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How important is enforcing standard mathematical language and/or notation?

Today, I was questioned by a writing instructor as to how vital it is to correct students when they explain something using nonstandard mathematical language. The example we discussed was correcting a student who says “x two” to indicate $x^2$ (instead of “x to the power of two” or “x squared”). I said that my primary role is to first determine what a student means, and if they mean $x\times x$, then I will almost always correct the language as saying such things may be misinterpreted as $x \times 2$.

This question came to me in the context of a discussion with other faculty about inclusive teaching practices, where a method shared involved allowing classes some freedom in creating and using their own language for items/principles/etc discussed in the topic. The perception was that allowing (and even encouraging) students to do this would build community in the classroom and let students attach some personal meaning to the items discussed. The instructors speaking favorably about this were from certain Humanities departments (writing, film studies, history). I was the only instructor from math or science in this conversation.

This is of interest to me because while my students must be able to communicate mathematics with others once they leave my class, I wonder if allowing some creativity here might do something positive for their overall experience studying math.

I have definitely had moments in certain classes where a student will (usually as a joke) propose a name for something — a solution technique, a common pitfall, a type of function — and the class adopts it. This happens when I don’t know of any standard terminology for it, so this class name may stick around for the term. I’m actually really happy when this happens organically. What I haven’t done is encourage playfulness with things that already have names.

So — is there any place in mathematics for student-created terminology and/or notation? How have you (or would you) allow it?

Finally, I understand what a huge setup for failure this would be if we didn’t enforce any language or notation — students would go to some other class and be completely lost, not recognizing basic things as typically presented. If you don’t like this idea in principle, that is totally fine. I am just wondering if there’s any place in modern mathematics for this kind of freedom for students.

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    $\begingroup$ Regarding "x two", this sounds to me more of an ESL issue than typical student misuses (e.g. "solve the trig identity"). When reciting $E = mc^2$ (essentially a generally known cultural phrase -- a cliché, even if many people have no idea what it means), would they say "E equals m c two"? And what about reciting $\pi r^2$ for the area of a circle? However, your question raises interesting issues to consider. (+1) $\endgroup$ – Dave L Renfro Feb 7 at 7:47
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    $\begingroup$ I really, really like this question, and hope to see some interesting answers. However, I imagine that those answers may differ (perhaps significantly) depending on the grade level of the students. In middle school and high school, allowing some freedom and creativity in notation will help students enjoy the experience more. But, for some reason I don't see myself doing this all that often in a university setting. $\endgroup$ – Brendan W. Sullivan Feb 7 at 16:29
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    $\begingroup$ "Inclusive practices" is yet another useless activity that bureaucrats of education came up with. This is why ed schools have such a bad rap - instead of teaching real pedagogy and content of the subject they push vague equality ideas. Resist if you can. As for terminology, of course it should be consistent, and it should be taught, this is what lectures are for, because textbooks usually do not help with correct spelling. To me, "x two" means $x_{2}$. The least your student can do is say "x power two", which would be acceptable to me. $\endgroup$ – Rusty Core Feb 7 at 18:12
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    $\begingroup$ In The Number Devil, rutabaga is used to mean root, etc. It makes for fun reading. But some readers may be confused by it. I like asking the question the way @Milo_Brandt did - how do we help students communicate about mathematics effectively? $\endgroup$ – Sue VanHattum Feb 9 at 3:38
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    $\begingroup$ You said the discussion was with humanities teachers, but I'm sure there were no foreign language teachers present. Math is a foreign language. If you wanted to know how to best do allow creativity, I would suggest talking to foreign language teachers. X two is clearly very ambiguous and not an example of creativity in my opinion. However I think creativity used appropriately can liven up your class and engage your students. $\endgroup$ – Amy B Feb 9 at 20:34
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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 level, efficacy is a way easier standard for students to internalize than an arbitrary standard set of language.

