79
A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol "=" is essentially operational, not relational — they think it means "the next step" or "the answer" or is an instruction to perform some operation. Knuth et al. ("The importance of equals sign understanding in the middle grades", Mathematics ...
49
It's purely a matter of how we choose to define the notation. The main reason for it is that it lets us write polynomial expressions (which are extremely common) without parentheses, e.g., $x^3 + 3x^2 y - 41x + 2z$ rather than $(x^3) + (3(x^2)y) - (41x) + (2z)$.
However, what really matters is that the notation is clear and unambiguous, so expressions like $...
45
Unrelated to US, in Germany the notation through school and university was quite consistent:
$\ln$ using base $e$
$\log$ using base $10$
$\log_x$ using base $x$
I may not know enough about the US education system, but if the definitions are not clearly homogeneous throughout the country, I would teach all three at school plus add a note that in some places ...
32
The different notations are to a considerable extent outgrowth of slightly different ways of looking at "the derivative." To me these different ways is the important thing to discuss, the notation is a by product. And, I think it is quite useful to spend some time on discussing these different ways and to mention the different notations in that context. (...
29
A colleague of mine includes on all his tests a line that reads (something like) "you will be graded on what you actually write down, not what I think you may have meant by what you wrote." He spends time carefully discussing what this means (examples like the one you gave are among them) before and after testing times. Normally after the second round (of ...
29
It is actually wrong to say that parenthesis means multiplication. In $(2)(5)$ it is the lack of an operator between the parenthesis that implies multiplication, NOT the parenthesis. The parenthesis are "needed" because $25$ means the number twenty-five, the parenthesis are purely for grouping.
"No operator means multiplication" is an extremely common ...
25
To use $$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx}$$ is mathematically correct if $dx$ is the name for a real variable. (If it should be something else it needs to be made clear what it should be.)
It would also be correct to say
$$f'(x)=\lim_{\text{small}\to 0}\frac{f(x+\text{small})-f(x)}{\text{small}}$$ with the understanding that $\text{small}$ is the ...
25
Personally, I don't think we attend to this sufficiently in lower-level mathematics (where it's actually needed most). Students need that vocabulary to interface with books, future teachers, tutors, other students, etc. I run questions on it in weekly quizzes; and if I had my druthers, it would be a major component of all tests (in addition to application-...
24
Consider the following observations:
People shorten things with increasing frequency of usage. For example, the most frequent words are short. This is a way of making communication more effective, e.g. see Huffman coding. For example this list has its first 5-letter word at 39th place (first three are for comparison):
$$\begin{array}{c|c|c|c}
\textbf{place} ...
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
I think that, depending on the maturity level of the students, you could just talk to them about why writing $\frac{1}{3}x$ rather than $1/3x$ makes it much clearer what you mean. They should understand.
Remind them that the purpose of writing anything, including mathematics, is to clearly convey an idea to the reader, and the notation $1/3x$ is ambiguous: ...
24
You might remind them that $y$ is just a name for a number. When they draw a plot, they draw a bunch of points: maybe $y=3$ here, $y=5$ there, and $y=-2$ over there. But at some point (no pun intended) we want to talk about the entire shape: we want to say that $f$ is symmetric, that $f$ is concave, that $f$ has an asymptote. We can't do that with $y$; ...
23
Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples:
The function that maps every non-empty list to its first element.
The function that maps every finite set to its size.
The function that maps color names to RGB triples.
The function that maps days to sunrise times at a ...
22
You are wrong about undergraduate courses always treating $\log$ as $\ln$. To my memory, all of my undergrad chemistry and physics (not just general, but majors texts), engineering, calculus, diffyQs, and engine maths books use $\ln$\ for natural log and log for base 10. Add onto that everything I've seen professionally in oil exploration and the military. ...
21
It was mentioned in other answers that having a more sloppy notation is better to not complicate the communication. This is okay for people who had really understood the concepts of mathematics.
