crumbs!

 
 
 

One of my students recently said to me, "I'm not good at math because I'm really slow." Right then and there, she had voiced what is one of many misconceptions that folks have about math.

But friends, speed has nothing to do with one's ability to do mathematics. In particular, being "slow" does not mean you do not have the ability to think about, understand, or enjoy the ideas of math.

Let me tell you....

Last weekend, I spent seven (SEVEN!) hours trying to understand two-and-a-half (TWO-AND-A-HALF!) sentences of a page-long proof. (My brain just couldn't make it to the end of the third sentence.) And this past weekend, it took me two (TWO!) full days (DAYS!) to understand three (THRE--okay, I'll stop shouting) little equations.

But now I finally understand the proof, and now I know what those little equations really mean. Even better, I understand them both thoroughly because of (not in spite of) all those hours that I spent with them.

So, as I wanted to tell my student:

Who cares how long it takes you?

We can never focus on and enjoy the mathematical scenery around us
if we're too busy concerned with racing our neighbors. 

Don't forget, dear friends: 

Speed is not equivalent to mathematical ability.

 
 

 

On a very related note, I am currently preparing for my oral exam*, so free time is a rarity these days. For that reason, I'm going to take a little break from blogging over the next few months. But I do hope to have lots of cool things to write about when my exam is over! In the mean time, I plan to continue sharing math-y things on Facebook, Twitter, and Instagram. See you in a few months!

*This means I'm learning about a specialized topic now, and will soon give an oral presentation (a seminar talk) in front of some faculty members. Afterwards, I'll be asked a bunch of questions about the topic to ensure I'm ready to move on to the dissertation stage of grad school. No pressure, really ;)

crumbs!

 

Physicist Freeman Dyson once observed that there are two types of mathematicians: birds -- those who fly high, enjoy the big picture, and look for unifying concepts -- and frogs -- those who dwell on the ground, find beauty in the scenery close by, and enjoy the details.

Of course, both vantage points are essential to mathematical progress, and
I often tend to think of myself as more of a bird.
(I'm, uh, bird-brained?)

Well one day I complained to my advisor, John Terilla -- you can see him doing cool math in that video above! -- about one of my classes which at the time felt a bit froggy:

 

Me: I'm having a hard time enjoying this class. I don't want to do a bunch of detailed computations on these objects. I'd rather spend my time learning category theory. I want to soar high and study the full landscape!

John: Well, yes, the sky is nice. But the ground -- and the ocean -- is nice, too. In fact, the best place to be is where the sky and the ocean meet, where the tips of the waves turn white.

That's the only place where you can go surfing.

That's where you want to be.


 

Snap, y'all.

I think I've a new respect for the little details.

Let's go surfing!

 

Group Elements, Categorically

Group Elements, Categorically

On Monday we concluded our mini-series on basic category theory with a discussion on natural transformations and functors. This led us to make the simple observation that the elements of any set are really just functions from the single-point set {✳︎} to that set. But what if we replace "set" by "group"? Can we view group elements categorically as well?

The answer to that question is the topic for today's post, written by guest-author Arthur Parzygnat.

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What is a Natural Transformation? Definition and Examples

What is a Natural Transformation? Definition and Examples

I hope you have enjoyed our little series on basic category theory. (I know I have!) This week we'll close out by chatting about natural transformations which are, in short, a nice way of moving from one functor to another. If you're new to this mini-series, be sure to check out the very first post, What is Category Theory Anyway? as well as What is a Category? and last week's What is a Functor?

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What is a Functor? Definition and Examples, Part 1

What is a Functor? Definition and Examples, Part 1

Next up in our mini series on basic category theory: functors! We began this series by asking What is category theory, anyway? and last week walked through the precise definition of a category along with some examples. As we saw in example #3 in that post, a functor can be viewed an arrow/morphism between two categories.

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Introducing... crumbs!

Introducing... crumbs!

Hello friends! I've decided to launch a new series on the blog called crumbs! Every now and then, I'd like to share little stories -- crumbs, if you will -- from behind the scenes of Math3ma. To start us off, I posted (a slightly modified version of) the story below on January 23 on Facebook/Twitter/Instagram, so you may have seen this one already. Even so, I thought it'd be a good fit for the blog as well. I have a few more of these quick, soft-topic blurbs that I plan to share throughout the year. So stay tuned! I do hope you'll enjoy this newest addition to Math3ma.

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What is a Category? Definition and Examples

What is a Category? Definition and Examples

As promised, here is the first in our triad of posts on basic category theory definitions: categories, functors, and natural transformations. If you're just now tuning in and are wondering what is category theory, anyway? be sure to follow the link to find out!

A category 𝖢 consists of some data that satisfy certain properties...

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What is Category Theory Anyway?

What is Category Theory Anyway?

A quick browse through my Twitter or Instagram accounts, and you might guess that I've had category theory on my mind. You'd be right, too! So I have a few category-theory themed posts lined up for this semester, and to start off, I'd like to (attempt to) answer the question, What is category theory, anyway? for anyone who may not be familiar with the subject.

Now rather than give you a list of definitions--which are easy enough to find and may feel a bit unmotivated at first--I thought it would be nice to tell you what category theory is in the grand scheme of (mathematical) things. You see, it's very different than other branches of math....

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#TrustYourStruggle

#TrustYourStruggle

If you've been following this blog for a while, you'll know that I have strong opinions about the misconception that "math is only for the gifted." I've written about the importance of endurance and hard work several times. Naturally, these convictions carried over into my own classroom this past semester as I taught a group of college algebra students.

Whether they raised their hand during a lecture and gave a "wrong" answer, received a less-than-perfect score on an exam or quiz, or felt completely confused during a lesson, I tried to emphasize that things aren't always as bad as they seem...

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A Quotient of the General Linear Group, Intuitively

A Quotient of the General Linear Group, Intuitively

Over the past few weeks, we've been chatting about quotient groups in hopes of answering the question, "What's a quotient group, really?" In short, we noted that the quotient of a group G by a normal subgroup N is a means of organizing the group elements according to how they fail---or don't fail---to satisfy the property required to belong to N. The key point was that there's only one way to belong to N, but generally there may be several ways to fail to belong.

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