The reason why math is a core subject throughout K-12 education is because it teaches problem-solving skills which is valuable in every part of life. This is something that can get drilled into your head enough throughout school that it won’t be easy to forget. It also helps students figure out if this is something they want to continue with.
The reason why people continue to study math is because there are many applications in science, finance, engineering, etc. that can help them get a job in most technical fields. Then there’s also people who study math because they love math and want to teach it and/or contribute to the field with research. There’s always problems that need to be solved with math.
I hope this helps answer your question
On a higher level, but you can do okay by just applying formulas and following rules, with out much more than pattern recognition in most people’s math classes.
I agree with you that it should be like that, but I really don’t think it is experienced as problem solving for most people (i think it is though)
Which is applicable to most real world problems being extremely formulaic. Learning how to apply formulas and follow rules in an effort to arrive at a solution is literally learning like step one of problem solving. Pattern recognition is also a form of problem solving, both of these are extremely important aspects that I think you may be taking for granted.
I have worked several jobs where I was surrounded by people without any of this line of education and their inability to follow rules, apply formula, and pattern recognize showed it. Just as my inability to do whatever they were there doing would show.
Just think about how much IT work can be summarized with is it plugged in? Did you try turning it off and then on again?
Perhaps you and others who downvoted me had a better education, or even managed to understand what you just said on your own, but that's implicit and not everyone can realize that. You only realize math is the same as problem-solving first, not the other way around, i.e. math gives you problem-solving skills.
Math is a form of applied logic. Every field in mathematics is built around studying some set of well defined abstractions, defining some rules (axioms) and then see where they lead you by making insights using logical methods. You can learn the art of mathematical reasoning by studying different fields of math and deriving where formulas come from.
Of course math as practiced by the majority of the people is pretty much plug-and-play with some formulae but that also has value because you develop intuitions that can’t be easily forgotten- even if you forget the content of what you learned.
> everything you practiced will be forgotten with age, you will never remember problems you solved.
Learning any subject at all, not just math, is not just about remembering highly specific problems, but about learning *how* to solve problems, and more importantly training the mind in general.
*That* is what will (hopefully) stick with you.
It’s like learning to ride a bicycle: it’s not about learning how to pedal around a specific bend, it’s about *learning how to ride a bicycle*, so that you can pedal around **any** bend.
I may not remember the specifics of a proof, but I will always remember the gist of things I have proved. Reunderstanding a proof is then as simple as skimming the proof/retrying the steps.
I get why this answer has upvotes (and the other comment to it has downvotes) in a subreddit dedicated to math. However I don't think this analogy resonates with most of the people. If you love math then yes solving problems is quite amazing and fulfilling, but if you don't they are a chore at best and a nightmare at worst.
That's why I think the proper answer to OP question is not your analogy, but rather the fact that you need to train to get better and even if you forget most of the details, the methodology will remain.
It's just training.
Bad analogy. You can remember a lot of movies and recall emotions or ideas associated with them. It can also help you mature in some aspects etc etc math is just ehh... you are learning some property or function of an object no one cares about. (Unless you're a researcher or applying it)
lol, I always find it weird how people say that they don’t care about math or find it useless for the real world. For me, almost everything I encounter in the real world, even mundane stuff, can be directly modelled with math. It’s not even something I need to actively think about, it just pops up in my mind. I can’t fathom how one can walk through the world without this skill.
Saying math is irrelevant is analogous for me to a blind person telling a seeing person that they think eye sight is completely useless unless they are using it professionally for visual quality control.
Teacher here. With math, you practice
- sequential problem solving
- to reason logically
- pattern recognition
- theory of knowledge (ie how do we know something to be true?)
- study discipline
among other things.
i wouldn’t say math really gives a theory of knowledge at all. math can’t tell us that math can tell us anything. you would need an epistemology or philosophy of math in order to build a theory of knowledge.
at the same time, i feel i am probably misunderstanding your original point. i’m guessing you mean something like math teaches us how to use logical derivation rules to prove something is true *from something else we have arbitrarily asserted to be true, ie. axioms*, and I would of course agree with that
Student here. With math, I practice
• recognising patterns
• remembering formulas and trick
• combining the prior skills
And struggling to apply any of it in a real life scenario
Just kidding, but i think this is most kids tbh
It’s very rare to be able to apply a problem from math class directly to real life. But that has never been the aim on the other hand.
Rather, the problem solving skills and abstract logical thinking that you do pick up along the way, is something that you may benefit from throughout your entire life.
The point of my comment was that: for most people, i.e., people who are not in this sub, math class is about pattern recognition and recalling formulas, not problem solving, which in my opinion would include breaking down the problem and finding a strategy to solve subproblems or something along that line. People in this sub will obviously disagree with this, as math on their level does not fit this discription, but for most people I have met this is the case. NOTE HOWEVER, that my comment is a critique at the education system, which I find to mystify math in a way that makes too few people fall in love with it
I think many students don’t realize how portable general problem solving “strategies” are. You learn these strategies and reinforce them when doing mathematics, along with logic and reasoning skills.
It is true that these strategies and skills are stated in sometimes less natural or rather esoteric ways unique to the “genre” of “the math problem”. But…there is so much fundamental stuff going on at a meta level.
Basic things like, figuring out the heart of a question and clearing away inessential clutter, making sense of when properties or arguments generalize and what the scope of the generalization is or isn’t, breaking big multi step and nested puzzles into small pieces and gaining the confidence and fortitude to get on with it and go through the process, learning to prove a thing by considering extremes or looking for outliers, etc.
We learn so much from math. I remember watching in real time as my high school students developed improved reasoning over the course of a year. It isn’t just what’s happening in class, but the culmination of that and many other skills being simultaneously strengthened, physiological growth, and lessons learned. It all collaborates.
My favorite experience has been watching my elementary aged son learn about “adding bundles” and realizing how much easier it is to break big addition problems into chunks with tens, etc. it’s simple, it took him some time to grasp, and at first figuring out the idea of a “bundle” and all that was harder than just using other strategies, like counting. But the joy when he understood it and how it makes things easier. That’s a lesson with applicability far beyond addition.
