Very thought provoking video essay. Many thoughts but I keep only the following one: Math education is messed up but the 5 answers that you provide do not go deep enough to the root of the problem. Usually when people give different explanations to the same phenomena, they are missing some more fundamental fact. To me, the more fundamental fact goes beyond math education, and it is about the education system as a whole. To keep it short, "education system" is a misnomer as it is not designed to educate people but to indoctrinate them into the work force (A similar idea is expressed in "The case against education" by Bryan Caplan). Math is the most tragic casualties of the "education system". Paul Lockhart's "A mathematician's lament" expresses this sentiment in the following quote: "Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done- I simply wouldn’t have the imagination to come up with the kind of senseless, soul- crushing ideas that constitute contemporary mathematics education. "
> Usually when people give different explanations to the same phenomena, they are missing some more fundamental fact. Great observation and couldn't agree more. I've felt a strange combination of joy and sadness rediscovering and learning mathematics from scratch as an adult, because it's a beautiful thing that I once thought was so ugly.
As I'm sure you know, the idea that public schools by and large are meant to train children for regimentation was part of mainstream 19th and early 20th century education theory, at least in the United States. They were explicit in the idea that only the children of a small minority were to be taught in a constructive way; the children of hoi polloi on the other hand were to be warehoused and trained to follow directions.
The way that made the most sense to me as a kid, was the number line, essentially One Dimensional Vectors. We left + off, but we assume all numbers are positive unless negative, and the sign is direction. While I didn't know it at the time, doing basic maths as vector addition made the most sense as a 2nd grader.
I think the number line is a wonderful visualisation of the continuity of integers, and can work really great for people who are good at imagining math in more abstract terms! The issue I see is that students often have trouble looking at abstract or compact notation and mentally converting that to number line concepts. It's sort of related to the concept I talk about right at the end of the video of the notation being "compressed" and so much of math education relying on the students ability to read the notation, mentally uncompress it, then solve that.
Yeah, I remember my first step of any maths exam was expanding the problems out into their most verbose format. Heck to this day I'm a serial brace abuser, though part of that is not quite trusting the compiler.
> Heck to this day I'm a serial brace abuser Haha yes, me too. Especially with binary / boolean operators I can never remember the operator precedence. It's probably telling that it's just easier to insert parenthesis everywhere...
The classic way of representing the inverse trig functions fail the non-ambiguity test. I'm sure I'm not the only person who's struggled with sin^(-1)x because I thought it meant the reciprocal, like everywhere else. btw, if I hadn't gone on to learn more math after high school, if you'd shown me a problem with sigma notation I would have told you that we didn't learn that in high school. My own experience was that math education was SLOOOOOOOOW. So I'd mess around with what we were working on over and over again, only to discover that I had merely spoiled the next few parts of the unit.
Yep it's a classic situation in math - in game dev we deal with things like quaternion rotation of a vector, which is referred to as "multiply" - and of course the matrix-vector product involved looks nothing like what you would intuitively think of as "multiply" if you had only learned multiplication of numbers. It's just considered convenient to not have to invent a new term for it, and to just know that "multiplication" means something totally different with different operands.
@@nicbarkeragain It's not a matter of being too lazy to invent new words. Those things are all called multiplication because they're NOT totally different! They share a certain set of algebraic properties, such as associativity and distributivity, that allow you to manipulate equations containing them in familiar ways.
Just to nitpick, at 23:00 the colon has a different meaning than division. 6:2 would mean there are 6 of thing A and two of thing B, meaning 8 things total, so it would be equivalent to 3/4. This notation is used so you can chain things together for example red, green and blue marbles could be shown as 12:8:10 for a total of 30 marbles. I've never found it useful in math and only really seen it in gambling odds and the confusion further proves your point :)
You're certainly right in some interpretations, but it also depends on whether the question is asking for the ratio as a "part to part" or "part to whole". 6:8 of "Part to part" means 6 of one thing and 8 of the other (14 parts total), whereas 6:8 "Part to whole" means that there are 6 of one thing, and 2 of the other (8 parts total). I agree, very confusing.
My main issue isn't with the notation itself, it generally just boils down to loops and variables. The thing that I consistently battle with is that when people provide these formulae the rarely actually tell you what any of the variables actually ARE! If its wikipedia, or a paper or even some gamedev blog, so many sources just notate a formula but dont include a damn key table. You're just to magically know what "J" or "Ø" is in that specific context through the power of sunshine and telepathy
35:48 The reason that particular question was so controversial isn't that people didn't learn order of operations properly, it's that it's written in a screwy way in the first place. The way actual mathematicians write things, the issue of the relative precendence of multiplication and division simply doesn't arise.