For instance, it is worth correcting "$x$ two" because this language does not convey what operation is being used - and it might even convey a lack of understanding of the fact that the notation $x^2$ refers to an operation. I think a reasonable response to such notation as a teacher is to point out that the student had something (exponentiation!) in mind that they didn't succeed in communicating and to give them the tool to communicating that (e.g. the language "$x$ squared" or "$x$ to the power of $2$" or however you want to phrase it). I would give similar feedback to a student who was manipulating a large expression and wrote something like $e^{(x+1)^2=x^2+2x+1}$ because I would want them to understand that they are really separately noting the equation $(x+1)^2=x^2+2x+1$ and then substituting that into a larger expression - and to ensure that they don't treat equality signs merely as the way to express a chain of simplifications. These sorts of language issues are usually easy to resolve, but should get the same attention that a mathematical issue would since they reflect mathematical structure.

If a student said something like "We take $x$ and plus it with $5$", I might point out that the usual way to say that is "We add $5$ to $x$", but this is a lot lower of a priority than the previous example - I'd be inclined to let it slide in spoken mathematics, but would correct it in anything written. A similar example is that a student once, while explaining her proof to me, used the phrase "we take $x$ and two-thirds it over to $y$" to refer to a weighted average - which was perfectly clear while we were both looking over her diagram of the process and more enjoyable than using technical language, even if I wouldn't like it in writing. These are more explicitly issues of poor notation - which should be dealt with wherever good notation is expected, but not confused with more important conceptual issues. (Just to emphasize: I am saying that there can comfortably be two standards of communication here - in class, it's often worth setting aside conventions to avoid distracting the class from the big picture and to avoid possibly alienating students purely based on language, but it's important that you at least sometimes ask students for composed work so that they have an opportunity to learn conventional phrasings and don't later feel like they were denied this knowledge)

This said, looking at small uses of language misses the point: the primitive concepts of mathematics have fixed names and students should learn to use these primitives properly. The question of nonstandard language is not about terms like "squared" but rather about the concepts students might wish to build on top of them. If you expect students to be able to communicate effectively, that means that they have to explain - in a human language - what they're doing and that means that suddenly we should be talking about students producing sentences such as "We begin by isolating $x$." and putting these sentences together into paragraphs (along with equations and formal manipulations) - and then explaining what they're up to at a high level, in the same sense that a writing teacher would demand "topic sentences". There's suddenly a lot of room for creativity once you ask students to communicate at this level - and there's room for idiosyncrasy too as a class comes across methods and explanations that appeal particularly to them. Focussing on these larger blocks of language is also something that I've found to help weaker students since it gives them the tools to explain understanding that may have been hard won for them - and the ability to explain is something that they may not have felt included in before.

There really is a danger of a teacher overreaching into this territory of higher abstraction - I had plenty of frustration as a student when teachers insisted that a concept not only be correct, but always understood and phrased in the teacher's language; it's better to push the student to be able to explain their mathematical thinking process - for instance, productive comments might look like "Your equations are correct, but hard to follow; could you include more writing about why you did the manipulations you did?" or "This would be clearer if you included a worked example" or "Could you draw a diagram of this step to help the reader?" or "You could phrase this more clearly by writing this last equation and then saying 'by taking the square of both sides'." The goal in such teaching would be that every student can communicate in a way that is clear - and while this involves standard notation for the details of mathematical rigor, there's not much to prescribe beyond that.


As a small side note, there is some language such as the names of various theorems where one could argue that the standard names might be better replaced by non-standard ones - and where using the standard notation doesn't actually reflect a better conceptual understanding. For instance, if you refer to Bezout's identity as "axby", you suddenly are using more descriptive notation than standard mathematics and you get to say a fun word. If someone in your class makes an insightful question, you can call it "So and so's conjecture" to give some ownership where it is due until the class arrives at whatever conclusion there is to be had. I put this as a footnote because, while I've found this to work well when teaching higher mathematics to high schoolers over summer (although it sometimes annoys students with more prior knowledge than their classmates), I've also had experiences as a student where teachers have found a flashy new way of teaching like this and implemented all the superficial furnishings of it - like using different names for things - but not actually done anything to make students feel the sort of ownership reflected in a class-specific vocabulary - and these experiences have been frustrating and felt insincere (or worse: have allowed a subset of the class to create notation that bewilders and excludes others). It's not something I would do for teaching more basic content or for teaching a class with an especially large range of comfort with mathematics. Basically: even in cases where the language isn't important, that's not where a mathematics teacher should look to improve their teaching - rather, they should look to the larger picture of communication.