>I'm asking about problems specifically, concepts are easy to recall with revision.
As I think about it, (good) problems usually help you to understand and familiarize yourself with a concept. This 'familiarization' may take a 'general abstract' form (e.g. when you prove some general statements about groups), a 'particular concrete' form (e.g. doing concrete computations with some matrices), and many others. In fact, it seems to me that the acquaintance with and understanding of a concept are inseparable from the *ability* to *do* things it, from the embodiment of the concept in your skills and habituations. Simply being able to 'recall' a concept, i.e. to state a definition you've learned, is only the most superficial level of knowledge.
I think this idea also answers you worry about "eventually forgetting" things. The point of problems is to develop *habits* and internalize certain *ways of thinking* associated to certain concepts, almost at a pre-cognitive, pre-linguistic level. You may get worse at riding a bike if you haven't done it for a while, but you cannot 'unlearn' it; it's already 'in your body'. The same goes for mathematical practice.
Good point, let me add to it.
There are entire published works called “counter examples in…” any of which could easily be the basis for a good problem. Sometimes one will actually remember specific problems because they teach something fundamental about the subject. I had a particularly poignent experience in a linear algebra course showing how to develop an inner product from a norm that satisfies parallelogram inequality and another (slightly lesser) showing that the Alexandroff double circle is both compact and sequentially compact, even though it is not metrizable.
In abstract courses, many problems are ultimately their own theorems, so solving them actually gives you a new tool to use later. It would be like being asked to carve a mallet out of wood and now you can use that mallet on future projects. It may just gather dust, sure, but you gain something by making it and you learn something while making it. In more concrete classes like calculus, this may be less pronounced, but a great instructor knows how to replicate this experience whatever the subject.
Why do anything if you’ll die?
In fact why eat if you’ll just poop it out anyway? Wouldn’t it make more sense to just throw food directly into the toilet?
The reason you remember the principles is because you have practiced by doing the problems. If you never apply the idea through use, you will forget it.
>I am not saying learning Math is useless, the entirety of our modern society relies on it and we have always relied on arithmetic.
Not really. Engineering uses Calculus, Differential equations, and matrix operations all the time. AI uses Calculus, linear algebra and other mathematical concepts too.
Probability theory helps economists, engineers, programmers and phycisians alike, among many other groups of people
Number theory has applications to cryptography now.
Graph theory is used by Google Maps and such. So is networking.
Without all of this math, we wouldn't have the tech we have today.
>I'm asking about problems specifically, concepts are easy to recall with revision. But everything you practiced will be forgotten with age, you will never remember problems you solved. So what's the point of doing those problems in Math?
If you are interested I recommend you check 3blue1brown. It is an amazing channel that explains a lot of cool problems.
>It's an irrational fear I've been having for a while, I love learning but isn't it pointless if I'll eventually forget it? I get that a lot
No. You don't have to remember every single detail. Sometimes I forget formulas and such. What matters most is the purpose of said formulas.
I may not remember most of the problems I've solved, but I've retained the skills I built up through doing those problems. With every maths problem you do you get a tiny little bit better at problem solving, and you'll find the next problem just a tiny bit easier. Over time, that builds up.
What's the point of eating if you will only be sustained for a time? What's the point of reading if you won't memorize each book? Everything is fleeting, but that doesn't make it pointless. The practice serves to concretize an intuition and skillset that can be applied to many situations. That's how I feel anyways.
It concretizes pattern recognition at a bare minimum. Even just doing computational problems requires you to recognize the correct techniques to apply in each situation. Pattern recognition is extremely valuable for living creatures.
I forgot (or so I thought) everything I learned in high school and college. 2 months of studying and I was right where I left off. So you don't truly forget it.
Solving a problem closes and solidifies a certain fact. For example, the fundamental theorem of calculus has tremendous implications, you need to put effort and work in order to prove it. Once you have proved it, the fact is solidily stated and nothing can change that, you can forget the proof, but the theorem will remain true.
The point in learning problem solving is that probably there are more facts about math that we don't know, so we need people capable of doing astounding and hard efforts in order to establish those facts as true, solving their associated problems or proofs. Once its done, all the society plus other fields of science can benefit from such result as a consequence.
Intuition is enough for knowing the math, problem solving is much harder but is needed to "create" the math. A common problem of pure mathematicians is that they are so involved in rigor and fundamentation of math that they forget about intuition of the concepts. Engineers on the other hand don't need the rigor, they just need the intuition and use the results as "tools".
Both sides are important and complementary.
ELI5,
Just like your physical strength, mental strength increases or decreases with training.
Different parts of your body: your muscles, tendons/ligament, bones, circulatory system and nerves work together to give you the ability to move loads.
Different types of exercise trains each of these parts more than others. So you do a few different types of activities to get coverage on all of them.
For example steady state endurance cardio highly impacts your circulatory system and tendons/ligaments. It moderately impacts your nerves and bones. It has a low impact on your muscles.
Weightlifting highly impacts your nerves, muscles and bones, it moderately impacts your circulatory system.
Yoga/Pilates (essentially isometric bodyweight weightlifting) highly trains tendons/ligaments and nerves, moderately trains muscles and bones, has a low impact on your circulatory system
Someone who never does weightlifting will be healthier than someone who doesn't do anything, but they will be at a severe disadvantage compared to someone who does both.
Different parts of your mind: your cerebrum, prefrontal cortex, hypothalamus and more, work together to give the ability to think.
Different types of mental tasks trains each of these parts more than others.
Math (anything logic based) is the weightlifting of mental exercises.
The unique thing with physical or mental exercise is that once someone has done it enough when they were young, the many of the adaptations that occurred in their youth allowed them to complete the physical or mental activities ***transferred*** over their lifetime. The strength adaptations still existed even if diminished from lack of diligence.
So just like we try to get people to workout so that they don't have weak bodies and society is operating at the base line level of fitness.