One thing that's refreshing about university level mathematics, is that often you start off using the bulkier more explicit notation and then you prove certain properties which allow you to remove that notation, it's like leveling up in a video game.
I think mathematical notation is a lot like musical notation. They've both been developed over centuries to fit a lot of information into a small space in a readable way. They have their flaws, but it's hard to find anything simpler that does the same job.
Just a note, BEDMAS/PEMDAS is supposed to be unambiguous. Thought, it is an additional case of something to be remembered, like you highlight many times. Division and Multiplication are on the same level of precedence, and Addition and Subtraction are both on their same level of precedence. They are simply performed left to right if the operations are on the same level or precedence. I'm sure you and everyone reading know this, but this was included when I was taught PEMDAS and it bugs me when it seems to be omitted from other people's education
Yeah I totally know what you mean, it's a shame it's taught in such an inconsistent way - all you have to do is look at the comments of one of those "viral math problem" posts to see how confused people are 🥲
I had some other alternative in mind concerning ‘PEMDAS’, ‘BODMAS’, ‘PEDMAS’ or whatsoever... or even the advocated ‘P’. One could follow the concept of ‘replace → parentheses’, shortened ‘RP’. In this context, replacing means that every non-commutative expression, which actually only stands for an abbreviated notation of a more elementary operation with the respective inverse element, gets replaced in the whole term in the first place. I.e.: ‘÷𝑥’ becomes ‘×𝑥⁻¹’ and ‘−𝑥’ becomes ‘+(−𝑥)’. For the viral mathematical example problem given here, this would mean: 6÷2(1+2) = 6×½(1+2) = ((6×½)(1+2)) = (3×3) = 9 or 6÷2(1+2) = 6×½(1+2) = (6×(½(1+2))) = (6×(½×3)) = (6×(3/2)) = 9
Thank you for this awesome explanation. I would be very interested to learn more about your insights into the fundamentals of learning. I appreciate your contribution!
Thanks! I'm super interested in education in general, there are a few more videos on my channel on the topic 🙂 I also make educational games, you can read a bit more if you like at delphiniumgames.com
This reminds me of the abstraction "debate" in software somewhat. Abstractions are useful, and *mandatory* to decrease the cognitive load of very complex problems. The issue becomes when abstract concepts are used without the understanding of what they're representing. What does an array represent? What does "arrayness" imply? Math is taught as a set of rules, but it's not that, it's a way to *describe* things, it's about representing concepts. It's a language, a way to express a mental mode, but it's not taught as one.
Brilliant video, I came across the same idea when learning how to use a Reverse Polish Notation calculator. I was already decent at math, but once I understood how it worked math became way easier.
I had not heard of the Reverse Polish Notation, was a very interesting read! I think doing the same kind of math that you already know, but presented in a radically different notation kind of "unlocks" something in your brain. It's easy to get tunnel vision and think of everything in terms of the notation that you know, rather than what it represents.
That's part of it for sure, but I particularly liked how stack notations like RPN remove order of operations from your mental load. You don't have to worry about the order, just line up the vales with their operations and watch the dominoes fall into place.
I love how you've formulated the video thus far in (35 minutes or so), but I do have a critique. This is not to disparage you or anything of course, I whole-heatedly agree with your sentiment and rationale; however, at the point in the video where you discuss combining the denominators as information-hiding, I think it is good to contest this. I was thinking about it and concluded that "Information hiding" is a particular type of abstraction, where that is in general a desirability of mathematics and finding the deeper patterns of things. Information hiding, in particular, is abstraction which does not lead to further insights being made and does not significantly contribute to the readability of the problem. In the case of, and forgive me for trying to put math in a plain-text comment (another reason, in our internet age, why so many dislike it), (x + 1)/x = x/x + 1/x, I believe that this is a useful abstraction 'rule' to present and enforce critical thinking about. The reason I believe this is not information hiding so much as a useful re-framing is that it is useful for a person doing the math to be able to group and un-group the terms in the numerator. For example, given a sum of seemingly different fractions with complex (in the colloquial sense) denominators, we may at first understand the problem as a set of quantities divided by differing amounts and then summing together; however, if we find that the denominators of the different fractions are actually equivalent expressions, then we can frame the problem as the sum of some number of quantities which is then divided by a 'final' quantity. Of course, as I paint it here, this is taking a very action or verb oriented approach, but I don't believe it is dependent on that- this is just the easiest way for me to express this difference in (useful) abstraction and (inhibitory) information hiding. Again, though, thank you for making this video! It's the first one I've seen of yours and I'll be sure to watch your others next (in particular, I'm eyeing the "Practical CPU Optimization..." video). Cheers!
