What It's Like - Saturday Morning Breakfast Cereal

I remind my freshman undergrads that at least once a week. Words are hard.

Why? It makes perfect sense.

The actual reason this is true is because what "..." means is actually to take the limit as you add more digits, and the limit is 1. Not to say you're wrong about mistrusting your math teachers, after all none of my teachers that mentioned this fact actually defined the "..." and basically explained it with unjustified manipulation of infinite series.

Definitely. I was perfectly, delightfully indifferent toward math until high school. Then I started to hate it because I had the most incompetent, unstable, condescending teachers imaginable. They just figured I was stupid, so I figured I was stupid.

Then I got to college, took a basic, basic calc class with a professor I really loved, and now it's my major.

But I got lucky. Everybody has the capability for mathematical literacy. But for some reason, our society tells a lot of its members that they, from an early age, were not meant to understand one of the most beautiful of all human innovations, right up there with music, film, and the Reuben sandwich.

The best teacher I ever had was my high school calculus teacher. He was tough by fair. Years later, I learned he was also teaching us the beginnings of linear algebra, analysis, and statistics.

I guess you have to work with what you're given, but try to broaden the scope for your students. Spark a bit of curiosity in them.

I totally agree that our math education is terrible, and it won't be solved by anything like common core or more standardized tests. I think the core problem is that our curriculum is designed by mathematically illiterate people. To them, math is a subject. To me, math is everything and all encompassing. If you want to know any topic, from linguistics to economics to physics, you have to learn math first. It is a grave mistake to not give all kids the opportunity to have a basic understanding of every type of math, so at the appropriate time for studying these other topics, they will already have the background necessary. Also, in terms of societal decision making, statistical fluency alone would transform many of the commonly held opinions.

Oh, thought you were the friend and invited. 😏

Please kind sir, can you tell me how the quantums make the homeopathy work?

Yeah, that just leads to disappointment on both sides. A lot of people also confuse cosmology and cosmetology with similar results.

Which is the primary reason to say astrophysics. The freaking facepalm when this happens.

The only person who tried to talk to me on a flight was a drunk man who thought my dad was my brother.

(I was probably around 15 at the time).

RIP programmers

I use the terminal and don't like talking to people. Maybe next time I get on a flight I should make my text colour green and run some scary looking commands that's actually just me updating my system.

Well Tyson is an entertainer. An entertainer with the knowledge of a physicist. It doesn't take that much knowledge to understand why textbooks in certain states shouldnt be anti-evolution. So as a public figure that many recognize as a smart guy, Tyson can share his opinions on the subject. The problem occurs when he presents himself as an expert in that field simply because he is a smart man. The other problem occurs when we assume that he isn't an expert in the field of high school education therefore he shouldn't have an opinion on it. And that his opinion is invalid or overreaching. That he should keep to Astrophysics instead.

That's quite right, but I think his concern is that one must learn a lot of mathematics before they reach a level of literacy and intuition that a mathematician would have. The cost of teaching mathematics in a creative manner is that it would take much more time to cover the same material.

I will say this: the number one obstacle towards real learning in any subject -not just math- is grading. At all times in my teaching experience, the biggest block in my students capability was the inability to focus on the math itself, instead being set on working towards the test. This is not their fault. Modern educators need to understand this and reinvent our evaluation system from the very beginning. It all needs to be chucked out, burnt and forgotten forevermore because it is pedagogically noxious.

Hmm, it's interesting that you say that. I always had the Mark Twain perspective on education('never let schooling get in the way of your education'), so maybe that was a stroke of luck for me.

The great part is that mathematicians don't stop to think about it: we're establishing an order isomorphism between the space of "math cognisance(s)" and the numbers 0-10! I think it's unlikely that the first set even possesses a meaningful order. This is kind of a joke but I really do believe grades are useless (for pedagogy).

I'm a believer in developing exams that students train themselves to pass. There are no grades or other evaluations. You can take the exams as frequently as you want, and you can go as far as you want. The role of the teacher becomes almost like a tutor, almost like a babysitter. We should have online material that can almost completely cover necessary instruction.

Well I don't have any well formed theories currently but I am working as a math tutor so someday I will. I think your theory is a little bit radical but I would lean towards not compartmentalizing math so much and have it form a part of other subjects.

I completely realize these are radical ideas. If I were somehow in governmental power, I'd be more pragmatic about making changes, but if I were an entrepreneur, I would pursue them to the fullest extent.

Wow, that was the best argument for common core that I have read to this date. Now I feel like part of the problem. There may be things I have taken for granted due to my own intuition.

However, as for this topic of division specifically, what sort of intuition should you be getting that many aren't? Maybe I'm not even getting this one, but it's not like fractions are a problem for me. I feel comfortable just knowing long division as an algorithm; I don't feel like I'm missing anything.

Since you are interested in this topic, I'll throw another idea your way. This probably represents the most original idea that I have. I didn't declare this thought and then set out to satisfy it. It slowly developed in me, organically, until I was following it to the 'T' and looking for further ways to optimize.

Are there any efforts to take what we know about machine learning and apply it to human learning?

I've come to develop very large graphical databases of notes because the relationships between nodes contain exponentially more data than simple sequences ever could. Due to my current program limitations, the database is functionally equivalent to nested sequences, as no node ever has more than one parent, but the visual representation as a graph is dramatically superior to a normal document format. This has improved my memory, helped me link entirely separate subjects (and learn interdisciplinary material much quicker), and given me the tools to develop an inside-out knowledge of a given topic very quickly (for instance, I covered all of a computer science B.S. curriculum in around a year).

This, in itself, is form representing function. Our brains are neural nets, and seeing graphs gives it a 1 to 1 representation to remember and to algebraically modify for further knowledge. In a way, it's like making the computer screen a source of RAM and hard drive for your brain, moreso than a document ever could; the analogy might be that you are increasing the bus width dramatically.

However, you could further take ideas from machine learning. You could start thinking about how you could feed your brain large amounts of data to process and develop rules for. As you develop the rules, you place them intelligently into your database. You could also try to start developing better deductive logic. For instance, a machine vision neural net that is only trained on recognizing apples may fail when you try to expand it somewhere down the line to recognizing cars. However, if you start at a more meta-knowledge level before ever introducing apples or cars, you might be able to develop a machine vision neural net that processes apples and cars quite well. The same applies to brains. Remember, we are neural nets too. Instead of teaching people specific subject after specific subject, we might need to teach more meta-knowledge. We might also want to teach a higher percentage of abstract concepts first (ie math before science and liberal arts). This should lead to more balanced thinkers (meaning: you specialize in a career later in life, but you still know a good deal about the world, making you self-sufficient, flexible in career changes, and knowledgeable voter -- an always important trait for a healthy democracy) and more creative thinkers (meaning: you are studying one specific field, but you can see how that field is placed in the grander scheme, you can apply methods used in other fields as it is fruitful, and you can see more alternative branches to thought more quickly).

I must disagree. The issue largely eschews math itself and has more to do with administration, grades, testing, etc.

What if math is discrete?