kitchen table math, the sequel: question for Katharine

Thursday, September 1, 2011

question for Katharine

I've been meaning to post something about the wording of Venn diagram problems for a while now.*

e.g.:
Thirty students take geometry, and 25 students take Spanish.
I have finally begun to divine fairy rapidly that what this sentence really means is: 
Thirty students take geometry, and 25 students take Spanish. Of those students, some may take both geometry and Spanish. Then again, maybe not.
I divine it, but I don't like it. I would never, ever write the first sentence if what I meant was the second sentence(s). (So no one's going to hire me to write Venn diagram problems any time soon, but never mind.)

I've called this post "question for Katharine" because I'm pretty sure that, as a writer, I'm following an implicit rule I know but can't name.

I might write the first sentence if I meant:
Thirty students took geometry, and another 25 students took Spanish.
Actually, I probably would write the more explicit two-sentence passage because I specialize in spelling things out. But I wouldn't think it was wrong if someone else wrote Thirty students take geometry, and 25 students take Spanish when what they meant was Thirty students take geometry, and another 25 students take Spanish.

But I perceive Thirty students take geometry, and 25 students take Spanish as an incorrect way to express Thirty students take geometry, and 25 students take Spanish. Of those students, some may take both geometry and Spanish. Then again, maybe not.

In short, according to my non-conscious rulebook: 
Thirty students take geometry, and 25 students take Spanish
Thirty students took geometry, and another 25 students took Spanish.

but

Thirty students take geometry, and 25 students take Spanish
Thirty students take geometry, and 25 students take Spanish. Of those students, some may take both geometry and Spanish. Then again, maybe not. 
Why is that?

I started to think it through the other day, but then it struck me that Katharine, who is a linguist, may already know.

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* awhile? a while? I'm going to figure that out soon.

21 comments:

GoogleMaster said...

My unconscious rulebook is exactly opposite yours. To me, "Thirty students take geometry, and 25 students take Spanish." means that of the universe of students involved in the problem, 30 are taking geometry; also, of the universe of students involved in the problem, 25 are taking Spanish. The statement implies nothing about the relationship between the number of geometry-takers and the number of Spanish-takers, unless the only Spanish class is scheduled at the same time as the only geometry class, but then if that were the case, then the problem would have said so.

When I read "Thirty students take geometry, and 25 students take Spanish.", I know a Venn-diagram type of problem is coming up.

Maybe my rulebook is more literal than yours, or maybe I have worked (for fun, usually) many more problems that start out this way. ;-)

Katharine Beals said...

My gut sense is that it's all about context. In the context of a Venn Diagram problem, it means possibly overlapping; in answer to "how many students are taking each class?", posed by someone who wants to make sure there are enough text books for each class, it also means this.

But in an "unmarked", generalized context, I'm not sure I have a clear of what sentence means (though, lacking a clear sense, I think I default to the less restrictive interpretation, which, again, includes possible overlapping. Perhaps part of the problem is that it's hard to imagine a context for this sentence. It's easier to imagine one for "30 students took Spanish, and 25 took French", and in this context one would tend to assume no overlap. Geometry and Spanish don't seem like either-or optios (unless, as Googlemaster notes, they're offered at the same time), but French and Spanish (in as much as students choose only one foreign language) do.

Perhaps I'll do what many linguists I know would do at this point: take an informal poll and get back to you!

MagisterGreen said...

In the absence of any markers in the sentence that expressly or implicitly denote exclusivity, such as "while" or "another" I wouldn't say that you could automatically assume that the two activities, Geometry and Spanish, are mutually exclusive. Thus you have to allow for the possibility that there is overlap among the students. Then again, maybe not.

In the context of making a Venn diagram, the statement allows the possibility of overlapping. In the context of a normal(-ish) conversation, I'd say it's just opaque English.

Catherine Johnson said...

Forgot to mention: this problem relates to an earlier post:
http://kitchentablemath.blogspot.com/2011/08/question.html

Catherine Johnson said...

I'm still too thick-headed to think it through, but maybe it's simply that the problem as written doesn't follow the 'rule' of explicit writing -- of saying what you mean clearly & not being intentionally ambiguous.

But I think, too, there's an issue about what can be left unsaid....

Of course it's true that the sentence doesn't tell us anything one way or the other about the possible intersection of the sets, but it seems quite wrong to me **not** to tell us anything about it -- and, since it doesn't, my default assumption is that these are indeed two separate sets.

Catherine Johnson said...

unless the only Spanish class is scheduled at the same time as the only geometry class, but then if that were the case, then the problem would have said so.

But why is this the case?

Why is this the kind of thing that, at least for you, has to be said?

