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See, that’s what the app is perfect for.

Sounds perfect Wahhhh, I don’t wanna
visualizingmath
visualizingmath

“Impossible Figures” Minimalist Posters by  Éric Le Tutour

An Impossible Figure is an optical illusion in which a physically impossible three-dimensional object is depicted in two-dimensions. Humans have a natural tendency to try to interpret drawings as three-dimensional objects, which is why when viewing an impossible figure, you may feel confused or find the image unsettling! 

Reutersvärd’s Triangle and the Penrose Triangle were first created by graphic artist Oscar Reutersvärd, the Impossible Cube (the kind of Necker Cube portrayed on the poster) was invented by M.C. Escher for his artwork Belvedere, and the Penrose Stairs were created by psychologist Lionel Penrose and his mathematician son, Roger Penrose. (The Penrose pair also created the Penrose Triangle independently from and later than Reutersvärd.) These inventors clearly show that “impossible objects are of interest to psychologists, mathematicians and artists without falling entirely into any one discipline”. Source. 

More pictures of these awesome minimalist posters can be found here. Find more work by Eric Le Tutour at his site www.ericletutour.fr/!

hurricane-euler

How To Study for a Math Test

prolific-euler

(i.e., what I should be doing instead of writing this)

Note: this takes hours. I know, because I do it. And it works, at least for me. Also feel free to ignore this because I’m pretty much just making myself a to-do list. Or feel free to make additions. 

1. Get a notebook, or some notebook paper, the backs of old worksheets if you’re me, or something else to write on. This will be your study sheet(s), and it is very important. 

2. Go through your class notes. Read them, thoroughly, and write down anything that you’re not 100% sure you’ll remember on the study sheet. 

3. Reread lecture slides from class, if you can get them. Do the same as with your notes. 

4. Read the textbook. Research has shown that just reading and highlighting isn’t particularly useful for retention, so mostly just skim for stuff (theorems, formulas, different problem-solving strategies, etc.) that didn’t make it into your notes. Write these down too. 

5. Look through your old homework. If you’re anything like me this is hell on the ego, but you need to know what you got wrong so you can get it right on the test. If necessary write stuff down (ex: don’t forget dx and +C on integrals).

6. Rewrite the study sheet. Consolidate and rephrase the information in whatever way makes the most sense to you. Be completely certain you understand everything. Organize it however works best for you - some people like colorful pens or highlighters, bullet points, whatever. Rewriting has the added benefits of helping you memorize it, and hopefully now everything you need to learn is on that sheet. 

7. Do practice problems. If your teacher provides some, do those. Do problems from the textbook that are similar to problems from class or homework - often, those are what end up on the test. Do as many as you can. With as much variety as possible - you want to be ready for anything. If you figure out something you didn’t know before, add it to the study sheet. If you have to look up a formula or something to do a problem, add it to the study sheet. 

8. Possibly rewrite the study sheet again. The goal is to have everything you could possibly need to know in one place. Right before the test, make another one with only the things you don’t know. Spend as much time as possible staring at this. 

9. Pass the test, then either burn or frame your study sheet. 

hurricane-euler

Gambler’s Fallacy

bisekibun

This is nothing new; in fact, the idea is 103 years old, but it goes as follows: the probability of success, for a random event, is unaffected by the number of losses seen prior. This seems like it should make sense, but it’s surprisingly hard to swallow when you’re the one losing a bet.

If you flip a fair coin 100 times, and somehow all 100 of those flips turn out to be on heads (unbelievable, right?), what’s the probability that your next flip will be tails. It’s still just 50%. No matter how many times you flip a coin on heads in a row, or roll a six on a die, or randomly pick a 5 of Spades from a deck of cards, the probability is still just ½, 1/6 or 1/52 respectively. The chances haven’t changed, but you might think you’re more likely to get get something new. This is the Gambler’s Fallacy, which almost always applies to a small number of trials.

This seems to conflict with the Law of Large Numbers, which states that the number of times you get a result should be equal to it’s probability, when you’ve taken a significantly large number of trials. For example, if I flip a coin 100 times, we expect that 50% of those flips should be heads, just likes the chance of any one flip being heads is also 50%.

The Law of Large Numbers is only an expectation, it doesn’t have to hold true. It’s perfectly possible that all 100 flips of the fair coin are heads. Maybe the issue is that 100 trials isn’t enough. So what number of trials is significant enough? 100? 100,000? Infinity?

Maybe it’s a fallacy in it’s own right to think that half of 1,000,000,000,000,000,000 coin flips will be heads. What do you think?

