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edited Feb 8, 2020 at 5:27

If a student said something like "We take $x$ and plus it with $5$", I'd be inclined toI 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)

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, I don't think that's not where a mathematics teacher should look to improve their teaching - rather, they should look to the larger picture of communication.

If a student said something like "We take $x$ and plus it with $5$", I'd be inclined to 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.

As a 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. Basically: even in cases where the language isn't important, I don't think that's where a mathematics teacher should look to improve their teaching - rather, they should look to the larger picture of communication.

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)

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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created Feb 7, 2020 at 22:53

Milo Brandt

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.

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.