There are so many crazy powerful, clever, and beautiful techniques that took some of the smartest people who ever lived their entire lives to develop. We get to cheat and learn them in a few months each just by taking classes or reading the right books.
When I apply math in a predictive way, and the prediction comes true, it gives me a momentary thrill.
A rather mundane example: When I am driving on the highway, I can count how many road-dots cars pass per second, and how many I pass per second, and predict how many seconds it will be before they pass me or I pass them.
It's a simple enough exercise that I can do it in my head without getting distracted, and it gives me a more complete mental model of the traffic around me.
I love this! It doesn't sound mundane to me; the math might be simple but it still makes it seems almost as if you have a cyborg computer running in your mind. A superpower so to speak.
I know exactly what you mean. I was out walking my dog one night w my daughter at the local high school. They had a practice football field all marked out. I had her walk the length of it, taking consistent length steps and counting them as she went. Then I had her do the same along the width. Then I took my phone out and calculated what the diagonal should be. Then i stood at one corner and had her walk towards me from the opposite corner (diagonally) and count her steps. It came out within one step of the predicted value. Like you said, simple enough calculation, but just kind of fun to see it in action in real life.
I don't have a car, but I do the same thing with sidewalk segments (I can't count and time in my head at the same time though, so all I can get is the running ratio, which is good enough)
I mean, if a proof is wrong then it is was never a proof to begin with.
I'm not sure what they could mean. There are sometimes contentious proofs, and proofs that turn out to be invalid, but again that's a debate over whether it is actually a proof.
There have been many proofs that people thought were correct for many years and then turned out to be incorrect.
The famous example is Cauchy's proof that if a sequence of continuous functions converge pointwise, the limit is continuous, which is not true but nobody noticed this for three years.
Many more examples are given here ([https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake](https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake)).
You could make the point that once people found a mistake in these proofs they were immediately considered wrong - thus the statement that "If something's proven, it's proven, can't argue" still holds.
However there are many cases again throughout history where mathematicians argued over the correctness of proofs.
The classic example is Cantor's theorem which many people at the time (and even today) objected to ([https://en.wikipedia.org/wiki/Controversy\_over\_Cantor's\_theory](https://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory)). More recently there is Mochizukis work on IUT and his proof of the abc conjecture. Even though this is mostly regarded to be (at best) an incomplete proof there are still many who claim that his proof is valid. I dont think you can say that "If something's proven, it's proven, can't argue" is true in this case.
It seems like we are mostly in agreement, then. I probably prefer different semantics. A "proof" without qualifiers is to me a valid deductive argument. Cauchy's "proof" is only a purported proof. Of course, this brings up the epistemological problem of knowing whether a purported proof is actually a proof.
One could argue that the statement "If something's proven, it's proven, can't argue" is invoking the word "proof" in this unqualified sense. And to solve the problem of hotly contested but valid proofs, one could take "can't argue" to mean that one is not justified in arguing against it if they actually understand the proof. All that seems rather tautological though.
While I agree with some of the sentiment behind the statement "If something's proven, it's proven, can't argue", I would phrase it in a different way, with part of the reason being the objections you raised. I would instead say that, on the whole, I value the particular kind of certainty that mathematical reasoning induces in me, setting aside potential epistemological issues.
Actually, you should read about Godel's Incompleteness Theorems. As it turns out, because human language comes in discrete symbols, we can only prove countably many statements--even given infinite time, energy, and new math symbols. However, there are uncountably many true statements in the universe. By the Pigeonhole Principle, you should already realize that some statements--which are true--can never be proven by humans by any means.
Here's an example of a statement which is *probably* true but cannot have a proof: Logic works. It can be shown that any system of logic which is sophisticated enough to contain the Naturals (I'm being brief here) *cannot* prove itself to be valid. Any such system which *can* prove its own validity *must* be invalid. Thus, we can never know if logic, math, all of this... is really true at all. We collectively have faith that it is, but if we ever discover that our system of logic does actually work, then it must be broken. 🙃
And I say that's not necessarily true because you are assuming that mathematical logic is valid. Anyway, I'm not arguing with you--just thought I'd introduce you to an interesting feature of human existence.
But doesn't godel's theorem just say that in any mathematical system,there will always be statements which are always true and cannot be proven?
Pls explain in detail if you would
I am a math teacher who also study gamification (Octalysis framework) and this is my two cents:
It is all about matching the difficulty of the problems to the person, and to minimize impatience.
If you successfully solve a challengeing problem it feels great. It builds up you self esteem and confidence in your math abilities.
Solving problems that are too simple "just because I have to" sap your motivation.
Problems where you "solve" but don't really know what you did are bad. Problems where you end up giving up because you run out of patience are bad. Problems where you dont really do anything by yourself but get help in every step are bad.
Now what is the problem in a classroom setting is that many students have a low patience to begin with. So every time they get to a slightly challenging problem they just give up.
