Tag Archives: doing math

On not giving up

3 Dec

On my very first day of high school, my calculus teacher had me take a pre-test to make sure I belonged in the class.  I was the first freshman of that ridiculously excellent school’s history to be in calculus, which was entirely due to the INCREDIBLE program at the University of Minnesota, UMTYMP– if anyone you care about is a 10-12 year old in the greater Twin Cities area, I highly highly suggest checking out the program.  The next night my parents were extremely concerned about me, as I was in tears over my calculus homework.  I thought that I wasn’t good enough and that it was too hard for me, rather than that perhaps I was less prepared or mathematically mature than expected (see my post on whether we should do math at all for more thoughts in this direction).  I still ended up rocking both years of AP calculus, and doing okay at Cal State Fullerton, where I took a few more math courses during my remaining years of high school.

I made this picture all by myself

I made this picture all by myself

It’s been over a decade since my first math tears, and about three years since my last ones- I struggled so much over integrals over contours in my complex analysis course, and definitely had tears spring to my eyes when I finally lugged myself over to the math tutor (incidentally, I had some beer with him over the summer at a conference and he is awesome) to ask about one particular problem.  I was ridiculously heartened when he didn’t immediately spit out an answer, and told me that “the most important thing is to stay calm,” which kept me from nervous breakdowning at a total stranger.  When handing back exams to my calculus students, I’ve definitely noticed a few hastily wiped tears and debated on commenting and potentially embarrassing them, or letting it go and feeling like a callous ass for the rest of the day.  Now I tell them the story in the next paragraph.

A few months ago, after doing that mini-triathlon, I saw a groupon for Crossfit and thought I’d try out what I affectionately call the “cult.”  I’ve mentioned before that I’m incredibly weak, and I’ve definitely noticed this during my crossfit classes- I’m an okay runner, but in terms of strength I’m last, and definitely lag behind the pregnant woman (who is a beast).  Until yesterday I’ve been completely at peace with this, because I can tell I’m getting stronger and improving and feeling pretty good, even when the trainers have to set up separate stations for me because I can’t do a push up with good form.  At my last session we had a coach I hadn’t met before who started us all with the same strength, and gave us a few minutes to fiddle with it and add more weight as needed.  It is clear to me now that the burden was on me to speak up or simply do my thing and give myself less weight after trying out the initial condition, but I was too shy/ashamed to help myself do my best, and I ended up sabotaging myself and finishing my workout with many, many tears because I felt inadequate and incompetent.  I’m glad that I finished, but the coach had to lessen my weight partway through and I did one less round than the other women.

The moral is that we aren’t all born with the same strengths.  As my boyfriend said when comforting me that night, some of these women in this class have been playing sports since middle school, when I was doing Destination Imagination and Future Problem Solvers, or at least did athletics in high school, when I was playing clarinet in the marching band.  And that’s OK.  Those women will be lifting more than me, and I will be doing the best I can with what I have.  Not everyone in my calculus section is going to be an A student- they never learned algebra and are working two jobs to support themselves through college, so don’t have the time to master algebra and pass their other actual classes.  That’s OK.  What’s important is that they finish the workout and give it the best they have.  And tears are OK too, they happen to all of us.

Brainstorming, also, Cantor sets

10 Jun

So I studied for those prelims for a few months (maybe two) on my own, reading old notes, highlighting things, rewriting relevant proofs, running through my homeworks, attempting extra exercises from the textbooks, the usual extreme studying.  I don’t know about you, but I am SO MUCH BETTER with other people when doing just about anything.  Knowing someone else will eat my baked goods makes my baked goods better.  Being accountable to a colleague makes me justify my ideas more and I write down fewer false things.

