From the Pleasure of Finding Things Out. I love the fact that he "outs" algorithms as stuff that can be used to help kids get the answer without knowing what they are doing.
"A series of steps by which you could get the answer if you didn't understand what you were trying to do." Best summary of our educational system that I've ever heard.
I think school is for educating idiots up to a basic level so they can do 19th century jobs. Honestly you can't get any stem job that I know on a public school education. And that's not counting the fact that they demand diplomas now for everything. I'm saying the curriculum itself is not for smart people.
In fourth grade I was assigned math and my father taught me how to do it in a different way, I was punished went with him to the principal's office where the principal tried to make my father out to be uneducated of how math was done, after giving my dad a series of problems to which he solved faster than the principal. The principal said he didn't care, If i didn't do it their way I was failing. Well I failed it their way and those Devil motherfukers passed me anyways. The u.s. education system in a nutshell unless your in a wealthy neighborhood where in private schools a different way is seen as creative and innovative. But in public school you're trained to be another one of the brainless sheep.
OMG yes sir! Most teachers dont even understand what they are trying to teach, they only repeat what they learned themselves, how they got their good grades etc. What they are teaching you, is what you need to know in order to pass the tests, so that it seems like the studens have learned something. On the paper, every country has great students.
What he's talking about when he says "by arithmetic" is what they mean by solving something by _inspection._ I can tell by just looking at it that x is 4 in the equation 16-x =12. I don't need to "subtract 16 from both sides" and then "divide both sides by negative 1" to solve for x. I can do it by inspection rather than by plodding, pedestrian, algebraic steps. The problem is that as equations get bigger and more complicated, you *have* to use these mindless algebraic steps. You can't see intuitively what _x_ is by "using arithmetic." You need the mechanics and the discipline of algebra. It's actually really cool to see it work (to solve word problems, for example) sometimes, especially when you'd have no fucking clue what x might be and then algebra works like magic. It's a powerful tool we use, not because we "don't understand what we're doing" (i.e. deducing an unknown's value), but because the task is far too big for our intuition and "inspection" alone.
you nailed it. even im more advanced education systems, I think it is very difficult to teach "inspection" to children who struggle with math. who have a very limited "intuitive" and quick approach to understanding it. maybe there are better methods of teaching still to be found.
The school should say that doing operations on x changes the value of x; we should therefore work backwards to deduce x from 17. Let the student figure out that subtracting 9 reverses the +9, and that dividing by 2 reverses the multiplier. Let the student figure out the algorithm and give them a hint if they are lost. That way, one day the student will see a ^3, and he will realize that he must reverse this ^3. Should he truly understand what it means to reverse, he will even realize on his own that ^2 has both a positive and a negative result once reversed. Teaching the student to fish rather than giving them a fish, essentially. There is too much of giving kids a series of if-then statements to memorize. If you see [pattern] apply [algorithm]. Yes, that is how things are solved, but no, that is not the way to teach it.
@@georgebrantley776 "The school should say that doing operations on x changes the value of x" They definitely shouldn't say this since the value of x *doesn't* change as you solve the problem. 2x + 7 = 15 2x = 8 x = 4 The value of x was 4 in every step. But otherwise I agree.
Child prodigies aren't smarter than everyone else they just start faster. By puberty everyone else is caught up and they go on to lead quite ordinary lives.
Indeed. It's a shame that I never encountered this concept until I started studying calculus. And then I fought it. I resented the idea that the teacher judged me not by whether I got the right answer but whether I took the right approach. This blew my mind. It should have been taught from a much younger age.
Absolutely. But only a fool (like many in the "top" comments) takes what Feynman says here seriously. All he's saying is that he found the principles of algebra easy -- but not everyone is a genius. Even if you are a genius, try mastering calculus without mastering algebra first. Einstein's most famous work employs a helluva lot of algebra and linear algebra. Oh, and Feynman's too!
True, but he is saying that too. I like Feynman, but he is not exactly what one would call humble. I read the book about his life once, and the „i am smarter than the rest“ mentality is everywhere.
A right answer for the wrong reason is better than a wrong answer for the right reason. A right answer is still a right answer, no matter how you got to it. Success is determined by the end result, not how you got there.
What really helped me understand math in high school was starting to use a computer. I wrote simple graphing programs... which helped to visualize, a key thing in math. With a few lines of calculations I used Newton's method to find roots of a function -- it literally just follows the slope to where the line reaches zero. This showed me there's a simple method to solve (almost) anything, for the first time I felt in control.
Exactly. I also felt elated by this same discovery years ago. The future of math and physics education is graphical computers. It's so powerful... I wonder why schools doesn't implement that...
@@joaocastro5871 In my Algebra I class in 1999-2000, they rolled out a cart of TI-92 graphing calculators for us to use every now and then. Our teacher also had blocks we would play with, there was also a transparent version of the blocks for the overhead projector. This was at a public school in a largely upper middle class town, where we neglected the roads because we put all the tax money into the schools. I don't know how much those calculators must have cost. I think they were only for the kids who started algebra early, not sure about that though. I remember really feeling like the concepts clicked when I was playing with the blocks and the graphs.
He was hands-down one of the best professors ever. He had already written the books, so every class was telling stories and giving examples. We could read the books, but the live visualizations he gave were 10X more valuable. A great man.
Some of his university lectures are actually in youtube. He’s a joy to watch and listen to. All he had was chalk and a blackboard, and a unique talent for explaining complex ideas.
"A series of steps by which you could get the answer if you didn't understand what you were trying to do." So true! I once had a teacher, she honestly had no idea, not the slightest clue, what was going on. Absolutely zero ability to figure anything out. But she had memorized the steps! By this means something useful and interesting is transformed into a useless chore. The exact opposite of Feynman. What a mind.
Charles Miller Those students who didn’t really understand the problems but memorized the steps of solutions, they cannot do the job in real life, so they become teachers, and their way of studying ruined many students who try to really understand the problems.
they don't even need to memorize the steps. All the teachers manual that comes with the text book, not only provides the answer but also ever step needed to solve the problem.
not every person is capable of being a genius. Some people just enjoy teaching, and let the students grow surpassing themselves. Let the people make a living without passing judgement
@@Kapiwolf123 You don't need to be a "genius" to understand elementary algebra. You also don't need to be a "genius" to be a decent teacher - you _do_ need to be taught or learn it the right way, which is too rare, and perhaps you also need a faculty for conveying things effectively to your students. That it took a "genius" to come to this realization is more a reflection on just how badly broken the system really is. I'd also say if you aren't good at teaching, that means it is _bad_ to try it, because bad teachers _hurt_ their students.
You try to explain the ideas and logic behind something, the student want the easy to follow formula. And when school demands that the students get good results on a certain test after each semester, a test which relies on student knowing some easy rules and can plug in numbers to get correct answer. What do you do as the teacher? Do you take every fight every day with your students and principal on this matter, or do you sometime to what the principal and the student want?
well said...it's not so hard to figure out the rules - most card games have more rules than Algebra...Feynman was a great thinker and many felt he was a great professor, but here he is just bragging and "geezin" about the past
The principle is not well working systems. The principle is understand methods. If a system works, there is little to no need to understand it. Just make sure it functions. If the method is understood, it can be taught to others and possibly used elsewhere.
@@7788Sambaboy With Feynman it was always a little bit of lack of self knowledge. That's not a bad thing, but he forgot that it looks like magic to those of us with out the back ground story.
What the school called doing it by algebra would be better described as doing it with a specific algoritm they deem important (true or not). A lot of math until higher levels is learning algoritms, to the point where students think that that is what math is, calculating stuff. Which is too bad.
xyhmo I don’t see the point of the ways that they teach algebra, they taught them set of rules like a machine without the fundamental grasp of numbers. Which in futile in effort and it’s comparable to programming a computer to recite what it was taught and not gain anything useful knowledge out of it.
To be fair, applied math pretty much stays that way all the way through, as well as all of the sciences. Only in pure math can the beauty and artistry of the medium be fully appreciated.
That was always my problem in high school. The teachers would always expect you to do things the way they had shown you, and would discourage critical thinking and figuring out "easier" ways of doing things. College level chemistry involves some pretty intimidating math with all their units and conversions. Soooo much easier to just leave the units out of the math and just stick on the proper unit once you've got your number.
I can tell from first hand experience that it can be hell for a teacher to have a student who is capable of understanding the course material at a higher level than the teacher. However, I haven’t met a single teacher who resents such a situation. Most of them cherish it. Imagine at one instance the student knows very little about the subject. The teacher explains it, and within an hour the student is at a higher level than the teacher in his understanding of the subject. Most teachers enjoy it because they encounter it only a few times during their teaching career.
I suffered badly in school because I only solved things in my head and produced an answer without all the unnecessary framework they were going on about. Every other year I’d go from a teacher that seemed to think I was the golden ticket because they were passionate about mathematics to a teacher just going through the motions who just wanted me to do what the curriculum required. I almost abandoned math because of this and thankfully when I got to calculus I ended up with a teacher who was the most passionate of my grade school teachers and he sent me in a trajectory towards higher math. It’s luck that this happened and I could have just as easily gone the opposite direction with another person trying to do their job and not be bothered by things that were not in the expected way of producing the answer. Dealing with this battle of attrition and breaking through to calculus is important because it takes a certain amount of passion to even teach it so anyone out there with similar issues just keep going and you’ll get to where answers are key and not how you produce it. As long as you can provide a tangible proof your answer is valid. I understand the frustration of the teachers that I dreaded now though, it’s just a part of the mathematics adventure.
@@timothywilson6138 I am surprised. If you were smart enough to solve for the answer in your head, I would assume that you would have figured out what is needed from you to get a good grade. I was always able to figure out “the system” to get what I wanted from the society (family, school, workplace, etc.). I always looked at the relations between myself and the rest of the world as an optimization problem to be solved. I was quite successful optimizing my life - socially, emotionally, financially.
