#teaching mathematics

[reality-detective]

Mathematics 🤔

#pay attention #educate yourselves #educate yourself #knowledge is power #reeducate yourself #reeducate yourselves #think about it #think for yourselves #think for yourself #do your homework #do some research #do your own research #question everything #ask yourself questions #mathematics #math #teach right #school system #educational system

[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.

#education #isopod #desk #work #joyful clutter #things that make me happy #mathematics #math jokes #teaching #school #silly life #bugs #bugblr #insects #invertebrates #cute

[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?

#math #mathematics #math teaching

[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.

#mathblr #mathematics #set theory #diagram #topos theory #diagonalization #topology #incompleteness #logic #nLab #Lawvere #fixed point #theorem #teaching #paradox #halting problem #math #phdblr #Yanofsky #Cantor #Tarski #Gödel #Russell #philosophy #category theory

[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!!

#mathematics #mathblr #education #math #teaching

[prettyboysdontlookatexplosions]

For better and for worse, [Sondheim's] is the most systematic and unsentimental mind that has ever addressed itself to the American musical—the sort of mind one might more easily imagine designing particle accelerators, or computer viruses too wily to destroy. “The first music teacher I had at Williams College was a man named Robert Barrow,” he says. “And everybody hated him because he was very dry, and I thought he was wonderful because he was very dry. And Barrow made me realize that all my romantic views of art were nonsense. I had always thought an angel came down and sat on your shoulder and whispered in your ear ‘dah-dah-dah-dum.’ Never occurred to me that art was something worked out. And suddenly it was the skies opening up. As soon as you find out what a leading tone is, you think, Oh my God. What a diatonic scale is—Oh my God! The logic of it. And, of course, what that meant to me was: Well, I can do that. Because you just don’t know. You think it’s a talent, you think you’re born with this thing. What I’ve found and what I believe is that everybody is talented. It’s just that some people get it developed and some don’t.”

(x) on the one hand, it's unbelievably funny that sondheim is like, "everyone would be thrilled to learn about leading tones and diatonic scales. the fact that i felt this way about them has nothing to do with me having any kind of talent. this is just the normal way for people to respond." but on the other hand i truly love so much knowing that one of the great creative luminaries of his century agreed with me that Teaching People Stuff Is Good Actually.

#one way i'd like started souring on things and a thing i still have a grudge over #was how often i'd hear or read people talking about the importance of teaching math in a way that nurtured mathematical creativity #meaning like: lots of problem solving with open ended problems not a lot of sitting down and doing your facts #and this made me feel very crazy bc i had taken piano lessons as a kid #(a 'purely' 'creative' pursuit) #and like... MUCH of my memory of learning to play the piano was. uh. #boring uncreative rote repetition in the service of building automaticity #and i mean. i was not a good piano student and i have no way of assessing my piano teacher's skill #but it's difficult for me to imagine a path to being a good pianist that does not run throough a lot of scales #or a way of even just memorizing a single piece that doesn't involve playing it until you're blue in the face #anyway. this isn't really about piano. it's about like #this notion that teachers had of capital-C Creativity #that felt totally separated from like the specifics of what i knew it meant to pursue like. many kinds of creative pursuits. #in a way that was like... what are you basing this on. where are you getting this idea. what does this look like in your head. #most of you are not people who consider yourselves to be particularly creative math thinkers #so what makes you feel like you know what it takes to turn someone into a creative math thinker

[greetings-inferiors]

Hey, I realize you do like maths. As someone who didnt go through highschool but got a highschool degree with only REALLY BASIC maths knowledge, I wanna ask: - Any advice or recommendations for someone who wants/needs to catch up/go from elementary to highschool maths ASAP many years after dropping the subject?

It seems to be an interesting subject but I had teachers that were so bad at teaching and so good at shaming and traumatizing that it blocked me and made me avoid maths like the plague, I do want to start over with maths and try again while making it a good experience this time, I need advice. Pls help. (anonymous cause embarrassed to admit I can barely get around with the basic 4 operations and begin getting lost when it goes into fractions, decimals, porcentages etc, and as a college student I should know advanced stuff like factoration and complex expressions by now)

I am incredibly blessed with the fact that I love maths, and had great teachers. I don’t really know how to get good at maths because by the time I was actually conscious about liking maths, I was already pretty good at it. I never had that thing of having to be better, because I’ve always just been good at it, and the things that I didn’t know I enjoyed learning so I just learnt them.

