of everything. Where mathematics comes from, by George Lakoff and Rafael E. Núñez. Basic Books, 2000. Exploring the role of ...


21

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 that's more substantive than "read an equation out loud" where there's only really one right way and not much opportunity for change - but, even on the small ...


21

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 undergraduates who have not had a theoretical math course. In my experience, students do not naturally think of geometric figures as sets of points. If $P = (-1,-1)$, $Q =...


20

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 there is an argument to be made for the term linguistically: a negative number is sort of the opposite of having a whole thing. But ultimately, there's not much ...


19

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 & [\text{apply $x \circ -$}] \\ y &= x \circ z & [\text{simplify the involution}] \text{.} \end{align*} I would be shocked to see anyone use that term ...


18

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 Latin radix. A longer quote from W. W. R. Ball, A Short Account of the History of Mathematics (4th edition, 1908): The algebra of Alkarismi holds a most ...


18

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. I'd be ok with a student's answer to "What is the y-intercept?" to be simply the y value, or the $(0,y_0)$ point. If a teacher prefers one, you can ask ...


18

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 theorem of calculus. So the course includes limits, the definition of the derivative, techniques and applications of the derivative including trigonometric and ...


16

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 applications of the ring or field axioms. Solving equations can be seen in the framework of algebraic or analytic geometry as studying the solution sets in some ...


16

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 large magnitude." If I am being a little less careful (which I often am when discussing asymptotic behaviour of, for example, rational functions), I'll generally ...


15

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 positive sense. If you say "obviously", then it has a message that "it better be obvious". And if it is for someone not clear, then it is an evaluation that this ...


15

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 to them at this point To intentionally make fun of something you know they thought was stupid (e.g. high school? if you're teaching a proofs course) And, ...


15

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 appropriate. I'd use compute only for numerical integration methods. evaluate and find are the two verbs that are used in textbooks and exams that I've come across. I ...


14

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 in favor of redundancy and details for such things.


14

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, remember that languages change and evolve as speakers of the language (for example) start using new words, stop using old words, alter pronunciation, and play with ...


14

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 numbers are packed in bunches of three or how decimal places are separated from the integral part). Focusing on a single standard will make it harder for students ...


14

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 show how you got your final answer if you want to get full credit for it. If the phrase "full credit" is not familiar to them, you may use "all the points" or ...


13

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 bird base or the water bomb base. Theorem: a folding procedure that produces something you wanted, such as the traditional crane. Corollary: a folding ...


13

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 $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$. link1, link2. From the links: The story goes that the Chicken McNugget Theorem got its name because in McDonalds, ...


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