Lex Fridman's interview with Wolfram helped me understand why Feynman is so good at teaching. Basically, Feynman didn't think doing calculations was that big of a deal (because they were easy for him). What was a big deal to Feynman was getting to a point where things were intuitive Maybe this is why he's such a good teacher. He never sacrificed sound reasoning and he always strove to make cold subjects more approachable
Prof Feynman is advocating here not against how it’s done but understanding WHY it’s done. This is also somewhat ironic because quantum mechanics contains a lot of not knowing WHY something is done. When he invented Feynman diagrams, he triggered a tremendous negative reaction ☺️
Actually I used to think like that and got into trouble, and then a professor advised me to learn the jut the basic methods of solving problems and only then I was able to solve the harder problems. It was a bit too late for me. The idea is some problem may require a lot of time to find the solution. In many cases, perhaps mathematicians have spent a lot of time finding the solutions. During an exam you don't always have time for that. To be effective and and get to performance you need find your own way but also learn how others solved many particular problems. If you always struggle to find the solution by yourself you will probably fail. To advance quickly in any field, use the knowledge of people who did it before you. This applies to any field. For example in computer science if you don't learn the recursive algorithm you may not be able to discover it by yourself.
Ya but when you have to “use algebra” to solve for much more complicated equations you can’t do it by just thinking, so they teach you on very basic problems first
Also like, most mathematics in the upper levels is _about_ formulating (and indeed, formalizing) those algorithmic rules. Anyone can pull a number or function from a database in their head and conclude that yes, the solution to x''-x'=0 is Ce^t+C2, but the mathematician would look at that and try to find rules s.t. all types of that equation can be solved using an algorithm. When you're dealing with infinite dimensional vector spaces or projective planes that cannot be accurately represented geometrically whatsoever, your numerical intuition fails completely as a tool to solve equations, and the algorithms are what's left. It's very clear that Richard Feynman has the mind of a physicist and not a mathematician; he's unused to relishing in the absurd, but it's in that strangeness that the applicability of his intuitive reasoning falls away, and only mathematical rigor remains.
....you do realize Feynman invented absolutely absurd mathematics to figure this stuff out, right? like, he was a physicist because he knew the math naturally and visualized this absurd stuff. and he's not talking about never teaching formula, he's talking about never teaching what the formula is or why it is for the sake of being the easiest way to teach. you can absolutely understand how higher formulas work by taking his approach because this is one of the guys that helped form how those higher approaches even work :/
I honestly don't think some naive set theory and the basics of groups, fields, rings, etc. would be beyond your average 6th grader. If you can understand the idea of putting chalk on a chalkboard, you can understand morphisms, and so much of algebra is like that (which is why I've always swayed towards the analysis end personally). Maybe we _should_ be teaching it to gradeschoolers. I doubt I was alone when growing up in finding ideas most easily digested when they moved from general cases inwards rather than the opposite, and it sounds to me like the trouble a lot of people have (Feynman's cousin included) has more to do with ambiguous language and definitions than anything, which is pretty much _exactly_ why modern, axiomatic mathematics was established in the first place. Might as well give them the real stuff.
Yea, fair enough. I guess the end-goal is more to give them the motivation to want to pick up Lang or Hungerford on their own, and memorizing group axioms without motivation probably won't help with that. I guess my thought was on whether or not you could overcome the tedium of the early "notation" phases in their mathematical careers by giving them glimpses of some of the more advanced stuff early on. I remember hating math in high school because I didn't know it was building to, what are probably, _the_ most profound and abstract ideas that us humans can comprehend. If all you encounter is "solve for x", you might start to think that that's all math is, and the only people who will enjoy it are inevitably the ones that won't do well when it moves past that point. I mean, to be honest, I don't think you reach "real" math until you've started taking courses in analysis, algebra, and non-elementary geometry (like differential or projective); and most people don't do that until their sophomore/junior years in undergrad (assuming they're pure math majors). Everything before that's kinda just build up; and it's one _hell_ of a long buildup.
I tried to help my daughter with her algebra in Jr High. She failed that homework assignment, because she did not do it in the same way the teacher showed in class. Her answer was correct, and she got there in fewer steps, but it was not per requirement. I was an engineer, so my interest was the fastest way to the correct answer. She understood, but she did it the teacher's way for the rest of that course. Later in college, she ended up tutoring calculus students. She was a business major, but she took the "for science and engineering majors" calculus and loved it. I guess it all worked out in the end.
In the 1960s new math disaster. No definition of the terms word used . No presentation of the math principle. My father taught me the principle and worked through example problems. Then went over the teachers method way so I would get the grade.
Two aspects of Feynman's persona, the way it comes over: danger and delightfulness. Here was a man who possessed a totally unfettered mind. He was not afraid to denounce all academic prizes and honours as so much baloney and to cutlass his way through BS where-ever he encountered it. Yet he also possessed a gentle and inclusive attitude. These traits are present in full measure in the Lectures On Physics that he left to posterity: iconoclasm, wedded to respect for those who went before, inform each and every one of those lectures. One helluva guy. ❤
He's kinda right. When you solve x+4=11, you don't need to make this steps. The answer is obvious. But when you're dealing with some kind of this: 3x + 7(13-2) = 5x * 16 - 50, the answer is not obvious. And these steps become useful, besides that, You can make a very good analogy with scales when dealing with such an eqations. When you think of them as of scales, these steps become very logical and intuitious. Sorry for mistakes, English is not my native.
his point was that people aren't independently thinking about how to solve a problem. The students approach the question by doing steps, which is just bad for mathematical thinking. Because when it comes down to a question which the steps aren't so obvious, what then? We should be taught not what to think but HOW to think.
It took me so long to unlearn that. For the longest time, being unable to find the right algorithm meant that I was immediately stuck on a problem. Learning to reason out a problem, breaking it into its component parts, and approximations, were ironically skills I learned through physics and computer programming classes rather than math class.
It took me oh about 40 years to figure out how statistics works and what they were trying to do, so when people make absurd statistical comments and I explain the fallacy of their logic, they get lost and call me stupid.
@@donaldkasper8346 discrete and probability aren't opposites though. in classical mechanics, there are no probabilities required when you know enough about a system, but energy and momentum are continuous values that can take any value. in quantum, momenta and energies of particles in certain situations can only take discrete values, but the position of a particle in a different system might be entirely up to probability.
I agree so much. The worst part is when students get to trig and calc, and have to start using critical thinking with math, they are extremely unprepared. When teaching calc, I have found for most students, the hardest part of the class is the algebra
Many people in their teen years do not like to think since they are not involved in situations that require such thought processes. But private schools require certain courses to be taught. Why is that?
Then you shouldn’t agree with feynman’s opinion. The hardest part of calculus is certainly the algebra, but to do more difficult (high school) algebra that appears in calculus, you need to understand 2x + 7 = 15 first. As in the subtract 7 and divide by two method that he was making fun of.
@@silomurphy4451 I think feyman forgot for a moment that most of us aren't anywhere as smart as he is. I think algebra as it now is the best way to introduce it to students
You can also substitute "precalculus" for "algebra" in this conversation. I remember the bulky and klutzy way we were shown how to solve difference quotients, and then w/ true calculus we were shown derivatives, and I thought, "I just wasted a semester of my life." I'm an engineer & surveyor now, and we still see this in mathematics education. Let them struggle w/ a crude way too "solve" a problem, and withhold the true & elegant way. Somehow the student is supposed to come to an epiphany w/ the struggle.
Precalculus was one of the most useless courses I ever took in college. Honestly I found it more challenging than Calc I just because of the funky ways to do things. Never used any of the stuff in that course once you knew certain things with derivatives and integrals.
Ok, Richard- you solved 2x+7=15 by using simple arithmetic. Now solve 0.1873x - 235.198 = 4285.19483 the same way. Basic Algebra is just using the rules of arithmetic and properties of equality to solve problems, in some cases very challenging problems. It's a pretty fundamental part of mathematics and related fields. I'm not saying a kid who struggles in algebra can't be successful in mathematics, but let's not malign algebra, since it is used all the time in diverse fields.
I always considered algebra a tool for your toolbox. Simple problems are often easy to solve without using algebra. Algebra is a strategy for solving problems. You may not need it but it is not horrible to have it available.
this is a weird video. He never actually discussed how he came to his answer, and the example he gives is very easy to do in your head. Not my favorite Feynman moment honestly. And you can very well "know what you're doing" and still apply basic algebraic techniques to find the unknown quantity.
The video isn't about how to do algebra it is about the danger of thinking getting the right answer the wrong way is dangerous -- something I'm constantly bombarded with from people who are terrible at math and wanna save the children or whatnot
shit he just explained why I was so screwed by Algebra in high school but ended up being able to do trig visually from the bridge of a ship. man. education. just damn.
In the past, I learn to solve with algorithms until the 8-9 year. Today, school math in germany teaches mathmatical competenced instead of algorithms. At the beginni g, this is much harder for the kids but it is the better way.
Algebra is really a number puzzle. If I said a number times two plus one is nine, what's the number? You would probably know the answer is 4, I hope. If I wrote, solve 2x+1=9, you could use some rules to find x. It's the same thing. Simple algebraic problems can be solved "by inspection", i.e. by just seeing the answer. As they get more complicated you need to learn rules to get the answer, but it's just a more complicated number puzzle. Like five times a number minus two, equals three times the same number plus four. It's not so obvious that the answer is 3, you could get it by messing around with numbers. But it's quicker to solve 5x - 2 = 3x + 4 for x, using some rules.
I was always good at math. I got an 800 in my SAT Math II and a 780 in my SAT Math I. The problem is that I don't think i ever knew what I was doing. I was solving the problems that I had already solved before, in the way that I was taught. I don't think I ever got to have an internalization with math, where I could solve a new problem. And I kind of blame my teachers for it now. Physics was different. I could think on my own once I understood how something worked. I don't think I EVER understood math. I still don't think I understand it.
Graham Black You're right. I agree with you completely. That really is the problem. If teachers weren't put under this weird pressure, maybe they could be more creative in their approaches. Everything is fucked.