The problem with having to relearn something is that you FEEL like you’re better than you are. I stopped learning Japanese for a few months, and when I come back to it, I’ll have to go over basic kanji again, my brain tells me that I know it, but I don’t. I need to go over the basics, but before I learnt the basics with the spark of learning pushing me through. Now I’ve got to essentially revise something I forgot. It sucks.

What I’d recommend, is by jumping into the deep end. There are lots of maths videos on YouTube, and they’re really interesting, but you won’t understand anything. But that’s fine, because the things you don’t understand, you can watch videos about those. And the parts of those you don’t understand you can research into that. It may not be the most efficient way to learn, but eventually you will. Trial by fire and all that, and it might be more fun because you’re looking at stuff that interests you! You’ll find that the simple stuff actually has rather complex and interesting explanations, which I find really cool.

If you want to relearn quickly, then you just have to study. It sucks, but that’s just how it is. I don’t know what elementary school is, I assume it’s 11-14, and high school probably means gcse, which is 15-16.

Some basic tips:

Think of the operators as logically as possible. When you see 5x15, literally think of 15 added together 5 times. Think of 6/20 as 6 lots of 1/20 (which itself is 0.05. Maybe even think of it as 1/2 times 1/10.) basically just think of the operators as simply as possible until you’re able to think of them as their own thing. Then you can start introducing indices, square roots, etc.

Don’t be afraid of using a calculator (learning how to use a calculator effectively will massively boost your mathematical literacy).

write everything down (don’t rely on your mental maths. If you literally have to do every single equation on a piece of paper (assuming calculators aren’t allowed), do it. Never trust your mental maths until you’re certain that you’ve got good mental maths. Seriously, 90% of mistakes come from trying to make a shortcut in your head and messing up. Many people, my self included in the past, see writing down your working out as a sign of weakness, it isn’t.

Try to avoid the divisor symbol as much as possible, it isn’t actually an operator, it’s shorthand for fractions (the dots are placeholders for the things in front and behind). Honestly, you should prioritise getting comfortable with fractions. They’re really useful, especially in algebra.

If you get good at algebra, you’ll be good at almost everything maths can throw at you. Being able to rearrange equations is a skill that you will literally never not use. It also helps you with regular number equations because you can think of the numbers as variables. It sounds weird or as if you’re complicating it, but it can help.

(A/B)*C=(A*C)/B. It’s surprising how useful it is, and how often I’ll forget about it lmao

Look into geometry! Everything you do in maths can and has been described with shapes. And for some people that can help them visualise it! If shapes help you with maths, look into shapes! Geometry!

Factorisation is essentially just the reverse of multiplication. (2*5*7)=70, therefore the prime factors of 70 are 2,5, and 7. The same applies to algebra. Just think of what could be multiplied together to make x^2+3x+2. And hey, there’s a really handy formula for finding out the factors of quadratics that I highly recommend memorising if you think you’ll need it!

And most of all: try and have fun! Basic maths can be very tedious, but think of it like learning a language. Once you get the alphabet (numbers) and grammar (operators) out of the way, you’ll start to see all of the complex words and phrases you can create, and understand. And, best of all, you will NEVER stop learning, so you may as well start now!