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    $\begingroup$ Is it common also in English to use 'plus' as a verb? In Swedish many pupils say "jag plussar" and "jag gångrar" (from the word "gånger" meaning "times") for "I add" and "I multiply". $\endgroup$ – md2perpe Feb 9 at 21:44
  • $\begingroup$ @md2perpe Yes, it seems like a fairly common mistake in English - I think it probably comes from the fact that $x+2$ is read as "x plus two" where it's not totally clear which part of speech "plus" is (it's technically a conjunction synonymous with "and"), but where it feels like we're describing the action of "add two to x." Maybe it's similar in Swedish? $\endgroup$ – Milo Brandt Feb 9 at 21:57
  • $\begingroup$ I think you're right. Also, Swedish and English are similar enough for the same things to happen. $\endgroup$ – md2perpe Feb 9 at 22:27
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    $\begingroup$ It seems to me that "We take x and plus it with 5" is perfectly correct math, it just isn't grammatically correct English. From @md2perpe comment it seems to be correct Swedish and I would guess that it is also correct in a few other languages. $\endgroup$ – quarague Feb 10 at 12:22
  • $\begingroup$ @quarague. I did not mean that it is correct Swedish. It's bad Swedish. If one does not remember the latinbased word "addera" (English: "add"), then one could use the pure Swedish phrase "lägga till" (directly translated "put to", meaning "add" or "append"). $\endgroup$ – md2perpe Feb 11 at 15:07
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Part of teaching mathematics is teaching how to communicate mathematics - both how to understand it, and how to write it clearly. Some of that is obviously complementary to teaching students how to solve math problems and understand mathematics; for instance, students who can't understand a question aren't going to be able to answer it correctly.

But it's not obvious that we should expect these skills to be learned perfectly in parallel. My understanding is that when teaching writing, the current approach prioritizes having students first learn to write fluidly, and only much later starts pushing students to improve spelling and grammar. I suspect our expectations in math should be similar: we should expect student's ability to write math conventionally to lag, possibly significantly, what they understand, and that's absolutely fine and developmentally normal.

Learning how to do mathematics and learning how to express mathematics really are two separate things, and it can be confusing to students to have them too mixed together. Students often can't tell the difference between mathematical reasoning and arbitrary notational rules, so being overly strict about how things are written distracts students from learning the math, and feeds the perception that math is a collection of unmotivated rules imposed by the teacher. (Particularly since many notational choices are motivated by concerns that come up much later, so students aren't in a good position to appreciate them when they're being introduced.)

We should certainly teach the standard notation, and we should certainly expect students to be able to read it. But allowing them to discuss and sometimes write it in non-standard terminology, as long as they understand that it's non-standard, could sometimes be beneficial.

Where I've had the time in a class (which is rarely), I've even been trying to make the distinction more explicit: to be more flexible about notation most of the time, but have a few assignments where the assignment is specifically to take a problem they've already solved and write it up more formally for their classmates. That gives students a change to practice correct notation, and emphasizes that correct writing is primarily about communicating with peers, not about satisfying the rules a teacher has laid down.