I am a teacher, in the UK, the pressure to get results is ridiculous. From everyone, all most kids want is an exam result. It makes the qualifications meaningless though, because we essentially over teach, which means kids often over achieve. They get qualifications that don't reflect their ability and effort, and for which they didn't learn the discipline to get. Then when they move on, they don't have a teacher there for them spoon feeding them.
Whole heartedly agreed, thats the approach I used to adopt when teaching my little brother mathematics. Equipping him with the basics then leaving him to think on his own about the problems at hand.
This resonated so much with me, in a different way. When i was studying algebra, i was sort of rebellious and didn't learn all the "algebra" steps my teacher taught, but used my own intuition. I felt confident enough in my reasoning skills to not pay attention to my math teachers. But when we got to graphing, my intuition broke down as things were too complicated, and my ego blocked any teacher lecture to help me. I eventually learned to balance reasoning with following rules overtime.
I really relate to your experience,I think it's a great idea to not try too hard to make everything intuitive and sometimes to just follow the rules,but I think everyone should try to every so often take a little piece of the subject and try to make it intuitive and sort of immediately obvious,I found many benefits in the long run to try to fully explain to myself stuff,but I couldn't manage it in the short term because I had to keep up with the pace,so for now I treat everything I don't understand as a black box and try to see how those black boxes relate to each other,this doesn't always work but I think you get the idea. And if between those black boxes you get something you fully understand it sort of spreads your knowledge to the other stuff.
Math teachers have never like me, even though I got an A+ in Geometry and an A in Algebra. Two years later I led a junior college debate team. Sixty years later I took a Conservation course taught by a math teacher. They STILL don't like me. I aLSO use my debate training and your intuition is actually a large BS filter. The math teacher doesn't seem to understand THAT.
with algebra you can find gotchas like dividing by zero - or finding multiple roots - that you might not find by just getting "the answer" in these simple examples. algebra is one of the simplest forms of coding & formal logic - which can give counterintuitive results
Algebra is faitly intuitive I think as long as you understand the principles. Once you understand the commutative property etc. and the reasons these rules exist, everything else just falls into place.
Most kids dont even care to wtf x is. Teach someone how to think is nice and all but is way harder then teaching some rules through with some may develop critical thinking. Imagine being a teacher and having 20 to 40 kids to teach how to think having only a few hours per week, its just impossible for the average school teacher. And thats not even considering that in most cases people prefer to learn by just following simple rules.
"A series of steps by which you could get the answer if you did not understand what you were trying to do (find x)". NO! It should be apparent that a variable in an algebraic equation is what the thing is about (finding x). The algorithms or steps to solve for x are just that. Feynman, obviously, was capable of inventing his own algorithms. Feynman here seems to think that many people encounter an algebraic equation and immediately go through some menu of algorithms, rather than consider the quality or nature or purpose of the variable they are solving for.
His opinion seems to be an either/or option about knowing how to use arithmetic to solve for “x” or follow the prescribed operational steps (i.e. properties or algebra) to find the answer blindly. I think a person can combine both abilities; I happen to think I’m one of them who does. 😊
Subtract 7 from both sides. You get 2x = 8. Divide both sides by 2. You get x = 4. Easy. That's how I was taught and that's how I teach my students. Of course, I am open to the possibility of other methods. I'm aware enough to know that there may be more than one way to get the correct answer.
@@flurbanmoran7797 I dare you to say that to the face of a mathematician. It's like saying balls are useless if it wasn't for basketball lol. True, a lot of math is inspired by physics but it's stupid to assume they're the same.
Oh, b*llsh!t. Of COURSE students know what they're trying to do. How else can they figure out which rule to apply at each step? There are different ways of solving problems, and some of them don't scale very well. If all you learn is how to solve simple problems by guessing, that won't help you solve more complicated problems in the future. And what is "solving by arithmetic", anyway? Feynman solved the problem by using the rules of algebra, just like everybody else does. At least that's how he would solve hard problems, and the problem in the video is hard enough for a teenager. The goal isn't to find the answer to a random equation, it's to learn how to use algebra so the kid can solve harder problems later in life. And how did Feynman learn mathematics in the first place? Did he reinvent algebra and calculus and complex analysis all by himself? No, someone showed him how to do it. There's nothing wrong with teaching young people how to do math. How else are they supposed to learn? Rediscover thousands of years' worth of discoveries all by themselves? Not even Richard Feynman could do that.
The great distinction between solving a problem to find the solution and searching for the solution to a problem. We need another Feynman in our lives.
He is right you know. Math has two pathways the algebraic part and the arithmetic/stat part. We are forced to do more of the algebraic path than others
One of the reasons why maths can be difficult for kids to learn. I remember repeating tables over and over and after doing so learnt absolutely nothing because it wasn’t memorable. I eventually learnt maths and the theoretical operations became something enjoyable. That’s the trick.
The reason for the"steps" is to lay the groundwork for a set of tools to employ against more complicated problems. This is often made quite clear to students paying attention.
Agreed. The wrote-learnt steps make it much easier when you get to complex problems, in just the same way that wrote-memorised times-tables allow you to quickly do anything in maths.
The way feynman solved the problem is actually from rote memory method of arithmatic. It is his method which is made up of arbitrary rules. The steps in algebra actually have a logic to them, and will help solve problems we have never faced before. If both sides are equal, then taking away equal amounts from both sides should lead us to the point where we know the value of x. I think feynman was being flippant here.
Try solving a standard variables on both sides problem without the properties of equality and only arithmetic. C'mon man. Such a cheap shot in the man of all those who struggle in math. The opposite of what he said is true: the biggest hindrance for students to learn how to solve an equation is that they only want to do it with arithmetic when a teacher is teaching the foundational properties of equalities, and then once you can no longer depend on arithmetic to solve an equation, students are lost.
How can we expect minimally paid, questionably educated teachers to teach math in a comprehensible manner? There will be very little genuine education in this country until our social infrastructure values people over profit.
I had a teacher who stood in front of the class and told us he was going to teach us the most useless things he could imagine ... and laughed ... 50 years later, anaerobic glycolysis is still the singularly most useless thing I had to memorize ...
Actually, trial and error is a perfectly valid method. It may not be the fastest way to determine the solution but, as it turns out, that is exactly how computers solve complex equations. And in the real world, most equations are too complex to solve by algebraic methods.
which is feynman's point that there are always many different ways of doing the same thing and getting the same results, making the whole education system only about 1 of many valid paths means many very fine mathematicians are considered dumb in math, while really it was the math teaching that was too dumb. it also needs to be noted that feynman wrote about students that said they just knew stuff and encouraged doing trivial stuff you already know, he isnt talking about not showing work, he's talking about a failure to teach the most basic understanding of what math is.
We should let students learn to do their own spelling and writing. Maybe every one have their own vocabulary words too. Personally I would my alphabet to have 33 letters.
What a great point. Public schools, at least when I went through k-12 twenty years ago, made mathematics seem like nothing more than another memorization game. The point behind all of it wasn't ever discussed, I knew that people didn't use trigonometry in their lives and there was no way I could afford college. So I yawned my way well beyond the requirements and didn't even take a math class my senior year. As I have experienced a growing curiosity about the natural world, as people do, I rediscovered math a few years back. Now I do Calculas for fun. If I'd understood the significance and power of mathematics in my youth I imagine my life would be much more satisfying. Thankfully I'm not in school so I can learn, lol
Writing out proofs was pretty useful, time consuming though. You had to show the reasoning behind the steps that you chose to solve the problem. After a hiatus, I was glad students are back to using this process, until it gets to be automatic.
Haha I was also thinking about that :P But I'm embarrassed to admit that no red flags and loud sirens went off in my mind saying "no way a 3 year old will learn algebra"
I think he misses the point. The point of doing algebra problems in school is to practice the particular technique or method being taught and understand its application. And for that purpose it is actually valuable to practice with problems for which there are easier solutions, so that you can, with a high level of confidence, check your work. (Sure, you could just print the answers at the back of the book, but misprints happen.) To question practicing algebra the way its done is like questioning a baseball player bothering with batting practice when they could just use a catapult to launch the ball out of the park.
The fact that Richard Feynman undoubtedly was a genius doesn't make him *always* right. The rules in algebra are not "false". They are correct. Moreover, they are absolutely necessary for all those who are not geniuses. And in fact, they are perfectly understandable if you, for instance, interpret x (the unknown) as a container of something like beans or lego pieces, and the numbers as coins you need to pay. I've read his "adventures of a curious character", and there are a number of similar situations where he simply doesn't get the hang of reality. For instance, he mentions a course in biology he was attending too, where he was supposed to make a talk about a cat (I believe it was), and he started out by naming the relevant bones of the cat skeleton. Every biologist in the class said "yes yes we know", and Feynman goes on to wonder why the hell they spend so much time by learning the bones of the skeleton by heart when any fool can look it up in a book! Now I do wonder how any biologist is going to do his daily work if he needs to get out a book every time he sees an animal's carcass...
I homeschooled our son. Once I talked to my husband about our boy's approach towards math. I complained that he always did the question in an unusual way. My husband told me a good intuition worth preserve. I never understood how my son understood math, because i never taught him anything directly about math. I gave him questions and we then compare our answers and approaches. His approach was barely ever the same with me. By the age of 8, he had done all k to 12 math, and started calculus. He is now studying math and CS in college. Math was not his passion, but he just has an very intuitive ability.
The word "Algebra" comes from a famous Mathematics textbook written by the Ancient Mathematician from Persia known as Al-khwarizmi. He is the Father of Algebra.
With all due respect, the whole idea behind 'algebra' is to learn the 'algebraic rules' ... Feynman is talking about problem solving at a higher order ... His pragmatic approach is typical of an *engineering* mindset ...
Public schools need separate educations systems: one for casual students and another for passionate ones. Or maybe dynamic schedule where the student can choose courses which he wants to attend.