Math is a creative endeavor.  People seem to think mathematicians are somewhere between human calculator, engineer, and crazy person.  But we don’t get grouped with, say, artist, writer, poet, all that often.  I’m going to quote some wikipedia here:

Plato did not believe in art as a form of creation. Asked in The Republic,[18] “Will we say, of a painter, that he makes something?”, he answers, “Certainly not, he merely imitates.”[16]

Now, as with many creative things, there are certain tools you can use to help with math.  Going on walks is a good thing.  Eating breakfast.  Naps, even.  But the thing I love, and the thing that helped me pass my prelims (whoooo), is brainstorming.  A few weeks ago I read an article that a facebook friend posted (isn’t it funny how you know what I’m talking about when I say “facebook friend” rather than “friend”?  In this case, a guy I went to high school with) about how brainstorming is basically stupid and broken.  And, given the parameters that the article offers, it’s pretty right: get in a group, generate lots of ideas, don’t be critical.  I don’t have a lot to say about brainstorming (though Jonah Lehrer for the new yorker and a guy from Stanford’s d.school do and these are both fascinating), but I do have things to say about problem solving with a group.

Brainstorming, as this three-step outline is, isn’t the way solving problems with a group should be done.  He’s on point with steps one and two, but step three should be different: be critical.  Fight for your ideas.  Fight other people’s wrong ideas.  I work with other grad students a lot, and I often don’t realize that I’m completely wrong until I’ve been talking about an idea for a few minutes.  I need them to fight me in order to learn and understand and be better.

The idea behind brainstorming is that if you’re critical of others’ ideas, they won’t want to share them and will clam up.  I absolutely felt that way for the first year or so of graduate school- I would feel that my peers shot me (personally) down, and that I didn’t have any good ideas.  What finally changed my mind and made me more combative was realizing that I do have good ideas, sometimes.  Not all the time, probably not even 50% of the time, but sometimes I’m right, I’m absolutely right and I can prove it if you-just don’t shoot me down or interrupt me or dismiss me.  These last three things still happen, and I’ve learned to fight back in a nonconfrontational manner- it’s easier in my case because everyone in the room just wants to reach a solution.  Some people have egos (I do too), but we learn to listen to each other to the extent we’re capable, and to make each other listen when we can’t extend ourselves to do so.

Basically, I agree with this guy: “Innovation Is About Arguing” is the first bit of his title.

On a totally different topic, let’s talk about the Cantor set.  It’s a pretty cool subset of numbers with lots of unintuitive properties.  Building it is fairly straightforward.  Look at a number line.  Focus only on the segment between 0 and 1.  Cut the segment into thirds, so you’ll have notches at 1/3 and 2/3.  Delete the middle third (1/3,2/3) so you end up with two segments.  Do this over and over again, deleting the middle third from each segment you have at any one point.  Look at the picture

 

I DID NOT MAKE THIS PICTURE it is from wikipedia.

And you just do that forever.  A few cool things about this set:

  • It contains no open intervals, since we’d cut out the middle third from any interval that showed up.
  • Despite that, no points in the Cantor set are isolated: if you take a point in it and look in a teensy neighborhood (like, \pm .00000001 teensy), you’ll still find other points from the Cantor set.
  • It’s uncountable, which means literally that you can’t count it , even if you had infinite time.  Counting numbers are countable (1,2,3…), and so are numbers that can be written as fractions (1, 2, 1/2, 3/4…), but all the numbers between 0 and 1 aren’t (this is Cantor’s diagonalization argument, which I’ll blog about some time).  You can show that the Cantor set is uncountable by making a function from it to the interval [0,1], which hits all the numbers between 0 and 1.  If it hit all the numbers between 0 and 1, we say the map is surjective or onto [0,1].  If it’s surjective, then for every number in [0,1], there’s a number in the Cantor set that maps to it, so the Cantor set is bigger than or equal to [0,1] in size.  Since [0,1] is uncountable, this means Cantor set is too.
  • If you consider the set C+C = \{x+y : x,y\in C \}, you get an interval, which is way incredible and unexpected because it contains no intervals- it’s like putting together two sets of discrete bread crumbs and magically having a loaf.

You think about that last bullet point, and I’ll do a post later proving it.  Hint: know how we can think of any number as a decimal expansion?  Well, you can also use base 3 and do a ternary expansion, so 1/3 = 0.1, 2/3 = 0.2, 4/9 = 1/3+1/9 = 0.11, etc.  The Cantor set, by the way we defined it, is all the numbers between 0 and 1 that contain no “2”s in their ternary expansion.

Hasta later!

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