I actually totally agree with you, and I went back and forth several times on including this section in the video, because in some ways it's one of those "oversimplifications" that I speak about negatively earlier on 😅 For those more fluent in algebra, it becomes natural and effortless to separate and combine fraction numerators in order to reveal opportunities for trivial simplification, use of identities etc. I really believe that one of the fundamental ideas that we fail to teach in math is that so many expressions are equivalent and can be substituted for one another. I have noticed however that often when first exploring the intersection of fractions and algebra, students are taught this nebulous concept of "cancelling" which involves identifying like terms in the numerator / denominator, then applying a specific mechanical process depending on the current layout of the fraction, without actually understanding what is mechanically enabling the "cancelling" in the first place. As a result I wanted to demonstrate that representing numerator addition in a normalized form as multiple fractions, the process of "cancelling" is often a trivial operation of dividing a term by itself, etc.
I’m curious to know actually how many people find notation as their main roadblock to learning math. I had the same experience of liking maths a a kid but falling off in my teens. I’m now a software developer but I still find compact notation difficult to parse, which is why a lot of functional languages are challenging for me to read despite having good understanding of functional programming. Single letter variable names melt my mind. Been a rude awakening having to learn linear algebra and some calculus for graphics programming as an adult
I think one of the best examples is that super viral twitter post Freya Holmer made where she explained that sigma notation and similar are just for loops. As a programmer you already understand it, it's just the notation that is frightening x.com/FreyaHolmer/status/1436696408506212353
Amazing video! I guess I’ve gotten so used to the notation that I completely understand it, but don’t see it’s errors in showing the true underlying meaning.
Thanks, and yes I was in exactly the same situation. Recently I built a computer algebra system - I assumed it would be pretty straightforward but really struggled at first. It made me realise that my "fundamental" understanding of math was based around tricks with the notation, rather than actual mathematics 🙂
I don't think, it's just math education that is messed up. Look at all sorts of UA-cam-videos. There seem to be only two kinds that involve math: Explicit math-videos and videos that try to explain something else and a little bit of math is necessary. Basically in every video of the latter kind the UA-camr says something like: "Warning: Now comes a bit of math. But please don't worry and close the video - I will explain it in simple terms.". And in most cases the math involved is really easy. Sure - I studied physics and therefore probably know more about math than 99% of people. But 99% of people seem to refuse to even try and understand (something simple). That also seems to go beyond math. I don't know, how often a UA-camr put a "my head exploded" or something like that in his video and all I could think was: "What is your problem?! It's not *that* hard to understand...".
There's probably some truth to this, but I don't think you can blame the notation too much. Look at how much people seem to hate word problems, getting rid of the equations just seems to make things more confusing. I think the main difficulty with math is that everything builds on previous things you're expected to have already learned. If you get confused by some bit of notation or don't understand some concept early on then you'll struggle with every subsequent step, which will make you fall even more behind and struggle even more. But each student will find different things easier or harder, and the real problem is that we don't take that into account with the standardized education system. If we introduced some really intuitive notation that everyone could learn in half the time, then schools would probably also start giving them half the time to learn it and then we'd basically be right back where we are now. What we really need is a more personalized system where everyone can learn at their own pace and spend more time on whatever they personally are struggling with.
Yes, exactly. I think a tricky problem in math is that people think you can't question the "instructions", like they are magic and perfect or something.
Commenting on an old video, but this counting-way of thinking things made me remember this wonderful Vihart video from a decade ago: ua-cam.com/video/N-7tcTIrers/v-deo.htmlsi=V29HWoutEV0bkF4k I thought I would share since I saw nobody else mention it.
I imagine its just as hard for the instructor when the student has no interest. So somebody has to change their attitude while the other finds a way of making it interesting. Not that you're not doing a great job. You are. But learning anything is all about attitude.
Yes, there is always an extent to which students are responsible for their own learning. Even the best teachers can only do so much. I think it's worth talking about how some students start very motivated in math as children, but seem to completely lose it by age 13-14. I'm sure there are a whole host of factors at play that result in that disenfranchisement.
