Minimalist Geometry/Art

Meet Tilman, an interactive and graphic designer in Germany, who has taken some time off from his usual work to create digital representations of geometrical shapes and properties. His tumblr presents a new piece each day and is worth checking out.

Calculus: There’s an app for that!

Every math teacher should go through a series of philosophical questions regarding the purpose of math and how it should be taught.

One of the more common dilemmas is usually: To allow the calculator or not allow the calculator?

Other dilemmas include: Homework, Worksheets, Memorizing the times table, Multiple-choice tests, Proficiency grading…

I came across something interesting the other day when using wolframalpha to check some work that one of the students I was tutoring was working on. Besides finding it hypocritical that I asked that student to work on the problem “the long way” and I just used a website to check the answer, I saw an interesting advertisement on the right side of the screen. After a few refreshes, I found another of a similar peculiarity.

The ad on the left was the first one I saw and it made me genuinely upset. To me, it looks like an app will just give you the answers! It probably won’t even tell you what the answers mean! Students will lose all motivation to actually learn the point of Calculus!

Of course, they don’t need an app to demotivate them to learn about Calculus. Most math teachers do a fine job of that on their own. It also upset me to think that soon computers will just do all the math and students will no longer need to learn it. But then I realized that this is a blessing in disguise! I’m CONSTANTLY asking students to explain their answers and what those answers mean. If I could take away the need for them to consistently get stuck or frustrated in doing the problem itself, we could focus on setting up the problem and then analyzing the meaning behind the answer. Assessments would no longer test if students could perform simple algebra or add fractions, it could test whether or not students could create accurate models and interpret their solutions! That’s what people do in the real world!

I know many teachers will disagree with me because fractions are just so important – crucial to life, really – but I don’t care. I like to live life on the edge.

If any of you are curious about what happens when you click those advertisements, here is a screen shot of the page. It certainly would make Calculus homework much easier, but only if teachers continue to ask students to perform operations. This app would actually be a blessing to those who take my approach. Look at all the ways the math can be represented! Oh, the possibilities!

Crazy Paper Things

Apparently there are lots of different “crazy paper things” floating around on YouTube. I have yet to make any so I’m not sure if they actually work or not but they do look awesome!

I’m sure there are more but these will definitely get you started.

Everybody Steals

While people have tried to pinpoint exactly who discovered what in mathematics, it’s impossible to truly give credit for certain discoveries. For example, we don’t know who the first person was that came up with the idea of zero. Even if we did it wouldn’t matter because there were people separated by great distances that developed mathematics separately from one another. It’s not fair to give credit to only those that worked on mathematics in a certain part of the world (although that’s what has happened anyway).

Confusing the history even more is the fact that mathematicians often accused each other of stealing their ideas, while other mathematicians did in fact steal others’ ideas (either from peers or their students). We don’t know for sure sometimes who did what because there can be conflicting accounts. It seems to have been fairly commonplace at the time.

I like this from Abstruse Goose – which came with the quote, “Good mathematicians copy; great mathematicians steal”:

It may seem harsh, but it’s not entirely incorrect in its assumption that Pythagoras stole the work of his students. It’s a commonly held belief that he in fact did do this – whether or not he murdered them is an entirely different account.

It also reminds me of a project called The Mathematics Genealogy Project that maps mathematicians’ relations to other mathematicians. For example, if you know someone who got their Ph.D. in mathematics, you can type in their name and see their advisor. Then you can click on the advisor’s name and see that person’s advisor. You can keep clicking and see who their “related to,” mathematically.

Wait, What?

Sometimes math can be used for trickery. There are card tricks that utilize math in order to amaze others. Then there are just some calculations that appear to hold true but really “hide” a small truth from people who may not see it.

Example:

a = x

(add a to both sides)

a + a = a + x

(combine like terms)

2a = a + x

(subtract 2x from both sides)

2a – 2x = a + x – 2x

(combine like terms)

2a – 2x = a – x

(factor out a 2)

2 (a – x) = a – x

(divide both sides by a – x)

2 = 1

Hmmm… That doesn’t make sense. There’s a mathematical fact that isn’t explicit here but with knowledge of it, you won’t be fooled by the conclusion here that 2 equals 1. Do you know what it is?

