This is one of the lowest percent corrects I've seen on a Question of the Day -- as low as the percent correct for the 3 people in an office question.
Why is that?
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They do what they do.
Thinking about schools and peers and parent-child attachments....I came across one of my favorite posts .
23 comments:
Very strange - I can only suppose that many people shut down and resort to wild guesswork when any aspect of infinity rears its head?
I agree with Amanda. People are scared of infinity.
I have no idea --- I asked because I wouldn't have predicted that this would be one of the most-missed questions.
Doing this problem, I had self-doubt based entirely in my experience of SAT questions: this one seemed too easy, so I kept assuming that I must be missing something.
I didn't consciously "stick with the facts" (I'll begin working on that for the first time tonight) -- and I wasn't writing anything down because I wasn't supposed to be doing SAT Questions of the Day.
Thinking about it now, I bet people add conditions to the problem - the same way some students end up adding equality of the two radii in the half-circle problem?
I bet that's it.
It's a little hard for me to see what information students would add, though.
gswp -- interesting.
What happens when a student encounters infinity in a problem -- ?
What answer would a student tend to come up with?
"What happens when a student encounters infinity in a problem -- ?"
I suspect that nothing good comes out of it.
One of my current pet projects is sorting out infinity and infintesimals ... partially because I'm becoming more and more convinced that a huge chunk of the crash-and-burn that often accompanies calculus is that calculus is the first class in which students must *really* understand these concepts.
[Of course, I don't think most kids who pass calculus actually *do* understand this ... they just learn how to "crank the wheel" as it were].
As nearly as I can tell, my Dolciani Algebra book has one paragraph on the fact that fractions are continuous and that this is different from integers. This is a pretty big change ... a given number doesn't have an "adjacent" number ... WTF? But this is glossed over and we move on. Then in calculus this becomes important and we have 12 years of using numbers with very different properties.
Infinity is like this, too, I think. Just in another direction. About six weeks ago I finally sorted out why you *can't* arrange Reals in the sort of diagonal-path that you can use for Rationals to demonstrate that the Rationals have the same type of infinity as the Integers. It isn't too hard, but it also isn't/wasn't taugh to me ... and if I look at BS math curricula it appears that it often isn't taught even to 4-year math majors :-(
So bringing these sort of questions in at a K-12 level probably is asking for trouble ... the concepts just haven't been taught (and if taught, haven't been taught well).
-Mark Roulo
I'll also add that I think the way sets are introduced often leads to problems later on. The common way (that I have seen) to introduce sets is with nice, small sets. Infinite sets have strange properties, but math has a lot of those sets (or at least seems to deal with them a lot), so the initial introduction to sets seems to create the groundwork for lots of fear and confusion later on.
-Mark Roulo
I'm going to be mulling the crash at calculus ----
I have a memory of Wu asking a question in class related to...something like the intersection of two infinite sets: is the intersection of two infinite sets smaller than each of the two sets?
(Allison & Cassy - if you're around - correct me if I'm wrong - thanks!)
I think there's also another confounder - since 7 and 13 are primes, it is easy for a student to think "well, there is 7 x 13 and no factors in common, so it must be just 1." But of course that leaves out 7x13x2 and 7x13x3, ad infinitum!
Cantor diagonalization is usually taught to math majors, though I can't remember where (too long since my undergrad days). I tried finding them in the course catalog here, but math classes are very lightly described, and the mathematicians don't bother putting detailed syllabuses on the web, so had trouble finding out which class they are in. I found the material eventually in the "introduction to proof" course, which is taught just after calculus here, and includes such basics as proof by induction.
CS students may get uncountability arguments in the corresponding "applied discrete math" class, but aren't guaranteed to, unless they take an elective in computability theory.
I showed my son Cantor diagonalization before he finished 4th grade, but I suspect it is one of those concepts that takes a while to sink in. I should check to see if he still remembers the argument about why real numbers aren't countable.
I seem to recall having an assignment wherein I had to prove the rationals were countable in a first-semester grad level analysis class. Maybe others encountered it sooner, but that's when I was first introduced to it.
Well, I'm sure I got Cantor diagonalization before grad school, but I can't remember whether I had it before entering college. Too long ago now!
If I did have it before college, it was not from the school curriculum. Probably from Martin Gardner's Mathematical Games column in Scientific American.
I saw it in my measure theory course. At MIT, measure theory was not a required course for vanilla math majors--you could take it or manifolds, iirc. It was not the analysis course based on Rudin that I took at Berkeley (though perhaps it was in that parallel course at MIT.)
The ugrad math measure theory was a required grad course for some subset of the EEs, though, so many were grad students seeing it for the first time.
I would have been unpersuaded by the diagonalization argument any earlier. Just because I demonstrated that one technique didn't allow me to write down all the reals wouldn't have convinced me that another way wouldn't have. I wouldn't have believed that the diagonalization argument accounted for all mappings to countability as a younger student. I needed a lot of force to convince me of that as a senior taking measure theory.
If I saw the diagonalization argument in computability theory, I've plum forgotten. I took that course twice, and taught it twice. I am sure there could have been a homework problem that diagonalized machines, but it made no impression on me at all.