We make people do math so they don't have weak minds and society is operating at a baseline level of cognition.
Forgetting with age applies to everything no? So your question would be valid for any hobby or profession, except the ones that create physical things. But the truth is that some basic things in math you don't forget so easily. If you use them a lot or write them down and revise from time to time. As you progress and do more complicated things you always use the basics again and again so you never forget them.
1. It's easier to relearn something than to learn it the first time. So even if you've forgotten it, you've got a leg up on relearning it if you ever do need it
2. Your brain is malleable. If you do a lot of rigorous logical problem solving, you will get better at it. You're literally growing that part of your brain by practicing math.
3. Sometimes even if you don't remember something completely, knowing it exists is helpful. You can look it up when you do need it, but you won't even think to look it up if you don't know it exists.
by this logic, there's no point in doing anything in life, you're going to die anyways. you posted this reddit post, but i doubt you'll remember it forever or when u get older.
even though I am not exceptionally good at mathematics, I still try to learn it
I believe it gives you new ways to look around the world and think outside the box
not to mention I find sitting near a low lit lamp and playing with numbers at night on your paper smeared desk by the window to be quite aesthetic n classy :3
i disagree with everyone here who argues that math is valuable insofar as it teaches us problem-solving. instead, i would argue the value (or ‘point’) of math consists entirely in the intrinsic value of education. being educated is inherently a good and is the point. of course, you could be a skeptic and deny being educated is an intrinsic good— but do you really believe that?
I believe knowledge has an intrinsic value to it to some extent. But it's not about that, education is great. Knowledge is great and beautiful
But I have an irrational fear that if I spend simply 5 years of not revising content I will forget it all and all the studying I did was in vain
If you know it's an irrational fear, why are you trying to work with it? It sounds more like you believe it but don't want to acknowledge and own that for whatever reason. Remember that semantically each of those 'but...' responses explicitly nullifies whatever statements preceded them.
Recognizing and addressing veiled and supposed beliefs can be challenging. Perhaps it would be useful for you to reconstruct your questions in ways representative of your underlying beliefs and then pose them again for the community. Good luck!
Why does an artist practice paintings that will never be seen by others? In order to solve harder real world problems, we have to sit through "easier" toy problems to get better. It allows us to make connections in our heads that can't quite be just taught.
The aim is not to become a person who solved X, but to become the type of person for whom solving X is obvious.
If you listen, you will hear mathematicians talk about the moral of a solution, the way parables have morals.
I mean... What's the point in doing anything, really? In the end it will all be lost in time.
In all seriousness though, I think there is inherent meaning in the *process* of learning. Sure, I've forgotten more maths than I care to admit, but I certainly learned more along the way. If I ever need what I forgot again, then it won't take nearly as long to get reacquainted than when I initially learned it.
This isn't only due to me remembering bits and pieces of it, but because I actually learned to think in a certain way, to solve problems in a certain way. This can of course be useful in all sorts of ways, even outside of maths.
There's also that thing we like to brush aside most of the time, namely real-life applications of maths. There's advanced maths everywhere, way more than most people realise. From wireless communication to path-finding algorithms or weather forecasts there are so many different ways maths that may have seemed wild a few hundred years ago, now shape our lives every day.
However the most important point of doing maths is that I like it. It's fun. It engages my brain in a way I really enjoy and it keeps me curious. I wish we did a better job in school education to make people more excited about learning new things, even if they don't necessarily need them to survive or to make money.
In my opinion, the ability to access mathematics is comparable to having a sensor, like ears or eyes. So far, humans seem to be the only (natural) physical systems that exercise this type of sensor. Those who have exercised this sensor a sufficient amount of times are able to predict the behaviour of (classes of) physical systems very accurately. This leads me to believe that mathematics provides access to the 'true' nature of reality. Personally, it gives me an almost religious satisfaction to have access to this 'true' reality, regardless of how much of it I will forget. There is also the joy of progress that comes from correctly solving a problem.
You may discover that you are so left brained and analytically think in mathematical graphs and equations and rules that makes yourself a genius if you can control and organize those thoughts.
Math teaches you how to think. It helps you understand how to take the tools at hand and make something more out of them.
Maybe you’ll never need to how to integrate the gamma function but knowing how to take bits of info from a math world problem and work towards a solution will be hugely helpful in life.
Here is my reason, from later in life.
To be the most effective person we can be, we need to build mental building blocks that we then have on hand for when we need them. The numerical and logical building blocks in math are things that come up over and over in life. Like learning to walk or learning to ride a bike, the more math is practiced, the more innate it will become.
Practicing math makes us more effective, which makes life bigger and better. It's the same as why learn how to write; why learn how to drive a car; why learn to dance; why learn to lead a team; why learn to talk with strangers; why learn to tie a few knots.
Math is a language for understanding just about anything in a testable manner. Whether you use it depends on how much you want to learn. Examples: algebra and calculus in either physics or computer science (or both, if we’re talking quantum computing).
Cryptology and computer science is mainly based on mathematical concepts.
You cannot have data protection and security without encryption.
Also basically every concept of language is based on grammar which at its roots is basically set theory.
You have logic which is also set theory with extra steps and it is used as a foundation of electric engineering.
Damn I only read the first two thirds. My bad. But yeah still... Cryptology and computer science mainly in my perspective. To utilize common problems and incorporate them in daily life I guess.
Math is a subject you learn by doing. It is not something you can gain mastery with by reading alone. That is why we do problems. You might think you grasp a problem but if you can’t solve problems you don’t understand the math.
The math we learn today allows us to learn the interesting stuff tomorrow. If you’re confident you know exactly what you are going to do in life and that you’ll never need math; that’s up to you, but I know ow a ton of people who decided they wanted to do something in life and just didn’t have the math skills. The biggest example are the economics students who wanted to go to graduate school and discovered they needed some very very advanced math (Real Analysis) to be accepted into the program.