As Feynman says, "... knowing the whole idea was to find out what X was, and it didn't make any difference how you did it." That doesn't mean he didn't use "algebra", it means he focussed on finding the answer. If he couldn't do it one way, he did it another. This was a trait that characterized his career, and led him to a Nobel Prize. I have no argument with that.
Well, the question is: could your cousin solve a much more complicated equation without algebra? Obviously he could not. That is what algebra was created to help you do.
Yea the problem is not teaching the algs. The problem is never giving the students context for why they would use this algorithm. Or why it works. And the right way to do that is to give them harder problems that you can't do in your head or via guess and check.
@@peterisawesomeplease One of the problems, though, is that one cannot reasonably be expected to work with harder examples and check their understanding with them until they have worked with easier examples. Yes, we should explain why one wants to do something, but one of the best ways to get a feel for how something works is to check how it works on easy examples.
@@MuffinsAPlenty Yea it's a bit of a dilemma sometimes. If you teach simple problems then students often miss the point of algebra since they can already solve basic problems in their head. But if you teach hard problems you will just intimidate them and leave them more confused.
@@peterisawesomeplease I find that students do sometimes, not only in algebra, confuse the algorithm and the underlying meaning of the result. However, that does not mean that these algorithms are not important, they are the only way in my opinion to tame the complexity of math as you dive deeper an deeper.
Just because he was a genius . . . that does NOT mean he was right about everything. Or that every single one of his opinions was infallible. No way. Seems to me he was somewhat of an ELITIST. Ya gotta be extraordinarily special . . . or you're nothing. Well that may or or may not be true.
I had considerable trouble with arithmetic. I have a slapdash approach to life that means to me that 99 is pretty much the same as 100, I constantly made silly little mistakes and since primary school math consisted of doing hundreds of long multiplication and long division sums (often in pounds, shillings and pence - with farthings) the chances of me actually getting one of these sums right was pretty negligible. Or in my system the same as zero. Getting close to the answer wasn't acceptable. You still got a big red 'X'. Imagine my joy when we turned to algebra! Those damn numbers went out the window. I didn't need to worry what five times seven was, or how many groats make a furlong. x+x = 2x and a * b = a.b what! how is that difficult? There were kids in my class scratching their heads and demanding to know what x was. The answer that it could be anything at all just seemed to drive them into a frenzy of disbelief. I never looked back.
Yes! I relate to this so hard. I have memories of literally slamming my head into the wall at not understand multiplication tables, but the second math went more abstract it became so simple. I'm happy other people had similar experiences.
Legit we were just told to memorize the quadratic formula and never told anything about why it works or anything. Just burn that thing into your head and use it whenever you need to factor a quadratic equation, even if you can easily do it by other means.
You will learn how to derive it in higher level maths, also high-school mainly rewards memorization but at university level you need to build a deeper understanding
@@Slyracoon666 Yeah I know how to derive it, it takes a bit of algebra to isolate x on one side but it's possible, I only know it because I read a lot of math in my spare time, and I use it to solve practical problems, it's just I never learnt it at school, which frustrates me because school is supposed to teach you these things but it doesn't. I guess the reason why lies in standardized testing incentivizing schools to drill random information without context into students' heads rather than actually teaching the subject, because teaching takes time and memorization gives better test results despite not actually teaching anything. It's exactly what Feynman is describing here, they never teach you _how_ to think, they just teach you _what_ to think, they just hand you a set of instructions of what to do without explaining to you what's actually happening. I think this is part of why most people hate math. Because they associate math with memorizing crazy formulas they'll rarely get use for. They're never exposed to math in a way that actually sparks curiousity, because they've never learnt to use math as a problem-solving tool. In many ways, school does the opposite of learning - it scares people _from_ learning, it's anti-learning.
Some students have the same problem with calculus. They know how to do the power rule to do derivatives and integrals ,but when it comes to understanding how to apply calculus later on, they haven’t a clue.
Kind of silly to knock it. It's just a tool. Use the tool and get the job done. I don't have a clue how a phone camera works, hardware or software, but I can follow the steps: point, compose, click a button. It's a good thing that it happens without thinking.
I sometimes teach math/s and I always keep this clip in mind. I'll show students the approved way, but if they have their way, I let them play. If you listen you learn. Students' resourcefulness and creativity is limitless.
I think hes wrong here. When algebra gets to more complicated stages and the answer isnt obvious at a first glance then the step by step solution is required. Learning it this way might be slower for simple questions but is the best way to solve difficult algebra.
I don't understand. There's literally only 1 way to solve 2x + 7 = 15 that's not obtusely convoluted, and it's to subtract 7 and divide by 2. 100% that's what the tutor told him to do, because there's no other conceivable way to go about it. What does Feynman mean his cousin couldn't accept that? What was his cousin trying to do instead? The rules behind subtracting 7 and dividing by 2 are a good thing; it's a systematic approach to solving more complex problems that you can't simply eyeball.
I think he’s right about doing algebra without thinking. But he’s wrong about the real difference between algebraic and arithmetical ways of finding the solution.
It's one thing to know what x is with some mental arithmetic but algebra is all about the steps of proving what x is, that's what math is all about, proving your answers.
You'd have to be pretty stupid to think you can solve all algebraic problems in your head, or that learning how to solve different types of equations is somehow not helpful lol
This was me with calculus. I could never understand what the answer meant in relation to the question and eventually gave up on the subject. If only I'd been confident enough to ask for help...
As with all maths, as Feynman says here, the trick is just to figure out what “x” is. That’s pretty much sums up maths. Once you grasp that, the rest should start falling into place. Of course, the child will answer, “But I don’t care what “x” is. It has no interest for me.” That’s when you know you have a fundamental problem that maths can’t solve.
For those who didn't realize, he says 3 years older. Not 3 years old.
After all these yrs. 😂
THANK YOU for that explanation!!! I can now reposition my jaw.
@@cufflink44 lol
Thanks for that.
I was like, "Huh?"
Olduh
I also had difficulty with algebra when I was 3 years old
poor guy
Yeah, and topology when you were 14.
3 years older
Jadin Andrews - Hey, me too! But by six, I had finally figured it out. Late bloomer.
you missed one r
"A series of steps by which you could get the answer if you didn't understand what you were trying to do."
Best summary of our educational system that I've ever heard.
I think school is for educating idiots up to a basic level so they can do 19th century jobs. Honestly you can't get any stem job that I know on a public school education. And that's not counting the fact that they demand diplomas now for everything. I'm saying the curriculum itself is not for smart people.
itsnotatoober you mean school curriculum or college/ diploma’s curriculum!
In fourth grade I was assigned math and my father taught me how to do it in a different way, I was punished went with him to the principal's office where the principal tried to make my father out to be uneducated of how math was done, after giving my dad a series of problems to which he solved faster than the principal. The principal said he didn't care, If i didn't do it their way I was failing. Well I failed it their way and those Devil motherfukers passed me anyways. The u.s. education system in a nutshell unless your in a wealthy neighborhood where in private schools a different way is seen as creative and innovative. But in public school you're trained to be another one of the brainless sheep.
Actually if you try to learn from the beginning to the end from theorem to theorem it makes sense. You need to know where the rules come from.
OMG yes sir! Most teachers dont even understand what they are trying to teach, they only repeat what they learned themselves, how they got their good grades etc. What they are teaching you, is what you need to know in order to pass the tests, so that it seems like the studens have learned something. On the paper, every country has great students.
What he's talking about when he says "by arithmetic" is what they mean by solving something by _inspection._ I can tell by just looking at it that x is 4 in the equation 16-x =12. I don't need to "subtract 16 from both sides" and then "divide both sides by negative 1" to solve for x. I can do it by inspection rather than by plodding, pedestrian, algebraic steps. The problem is that as equations get bigger and more complicated, you *have* to use these mindless algebraic steps. You can't see intuitively what _x_ is by "using arithmetic." You need the mechanics and the discipline of algebra. It's actually really cool to see it work (to solve word problems, for example) sometimes, especially when you'd have no fucking clue what x might be and then algebra works like magic. It's a powerful tool we use, not because we "don't understand what we're doing" (i.e. deducing an unknown's value), but because the task is far too big for our intuition and "inspection" alone.
Great comment. This deserves more likes
you nailed it.
even im more advanced education systems, I think it is very difficult to teach "inspection" to children who struggle with math. who have a very limited "intuitive" and quick approach to understanding it.
maybe there are better methods of teaching still to be found.
The school should say that doing operations on x changes the value of x; we should therefore work backwards to deduce x from 17. Let the student figure out that subtracting 9 reverses the +9, and that dividing by 2 reverses the multiplier. Let the student figure out the algorithm and give them a hint if they are lost.
That way, one day the student will see a ^3, and he will realize that he must reverse this ^3. Should he truly understand what it means to reverse, he will even realize on his own that ^2 has both a positive and a negative result once reversed. Teaching the student to fish rather than giving them a fish, essentially.
There is too much of giving kids a series of if-then statements to memorize. If you see [pattern] apply [algorithm]. Yes, that is how things are solved, but no, that is not the way to teach it.
@@georgebrantley776 I wholly agree with you, sir.
@@georgebrantley776 "The school should say that doing operations on x changes the value of x"
They definitely shouldn't say this since the value of x *doesn't* change as you solve the problem.
2x + 7 = 15
2x = 8
x = 4
The value of x was 4 in every step.
But otherwise I agree.
Forget Feynman his cousin was the real genius if he was in high school at 3 years old.
David Vickers he said "older" not "old"
Child prodigies aren't smarter than everyone else they just start faster. By puberty everyone else is caught up and they go on to lead quite ordinary lives.
Because they tend to be lazy, since they are driven by comparison to others, whereas successful smart people are driven by competing with themselves.
I heard the same thing, thought I was tripping
No, because of the way their brains develop.
"The idea is to know what you're doing, rather than to get the right answer." - Tom Lehrer
Indeed. It's a shame that I never encountered this concept until I started studying calculus. And then I fought it. I resented the idea that the teacher judged me not by whether I got the right answer but whether I took the right approach. This blew my mind. It should have been taught from a much younger age.
