Towards more meaningful math notation
Students struggle a lot with the way mathematics is written.
For example, most students don’t have too much of a problem with:
5(a + b) = 5a + 5b
Then they see this and it is also OK:
5(ab) = 5ab
In most cases you can substitute various values of a and b and the students can see that it works. Fair enough. Then the student does twenty (mind-numbing) examples of such bracket expansion and they feel they have got it.
Later, they come across things like:
sin(a + b)
And then their math teacher goes ape when the student expands it like:
sin(a + b) = sin a + sin b
Perfectly logical, in the minds of the student.
Similarly, it is logical to have the following, isn’t it?
log(a + b) = log a + log b
Oh, and then we have functions. You know, like this:
f(x)
Is that the same as
f × x? (That is, f multiplied by x?)
Why not?
I wish to propose an alternative notation for concepts where you cannot expand in the way you do with simple algebra. It might look something like this:
This would send a much clearer message to students that the particular function or operation does not work in the same way as simple algebra works.
Now, the proposed rectangle would be a nightmare given that we need to type mathematics (actually, everything is a nightmare when you are trying to type mathematics…).
So a more computer friendly option would be to (exclusively) use [ ] – square brackets – for such concepts, like this:
sin[x + y]
log[x + y]
f[x]
Would this work? Would it confuse everyone even more? I feel that if such a notation were to be universally adopted, then less confusion would arise.
[I wrote about notation before in Phase shift or Phase Angle?].
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11 Jun 2007 at 5:28 pm Link to this comment
Hi Zac,
As a math teacher, I agree with what you are saying. We have quite a few confusing notations that are conventionally used, but could be changed. Add to your (parentheses as function) vs. (parentheses as multiplication) problem the negative sign… 1) opposite of a number 2) subtract a number 3)as an exponenent, take the reciprocal of the base 4) f^-1(x), take the inverse of the function.
And of course, logs, which I wrote about on my site.
I doubt notations will change any time soon. I guess the best thing we can do right now is to recognize it as a problem and provid students with explicit instruction about avoiding these pitfalls.
12 Jun 2007 at 3:50 am Link to this comment
[...] gives examples of situations where students struggle with the way maths is written. The blogger proposes an alternative notation for the way parentheses are used in some functions. Share [...]
12 Jun 2007 at 6:57 am Link to this comment
Dan – thanks for your comment.
I fully agree with your point about raising to the power -1.
In my math site, I use “arcsin” and “arccos” etc for the inverse trigonometric functions. For example, see Trigonometric Functions of any Angle, where it says:
12 Jun 2007 at 12:39 pm Link to this comment
Hi,
I read your post with interest. I’m a graduate student in mathematics.
Indeed the mathematics computer software program Mathematica(TM) uses the square bracket notation for functions i.e. log(x+y) must be written as Log[x+y] with its syntax.
However, for later use, the round bracket notation is still be best(quickest) and easiest to use, esp in higher level maths, e.g. A-levels or engineering use. To a first timer it may be hard to get used to at first, but after repeated correction and usage, one can distinguish between a(b)=a.b and a(b) as a function.
12 Jun 2007 at 12:46 pm Link to this comment
Thanks, Uliang for your thoughts.
Yes, you are now used to it and so am I – but we have been dealing with higher level mathematics for some time. But there are thousands (millions?) of students who are clueless because of bad notation. They never get to see higher mathematics because they are so confused by something that is unnecessarily inconsistent.
Why is English so difficult to learn for non-native speakers? Because it is full of inconsistencies in spelling and grammar. Math has the same problems.
I’m just wondering – why do you feel that round brackets are quicker and easier? I would argue that the time taken to write ( ) and [ ] and { }, for that matter, are all much the same. But if their meaning was more intelligently applied, we would spend a lot less time in secondary school maths classrooms trying to explain why things work one way here and another way there, even though they look the same.
Heck, some students may even hate mathematics less, and decide to continue studying it!
15 Jun 2007 at 6:40 pm Link to this comment
This is a brilliant idea. When I was teaching high school I would also see students with an equation like cos(x) = blah try to “divide by cos”… which of course doesn’t make any sense, but by analogy with 5(x) = blah it makes a lot of sense. I’d never really thought clearly about the problem with notation before.