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    $\begingroup$ "the current approach prioritizes having students first learn to write fluidly, and only much later starts pushing students to improve spelling and grammar" — which is a wrong approach. Teaching of reading and writing in American schools is broken beyond repair. It is not unusual to see eight-grade students reading at second-grade level. "Learning how to do mathematics and learning how to express mathematics really are two separate things" — no, these are two sides of the same thing, like reading and writing are two sides of becoming literate, which is why they should be taught simultaneously. $\endgroup$ – Rusty Core Feb 7 at 18:31
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    $\begingroup$ @RustyCore: Reading and writing are closely related, but through primary and secondary school, students read at a level far behind what they're expected to write. Similarly, we might teach student to do math and write it simultaneously, but recognize that it's typical for their writing to lag their ability to do the math by a number of years. $\endgroup$ – Henry Towsner Feb 7 at 19:10
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    $\begingroup$ Students "read at a level far behind what they're expected to write"? Are you saying they are expected by their teachers to write much better than they read? These are the same teachers who taught them reading? Ideally, kids should learn how to read in the first semester of the first grade and then just refine their knowledge of grammar, syntax, style, etc. Whom I am kidding, teaching grammar or diagramming sentences is unheard of these days. Instead, I see subjects like "power literacy" in eight grade, and kids reading like 1980s text-to-speech machines. $\endgroup$ – Rusty Core Feb 7 at 19:25
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    $\begingroup$ @RustyCore: Of course it's an imperfect analogy, but it's the one you chose. I mean both, but the latter is more relevant - students will read compound sentences and see commas used well before they reliably use it themselves. The universal assertion "should be marked as an error", stripped of any context, is precisely what I disagree with. Expecting students to eventually use commas correctly does not require marking kindergartners wrong when they don't. $\endgroup$ – Henry Towsner Feb 7 at 22:16
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    $\begingroup$ @RustyCore I have said no such thing. As I keep saying, we should teach the rules with the material, but we should also sometimes wait until students have had more exposure to seeing the rule before we hold students accountable for applying it. $\endgroup$ – Henry Towsner Feb 7 at 22:43
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I very much appreciate these questions. I am in favor of a very strict style; or in other words, helping students to avoid errors and confusions in the standard language as much and as early as possible. The main thing that sticks in my mind is that the longer a bad practice persists, the "stickier" it is, and the harder it is to fix later on.

Comparing (a) learning a thing right the first time, vs. (b) learning a thing wrong and fixing it later, item (b) takes roughly three times as much effort as (a). That's because (b) entails (1) learning the thing wrong initially, (2) working to lose the erroneous idea/muscle memory, and (3) finally learning the thing right.

Anecdote: I was teaching a remedial algebra class and reviewing the arithmetic order of operations. As part of the discussion, it became clear to me and a young woman of color that she had flat-out been taught something totally incorrect at the high school level. She burst into tears and fled from the classroom, enraged that, "she'd been taught everything wrong and had to learn it all over again".

Now, that was an outlier reaction, but it's always stuck with me. She's right, and I respect her high level of intellectual honesty. Not bothering to get these fundamentals right early on is something like a brutality to the students in question. Not caring about it is an equity issue.

Let's compare to an analogous trend a few decades ago in English instruction; to get rid of directed instruction in phonics, in favor of a more flexible, exploratory, "whole word" learning of vocabulary. In short, the results were that such a removal of direct instruction was clearly detrimental. From a meta-study reported by Brady, Susan A. "Efficacy of phonics teaching for reading outcomes." Explaining individual differences in reading: Theory and evidence (2011): 69-96:

Overall, research reviewed in the NRP report indicates that students taught with systematic phonics instruction have better reading scores, whether measured at the end of the training period or at the end of the school year of instruction (Cohen's d = .44), Systematic phonics instruction was found to produce better reading growth than all of the types of nonsystematic or nonphonics instruction (i.e., basal programs, whole-language approaches, regular curriculum, whole word curriculum, and miscellaneous programs). Further, systematic phonics was found to be effective whether taught through individual tutoring (d = .57), through small groups (d = .43), or to the whole class (d = .39).

My impression is that likewise, the elimination of directed grammar instruction and correction is now producing evidence of serious corrosive effects; but I'm not up on that research and would need to go digging for specifics.

In summary, all the signs I can gather -- both from in-class experiences and decisive evidence from language-instruction research -- point toward it being better to get the fundamentals right sooner rather than later.

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It is of course important to give your students a good understanding of the customs and conventions that are commonly used in mathematics, but I think that creativity, imagination and flexibility of mind are just as important - as is the courage to think 'crazy' thoughts, such as "what is $\sqrt{-1}?$", "is infinity always the same?" and so on.

Maths is in some ways closer to art than to the 'hard sciences', and there is scope for a lot of playfulness. Mathematical jokes are very valuable in making the subject enjoyable, especially if they illustrate some clever, not obvious point; eg. something that is beguilingly plausible, but actually impossible. Or, from topology in particular, a counterexample to something that seems intuively true - there is a famous book about this: "Counterexamples in Topology" by Lynn Steen and J. Arthur Seebach, Jr.

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This reminds me of my question “Amplitude” of Tan and Cot functions which referenced a non-standard use of the word amplitude.

