How I Study for a Math Class

The other day a student came to visit my office hours for some homework help.  At the end of the visit, I wished her good luck with the course.  She mentioned that the class (beginning calculus) was going to take a lot of hard work and studying.  I didn't say anything to disagree with her, and I let her go on her way*.  I do, however, believe firmly in studying smarter, not harder, and that it doesn't take "a lot of work" to do well in a math class.

My Personal Plan

When I take a traditional math class (i.e. lectures, homework or quizzes, and exams), I do all of the homework by myself.  As I mentioned here as well, the struggle of doing this does more to cement things in my mind than anything else I could do.

How do I do a homework assignment that I don't already know how to do?  How do I figure it out?

Here's my plan of attack.

Before Working on Problems


I don't do anything before working on problems.  I start by working on problems.  Some people may start by trying to learn the material needed for doing a problem, and then they do the problems.  I don't like that approach.  To me, it's hard to learn math until I see how I can use it in a problem.  Therefore, I dive right into the problems and then go back later to see what I need to know.


How I Do a Math Problem

1. After reading the problem, make sure I know what the problem is saying.  On a simple problem this may be easy, but on a more complex problem it may take time just to understand the problem enough to proceed with it.
2. Once I understand what I need to figure out, I will be somewhere between knowing exactly what to do to solve the problem and having no clue whatsoever.  In the case that I have no clue, I look back at the math book.  In most any math class the material in the section being study is the key information needed to solve a problem.  I especially key in on examples given in the explanation portion.  Even up to beginning calculus, a lot of problems can be solved by just copying the steps of an example**.
3. If there are any more holes in my knowledge, I will look at either other sections of the book or on the Internet (Wikipedia is great for this).
4. If possible, I check my answer.  If I am incorrect, I go back through the problem again and carefully analyze what I am doing to see what I have done wrong.  If I simply made a careless error, I will probably catch it by doing this.  If I have a larger gap in understanding, I will have to basically re-do steps 1-3.

If I diligently and patiently follow the above steps, I do all of my homework problems by myself.  If a problem is hard, I often repeat these steps in between doing a lot of thinking.

Follow Up


There are a few natural ways I follow up on what I have learned from a homework assignment.  First is going to class.  Seeing the teacher use the material I have worked on can really help make that material be even clearer for me.  Watching someone who knows what they're doing do math for some reason conveys more information than just reading it in a book.

Another thing is to see homework solutions.  For a problem I did correctly, this can confirm my knowledge of it or even show me an easier way to do things.  For a problem I did incorrectly, this can help show me what I need to do later.  In the case that I did something incorrectly, unless I made a small, careless error, what I ought to do to is do more problems like the one I missed; just knowing about how to do something is a lot less effective than doing it and applying it myself.

What Kind of Skills Does this Require?


Being able to do this doesn't require extensive knowledge of a subject.  There are a few skills which get used, however.

1. Figuring out what a problem is saying is a skill.  I have a hard time describing how to do that because what I basically do is a lot of visualization and other mental gymnastics.
2. In looking at the book and looking at the examples, I need to be able to compare what's there to the problem I am working on.  Once I have found a match, I need to have the skill of taking the example and picking out the essential steps which I then have to transfer to my problem.  Basically, this skill is recognizing and reproducing patterns.
3. Doing a problem itself relies on a lot of math skills.  For example, a calculus problem could involve techniques from high school algebra, geometry, trigonometry, and pre-calculus, sometimes all in the same problem.

My opinion is that these skills are developed by doing.  In my mind it's like learning to play a piano: I can't just read a book or have someone explain to me how to play a piano, I need to put in a lot of practice time actually playing the piano.  In my opinion, therefore, if someone is deficient in these areas, they can improve only through a lot of practice doing problems.

The Benefits

In my experience, doing this means I don't have to study that much for a math class.  Once I have done my homework, I have learned the material.  Until I have done my homework, I don't really know the material that well.  That's what it comes down to.  (I just wish some other people got that too.)  If I did my homework correctly, the only extra studying that I end up doing is the short time I spend refreshing my memory of something I learned so I can take a test.  If I didn't do my homework 100% correctly, I figure out what I did wrong and then put in a modest amount of time on practice problems***.  Either way, I don't end up investing that much time into a math class.  (talk about testing)

To someone who may not be "good" at math, the benefits to focusing on doing problems is the same.

Notes:

* = I didn't argue with her at all because, among other things, she wouldn't have believed me.

** = If this is the case, however, then the problem becomes too easy.  As a way of cementing my knowledge, I will try to do subsequent, similar problems from memory.
*** = To get practice problems, I would look to a few sources.  I would do similar problems from my math book for the class.  I would look on the course website for old midterm exams.  I would google the topic I am having trouble with to see what I could get.

The Shame of College Sports: Excellent Read

Here is the link.  I thought it was a very interesting read.  Here are the main points I took away from it.

1. The NCAA is essentially a cartel.  They use their monopoly position to fix the price at which a young athlete may play college sports.  That price is that an athlete may receive a year-to-year full scholarship and nothing more.
2. In many ways, the NCAA is exploiting the athletes.  Here are some examples:
• If a player doesn't perform on the field, he can have his scholarship revoked the next year, and if he doesn't have the money to continue at the university, he will not be able to finish his education there. 
• If a player wants to switch schools, he must sit out a full year before being able to play again.
• If a player is injured playing for his school, the school is not liable.  There is no workman's compensation in college sports.
• The player can't make any outside money off of his image even though his school can use his image to make money.
• The NCAA can punish an athlete for an infraction of their rules without proper due process.  It picks and chooses its battles seemingly at random.
3.  The NCAA makes TONS of money.  Because of items 1 and 2, the students see basically none of it.