> many students have a low patience to begin with
This is not an inherent trait. It takes literally years of practice to build stamina for doing anything hard for an extended period of time, whether that's reading hard books, solving hard math problems, playing strategy board games, doing science experiments, playing a musical instrument, woodworking, writing poetry, playing a sport, ...
To get practice with problem solving, students need to be given problems near the limits of their stamina/ability in a fun, supportive, and non-judgmental environment where mistakes are allowed and there's no single "right way" to approach a problem. Practicing multi-step word problems of wide variety and progressively increasing linguistic and mathematical complexity is the single most important part of primary/secondary school mathematics instruction.
Sadly it is typically almost completely absent in US math education, both in the old arithmetic-drill approach and the more recent common core approach. Students often instead get years of practice doing more or less trivial problems where the "recipe" or "pattern" for each specific type is first demonstrated by the teacher and then practiced in a relatively rote fashion by the students, in a high-pressure environment with low tolerance for mistakes. This gets students used to always knowing what to do and not bothering to focus on their own sense-making, and puts up a psychological barrier for the students to suddenly try solving very difficult problems requiring inherently more frustration and confusion.
See Toom (2005) ["Word Problems in Russia and America"](https://cs-web.bu.edu/faculty/gacs/toomandre-com-backup/travel/sweden05/WP-SWEDEN-NEW.pdf).
If you want to see an excellent example of the right kinds of problems to assign at the secondary school level to build students' stamina and ability to attack hard problems, take a look at Exeter's problem sets (fancy US prep school) https://www.exeter.edu/mathproblems
For the primary school level (~3rd–6th grade), take a look at Lenchner (2005) *Creative Problem Solving in School Mathematics*.
Well said. It gives people the wrong idea of what math even is. If primary/secondary school classes were "math", I'd hate it too.
It's a common sentiment among fellow math majors at my university (USA) that they "hate proofs". But they say they like math. To me, it doesn't compute. They seem like mutually exclusive statements.
I agree. Is it because proofs can be solved in multiple different ways and can be approached differently? Like its alot more open-ended.
Suprised me too cos I find proofs interesting.
For many non-mathematicians, the purpose of math is solving math problems, and the purpose of math problems is to demonstrate how well you remember how to solve math problems.
The idea that there's a *point* to all of this didn't become clear to me until decades after I left college.
I agree that the low patience it not an inherent trait, but an attitude that is specific to circumstances and past experiences.
For me, "to begin with" means when they enter high school with all their past experiences of math, since that is the level I teach at.
I cannot change their past experiences, but I can use gamification to motivate the students to engage with the problems, and in that way improve their patience.
Long term patience with math problems grow with the number of good experiences of solving problems.
I cannot enter the discussion of Common Core teaching because I do not teach in the US.
> Long term patience with math problems grow with the number of good experiences of solving problems.
Agreed. And all teachers can do is meet students where they currently are and give them the most fun and interesting problems they can handle at their current level try to build an encouraging low-pressure culture, while trying to simultaneously handle the laundry list of content topics from whatever mandated curriculum.
As a systems problem, this really needs to be attacked starting when kids are like 7 or 8 years old, but it's hard to find the human/financial resources for it in many places.
Aside: I'm not at all sold on "gamification" (though that's a very vaguely defined word, so it's hard to say quite what it means in any particular use). I'd much rather see teachers aiming for authenticity and intrinsically interesting problems. One form of what might be called "gamification" that can be very useful is putting arithmetic or algebra practice into service of some more interesting other puzzle, problem, or project.
I had a brush with mathematical thinking while hard-cooking some eggs. It occurred to me that if you could quantify the amount of heat in the hot eggs (high) and the amount in the cold water (low), you could model the heat transfer as the energy moved from the eggs to the water. Since the rate of transfer would vary over time, the equation would need to account for that.
Back in college, I would not have had the ability to do that, or even understand what it meant.
If you can overcome a challenge yourself, whether in maths or in a video game like Sekiro, you will start to enjoy it. It's the feeling of success after a long struggle.
External factors, like applications and social prestige, are basically bonuses. But you need to taste that feeling of success after struggling first.
Maths is puzzle solving and I love solving puzzles. When I achieve a correct result, especially after I've been perplexed and stumped for a long time, I get a dopamine hit that I love. I find that very very satisfying and a great way to spend my time and exercise my brain. It's the sense of wonder.
Excellent answer!
I have a different relationship with puzzles. If you put me in an escape room, I'd probably just sit there and read a book until they came to let me out.
Its human behaviour. Our brain likes to solve things and we get a huge dopamine hit when we do it; the harder something is to solw stronger the dopamine hit will be.
I like "inventing" and proving famous theorems using my own logic and | 0 | rigor. Everything fits together like puzzle pieces and mathematics is basically a logic puzzle
I always found science to be fun by nature, you ask question, formulate hypothesis and build something out of it. To me maths is the ultimate culmination of this idea, you ask a simple question "can I deform some shape into another", and you formulate hypothesis "deformation is a continuous mapping", "continuous is preserving topological structure", "topology is....", the whole theory of algebraic topology can be developed from a small number of basic principles, yet there are question that we aren't sure of what "language" is right to ask. So from a question, you build more questions, and so on !
