Bill Shillito - now at @solidangles.bsky.social
@solidangles
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Math instructor at @OglethorpeUniv , PhD student at @GeorgiaStateU . Views my own. He/him. Pronounced SHILL-lit-toe. I like to divide by zero.
Marietta, GA
Joined July 2014
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Send us removal request.
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Pinned Tweet
It's time for the COR 314 Meme Review!
My Mathematics and Human Nature classes made memes related to the course for our last day today, and BOY DID THEY DELIVER.
Here are some of the memes, along with what we studied that prompted them.
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@graciegcunning
Maybe it's the people I follow, but all I'm seeing is people talking about how you're NOT dumb — you're asking the important things that many others else think about but never have the courage to ask. 🙂 You're EXACTLY the kind of student we love to have!
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that bus meme except it's about mathematics
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I THINK I GOT IT!
I've been trying to understand this for years, but the missing magic words that made it finally click were "inner product."
Thank you
@math_vet
!
I'll try to explain my understanding in a short thread in case it helps anyone. 🧵 (1/8)
Can someone please explain or recommend a resource that explains the intuition behind the Laplace transform?
Like, I believe it works and is very useful! But what I'm wondering is how the heck would someone have first thought to construct that particular integral out of nowhere?
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Can someone please explain or recommend a resource that explains the intuition behind the Laplace transform?
Like, I believe it works and is very useful! But what I'm wondering is how the heck would someone have first thought to construct that particular integral out of nowhere?
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Hot take: Not everything in math needs a "real-world application" (henceforth "RWA") to be worthwhile. That's a narrative we need to stop pushing.
I've been thinking about this after some conversations with a friend of mine. Thread below. (1/n)
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Not everything needs a “““real-world””” application.
And insisting on contriving one for every single topic does more harm than good.
What opinion about math education would have you like this?
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I don't know who needs to hear this, but the Pythagorean Theorem is NOT "a² + b² = c²."
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Just had a thought while studying for my Advanced Matrix Analysis qualifying exam (and screaming internally).
When I taught high school, a student in my Algebra II class had an idea for padding matrices with zeros to make any addition or multiplication defined.
Is this a thing?
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Wow. I didn't expect such a big response to my Laplace transform threads, but apparently they've sparked good discussion.
Maybe I should make more threads that explain the "why" behind things in mathematics people have been wondering about.
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Hey
#MathTwitter
! Here's a little something inspired by
@matthematician
's "is-it-a-sandwich" activity and
#HoagieHomies
everywhere. 🥪
What makes a number, a NUMBER?
Would love to hear your thoughts! 🙂
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Incorrect math on
@jeopardy
tonight: "In math, when the number of elements in a set is countable, it is this kind of set." Question: "What is finite?" To whoever wrote that question: Go look up what "countable" actually means in mathematics.
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Normalize writing diagonalized matrices as UWU⁻¹
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When math teachers say "simplify" and mean "make it more complicated"
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No one:
Absolutely no one:
My friends: KLEIN FUR-GROUP
(please forgive me)
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I taught a course in computational algebraic geometry this summer.
To high school students.
Here's how it went.
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What's something in math you've never really understood where it comes from or why it works?
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@math_vet
THIS. THIS WAS IT.
I knew the first part already, but the fact that it's an inner product makes so much more sense now!
You're essentially asking the question, "how well does f(t) 'align' with e^-st for various values of s?", right?
I think I finally actually GET it now! 😁
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Hey
@graciegcunning
! Math teacher here. Watched your video.
You're NOT stupid for wondering those things. 🙂
There's a bunch of us who agree "why would anyone need to know this" is a *totally* valid question, and you should absolutely NOT be lambasted for asking it.
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I think I'm starting to get the hang of writing these "un-Photomath-able" questions.
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"yOu cAn'T sUbTrAcT a lArGeR nUmBeR fRoM a sMalLeR nUmBeR"
lol watch me
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Middle school math teachers: "yOu CaN't UsE gUeSs AnD cHeCk!"
College math textbooks: "By inspection we see that x=2 is a solution."
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Meanwhile, we end up ignoring so many other important points of mathematics.
Noticing patterns.
Finding structure.
Making connections.
Evaluating options.
Those are important "applications" as well. They count. (7/n)
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Looking forward to getting my hands on the 9th edition of Stewart's Calculus textbook! I can only imagine what changed within four whole years that makes my 8th edition obsolete.