I have the feeling that understanding what a function is one of the those things students don't really understand. Common mistakes (and in my opinion all of them ...
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
You should tell them these two main benefits:
(1) Function notation is concise! For example, instead of writing "Find $y$ when $x=5$" one can simply write "Find $f(5)$" This becomes very appreciable when dealing with long or complicated problems asking for a lot of information. We also shorten things like this all the time. For instance, instead of writing $...
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 ...
20
To distinguish not too strictly between $f$ and $f(x)$ allows to operate more easily with functions built up from other functions.
For example, one might want to say things like:
Let us consider the function $\sin (3 x^2)$ on the intervall $[0,1]$.
This can be considered as sloppy in a formal sense, but I think it is still clear and in some contexts ...
20
It is interesting to see how computer algebra systems deal with this kind of thing. In Maple, for example, you can do the following:
Define f := x^3
Enter diff(f,x) to get the derivative
Enter subs(x=3,f) to evaluate.
Alternatively:
Define g := (x) -> x^3
Enter D(g) to get the derivative
Enter g(3) to evaluate
You can convert between the two idioms ...
20
When a student writes incorrect notation, ask them to read it out loud. I would say something like:
Something here doesn't look right, but we can fix it. Could you read this work out loud? I think probably you are not super familiar with this topic, and that's okay, but this can help us fix it.
I had success with this when dealing with a student who ...
19
To answer the ultimate question ("Can anybody explain where this writing tradition comes from?"): It's explicitly taught that way by many U.S. instructors and textbooks.
Examples: From the otherwise excellent Martin-Gay Prealgebra & Introductory Algebra (sec 1.5):
The × is called a multiplication sign... The symbols ∙ and () can also
be used to ...
19
Common knowledge
The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea.
Connections to other mathematics
The notation with AB, CA and BC might be something the students have used or will use in less analytical ...
18
In mathematics, $\log$ means natural logarithm. So, as you become a mathematician, sometime during that process you must learn this.
That standards document is for natural sciences, not for mathematics. I guess mathematicians are too independent-minded for such things.
This would be seen, for example, in complex analysis and in real analysis.
Note: ...
18
I would argue that we should never use $\log$ for $\log_{10}$ anymore, only warn that this was historically often done. Sticking to the ISO convention is probably safest:
$$\begin{aligned}
\log_{e} \equiv \ln
\\ \log_2 \equiv \operatorname{lb}
\\ \log_{10} \equiv \lg
\end{aligned}$$
I personally would use of the shorthands only $\ln$, and write out all ...
17
I disagree that it is "terribly harmful".
Do not prevent them from writing $(2)(5)$. Instead prevent them from writing things that are actually wrong.
Thinking that $\sin x$ is $\sin$ times $x$ can happen (and does happen) even without parentheses.
I agree $(2)(5)$ looks strange, but if they can write $(4-2)(4+1)$, what is wrong with $(2)(5)$?
17
Just because you've defined a meaning for $a < b < c$ does not mean that any mishmash of other relational operators becomes equally well-defined as notation. Stick with what you've defined for a chained equality and don't permit arbitrary, nonstandard, off-track jaunts.
For example, here's the treatment in Sullivan, College Algebra, Sec. 1.5. Note ...
16
What's wrong with this?:
$$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$
with $\frac{dt}{dt}=1$. It's what you would get if you had $f(x,y,z)$, except that $z=t$.
You do have to be a bit careful, though. You want to be clear that $f(x,t,t)=xt^2$ ...
16
My answer is probably not very useful when teaching in high school. I'll just mention here a few reasons why this definition is in fact a good one, and why it's a good idea to teach this formula at a university level mathematics course.
$\newcommand{\u}{\mathbf{u}}$
$\newcommand{\v}{\mathbf{v}}$
$\newcommand{\w}{\mathbf{w}}$
$\newcommand{\R}{\mathbb{R}}$
Let ...
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