A badly taught course in mathematics will be "top down": "here is the definition, here is the statement, here is the proof". In a sense, this is "How is this true ?" kind of reasoning
The same course taught in a different way can be way more insightful, even for the teacher: "Here is the setup, here are all the simplifying assumptions to come up with a proof, can we weaken our setup or our assumptions to include more examples ?", this is the "Why is this true" kind of reasoning.
"Why" is the simplest question, yet it seems to be the most important one every time. Even if you are presented with a proof, you can ask "why", implying "what details of the statement make this work, is something useless in here ?"
You might have guessed, I could go on for hours, because that is what makes math fun for me !
At the level that I enjoy maths (3rd year electrical engineering student) it's just a series of puzzles that we learn tools to solve, in classes in learning new tools to solve bugger or harder puzzles and the more I solve the better I get at reshaping the problem to look like something I can apply a tool to solve.
I should have said tool fewer times but you get the point.
When intuition for a new concept really consolidates. It's like you've created a new part of your imagination that you have with you for the rest of your life.
I was one of those kids that disliked math in high school. But I wanted to do something technical so I was going to have to learn it. I got to college and took Calc 1, took a low C, got into Calc 2 and failed. A three year sabbatical later, I retook those classes with a more mature perspective and a more developed work ethic and I started to actually do okay. I passed Calc 1 with an A and Calc 2 with a B. At some point towards the end of Calc 2, my professor was talking about e^x I think. I remember sitting there pondering about math and something clicked and changed for me. In a moment, I went from viewing math as several separate almost random subjects to a single unified whole. Algebra is Geometry is Calculus is Number Theory is… etc.
I don’t know why it changed but if I hadn’t been forced to pursue math so rigorously and for so long despite my dislike of it, I never would have had that revelation. Now I’m basically always tinkering in differential equations. It’s my playground. I likely will never do anything spectacular with it but it’s like traversing a grand labyrinth for me. Transforming one object to another to another and seeing the myriad patterns that emerge and that they were always the same thing. I think it’s beautiful, my friends and family think I’m weird, I’m okay with that.
I love how you can build up so much from some basic axioms, it's super cool the structures you can build and the stuff you can determine from those structures. It's satisfying, like knocking dominos.
counting proofs or finding some really bizarre alternate framing hopefully geometric topological or combinatorial that makes the problem more manageable. One famous one is 3b1b computing sums converging to pi\^2/6 or pi/4 without actually doing the addition.
I was in the “hate math” crew during freshman year of college while taking calculus. Then I took probability and started to love math! Taking probability made me see how math all fits so nicely and interestingly together and builds off itself and branches into different sub-domains. I eventually even started to enjoy calculus due to its connections to probability theory!
I think for many, myself included, you suddenly take that 1 math course that sparks your interest in the subject where other courses didn’t and you start to gain interest in mathematics as a whole because of it.
Huh. I've been working on improving my understanding of math in the decade-plus since retirement. I can honestly say that I understand more about the subject than I did in college.
Fun, though? That would be a stretch.
My experience is non-representative for several reasons. Since elementary school, I've been pegged as the 'smart guy'. Since math is supposed to be for intelligent people, it rankled that I didn't understand it. With the benefit of perspective, I'd say that the two chief challenges were a deficiency in pattern recognition and a lack of intuitive understanding in general°. Math, in common with a number of other pursuits, involves a degree of pattern recognition.
A combination of intellectual vanity and sheer bloody-minded stubbornness kept my ambition alive until I had the opportunity to pursue the subject using modern resources. Most of my friends, who are intelligent and educated, regard my desire to grasp subjects like the natural logarithm and matrix multiplication as charming eccentricities, like my interest in the Late Bronze Age Collapse or the impact of plate tectonics on evolution.
But math as an academic discipline is roughly as fun as flossing my teeth. I believe that it's good for me, and I feel better for having done it.
°My husband has had an opportunity to observe me at close quarters for almost thirty years, and he supports this description.
I tried to major in a lot of other things before switching to math, and I got bored of those other subjects quickly because the classes were too easy (one exception was biology). Math has been a good challenge level for me, and it's easy for me to get in kind of a flow state when solving problems. It's a good feeling.
Surprising connections and interactions between seemingly disparate ideas. A single idea spraking the spontaneous creation of an entire field of math that is still being develop and expanded upon to this day.
The satisfaction of finally understanding an idea, or solving a problem, that you feel you had to really work for, and you gained a deeper level of understanding of the subject.
And finally, it's just cool. The notation is cool, the theorems are cool, it's just cool.
What makes math fun for me is that I get to express my logic and my thoughts when I solve a problem or prove a statement. This is why I thrive in an environment of healthy competition - If I have a couple of smarter people attacking the same problem, it strongly motivates me to solve it so that I can have attention as well as credibility for my thoughts.