Absolutely. But only a fool (like many in the "top" comments) takes what Feynman says here seriously. All he's saying is that he found the principles of algebra easy -- but not everyone is a genius. Even if you are a genius, try mastering calculus without mastering algebra first. Einstein's most famous work employs a helluva lot of algebra and linear algebra. Oh, and Feynman's too!
@@carlodave9 It’s easy to criticize another’s inability to do what come naturally to me.
True, but he is saying that too. I like Feynman, but he is not exactly what one would call humble. I read the book about his life once, and the „i am smarter than the rest“ mentality is everywhere.
A right answer for the wrong reason is better than a wrong answer for the right reason. A right answer is still a right answer, no matter how you got to it. Success is determined by the end result, not how you got there.
Such a cool video. Thank you for posting this:)
Hey from you...
What cool about this he shit on algebra my love❤
See the whole documentary, 'the pleasure of finding things out'
What really helped me understand math in high school was starting to use a computer. I wrote simple graphing programs... which helped to visualize, a key thing in math. With a few lines of calculations I used Newton's method to find roots of a function -- it literally just follows the slope to where the line reaches zero. This showed me there's a simple method to solve (almost) anything, for the first time I felt in control.
3blue1brown made a great video about Newton's algorithm
If I understood the imagining component of math, it would've changed my life. Now I'm like 40 and just getting to it.
Hey can you share more about the programs you wrote because I also really like programming and looking to get more into maths and graphics
Exactly. I also felt elated by this same discovery years ago. The future of math and physics education is graphical computers. It's so powerful... I wonder why schools doesn't implement that...
@@joaocastro5871 In my Algebra I class in 1999-2000, they rolled out a cart of TI-92 graphing calculators for us to use every now and then. Our teacher also had blocks we would play with, there was also a transparent version of the blocks for the overhead projector.
This was at a public school in a largely upper middle class town, where we neglected the roads because we put all the tax money into the schools. I don't know how much those calculators must have cost. I think they were only for the kids who started algebra early, not sure about that though.
I remember really feeling like the concepts clicked when I was playing with the blocks and the graphs.
He was hands-down one of the best professors ever. He had already written the books, so every class was telling stories and giving examples. We could read the books, but the live visualizations he gave were 10X more valuable. A great man.
What a blessing to take classes with him! I'd love to hear stories, if you feel like sharing.
Some of his university lectures are actually in youtube. He’s a joy to watch and listen to. All he had was chalk and a blackboard, and a unique talent for explaining complex ideas.
@@MasterKoala777 those are the best teachers
Huh? You had him as a professor?
@@jonathanyoung2588 ruclips.net/video/EYPapE-3FRw/видео.html
Notice how he has provided grad students to ensure these 101 students aren't
left behind
0:01 "My cousin at that time who was 3 years old and was in high school..."
"A series of steps by which you could get the answer if you didn't understand what you were trying to do." So true! I once had a teacher, she honestly had no idea, not the slightest clue, what was going on. Absolutely zero ability to figure anything out. But she had memorized the steps! By this means something useful and interesting is transformed into a useless chore. The exact opposite of Feynman. What a mind.
Charles Miller Those students who didn’t really understand the problems but memorized the steps of solutions, they cannot do the job in real life, so they become teachers, and their way of studying ruined many students who try to really understand the problems.
they don't even need to memorize the steps. All the teachers manual that comes with the text book, not only provides the answer but also ever step needed to solve the problem.
not every person is capable of being a genius. Some people just enjoy teaching, and let the students grow surpassing themselves. Let the people make a living without passing judgement
@@Kapiwolf123 You don't need to be a "genius" to understand elementary algebra. You also don't need to be a "genius" to be a decent teacher - you _do_ need to be taught or learn it the right way, which is too rare, and perhaps you also need a faculty for conveying things effectively to your students. That it took a "genius" to come to this realization is more a reflection on just how badly broken the system really is. I'd also say if you aren't good at teaching, that means it is _bad_ to try it, because bad teachers _hurt_ their students.
You try to explain the ideas and logic behind something, the student want the easy to follow formula.
And when school demands that the students get good results on a certain test after each semester, a test which relies on student knowing some easy rules and can plug in numbers to get correct answer.
What do you do as the teacher?
Do you take every fight every day with your students and principal on this matter, or do you sometime to what the principal and the student want?
Richard Feynman is a boss.
He is the boss of bosses.
True.
Was
Infinity, compared to you, everyone "is a boss." 😂
He's the final boss but when you get to him, he is like, "you got here just in time!"
Nevertheless, the algebraic method is still intuitive-friendly if one makes the very small effort to understand why the rules work so well..
well said...it's not so hard to figure out the rules - most card games have more rules than Algebra...Feynman was a great thinker and many felt he was a great professor, but here he is just bragging and "geezin" about the past
The principle is not well working systems. The principle is understand methods. If a system works, there is little to no need to understand it. Just make sure it functions. If the method is understood, it can be taught to others and possibly used elsewhere.
@@7788Sambaboy With Feynman it was always a little bit of lack of self knowledge. That's not a bad thing, but he forgot that it looks like magic to those of us with out the back ground story.
@@ValeriePallaoro well said.
@@7788Sambaboy but you weren't graded for understanding the rules, you were graded for memorizing them. That's the whole point.
Ohh Richard.. You left too soon.. :(
There are other teachers like sal khan and eddie woo that are keeping these ideas alive in the way they teach! Feynman is not gone!
At 69, on 15tg Feb ;P
@bumboni 'twas tough the day after valentine's day to live.
But it’s not hard to understand why you subtract 7 from both sides or divide by 2
What the school called doing it by algebra would be better described as doing it with a specific algoritm they deem important (true or not). A lot of math until higher levels is learning algoritms, to the point where students think that that is what math is, calculating stuff. Which is too bad.
xyhmo I don’t see the point of the ways that they teach algebra, they taught them set of rules like a machine without the fundamental grasp of numbers. Which in futile in effort and it’s comparable to programming a computer to recite what it was taught and not gain anything useful knowledge out of it.
To be fair, applied math pretty much stays that way all the way through, as well as all of the sciences. Only in pure math can the beauty and artistry of the medium be fully appreciated.
I'd argue that that is the role of all of math, to build relationships between previously unrelated objects.
So you're saying Math needs you to convince it's researchers to think about relationships on a large scale?
That was always my problem in high school. The teachers would always expect you to do things the way they had shown you, and would discourage critical thinking and figuring out "easier" ways of doing things. College level chemistry involves some pretty intimidating math with all their units and conversions. Soooo much easier to just leave the units out of the math and just stick on the proper unit once you've got your number.
I can tell from first hand experience that it can be hell for a teacher to have a student who is capable of understanding the course material at a higher level than the teacher. However, I haven’t met a single teacher who resents such a situation. Most of them cherish it. Imagine at one instance the student knows very little about the subject. The teacher explains it, and within an hour the student is at a higher level than the teacher in his understanding of the subject. Most teachers enjoy it because they encounter it only a few times during their teaching career.
Bob Brown @ There are also teachers who RESENT having pupils who are smarter than THEY are....!
@@irenehartlmayr8369 that's true ,but there are teachers that resent students for a lot of things.
I suffered badly in school because I only solved things in my head and produced an answer without all the unnecessary framework they were going on about. Every other year I’d go from a teacher that seemed to think I was the golden ticket because they were passionate about mathematics to a teacher just going through the motions who just wanted me to do what the curriculum required. I almost abandoned math because of this and thankfully when I got to calculus I ended up with a teacher who was the most passionate of my grade school teachers and he sent me in a trajectory towards higher math. It’s luck that this happened and I could have just as easily gone the opposite direction with another person trying to do their job and not be bothered by things that were not in the expected way of producing the answer. Dealing with this battle of attrition and breaking through to calculus is important because it takes a certain amount of passion to even teach it so anyone out there with similar issues just keep going and you’ll get to where answers are key and not how you produce it. As long as you can provide a tangible proof your answer is valid. I understand the frustration of the teachers that I dreaded now though, it’s just a part of the mathematics adventure.
@@timothywilson6138 I am surprised. If you were smart enough to solve for the answer in your head, I would assume that you would have figured out what is needed from you to get a good grade. I was always able to figure out “the system” to get what I wanted from the society (family, school, workplace, etc.). I always looked at the relations between myself and the rest of the world as an optimization problem to be solved. I was quite successful optimizing my life - socially, emotionally, financially.
Exactly true!!The best teacher we never had,not even today!!
miss him very much
Lex Fridman's interview with Wolfram helped me understand why Feynman is so good at teaching. Basically, Feynman didn't think doing calculations was that big of a deal (because they were easy for him). What was a big deal to Feynman was getting to a point where things were intuitive
Maybe this is why he's such a good teacher. He never sacrificed sound reasoning and he always strove to make cold subjects more approachable
Just like in market economics... Big words, little theory.
@@vif3182 Comparing theoretical physics to market economics is the most insane take I have ever seen.
I’m still trying to find my X. She left with my favorite hunting dog!
did you try subtracting 7?
Prof Feynman is advocating here not against how it’s done but understanding WHY it’s done. This is also somewhat ironic because quantum mechanics contains a lot of not knowing WHY something is done. When he invented Feynman diagrams, he triggered a tremendous negative reaction ☺️
It's not ironic. We always search for why in quantum mechanics, it's just that "why" isn't so clear
0:58 His mood changed so unbelievably fast
it stops just when he's about to say something interesting
It was all interesting. The sad thing is that it stops.
Actually I used to think like that and got into trouble, and then a professor advised me to learn the jut the basic methods of solving problems and only then I was able to solve the harder problems. It was a bit too late for me. The idea is some problem may require a lot of time to find the solution. In many cases, perhaps mathematicians have spent a lot of time finding the solutions. During an exam you don't always have time for that.
To be effective and and get to performance you need find your own way but also learn how others solved many particular problems. If you always struggle to find the solution by yourself you will probably fail.