I mean, what if calculus has changed? Maybe the derivative of x² is 3x now and I'm behind the times!
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⬜️⬜️⬜️🟥
⬜️⬜️⬜️⬜️
⬜️🟥⬜️⬜️
⬜️⬜️⬜️⬜️
⬜️⬜️🟥⬜️
🟦⬜️⬜️⬜️
⬜️⬜️⬜️⬜️
⬜️⬜️🟦⬜️
🟥⬜️⬜️⬜️
⬜️⬜️⬜️⬜️
⬜️🟥⬜️⬜️
⬜️⬜️⬜️⬜️
⬜️⬜️🟥⬜️
⬜️⬜️⬜️🟦
⬜️⬜️⬜️⬜️
⬜️⬜️🟦⬜️
⬜️⬜️⬜️🟥
⬜️⬜️⬜️⬜️
🟥⬜️⬜️⬜️
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Realization of the day:
The converse, inverse, and contrapositive transformations on logical implications, along with an "identity" transformation that just returns the original statement, form a group that's isomorphic to the Klein 4-group.
🤯 V₄ IS EVERYWHERE! 🤯
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Idea: Honest titles for math classes
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@matthematician
Never have I agreed so much with the answer while disagreeing so much with the method.
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We recognize that many of our mathematicians actively use other number systems. However, we will no longer allow free promotion of number systems other than the real numbers on Twitter.
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I started a thing: "Helpful questions."
Contributions or wording suggestions welcome.
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So uhh ... can someone explain how the heck this happened out of absolute nowhere, on a video I published like 3 years ago? 🤯
Did something happen that all of a sudden made people hugely interested in projective geometry?
. . . what do I do now?
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As I plan my Calculus I course for this Spring, I never cease to be amazed by the skill with which textbooks like Stewart manage to utterly kill any kind of intuition or motivation for the subject by hiding it behind a deluge of symbolic manipulations and formal fussiness.
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I blame grades.
Being a math professor is like, "Here is this carefully crafted beautiful piece of math that is the epiphany of centuries of intellectual investigation."
Student: is this on the test?
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mfw Americans call maths "math"
mfw Americans call a trapezium a "trapezoid"
mfw Americans call a triple pointywocket a "triangle"
mfw Americans call a Newtonian slopesy-wopesy a "derivative"
mfw Americans call a flippityfraction a "reciprocal"
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This also explains what's going on with Fourier transforms! There, we're testing "alignment" against exponentials that are equivalent to sinusoids, since e^iθ = cos θ + i sin θ.
I learned to calculate those when I took PDE, but now I actually *understand* the transform. (7)
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I guess you could call games a RWA in themselves, but I think that would be missing the point.
All I'm saying is, let's stop reducing mathematics to just RWAs, and instead celebrate *all* the ways that math helps us all be better thinkers. (15/15)
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Just got accepted to the PhD program at
@GeorgiaStateU
! 🥳🎉 LET'S GOOOO!!
Going for the PhD in Mathematics with a concentration in Collegiate Mathematics Education. Starting this fall, and will be a graduate teaching assistant as well!
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So yeah, I *THINK* this hopefully explains the WHY behind Laplace transforms and not just the what or how.
If anything I've said seems off, let me know! 😅 (8/8)
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When we calculate the Laplace Transform of a function f(t), we're asking, "how well does f(t) 'align' with e^-st for various values of s?"
The resulting function of s (a complex variable) help answer this question for many different values at once. (5)
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Has anyone ever done activities on HOW to ask good questions in math class?
I've noticed that students who don't understand something often don't really know how to articulate what they don't understand. It might help to have tips for framing what sorts of things to ask.
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There's so much cool math to learn after calculus!
Linear algebra is AWESOME
Number theory is CLASSIC
Abstract algebra is MIND-BLOWING
Combinatorics is PLAYFUL
Complex analysis is BEAUTIFUL
Real analysis
Topology is FASCINATING
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You know the adults-talking sound in Peanuts cartoons? Where it just sounds like "🎺🎺🎺🎺🎺"?
I'm pretty sure that when textbooks say "IF this AND this THEN formula," lots of students just hear "🎺🎺🎺🎺🎺 formula."
And that's how you get errors like this.
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To be clear, I'm not at all asking how to use it. I know that the process is:
* Take ODE
* Rephrase to algebraic equation with Laplace transform
* Solve algebraic equation
* Turn back into solution to ODE
But like ... where did it COME from?