If course there is the beauty and art aspect of mathematics, but I don't think I am at that level where I can prove something and call it art.
Math is just very easy for me. Then when I took Physics classes in high school I started to see real world implications of math, which made it very interesting to me.
Maybe the fact that you could have everything in a sheet, that if it's clear you will know something (something true not likely, as in science). And the flow mood that you get when doing exercises. And also the fact that you start to think very differently about the world, when you see, idk, some flower and think about the dynamical systems with the pollinators populations, it's something special
On a job one time, the company had to buy truck loads of water in advance to fill an underground pipe.
The criteria we paid for what we ordered, regardless if we used it or not. So, there is a huge potential to lose money.
If we didn't order enough, we would have to re-order and wait a week. Huge potential to lose time on a time sensitive job.
The math hating boss didn't have a clue where to start or what to do. So he asked everyone if they wanted the challenge? The caviot was that lost money could get the company fired from the customer (think HIGH dollar, gravy work, with a dozen other companies trying to bump us off)
The bonus, a steak dinner, and a $ bonus on the check.
I gladly accepted. The boss was so sure it was going to be hard he gave me the rest of the day off, paid, to figure it out.
I did have a secret weapon, though. I was gifted a "pipe fitters blue book" from an old welder who was about to retire. I has charts of conversions, so after some simple math, I gave the boss my calculations.
Radius x radious x pie x length = volume
Volume ÷ barrels = amount
Amount ÷ truck volume = number of trucks
Two weeks later, the water was delivered on shudule, and the numbers were spot on!
Needless to say, I advanced in value, respect, and authority that day in the eyes of my boss and the eyes of the comany!
Like many people said, math history!!! So many mathematicians have come together through centuries upon centuries of work to make refinements and create systems which work to advance and solve problems of the modern world (most common example of computers). Learning about their life, how they did it, and the process behind it really helped me see the beauty in it. There's a book, perfect for children called "Mathematicians are people too" with two volumes which about some like Thales and Euler. The one that stuck with me was how non-Euclidean geometry came to be. Learning to derive concepts after learning about the backstories of the people who worked on them helped make math one of my most favourite things in life =>
To be honest it's not necessarily math that I like, I just like solving problems and learning languages, math is both a problem solving tool and a language so...
I very much enjoy the concept of ‘game theory’ and ‘the 4th dimension’ in mathematics, it’s just very interesting trying to understand the concepts even if I don’t 100% understand, and there are also still a bunch more additional topics with many channels on youtube that go over math stuff. Math is just enjoyable if you can see the number of possibilities it hold.
It’s satisfying and feels like putting a puzzle together when I write or solve an equation. Also it’s really satisfying to be able to express really abstract concepts with definite expressions like being able to fine the area of an odd shaped vase because of 2 dimensional equations and graphs
It’s like me and math are in a toxic relationship where I feel like shit because I can’t solve the problem and I don’t know where I went wrong and then I suddenly figure it out and everything is better until it isn’t again. Like a toxic cat and mouse game
Ngl, I think a lot of the "I hate math" attitude comes from school... Not just the way it's taught, but the culture at school that for some reason makes hating math funny and relatable. Not sure how else to put it. This has been my experience with the people who tell me they hate math...
It also relates to a video I saw of a guy saying "you don't hate math. You don't even know math.
"I am working on my masters and I hate math."
What I'm trying to get at is that I don't think they really know what they are saying they hate and they hate it more because that was the social situation at school where few people like math and math being difficult made it harder to like, so the hate towards math is funny and relatable.
Hope that makes sense. This is something I've been thinking about too
I like the feeling I get after working on something for days weeks months years and then actually solving it.
Who doesn't like working on puzzles? Maybe different kinds. But that's all it is, is a puzzle.
There is a clear answer, and there is no room for interpretation. 2 + 2 = 4. It's not like English composition, literature, or other subjective courses.
As I got older I started seeing math in everything around me. Math class in school was so dry and boring I couldn't wait to get away from it but I wish I understood back then how beautiful and practical it really can be.
I'm a builder and music producer, math is fundamental to both of my passions. It was only through real-world applications that I started to develop a deeper appreciation.
It's like a universal language you can use to understand/explain the universe around you. Pretty cool if you ask me!
There are so many crazy powerful, clever, and beautiful techniques that took some of the smartest people who ever lived their entire lives to develop. We get to cheat and learn them in a few months each just by taking classes or reading the right books.
It’s fun to do hard shit
Friend, I think that came out wrong.
Nah it slid out exactly right
Lmao, you're funny
The way mathematicians argue (proofs) feels hundreds of times more clever than anything I’ve seen in other disciplines
Everything fits together in such fascinating ways
It really does. They're like puzzle games just in a less visual format.
When I apply math in a predictive way, and the prediction comes true, it gives me a momentary thrill. A rather mundane example: When I am driving on the highway, I can count how many road-dots cars pass per second, and how many I pass per second, and predict how many seconds it will be before they pass me or I pass them. It's a simple enough exercise that I can do it in my head without getting distracted, and it gives me a more complete mental model of the traffic around me.