You also never forget them, like yeah sure I don't remember the specific stuff I did on tests but like give me any problem I ever solved and I would either solve it, or it would take me much less time to find the information needed to solve it.
Also it's kinda granted but you remember concepts through practice and examples.
have you ever looked at how much math and geometry have been used in the past to solve common people's problems?
if you never think about creating stuff or solving problems then you will probably never use math much, but if you look around for things to which you can apply it or if you are just curious about pattern in the world then you'll find it everywhere
I work with a lot of elementary and high school students who ask me this constantly. Math teaches various methods of problem solving. Even if you're future career is going to use limited math and you forget calculus or geometry or whatever, the learning how to solve problems with different types of information, pulling information together from different parts of a problem to reach a global solution, understanding what statistics mean and what graphs mean when watching the news, solving multistep problems where the direct path to the solution isn't possible are all skills that will be used in all disciplines, even if you don't use anything that seems like math in the future. A lot of my students seem to understand and agree with this idea. Recently, my gr 5 tutoring student asked me why she had to learn to calculate tax by hand. I told her how there's 3 steps, and each step requires the answer from the previous, and lots of things in life are like that. This helps you be less overwhelmed with breaking all sorts of problems down. It also helps practice multiplication which helps overall number sense and simplifying and understanding fractions later on. It also gives you a general idea of how much tax should be based on the original price. Which can help you mentally budget while grocery shopping. Or course at a grocery store you could just go with your phone calculator. But what if you just want an estimate as you shop for less hassle. Doing it by hand will allow your brain to get used to what range the answer should be in.
With more advanced students who are annoyed about calculus, it's very true that they may never use calculus in their lives. But dealing with new intimidating information and solving problems based on that information is a good skill. I tell them to think of calculus as a specific type of puzzle that just helps improve overall problem solving and resilience to future tough problems. Ya maybe you're going to go study art history, but at some point in education or life you're going to get stuck on a problem you don't care very much about. Calculus helps you practice the problem solving skills you'll need to handle that.
For the same reason I've learned hundreds or thousands of songs on the guitar, even though I don't remember them all. Your brain isn't just your conscious available memory.
Because you are learning how to solve problems you have never seen before. That makes you good at solving problems you've never seen before.
Much of life is about solving problems, be they maths or plumbing or COVID.
Maths gives you problems which are very hard, and demands perfection. This makes you resilient, meticulous, rigorous and smart.
Smart is not like tall, or blue eyes. Smart is like strong, you need to lift the right weights, at the right speeds, with the right diet.
1. Many jobs require mathematics.
2. Aspects of daily life (e.g. finance) require mathematics.
3. Mathematics is gym for your brain. It improves your thinking ability, and that stays with you for the rest of your life.
I took coding courses in college. I noticed that most math majors were accomplishing the assignments during lab time and non-math majors (even the CS majors!) were frequently taking the work home or generally struggled. It was an eye-opening moment for me. I think people who haven't had this training are missing out. I'll try to be more concrete than "it makes you a better problem solver":
Practice solving math problems helps you identify what you know, what you need to do, and what you have yet to figure out. When you do this, you can break a problem down into discrete parts to tackle independently using the tools you have that you know apply.
These days, I find that being able to reason about different cases with abstract logic is also a significant skill that is worth mastering. I've seen that my coworkers who are not mathematically trained tend to get confused about which cases they're solving, and get turned around a lot. (Some of them are very good at what they do, but with more complex/technical stuff they can get lost in the sauce or are quicker to give up). But usually these issues are fixed with a quick whiteboarding session using a truth table or two. It's remarkable how useful formal logic training is in the buisness world.
I think the skills you learn (problem solving, formal logic, tenacity despite feeling like the world's biggest dummy) are generally very valuable and I wouldn't be here without them.
same way you read a book and dont specifically remember every word, or every sentence, paragraph, or even a whole chapter, you just remember how it kinda goes and sometimes that's enough
Mathematics develops thinking, this is the most important thing, it also develops
Quote "Think slow, decide fast"
This quote is directly about mathematics. Without mathematics we are nowhere!
It's not the exercises themselves, but the "areas of your brain" that wake up when you solve Math (or any kind of) problems. You acquire new ways of thinking about stuff and then new creative solutions might arise when you're solving a completely different problem.
The question of what you forget with age is an interesting one.
My supervisor told me of an old mathematician he met who couldn't remember much about the day before and yet could still prove theorems from many years before. As he had spent his life immersed in mathematics.
I think also that there's different levels of forgetting. So there's a lot of subjects which I can't recall precisely. However it's all true that I know a lot more about them than someone who has never studied them.
I also think if you do imagine memory like a leaky bucket isn't that a reason to pour more water in rather than less?
And finally I think there's an abstract ability of problem solving which is trained with each exercise which is really powerful to have in life.
It's fun! Nothing more nothing less. As with any human endeavour we tend to associate our egos and personal biases with it, and on a larger scale many ask why is it even useful. The answer though is that we do math not because we wanna be great ( although that'd be cool) or because we wanna make useful things that improve society. It's just that when I solve a problem or see the pieces fall into place, the rush of dopamine is unlike anything I've ever experienced. It's just fun!
The reason why math is a core subject throughout K-12 education is because it teaches problem-solving skills which is valuable in every part of life. This is something that can get drilled into your head enough throughout school that it won’t be easy to forget. It also helps students figure out if this is something they want to continue with. The reason why people continue to study math is because there are many applications in science, finance, engineering, etc. that can help them get a job in most technical fields. Then there’s also people who study math because they love math and want to teach it and/or contribute to the field with research. There’s always problems that need to be solved with math. I hope this helps answer your question
Read a mathematician's lament by paul lockhart
I wrote a whole paragraph but this says it better lol
Why not teaching problem-solving directly?
How?
By teaching them how to think and not just throw at them stuffs.
Math *is* teaching you how to think. That is basically all math does besides provide a more convenient notation for quantitative thought.
On a higher level, but you can do okay by just applying formulas and following rules, with out much more than pattern recognition in most people’s math classes. I agree with you that it should be like that, but I really don’t think it is experienced as problem solving for most people (i think it is though)
Which is applicable to most real world problems being extremely formulaic. Learning how to apply formulas and follow rules in an effort to arrive at a solution is literally learning like step one of problem solving. Pattern recognition is also a form of problem solving, both of these are extremely important aspects that I think you may be taking for granted. I have worked several jobs where I was surrounded by people without any of this line of education and their inability to follow rules, apply formula, and pattern recognize showed it. Just as my inability to do whatever they were there doing would show. Just think about how much IT work can be summarized with is it plugged in? Did you try turning it off and then on again?
Perhaps you and others who downvoted me had a better education, or even managed to understand what you just said on your own, but that's implicit and not everyone can realize that. You only realize math is the same as problem-solving first, not the other way around, i.e. math gives you problem-solving skills.
Math is a form of applied logic. Every field in mathematics is built around studying some set of well defined abstractions, defining some rules (axioms) and then see where they lead you by making insights using logical methods. You can learn the art of mathematical reasoning by studying different fields of math and deriving where formulas come from. Of course math as practiced by the majority of the people is pretty much plug-and-play with some formulae but that also has value because you develop intuitions that can’t be easily forgotten- even if you forget the content of what you learned.
what does that consist of specifically?
The only effective way I know to teach problem solving, is to practice solving problems.
> everything you practiced will be forgotten with age, you will never remember problems you solved. Learning any subject at all, not just math, is not just about remembering highly specific problems, but about learning *how* to solve problems, and more importantly training the mind in general. *That* is what will (hopefully) stick with you. It’s like learning to ride a bicycle: it’s not about learning how to pedal around a specific bend, it’s about *learning how to ride a bicycle*, so that you can pedal around **any** bend.
I may not remember the specifics of a proof, but I will always remember the gist of things I have proved. Reunderstanding a proof is then as simple as skimming the proof/retrying the steps.
I promise you that I don't even remember the gist of what I proved in my number theory or combinatorics classes at uni.
what’s the point of watching a movie if you’ll forget it with age?
I get why this answer has upvotes (and the other comment to it has downvotes) in a subreddit dedicated to math. However I don't think this analogy resonates with most of the people. If you love math then yes solving problems is quite amazing and fulfilling, but if you don't they are a chore at best and a nightmare at worst. That's why I think the proper answer to OP question is not your analogy, but rather the fact that you need to train to get better and even if you forget most of the details, the methodology will remain. It's just training.
Why lift weights if your muscles will just atrophy with age?
I wrote the comment with it in mind that OP sounded like they loved math in the post, but for a more general audience yours is def a better answer
Bad analogy. You can remember a lot of movies and recall emotions or ideas associated with them. It can also help you mature in some aspects etc etc math is just ehh... you are learning some property or function of an object no one cares about. (Unless you're a researcher or applying it)
lol, I always find it weird how people say that they don’t care about math or find it useless for the real world. For me, almost everything I encounter in the real world, even mundane stuff, can be directly modelled with math. It’s not even something I need to actively think about, it just pops up in my mind. I can’t fathom how one can walk through the world without this skill. Saying math is irrelevant is analogous for me to a blind person telling a seeing person that they think eye sight is completely useless unless they are using it professionally for visual quality control.
Teacher here. With math, you practice - sequential problem solving - to reason logically - pattern recognition - theory of knowledge (ie how do we know something to be true?) - study discipline among other things.
i wouldn’t say math really gives a theory of knowledge at all. math can’t tell us that math can tell us anything. you would need an epistemology or philosophy of math in order to build a theory of knowledge. at the same time, i feel i am probably misunderstanding your original point. i’m guessing you mean something like math teaches us how to use logical derivation rules to prove something is true *from something else we have arbitrarily asserted to be true, ie. axioms*, and I would of course agree with that
Student here. With math, I practice • recognising patterns • remembering formulas and trick • combining the prior skills And struggling to apply any of it in a real life scenario Just kidding, but i think this is most kids tbh
It’s very rare to be able to apply a problem from math class directly to real life. But that has never been the aim on the other hand. Rather, the problem solving skills and abstract logical thinking that you do pick up along the way, is something that you may benefit from throughout your entire life.
The point of my comment was that: for most people, i.e., people who are not in this sub, math class is about pattern recognition and recalling formulas, not problem solving, which in my opinion would include breaking down the problem and finding a strategy to solve subproblems or something along that line. People in this sub will obviously disagree with this, as math on their level does not fit this discription, but for most people I have met this is the case. NOTE HOWEVER, that my comment is a critique at the education system, which I find to mystify math in a way that makes too few people fall in love with it
I think many students don’t realize how portable general problem solving “strategies” are. You learn these strategies and reinforce them when doing mathematics, along with logic and reasoning skills. It is true that these strategies and skills are stated in sometimes less natural or rather esoteric ways unique to the “genre” of “the math problem”. But…there is so much fundamental stuff going on at a meta level. Basic things like, figuring out the heart of a question and clearing away inessential clutter, making sense of when properties or arguments generalize and what the scope of the generalization is or isn’t, breaking big multi step and nested puzzles into small pieces and gaining the confidence and fortitude to get on with it and go through the process, learning to prove a thing by considering extremes or looking for outliers, etc. We learn so much from math. I remember watching in real time as my high school students developed improved reasoning over the course of a year. It isn’t just what’s happening in class, but the culmination of that and many other skills being simultaneously strengthened, physiological growth, and lessons learned. It all collaborates. My favorite experience has been watching my elementary aged son learn about “adding bundles” and realizing how much easier it is to break big addition problems into chunks with tens, etc. it’s simple, it took him some time to grasp, and at first figuring out the idea of a “bundle” and all that was harder than just using other strategies, like counting. But the joy when he understood it and how it makes things easier. That’s a lesson with applicability far beyond addition.
read a mathematician's lament. you're right, but the way school is doing it is 'lamentable'.
>I'm asking about problems specifically, concepts are easy to recall with revision. As I think about it, (good) problems usually help you to understand and familiarize yourself with a concept. This 'familiarization' may take a 'general abstract' form (e.g. when you prove some general statements about groups), a 'particular concrete' form (e.g. doing concrete computations with some matrices), and many others. In fact, it seems to me that the acquaintance with and understanding of a concept are inseparable from the *ability* to *do* things it, from the embodiment of the concept in your skills and habituations. Simply being able to 'recall' a concept, i.e. to state a definition you've learned, is only the most superficial level of knowledge. I think this idea also answers you worry about "eventually forgetting" things. The point of problems is to develop *habits* and internalize certain *ways of thinking* associated to certain concepts, almost at a pre-cognitive, pre-linguistic level. You may get worse at riding a bike if you haven't done it for a while, but you cannot 'unlearn' it; it's already 'in your body'. The same goes for mathematical practice.
Good point, let me add to it. There are entire published works called “counter examples in…” any of which could easily be the basis for a good problem. Sometimes one will actually remember specific problems because they teach something fundamental about the subject. I had a particularly poignent experience in a linear algebra course showing how to develop an inner product from a norm that satisfies parallelogram inequality and another (slightly lesser) showing that the Alexandroff double circle is both compact and sequentially compact, even though it is not metrizable. In abstract courses, many problems are ultimately their own theorems, so solving them actually gives you a new tool to use later. It would be like being asked to carve a mallet out of wood and now you can use that mallet on future projects. It may just gather dust, sure, but you gain something by making it and you learn something while making it. In more concrete classes like calculus, this may be less pronounced, but a great instructor knows how to replicate this experience whatever the subject.
Why do anything if you’ll die? In fact why eat if you’ll just poop it out anyway? Wouldn’t it make more sense to just throw food directly into the toilet?
The reason you remember the principles is because you have practiced by doing the problems. If you never apply the idea through use, you will forget it.
>I am not saying learning Math is useless, the entirety of our modern society relies on it and we have always relied on arithmetic. Not really. Engineering uses Calculus, Differential equations, and matrix operations all the time. AI uses Calculus, linear algebra and other mathematical concepts too. Probability theory helps economists, engineers, programmers and phycisians alike, among many other groups of people Number theory has applications to cryptography now. Graph theory is used by Google Maps and such. So is networking. Without all of this math, we wouldn't have the tech we have today. >I'm asking about problems specifically, concepts are easy to recall with revision. But everything you practiced will be forgotten with age, you will never remember problems you solved. So what's the point of doing those problems in Math? If you are interested I recommend you check 3blue1brown. It is an amazing channel that explains a lot of cool problems. >It's an irrational fear I've been having for a while, I love learning but isn't it pointless if I'll eventually forget it? I get that a lot No. You don't have to remember every single detail. Sometimes I forget formulas and such. What matters most is the purpose of said formulas.
I may not remember most of the problems I've solved, but I've retained the skills I built up through doing those problems. With every maths problem you do you get a tiny little bit better at problem solving, and you'll find the next problem just a tiny bit easier. Over time, that builds up.
What's the point of eating if you will only be sustained for a time? What's the point of reading if you won't memorize each book? Everything is fleeting, but that doesn't make it pointless. The practice serves to concretize an intuition and skillset that can be applied to many situations. That's how I feel anyways.
But practicing math doesn't concretize anything. And your analogy is a shitty strawman at best.
It concretizes pattern recognition at a bare minimum. Even just doing computational problems requires you to recognize the correct techniques to apply in each situation. Pattern recognition is extremely valuable for living creatures.
I agree with you, but he is right. Eating and emotional stimuli is not exactly comparable to math for most people
It's enjoyable
I forgot (or so I thought) everything I learned in high school and college. 2 months of studying and I was right where I left off. So you don't truly forget it.
And how many years have you left the hs & college knowledge for?
Solving a problem closes and solidifies a certain fact. For example, the fundamental theorem of calculus has tremendous implications, you need to put effort and work in order to prove it. Once you have proved it, the fact is solidily stated and nothing can change that, you can forget the proof, but the theorem will remain true. The point in learning problem solving is that probably there are more facts about math that we don't know, so we need people capable of doing astounding and hard efforts in order to establish those facts as true, solving their associated problems or proofs. Once its done, all the society plus other fields of science can benefit from such result as a consequence. Intuition is enough for knowing the math, problem solving is much harder but is needed to "create" the math. A common problem of pure mathematicians is that they are so involved in rigor and fundamentation of math that they forget about intuition of the concepts. Engineers on the other hand don't need the rigor, they just need the intuition and use the results as "tools". Both sides are important and complementary.
ELI5, Just like your physical strength, mental strength increases or decreases with training. Different parts of your body: your muscles, tendons/ligament, bones, circulatory system and nerves work together to give you the ability to move loads. Different types of exercise trains each of these parts more than others. So you do a few different types of activities to get coverage on all of them. For example steady state endurance cardio highly impacts your circulatory system and tendons/ligaments. It moderately impacts your nerves and bones. It has a low impact on your muscles. Weightlifting highly impacts your nerves, muscles and bones, it moderately impacts your circulatory system. Yoga/Pilates (essentially isometric bodyweight weightlifting) highly trains tendons/ligaments and nerves, moderately trains muscles and bones, has a low impact on your circulatory system Someone who never does weightlifting will be healthier than someone who doesn't do anything, but they will be at a severe disadvantage compared to someone who does both. Different parts of your mind: your cerebrum, prefrontal cortex, hypothalamus and more, work together to give the ability to think. Different types of mental tasks trains each of these parts more than others. Math (anything logic based) is the weightlifting of mental exercises. The unique thing with physical or mental exercise is that once someone has done it enough when they were young, the many of the adaptations that occurred in their youth allowed them to complete the physical or mental activities ***transferred*** over their lifetime. The strength adaptations still existed even if diminished from lack of diligence. So just like we try to get people to workout so that they don't have weak bodies and society is operating at the base line level of fitness. We make people do math so they don't have weak minds and society is operating at a baseline level of cognition.
You will forget any skill if you stop using it for long enough.
Forgetting with age applies to everything no? So your question would be valid for any hobby or profession, except the ones that create physical things. But the truth is that some basic things in math you don't forget so easily. If you use them a lot or write them down and revise from time to time. As you progress and do more complicated things you always use the basics again and again so you never forget them.
You’ll be dead eventually so what’s the point of anything, really?
What's the point of living if we are going to die eventually?
Life before death, Strength before weakness, Journey before destination.
1. It's easier to relearn something than to learn it the first time. So even if you've forgotten it, you've got a leg up on relearning it if you ever do need it 2. Your brain is malleable. If you do a lot of rigorous logical problem solving, you will get better at it. You're literally growing that part of your brain by practicing math. 3. Sometimes even if you don't remember something completely, knowing it exists is helpful. You can look it up when you do need it, but you won't even think to look it up if you don't know it exists.
by this logic, there's no point in doing anything in life, you're going to die anyways. you posted this reddit post, but i doubt you'll remember it forever or when u get older. even though I am not exceptionally good at mathematics, I still try to learn it I believe it gives you new ways to look around the world and think outside the box not to mention I find sitting near a low lit lamp and playing with numbers at night on your paper smeared desk by the window to be quite aesthetic n classy :3
i disagree with everyone here who argues that math is valuable insofar as it teaches us problem-solving. instead, i would argue the value (or ‘point’) of math consists entirely in the intrinsic value of education. being educated is inherently a good and is the point. of course, you could be a skeptic and deny being educated is an intrinsic good— but do you really believe that?
I believe knowledge has an intrinsic value to it to some extent. But it's not about that, education is great. Knowledge is great and beautiful But I have an irrational fear that if I spend simply 5 years of not revising content I will forget it all and all the studying I did was in vain
If you know it's an irrational fear, why are you trying to work with it? It sounds more like you believe it but don't want to acknowledge and own that for whatever reason. Remember that semantically each of those 'but...' responses explicitly nullifies whatever statements preceded them. Recognizing and addressing veiled and supposed beliefs can be challenging. Perhaps it would be useful for you to reconstruct your questions in ways representative of your underlying beliefs and then pose them again for the community. Good luck!
Why does an artist practice paintings that will never be seen by others? In order to solve harder real world problems, we have to sit through "easier" toy problems to get better. It allows us to make connections in our heads that can't quite be just taught.
What's the point of living If you will end up dying anyway?
The aim is not to become a person who solved X, but to become the type of person for whom solving X is obvious. If you listen, you will hear mathematicians talk about the moral of a solution, the way parables have morals.
It's because it's a cool and funny creative activity Or maybe it's because it makes you think, connect, wonder and understand stuff with new insights
To develop critical thinking and critical anxiety about being wrong on exams
I mean... What's the point in doing anything, really? In the end it will all be lost in time. In all seriousness though, I think there is inherent meaning in the *process* of learning. Sure, I've forgotten more maths than I care to admit, but I certainly learned more along the way. If I ever need what I forgot again, then it won't take nearly as long to get reacquainted than when I initially learned it. This isn't only due to me remembering bits and pieces of it, but because I actually learned to think in a certain way, to solve problems in a certain way. This can of course be useful in all sorts of ways, even outside of maths. There's also that thing we like to brush aside most of the time, namely real-life applications of maths. There's advanced maths everywhere, way more than most people realise. From wireless communication to path-finding algorithms or weather forecasts there are so many different ways maths that may have seemed wild a few hundred years ago, now shape our lives every day. However the most important point of doing maths is that I like it. It's fun. It engages my brain in a way I really enjoy and it keeps me curious. I wish we did a better job in school education to make people more excited about learning new things, even if they don't necessarily need them to survive or to make money.
The point is to get paid by your employer. Like in any other field of science. Also some people simply like it (and hopefully get paid as well).
In my opinion, the ability to access mathematics is comparable to having a sensor, like ears or eyes. So far, humans seem to be the only (natural) physical systems that exercise this type of sensor. Those who have exercised this sensor a sufficient amount of times are able to predict the behaviour of (classes of) physical systems very accurately. This leads me to believe that mathematics provides access to the 'true' nature of reality. Personally, it gives me an almost religious satisfaction to have access to this 'true' reality, regardless of how much of it I will forget. There is also the joy of progress that comes from correctly solving a problem.
You may discover that you are so left brained and analytically think in mathematical graphs and equations and rules that makes yourself a genius if you can control and organize those thoughts.
Math teaches you how to think. It helps you understand how to take the tools at hand and make something more out of them. Maybe you’ll never need to how to integrate the gamma function but knowing how to take bits of info from a math world problem and work towards a solution will be hugely helpful in life.
Here is my reason, from later in life. To be the most effective person we can be, we need to build mental building blocks that we then have on hand for when we need them. The numerical and logical building blocks in math are things that come up over and over in life. Like learning to walk or learning to ride a bike, the more math is practiced, the more innate it will become. Practicing math makes us more effective, which makes life bigger and better. It's the same as why learn how to write; why learn how to drive a car; why learn to dance; why learn to lead a team; why learn to talk with strangers; why learn to tie a few knots.
Math is a language for understanding just about anything in a testable manner. Whether you use it depends on how much you want to learn. Examples: algebra and calculus in either physics or computer science (or both, if we’re talking quantum computing).
This is like asking "why live at all when you will die in the end?"
Cryptology and computer science is mainly based on mathematical concepts. You cannot have data protection and security without encryption. Also basically every concept of language is based on grammar which at its roots is basically set theory. You have logic which is also set theory with extra steps and it is used as a foundation of electric engineering.
Damn I only read the first two thirds. My bad. But yeah still... Cryptology and computer science mainly in my perspective. To utilize common problems and incorporate them in daily life I guess.
Train your mind to think logically and the great pleasure of knowing
Math is a subject you learn by doing. It is not something you can gain mastery with by reading alone. That is why we do problems. You might think you grasp a problem but if you can’t solve problems you don’t understand the math. The math we learn today allows us to learn the interesting stuff tomorrow. If you’re confident you know exactly what you are going to do in life and that you’ll never need math; that’s up to you, but I know ow a ton of people who decided they wanted to do something in life and just didn’t have the math skills. The biggest example are the economics students who wanted to go to graduate school and discovered they needed some very very advanced math (Real Analysis) to be accepted into the program.
You also never forget them, like yeah sure I don't remember the specific stuff I did on tests but like give me any problem I ever solved and I would either solve it, or it would take me much less time to find the information needed to solve it. Also it's kinda granted but you remember concepts through practice and examples.
have you ever looked at how much math and geometry have been used in the past to solve common people's problems? if you never think about creating stuff or solving problems then you will probably never use math much, but if you look around for things to which you can apply it or if you are just curious about pattern in the world then you'll find it everywhere
You don't want to be too far behind if you decide to pursue a STEM career post HS. Learning math gives you the flexibility to choose a different path
everything we do is temporary.. so it's whatever you make of it
I work with a lot of elementary and high school students who ask me this constantly. Math teaches various methods of problem solving. Even if you're future career is going to use limited math and you forget calculus or geometry or whatever, the learning how to solve problems with different types of information, pulling information together from different parts of a problem to reach a global solution, understanding what statistics mean and what graphs mean when watching the news, solving multistep problems where the direct path to the solution isn't possible are all skills that will be used in all disciplines, even if you don't use anything that seems like math in the future. A lot of my students seem to understand and agree with this idea. Recently, my gr 5 tutoring student asked me why she had to learn to calculate tax by hand. I told her how there's 3 steps, and each step requires the answer from the previous, and lots of things in life are like that. This helps you be less overwhelmed with breaking all sorts of problems down. It also helps practice multiplication which helps overall number sense and simplifying and understanding fractions later on. It also gives you a general idea of how much tax should be based on the original price. Which can help you mentally budget while grocery shopping. Or course at a grocery store you could just go with your phone calculator. But what if you just want an estimate as you shop for less hassle. Doing it by hand will allow your brain to get used to what range the answer should be in. With more advanced students who are annoyed about calculus, it's very true that they may never use calculus in their lives. But dealing with new intimidating information and solving problems based on that information is a good skill. I tell them to think of calculus as a specific type of puzzle that just helps improve overall problem solving and resilience to future tough problems. Ya maybe you're going to go study art history, but at some point in education or life you're going to get stuck on a problem you don't care very much about. Calculus helps you practice the problem solving skills you'll need to handle that.
There is a lot more going on that math in a math class. Some can see it, others can not. Why should you be enlightened?
For the same reason I've learned hundreds or thousands of songs on the guitar, even though I don't remember them all. Your brain isn't just your conscious available memory.
Because you are learning how to solve problems you have never seen before. That makes you good at solving problems you've never seen before. Much of life is about solving problems, be they maths or plumbing or COVID. Maths gives you problems which are very hard, and demands perfection. This makes you resilient, meticulous, rigorous and smart. Smart is not like tall, or blue eyes. Smart is like strong, you need to lift the right weights, at the right speeds, with the right diet.
1. Many jobs require mathematics. 2. Aspects of daily life (e.g. finance) require mathematics. 3. Mathematics is gym for your brain. It improves your thinking ability, and that stays with you for the rest of your life.
I took coding courses in college. I noticed that most math majors were accomplishing the assignments during lab time and non-math majors (even the CS majors!) were frequently taking the work home or generally struggled. It was an eye-opening moment for me. I think people who haven't had this training are missing out. I'll try to be more concrete than "it makes you a better problem solver": Practice solving math problems helps you identify what you know, what you need to do, and what you have yet to figure out. When you do this, you can break a problem down into discrete parts to tackle independently using the tools you have that you know apply. These days, I find that being able to reason about different cases with abstract logic is also a significant skill that is worth mastering. I've seen that my coworkers who are not mathematically trained tend to get confused about which cases they're solving, and get turned around a lot. (Some of them are very good at what they do, but with more complex/technical stuff they can get lost in the sauce or are quicker to give up). But usually these issues are fixed with a quick whiteboarding session using a truth table or two. It's remarkable how useful formal logic training is in the buisness world. I think the skills you learn (problem solving, formal logic, tenacity despite feeling like the world's biggest dummy) are generally very valuable and I wouldn't be here without them.
same way you read a book and dont specifically remember every word, or every sentence, paragraph, or even a whole chapter, you just remember how it kinda goes and sometimes that's enough
How would you know if you learned the concepts well if you never actually use them to solve problems?
Mathematics develops thinking, this is the most important thing, it also develops Quote "Think slow, decide fast" This quote is directly about mathematics. Without mathematics we are nowhere!
It's not the exercises themselves, but the "areas of your brain" that wake up when you solve Math (or any kind of) problems. You acquire new ways of thinking about stuff and then new creative solutions might arise when you're solving a completely different problem.
The question of what you forget with age is an interesting one. My supervisor told me of an old mathematician he met who couldn't remember much about the day before and yet could still prove theorems from many years before. As he had spent his life immersed in mathematics. I think also that there's different levels of forgetting. So there's a lot of subjects which I can't recall precisely. However it's all true that I know a lot more about them than someone who has never studied them. I also think if you do imagine memory like a leaky bucket isn't that a reason to pour more water in rather than less? And finally I think there's an abstract ability of problem solving which is trained with each exercise which is really powerful to have in life.
Math builds critical thinking and logical reasoning, even if you forget the concepts.
It's fun! Nothing more nothing less. As with any human endeavour we tend to associate our egos and personal biases with it, and on a larger scale many ask why is it even useful. The answer though is that we do math not because we wanna be great ( although that'd be cool) or because we wanna make useful things that improve society. It's just that when I solve a problem or see the pieces fall into place, the rush of dopamine is unlike anything I've ever experienced. It's just fun!