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Math is incredibly useful; probably everyone here knows that. But it's actually fine that most people don't use it. What we really need is for every student to

by rland 5y ago

Math is incredibly useful; probably everyone here knows that. But it's actually fine that most people don't use it. What we really need is for every student to realize why math is useful -- so that if they want to, they can go learn it so that they can make some kind of technological contribution to the world. The fact is, you need to be good at math to make some new fundamental breakthroughs in technology. Those breakthroughs benefit everyone, even those who don't know any math at all. But you won't really get why that's the case until you know enough about sines and cosines to make sense of modern technological development. The real issue is that we just teach the math, hoping that students will understand on their own that it's fundamental. What we should do instead is try to instill that sense of wonder about the world that inspires people to study math. We don't need every person to know calculus, we just need everyone to understand why calculus is useful -- then they can be inspired to go further. Educators say that they're trying to do this, but I'm sorry "how many meters of fence does Farmer Bob need to keep the sheep inside the pen" is not inspiration -- it's drivel, no better than "find the perimeter given a=5, b=6." If every person understood why math is so useful, we would have many more people who are motivated to work on improving the state of things.

Shared404 5y ago
> it's drivel, no better than "find the perimeter given a=5, b=6." Not only no better, but actually worse - if you add a word problem around the problem that doesn't fundamentally add anything or make the problem easier to understand [0] all you've added is another layer of text to parse through. [0] Sometimes it can be helpful. Adding a relevant layer of physicality to help reason about the problem can be nice, especially in calculus.
- netizen-936824 5y ago
  Adding the extra physical descriptions to calculus actually helped me better understand what the equations meant and represented. Definitely beneficial in some scenarios to some people
  - Shared404 5y ago
    Same here - that experience is why I aded that exception. Until Calculus/Physics though, I don't think there's much added benefit - except maybe an occasional "here's how algebra helps while grocery shopping".
    - netizen-936824 5y ago
      One other thing I find super helpful is understanding exponential growth and decay. That gets used in quite a few places besides just the sciences
    - jacobr1 5y ago
      When I tutored elementary kids, physical examples really helped with fractions.
      - Shared404 5y ago
        That makes sense. Seems reasonable that as you introduce new concepts having a physical equivalent would make it easier.
- Jarwain 5y ago
  I think that act of translation is pretty helpful; it's not often irl math is presented nicely, but rather as a scenario that one has to solve. Translating a scenario into a math problem is practice for that, I think. To try and inspire people, though, maybe word problems could be written to be more relevant
  - after_care 5y ago
    I always found the context in which I’m solving a word problem to be the most critical piece to accurately solving it —- and the reason they aren’t that great. Oh, I’m taking a test it over polynomial expressions —- I already know how I’m going to solve whatever word problems I run across.
- fatnoah 5y ago
  > [0] Sometimes it can be helpful. Adding a relevant layer of physicality to help reason about the problem can be nice, especially in calculus. I think you've hit the nail on the head here. I had tremendous difficulty with Calculus in High School. Everything was introduced via mathematical derivation an proof. I'd gone over halfway through the year with Cs and Ds and, even though it was almost 30 years ago, I remember the moment it clicked. I was sitting the library working on some problem about pools draining and "OMG, it's just about how fast things are changing" popped into my head. That made derivatives, area under a curve, and all of those things make sense to me. Ironically, I ended up being the only person in my class to get a 5 on the AP test and then ended up as a EE major, which was very calculus heavy.
- huetius 5y ago
  This rings true to me. I was only able to get an intuition for calculus when I was able to conceive of it in geometric terms. I later learned that this was how Newton originally derived it. This helped me to “trust” it in more advanced contexts where geometric intuition becomes exceedingly difficult.
paulpauper 5y ago
>What we really need is for every student to realize why math is useful -- so that if they want to, they can go learn it so that they can make some kind of technological contribution to the world. That is what word problems try to instill.
- Broken_Hippo 5y ago
  Poorly. Many word problems I had were't actually reflective of real world, everyday living. You know, the sort of things folks working a minimum wage job might encounter. I find it absolutely amazing that we haven't done this better since it has been an issue for literally decades.