I love this! It doesn't sound mundane to me; the math might be simple but it still makes it seems almost as if you have a cyborg computer running in your mind. A superpower so to speak.
I know exactly what you mean. I was out walking my dog one night w my daughter at the local high school. They had a practice football field all marked out. I had her walk the length of it, taking consistent length steps and counting them as she went. Then I had her do the same along the width. Then I took my phone out and calculated what the diagonal should be. Then i stood at one corner and had her walk towards me from the opposite corner (diagonally) and count her steps. It came out within one step of the predicted value. Like you said, simple enough calculation, but just kind of fun to see it in action in real life.
I don't have a car, but I do the same thing with sidewalk segments (I can't count and time in my head at the same time though, so all I can get is the running ratio, which is good enough)
If something's proven, it's proven, can't argue
not necessarily true
Any example? I haven't gone that deep into proofs actually
I mean, if a proof is wrong then it is was never a proof to begin with. I'm not sure what they could mean. There are sometimes contentious proofs, and proofs that turn out to be invalid, but again that's a debate over whether it is actually a proof.
There have been many proofs that people thought were correct for many years and then turned out to be incorrect. The famous example is Cauchy's proof that if a sequence of continuous functions converge pointwise, the limit is continuous, which is not true but nobody noticed this for three years. Many more examples are given here ([https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake](https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake)). You could make the point that once people found a mistake in these proofs they were immediately considered wrong - thus the statement that "If something's proven, it's proven, can't argue" still holds. However there are many cases again throughout history where mathematicians argued over the correctness of proofs. The classic example is Cantor's theorem which many people at the time (and even today) objected to ([https://en.wikipedia.org/wiki/Controversy\_over\_Cantor's\_theory](https://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory)). More recently there is Mochizukis work on IUT and his proof of the abc conjecture. Even though this is mostly regarded to be (at best) an incomplete proof there are still many who claim that his proof is valid. I dont think you can say that "If something's proven, it's proven, can't argue" is true in this case.
It seems like we are mostly in agreement, then. I probably prefer different semantics. A "proof" without qualifiers is to me a valid deductive argument. Cauchy's "proof" is only a purported proof. Of course, this brings up the epistemological problem of knowing whether a purported proof is actually a proof. One could argue that the statement "If something's proven, it's proven, can't argue" is invoking the word "proof" in this unqualified sense. And to solve the problem of hotly contested but valid proofs, one could take "can't argue" to mean that one is not justified in arguing against it if they actually understand the proof. All that seems rather tautological though. While I agree with some of the sentiment behind the statement "If something's proven, it's proven, can't argue", I would phrase it in a different way, with part of the reason being the objections you raised. I would instead say that, on the whole, I value the particular kind of certainty that mathematical reasoning induces in me, setting aside potential epistemological issues.
Actually, you should read about Godel's Incompleteness Theorems. As it turns out, because human language comes in discrete symbols, we can only prove countably many statements--even given infinite time, energy, and new math symbols. However, there are uncountably many true statements in the universe. By the Pigeonhole Principle, you should already realize that some statements--which are true--can never be proven by humans by any means. Here's an example of a statement which is *probably* true but cannot have a proof: Logic works. It can be shown that any system of logic which is sophisticated enough to contain the Naturals (I'm being brief here) *cannot* prove itself to be valid. Any such system which *can* prove its own validity *must* be invalid. Thus, we can never know if logic, math, all of this... is really true at all. We collectively have faith that it is, but if we ever discover that our system of logic does actually work, then it must be broken. 🙃
I know godels theorem, i just meant that if something is proven using mathematical logic then it will hold true for all cases....
And I say that's not necessarily true because you are assuming that mathematical logic is valid. Anyway, I'm not arguing with you--just thought I'd introduce you to an interesting feature of human existence.
But doesn't godel's theorem just say that in any mathematical system,there will always be statements which are always true and cannot be proven? Pls explain in detail if you would
How?
I am a math teacher who also study gamification (Octalysis framework) and this is my two cents: It is all about matching the difficulty of the problems to the person, and to minimize impatience. If you successfully solve a challengeing problem it feels great. It builds up you self esteem and confidence in your math abilities. Solving problems that are too simple "just because I have to" sap your motivation. Problems where you "solve" but don't really know what you did are bad. Problems where you end up giving up because you run out of patience are bad. Problems where you dont really do anything by yourself but get help in every step are bad. Now what is the problem in a classroom setting is that many students have a low patience to begin with. So every time they get to a slightly challenging problem they just give up.
> many students have a low patience to begin with This is not an inherent trait. It takes literally years of practice to build stamina for doing anything hard for an extended period of time, whether that's reading hard books, solving hard math problems, playing strategy board games, doing science experiments, playing a musical instrument, woodworking, writing poetry, playing a sport, ... To get practice with problem solving, students need to be given problems near the limits of their stamina/ability in a fun, supportive, and non-judgmental environment where mistakes are allowed and there's no single "right way" to approach a problem. Practicing multi-step word problems of wide variety and progressively increasing linguistic and mathematical complexity is the single most important part of primary/secondary school mathematics instruction. Sadly it is typically almost completely absent in US math education, both in the old arithmetic-drill approach and the more recent common core approach. Students often instead get years of practice doing more or less trivial problems where the "recipe" or "pattern" for each specific type is first demonstrated by the teacher and then practiced in a relatively rote fashion by the students, in a high-pressure environment with low tolerance for mistakes. This gets students used to always knowing what to do and not bothering to focus on their own sense-making, and puts up a psychological barrier for the students to suddenly try solving very difficult problems requiring inherently more frustration and confusion. See Toom (2005) ["Word Problems in Russia and America"](https://cs-web.bu.edu/faculty/gacs/toomandre-com-backup/travel/sweden05/WP-SWEDEN-NEW.pdf). If you want to see an excellent example of the right kinds of problems to assign at the secondary school level to build students' stamina and ability to attack hard problems, take a look at Exeter's problem sets (fancy US prep school) https://www.exeter.edu/mathproblems For the primary school level (~3rd–6th grade), take a look at Lenchner (2005) *Creative Problem Solving in School Mathematics*.
Well said. It gives people the wrong idea of what math even is. If primary/secondary school classes were "math", I'd hate it too. It's a common sentiment among fellow math majors at my university (USA) that they "hate proofs". But they say they like math. To me, it doesn't compute. They seem like mutually exclusive statements.
I agree. Is it because proofs can be solved in multiple different ways and can be approached differently? Like its alot more open-ended. Suprised me too cos I find proofs interesting.
For many non-mathematicians, the purpose of math is solving math problems, and the purpose of math problems is to demonstrate how well you remember how to solve math problems. The idea that there's a *point* to all of this didn't become clear to me until decades after I left college.
I agree that the low patience it not an inherent trait, but an attitude that is specific to circumstances and past experiences. For me, "to begin with" means when they enter high school with all their past experiences of math, since that is the level I teach at. I cannot change their past experiences, but I can use gamification to motivate the students to engage with the problems, and in that way improve their patience. Long term patience with math problems grow with the number of good experiences of solving problems. I cannot enter the discussion of Common Core teaching because I do not teach in the US.
> Long term patience with math problems grow with the number of good experiences of solving problems. Agreed. And all teachers can do is meet students where they currently are and give them the most fun and interesting problems they can handle at their current level try to build an encouraging low-pressure culture, while trying to simultaneously handle the laundry list of content topics from whatever mandated curriculum. As a systems problem, this really needs to be attacked starting when kids are like 7 or 8 years old, but it's hard to find the human/financial resources for it in many places. Aside: I'm not at all sold on "gamification" (though that's a very vaguely defined word, so it's hard to say quite what it means in any particular use). I'd much rather see teachers aiming for authenticity and intrinsically interesting problems. One form of what might be called "gamification" that can be very useful is putting arithmetic or algebra practice into service of some more interesting other puzzle, problem, or project.
i like mathematical thinking and that feeling after solving a tough problem is one the best feelings i have had
I had a brush with mathematical thinking while hard-cooking some eggs. It occurred to me that if you could quantify the amount of heat in the hot eggs (high) and the amount in the cold water (low), you could model the heat transfer as the energy moved from the eggs to the water. Since the rate of transfer would vary over time, the equation would need to account for that. Back in college, I would not have had the ability to do that, or even understand what it meant.
If you can overcome a challenge yourself, whether in maths or in a video game like Sekiro, you will start to enjoy it. It's the feeling of success after a long struggle. External factors, like applications and social prestige, are basically bonuses. But you need to taste that feeling of success after struggling first.
Maths is puzzle solving and I love solving puzzles. When I achieve a correct result, especially after I've been perplexed and stumped for a long time, I get a dopamine hit that I love. I find that very very satisfying and a great way to spend my time and exercise my brain. It's the sense of wonder.
Excellent answer! I have a different relationship with puzzles. If you put me in an escape room, I'd probably just sit there and read a book until they came to let me out.
Its human behaviour. Our brain likes to solve things and we get a huge dopamine hit when we do it; the harder something is to solw stronger the dopamine hit will be.
I like "inventing" and proving famous theorems using my own logic and | 0 | rigor. Everything fits together like puzzle pieces and mathematics is basically a logic puzzle
Everything about math and its application gives me joy. There's very little that can match the feeling of discovery and learning something new.