@@nicbarkeragain My problem started around 4th grade. I never talked about 8t with snyone, and nobody seemed to notice. I did excelet until 4th grade. then something happened. I lost all confidence in all subjects. I felt like I was absent the day that they explained the key to everything. it wasnt until I went to continuation and THEY wanted to know what made me tick. Had I not gone there, I would have literally gone through life thinking teachers dont care. Understand that it wasnt just the students there that were rejects. the school district sent the teachers there as well. There was nothing wrong with them except that they just didn't fit their "niche" (?) Either overweight, kinda sloppy or laughed at odd times. But I learned more from them than the yuppy school I previously attended. Students and faculty worked on a first name basis. It was pretty much self disciplined. You either participated and graduated or you didnt. And if you didnt show up they made that decision for you. If you got caught smoking pot (,or any dope) once, you were out. you got caght fighting youre out. Cigarettes were fine just keep them away from class. My folks never thought I would pull it off let alone get the grades I got.
@@samueldeandrade8535 while I agree in principle, we also can't demand teachers be perfect in reaching all disinterested students in an environment with limited time and difficult work conditions. I don't think any of the teachers, students or many other factors are completely to blame, as with many things it's a mixture rather than definitive 🙂
The best you can do is appeal to their interests and try to show them how math applies to it. But that's not always realistic in a chaotic classroom with dozens of students.
@@nicbarkeragain@15:49 (- 3 (- 0 7))? See "Programming is (should be) fun" by Gerald Jay Sussman (also on UA-cam). You'll thank me later. I found this presentation long and without any clarification. So unhelpful even in the mentioned example.
i don't wanna watch the whole thing but i don't think math concepts are hard to learn at all, although intuitively mastering and understanding them is.
My take-away from this video is... ...that I'm not a math person. Not because I lack the ability to think but because I'm afraid of it, emotionally. I hate math. Fuck math. I'm not a math person.
Honestly I failed practical math three times in high school. I simply refused to try and understand As a result I had to finish school in a continuation school.Which turned out one of the best things that happened to me. The classes were smaller, for starters. The instructors had one goal. Get the student legitimately graduated. The math instructor talked with me first. Then tested me on what I did know. Moved up to the next level and made sure I understood it. But then came the most important part, and without it everything learned would be all for nada. PRACTICE. It wasnt cuz I was dumb or because it was too hard. The lack of interest and nothing making it worth remembering is the hole I kept falling through. So his remedy? Practice at least an hour every night. First, write out the operations in case you forget then pull random problems out of the book and solve them. I got an A in the class. The first A I ever got in school, except for PE. Mr. Bueller, McNally High School. I owe you!!! 🦾
So glad to hear that you had a teacher who really took the time to connect with you personally and understand what it was that you needed. Regarding practise I completely agree with you - I was lucky to grow up with parents who encouraged me to learn a musical instrument, which makes you realise that practise is absolutely essential to improving. Math is part of a small group of school subjects which are very heavily dependent on repetition to improve - foreign languages are another prime example. Which can be very problematic if the student is expected to do all that practise in their own time as homework, but they aren't motivated or don't have a home life which is conducive to it.
@@nicbarkeragain Thanx. I noticed the keyboard back there. My parents encouraged me too but I think they regreted later as I was going to be a Rockstar, dontcha know? lol. I devoted too much time trying to be Jimmy Page . a lot of mispent youth there. Dont get me wrong I play fairly well. But I didnt know you pretty much had to kill a few people to get somewhere. So spent alot of time working lighting and PAs in nightclubs. Calous fingers for nothing. 😡 lol I dunno...Worst mistakes are the best lessons. take care 👍
@@siggyretburns7523 I was absolutely obsessed with electric guitar when I was younger too (it's been almost 20 years now since I started) but after all that I don't consider any time spent playing music to be wasted 🙂
@@nicbarkeragain Well it wasnt that I wasted time with the guitar. More like I wasted time not practicing . Dreams do come true, but not by dreams alone. I went for years afterwards not even picking one up. Outside of rebuilding calluses it all comes right back very quickly.
Very thought provoking video essay. Many thoughts but I keep only the following one:
Math education is messed up but the 5 answers that you provide do not go deep enough to the root of the problem. Usually when people give different explanations to the same phenomena, they are missing some more fundamental fact. To me, the more fundamental fact goes beyond math education, and it is about the education system as a whole. To keep it short, "education system" is a misnomer as it is not designed to educate people but to indoctrinate them into the work force (A similar idea is expressed in "The case against education" by Bryan Caplan). Math is the most tragic casualties of the "education system". Paul Lockhart's "A mathematician's lament" expresses this sentiment in the following quote:
"Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done- I simply wouldn’t have the imagination to come up with the kind of senseless, soul- crushing ideas that constitute contemporary mathematics education. "
> Usually when people give different explanations to the same phenomena, they are missing some more fundamental fact.