#text post #randyposting #maths #Randy tries to be a teacher #ask #ask answered #anon ask #I hope this helps! #also don’t worry I wouldn’t judge anyone for being bad at maths #I can understand how important a good teacher is #math #mathematics #mathblr #if anyone wants to pitch in that’d be great!!! #teaching #maths teacher #teach #tutor #tutoring #help #assistance #maths help #maths problem #math help #math problem #learning #anon asks #long post

[simplydnp]

very excited to see fellow Dan and Phil fan in stem

i know that i have a degree in dan and phil studies but genuinely my true passion is actually mathematics and i am So sorry

#like. i must be so clear: i am a Nerd. with pride #like. ya girl has never cut any class ever in her life. and i watch math videos for FUN #i am helping my sister with a university math class that i never took. on a subject i intentionally avoided (geometry my beloathed) #and i am doing her homework with her without reading the textbook. for FUN. and i get them right. #i am Built for mathematics okay i love her dearly #you teach me a math concept once. show me it *one* time. and i understand it. #i will Ride for my girl math okay. she's awesome. and cool. and useful. #but i do totally understand if she scares you. but i promise she's actually really nice. so dont be mean to her #my last ever math course. was an advanced proofs class. that has a rep of being difficult. it had 2 midterms. a long final. not openbook. #i finished the course with over a hundred percent. 108 and 110 on each midterm. 100 on the final. 110 on assignments. #and i fucking adored every moment in class. and the homework. and studying. such a cool goddamn course. #i just. really like math okay. #dnp #c.text

[artistic-arteries]

I wish we taught kids math games to help them practice addition, subtraction, multiplication, and division in their heads.

I just started playing Cult of the Lamb and I loved the knucklebones mini game so much. It's addition and simple multiplication and would be great for those just starting to learn multiplication. (Play here, wiki here)

Another good one is sticks, its just addition so it's great for first and second graders. (Wikipedia entry here)

#math #mathematics #math education #early education #elementary school #primary school #teaching math #math teaching tools

[scifigeneration]

3 reasons we use graphic novels to teach math and physics

by Sarah Klanderman, Assistant Professor of Mathematics at Marian University and Josha Ho, Adjunct Professor of Mathematics and Computer Science at Marian University

Post-pandemic, some educators are trying to reengage students with technology – like videos, computer gaming or artificial intelligence, just to name a few. But integrating these approaches in the classroom can be an uphill battle. Teachers using these tools often struggle to retain students’ attention, competing with the latest social media phenomenon, and can feel limited by using short video clips to get concepts across.

Graphic novels – offering visual information married with text – provide a means to engage students without losing all of the rigor of textbooks. As two educators in math and physics, we have found graphic novels to be effective at teaching students of all ability levels. We’ve used graphic novels in our own classes, and we’ve also inspired and encouraged other teachers to use them. And we’re not alone: Other teachers are rejuvenating this analog medium with a high level of success.

In addition to covering a wide range of topics and audiences, graphic novels can explain tough topics without alienating student averse to STEM – science, technology, engineering and math. Even for students who already like math and physics, graphic novels provide a way to dive into topics beyond what is possible in a time-constrained class. In our book “Using Graphic Novels in the STEM Classroom,” we discuss the many reasons why graphic novels have a unique place in math and physics education. Here are three of those reasons:

Explaining complex concepts with rigor and fun

Increasingly, schools are moving away from textbooks, even though studies show that students learn better using print rather than digital formats. Graphic novels offer the best of both worlds: a hybrid between modern and traditional media.

This integration of text with images and diagrams is especially useful in STEM disciplines that require quantitative reading and data analysis skills, like math and physics.

For example, our collaborator Jason Ho, an assistant professor at Dordt University, uses “Max the Demon Vs Entropy of Doom” to teach his physics students about entropy. This topic can be particularly difficult for students because it’s one of the first times when they can’t physically touch something in physics. Instead, students have to rely on math and diagrams to fill in their knowledge.

Rather than stressing over equations, Ho’s students focus on understanding the subject more conceptually. This approach helps build their intuition before diving into the algebra. They get a feeling for the fundamentals before they have to worry about equations.

After having taken Ho’s class, more than 85% of his students agreed that they would recommend using graphic novels in STEM classes, and 90% found this particular use of “Max the Demon” helpful for their learning. When strategically used, graphic novels can create a dynamic, engaging teaching environment even with nuanced, quantitative topics.

Combating quantitative anxiety

Students learning math and physics today are surrounded by math anxiety and trauma, which often lead to their own negative associations with math. A student’s perception of math can be influenced by the attitudes of the role models around them – whether it’s a parent who is “not a math person” or a teacher with a high level of math anxiety.