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That feeling when you realize a connection between two things in completely different areas of math:
Lagrange interpolating polynomials in numerical analysis, and the Chinese Remainder Theorem in number theory.
Mind = blown.
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@ProfKinyon
Hmm. My answer would be that {1,2,3} seems to be a subgroup isomorphic to ℤ₃, but that's impossible because the group order is 7, and that violates Lagrange's Theorem.
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Sneak peek at next calculus video ... should be up by tomorrow!
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Multiplication: "Old-School" Way vs. "Common Core" Way.
(Yes this example uses negative numbers. That's because learning based on UNDERSTANDING gives you flexibility you can take to future classes, rather than having to learn new rules for every separate kind of problem.)
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Being a rhythm game player and a math teacher means constantly looking for new places to hide the Konami number. 5⃣7⃣3⃣
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Look what the kids are being taught nowadays! 😂 Can you believe it?
#CommonCore
#Multiplication
#AreaModel
#ITeachMath
#MTBoS
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Don't mind me, just prepping notes for Abstract Algebra tomorrow
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The world if function notation were (x)f instead of f(x)
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P(x,y) = x buffalos y
B = the set of buffalo in Buffalo
The buffalo⁸ statement can be phrased as:
∀a∈B, ∃b∈B:P(b,a)→∃c∈B:P(a,c)
🦬 🦬 🦬 🦬 🦬 🦬 🦬 🦬
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I don't care about contrived "word problems" that "use" polynomials.
I do care, however, that polynomials act a whole lot like integers. That insight is a great example of using what you do know to explore what you don't know (yet). (8/n)
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Do British Python programmers start their classes with "def __innit__"?
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You might not "need" to logarithmically differentiate god-awful functions to build a bridge or whatever.
But the idea that logs make hard problems easier? THAT's powerful.
To borrow
@ddmeyer
's phrasing again: It's not "real world", but it's real MATH. (9/n)
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I'm always reminded of
@ddmeyer
's posts about pseudocontext.
When you insist on RWAs in every lesson, you end up with my favorite classic example: "using" special right triangles to find the right size bandana for your clearly confused doggo. (5/n)
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@3blue1brown
I hope it eventually gets back to her how many math/science people (especially teachers) agree that her concerns are valid.
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Tutoring a HS geometry student, and they have to do this proof.
Asked student to explain in their own words what's going on. Response: "They add up to 180°, and they're the same, so they each have to be 90°."
Why can't that be enough at this level?
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As of last night, I finally have a degree in math! 🥳
Big shoutout to
@MathProfTom
(left) and
@DrChadWiley
(right) for being a big part of the reason I came to
@emporiastate
. They're great professors and awesome people.
Onward to doctorate!
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You can test how well two FUNCTIONS "align" with each other by replacing summation with integration, giving a continuous analog of the dot product.
In fact both of these are specific instances of inner products, which we're discussing in two of my PhD classes this semester. (4)
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Sent to me by a coworker! 😂
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Matrix question!
Why should I care about simultaneous upper triangularizability or simultaneous diagonalizability?
I’ve learned a bunch of theorems about these concepts but I have zero idea about why they’re important.
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Yes it gets abstract sometimes. But that's not a bug in math, that's a feature.
Being abstract means you play with the IDEAS, and your conclusions can then be applied (if necessary) to unexpected things later.
My favorite example is with knot theory. (10/n)
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So, dot products!
I'm teaching Calculus III this semester, and I tell my students they test "alignment."
If u · v is positive, then u and v "point in similar directions".
Negative? "Roughly opposite directions".
Zero? "Perpendicular/unrelated." (2)
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@hels
College math instructor here. My thought: make sure that both baskets have the same number of fruit. You could do this by adding 3 oranges and 2 bananas to the right, but you could also add FOUR oranges and 2 bananas, and then an extra orange to the left!
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But then some time later, people discovered DNA. And found that knots in DNA explain the behavior of mutations and such.
Who do you think they turned to in order to analyze these knots?
All that abstract playing ended up answering questions that hadn't been asked yet. (12/n)
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Might have my Calculus I class figure out derivatives of sin, cos, exp, and ln just from playing with graphs, spend more time with modeling, and make the limit proofs an optional exploration.
Would that get me excommunicated from the church of mathematics for heresy? 😂 (1/4)
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Games are the ultimate form of abstraction. We don't have to have some "practical" purpose for moving pieces around a board.
