Allowing nonstandard mathematical language and/or notation

Question

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.

Answer 1

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.

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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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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).

Answer 6

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.