  - arvinsim 5y ago
    I enjoyed math in my Physics classes because of the real life context it is used in.
    - jacobr1 5y ago
      I read about some schools that offer hybrid physics+calculus classes for exactly that reason. Why even bother to separate the subjects. Other schools have experimented with this like project based learning - build a robot or whatever - and then add in tutoring to help close the knowledge gaps that aren't getting picked up. There are plenty of ways to structure learning beyond the Prussian model.
  - unethical_ban 5y ago
    I had an eastern European physics teacher in the 2000s. Our classes had word problems about baseballs thrown from the top of buildings to calculate acceleration/speed. I remember him saying "back home all our kinetic physics problems like this were about artillery". Seemed like some good real world application. Heh.
    - inter_netuser 5y ago
      It is. MIT is the public research arm of the DoD. Most of fundamental physics research is military connected apnea way or another.
hnmm23 5y ago
Any good resources that help instill the "why"?
- eternityforest 5y ago
  I'm working on a reference guide here for exactly that: https://github.com/EternityForest/AnyoneCanDoIt/blob/master/MathELI5.md https://github.com/EternityForest/AnyoneCanDoIt/blob/master/... It's basically "Here's some cool stuff math can do, and interesting historical anecdotes, and how to make a computer do it without actually understanding any of it, and here's what you could do if you actually understood it" There may be errors because... I don't actually understand it and can't derive it myself, so I'm relying entirely on sources. But I think there's value in a text on math by someone with no talent for it, since most books even for beginners assume people have a certain level of ability and treat things i would call very hard as obvious. I don't have any excercises or material on how to actually DO the math, just how to use a CAS solver, I leave the actual teaching of math to those who understand it and focus on showing how it's relevant at all to those of us who don't want to go back and study for two years.
- cknakal 5y ago
  This website has helped me out a lot with understanding the ideas that math represents: https://betterexplained.com/ https://betterexplained.com/
- munk-a 5y ago
  I'd suggest the youtube channel 3Blue1Brown - the teacher that runs the channel likes to pull problems into physical space and applications often.
alexashka 5y ago
> Math is incredibly useful For what? It isn't useful for teenagers who are trying to get laid and be cool, at all. It also isn't useful for what 99% of humans end up doing, unless by math you mean basic arithmetic, enough to add and subtract on your 'smart' phone. Math isn't useful. People who are amazing at math and are interested in it - they can be useful, but mostly not. What they don't teach in school is how the real world works, because it is fucking outrageous. What they teach is how to go be a cog in a machine, run by idiots. They don't teach it by telling you, they teach it by making you sit still, listen and do arbitrary, idiotic tasks for 12 years. Sure they teach you a thing or two along the way, but anyone would learn more from doing anything else they were interested in for 12 years. That's the dirty little secret. If you let people do what they are interested in, they'll learn math, programming, exercise, health, whatever, because it'll be necessary to achieve their goals. But you can't make people who've had a taste of achieving their goals slave away their whole life to help some idiot achieve theirs. Plus most parents are fearful idiots who never treat children like human beings, they treat them like their personal pet project whose goal is to line up with society's idea of success. Pathetic. Oh well, that's people for you :)
- foxes 5y ago
  The way the world and universe actually works seems to be described by maths you probably have no hope of understanding with that attitude.
- waynesonfire 5y ago
  You're an idiot. Math is everywhere. You can be ignorant of it, it won't kill you. But you're going to have a hard time in just about every endeavour I can think of.
  - alexashka 5y ago
    Yup, humans have been struggling for millions of years. Then math finally came to the rescue.
    - waynesonfire 5y ago
_y5hn 5y ago
"Why" won't meet the KPIs and metrics though.
- jrumbut 5y ago
  What I find interesting is that so many schools don't meet the KPIs and metrics anyway despite stripping away most of the educational experience that isn't test prep. If that was yielding great test scores it would be one thing, but even that isn't happening in many states (or any state that I've looked at but I only pay attention to a few).