While you’re working on that, check out Math-Fail.com for some math-related funnies.

NASA Has Space Math

Here’s just a quick heads up on the awesomeness that is NASA. They have a page dedicated to math! My favorite are the “Problem Archives” where teachers can present real data, pictures, graphs, etc. from NASA to their students in an already lesson-friendly PDF. There’s nothing better than supplementing a textbook with this kind of stuff!

Space Math @ NASA

Please Excuse My Dear Aunt Sally

The reason your math teacher made you (most likely) remember PEMDAS, or “Please Excuse My Dear Aunt Sally,” is because without it we wouldn’t be able to properly read mathematics. For example, you can read the sentence, “The quick brown fox jumps over the lazy dog,” and picture what it is trying to convey. There’s a dog, he or she is lazy, maybe laying down. A fox, which is the color brown, jumps over the dog and is doing this at a fast speed. Everyone reads this sentence and understands it in the exact same way. We know which adjectives describe which nouns and what the verb is referring to. We know the fox jumped, not the dog. We know that the word “brown” describes the color of the fox. We learned the rules of English in order to be able to read, write, and communicate so that everyone knows exactly what we are trying to say. The order of operations is the equivalent of this, but for mathematics.

Without the order of operations, everyone would do calculations however he or she wanted and everyone could get different answers. If I say to you, “First add two and eight, then divide your sum by five,” I could write it out as: (2+8)÷5 or (2+8)/5. If you follow the order of operations properly, you will get the solution I was referring to. If you don’t, you don’t understand what I’m trying to say. I also use the knowledge of the order of operations to write those expressions. I couldn’t write 2+8÷5 because the answer you would get, following the correct procedures according to the order of operations, would not be the one I was referring to. The order of operations allows everyone to be able to communicate mathematics correctly, without further explanation or confusion.

PEMDAS stands for: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction and that’s the order you’re supposed to do the mathematics (sort of). You’d think math teachers would love PEMDAS, and I’m sure you thought they did, but most math teachers have a huge problem with it. It makes their students think that multiplication comes before division and addition before subtraction when in fact multiplication and division are essentially the same operation, as is addition and subtraction. With two small examples, let me explain. 2-1 (two minus one) is mathematically equivalent to 2+(-1) (two plus negative one). 4×2 (four times two) is mathematically equivalent to 4÷½ (four divided by one half). If we have a longer expression for which we will use PEMDAS to simplify, like 4÷½-2+4×2, and we don’t use PEMDAS correctly, then when I rewrite the expression but keep it mathematically equivalent, like 4×2+(-2)+4÷½, you’d get a different solution even though both expressions are mathematically equivalent and give you the same solution if you follow the order of operations correctly.

Ok, that was a lot of boring math stuff and I apologize, so here’s an awesome graphic design I found on shirt.woot! to lighten up this post a little. Enjoy!

You can use the order of operations to read this expression!

Just Cut, Fold, and Paste

So there are these things called “nets” in math. I actually didn’t even know what the term “net” was referring to until I started student teaching and the Geometry students starting asking me what it was. Their textbook was asking them to draw a net for a given 3D shape.

A net is a two-dimensional figure that can be folded into a three-dimensional object.

-NCTM

If you were to unfold a certain three-dimensional polyhedron, you could cut the shape at its edges and get pieces in which you could then lay flat. For example, take this pyramid:

Imagine it has a square bottom, it is made of paper, and you can hold it in your hand. Think about what it would look like if you unfolded it, if you could take the faces and lay them all flat.

This is probably the most common way to think about the above pyramid unfolded but there are more. Consider it a challenge to discover another.

Here are some additional shapes to think about:

You can also work the other way. Given a net, you could build the 3D shape. There are many print outs available online which include tabs on the nets to make it easier to glue or tape these shapes together. Here’s an example.

The cube has many different nets that all represent the cube. I’ve seen questions on standardized tests that tests an individual’s spacial skills by giving them different options and asking which is or isn’t a valid net.

For example: Which of the below nets will build a cube?

One group of professors and students took this idea a little further. They started building nets out of fabric and zippers, trying to see which nets they could construct that would use only one zipper to build the corresponding 3D object.

Check it out!

You can check out other math related art at the Bridges Math Art Galleries webpage.