I think it’s entirely feasible to teach kids the bracket notation, and then inform them later that most people just write parentheses — by that point I don’t think it would be too hard for students to transition to always *writing* parentheses but *thinking* brackets in their head.
16 Jun 2007 at 4:13 am Link to this comment
Thanks, Brent for your comment.
I was suggesting a permanent change, but you have a good point. Perhaps students could start with square bracket notation and move on to the plethora of existing notations later.
16 Jun 2007 at 5:10 am Link to this comment
What about calculus, Zac?
There are several ways of writing differentiation, and they mostly mean the same thing:
dy/dx
y’
f’
f’(x)
x dot (I mean x with a dot above, meaning dx/dt)
Surely this notation needs tidying up, too?
16 Jun 2007 at 6:02 am Link to this comment
I agree, Steven.
When students start differentiation, they are nearly always dragged through “first principles”.
The notation is usually horrendous:
lim Δx → 0 {(f(x + Δx) − f(x))/Δx}
Say what? Not only do the students have to cope with a challenging new concept (“gets closer and closer to but doesn’t actually get there”), but a horrible notation.
I have used “h” instead of Δx in the section in the math site: Derivative by First Principles.
17 Jun 2007 at 12:08 pm Link to this comment
By the way, this is Mathematica’s notation. “()” indicates order of operations, “[]” indicates application of functions, “{}” indicates an ordered pair or a list. So
ParametricPlot[{Sin[t], 2*(Cos[t]+1)}]
would plot an ellipse whose x-coordinate is Sin[t] and whose y-coordinate is 2*(Cos[t]+1). (Also, “[[]]” indexes into a vector or array, but pedagogically we want to discourage students from thinking of an array and a function as different objects.)
I find that I have very little trouble making these distinctions, even when I’ve been away from the software for several months, so I think it would be easy for experienced mathematicians to switch to the new notation.
19 Jun 2007 at 12:30 am Link to this comment
This is an interesting idea that could also help students keep the concepts of “function” and “argument” clearer. On the other hand, I feel that if we were more consistent with function notation, we may not need such changes. My personal pet peeve is with the notations used for the trig functions. In general, for a function ƒ and its argument x, we write the value as ƒ(x). But for the function sine we often just write sin x. Worse, ƒ2(x) usually means ƒ(ƒ(x)), but sin2(x) means [sin(x)]2, and then it’s not at all clear what sin-1(x) should mean.
20 Jun 2007 at 2:48 pm Link to this comment
I live and teach Mathematics in South Africa, and I am so glad to here that maths teachers elsewhere have the same problems as I have. A brilliant idea, to change the notation.
21 Jun 2007 at 12:57 am Link to this comment
Hi Rika and thanks for your comment. Math students seem to face the same algebra, trigonometry and logarithm problems world-wide, and I am sure notation is a big part of the mystery – and it need not be so.
22 Jun 2007 at 11:29 am Link to this comment
I’m not convinced about the need for a change. If you like, you can think of the notation 5(.) as representing the function “multiply-by-5″. OK, that’s not how you would think about it when learning multiplication (or teaching it), but the point is that one can make it consistent. In any case, I don’t see how changing the notation helps. Even thinking purely in terms of functions, we know there are functions for which f(x+y)=f(x)+f(y), and other functions where this does not hold, so why would the confusion go away by changing to brackets instead of parentheses?
22 Jun 2007 at 1:36 pm Link to this comment
You’re right, jk about functions of the form f(x+y)=f(x)+f(y).
What I am suggesting is more like a “flag” to indicate to novice users that a particular item will most likely behave differently to some other item.
I’m still thinking about how you could ever make “5(.)” consistent with any of the existing inconsistencies in math notation…
15 Jul 2007 at 3:44 am Link to this comment
this may make things easier at the beginning, but eventually if the student starts using generic operators, this could lead to confusion, as some are linear and some aren’t. I think it may be better to always use x*(a+b) in cases where there is likely to be confusion with multiplication, and stress the difference when functions/operators are introduced, rather than to change the function notation.
24 Oct 2007 at 7:38 pm Link to this comment
It seems better to simply use multiplication sign. Reserve juxtaposition for function application.
25 Oct 2007 at 11:51 am Link to this comment
That sounds like a good compromise to me, Alexey.