Goodbye Freebs

I previously blogged about receiving a free Google Chromebook back in December.  Since it was free, I named it "Freebs".  Since December, I had some good times with Freebs.  I enjoyed playing around with the new ChromeOS.  I took it to school to do work with.  It was light and fast.  A few months later, I figured out how to put Ubuntu on it, and then Freebs became a more ordinary laptop.  I did a lot of tinkering around with good old Freebs.  As of late, since my laptop's hinge broke, I was using Freebs full time for my summer work.

Freebs on his first day, next to my larger laptop.

There were problems, however.  Early on, the touchpad broke.  My theory is that the wiring burned out.  I had been using a wireless mouse, which was less than ideal.  Also, although I could live with it, Freebs had a 16 GB hard drive, which is tiny for a laptop.

For my birthday, my mom gave me a new laptop.  It's Dell's convertible netbook tablet.  Here's a stock picture of what they look like:

Although it doesn't give the touch experience that an iPad gives, it's a good netbook and there are a couple of tasks where it's really useful to be able to turn it into a tablet.  Specifically, taking "ink" notes is really useful, and reading articles on the Internet in tablet mode is a distinctly more pleasant experience than reading them on a laptop.  It eventually became clear that this device was going to be my go-to take to school/browse the Internet quickly laptop.  Since that was what Freebs was, he suddenly had no purpose in my great tower of gadgets.

Logically, once I realized that I didn't need Freebs, I had to sell him.  I don't need clutter and I prefer to save money.  The thought of it made me sad, however.  Although Freebs was just a laptop with a broken touchpad, it felt like he was a part of the family.  We gave him a name and referred to him as a he.  When I listed Freebs on Craiglist to sell, it felt like I was listing my loyal dog to sell because I got another younger, better dog.

It turned out that I was able to sell Freebs literally within hours of listing.  I got my cash money, but I felt sad about it.  There is no rational reason to feel sad about it because at the point I am at Freebs was just an inanimate object and, even worse, clutter.  I still felt sad about it.

Goodbye Freebs!

Rant About the NCAA

In the news today was Ohio State and how it is vacating all of its wins from last season.  There are often stories in the news about people and schools being punished for breaking the NCAA rules.  Here are my thoughts on the NCAA rules.

Should anyone be surprised that a lot of the top teams are cheating the system?  If there were no rules against paying players, the universities would offer up lots of money to the top players so that their teams would be better and so that they would bring in more money.  There is a strong economic incentive to pay players.  

I think the NCAA is a weird system.  For football and men's basketball, there is a lot of public interest and so a school's team in these sports will generate a lot of revenue.  The rules then say that the profits don't go to the players but just go to the university; the players can't so much as parlay their fame and make money on the side; basically the NCAA is saying that not only will the schools keep the lions' share of the money, the players under no circumstances will make a lot of money in any other way.

As a student at a university, I would be somewhat happier if that money was pumped back into the academic system so that the average student didn't have to pay as high of tuition.  As I understand it, however, the money actually goes to the other many NCAA sports on campus.  In virtually all cases, no other sports programs make a profit.  In part, this is because a very small percent of people on campus actually care about those programs.  The big winners in this system are the athletes.  They get full-ride scholarships and they get to play their sport of choice for free.  It seems to me that most anyone would rather do that then work a crummy college job to pay their way through school.  I don't agree with this.  From an economics point of view, it's equivalent to the average student paying higher tuition so that these athletes can get a sweet gig.  With all due respect to those people who are NCAA athletes, I don't want to pay your tuition.

I feel that college sports should be privatized.  College and serious, high-level sports are two different things.  College sports is basically entertainment consumed by the masses, not just college students.  The United States is one of the only countries in the world that mixes school and sports we do; personally, I think the rest of the world has it right.  I believe the college sports money should go to those who actually earn it: the football players and the men's basketball players.  As for other sports, why can't the athletes themselves pay their own way?  In a privatized system they could be subsidized by sponsorships (like the Olympic athletes.)

I Switched to Linux

Within the past few months I have switched my main computer operating system from Windows to Linux.  Specifically, I have Arch Linux installed on my laptops.   I wish someone would've told me about how great Linux was sooner because I like it better than Windows.

The thing I like most about Linux is all the tinkering that I can do with it.  From my perspective, it has been much easier to learn how to dig into the inner workings of my Linux system than on my old Windows system (I have no idea how to really tinker with Windows).  This makes me feel that I have more control over my computer than I did before with Windows.  Further, just by its nature, there is a lot more customization that can be done to Linux to make it suit exactly what I want and need.  There are a lot of different options for Linux systems, ranging from the conventional to the new and radical.  To me, this makes Linux and its various distributions far more interesting than Windows ever was.

 

There are a couple other things worth mentioning.  First, my laptop has a noticable performance boost in using Linux over Windows.  I also don't have to deal with bugs that would pop up in Windows.  Second, Linux is free to install and use; not only that, it has a ton of free software applications; I actuallly have better applications on my Linux machine than I did on my Windows because it has been so easy to find high-quality free applications; also, it is super easy to install and remove applications on Linux.