#teaching mathematics

[reality-detective]

Mathematics 🤔

[maths-screaming]

Things That Are and Are Not Changing as a Result of Re-Teaching Myself Basic Math

Things That Are Changing

I am less anxious about basic math.

I have more tools for doing basic arithmetic problems.

I'm better at identifying which will be the fastest or easiest tool for any given problem.

I can more quickly and easily ID when an exact answer is needed or when an estimate will suffice.

I'm marginally better at noticing when an answer can't be correct.

Things That Are Not Changing

I still transpose numbers frequently.

I still transpose operations frequently (adding when I should subtract, dividing when I should multiply, etc.)

I still have initial anxiety when looking at a math problem, before the "oh yeah, I have more tools for addressing this now" kicks in.

I still frequently mix up my right and my left.

My sense of direction is still bad.

I cracked Maths - No Problem! Textbook 4A today, putting me halfway through the series. I'm making this list for future reference, because I suspect the things that aren't changing will continue to not change.

Better math education won't change the fact that I have dyscalculia. I didn't expect it to, but I also didn't know what it would or wouldn't change. When I started this, I didn't know where my dyscalculia ended and my poor math education or math anxiety began.

Still, if we can fix "poor math education" and "math anxiety," I'll be much further ahead than when I started - and more willing to live with the dyscalculia.

[wronghands1]

[datapilled]

Going to go on a little rant but my god I'm sick of how undervalued maths is. I hate that for the past 10 years, including primary school, whenever I tell people I like maths I get asked if I'm insane. I hate that in a country that has made major contributions to maths, there's barely any effort to keep up that history and support new mathematicians.

I hate that I can't criticise people who also do stem that go on and on about how everything relating to maths is stupid and useless because "that's just how they feel"...Even when they're doing computer science. And said computer scientists complain every. Single. Time. When a maths equation appears.

I hate that I can't point out how it's weird how people demonize mathematics itself instead of how it's taught, which is where the problem usually lies. I hate that I'm expected to drop mathematics entirely to go into a "proper" job like programming instead of research into population models and simulations. I hate how maths is being co-opted by people making points about "how simple the world is" when maths is nothing like they portray it to be. I hate that maths is seen as a field only of white, cishet men when there's been so many women, poc and queer people that have done groundbreaking work.

I hate that we're in a world where mathematics is so important to everyday life, and yet the idea of actually studying maths and continuing research into it is frowned upon.

[futurebird]

no it is not “junk”

i need all of these to do my job!

YES even the plastic isopod. (his name is TIMOTHY, OK?)

Timothy is critical to my lesson planning process.

[alhaithamexplains]

[katzensilber]

Man, this is why I always hated math class.

Khan Academy gave me this easy (or so I thought) problem to solve:

"A factory makes toys that are sold for $10 a piece. The factory has 40 workers, and they each produce 25 toys a day. The factory is open 5 days a week. What is the total value of toys the factory produces in a day?"

I said to myself, okay...

10 times 40 times 25 equals 10,000. That means the factory produces $10,000 worth of value per WORKING day.

10,000 times 5 equals 50,000. And then there are two days per week (the weekend) when no value is produced. So, per week, the total value produced is still only $50,000.

And finally, there are seven days in a full week, so $50,000 divided by 7 equals roughly $7,142.86. That's the total value of toys produced per day. Not "per work day", but "per day", as it specifically says in the problem.

But what does the teacher say as he guides us through it? 10 times 40 times 25 equals 10,000, so the answer is 10,000.

That's it. The end.

He even acknowledges: "So, you might be thinking, 'Hey wait, we didn't use all the information! We didn't use the "five days per week" information!' And yeah, as it turns out, that was information we didn't need."

'As it turns out'?! Are you kidding me right now?! I carefully read the question, precisely calculated exactly what it was asking for, taking all of the provided information into account... and now you're saying I'm supposed to have somehow magically known to ignore one of those pieces of information in order to make it easier for myself?

Just admit that your problem is worded badly, dude. UGH this kind of shit drives me crazy. Why can't people use words clearly >:(

[sanktpolypenbourg]

As I said in a previous post, I have deep sympathy for the frustration of people who are good at math when they see math so almost universally hated by children and adults

And again and again, they try to explain that math is very much within everyone's reach and can be fun and, at least in western countries, education was to blame, messing up this very doable and fun thing by teaching it wrong

But I still gotta wonder - why math? If it is really just education messing this up, why does it mess up so much with math, specifically? I'm sorry but I still cannot shake the sense that even if it's just bad teaching, math is especially vulnerable to bad teaching.

Or is it maybe just that math is the only truly exact science, so there is no margin of error, so unlike every other field where you can sortof weasel around and get away with teaching and retaining half-truths and oversimplifications and purely personal opinions, math is unforgiving with the vague and the incorrect?