Graphic novels can help make math more accessible not only for students themselves, but also for parents or students learning to be teachers.

In a geometry course one of us (Sarah) teaches, secondary education students don’t memorize formulas and fill out problem sheets. Instead, students read “Who Killed Professor X?”, a murder mystery in which all of the suspects are famous mathematicians. The suspects’ alibis are justified through problems from geometry, algebra and pre-calculus.

A peak inside the mathematical graphic novel ‘Who Killed Professor X?’.

While trying to understand the hidden geometry of suspect relationships, students often forget that they are doing math – focusing instead on poring over secret hints and notes needed to solve the mystery.

Although this is just one experience for these students, it can help change the narrative for students experiencing mathematical anxiety. It boosts their confidence and shows them how math can be fun – a lesson they can then impart to the next generation of students.

Helping students learn and readers dream big

In addition to being viewed favorably by students, graphic novels can enhance student learning by improving written communication skills, reading comprehension and critical literacy skills. And even outside the classroom, graphic novels support long-term memory for those who have diagnoses like dyslexia.

Pause and think about your own experience – how do you learn about something new in science?

If you’re handed a textbook, it’s extremely unlikely that you’d read it cover to cover. And although the internet offers an enormous amount of math and physics content, it can be overwhelming to sift through hours and hours of videos to find the perfect one to get the “aha!” moment in learning.

Graphic novels provide a starting point for such a broad range of niche topics that it’s impossible for anyone to be experts in them all. Want to learn about programming? Try the “Secret Coders” series. Want to understand more about quantum physics? Dive into “Suspended in Language: Niels Bohr’s life, discoveries, and the century he shaped.” Searching for more female role models in science? “Astronauts: Women on the Final Frontier” could be just what you’re looking for.

With all that they offer, graphic novels provide a compelling list of topics and narratives that can capture the attention of students today. We believe that the right set of graphic novels can inspire the next generation of scientists as much as any single individual can.

This article is republished from The Conversation under a Creative Commons license. Read the original article.

#science #mathematics #Physics #education #Graphic novels #comics #teaching math #teaching science #Physics education #Math anxiety #Youtube

[forgottenbones]

Teaching an idiot basic maths | Blackadder - BBC

#blackadder #a bit when they try teaching me mathematics... #a tad more complex than this I must add #''the renaissance was just something that happened to other people.''

[riemmetric]

I woke up yesterday with the overwhelming feeling of I love math. The Gauss Bonnet theorem and the theorem of surface classification have that effect on me, I suppose.

#mathblr #sorry if the second one is not named correctly english is my second language #mine: mathematics #the problem session class i teach this week is about them

[wronghands1]

#parity #odd #even #numbers #math #mathematics #wordplay #equals #division #teaching #wronghands #john atkinson #cartoon #webcomic #math humor #funny

[strawhatboy]

I will start teaching on monday! so excited, wish me luck besties <333

#I'll be teaching portuguese mathematics and english. my students are like 11 and seem to be very kind! #two of them need individual lessons because apparently they like each other #and can't concentrate during the classes when they are in the same environment skdkalsdh so cute

[asoulunbound]

Don't tell me why, but I like the idea that Alex would help Spencer get a teaching position at Harvard somewhere down the line. I love the idea of them being colleagues again and Spencer moving to Boston with his family. 🥰🥰 Although he would miss the rest of his team in DC.

#&& this is calm and it’s doctor (Spencer): headcanon #(he would teach mathematics!)

[mainlysarcastic]

The relationship between math and philosophy is so fucking cool and annoying

Like why’d school have to kill my understanding and love of math ?? My college education falls in the realm of philosophy but the more math/physics I learn now as an adult (thx science YouTube channels for sharing educational things in an engaging way) the more I see how philosophy and mathematics are actually like two sides of the same coin

It’s so fucking beautiful and confusing

#mathematics #philosophy #echo thoughts #I’m angry at the educational system making us think math is boring #and for teaching it in such unengaging ways #also especially fuck common core cause my school switched when I started HS & it sucked ass going from one Major curriculum to another #anyways #education #science rules