17 Jan 2008 at 7:11 am Link to this comment
I think this is such a great topic, so I’m going to bring it up again. It seems like much of these notations came about just to save memory on devices such as graphing calculators. Luckily, I know most of the rules and exceptions, so I never really have problems with this. I have noticed fellow classmates having problems, though. I have used websites which use brackets for functions and parentheses for order of operations; I thought it was a good idea. So, I do agree that there should be a standardized set of math notations.
Unfortunately, this is very similar to the English standard versus the Metric system. I’m sure we can all agree that there is practically no advantage in using standard measurements. This changes nothing, though. The entire nation is still stuck with this terrible system and is barely doing anything to change it.
I would love for these changes to happen (adopting the Metric system and creating a better system of math notations), but I doubt I will see that day.
17 Jan 2008 at 11:05 am Link to this comment
Hi Magick and thanks for your response.
I’m a little confused by this statement: “I’m sure we can all agree that there is practically no advantage in using standard measurements.”
The rest of your statement appears to contradict this.
A quick story for you. In the early 1970s, Australia bit the bullet and required, by law, the conversion to the metric system. There was grumbling and some inconvenience for a while, but it happened. And it made life much easier for math students and their teachers.
Schools had been teaching both systems during the 60s and then only metric in the 70s, so the younger generation didn’t have too many problems.
It seems extraordinary to me that the US is still dragging its feet on adopting the metric system. But then, when you look at other things like recognizing global warming, it starts to look consistent.
Footnote: Some strange anomalies occurred in Australia. Mechanics had to have 2 sets of spanners when working on some cars, where (say) the body was made in Australia (metric) and the engine came from the US (Imperial system).
How crazy is that?
15 Mar 2008 at 7:40 pm Link to this comment
nice idea ,, actually i didnot get confused before because i understand from were each case comes ,, any way but its really usefull for other students that they hate math .. lol they are looking for any thing to say math is hard and confsing
21 Mar 2008 at 8:31 am Link to this comment
Hi Sara and thanks for your input.
I am in the process of writing a more detailed proposal about an alternative math notation and I hope to publish it soon.
8 Apr 2008 at 3:25 am Link to this comment
I am really upgraded each day I have my lessons from intlmaths.PLEASE I need need more tutorials on trigonometry integrals, partial integrals, integration by parts, reduction formula and its application, multiple integration. Please send a step by step solution to some problems in each case.
Thanks
8 Apr 2008 at 4:48 am Link to this comment
Hi Living St Vincent. I am adding to the Interactive Mathematics site all the time. I will keep your requests in mind for future revisions.
19 Nov 2008 at 1:28 pm Link to this comment
why not preserve the parenthesis for function application
and use square brackets for grouping?
19 Nov 2008 at 2:10 pm Link to this comment
Hi Shiva. Square brackets are suggested as an attempt to be consistent with Mathematica’s notation.
Personally, I think either is fine, as long as they are different!
2 Jan 2009 at 5:59 pm Link to this comment
It is of note that the square brackets are used to denote array access in modern high level programming languages ( arrays = vectors / matrixes ). This could also lead to some ambiguity but I think the general gist of what you are saying makes sense, and I’m no expert in math.
21 Mar 2009 at 5:41 am Link to this comment
I disagree with you notation because it’s moving the problem around.
If a student expand “sin(a+b)” in “sin a+sin b” that mean that this student don’t understand that sin(a+x) means the function sin() applied to “a+x”.
In other words, you could explain that “+”,”-” etc are in fact binaries function. So you can write “a+b” as “s(a,b)”.
Then when you see “sin(a+b)” this is like “sin(g(a,b))” and there is no reason to distribute.
So when you write “5(a+b)=5a+5b” it’s like : “p(5,s(a,b))=s(p(5,a),p(5,b))”
If they are able to write it that way, they should avoid the error in the future.
21 Mar 2009 at 7:02 am Link to this comment
@Programaths: If they are able to write it the way you suggest, they would not be the ones having trouble with the expression sin(a+b).
Our aim is to make life simpler, not even more confusing.
12 May 2009 at 7:37 pm Link to this comment
I completely agree with the proposed changes to the notation. In my experience, kids most often give up on math before they manage to see past the confusing/inconsistent notation.
I’m definately going to make use of this in the future.