Great observation and couldn't agree more. I've felt a strange combination of joy and sadness rediscovering and learning mathematics from scratch as an adult, because it's a beautiful thing that I once thought was so ugly.
As I'm sure you know, the idea that public schools by and large are meant to train children for regimentation was part of mainstream 19th and early 20th century education theory, at least in the United States. They were explicit in the idea that only the children of a small minority were to be taught in a constructive way; the children of hoi polloi on the other hand were to be warehoused and trained to follow directions.
@@jessejordache1869 more people should make aware of this
The way that made the most sense to me as a kid, was the number line, essentially One Dimensional Vectors. We left + off, but we assume all numbers are positive unless negative, and the sign is direction. While I didn't know it at the time, doing basic maths as vector addition made the most sense as a 2nd grader.
I think the number line is a wonderful visualisation of the continuity of integers, and can work really great for people who are good at imagining math in more abstract terms! The issue I see is that students often have trouble looking at abstract or compact notation and mentally converting that to number line concepts. It's sort of related to the concept I talk about right at the end of the video of the notation being "compressed" and so much of math education relying on the students ability to read the notation, mentally uncompress it, then solve that.
Yeah, I remember my first step of any maths exam was expanding the problems out into their most verbose format. Heck to this day I'm a serial brace abuser, though part of that is not quite trusting the compiler.
> Heck to this day I'm a serial brace abuser
Haha yes, me too. Especially with binary / boolean operators I can never remember the operator precedence. It's probably telling that it's just easier to insert parenthesis everywhere...
The classic way of representing the inverse trig functions fail the non-ambiguity test. I'm sure I'm not the only person who's struggled with sin^(-1)x because I thought it meant the reciprocal, like everywhere else.
btw, if I hadn't gone on to learn more math after high school, if you'd shown me a problem with sigma notation I would have told you that we didn't learn that in high school. My own experience was that math education was SLOOOOOOOOW. So I'd mess around with what we were working on over and over again, only to discover that I had merely spoiled the next few parts of the unit.
Yep it's a classic situation in math - in game dev we deal with things like quaternion rotation of a vector, which is referred to as "multiply" - and of course the matrix-vector product involved looks nothing like what you would intuitively think of as "multiply" if you had only learned multiplication of numbers. It's just considered convenient to not have to invent a new term for it, and to just know that "multiplication" means something totally different with different operands.
@@nicbarkeragain It's not a matter of being too lazy to invent new words. Those things are all called multiplication because they're NOT totally different! They share a certain set of algebraic properties, such as associativity and distributivity, that allow you to manipulate equations containing them in familiar ways.
Thanks again Nic! This is superbly put together, brilliant content as always.
Thanks J ❤️
36:50 lisp was the answer all along
“People pay the same amount of reverence to mathematical notation as they do to math. And that’s the problem.”
👏👏👏👏👏👏👏👏👏👏👏👏
Bravo
Just to nitpick, at 23:00 the colon has a different meaning than division. 6:2 would mean there are 6 of thing A and two of thing B, meaning 8 things total, so it would be equivalent to 3/4. This notation is used so you can chain things together for example red, green and blue marbles could be shown as 12:8:10 for a total of 30 marbles. I've never found it useful in math and only really seen it in gambling odds and the confusion further proves your point :)
You're certainly right in some interpretations, but it also depends on whether the question is asking for the ratio as a "part to part" or "part to whole". 6:8 of "Part to part" means 6 of one thing and 8 of the other (14 parts total), whereas 6:8 "Part to whole" means that there are 6 of one thing, and 2 of the other (8 parts total). I agree, very confusing.
My main issue isn't with the notation itself, it generally just boils down to loops and variables. The thing that I consistently battle with is that when people provide these formulae the rarely actually tell you what any of the variables actually ARE!
If its wikipedia, or a paper or even some gamedev blog, so many sources just notate a formula but dont include a damn key table. You're just to magically know what "J" or "Ø" is in that specific context through the power of sunshine and telepathy
35:48 The reason that particular question was so controversial isn't that people didn't learn order of operations properly, it's that it's written in a screwy way in the first place. The way actual mathematicians write things, the issue of the relative precendence of multiplication and division simply doesn't arise.