Source: bisekibun
hurricane-euler
trebled-negrita-princess

It’s kinda sad how children automatically label themselves as “dumb” when they can’t do or understand MATH… Seeing how the world puts so much emphasis on MATH, nobody gets credit when they’re amazing at writing, or music, or ART… Because the only thing that really matters is the fucking quadratic formula, right?

thatmathblog

Alright, this really kinda needs to be addressed….

A few points:

1. Math is not emphasized enough, if anything, and students are often taught the wrong way due to ignorance.

2. Math is not just the quadratic formula. Math and logic are, in a lot of ways, the entire basis of our existence. They’re the essence of our communication as individuals.

3. Not a single teacher I ever worked with (including myself) ever said that being good at math is what made you smart, and that you can’t be smart without knowing math. Those who say so are very far and few between.

1. Math isn’t taught correctly in school. No, I don’t mean the new-fangled Common Core “I don’t understand my kid’s arithmetic!” stuff. Math itself is about problem solving and critical thinking. Not memorizing formulas and taking an algorithmic process through things. Students are often told what to do, how to do it, and when to do it, but never given any understanding of why.

For example, I was never taught the derivation of the quadratic formula. Turns out, it’s quite easy. 

Often times, the explanations are just plain wrong or begging the question. A common response to “Why can’t you divide by 0?” is “Because 1/x goes to infinity at 0”. Okay….but that doesn’t explain why it can’t be infinity. The graph never shows the entire thing. It’s not definitive in the case of “Well, as you see, the line y = 5+x is 5 at x = 0”. Turns out it’s a simple contradiction and it can be explained to 6th graders. (Suppose we could divide by 0: what’s 0/0? Well:

0/0 = 0 * (1/0) = 0, since anything times 0 is 0.
0/0 = 1, since anything divided by itself is 1.

1 = 0/0 = 0 * (1/0) = 0, and thus, if we could divide by 0, we’d have 0 = 1.)

Just think about it: when was the last time a math teacher of yours had a satisfactory answer to a question?

Not only that, but you are often taught things that really aren’t too applicable in your daily lives. But, this itself is a misdirection, since anyone who goes into any sort of STEM field is going to be using this kind of thing daily, even if indirectly.

What the average person needs in a proper applied math course (basically math NOT for someone who will need college math…well, frankly for everyone including them), would be a course to teach basic accounting, estimating when they need to stop for gas, calculating interest, mental math tricks, etc. But that’s neither here or now.

This breeds contempt and ignorance. By not getting a proper mathematical understanding and proper teaching, people who DON’T grow up to love it the proper way hate it. Math to them is either just a random tool they use or just “Dumb and stupid”. This leads to bad policy towards math and leads to cultural ignorance. This even leads to the false belief that math can only be taught with rote drills.

2. Math is not just the quadratic formula. Math is almost literally everything. Math can be used for almost literally anything. Math and logic together form your entire basis for organizing and understanding your world.

Communicating with people and making implicit understanding of their statements based on context? Inference, which is generally logic. Estimating how far you have until you have to stop for gas again? Math. Figuring out your bills? Math. Cooking? Math. Even the rest of the STEM fields (and some non-stem) are math: Sociology is applied Psychology which is applied Biology which is applied Chemistry which is applied Physics which is applied math. Computer science is math. Your computer is thanks to a long line of mathematicians who figured out the logic necessary to, well, do computational logic. Thank Rear Admiral Grace Hopper, who wrote the first modern compiler. Thank Ada Lovelace, who more or less invented software. Thank Claude Shannon for inventing information theory and proving that electronic switches can model and perform logic.

This is all math.

You name me a topic, and I’ll point you at rigorous math studying that topic.

3. Not even math teachers think you need to know math to be smart. I should know, I used to be one. Math teachers (who aren’t professors of abstract disciplines in a university) know their topic is going to be the stepping stone to another STEM field or related activity. They may think you need math to understand that topic (you do), or that you need math to understand your world (you really do), but they don’t think you need math to be smart. They just think you need math to be less ignorant.

I hated math as a kid. Absolutely hated it with a passion. I sucked at it and barely passed AP Calc my senior year (which I only got into because like 5 other kids moved and I was put into it without my choice). I only started liking it when I was taught properly by good teachers at my university. I only got an appreciation when I was allowed to explore and ask questions and try and understand why things happened. Ultimately, that’s what math is: trying to understand. It’s not the quadratic formula, it’s not the Pythagorean formula, it’s not statistics (obligatory statisticians aren’t real mathematicians joke here), it’s the process of trying to understand something and why it happens.

And, I know for a fact, this is why most people think they’re “too dumb” for math.

Source: bando--grand-scamyon