21 Aug 2009 at 5:51 pm Link to this comment
When writing by hand, I like to distinguish function abbreviations (such as sin, log, ln, f) from variables by writing them in a script style, as in formal handwriting. Variables are simply printed in simple san-serif style as always. When typed, I’ve seen that we are supposed to put variables in italics to distinguish them as variables, though I haven’t bothered as I don’t care for the look.
As for derivative symbols, a neat alternative I found is to use subscript of the variable that the derivative is with respect to. So, for a function f(x) (I’d have made the f in script style if I could), its derivative would be fx(x) (I’d have made the first x subscript on the f here if I could). The chain rule can be written thusly: given y=f(g(x)), the derivative of y with respect to x could be written f[subscript g][multiplication dot]g[subscript x](x). Anyway it’s more of an exercise for my own satisfaction to make concisely written notes, as students would be learning the other notations for decades to come I imagine.
21 Aug 2009 at 9:37 pm Link to this comment
Thanks for your inputs, Nicks.
I also like subscript notation for derivatives, but as you just found, it is a bother to type it on the Web.
For reference, you can type your
f[subscript g][multiplication dot]g[subscript x](x)
as
f<sub>g</sub>·g<sub>x</sub>(x)
and it will look like the following, as you intended:
fg·gx(x)
26 Oct 2009 at 1:41 pm Link to this comment
It seems to me that there’s actually nothing wrong with the notation, per se. There’s an unspoken assumption that is not stated for the benefit of students until maybe 2nd or 3rd year of college, if you’re lucky.
Linear functions can be distributed out of their parentheses. This is why you can write int(x+y) = int(x)+int(y) or laplace(x+y)=laplace(x)+laplace(y). However, it’s critical to math literacy to recognize that one of the things you have to know about every function is whether or not it’s a linear operation on the argument. The four basic operations are all linear–but when you cross into college nobody informs you of when that stops happening. That’s the issue.
27 Oct 2009 at 1:47 am Link to this comment
I am teaching maths for computer scientist at university level. We use the mapleTA system based on the computer algebra system Maple. One problem is that such a system uses a strict functional notation whilst students are used to an operator notation.
For example Maple interprets
sin (x+1)^5 as raising to power five the result of sin applied to x+1.
However in operator notation sin (x+1)^5 is interpretated as computing the sin of (x+1)^5.
Operator notation is undoubtably the notation of choice for the standart arithmetic operations such as *,+,^ etc
No one (except computer scientist in some cases) would write
+(1,*(2,3)) for 1+2*3.
However, my feeling is that the operator notation should be used only for elementary operations. If you start using unary operation notations like sin x, ln x, after introducing sin and ln as functions and you did not give an explanation that operators are associated with functions that can be expressed by a particular notation (which would go too far at low level maths) then you are potentially confusing students.
And then you can confuse them even more when you consider composition: Does sin^(-1)x means arcsin(x) or 1/sin(x)?
If you only use pure functional notation then
sin^(-1) (x) means arcsin(x), sin (x) ^(-1) means 1/sin(x), and sin (x^(-1)) means sin(1/x).
27 Oct 2009 at 3:22 pm Link to this comment
Thanks, Chardaire. I think we should never use “^(-1)” when referring to the function sin(x).
So arcsin(x) (or commonly asin(x)) should be used rather than sin^(-1) (x).
Also, csc(x) (or perhaps cosec(x)) should be the convention and we should avoid any use of “sin (x) ^(-1)”.
The Maple interpretation of “sin (x+1)^5″ is unfortunate and you’re right, that would cause more confusion.
12 Nov 2009 at 4:20 pm Link to this comment
Thanks for this great knowdge…i like it very much.
Actually i found these types of problems in my life.
According to me these types of symbles are very important for mathematics….Thanks again…!!!
15 Nov 2009 at 9:35 pm Link to this comment
very bad suggetion step wise ans of every ques must be given
15 Nov 2009 at 10:15 pm Link to this comment
Akky – what is it about the suggested change in notation that you do not like?
I’m not sure what the second part of your response is referring to. A change in notation does not necessarily imply any change to the amount of explanation given in a solution, surely?
12 Dec 2009 at 1:20 am Link to this comment
ok here is my deal, why is there so many big words that people cant understand? im 15 and i understand everthing fully. The notation is perfect!