One thing that's refreshing about university level mathematics, is that often you start off using the bulkier more explicit notation and then you prove certain properties which allow you to remove that notation, it's like leveling up in a video game.
I think mathematical notation is a lot like musical notation. They've both been developed over centuries to fit a lot of information into a small space in a readable way. They have their flaws, but it's hard to find anything simpler that does the same job.
Just a note, BEDMAS/PEMDAS is supposed to be unambiguous.
Thought, it is an additional case of something to be remembered, like you highlight many times.
Division and Multiplication are on the same level of precedence, and Addition and Subtraction are both on their same level of precedence.
They are simply performed left to right if the operations are on the same level or precedence.
I'm sure you and everyone reading know this, but this was included when I was taught PEMDAS and it bugs me when it seems to be omitted from other people's education
Yeah I totally know what you mean, it's a shame it's taught in such an inconsistent way - all you have to do is look at the comments of one of those "viral math problem" posts to see how confused people are 🥲
I had some other alternative in mind concerning ‘PEMDAS’, ‘BODMAS’, ‘PEDMAS’ or whatsoever... or even the advocated ‘P’. One could follow the concept of ‘replace → parentheses’, shortened ‘RP’.
In this context, replacing means that every non-commutative expression, which actually only stands for an abbreviated notation of a more elementary operation with the respective inverse element, gets replaced in the whole term in the first place. I.e.: ‘÷𝑥’ becomes ‘×𝑥⁻¹’ and ‘−𝑥’ becomes ‘+(−𝑥)’.
For the viral mathematical example problem given here, this would mean:
6÷2(1+2) = 6×½(1+2) = ((6×½)(1+2)) = (3×3) = 9
or
6÷2(1+2) = 6×½(1+2) = (6×(½(1+2))) = (6×(½×3)) = (6×(3/2)) = 9
Thank you for this awesome explanation. I would be very interested to learn more about your insights into the fundamentals of learning. I appreciate your contribution!
Thanks! I'm super interested in education in general, there are a few more videos on my channel on the topic 🙂 I also make educational games, you can read a bit more if you like at delphiniumgames.com
This reminds me of the abstraction "debate" in software somewhat.
Abstractions are useful, and *mandatory* to decrease the cognitive load of very complex problems.
The issue becomes when abstract concepts are used without the understanding of what they're representing.
What does an array represent? What does "arrayness" imply?
Math is taught as a set of rules, but it's not that, it's a way to *describe* things, it's about representing concepts. It's a language, a way to express a mental mode, but it's not taught as one.
Brilliant video, I came across the same idea when learning how to use a Reverse Polish Notation calculator. I was already decent at math, but once I understood how it worked math became way easier.
I had not heard of the Reverse Polish Notation, was a very interesting read! I think doing the same kind of math that you already know, but presented in a radically different notation kind of "unlocks" something in your brain. It's easy to get tunnel vision and think of everything in terms of the notation that you know, rather than what it represents.
That's part of it for sure, but I particularly liked how stack notations like RPN remove order of operations from your mental load.
You don't have to worry about the order, just line up the vales with their operations and watch the dominoes fall into place.
I love how you've formulated the video thus far in (35 minutes or so), but I do have a critique. This is not to disparage you or anything of course, I whole-heatedly agree with your sentiment and rationale; however, at the point in the video where you discuss combining the denominators as information-hiding, I think it is good to contest this. I was thinking about it and concluded that "Information hiding" is a particular type of abstraction, where that is in general a desirability of mathematics and finding the deeper patterns of things. Information hiding, in particular, is abstraction which does not lead to further insights being made and does not significantly contribute to the readability of the problem. In the case of, and forgive me for trying to put math in a plain-text comment (another reason, in our internet age, why so many dislike it), (x + 1)/x = x/x + 1/x, I believe that this is a useful abstraction 'rule' to present and enforce critical thinking about. The reason I believe this is not information hiding so much as a useful re-framing is that it is useful for a person doing the math to be able to group and un-group the terms in the numerator. For example, given a sum of seemingly different fractions with complex (in the colloquial sense) denominators, we may at first understand the problem as a set of quantities divided by differing amounts and then summing together; however, if we find that the denominators of the different fractions are actually equivalent expressions, then we can frame the problem as the sum of some number of quantities which is then divided by a 'final' quantity. Of course, as I paint it here, this is taking a very action or verb oriented approach, but I don't believe it is dependent on that- this is just the easiest way for me to express this difference in (useful) abstraction and (inhibitory) information hiding.