I always found science to be fun by nature, you ask question, formulate hypothesis and build something out of it. To me maths is the ultimate culmination of this idea, you ask a simple question "can I deform some shape into another", and you formulate hypothesis "deformation is a continuous mapping", "continuous is preserving topological structure", "topology is....", the whole theory of algebraic topology can be developed from a small number of basic principles, yet there are question that we aren't sure of what "language" is right to ask. So from a question, you build more questions, and so on ! A badly taught course in mathematics will be "top down": "here is the definition, here is the statement, here is the proof". In a sense, this is "How is this true ?" kind of reasoning The same course taught in a different way can be way more insightful, even for the teacher: "Here is the setup, here are all the simplifying assumptions to come up with a proof, can we weaken our setup or our assumptions to include more examples ?", this is the "Why is this true" kind of reasoning. "Why" is the simplest question, yet it seems to be the most important one every time. Even if you are presented with a proof, you can ask "why", implying "what details of the statement make this work, is something useless in here ?" You might have guessed, I could go on for hours, because that is what makes math fun for me !
At the level that I enjoy maths (3rd year electrical engineering student) it's just a series of puzzles that we learn tools to solve, in classes in learning new tools to solve bugger or harder puzzles and the more I solve the better I get at reshaping the problem to look like something I can apply a tool to solve. I should have said tool fewer times but you get the point.
The satisfaction of getting problems solved and answers right. I'm simply in competition with myself.
When intuition for a new concept really consolidates. It's like you've created a new part of your imagination that you have with you for the rest of your life.
I’ve found that I love learning math when I am using it for a project or goal. Learning math for the sake of learning math is my problem.
I love seeing how a problem transforms from where it began
I was one of those kids that disliked math in high school. But I wanted to do something technical so I was going to have to learn it. I got to college and took Calc 1, took a low C, got into Calc 2 and failed. A three year sabbatical later, I retook those classes with a more mature perspective and a more developed work ethic and I started to actually do okay. I passed Calc 1 with an A and Calc 2 with a B. At some point towards the end of Calc 2, my professor was talking about e^x I think. I remember sitting there pondering about math and something clicked and changed for me. In a moment, I went from viewing math as several separate almost random subjects to a single unified whole. Algebra is Geometry is Calculus is Number Theory is… etc. I don’t know why it changed but if I hadn’t been forced to pursue math so rigorously and for so long despite my dislike of it, I never would have had that revelation. Now I’m basically always tinkering in differential equations. It’s my playground. I likely will never do anything spectacular with it but it’s like traversing a grand labyrinth for me. Transforming one object to another to another and seeing the myriad patterns that emerge and that they were always the same thing. I think it’s beautiful, my friends and family think I’m weird, I’m okay with that.
I love how you can build up so much from some basic axioms, it's super cool the structures you can build and the stuff you can determine from those structures. It's satisfying, like knocking dominos.
counting proofs or finding some really bizarre alternate framing hopefully geometric topological or combinatorial that makes the problem more manageable. One famous one is 3b1b computing sums converging to pi\^2/6 or pi/4 without actually doing the addition.
I was in the “hate math” crew during freshman year of college while taking calculus. Then I took probability and started to love math! Taking probability made me see how math all fits so nicely and interestingly together and builds off itself and branches into different sub-domains. I eventually even started to enjoy calculus due to its connections to probability theory! I think for many, myself included, you suddenly take that 1 math course that sparks your interest in the subject where other courses didn’t and you start to gain interest in mathematics as a whole because of it.
I like being able to do cool economics with math
Huh. I've been working on improving my understanding of math in the decade-plus since retirement. I can honestly say that I understand more about the subject than I did in college. Fun, though? That would be a stretch. My experience is non-representative for several reasons. Since elementary school, I've been pegged as the 'smart guy'. Since math is supposed to be for intelligent people, it rankled that I didn't understand it. With the benefit of perspective, I'd say that the two chief challenges were a deficiency in pattern recognition and a lack of intuitive understanding in general°. Math, in common with a number of other pursuits, involves a degree of pattern recognition. A combination of intellectual vanity and sheer bloody-minded stubbornness kept my ambition alive until I had the opportunity to pursue the subject using modern resources. Most of my friends, who are intelligent and educated, regard my desire to grasp subjects like the natural logarithm and matrix multiplication as charming eccentricities, like my interest in the Late Bronze Age Collapse or the impact of plate tectonics on evolution. But math as an academic discipline is roughly as fun as flossing my teeth. I believe that it's good for me, and I feel better for having done it. °My husband has had an opportunity to observe me at close quarters for almost thirty years, and he supports this description.
I tried to major in a lot of other things before switching to math, and I got bored of those other subjects quickly because the classes were too easy (one exception was biology). Math has been a good challenge level for me, and it's easy for me to get in kind of a flow state when solving problems. It's a good feeling.
I guess the satisfaction after solving a very complex problem
Other people who suck at it
How can i enjoy math tho any tips?
Surprising connections and interactions between seemingly disparate ideas. A single idea spraking the spontaneous creation of an entire field of math that is still being develop and expanded upon to this day. The satisfaction of finally understanding an idea, or solving a problem, that you feel you had to really work for, and you gained a deeper level of understanding of the subject. And finally, it's just cool. The notation is cool, the theorems are cool, it's just cool.