To advance quickly in any field, use the knowledge of people who did it before you. This applies to any field.
For example in computer science if you don't learn the recursive algorithm you may not be able to discover it by yourself.
Ya but when you have to “use algebra” to solve for much more complicated equations you can’t do it by just thinking, so they teach you on very basic problems first
adam aguiar this
Also like, most mathematics in the upper levels is _about_ formulating (and indeed, formalizing) those algorithmic rules. Anyone can pull a number or function from a database in their head and conclude that yes, the solution to x''-x'=0 is Ce^t+C2, but the mathematician would look at that and try to find rules s.t. all types of that equation can be solved using an algorithm. When you're dealing with infinite dimensional vector spaces or projective planes that cannot be accurately represented geometrically whatsoever, your numerical intuition fails completely as a tool to solve equations, and the algorithms are what's left. It's very clear that Richard Feynman has the mind of a physicist and not a mathematician; he's unused to relishing in the absurd, but it's in that strangeness that the applicability of his intuitive reasoning falls away, and only mathematical rigor remains.
....you do realize Feynman invented absolutely absurd mathematics to figure this stuff out, right? like, he was a physicist because he knew the math naturally and visualized this absurd stuff.
and he's not talking about never teaching formula, he's talking about never teaching what the formula is or why it is for the sake of being the easiest way to teach. you can absolutely understand how higher formulas work by taking his approach because this is one of the guys that helped form how those higher approaches even work :/
I honestly don't think some naive set theory and the basics of groups, fields, rings, etc. would be beyond your average 6th grader. If you can understand the idea of putting chalk on a chalkboard, you can understand morphisms, and so much of algebra is like that (which is why I've always swayed towards the analysis end personally). Maybe we _should_ be teaching it to gradeschoolers. I doubt I was alone when growing up in finding ideas most easily digested when they moved from general cases inwards rather than the opposite, and it sounds to me like the trouble a lot of people have (Feynman's cousin included) has more to do with ambiguous language and definitions than anything, which is pretty much _exactly_ why modern, axiomatic mathematics was established in the first place. Might as well give them the real stuff.
Yea, fair enough. I guess the end-goal is more to give them the motivation to want to pick up Lang or Hungerford on their own, and memorizing group axioms without motivation probably won't help with that. I guess my thought was on whether or not you could overcome the tedium of the early "notation" phases in their mathematical careers by giving them glimpses of some of the more advanced stuff early on. I remember hating math in high school because I didn't know it was building to, what are probably, _the_ most profound and abstract ideas that us humans can comprehend. If all you encounter is "solve for x", you might start to think that that's all math is, and the only people who will enjoy it are inevitably the ones that won't do well when it moves past that point.
I mean, to be honest, I don't think you reach "real" math until you've started taking courses in analysis, algebra, and non-elementary geometry (like differential or projective); and most people don't do that until their sophomore/junior years in undergrad (assuming they're pure math majors). Everything before that's kinda just build up; and it's one _hell_ of a long buildup.
I tried to help my daughter with her algebra in Jr High. She failed that homework assignment, because she did not do it in the same way the teacher showed in class. Her answer was correct, and she got there in fewer steps, but it was not per requirement. I was an engineer, so my interest was the fastest way to the correct answer. She understood, but she did it the teacher's way for the rest of that course. Later in college, she ended up tutoring calculus students. She was a business major, but she took the "for science and engineering majors" calculus and loved it. I guess it all worked out in the end.
Cool story!
Middle school teacher: still failed her 😎
In the 1960s new math disaster. No definition of the terms word used . No presentation of the math principle.
My father taught me the principle and worked through example problems. Then went over the teachers method way so I would get the grade.
Feynman: "My cousin was 3 years old at that time and was in high school"
Me: "WHAAAAT?!" Rewinds x2
ranka shanka "..three years older". It's his New York accent. I had to listen closely, too. But, yeah, it sounds so similar.
Two aspects of Feynman's persona, the way it comes over: danger and delightfulness. Here was a man who possessed a totally unfettered mind. He was not afraid to denounce all academic prizes and honours as so much baloney and to cutlass his way through BS where-ever he encountered it. Yet he also possessed a gentle and inclusive attitude. These traits are present in full measure in the Lectures On Physics that he left to posterity: iconoclasm, wedded to respect for those who went before, inform each and every one of those lectures. One helluva guy. ❤
Feynman was able to see almost anything in a unique way. In his mind he could see a frantic particle trying to decide where to go next
He's kinda right. When you solve x+4=11, you don't need to make this steps. The answer is obvious. But when you're dealing with some kind of this: 3x + 7(13-2) = 5x * 16 - 50, the answer is not obvious. And these steps become useful, besides that, You can make a very good analogy with scales when dealing with such an eqations. When you think of them as of scales, these steps become very logical and intuitious.
Sorry for mistakes, English is not my native.
his point was that people aren't independently thinking about how to solve a problem. The students approach the question by doing steps, which is just bad for mathematical thinking. Because when it comes down to a question which the steps aren't so obvious, what then? We should be taught not what to think but HOW to think.
127/77
Bro you type better English than some native English speakers in America lmao.
@@josh1234567892 It's called "fishing for compliments". :)
@@martinmiltonmonson no it is,129/297
It took me so long to unlearn that. For the longest time, being unable to find the right algorithm meant that I was immediately stuck on a problem. Learning to reason out a problem, breaking it into its component parts, and approximations, were ironically skills I learned through physics and computer programming classes rather than math class.
It took me oh about 40 years to figure out how statistics works and what they were trying to do, so when people make absurd statistical comments and I explain the fallacy of their logic, they get lost and call me stupid.
@@donaldkasper8346 just wait until they start talking about probability
@@onebeets wait until 1 is 100% less than 2!!!!
@@onebeets There are two branches of physics. Quantum physics where everything is discrete. And the other, where everything is a probability.
@@donaldkasper8346 discrete and probability aren't opposites though. in classical mechanics, there are no probabilities required when you know enough about a system, but energy and momentum are continuous values that can take any value. in quantum, momenta and energies of particles in certain situations can only take discrete values, but the position of a particle in a different system might be entirely up to probability.
I agree so much. The worst part is when students get to trig and calc, and have to start using critical thinking with math, they are extremely unprepared. When teaching calc, I have found for most students, the hardest part of the class is the algebra
of 990 HS grads we had a grand total of 17 in intro. to Trig. and Calculus: 16 scholarship students and ME. the shakeout was obvious in the numbers.
Many people in their teen years do not like to think since they are not involved in situations that require such thought processes. But private schools require certain courses to be taught. Why is that?
Then you shouldn’t agree with feynman’s opinion. The hardest part of calculus is certainly the algebra, but to do more difficult (high school) algebra that appears in calculus, you need to understand 2x + 7 = 15 first. As in the subtract 7 and divide by two method that he was making fun of.
@@silomurphy4451 I think feyman forgot for a moment that most of us aren't anywhere as smart as he is.
I think algebra as it now is the best way to introduce it to students
I am a normal teen, when I see Richard Feynman , I click the video.
You can also substitute "precalculus" for "algebra" in this conversation. I remember the bulky and klutzy way we were shown how to solve difference quotients, and then w/ true calculus we were shown derivatives, and I thought, "I just wasted a semester of my life." I'm an engineer & surveyor now, and we still see this in mathematics education. Let them struggle w/ a crude way too "solve" a problem, and withhold the true & elegant way. Somehow the student is supposed to come to an epiphany w/ the struggle.
Precalculus was one of the most useless courses I ever took in college. Honestly I found it more challenging than Calc I just because of the funky ways to do things. Never used any of the stuff in that course once you knew certain things with derivatives and integrals.
Ok, Richard- you solved 2x+7=15 by using simple arithmetic. Now solve 0.1873x - 235.198 = 4285.19483 the same way. Basic Algebra is just using the rules of arithmetic and properties of equality to solve problems, in some cases very challenging problems. It's a pretty fundamental part of mathematics and related fields. I'm not saying a kid who struggles in algebra can't be successful in mathematics, but let's not malign algebra, since it is used all the time in diverse fields.
Dawg you're trying to talk to a deadman 💀
Series of steps. Al Kwarizimi. For whom the "algorithm" is named...
I always considered algebra a tool for your toolbox. Simple problems are often easy to solve without using algebra. Algebra is a strategy for solving problems. You may not need it but it is not horrible to have it available.
EXACTLY
this is a weird video. He never actually discussed how he came to his answer, and the example he gives is very easy to do in your head. Not my favorite Feynman moment honestly. And you can very well "know what you're doing" and still apply basic algebraic techniques to find the unknown quantity.
He made it easy so that layman can understand
The video isn't about how to do algebra it is about the danger of thinking getting the right answer the wrong way is dangerous -- something I'm constantly bombarded with from people who are terrible at math and wanna save the children or whatnot
If he was my teacher I'd be a mathematician.
shit he just explained why I was so screwed by Algebra in high school but ended up being able to do trig visually from the bridge of a ship. man. education. just damn.
Classic Feynman response! Direct, piercing and accurate!
In the past, I learn to solve with algorithms until the 8-9 year. Today, school math in germany teaches mathmatical competenced instead of algorithms. At the beginni g, this is much harder for the kids but it is the better way.
That's the way I was taught algebra, to find x. How else is one supposed to do it? Somebody teach me.
Algebra is really a number puzzle. If I said a number times two plus one is nine, what's the number? You would probably know the answer is 4, I hope. If I wrote, solve 2x+1=9, you could use some rules to find x. It's the same thing. Simple algebraic problems can be solved "by inspection", i.e. by just seeing the answer. As they get more complicated you need to learn rules to get the answer, but it's just a more complicated number puzzle. Like five times a number minus two, equals three times the same number plus four. It's not so obvious that the answer is 3, you could get it by messing around with numbers. But it's quicker to solve 5x - 2 = 3x + 4 for x, using some rules.