Badly misleading question (surprise!) because asking "how many integers" normally anticipates a finite number as an answer. While it's true that the cardinality of a countably infinite set is "more than 13" it's also more than 71, or 315,000,000 or any other finite number. The best one can say about (E) is that it isn't wrong, and all the other answers are. It isn't right.
FYI, my touchstone is when a student can answer both:
1) Why the rationals diagonal trick *doesn't* work for reals, and
2) Why the reals diagonalization doesn't apply to rationals.
It has bothered me for a long time that the two "proofs" looked so different. There are, of course, reasons, but (again) they don't seem to be taught :-(
-Mark Roulo
"Just because I demonstrated that one technique didn't allow me to write down all the reals wouldn't have convinced me that another way wouldn't have."
You were correct to be skeptical. My current view (and I think I'm correct) is that there is a way to write down all the reals *YOU* will ever see and then apply the rationals trick to them. Ta-da.
Unfortunately, it turns out that there exist reals that one cannot "write down" in any finite way. The diagonalization trick depends on this.
So ... computable reals have same cardinality as the integers. Non-computable reals have a larger cardinality. Non-computable reals are very strange ... mathematicians (well, some mathematicians) claim that they exists, but we can't write down (this is mostly correct) any of them, even with arbitrary notation. Pi is computable. So is e. Etc. In short, every number you have ever seen in your life (and ever will see) is computable.
*Every* explanation of the two "proofs" leaves out this little tid-bit.
-Mark Roulo
"I have a memory of Wu asking a question in class related to...something like the intersection of two infinite sets: is the intersection of two infinite sets smaller than each of the two sets?"
The answer is, "it might be." Depends on the two sets.
Which kinda gets back to my complaint about sets. Any my complaint about holding off on infinities (especially if you are going to test them!).
-Mark Roulo
Mark, that wasn't even my point, but it's a good one. My point was just that as a student, when are proofs of impossibility really just lack of imagination? You can't use the proof that sqrt(2) is irrational to prove that sqrt(4) isn't. And what if the proof is wrong? So how do I really know that I've covered all proofs when I hit a contradiction? There's an entire branch of math called constructivism based on this point, actually. The original constructivism!
But to your greater point, Wu talks about this in a paper that has gone largely unnoticed. Basically, the undergraduate math major is a guided discovery project where it is okay that students don't understand, because they'll get it again the next time around in grad school. And if they don't get it during coursework in grad school. they'll figure it out when teaching ugrads, or during their quals. Trust the spiral!
Except most math majors never go to grad school at all, and so they will never see it again. And many of those that don't will end up teaching k-12, so the teachers will have these incorrect notions in the head that have never been corrected, and lo, will pass them on to future generations.
It is part of why he thinks teaching the material of how to understand school mathematics is so important--because this is what most math majors will end up doing, so we should be sure they do it well.
*Every* explanation of the two "proofs" leaves out this little tid-bit.
I got the countability of algebraic numbers and of computable numbers somewhat later than I got the countability of rationals and uncountability of reals---probably in my grad school class on complexity theory, but it is covered in courses on comptability, so you can't claim that *every* presentation ignores it.
"My point was just that as a student, when are proofs of impossibility really just lack of imagination?"
For K-12, I'm actually okay (mostly) with an explanation of the proof, but not getting too hung up on the student being actually able to verify it. My belief back in high school was that a sufficiently clever teacher (or my friend James), could come up with an incorrect proof that I would be unable to disprove.
Still, I think the diagonal walk through the rationals is something that a student can understand, and the proof here is that one *CAN* create a mapping.
The problem comes with the reals. My basic complaint here (I hope I'm not repeating myself) is that the proof that the reals has a different cardinality from the integers is wildly different from the proof that the rations have the same cardinality.
With no explanation for why you can't do a diagonal walk through the reals. *THIS* is my complaint about the whole way that this is presented and taught (along with my bitching about how late it is taught). I want an explanation, even if it isn't rigorous.
For my purposes, I think the cantor's diagonal on the reals needs to be accompanied by an explanation for why the diagonal walk doesn't work. And this is what I don't see presented very often (well, "never" for me ... gasstationwithoutpumps appears to have gotten this explanation so it does happen somewhere).
I can live without a super rigorous proof for why you can't do a diagonal walk (for K-12), but I really think some sort of explanation is called for.
-Mark Roulo
"You can't use the proof that sqrt(2) is irrational to prove that sqrt(4) isn't."
For what it is worth, I think that I have a proof for the irrationality of sqrt(2) that (kinda) sheds light on why sqrt(4) isn't irrational. I'd like to see more proofs with accompanying insight ... and they may well exist! I'm not a math person, so I suspect that I've missed lots of stuff that I'd like to know.
-Mark Roulo
What makes the problem hard for me--and why I would have missed it--is that it uses terms that I no longer remember. So I do have to resort to wild guesses because I can't, for the life of me, interpret what this question is asking me to do [smile].
That's what happens when you no longer use math... you lose it. I could still do the computations if I could remember what everything meant [smile]. At least, I'm going to tell myself that!
~Luke
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