Again, though, thank you for making this video! It's the first one I've seen of yours and I'll be sure to watch your others next (in particular, I'm eyeing the "Practical CPU Optimization..." video).
Cheers!
I actually totally agree with you, and I went back and forth several times on including this section in the video, because in some ways it's one of those "oversimplifications" that I speak about negatively earlier on 😅
For those more fluent in algebra, it becomes natural and effortless to separate and combine fraction numerators in order to reveal opportunities for trivial simplification, use of identities etc. I really believe that one of the fundamental ideas that we fail to teach in math is that so many expressions are equivalent and can be substituted for one another.
I have noticed however that often when first exploring the intersection of fractions and algebra, students are taught this nebulous concept of "cancelling" which involves identifying like terms in the numerator / denominator, then applying a specific mechanical process depending on the current layout of the fraction, without actually understanding what is mechanically enabling the "cancelling" in the first place. As a result I wanted to demonstrate that representing numerator addition in a normalized form as multiple fractions, the process of "cancelling" is often a trivial operation of dividing a term by itself, etc.
I’m curious to know actually how many people find notation as their main roadblock to learning math. I had the same experience of liking maths a a kid but falling off in my teens. I’m now a software developer but I still find compact notation difficult to parse, which is why a lot of functional languages are challenging for me to read despite having good understanding of functional programming. Single letter variable names melt my mind. Been a rude awakening having to learn linear algebra and some calculus for graphics programming as an adult
I think one of the best examples is that super viral twitter post Freya Holmer made where she explained that sigma notation and similar are just for loops. As a programmer you already understand it, it's just the notation that is frightening
x.com/FreyaHolmer/status/1436696408506212353
Amazing video! I guess I’ve gotten so used to the notation that I completely understand it, but don’t see it’s errors in showing the true underlying meaning.
Thanks, and yes I was in exactly the same situation. Recently I built a computer algebra system - I assumed it would be pretty straightforward but really struggled at first. It made me realise that my "fundamental" understanding of math was based around tricks with the notation, rather than actual mathematics 🙂
Could you talk about calculus?
I don't think, it's just math education that is messed up. Look at all sorts of UA-cam-videos. There seem to be only two kinds that involve math: Explicit math-videos and videos that try to explain something else and a little bit of math is necessary. Basically in every video of the latter kind the UA-camr says something like: "Warning: Now comes a bit of math. But please don't worry and close the video - I will explain it in simple terms.". And in most cases the math involved is really easy. Sure - I studied physics and therefore probably know more about math than 99% of people. But 99% of people seem to refuse to even try and understand (something simple). That also seems to go beyond math. I don't know, how often a UA-camr put a "my head exploded" or something like that in his video and all I could think was: "What is your problem?! It's not *that* hard to understand...".
There's probably some truth to this, but I don't think you can blame the notation too much. Look at how much people seem to hate word problems, getting rid of the equations just seems to make things more confusing.
I think the main difficulty with math is that everything builds on previous things you're expected to have already learned. If you get confused by some bit of notation or don't understand some concept early on then you'll struggle with every subsequent step, which will make you fall even more behind and struggle even more. But each student will find different things easier or harder, and the real problem is that we don't take that into account with the standardized education system. If we introduced some really intuitive notation that everyone could learn in half the time, then schools would probably also start giving them half the time to learn it and then we'd basically be right back where we are now. What we really need is a more personalized system where everyone can learn at their own pace and spend more time on whatever they personally are struggling with.
Check out the book "The Realm of Algebra" by Issac Asimov
Thanks for the recommendation!
Excellent explanation. It's like operating VCRs. If the instructions are confused, so are the users.
Yes, exactly. I think a tricky problem in math is that people think you can't question the "instructions", like they are magic and perfect or something.
Commenting on an old video, but this counting-way of thinking things made me remember this wonderful Vihart video from a decade ago: ua-cam.com/video/N-7tcTIrers/v-deo.htmlsi=V29HWoutEV0bkF4k
I thought I would share since I saw nobody else mention it.
Can you make the same video to explain Category Theory notation?
I would have to spend a lot more time learning and becoming comfortable with category theory to be able to do that 😅
I imagine its just as hard for the instructor when the student has no interest. So somebody has to change their attitude while the other finds a way of making it interesting. Not that you're not doing a great job. You are. But learning anything is all about attitude.