(I'm not disagreeing with you! I'm asking what the rule is that makes you believe some things should be said & others don't have to be said...)

Also, I'm wondering why we have different rules.

Probably because you've spent a lot more time reading Venn diagram problems!

Linda Seebach said...

I think it's a math thing; assume nothing that is not explicitly stated.

In the lateb19th century mathematicians began to look anew at Euclidean geometry, and not only did they discover that the fifth postulate (about parallel lines) described only one of several possible geometries, all equally consistent, but also that Euclid had made a lot of assumptions he didn't acknowledge that also neded to be axioms. See, e.g. David Hilbert.

Glen said...

Out of context, there is an implicit contrast: "Two students do X, and three students do Y" is a common sentence pattern in English, which implies that you are talking about five students and how they divide up. If X and Y aren't obviously alternatives, this form implies that they are. "Two students take Spanish and three take French" would be natural English if talking about five students. If talking about four students, it would be odd. "Two students take Chinese, and three students are hispanic" might prompt an exasperated, "What, hispanic students can't take Chinese?", because it does seem to contrast X, taking Chinese, with Y, being hispanic.

If you didn't mean to contrast one group of students from another, you would probably say it differently. For example, "The geometry class has 30 students, the Spanish class has 25, and some students could be in both classes."

But in the context of a math problem, all of that changes. Math problems written in natural language still require you to make disambiguating assumptions--it is still natural language, after all--but they want to you put more weight on what is literally said and less on other factors ("bayesian priors").

In such a context, you are trained to interpret "Two students do X, and three students do Y," without assuming two disjoint groups. You learn to be literal and legalistic in a math problem context, which is a context-based re-weighting of the factors involved in interpretation of language.

Catherine Johnson said...

I think it's a math thing; assume nothing that is not explicitly stated.

That's the problem for me: I am being asked to assume "for the sake of argument" that some students may be taking both:

Of those students, some may take both geometry and Spanish. Then again, maybe not.

Those last two sentences aren't explicitly stated, either.

Allison said...

to be clear, your "another" sentence is COMPLETELY WRONG!

"30 students took Spanish. Another 25 took French."

That "another" means *a set different from the 30 you mentioned*, so it NEVER includes the 30. That's NOT what they are saying.

They are trying to say "there are 30 students who are taking geometry. There are 25 students who are taking Spanish."

Your brain is linking where there is no explicit link. I think this is an issue of logic--the English doesn't make any link between the two either, actually.

The answer is that math is not English. Math is math, and math is precise in a way that verbal languages are not. That is why in math, we use symbols rather than English words, so that it's clear what we mean. (And why the focus on words in k-8 or 9-12 math is so terrible, because it increases the cognitive load and confuses the students!)

As someone is better at math, they are better able to speak in a way that is explicit, but they are also better at THINKING more precisely, and recognizing only what's specifically spelled out.

Allison said...

-- a rule in English of saying what you mean clearly & not being intentionally ambiguous.

But the statement "30 students take geometry. 25 take spanish" is CLEAR and not intentionally ambiguous to me.

To say anything longer or with more words would be confusing. this is precisely what is meant. there are 30 students taking spanish. there are 25 students taking geometry. you don't know anything about the linkage between them yet. To say "some may take both, maybe maybe not" is PRECISELY what the first two sentences say but without your extra sentence baggage.

Catherine Johnson said...

But the statement "30 students take geometry. 25 take spanish" is CLEAR and not intentionally ambiguous to me.

I can guarantee you it's not CLEAR (all caps!) to others.

That's what I'm asking about.

Catherine Johnson said...

to be clear, your "another" sentence is COMPLETELY WRONG!

"30 students took Spanish. Another 25 took French."

That "another" means *a set different from the 30 you mentioned*, so it NEVER includes the 30. That's NOT what they are saying.


hmmmm...not sure whether I've written the post clearly.

I'm saying exactly what you've just said. Or trying to.

I'm saying that "30 students take geometry, and another 25 take Spanish" means two separate sets.

I'm saying that the word "another" means *a set different from the 30 you mentioned*.

Catherine Johnson said...

To say "some may take both, maybe maybe not" is PRECISELY what the first two sentences say but without your extra sentence baggage.

"extra sentence baggage"?

nope, wrong

A good writer works with what he or she knows readers will take away from his or her prose.

Omitting "extra sentence baggage" because your reader ought to understand what you're saying is bad writing.

Catherine Johnson said...

Out of context, there is an implicit contrast: "Two students do X, and three students do Y" is a common sentence pattern in English, which implies that you are talking about five students and how they divide up. If X and Y aren't obviously alternatives, this form implies that they are. "Two students take Spanish and three take French" would be natural English if talking about five students.

That's it!!

Thank you!!!

