LEARN The Electoral College (from a young mathemetician's perspective)

Is this you: "WTF happened this election? DOWN WITH THE ELECTORAL COLLEGE... I think!?"?


If yes, read on. If you don’t understand the electoral college, definitely read on. If not, whatever, read it anyway.


Donald Trump won the 2016 election with the greater number of electoral college votes *shudders*. Hillary Clinton, however, received the greater number of direct votes *shudders*. I'm sure a few of you are thinking that the electoral college is rigged or flawed or unnecessary. Let's learn why or why not. By looking closely at what the electoral college is from a mathematician's point of view, we can see what the political implications are. Maybe it'll give you a deeper argument to prove your point or disprove your friend's point.


This has only happened 5 times. In 1824 John Quincy Adams beat Andrew Jackson, in 1876 Rutherford B. Hayes beat Samuel J. Tilden, in 1888 Benjamin Harrison beat Grover Cleveland, and in 2000 George W. Bush beat Al Gore. In each of these cases, the loser won the popular vote, but lost the election. There is no obvious pattern we can see that say when these kind of elections are likely to occur. One thing to note is that Republicans (as we know of the party today) have never been victim to losing in this way, but they so happen to be the party that tend to agree with the values behind the electoral college. 5 times isn't that many. Assuming that there is no pattern of this kind of event occuring, this means that there is only an 8.9% probability (so far) that in a given election, this situation will occur.


What is the electoral college?


Well, each state gets to directly elect a number of Representatives that is proportionate to the population in that state. These people, and the state's two senators are the electors for each state. After counting the votes from people within each state, the state gives all of their electoral votes to the most popular candidate. Regardless of if 51% or 99% of the population within a state votes for one candidate, 100% of the electoral votes go to the candidate. THAT is why there is sometimes a difference between the candidate with the popular vote and the electoral college vote. Let's look at this on a small scale:


Montana and The District of Columbia each have three electoral votes (meaning they have very similar population sizes). In the district of Columbia, 93% of people voted for Clinton, so they used their 3 electoral votes for her. Montana, 57% of people voted for Trump, so they used their 3 electoral votes for him. According to the electoral college there would be a tie because there are equal electoral votes. BUT Clinton won by a HUGE margin in the District of Columbia, but in Montana, Trump won by a small margin. So there actually were more people who voted for Clinton, but the electoral college doesn't account for HOW MUCH a candidate wins by. Elections like this occur when some states REEEEAAALLLYYYY don't want a candidate to win, but the states that vote for that candidate only SORTA want that candidate to win.


The argument comes down to: Are all people created equal or are all states created equal? If both are created equal... is the electoral college the best way to reflect that?


Let's look at the simpler argument first. Direct election based on popular vote, would represent that "we hold these truths to be self-evident: that all men are created equal." By this, I mean that every person's vote has equal impact on the outcome of an election. If all men are created equal, and more PEOPLE are voting for Clinton than Trump, OBVIOUSLY she should win, right?


Well... maybe not. Do you think state government is important? Well, I think so! New York is different than Texas in more ways than one. Even if you think that different states all have the same culture (I disagree, but sure), you have to agree with me that different states want different outcomes from an election, based on geographical, historical, and economical concerns (among other concerns as well). Think of the electoral college like two separate direct elections. Each state takes the votes from the people and pick a candidate. Then the nation takes the votes from each state, accounting for population size, and picks the final candidate. This means that even if more people are unhappy, more populations of people are happy. This does not mean that more states are happy, per se, but it means that more of the state's weighted by their population sizes are happy. So it's sorta a weird combination of state and individual people happiness.


If you take away anything from this blog post, let it be that EVEN IF YOU DISAGREE WITH DIRECT ELECTION OR THE ELECTORAL COLLEGE, THERE IS A LEGITIMATE ARGUMENT FOR EITHER CASE.


Here is a video of me walking through the question: Is the current electoral college representing all states equally? using math to prove it.

(I misspelled Minnesota and I accidentally wrote Denver instead of Delaware... I'm a math person, not an English, History, or Politics person. Sorry)

If you liked this- try this politics blog: Thoughts on Trump and Clinton from a Fed Up 19 Year Old!

Dumb Arguments that I've Seen:


People often say that we shouldn't eliminate the electoral college because it would be hard to. The fact that it would be difficult to amend the constitution, is NOT a reason to keep it the same. I think America should strive to be the best form of government that it can be and that means change may be required. So scratch that argument. I think it's invalid and irrelevant.


People often say that we should get rid of the electoral college because it's an outdated system... no, actually, it is a timeless idea (unless you want to argue that all states are exactly the same). And after reading why the electoral college exists, you should understand that the reason it is still around is much more complex than just "it's convenient and people were more illiterate before".

LEARN What Even is Pi?

Pi (π) is an irrational number representing the ratio of the circumference of a circle to its diameter.

