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.

David the Gnome

My hat is conical!

When I was very young I watched a show called David the Gnome. As you might guess, the show was about David and he was a gnome. Not a terribly creative title, but when I was young I was captivated.

The Wikipedia article on David the Gnome says that the American version was actually a dubbed Spanish version that was based on a book written by a Dutch author. So apparently, in addition to being a gnome, David was an ambassador to the UN.

In one of David's adventures, he came across a chicken with six chicks. Tragically, the chicken could only count to three, and the chicks kept getting lost without the mother hen even knowing. Although a negligent parent, you have to give the bird bonus points for being able to count at all. [footnote 1]

David, recognizing the problem as a serious one, taught the mother that she should arrange her chickens in two groups of three. The mother could count to three, and she could do it twice, thus she could keep track of all her baby chicks. [footnote 2]

I think that the simple brilliance of this might have been lost on even the creators of the show. David’s poultry grouping insight alludes to many fundamental concepts in mathematics. The fact that every whole number can be represented with a unique product of its prime factors (like 6 = 2×3) is called the fundamental theorem of arithmetic. Had the mother had a prime number of chicks (like five or seven) David’s solution wouldn’t have been so simple because neither 5 nor 7 can be written as a product of smaller numbers.

The chicken could even keep track of 12 chicks, though it's a little trickier. You couldn't put the chicks in 2 rows of 6 (mama chicken can't count to 6), nor could you put them in 3 rows of 4 (4 is still too big), but you could put the chicks in three 2 by 2 groups:

You could also organize a peep of 18, or 27. If you're clever, you can do 16 too.

In addition to the fundamental theorem of arithmetic, David alluded to a tool often employed by mathematicians: when confronted with a difficult problem, reduce it to a problem that’s already been solved. Rather than teach  the mama chicken how to count higher, most likely an impossible task, he had the chicken do the something simple twice.

David applied simple and elegant mathematical thinking to a life threatening situation and his solution is profound, yet straightforward enough for an animated chicken to understand. He's my hero.

-Nick

Footnotes
[1] There is actually some anecdotal evidence that some birds can count as high as three. (Chapter 1, fourth paragraph)
[2] I like to imagine that before David came across the chicken, there were at least a dozen more chicks, all dead now due to negligence.

Fret Spacing on a Guitar

As you look down the neck of a guitar, the frets get closer and closer. What is the rule for deciding how close the frets should be?

Preliminary Information:

1. A string halved in length vibrates at twice the frequency (physics).
2. An octave is the interval between one musical pitch and another with half or double its frequency (Wikipedia article on octaves).
3. There are 12 notes in an octave (music theory).

To go up one octave, you have to halve your string. To go up another octave you must quarter your string. In general, to go up t octaves you need to have a string of length 

To have our length function, l, go up by twelfths of an octave (notes) rather than octaves, we need to adjust it to be: