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I'm a math PhD who has been asked to interview a high school student and determine what he/she is interested in and how strong the student is.

Usually I would want them to talk as much as possible about what classes they've taken and what parts of the classes they like the best. Generally I'd like to allow them to get comfortable talking about what they like rather than what they think I'd like. However, that might not work this time as the student is a non-native english speaker and may get less comfortable as I try to get him/her to talk more.

Thus far I'm thinking about watching how he/she solves different problems in different areas of math, but I would like to know if anyone has done this before or has any suggestions for evaluating a student who might not be comfortable talking much.

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Dear Curious, would trying to find a mathematically adept translator be impractical? –  J. M. Oct 17 '10 at 16:18
    
Well it's not as if this person can't speak english. I just know that it can be uncomfortable for non-native speakers to attempt to talk at length. In any case, while it wouldn't be completely impractical to find someone who spoke the language, it would be quite impractical to pawn my work off onto someone else every time I had to deal with a non-native speaker. –  Curious in the USA Oct 17 '10 at 16:37
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It is customary to mark soft questions like these as Community Wiki. –  Aryabhata Oct 17 '10 at 17:26
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I'm not trying to be snarky here (really!), but: is it so clear how to assess the mathematical ability of a teenager whose native language is English (or whatever yours happens to be)? It's not to me... –  Pete L. Clark Oct 17 '10 at 21:04
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Don't underestimate the kid. If he has been in an English speaking country for more than 3 or 4 years, expect him to have picked up all the necessary jargon. –  Willie Wong Oct 17 '10 at 21:25
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2 Answers

Well, I'm a non-native English speaker, so I can provide some insight.

I think that one of the first problems would be names and notation in general. As far as I can see, there is no easy way to solve this unless you already know the customs of the country the student is coming from, but your education will probably make it easier to guess what means what. For example in my country we use $log_{base}number$ for general case logarithms, $lg$ for $log_{10}$ and $ln$ for $log_e$. Another example would be popular sets. In some countries 0 is a natural number and in some other countries it isn't. Another example would be popular trigonometric functions. In my country for example secant and cosecant are considered deprecated and aren't used. Same thing with grads as angle measurement units.

Problems with this part can be especially tricky because they can stay undetected for a long time. I've seen some math problems whose solutions depend on whether zero is a natural number or not. Another example here can be integrals. As far as I can see the most popular name in English for operation opposite of finding a derivative is antiderivation. In my area that's called indefinite integration and the indefinite part if often omitted.

Still I think that most of the notational problems can be solved with Wikipedia.

Next part would be names of fields of mathematics and popular theorems. How big problems this is going to be is closely related to which language terms come from. Sometimes terms are going to be shared between languages and sometimes there will be no direct translation. For example in my language, there is no term for calculus. Instead we use mathematical analysis, which as far as I know is superset of calculus.
Also in some cultures some theorems are named after famous mathematicians and in some other cultures they are named after what the theorem describes. I can't find any examples right now, but I know I encountered such problems when asking few questions here.

Another problem could be non-existence of a concept. For example I was unable to find any mention of Ivan Vsevolodovich Meshcherskiy's equation (original article here) for bodies with variable mass in any English language web-sites. Here it's considered one of the fundamental rocketry equations, right next to Tsiolkovsky's formula. While the example is related more to physics than mathematics, be prepared to find similar examples in mathematics if the student is coming from a different culture.

Since you said that student knows some English, here's my recommendation: Try starting a conversation about some topic in English and see if student is familiar with it. If it doesn't ring a bell, be ready to provide simple examples related to the area. I sometimes had problems finding exact name for some field of mathematics but immediately recognized it as soon as I saw a few simple problems. Another good idea would be to make a list of alternative names for every operation and see if some of them are familiar.

Yet another important thing is to have at least some familiarity with mathematics program which the student followed. Sometimes focus of what's studied in classes can be a bit different than what's written on paper. I remember now back when I was in secondary school we had formal mathematical proofs in books and they were supposed to be a part of the curriculum, but teachers in schools just decided that proofs are not important and did not teach them. Also in some schools less attention was payed to proper use of terminology. I know some of my friends who didn't know what was derivative and it's graphical representations, but knew that $x^2{'}=2x$ and same thing for integrals. Another point can be some obvious areas which may have been skipped. When was in secondary school, we didn't learn almost any probability and statistics and did very little work with percentages. On the other hand I see that in some other countries it's expected that someone who has graduated from primary school can without any problems calculate his interest rate.

I can't think of anything else to add at the moment...

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Seeing textbooks or other material, in any language, from which this person has learned mathematics, would be useful and relatively easy to interpret for an Anglophone mathematician. If these are not available then equivalents may exist on the internet. Pointing to material in books in English and seeing if it is familiar may also reveal something.

At the high end, the International Mathematical Olympiad problems are available online in dozens of languages, probably including that of the student's native country. This could at least tell you whether the student has heard of some concept used in a problem, and you would have a translation into English so you know you are discussing the same thing.

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