I work in natural language; my entire career is about using language as my reader expects it to be used ---- and, interestingly, the Venn diagram problems are far and away the most difficult problems for me to 'read' in the way that math is read.

Catherine Johnson said...

Glen - if you're around - I don't quite follow your point about Hispanic vs. Chinese ---

Are you saying there is something about parallelism and 'category equivalence' (apples to apples) "30 students take geometry, and 25 students take Spanish" that makes it seem as if these are two sets in natural language?

(Not sure what the correct term is for 'category equivalence'....)

I think there's something about the parallel grammatical structure that pushes me towards a disjoint sets reading....

I'm back in the thick of planning my composition course; I'll have to notice how textbooks explain parallelism.

Anonymous said...

I looked in the book I'm teaching from this year for Venn diagram problems, and they are careful not to use "and".

For example:

"A class of 30 music students includes 13 who play the piano, 16 who play the guitar, and 5 who play both the piano and the guitar."

Allison said...

-- I am being asked to assume "for the sake of argument" that some students may be taking both:

No, you're wrong. You're not being asked to assume anything.

You're being told *NOT TO ASSUME ANYTHING*.

You are doing the assuming. You're assuming the two sets are linked. But the sentence "30 take X, 25 take Y" does not entail any such relationship. So that's why the extra sentences are unnecessary: no assumption about connection is being made.

Allison said...

---Omitting "extra sentence baggage" because your reader ought to understand what you're saying is bad writing.

In a proper math problem, everything you need to know is *given to you*. So a math writer can make extremely strong assumptions about what a math reader "ought to know": exactly what he tells them.

But we have to take short cuts. We can't begin every problem redefining the integers, 0, the empty set, that A=A, etc. We assume a body of knowledge as prior knowledge. This assumes you have agreed on what the words mean. More it means you have to agree to MAKE NO ASSUMPTIONS beyond the specific math you've been taught (like, say, you can assume we've defined 0, and equals, and the integers, etc...).

If you can't assume someone is mathematically versed enough to understand the concept you're at, the amount of extra sentences you need would fill volumes of textbooks--so you can't write down all of the extra "this means don't assume the universe of possibilities" sentences to clarify. Clarity comes from working with only writing down the set of truths you need to solve the problem at hand.

Glen said...

Glen - if you're around - I don't quite follow your point about Hispanic vs. Chinese ---

Are you saying there is something about parallelism and 'category equivalence'...?


Maybe. This is what I'm thinking, FWIW:

Placing two statements in apposition in this way (statement1 and statement2) suggests, without explicitly claiming, a relationship between them.

Interpreting such an utterance obliges us to look for the relationship. Context may make it obvious. If not, the semantics of the statements themselves may make the relationship obvious.

"The drought is continuing, and we're having trouble paying our bills." Even without context, it seems most likely that this means A is causing B, even though it doesn't explicitly say so.

"Two students are studying Chinese, and three are studying French." Out of context, semantic knowledge suggests that this is not A causing B but A being contrasted with B. Semantic information makes the difference.

What if we have no semantic information, only syntax? "A is doing X, and B is doing Y." That's the pattern of both the drought example and the foreign language example but, without the semantic clues, it seems to imply contrast, not causality. So, maybe, this pattern itself implies contrast by default but is easily overridden by semantic and contextual information when that information is available.

So what about "Two students study Chinese, and three are Hispanic"? Since these two statements are bundled as if they were related, and there is no plausible causality (people studying Chinese won't make other people Hispanic), and we don't know anything about context, it could easily sound like a contrast is being made between STUDYING Chinese and BEING Hispanic, which suggests that the speaker could be assuming that if you're Hispanic, you don't study Chinese.

I think there's something about the parallel grammatical structure that pushes me towards a disjoint sets reading

I think so, too, and I think the reason is simplicity. Some researchers have found something akin to Occam's Razor emerging from both bayesian mathematical models and human psychological studies. It seems that we find simpler explanations more plausible. I'm thinking that "two people do X, and three people do Y" sounds like five people divided into two disjoint groups, because that's a simpler explanation of the "data" (the statement) than some model with two groups PLUS potential and unknown overlap.

Anyway, I find the simplicity explanation most plausible, because it's the simplest.

Lsquared said...

I see how the the parallel grammatical structure (especially when the two statements are in the same sentence) makes you think of disjoint sets. It's also difficult because there's no context. I'll bet that if it were a middle school touting how popular their academic clubs were and they said: there are 30 students in the after school math club, 25 in the French club, and 35 in the science club, you would be thinking to yourself--"I wonder if it's the same involved parents and kids that are making all 3 clubs happen?" That language usually means that the speaker intends you to believe that the groups are disjoint, even though they never actually said that they were.