I was always good at math. I got an 800 in my SAT Math II and a 780 in my SAT Math I. The problem is that I don't think i ever knew what I was doing. I was solving the problems that I had already solved before, in the way that I was taught. I don't think I ever got to have an internalization with math, where I could solve a new problem. And I kind of blame my teachers for it now.
Physics was different. I could think on my own once I understood how something worked. I don't think I EVER understood math. I still don't think I understand it.
Teacher's have too much pressure on them to get exam results.
Graham Black You're right. I agree with you completely. That really is the problem. If teachers weren't put under this weird pressure, maybe they could be more creative in their approaches. Everything is fucked.
I am a teacher, in the UK, the pressure to get results is ridiculous. From everyone, all most kids want is an exam result. It makes the qualifications meaningless though, because we essentially over teach, which means kids often over achieve. They get qualifications that don't reflect their ability and effort, and for which they didn't learn the discipline to get. Then when they move on, they don't have a teacher there for them spoon feeding them.
To me, it's just Feyman's viewpoint, his perception. The rules for solving 2x + 7 = 15 are perfectly alright.
Whole heartedly agreed, thats the approach I used to adopt when teaching my little brother mathematics. Equipping him with the basics then leaving him to think on his own about the problems at hand.
This resonated so much with me, in a different way. When i was studying algebra, i was sort of rebellious and didn't learn all the "algebra" steps my teacher taught, but used my own intuition. I felt confident enough in my reasoning skills to not pay attention to my math teachers. But when we got to graphing, my intuition broke down as things were too complicated, and my ego blocked any teacher lecture to help me. I eventually learned to balance reasoning with following rules overtime.
I really relate to your experience,I think it's a great idea to not try too hard to make everything intuitive and sometimes to just follow the rules,but I think everyone should try to every so often take a little piece of the subject and try to make it intuitive and sort of immediately obvious,I found many benefits in the long run to try to fully explain to myself stuff,but I couldn't manage it in the short term because I had to keep up with the pace,so for now I treat everything I don't understand as a black box and try to see how those black boxes relate to each other,this doesn't always work but I think you get the idea. And if between those black boxes you get something you fully understand it sort of spreads your knowledge to the other stuff.
Math teachers have never like me, even though I got an A+ in Geometry and an A in Algebra. Two years later I led a junior college debate team. Sixty years later I took a Conservation course taught by a math teacher. They STILL don't like me. I aLSO use my debate training and your intuition is actually a large BS filter. The math teacher doesn't seem to understand THAT.
@@thedwightguy Get over yourself, dude.
@@GlazeonthewickeR really ikr! Fr
with algebra you can find gotchas like dividing by zero - or finding multiple roots - that you might not find by just getting "the answer" in these simple examples. algebra is one of the simplest forms of coding & formal logic - which can give counterintuitive results
Algebra is faitly intuitive I think as long as you understand the principles.
Once you understand the commutative property etc. and the reasons these rules exist, everything else just falls into place.
Most kids dont even care to wtf x is. Teach someone how to think is nice and all but is way harder then teaching some rules through with some may develop critical thinking. Imagine being a teacher and having 20 to 40 kids to teach how to think having only a few hours per week, its just impossible for the average school teacher. And thats not even considering that in most cases people prefer to learn by just following simple rules.
Very simplistic attitude from Feynman. The method must be taught so solve more difficult problems.
The principle must be learned so you can solve problems.
He totally missed what and why you learn algebra
"A series of steps by which you could get the answer if you did not understand what you were trying to do (find x)". NO! It should be apparent that a variable in an algebraic equation is what the thing is about (finding x). The algorithms or steps to solve for x are just that. Feynman, obviously, was capable of inventing his own algorithms. Feynman here seems to think that many people encounter an algebraic equation and immediately go through some menu of algorithms, rather than consider the quality or nature or purpose of the variable they are solving for.
His opinion seems to be an either/or option about knowing how to use arithmetic to solve for “x” or follow the prescribed operational steps (i.e. properties or algebra) to find the answer blindly. I think a person can combine both abilities; I happen to think I’m one of them who does. 😊
@Hello there, how are you doing this blessed day?
0:46 "I learned algebra, fortunately, by not going to school." - Richard Feynman
Subtract 7 from both sides. You get 2x = 8. Divide both sides by 2. You get x = 4. Easy. That's how I was taught and that's how I teach my students. Of course, I am open to the possibility of other methods. I'm aware enough to know that there may be more than one way to get the correct answer.
They teach you as set of rules in school they don't want to teach meaning.
Very true........well said..
You want meaning go to the bible. But seriously, math has no meaning without physics.
Flurban Moran you would be partly right. However, business and accounting gave rise to math long before physics did, civilization-wise.
Really? Proof?
@@flurbanmoran7797 I dare you to say that to the face of a mathematician. It's like saying balls are useless if it wasn't for basketball lol. True, a lot of math is inspired by physics but it's stupid to assume they're the same.
Oh, b*llsh!t. Of COURSE students know what they're trying to do. How else can they figure out which rule to apply at each step? There are different ways of solving problems, and some of them don't scale very well. If all you learn is how to solve simple problems by guessing, that won't help you solve more complicated problems in the future.
And what is "solving by arithmetic", anyway? Feynman solved the problem by using the rules of algebra, just like everybody else does. At least that's how he would solve hard problems, and the problem in the video is hard enough for a teenager. The goal isn't to find the answer to a random equation, it's to learn how to use algebra so the kid can solve harder problems later in life.
And how did Feynman learn mathematics in the first place? Did he reinvent algebra and calculus and complex analysis all by himself? No, someone showed him how to do it. There's nothing wrong with teaching young people how to do math. How else are they supposed to learn? Rediscover thousands of years' worth of discoveries all by themselves? Not even Richard Feynman could do that.
The great distinction between solving a problem to find the solution and searching for the solution to a problem. We need another Feynman in our lives.
Despite not being a biologist, his critical approach to problem solving could really have helped with things like the present batflu.
He is right you know. Math has two pathways the algebraic part and the arithmetic/stat part. We are forced to do more of the algebraic path than others
One of the reasons why maths can be difficult for kids to learn. I remember repeating tables over and over and after doing so learnt absolutely nothing because it wasn’t memorable. I eventually learnt maths and the theoretical operations became something enjoyable. That’s the trick.
The reason for the"steps" is to lay the groundwork for a set of tools to employ against more complicated problems. This is often made quite clear to students paying attention.
Agreed. The wrote-learnt steps make it much easier when you get to complex problems, in just the same way that wrote-memorised times-tables allow you to quickly do anything in maths.
The way feynman solved the problem is actually from rote memory method of arithmatic. It is his method which is made up of arbitrary rules.
The steps in algebra actually have a logic to them, and will help solve problems we have never faced before.
If both sides are equal, then taking away equal amounts from both sides should lead us to the point where we know the value of x. I think feynman was being flippant here.
Try solving a standard variables on both sides problem without the properties of equality and only arithmetic. C'mon man. Such a cheap shot in the man of all those who struggle in math. The opposite of what he said is true: the biggest hindrance for students to learn how to solve an equation is that they only want to do it with arithmetic when a teacher is teaching the foundational properties of equalities, and then once you can no longer depend on arithmetic to solve an equation, students are lost.
Derek, I am at complete loss as to why sensible comments like yours are getting so little attention here.
How can we expect minimally paid, questionably educated teachers to teach math in a comprehensible manner?
There will be very little genuine education in this country until our social infrastructure values people over profit.
I had a teacher who stood in front of the class and told us he was going to teach us the most useless things he could imagine ... and laughed ... 50 years later, anaerobic glycolysis is still the singularly most useless thing I had to memorize ...
Over quadratic formula ?
@@kathanshah8305 - I've use the quadratic formula ... very handy for solving acceleration problems ...
Richard Feynman is an awesome human being.
Yeah but how are you going to solve a quadratic equation using trial and error?
Actually, trial and error is a perfectly valid method. It may not be the fastest way to determine the solution but, as it turns out, that is exactly how computers solve complex equations. And in the real world, most equations are too complex to solve by algebraic methods.
David Petro
It's valid, just not efficient in certain situations.
which is feynman's point that there are always many different ways of doing the same thing and getting the same results, making the whole education system only about 1 of many valid paths means many very fine mathematicians are considered dumb in math, while really it was the math teaching that was too dumb.
it also needs to be noted that feynman wrote about students that said they just knew stuff and encouraged doing trivial stuff you already know, he isnt talking about not showing work, he's talking about a failure to teach the most basic understanding of what math is.
bananian that’s how equations of degree n>4 are solved.
Any examples on how to use trial and erroe?
We should let students learn to do their own spelling and writing. Maybe every one have their own vocabulary words too. Personally I would my alphabet to have 33 letters.
In Latvia we do have 33 letter alphabet, and it does not include any of q,w,x,y.
I want to see Joaquin Phoenix playing Richard Feynman
What a great point. Public schools, at least when I went through k-12 twenty years ago, made mathematics seem like nothing more than another memorization game. The point behind all of it wasn't ever discussed, I knew that people didn't use trigonometry in their lives and there was no way I could afford college. So I yawned my way well beyond the requirements and didn't even take a math class my senior year. As I have experienced a growing curiosity about the natural world, as people do, I rediscovered math a few years back. Now I do Calculas for fun. If I'd understood the significance and power of mathematics in my youth I imagine my life would be much more satisfying. Thankfully I'm not in school so I can learn, lol
Writing out proofs was pretty useful, time consuming though. You had to show the reasoning behind the steps that you chose to solve the problem. After a hiatus, I was glad students are back to using this process, until it gets to be automatic.
His cousin was 3 years old and in high school? ahah
"3 years older"
David Petro I know but it sounded like that haha
Haha I was also thinking about that :P But I'm embarrassed to admit that no red flags and loud sirens went off in my mind saying "no way a 3 year old will learn algebra"
maybe he meant it figuratively? like he was a high schooler who had the thinking capacity of a 3 year old. idk
I think he was 3 years old. not his cousin
I think he misses the point. The point of doing algebra problems in school is to practice the particular technique or method being taught and understand its application. And for that purpose it is actually valuable to practice with problems for which there are easier solutions, so that you can, with a high level of confidence, check your work. (Sure, you could just print the answers at the back of the book, but misprints happen.)
