First World Problems

That is one really ugly red car.

Three Page Comic

I really hope I caught all the spelling errors. I probably didn't.

Note Taking in Math Class

I'm a math undergrad with a real passion for mathematics (enough to try to maintain a blog about it). As of today, I'm taking four math classes, have a gpa of 3.9, and spend a good amount of my time tutoring. Also, I don't take notes.

I'm not sure what place note taking has in a math classroom. The books that I spent more than $100 a piece on (way too much!) cover all the same content that my classmates' notes do. If I forget the definition of the Laplace Transform, then I don't need notes to look back on, I have a book. Failing that, I have Wikipedia, Wolfram's Math World, and Paul's Online Math Notes. If I need to have problem worked out, I have Khan Academy or any other of a variety of YouTube videos. My school, and I imagine every other school in the world, is packed with thousands of books, many of which are about differential equations. If I need help with Laplace Transforms, or any other topic in math, I have a plethora of sources to reference. Why then, on top of all that, should I take notes?

Further, I think taking notes in math class have negative consequences. Sometimes I look around and notice my classmates too absorbed in their note taking to actually be paying attention in class. The professor might add a bit of interesting information verbally, and my classmates are often too busy copying what's on the board to hear it. Also, there is the problem of divided attention: if your attention is being put into your notes, you are not working on comprehending the material. I assume that these students go back over their notes at a later time and try to make sense of the material then, but that has to be very tough when there is no professor to offer insights.

So why do so many students take notes? I suspect it's because of years of programming by high school and elementary school teachers. I also think it's because students have developed skills that are appropriate for other classes (note taking is very valuable in English or history class) and mistakenly believe that those good habits will translate to their math classes. I think all of this negatively impacts math students' education.

Wrong vs. Not Helpful

In solving a problem, students inevitably make mistakes. This is how we learn. However, there is a huge difference between being wrong and being not helpful.

Wrong

Suppose a student is asked to solve the following for x:

Seeing the 2 next to the x, a student may try to divide both sides by 2. This can be done correctly, but let's assume that the distributive rule is momentarily forgotten:

This leads to the incorrect answer of x = 1, whose falsity can be demonstrated by substituting 1 in for x in the original equation (which would give 6 = 10).

Not Helpful

Now imagine that the student takes the same equation, and subtracts 2x from both sides, giving:

This is not wrong, subtracting 2x is not a violation of any mathematical rule, but neither is it helpful. A student who attacks a problem like this may still be in need of assistance, but a different type of assistance from before.

My Point

This distinction is obvious to educators. It is not, however, necessarily obvious to students. And it needs to be made obvious to students. There is a rampant misconception of math that there is a right way and a wrong way of doing math, and if you're not not doing it correctly, you're wrong. (I previously wrote about that here.) Sometimes you're doing something useless that is not necessarily wrong. And that is ok.

I think a good way to get this distinction across is by way of analogy. Consider chess: there is a notable difference between a wrong move (moving a rook diagonally) and a foolish one (exposing yourself to checkmate).

Six Study Tips for Math

Math is Hard. It really, really is. There are thousands of tips on how to succeed in math class, here are some that I think are especially important.

Create Your Own Problems

Solving someone else's math problems demonstrates and ability to follow rules. Creating your own problems shows an understanding of the subject. It also forces creativity, an essential (and often overlooked) skill in mathematics.

Compare and contrast problems

Post game Analysis is so underrated. After doing 5 or 6 problems, ask yourself: which of these was the hardest? The easiest? What did all the problems have in common? What made them unique?

Don't Do Your Homework All at Once

If you decided to start training for a marathon, you wouldn't start by going outside and just running for four hours straight. Instead, you would run a little each day until you got into shape. Math is no different. Study or do homework for 20 minutes to half hour at a time. Then read a book, or make dinner, or call a friend... whatever. Just don't sit and do math for hours on end, it's counterproductive.

Do math in your downtime

Play math games; practice your arithmetic. I like factoring numbers in my head (the time on a digital clock, licence plates), it keeps me sharp, and it strengthens my ability to visualize difficult problems.

Read the Textbook

Yes, I know it's poorly written, all text books are. And no, you're not going to understand after one read through. But reading the text will at least familiarize yourself with the vocab and notation.

Later, when getting help (in class, from a tutor, from a friend) reflect on what you read. Why didn't you understand it when you read it? Do you understand it now? Over time you will learn to read math books.

The Internet is Awesome

Read the Wikipedia article on what ever it is you're studying. Look at the pictures. Google the subject, or watch lectures on YouTube. The more versions of a lesson you get, the more likely you are to gain insight.

Math: Discovered or Invented?

In my previous post, I offered up two classes of truths: Discovered Truths (The Earth is round) and Created Truths (Darth Vader is Luke Skywalker's Dad). Then I asked what class mathematical truths (2+2=4) belong to. Here are some possible answers:

Math is Discovered

Math, like science, is something we discover. Before there was anyone to count them, one dinosaur and one dinosaur made two dinosaurs. Before we had a word for 50, it was still the sum of two squares in two different ways (7²+1² = 5²+5² = 50).

Math is Created

Math is an art and, like the arts, is a creation. We create the truths in math. Mathematical objects like triangles and the number 17 are created by mathematicians, thus any truths about them are constructed.

Math is A Priori

Unlike the arts, which are dependent on our imaginations, or the sciences, which are dependent on the physical universe, math is A Priori, independent of everything. If there were no physical universe, 1+1 would still equal 2 (though there would be no way to express it). Math does not fit into the categories of "Discovered Truths" and "Invented Truths," Rather, they get their own category: A Priori Truths.

There is No Truth

Truth is a convenient fiction. Language is just a series of meaningless sounds (or symbols) that are used to invoke specific behaviors in others (or oneself). But just because those sounds/symbols succeed in creating reactions in others, doesn't mean they actually mean anything. There is no meaning therefore, there is no truth [footnote 1].

All Truths Are Created

We create our language(s), we express all of our truths with language, therefore we create all of our truths. Any "truth" of science (or math) is a construct of our minds and hence a created truth.

All Truths Are Discovered

Every possible linguistic expression is essentially a number (See Jorge Borges, Library of Babel), all numbers exist independently of us, so any truth expressible in language exists independently of us. More on this soon.


[1] I am very much aware of the irony of using language to express the meaninglessness of language.

What Is Your Math Philosophy?

http://www.gocomics.com/calvinandhobbes/2011/05/31/

Truth is a tricky thing. Some things are true because we say so, others are true independent of us. Lets call these two classes of truths Created Truths and Discovered Truths. Some examples of Created Truths:

• In chess a bishop moves diagonally.
• The ninth word in Never Gonna Give You Up is rules.
• The main character in the above comic strip is named Calvin.
• Darth Vader is Luke Skywalker's father.
• Peter Parker is Spiderman.

All of these things are true because someone said so (the creator(s) of chess, Rick Astley, Bill Watterson, George Lucas, and Stan Lee).

Some examples of Discovered Truths:

• The Earth revolves around the sun.
• Water and ice have the same molecular structure.
• Your eye color is determined by your genes.
• Fire burns wood.
• Iron is attracted to magnets.

Each of these is true independent of us. No one decided that the Earth revolves around the sun, it just does.


Now consider Mathematical Truths. Some examples:

• 2 + 2 = 4
• There is no largest prime number
• 1729 can be written as the sum of two cubes in two different ways
• The graph of y = x²-2x+3 has a minimum at (1,2).
• The area of a circle with radius r is πr²

Are these truths discovered or created?  More on this soon.