# Teaching words with multiple meanings in Mathematics and English

We want to teach about words that have multiple meanings, in mathematics and English for middle school children (e.g., function, root, or, volume, angle, constant). This is for students who are learning English as a second language. That is, how do you teach words that have different connotations or meanings in mathematics compared to English as taught in literature class.

Other conversation on this forum have identified these words, creating a nice reference list. Examples of vocabulary that have different meanings in Mathematics compared to "everyday" English

Now I am asking how to teach these words with multiple meanings.

Personally, the perspective I recommend is kind of the reverse. Despite the linked thread, I find that most of the time, the mathematical term has some reason why it was picked for the mathematical concept in question. It wasn't picked in an arbitrary or malicious fashion. At some point when the practitioner first used it, it seemed like the best English word for the concept in question.

Of course, natural language has ambiguity and multiple meaning to most words: but if you dig into the English dictionary, usually one of the several senses for a word is directly applicable to the math concept. As someone who does a lot of reading/writing as well as math instruction (and usually wants to dig into the history of math concepts before teaching them), the English connotation is clear, and I get a bit frustrated that my students' English isn't strong enough for that to be immediately evident -- as well as that there's no space in normal books for that connection, nor time in class sessions.

So: On reading the OP's question my heart kind of leaped for joy a tiny a bit at having the opportunity to make these connections clear. I already spend time in some cases reflecting explicitly on the English meaning of some more difficult terms as I introduce them in my college courses -- e.g.: predicate, relation, equivalence -- but I often wish wistfully I had time to do that more comprehensively.

Let me pick 3 semi-random examples from the linked thread of "different meanings". For convenience I'm using dictionary.com as my reference.

• Even: The primary definition in English is "level; flat; without surface irregularities; smooth". Now, this same term has been used since ancient Greece (written ζυγός αριθμός). The point is that the number is evenly divisible by 2, that is, when halved, there are no "irregularities" in the way of any awkward remainder or fraction in the result.

• Series: Primary definition here is "a group or a number of related or similar things, events, etc., arranged or occurring in temporal, spatial, or other order or succession; sequence". Clearly that's appropriate for both a run of TV show and an ordered list of values being summed. (The only nuance being that in English series and sequence are totally synonymous, where in math one does need to remember they're a bit more specific.)

• Volume: In this case the relevant sense is a bit further down the list: "a mass or quantity, especially a large quantity, of something: [e.g.] a volume of mail". So this sense of "filling an amount of space" is definitely present in English usage. Consider the English word voluminous, "forming, filling, or writing a large volume or many volumes" which has the same sense of filling-space, but is not (to my knowledge) used in mathematics.

A few other things that I find my students respond quite well to when I take a momentary historical detour: (1) the fact that the radical symbol comes from an over-time transmogrified "r", being an abbreviation for radix/root, (2) the fact that our summation symbol $$\Sigma$$ (Sigma) is the Greek equivalent to our capital "S", the obvious choice to abbreviate "summation" (and you can easily draw out the graphical evolution from one to the other; mostly, drop the lower bar), (3) likewise for capital $$\Pi$$ (Pi), which is the analog to our capital "P", the obvious choice to abbreviate "product".

So in short, you should look at this class as a huge opportunity -- not to highlight the differences, but to underscore the similarities of the meanings between the English and mathematical terms, that would in many cases by cryptic to people with weak English vocabulary. The more research and connections you can make like that, the stronger it should be in your students' minds, and you get to serve two goals at once with this project. In some sense I'm even a tiny bit envious of your opportunity. Good luck!

One very practical approach is just to teach the math in close to 100 percent English. Math is actually easier than literature or history to learn in a foreign language. So don't do it bilingually. Explanations, sure. Fine. But nominally 100 percent immersion in math class. This also helps to make clear just subliminally that you are using terms in their math context such as line or area...no differently than for English language only students who can tell by context when a word is used for a mathematical purpose as opposed to essay writing usage.

For instruction in English itself, this is a normal second language class. History may be bilingual, if students are weak. And if at all possible, avoid assigning classes on a third language. My sister did immersion year in German high school. Math was easiest. Well after Englush. Then science was next...easy except for occasional vocabulary. German lit and history were hardest because of the verbal content. Well...except for Latin. That was just a bridge too far and they eventually acceded to letting her drop it. Just too hard to learn a third language in a second language.

• Many thanks for your response to my question about how to teach words with multiple meaning for ELL. I will consider 100% immersion except for explanations. Since so many methods exist for teaching vocabulary for ELL (word banks, examples, develop explanations, etc.) I thought many ways would exist for teaching words with multiple meanings. Jun 15 at 21:12
• I would avoid stressing multiple meanings and nuances for students that are just learning. Better to handle them as they come up. It is a common fallacy on this board, that people struggle because of inadequately precise definitions. And if we were just super precise and thus complicated, students would do better. But this is wrong. Humans are animals. We have limited brains. We do better if we learn things at a simple level first. Including over simple. There is still a benefit almost always from the simple version. Then pick up nuances later, building on something. Jun 16 at 11:22
• As far as the differences of math. I think you will be pleasantly surprised that students will know when they are in math class, doing math stuff, to use math implications of words. In the cases they don't, deal with it ad hoc. But please don't overcomplicate the math class by dwelling on terminology subtleties as opposed to building number and equation manipulation skills. And for God's sake, don't mess up a normal ESL class by discussing math nuances while teaching normal usage. In that class, you should stress most common usage. They can deal with math class variances in math class. Jun 16 at 11:29
• Don’t overlook the possibility of using Esperanto as a bridge language. Jun 19 at 8:55