Yes, there is always an extent to which students are responsible for their own learning. Even the best teachers can only do so much. I think it's worth talking about how some students start very motivated in math as children, but seem to completely lose it by age 13-14. I'm sure there are a whole host of factors at play that result in that disenfranchisement.
@@nicbarkeragain My problem started around 4th grade. I never talked about 8t with snyone, and nobody seemed to notice. I did excelet until 4th grade. then something happened. I lost all confidence in all subjects. I felt like I was absent the day that they explained the key to everything. it wasnt until I went to continuation and THEY wanted to know what made me tick. Had I not gone there, I would have literally gone through life thinking teachers dont care.
Understand that it wasnt just the students there that were rejects. the school district sent the teachers there as well. There was nothing wrong with them except that they just didn't fit their "niche" (?) Either overweight, kinda sloppy or laughed at odd times. But I learned more from them than the yuppy school I previously attended. Students and faculty worked on a first name basis. It was pretty much self disciplined. You either participated and graduated or you didnt. And if you didnt show up they made that decision for you. If you got caught smoking pot (,or any dope) once, you were out. you got caght fighting youre out. Cigarettes were fine just keep them away from class.
My folks never thought I would pull it off let alone get the grades I got.
We can't demand students to have interest.
@@samueldeandrade8535 while I agree in principle, we also can't demand teachers be perfect in reaching all disinterested students in an environment with limited time and difficult work conditions. I don't think any of the teachers, students or many other factors are completely to blame, as with many things it's a mixture rather than definitive 🙂
The best you can do is appeal to their interests and try to show them how math applies to it. But that's not always realistic in a chaotic classroom with dozens of students.
Forth? Lisp?
😅 not quite sure what you're getting at here
@@nicbarkeragain@15:49 (- 3 (- 0 7))?
See "Programming is (should be) fun" by Gerald Jay Sussman (also on UA-cam). You'll thank me later.
I found this presentation long and without any clarification. So unhelpful even in the mentioned example.
im bad at math like im bad a vim motions....
so good!
i don't wanna watch the whole thing but i don't think math concepts are hard to learn at all, although intuitively mastering and understanding them is.
My take-away from this video is...
...that I'm not a math person. Not because I lack the ability to think but because I'm afraid of it, emotionally. I hate math. Fuck math. I'm not a math person.
Honestly I failed practical math three times in high school. I simply refused to try and understand
As a result I had to finish school in a continuation school.Which turned out one of the best things that happened to me. The classes were smaller, for starters. The instructors had one goal. Get the student legitimately graduated.
The math instructor talked with me first. Then tested me on what I did know. Moved up to the next level and made sure I understood it. But then came the most important part, and without it everything learned would be all for nada. PRACTICE. It wasnt cuz I was dumb or because it was too hard. The lack of interest and nothing making it worth remembering is the hole I kept falling through. So his remedy? Practice at least an hour every night. First, write out the operations in case you forget then pull random problems out of the book and solve them.
I got an A in the class. The first A I ever got in school, except for PE.
Mr. Bueller, McNally High School. I owe you!!! 🦾
So glad to hear that you had a teacher who really took the time to connect with you personally and understand what it was that you needed. Regarding practise I completely agree with you - I was lucky to grow up with parents who encouraged me to learn a musical instrument, which makes you realise that practise is absolutely essential to improving. Math is part of a small group of school subjects which are very heavily dependent on repetition to improve - foreign languages are another prime example. Which can be very problematic if the student is expected to do all that practise in their own time as homework, but they aren't motivated or don't have a home life which is conducive to it.
@@nicbarkeragain Thanx. I noticed the keyboard back there. My parents encouraged me too but I think they regreted later as I was going to be a Rockstar, dontcha know? lol. I devoted too much time trying to be Jimmy Page . a lot of mispent youth there. Dont get me wrong I play fairly well. But I didnt know you pretty much had to kill a few people to get somewhere. So spent alot of time working lighting and PAs in nightclubs. Calous fingers for nothing. 😡 lol I dunno...Worst mistakes are the best lessons.
take care 👍
@@siggyretburns7523 I was absolutely obsessed with electric guitar when I was younger too (it's been almost 20 years now since I started) but after all that I don't consider any time spent playing music to be wasted 🙂
@@nicbarkeragain Well it wasnt that I wasted time with the guitar. More like I wasted time not practicing . Dreams do come true, but not by dreams alone.
I went for years afterwards not even picking one up. Outside of rebuilding calluses it all comes right back very quickly.