Places like UChicago teach us to ask “what’s next” for our own advancement, to do this now so we can get to that later. I learned to ask “what’s next” to be as useful as possible to as many kids as I have in front of me.

Making concept maps are incredibly important in learning mathematics, far better than just a long list of isolated examples. I look at a key concept in Linear Algebra - Linear Independence - and make my own little concept map about it. You can be creative when you make your own, this is just what popped into my head, but the point is to always be thinking about the connections between different ideas first and foremost. ************************************************** Now it's your turn: 1) Summarize the big idea of this video in your own words 2) Write down anything you are unsure about to think about later 3) What questions for the future do you have? Where are we going with this content? 4) Can you come up with your own sample test problem on this material? Solve it! Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. *************************************************** Want more ideas for learning math effectively? ►How to Watch Math Videos: https://www.youtube.com/watch?v=Ljq5y3ksk8o ►5 Tips to Make Math Practice Problems Effective: https://www.youtube.com/watch?v=LPH2lqis3D0 **************************************************** ►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho **************************************************** Course Playlists: ►Calculus I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ►Calculus II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdQCBjYswqbn7LxL1pW4cW4 ►Discrete Math: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►Linear Algebra: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ***************************************************** ►Follow me on Twitter: http://twitter.com/treforbazett ***************************************************** This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati. BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett

Although project-based learning holds some place in a large percentage of edtech startups today — potentially because the lack of administrative burden makes it more economically efficient — I think the lasting value of this type of education is largely overhyped. …they often attempt to pack too many disparate kinds of learning into the project-based model. In doing so, they can inadvertently make it harder to master fundamental concepts by strengthening strengths and not pushing students to improve their weaknesses. The general level of excitement over revisiting the foundations of education today is super interesting, but it could also be very easy for us to reinvent the wheel in a worse way.

25th of March, 2021. Part of the Topos Institute Colloquium. ----- Abstract: The climate crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that no physical quantity can grow exponentially forever. This transformation may affect mathematics — and be affected by it — just as dramatically as the agricultural and industrial revolutions did. After a review of the problems, we discuss how mathematicians can help make this transformation a bit easier, and some ways in which mathematics may change.

Dropbox mirror: https://www.dropbox.com/s/bosace9japm3s5j/Gian-Carlo%20Rota%27s%20Lessons%20I%20Wish%20I%20Had%20Been%20Taught.pdf?dl=0

Kevin Buzzard's invited talk at CICM arguing that getting undergraduate mathematics formalised in theorem provers is useful and important. Slides at http://wwwf.imperial.ac.uk/~buzzard/one_off_lectures/ug_maths.pdf

I think one of the reasons Zoom school horrified so many people, is not because it was a big break in how education works but because it brought the reality of how education works, to the forefront it highlighted the American classroom as the consumer audience structure it is

If students don’t like what’s been pushed onto them, they’ll go around it. The only way to help students is to build for them, and the only way to do that is to watch them try to learn.

That’s the value of person-to-person teaching that’s irreplaceable in textbooks and MOOCs – the human ability to recognize a problem someone doesn’t even know they have.

If this is really how math is done, then why isn't it taught that way at every level?

I’ve come here for a better life, and for the lack of a better word, all I can feel sometimes is that I’ve bet on the wrong horse.

I typically engage students in this model by asking them to draw their own Comfort/Chaos Circle. This is a great activity to get students to check with themselves and communicate where they are with me.

Thank you for 2^21 subscribers. Store: http://3b1b.co/store ↓↓Links to things discussed below ↓↓ The Bit Player https://thebitplayer.com/ The Idea Factory https://amzn.to/325PY6W The Information, A History, A Theory, A Flood https://amzn.to/2QhfUtR You can read about Lorenz in "Chaos", also by James Gleick https://amzn.to/3tcXtVF Anthropocene reviewed https://www.wnycstudios.org/podcasts/anthropocene-reviewed Hardcore history https://www.dancarlin.com/ The Numberphile podcast https://www.numberphile.com/podcast Recent Quanta article on the eigenvector discovery https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/ Channel supported by http://3b1b.co/thanks ------------------ Animations on this channel are largely made using manim, a scrappy open source python library. https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/a... Stream the music on Spotify: https://open.spotify.com/album/1dVyjw... If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown... Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown

Tips on problem-solving, with examples from geometry, trig, and probability. Past episodes with integrated quizzes: https://itempool.com/c/3b1b Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev Home page: https://www.3blue1brown.com Brought to you by you: https://3b1b.co/ldm-thanks Huge huge thanks to Ben Eater: https://www.youtube.com/user/eaterbc And Cam Christensen, creator of ItemPool: https://itempool.com/ Notes by Ngân Vũ: https://twitter.com/ThuyNganVu/status/1265480770832855040 Mistakes: 50:35, there should be a dx in the integral 54:40, if you notice the mistake here and are inclined to complain, keep watching ------------------ Video timeline (thanks to user "noonesperfect") 0:34 9-Problem Solving Principles/Tip 1:15 Question 1 (Probability) 2:08 Who is Ben Eater? 4:25 Inscribed angle theorem, θL=2*θs 5:58 Tip-1 7:48 Tip-2 8:16 Question 2 9:34 Answer 2 10:29 Tip-3 15:17 Tip-4 22:48 Question 3 25:56 Answer 3 (Marked incorrectly, Answer: Option D) 26:31 Answer 1 27:28 Explanation for Q1, Floor function 30:38 Tip-5 32:53 Tip-6 33:36 Question 4 34:43 Answer 4 36:37 Question 5 38:10 Answer 5 41:48 Probability graph in Desmos 44:08 Revisiting an alternating series sum for ln(2) 47:29 Tip-7 51:08 Tip-8 55:23 Audience questions through tweets 57:28 Tip-9 58:29 Python programming for various probability question 1:04:31 Arts created using Desmos graph tool with mathematical expressions 1:05:54 Thank you, appreciation to the team and all. ------------------ The live question setup with stats on-screen is powered by Itempool. https://itempool.com/ Curious about other animations? https://www.3blue1brown.com/faq#manim Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown

The remarks below started as a first draft and incorporate feedback from half-a-dozen colleagues, mentors, and friends who push my thinking, every day.

the mark of a good education would shift from what school you attended to which professors you learned from. You could stay on one campus for four years, or move around every semester to get the best professors in person. Totally up to you. Similar to how Minerva has their students change campuses each semester, you could jump around the country, or world, to get the professors you want to learn from in person.

Simulate discovery whenever possible. For example, to teach the importance of numbers, first, imagine a world without numbers. Now give the students a problem to solve, like keeping track of the number of sheep in their herd in this world without numbers. The student needs the same opportunity to build their own abstractions as the Expert got. Without this, they can never gain true understanding.

With these two new models for schools in place (decision schools + trade schools), we'd also have the benefit of easy mobility later in life. Want to make a change and not sure what comes next? Go to Decision School, decide on a new path, then go to another trade school that can get you into that field.

For Gumroad's monthly financial updates, follow @shl on Twitter. Feedback? Send it to sahil@gumroad.com. Our third open board meeting! To find out about the next one we do, follow me at gumroad.com/sahil/follow

We have a part our school day dedicated to tutoring and reteaching. Every kid deserves to go to a school where they are always safe and never bored. There is a tremendous amount of intellectual labor that goes into good teaching. Every great teacher knows this and we think that in order to be a great school, we should provide and protect our teachers’ time to do it. We will not be back in the same building until next school year, but the school is not the building. We aren’t safe from the world, but we can feel the safety of teachers and classmates celebrating our successes and learning from our errors.

But you can tell a lot about a teacher by how they respond when students don’t succeed. Some will say, “What’s wrong with you?” Others will ask, “What’s wrong with me?” great teachers know three things: they know their subject, they know how to explain their subject, and they know their students. Knowing the history of a topic, the interesting problems, the common misconceptions, the multiple approaches, the links to other topics. And it’s about expert communication of all of that. They co-create educational experiences with their students because they realize that students hold key knowledge of their own that most don’t acknowledge or even recognize.

Isabel’s solution of pre-writing part of her talk and then filling in parts of it to give a more interactive feel was really excellent, and I plan to imitate it in the future.

You can’t help but wonder: if we’re forced for an entire year to forego the in-person benefits of these practices, what happens when life returns to normal? ​ It’s basic human nature to pursue positional scarcity, and to do whatever it takes to compete for it. The virus won’t change that.

I’m a better teacher when I focus on what students experience first, and my actions as a teacher second.

For them the videos were a grace they didn’t have to earn. I like to tell them the struggle is the more interesting place to be: because a healthy confusion is where the real learning begins. In fact, failing a student CAN be done with grace, so that the student understands their dignity has not been tarnished even though their work has been justly assessed---just as a parent can discipline her child if the child knows her love is unconditional. Grace is precisely what makes hard conversations possible, and productive, between people. But you have to extend the grace first. One of my students said to me, so gently: “Should I be terminally ill later in life, I would want my son to act as you have.”

Our second open board meeting! To find out about the next one we do, follow me at gumroad.com/sahil/follow

As opposed to a textbook, real maths is highly non-linear. ​ As we will see throughout the post, personalization (and the engagement inherent in it) is essential to the success of the lecture.] ​ [Before the third, I ask the class whether the first two alone are enough. If I get nods, I draw a random collection of dots and lines, with the lines not at all connected to the dots, and they see we need some statement of incidence.] ​ Since we will always draw constellations as a picture, we can just use the picture as our “function.” ​ Compare this to being given the definitions and propositions in the established mathematical language. To an untrained, uninterested student, this is not only confusing, but boring beyond belief! They don’t have the prerequisite intuition for why the definition is needed, and so they are left mindlessly following along at best, and dozing off at worst.

“Race to the Top; what a horrid metaphor for education. A race? Everyone is on the same track, seeing how fast they can go? Racing toward what? The top? The top of what? Education is not a race, it’s an amble. Real education only occurs when everyone is ambling along their own path.” —Peter Gray ​ 2. Always work (note, write) from your own interest, never from what you think you should be noting or writing. Trust your own interest.