Sounds simple right? But what if I told you that π just shows up all over the place, even when we aren't talking about circles? It shows up in electromagnetism, complex numbers, quantum mechanics, Fourier analysis, and basically anywhere where you have a loop! If we miscalculated π to be only a liiiitttllleee bit different, the way we have defined our ENITRE UNIVERSE would be indescribably different. That doesn't seem so simple anymore, does it?


This video is incredible. It's the best video I've seen (and... I've seen a lot of videos... it's not something I'm proud of) about π. It actually describes how versatile and crazy important π is, without focusing only on how cool irrational numbers are.

I'm sure it would be easy to convince you that π is interesting. If you still aren't convinced... then you didn't watch the video ;)


I'm sure it would be easy to tell you π matters because π is used all the time by really smart people (lawlz - not me)... Just like how we know astrophysicists matter, even if we have no clue how they work... But I want you to take a second, if you can to think about this: if you took a string and wrapped it around a circle to find the circumference of the circle, you will never be able to tell me the exact circumference without using π. The circumference of any circle must always be irrational, just like pi.


That is mind boggling to me because that means that you can't EVER measure a circumference exactly. WHATTTT? You can only CALCULATE circumference exactly (assuming you agree that π has been calculated accurately). That also means that you cant EVER know the exact volume or surface area of a sphere just by measuring it, and you can't EVER know the volume or surface area of a cylinder just by measuring it. If you don't think that's crazy, then... I dunno... go watch Cosmos or something.


So how did we find pi?


Well. We drew REALLY big circles in the sand and measured the circumference and diameter. You could also do some math! Gross, I know. There were many methods to finding pi, and many people contributed to finding this number over time, but since pi is irrational, we will never know all of it. No matter how hard we try... We can NEVER fully know all of life's secrets.


Wanna know something else important about π? I think π is one proof that math is discovered and NOT invented. Maybe this seems blantantly wrong or obviously true to you, but let me tell you, this is a hot debate in the math world (as hot as anything can get in the math world at least). There are strong arguments for either case, but I believe that math is discovered, we merely invent the terms. If you don't think that this is a fascinating debate, then I either did a bad job of explaining π, or you don't care about knowledge.


HOW does π prove such an abstract idea you ask? Well... as humans, we didn't invent circles, did we? No. They existed. Or at least the orbit of the Earth around the sun existed before we knew what π was. Shapes exist in nature. Patterns in nature can be described using shapes. We discovered a relationship between the properties of circles AND it shows up all over the place without us even guessing that it would.


"...π. Put a smile on your face ten miles wide. Looks so good bring a tear to your eye. Sweet... π"~Warrant (these are the real meaning behind their lyrics)


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Geometry: LEARN Area Formulas

In geometry class in high school, I was always given the area formulas for 2d and 3d shapes. But I had no explanation for how these formulas came to be. Understanding these formulas was never expected of me, but it was something that I expected to be taught. If you only came to this blog to look at the area formulas, I encourage you to read on and learn more math. Here is an image that showcases many different area formulas.





Let’s understand these formulas now.


How do we bring shapes from a simple form into a more complicated form: a 1d line to a 2d shape, or a 2d shape to a 3d solid? We multiply. By something. It’s as simle as that in ALL cases. The simplest case is a square.





We start with a line, and if we want to extend the line into a new dimension, we multiply by how much we want to extend it by, creating a rectangle (in this case a square). Then to turn a square into a cube, we multiply by how much we want to extend the square by. Notice that this is the same explanation for if you wanted to make a rectangle and a recangular prism.


You ponder, “Surely, it can’t be that simple!”


“Oh, but it can! And don’t call me Sherley.”


“Fine. Maybe for a square this works (that’s the easy one) but… what about circles?”









We start with a line, and if we want to extend the line into a new CIRCULAR dimension, we multiply by how much we want to extend it by. Circles, however use a special multiplier (you might have heard of pi). So to create a circle from a line (assuming the line is our radius), we multiply the length of the line times itself times π. To turn a circle into a cylinder, we multiply by how tall we want to extend the circle by just like we did with a square to a cube or a rectangle to a rectangular prism.


To make a circle a sphere… we multiply the area of or circle by (4/3) times the radius. But why 4/3? And what even is pi?!?! Find out next time!

LEARN The Pythagorean Theorem

The Pythagorean Theorem says that a²+b²=c².

In math classes, the problems you'd have to solve using the Pythagorean Theorem sound something like...

A painter has a 17-foot ladder and in order to reach the spot that he needs to paint, the top of his ladder needs to rest 15-feet up the wall. How far must the bottom of his ladder be from the base of the wall for BLAH BLAH BLAH?

Every time I see these kinds of problems, I think they're absolutely ridiculous! If I'm a painter, I'm not going to measure the distance my ladder needs to be from the wall, I'm just gonna rest it against the wall and move it around until it's in the right position! AND nobody uses ladders that rest against a wall anymore, we use the ladders that stand on their own. If you're not a physicist or a carpenter, you won't need to use the Pythagorean Theorem, so let's stop making word problems that lie to you.