[stupidvillainousposts]

Headcanon that Stan is actually really smart about certain subjects and accidentally spews out random information without realizing it (Bonus points if it turns his Nerdy Hillbilly on)

Dipper: I'm supposed to build a robot for school that helps explain my summer experiences this year. The coding and layouts are easy, but I don't think I'd be able to create a working model.

Mabel: Why don't you ask Old Man McGucket? He builds all sorts of stuff all the time.

-One Frustrating Phone Call Later-

Dipper, showing Fidds his layout: And that's essentially what I need done.

Fidds: Well, that don't look too difficult. Give me some scrap metal and a spit bucket and I'll have this built in no time!

Stan: *After Catching a Glimpse of Dipper's Plans* I dunno, old man. Scraps wouldn't be able to sustain the type of AI and intellect the kid is wanting to implement. *Leans Over Fidds* Taking the size, purpose, and necessary functionality into account, I'd say the best course of action is- *Goes on a Tangent About Machines and Quantum Mechanics (for Some Reason)*

Dipper: *Absolutely Stunned*

Fidds, inwardly: Fiddleford Hadron McGucket, you need this fat man and NOW.

[m---a---x]

Welcome to the premier of One-Picture-Proof!

This is either going to be the first installment of a long running series or something I will never do again. (We'll see, don't know yet.)

Like the name suggests each iteration will showcase a theorem with its proof, all in one picture. I will provide preliminaries and definitions, as well as some execises so you can test your understanding. (Answers will be provided below the break.)

The goal is to ease people with some basic knowledge in mathematics into set theory, and its categorical approach specifically. While many of the theorems in this series will apply to topos theory in general, our main interest will be the topos Set. I will assume you are aware of the notations of commutative diagrams and some terminology. You will find each post to be very information dense, don't feel discouraged if you need some time on each diagram. When you have internalized everything mentioned in this post you have completed weeks worth of study from a variety of undergrad and grad courses. Try to work through the proof arrow by arrow, try out specific examples and it will become clear in retrospect.

Please feel free to submit your solutions and ask questions, I will try to clear up missunderstandings and it will help me designing further illustrations. (Of course you can just cheat, but where's the fun in that. Noone's here to judge you!)

Preliminaries and Definitions:

B^A is the exponential object, which contains all morphisms A→B. I comes equipped with the morphism eval. : A×(B^A)→B which can be thought of as evaluating an input-morphism pair (a,f)↦f(a).

The natural isomorphism curry sends a morphism X×A→B to the morphism X→B^A that partially evaluates it. (1×A≃A)

φ is just some morphism A→B^A.

Δ is the diagonal, which maps a↦(a,a).

1 is the terminal object, you can think of it as a single-point set.

We will start out with some introductory theorem, which many of you may already be familiar with. Here it is again, so you don't have to scroll all the way up:

Exercises:

What is the statement of the theorem?

Work through the proof, follow the arrows in the diagram, understand how it is composed.

What is the more popular name for this technique?

What are some applications of it? Work through those corollaries in the diagram.

Can the theorem be modified for epimorphisms? Why or why not?

For the advanced: What is the precise requirement on the category, such that we can perform this proof?

For the advanced: Can you alter the proof to lessen this requirement?

Bonus question: Can you see the Sicko face? Can you unsee it now?

Expand to see the solutions:

Solutions:

This is Lawvere's Fixed-Point Theorem. It states that, if there is a point-surjective morphism φ:A→B^A, then every endomorphism on B has a fixed point.

Good job! Nothing else to say here.

This is most commonly known as diagonalization, though many corollaries carry their own name. Usually it is stated in its contraposition: Given a fixed-point-less endomorphism on B there is no surjective morphism A→B^A.

Most famous is certainly Cantor's Diagonalization, which introduced the technique and founded the field of set theory. For this we work in the category of sets where morphisms are functions. Let A=ℕ and B=2={0,1}. Now the function 2→2, 0↦1, 1↦0 witnesses that there can not be a surjection ℕ→2^ℕ, and thus there is more than one infinite cardinal. Similarly it is also the prototypiacal proof of incompletness arguments, such as Gödels Incompleteness Theorem when applied to a Gödel-numbering, the Halting Problem when we enumerate all programs (more generally Rice's Theorem), Russells Paradox, the Liar Paradox and Tarski's Non-Defineability of Truth when we enumerate definable formulas or Curry's Paradox which shows lambda calculus is incompatible with the implication symbol (minimal logic) as well as many many more. As in the proof for Curry's Paradox it can be used to construct a fixed-point combinator. It also is the basis for forcing but this will be discussed in detail at a later date.

If we were to replace point-surjective with epimorphism the theorem would no longer hold for general categories. (Of course in Set the epimorphisms are exactly the surjective functions.) The standard counterexample is somewhat technical and uses an epimorphism ℕ→S^ℕ in the category of compactly generated Hausdorff spaces. This either made it very obvious to you or not at all. Either way, don't linger on this for too long. (Maybe in future installments we will talk about Polish spaces, then you may want to look at this again.) If you really want to you can read more in the nLab page mentioned below.