To question practicing algebra the way its done is like questioning a baseball player bothering with batting practice when they could just use a catapult to launch the ball out of the park.
The fact that Richard Feynman undoubtedly was a genius doesn't make him *always* right. The rules in algebra are not "false". They are correct. Moreover, they are absolutely necessary for all those who are not geniuses. And in fact, they are perfectly understandable if you, for instance, interpret x (the unknown) as a container of something like beans or lego pieces, and the numbers as coins you need to pay.
I've read his "adventures of a curious character", and there are a number of similar situations where he simply doesn't get the hang of reality. For instance, he mentions a course in biology he was attending too, where he was supposed to make a talk about a cat (I believe it was), and he started out by naming the relevant bones of the cat skeleton. Every biologist in the class said "yes yes we know", and Feynman goes on to wonder why the hell they spend so much time by learning the bones of the skeleton by heart when any fool can look it up in a book! Now I do wonder how any biologist is going to do his daily work if he needs to get out a book every time he sees an animal's carcass...
Bruh do you know what a phone is...
@@letis2madeo995 huh?
I homeschooled our son. Once I talked to my husband about our boy's approach towards math. I complained that he always did the question in an unusual way. My husband told me a good intuition worth preserve. I never understood how my son understood math, because i never taught him anything directly about math. I gave him questions and we then compare our answers and approaches. His approach was barely ever the same with me. By the age of 8, he had done all k to 12 math, and started calculus. He is now studying math and CS in college. Math was not his passion, but he just has an very intuitive ability.
An extraordinarily gifted fellow.
algebra isn’t just 2x + 7= 15
Yeah, it is also classification of finite simple groups and other nice, simple stuff.
The word "Algebra" comes from a famous Mathematics textbook written by the Ancient Mathematician from Persia known as Al-khwarizmi. He is the Father of Algebra.
With all due respect, the whole idea behind 'algebra' is to learn the 'algebraic rules' ... Feynman is talking about problem solving at a higher order ... His pragmatic approach is typical of an *engineering* mindset ...
...but he was a physicist
@@Darninja99 For what it's worth, he actually started as an Engineering major.
Well, many pure mathematicians have an engineering mindset, don't they?
@@u.v.s.5583 Pure mathematicians = engineering?
@@u.v.s.5583 No, most pure mathematicians I feel would heavily disagree with what Feyman is saying here.
You have to appreciate that this comes from somebody who solved the pretty much the hardest algebra physics problem in the world
@authorization batman no, thank you
Public schools need separate educations systems: one for casual students and another for passionate ones.
Or maybe dynamic schedule where the student can choose courses which he wants to attend.
That's an ideal scenario that's not practical in real life lol
As Feynman says, "... knowing the whole idea was to find out what X was, and it didn't make any difference how you did it." That doesn't mean he didn't use "algebra", it means he focussed on finding the answer. If he couldn't do it one way, he did it another. This was a trait that characterized his career, and led him to a Nobel Prize. I have no argument with that.
Well, the question is: could your cousin solve a much more complicated equation without algebra?
Obviously he could not. That is what algebra was created to help you do.
Yea the problem is not teaching the algs. The problem is never giving the students context for why they would use this algorithm. Or why it works. And the right way to do that is to give them harder problems that you can't do in your head or via guess and check.
@@peterisawesomeplease Mathematics doesn't need a context. It IS the context.
@@peterisawesomeplease One of the problems, though, is that one cannot reasonably be expected to work with harder examples and check their understanding with them until they have worked with easier examples. Yes, we should explain why one wants to do something, but one of the best ways to get a feel for how something works is to check how it works on easy examples.
@@MuffinsAPlenty Yea it's a bit of a dilemma sometimes. If you teach simple problems then students often miss the point of algebra since they can already solve basic problems in their head. But if you teach hard problems you will just intimidate them and leave them more confused.
@@peterisawesomeplease I find that students do sometimes, not only in algebra, confuse the algorithm and the underlying meaning of the result. However, that does not mean that these algorithms are not important, they are the only way in my opinion to tame the complexity of math as you dive deeper an deeper.
The Clint Eastwood of science
Primary school teachers today would take the whole morning explaining this simple process - using diagrams, colouring in, and group discussions.
Just because he was a genius . . . that does NOT mean he was right about everything. Or that every single one of his opinions was infallible. No way. Seems to me he was somewhat of an ELITIST. Ya gotta be extraordinarily special . . . or you're nothing. Well that may or or may not be true.
I had considerable trouble with arithmetic. I have a slapdash approach to life that means to me that 99 is pretty much the same as 100, I constantly made silly little mistakes and since primary school math consisted of doing hundreds of long multiplication and long division sums (often in pounds, shillings and pence - with farthings) the chances of me actually getting one of these sums right was pretty negligible. Or in my system the same as zero. Getting close to the answer wasn't acceptable. You still got a big red 'X'.
Imagine my joy when we turned to algebra! Those damn numbers went out the window. I didn't need to worry what five times seven was, or how many groats make a furlong. x+x = 2x and a * b = a.b what! how is that difficult?
There were kids in my class scratching their heads and demanding to know what x was. The answer that it could be anything at all just seemed to drive them into a frenzy of disbelief.
I never looked back.
Yes! I relate to this so hard. I have memories of literally slamming my head into the wall at not understand multiplication tables, but the second math went more abstract it became so simple. I'm happy other people had similar experiences.
Yeah numbers suck ass
No homo I think he has a very sexy voice.
Especially at 1:11 - 1:16
Du wanna eat all of him
Legit we were just told to memorize the quadratic formula and never told anything about why it works or anything. Just burn that thing into your head and use it whenever you need to factor a quadratic equation, even if you can easily do it by other means.
You will learn how to derive it in higher level maths, also high-school mainly rewards memorization but at university level you need to build a deeper understanding
@@Slyracoon666 Yeah I know how to derive it, it takes a bit of algebra to isolate x on one side but it's possible, I only know it because I read a lot of math in my spare time, and I use it to solve practical problems, it's just I never learnt it at school, which frustrates me because school is supposed to teach you these things but it doesn't. I guess the reason why lies in standardized testing incentivizing schools to drill random information without context into students' heads rather than actually teaching the subject, because teaching takes time and memorization gives better test results despite not actually teaching anything. It's exactly what Feynman is describing here, they never teach you _how_ to think, they just teach you _what_ to think, they just hand you a set of instructions of what to do without explaining to you what's actually happening.
I think this is part of why most people hate math. Because they associate math with memorizing crazy formulas they'll rarely get use for. They're never exposed to math in a way that actually sparks curiousity, because they've never learnt to use math as a problem-solving tool. In many ways, school does the opposite of learning - it scares people _from_ learning, it's anti-learning.
I found "X." It's between "W" and "Y."
DrumWild no it's not, it's between Z and C
GTFO with your lame joke
Rookie mistake. Instead of al.gebra you're using al.phabet.
@@badhombre4942 Why isn't it right to use the alphabet instead of algebra?
@@AGriffith Because it misses the entire point of the question.
Some students have the same problem with calculus. They know how to do the power rule to do derivatives and integrals ,but when it comes to understanding how to apply calculus later on, they haven’t a clue.
In This Comment Section: people who failed algebra complaining that the system failed them
rightfully so in a lot of cases if you ask me.
Kind of silly to knock it. It's just a tool. Use the tool and get the job done. I don't have a clue how a phone camera works, hardware or software, but I can follow the steps: point, compose, click a button. It's a good thing that it happens without thinking.
I sometimes teach math/s and I always keep this clip in mind. I'll show students the approved way, but if they have their way, I let them play. If you listen you learn. Students' resourcefulness and creativity is limitless.
good!
Memorizing vs Understanding. Memorizing leads to pseudo knowledge whereas, understanding leads to knowledge.
the Number Line never 4get the number line (Richard's final quote)
I think hes wrong here. When algebra gets to more complicated stages and the answer isnt obvious at a first glance then the step by step solution is required. Learning it this way might be slower for simple questions but is the best way to solve difficult algebra.
I don't understand. There's literally only 1 way to solve 2x + 7 = 15 that's not obtusely convoluted, and it's to subtract 7 and divide by 2. 100% that's what the tutor told him to do, because there's no other conceivable way to go about it. What does Feynman mean his cousin couldn't accept that? What was his cousin trying to do instead? The rules behind subtracting 7 and dividing by 2 are a good thing; it's a systematic approach to solving more complex problems that you can't simply eyeball.
NOOOO. One different approaching is just putting some natural numbers in x. Then you can find the answer!
I think he’s right about doing algebra without thinking. But he’s wrong about the real difference between algebraic and arithmetical ways of finding the solution.
It's one thing to know what x is with some mental arithmetic but algebra is all about the steps of proving what x is, that's what math is all about, proving your answers.
Good for you!
The impossible temperature relevantly bang because brother-in-law evidently interest atop a hapless vinyl. lively, spooky deposit
The way they taught us was even worse - move a plus over to the other side and it becomes a minus, etc.
You subtract from both sides, but in doing so you use some axioms like a-a=0, 0-a=-a and so on :P
That's basically the same thing. I actually find this one easier
You'd have to be pretty stupid to think you can solve all algebraic problems in your head, or that learning how to solve different types of equations is somehow not helpful lol
This was me with calculus. I could never understand what the answer meant in relation to the question and eventually gave up on the subject.
If only I'd been confident enough to ask for help...
As with all maths, as Feynman says here, the trick is just to figure out what “x” is.
That’s pretty much sums up maths. Once you grasp that, the rest should start falling into place.
Of course, the child will answer, “But I don’t care what “x” is. It has no interest for me.”
That’s when you know you have a fundamental problem that maths can’t solve.