I bet if I asked someone what the Pythagorean Theorem is, they would say: "if I know the lengths of two sides of a right triangle, I can use the Pythagorean Theorem to find the third length."

But try thinking about it like this:

Imagine wanting so desperately to understand the world you live in, but you have no formulas to work with. You have no idea how to think about the space things occupy in two dimensions let alone in three. Some philosopher walks up to you and says that if you have two squares that take up different amounts of space, then there is always a square that you can create that takes up the same amount of space as those two squares combined. And if you line these squares up so that their corners are touching, they will always make a triangle. And not just any triangle, no matter what size your three squares are, one corner will be the same size.

It sounds a little more complicated now, doesn't it? If you don't understand what he's saying, or if it doesn't seem obviously true, wouldn't you want him to prove it? You wouldn't just take his word for it, would you? So... why do we all just accept it in our math classes?

The Pythagorean Theorem is simple mathematically, but it says so much about area, space and what it means to square a number.

Below is a video of my favorite proof of the Pythagorean Theorem (it's only 45 seconds and you don't need sound). This proof is so beautiful and simple that I don't even need to explain what's happening for you to see how it proves the Pythagorean Theorem (or at least one case in which it works).

But what I love most about this proof (because I'm a nerd) is that in order to use water inside of boxes to prove the Pythagorean Theorem, you've started talking about three dimensions, instead of two. Because water occupies three-dimensional space. So really this proof says that h(a²)+h(b²)=h(c²). This slight difference, which doesn't change the theorem mathematically, says WORLDS about how we can think about shapes and space.

The next post will get even deeper into this topic to talk more about how to think about space, area and what it means to square something!

Hate Love LEARN MATH

Hate Love LEARN MATH exists to help students engage foundational principles in mathematics in new ways.


When I say "math", maybe you recall disappointing grades, irrelevant word problems, and confusing material. But when I hear the word "math" I think of artistic proofs, Leonardo Da Vinci's Vitruvian Man, and fun puzzles. I love math, so I often wonder why so many people hate math. And I realized after tutoring students in college level math courses that it's because they were taught wrong. In order to meet curriculum requirements, math teachers have been forced to move too quickly through class material, skipping crucial lessons. Math is taught wrong because it doesn't focus on teaching students how to think about numbers and solving problems, but rather, getting them into college.Whether you hate math or love it, this blog will teach you new ways of thinking about mathematics.


Below is a three minute video showing the reactions that students have when asked to solve an impossible problem. This demonstrates how the current way of teaching math has failed.

Hate Love LEARN MATH exists to teach basic math principles in a way that makes students understand and see the value in math. Some math topics this blog will cover are:

LEARN The Pythagorean Theorem
Geometry: LEARN Area Formulas
LEARN What even is Pi?
LEARN The Electoral College (from a young mathematician's perspective)
LEARN The Fibonacci Sequence
LEARN What STEM Adults Want For Christmas
LEARN Why Artists Don't Need to Hate Math
Fun Math Puzzles to LEARN From Part 1
Fun Math Puzzles to LEARN From Part 2
Fun Math Puzzles to LEARN From Part 3
LEARN Area of a Circle Segment

If basic level math classes were taught at a pace that allows for deep engagement in topics, more people would understand math and they would actually want to take higher level math classes. Students who stumble upon Hate Love LEARN MATH looking for formulas and examples will engage the topics and understand math principles.

About Me

Hello, I’m Anna and I like teaching math.

I’m a junior at The King’s College studying Business Administration. I live in New Jersey and go to school in NYC. I am the faculty assistant for my college’s math department, which means I grade quizzes and tutor students to help with College Algebra, Pre-Calculus, Calculus I, Finite Math for Business, and Business Statistics Using Excel. Helping students understand math and get excited about it is one of the most rewarding things I’ve done. I love math and I want to help you understand math. And it’d be pretty cool if I could get you to love math too.

I believe that math exists, whether we discover it or not. I don’t believe that math is an invention. I believe that math that is invented is wrong and I believe that correct mathematical principles are simple compared to the vastness of the world that they describe. I believe that math and art are inseperable – there is art in every mathematical idea and there is math in every artistic creation.

It is a fact that there is math in the world around us, whether it exists merely by human invention or not. It would be a shame to discredit elegant explanations of the way the world works just because these explanations are described using numbers.


That’s my face. My puppers’ name is Sadie. And the fishies’ names are Io (named after Jupiter’s largest moon) and Lester (Lester is a black moore. His name is Lester so I can say “Les is moore”). Sunflowers are my favorite flower because they are bright and bold and they make me smile (I also appreciate the math in them, but I liked them before I had heard of Fibonacci and before I understood e).