This proof requires our category to be cartesian closed. This means that it has all finite products and gives us some "meta knowledge", called closed monoidal structure, to work with exponentials.

Yanofsky's theorem is a slight generalization. It combines our proof steps where we use the closed monoidal structure such that we only use finite products by pre-evaluating everything. But this in turn requires us to introduce a corresponding technicallity to the statement of the theorem which makes working with it much more cumbersome. So it is worth keeping in the back of your mind that it exists, but usually you want to be working with Lawvere's version.

Yes you can. No, you will never be able to look at this diagram the same way again.

We see that Lawvere's Theorem forms the foundation of foundational mathematics and logic, appears everywhere and is (imo) its most important theorem. Hence why I thought it a good pick to kick of this series.

If you want to read more, the nLab page expands on some of the only tangentially mentioned topics, but in my opinion this suprisingly beginner friendly paper by Yanofsky is the best way to read about the topic.

[catscomputerscreativewriting]

Listen, textbook, I admire the dedication to grifting my students into learning most of Constructive Logic before they've realized it. They are computer scientists after all. It's good for them. But... really? No explanation of formal proofs by contradiction in the undergraduate, Logic for CS textbook? You do include double negation elimination, which is an equivalent Classical axiom, but not, like, (A -> Bottom) -> ~A?

You mean I'm gonna have to derive that myself from the completely random and arbitrary Classical axioms you've decided to adopt and then neglected to prove the completeness of? Otherwise, all of these proofs will become extremely cumbersome and all of my students will swear off CS theory for the rest of their lives?

... ok fine, it'll be in the slides. T-T

[maths-screaming]

Commence the Screaming

This is a blog about innumeracy.

Innumeracy is like illiteracy, but for numbers. I have dyscalculia - or "math dyslexia," if you prefer something easier to say that more people also understand.

Like most people with dyscalculia, I've had it my whole life. I've also been functionally innumerate my whole life. I can count (usually); I can do basic addition and multiplication (kind of), subtraction (sometimes) and division (ehhh....). I hate games that involve counting money or keeping score. I can read an analog clock if you make me. I mix up my right and left more than random chance allows. I get lost a lot.

My math skills top out at "calculate a 20% tip on this bill" or "count change." I can't even reliably use a calculator for a lot of things, because calculators need the user to know what they're trying to get the calculator to calculate, and I can't always tell the thing the steps in the right order.

I'm 42, and for 42 years, I've hated, avoided, and feared basic math. I have three college degrees - not one requiring a single math course. I'm fascinated by several topics involving math, but I can't do them. The numbers knock me out of contention every time.

So why - now, when I have my own house, a stable job that requires only the occasional calculator arithmetic, a reliable car, and something resembling a savings account - do I care to change that?

Honestly...I don't know. There are certainly a few factors in play, including:

Most of my friends are math nerds, science nerds, and/or spatial-reasoning-artist nerds, and I want to appreciate their nerdery appropriately;

My students are aware that I'm a giant nerd who is interested in everything and thus love to ask me questions, and it hurts to admit to 15 year olds that I'm better at ancient Greek than at basic algebra;

I read Ben Orlin's Math With Bad Drawings and I want to be able to do the math, dang it. (I can already do bad drawings.)

But the biggest one is this:

Having learned what it is like to spend 40 years of one's life hating and fearing math, I don't want to spend another 40 years of my life hating and fearing math.

So...here we go. Middle-aged librarian tries to get slightly less bad with the numbers. Much screaming ensues. Ask me anything.

[hexiflexi]

Ok I love these little wizard memes:

HOWEVER!

One thing has really always bothered me about them is how people forget that the reason they feel disconnected from day to day life isn’t because school/work is inherently unfun and unwhimsical, but because capitalism has stripped any joy from these very human activities. For example, you like dnd subclasses? Hogwarts houses? Owl house covens? Any other subcategories of magic? You wanna have a specific type of magic that you learn and study in the wilderness?

Well you can! Wanna draw magic runes? Try geometry! Wanna get into potion making? Inorganic chemistry is right there! Wanna practice healing magic? Be a nurse/doctor! Wanna be an artificer? Get into engineering! Wanna practice divination? Try meteorology!

Basically my point is whimsifying life isn’t just for going to the library, seeing friends, and picking flowers; it’s also for the sciences and mathematics!

[rednecknerdguy]

Reading through “A Mathematician’s Lament” and stumbled on this quote.

Like the whole book is a revelation as to why I’m scared of solving the area of a regular polygon, but also it’s depressing because I have seen some of my former students go through the same anxiety I did and believe that math isn’t for them. Like. There’s a whole world out there for them! Math is beautiful but they will never get to see it because we cut out their chests before they had a chance to feel math the way it was meant to be felt!!

[alhaithamexplains]