Math has a language, one that should make communication on this topic pretty clear. When discussing notation or language, I frequently resort to the example "You are on the phone with a friend, pre facetime, audio only. As you describe your equation or geometric construct, will your friend be able to write down exactly what you are describing?

As Rusty commented, I hear x two as $x_{2}$ as well, although I often pronounce the latter as "X sub two".

The question that might remain is how much to push for perfection. I respect Ben, but it wouldn't have occurred to me that sin(x) isn't perfect notation. I think we agree that 3 minus 4 is -1 and is pronounced "negative 1", but do we correct a student who answers "minus 1"? That's not rhetorical, I'll answer. No, I don't. When in front of a class of 20 High School students, 16 of whom aren't raising their hands and I need to push them to do so, I'm not going to choose that level of pedantry and risk having the student go silent. I do, however, take the opportunity as some other time to show them that the TI84 we commonly use in class has a minus key and also a negative key. That can help get the point across.

On the other hand, there should be room for a bit of 'inside jokes' or phrases the class might use. When reviewing the 45/45/90 triangle ratios, a student observed "it's faster when going from hypotenuse to leg to simply multiply by $\sqrt{2}/2$ than to divide by $\sqrt{2}$ and rationalize. I was happy to see this aha moment, and the class named that manipulation "the Sam trick" when describing the steps later on. Obviously, that isn't going to spread around, and the students know it's a shortcut, if even that. (And I know that we shouldn't fear the radical in the denominator, but I am subbing this semester, not my call).

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    $\begingroup$ "This reminds me of my question “Amplitude” of Tan and Cot functions" --- And now I'm reminded that I never got back to writing an answer, as discussed in the comments. I'll see if I can do something about that in the next few days. I did do some additional research on this a while back. Among other things, I looked through many test prep guides and Official Guides in several bookstores. $\endgroup$ – Dave L Renfro Feb 9 at 20:15
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    $\begingroup$ Please don't "correct" students who say "minus 1". That is correct and acceptable both in a mathematical context and in an everyday context. $\endgroup$ – Rosie F Feb 9 at 22:04
  • $\begingroup$ I appreciate the input. So is there no distinction to be made between the two words, similar to mean/average being redundant ? $\endgroup$ – JTP - Apologise to Monica Feb 9 at 22:08
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    $\begingroup$ So is there no distinction to be made between the two words, similar to mean/average being redundant ? --- @Rosie F did not say or imply there is no distinction (that's a very high bar). Also, you used the word "redundant" incorrectly. $\endgroup$ – Dave L Renfro Feb 9 at 22:40
  • $\begingroup$ Synonymous ? This is the Math site, not the English site correct? (Kidding, thx for the correction) $\endgroup$ – JTP - Apologise to Monica Feb 9 at 22:56
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It's your job to teach standard notation and terminology, and to correct your students if they get it wrong. If they get it wrong repeatedly, it's your job to correct them repeatedly.

If a student comes out of an algebra class saying "x two" for $x^2$, then that student has been done an enormous disservice. They will be marked as mathematically illiterate by any person they encounter in the future.

I would actually argue for much stricter enforcement of style, but this is more a matter of opinion. I would correct students who wrote any of the following, as a matter of style:

$$\sin(\theta),\qquad \frac{a}{\frac{b}{c}},\qquad x2y.$$

They should also be learning the Greek alphabet, e.g., that $\rho$ makes the "r" sound and is not p, and so on.

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    $\begingroup$ I am at a loss for what you think sin(𝜃) should be corrected to. $\endgroup$ – Henry Towsner Feb 7 at 17:57
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    $\begingroup$ @HenryTowsner: The point about $\sin(\theta)$ is that the parens are not necessary because there is no ambiguity in the order of operations. Students write these unnecessary parens these days because of the influence of computer languages and calculators. $\endgroup$ – Ben Crowell Feb 7 at 18:34
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    $\begingroup$ I write $\sin(\theta)$ all the time, both for my students in class and sometimes on my own, as well. Am I mathematically illiterate? $\endgroup$ – Brendan W. Sullivan Feb 7 at 18:44
  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – quid Feb 10 at 23:17

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