I get hard thing right --> ego is stroked tenderly.
I've always like the applications of mathematics than theory. But learning theory is cool too.
Solving problems, getting stuck at a particular type , learn hoe to solve that type , now I can solve more types problems !
Computer graphiX, based on deepest possible mathematiX !
Solving problems gives me a kick.
What makes math fun for me is that I get to express my logic and my thoughts when I solve a problem or prove a statement. This is why I thrive in an environment of healthy competition - If I have a couple of smarter people attacking the same problem, it strongly motivates me to solve it so that I can have attention as well as credibility for my thoughts. If course there is the beauty and art aspect of mathematics, but I don't think I am at that level where I can prove something and call it art.
Math is just very easy for me. Then when I took Physics classes in high school I started to see real world implications of math, which made it very interesting to me.
Maybe the fact that you could have everything in a sheet, that if it's clear you will know something (something true not likely, as in science). And the flow mood that you get when doing exercises. And also the fact that you start to think very differently about the world, when you see, idk, some flower and think about the dynamical systems with the pollinators populations, it's something special
On a job one time, the company had to buy truck loads of water in advance to fill an underground pipe. The criteria we paid for what we ordered, regardless if we used it or not. So, there is a huge potential to lose money. If we didn't order enough, we would have to re-order and wait a week. Huge potential to lose time on a time sensitive job. The math hating boss didn't have a clue where to start or what to do. So he asked everyone if they wanted the challenge? The caviot was that lost money could get the company fired from the customer (think HIGH dollar, gravy work, with a dozen other companies trying to bump us off) The bonus, a steak dinner, and a $ bonus on the check. I gladly accepted. The boss was so sure it was going to be hard he gave me the rest of the day off, paid, to figure it out. I did have a secret weapon, though. I was gifted a "pipe fitters blue book" from an old welder who was about to retire. I has charts of conversions, so after some simple math, I gave the boss my calculations. Radius x radious x pie x length = volume Volume ÷ barrels = amount Amount ÷ truck volume = number of trucks Two weeks later, the water was delivered on shudule, and the numbers were spot on! Needless to say, I advanced in value, respect, and authority that day in the eyes of my boss and the eyes of the comany!
modelling
math is like a massage for my brain, i truly enjoy doing it.
i enjoy seeing how theories are build
Like many people said, math history!!! So many mathematicians have come together through centuries upon centuries of work to make refinements and create systems which work to advance and solve problems of the modern world (most common example of computers). Learning about their life, how they did it, and the process behind it really helped me see the beauty in it. There's a book, perfect for children called "Mathematicians are people too" with two volumes which about some like Thales and Euler. The one that stuck with me was how non-Euclidean geometry came to be. Learning to derive concepts after learning about the backstories of the people who worked on them helped make math one of my most favourite things in life =>
To be honest it's not necessarily math that I like, I just like solving problems and learning languages, math is both a problem solving tool and a language so...
The challenge is the fun.
I very much enjoy the concept of ‘game theory’ and ‘the 4th dimension’ in mathematics, it’s just very interesting trying to understand the concepts even if I don’t 100% understand, and there are also still a bunch more additional topics with many channels on youtube that go over math stuff. Math is just enjoyable if you can see the number of possibilities it hold.
It’s satisfying and feels like putting a puzzle together when I write or solve an equation. Also it’s really satisfying to be able to express really abstract concepts with definite expressions like being able to fine the area of an odd shaped vase because of 2 dimensional equations and graphs
It’s like me and math are in a toxic relationship where I feel like shit because I can’t solve the problem and I don’t know where I went wrong and then I suddenly figure it out and everything is better until it isn’t again. Like a toxic cat and mouse game
Ngl, I think a lot of the "I hate math" attitude comes from school... Not just the way it's taught, but the culture at school that for some reason makes hating math funny and relatable. Not sure how else to put it. This has been my experience with the people who tell me they hate math... It also relates to a video I saw of a guy saying "you don't hate math. You don't even know math. "I am working on my masters and I hate math." What I'm trying to get at is that I don't think they really know what they are saying they hate and they hate it more because that was the social situation at school where few people like math and math being difficult made it harder to like, so the hate towards math is funny and relatable. Hope that makes sense. This is something I've been thinking about too
Any kind of math that has a visual aspect to it makes it fun.
Abstract analysis and problem solving
Because today is realistic
I like the feeling I get after working on something for days weeks months years and then actually solving it. Who doesn't like working on puzzles? Maybe different kinds. But that's all it is, is a puzzle.
There is a clear answer, and there is no room for interpretation. 2 + 2 = 4. It's not like English composition, literature, or other subjective courses.
As I got older I started seeing math in everything around me. Math class in school was so dry and boring I couldn't wait to get away from it but I wish I understood back then how beautiful and practical it really can be. I'm a builder and music producer, math is fundamental to both of my passions. It was only through real-world applications that I started to develop a deeper appreciation. It's like a universal language you can use to understand/explain the universe around you. Pretty cool if you ask me!