We play games because they're fun, they make us think, and they bring us together.
They're an essential part of the human experience. (14/n)
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That intuition explains WHY dot products often show up with things like line integrals and surface integrals.
It also explains what's going on with the correlation coefficient in statistics — it's just the cosine of an angle found by taking a higher-dimensional dot product. (3)
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Every semester of teaching college so far, I've had to deal with students using Chegg. Somehow it doesn't get any easier. It drains me.
But still trying to remind myself it's not the students' fault that their education system revolves around the pursuit of The Almighty "A".
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You might have read "A Mathematician's Lament" by Paul Lockhart, in which he describes how absurd it would be to reduce music to correctly writing notes and stems, or reduce visual arts to practicing paint-by-numbers and mixing colors.
(2/n)
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Just found out I passed my first PhD qualifying exam (Collegiate Mathematics Education)! 🥳
Two more to go next year — Real Analysis and Matrix Analysis. 😱
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I would argue that the insistence on a RWA for everything does a similar thing. To me, that's like saying that the only reason to learn music is to write commercial jingles, or that the only reason to learn visual art is to paint walls. (3/n)
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I think it makes sense we'd test "alignment" against e^-st when doing differential equations because [1] they behave so well under differentiation and [2] they show up in all sorts of real-world applications. (6)
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Mathematicians started investigating knots because physicists thought elements were knotted ether.
Then those physics discovered atoms.
Mathematicians didn't care — they kept on playing around with knots, just for the joy of exploration! (11/n)
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Pseudocontext looks like RWAs on the surface, but the key thing is that nobody would actually ever use that math for that purpose.
And they're also often way too neat to be anything close to "math in the wild," to borrow the phrasing of my coworker Dr. Linda deCamp. (6/n)
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@Jeanvaljean689
That I can do! — e was postulated to answer the question of how to compute continuously compounded interest.
If you invest $1 at 100% interest compounded continuously, after 1 year, you'll have $2.71828... = e.
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Just got 195/200 on my graduate Vector Spaces final.
Abstract Vector Spaces is the class that convinced me I was bad at math in undergrad.
So I feel like I just defeated my longtime archnemesis. 😤
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As someone who has taught calculus to many different levels of student, this is utter bullshit. Yes there are a lot of confounding factors that can make it difficult. But supposed "cognitive ability" isn't one of them if you know how to teach (and reach) students.
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After some consideration I've made a slight revision.
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Is the syllabus the right place to throw shade at traditional Calculus textbooks? 😏
...yes. Yes it is.
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Just submitted my Ph.D. program application (after spending like a day hovering over the "submit" button and repeatedly going back to quintuple-check everything).
Here's hoping everything goes well! 🤞
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I've also been thinking about this a lot because of how I've been diving further into combinatorial game theory.
Games are some of the oldest math we have. (It's also never a bad time to point out that games in Africa are a MAJOR part of our mathematical heritage, btw.) (13/n)
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Here’s a thought: what would it take to DISINCENTIVIZE cheating on an exam instead of just trying to PREVENT it?
Like what if I gave, say, a Calc I final and let students use their cell phones for Desmos — how could I design it so students wouldn’t even feel the NEED to cheat?
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🤯
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Happy
#TauDay
— a celebration of the best circle constant! Anyone have plans to eat twice as much pi? 🍕🥧
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Maybe if we had better, more evocative words to use than "sine", "cosine", and "tangent", we wouldn't have had to rely on SOHCAHTOA in the first place.
Like maybe "height ratio", "width ratio", and "slope ratio", or something like that? Or maybe "far ratio" & "close ratio"?
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@LeviathanNow
I was gonna tag you once I got some responses! 😄
Also asked in a math-centric Discord server, apparently it's useful in K-theory, which I only vaguely know about ...
The quote was said, "Your Algebra II student was cooking!"
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What's your pithiest explanation of "why do inside transformations go backwards?"
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Minor pet peeve: claims that "multiplication isn't repeated addition."
Just because multiplication gets extended to a context where repeated addition breaks down does not mean that introducing it that way is a disservice.
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Repeat after me: Logical foundation is not the same thing as pedagogical foundation.
If you are teaching epsilon delta limits to your beginning calculus class please ask yourself why
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Goal for this year in College Algebra:
#EndSimplify
I want to be transparent about what students should do. Expand, contract, write without negative exponents, write WITH negative exponents. "Simplify" is subjective and overloaded.
Materials I'm working on attached.
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