  - _y5hn 5y ago
    The tests are made to fit a distribution. But test prep is so demotivating and inhumane, much less people are able to cope with it.
technothrasher 5y ago
> What we really need is for every student to realize why math is useful We need to do this for science education as well. The foundation of science that we need the population at large to be educated in is not the steps of the scientific method, how to dissect a pig, or the difference between a covalent and an ionic bond. It is instead logic, critical thinking, rationalism and empiricism, and why science is superior to "common sense". Yet for some reason our primary educational systems tend to focus on the what way more than this why.
- inter_netuser 5y ago
  common sense is just a form of implicit learning. many do a great job at it.
- dekhn 5y ago
  it's funny when you mention the difference between covalent and ionic bonds. my teacher in high school chemistry or physics told us cutting a piece of paper was a physical, not a chemical change (the textbook spent a lot of time differentiating those). It wasn't until many years later that I learned that cutting paper breaks hydrogen bonds. H-bonds are neither covalent nor ionic, but more like in-between (partial sharing of electron density by two different nuclei) but can be modelling (approximately) using ionic models (partial charges separated by a distance). Is breaking an h-bond physical, not chemical? I want to talk to my teacher.
  - hickimsedenolan 5y ago
    Not an expert, but it is a physical chance. H-Bonds are considered as a weak interaction, and making or breaking them (via phase transitions, or in this case breaking H-Bonds) do not change the chemical structure of the compounds or molecules, making them "weak" and "physical," because it is not that hard to do so.
    - hickimsedenolan 5y ago
      change*, not chance
m4thisfunaway 5y ago
The classic "Lockhart's Lament" paper makes a similar point, that math education would benefit from a major shift in pedagogy (PDF): https://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...
geoduck14 5y ago
>Educators say that they're trying to do this, but I'm sorry "how many meters of fence does Farmer Bob need to keep the sheep inside the pen" is not inspiration -- it's drivel As a counter point, I'm planning on adding drainage to my back yard. I actually am measuring how much pipe I need and how much water I can hope to drain out.
albrewer 5y ago
> "how many meters of fence does Farmer Bob need to keep the sheep inside the pen" is not inspiration -- it's drivel I recently had to do a line integral to figure out how much coiled pipe I could fit inside a trench of a given width (trying to lay out geothermal loops in my yard to see if they fit (they don't)). I came up with a parametric formula that approximated the coiled pipe shape and integrated to get a length formula. Advanced math comes up in the weirdest ways.
- hilbert42 5y ago
  "I came up with a parametric formula..." Parametric forumae and equations can be incredibly useful. I remember learning about them at school but we never applied them to any realworld examples. Even at the time I recognized that without practical examples our understanding of them would be problematic. Unfortunately, the problem with most mathematics training - especially so at high school - is the overly theoretical approach used in teaching it. It seems to me that at least in high school we should place much more emphasis on the practical application of mathematics. The theoretical approach is all very well for the 'gifted' and those with a mathematical bent but it's often of little help to many other students. I blame this mainly on those who write the textbooks and set the syllabuses. After all, they're the ones who are good at math so they project their theoretical worldview of mathematics onto everyone else. A classic example of the problem is the dozens of trigonometrical identities that we had to learn at school many of which were of no known practical use to us students at that stage of our training. Sure, at university they came into their own but by then one's already more adept at mathematics so they then start to make sense. The same can happen at university too, I recall the tedium of learning vector algebra and matrixes without them being of any seeming use. At the time, no one bothered to explain how incredibly useful they are when it comes to manipulating Maxwell's equations, etc. It's occurred to me it would be very useful if we had a website (and perhaps also in reference book form) devoted to providing mathematical solutions to thousands of realworld problems of a practical nature. For instance, problems could be grouped alphabetically so they're easy to find. Once located, one would find various solutions to the problems (each solution using a different mathematical technique, etc.). For instance typical entries could be: Interest, Compound Interest, Simple Moreover, the website could be made interesting if we also included additional mathematical information such as short biographies on famous mathematicians together with how they arrived at their discoveries, etc. Of course, the site would be graded so as not to scare off the mathematically timid, that way it would also useful to those who are more advanced in mathematics. I envision it as a short of Wiki for math solutions. This may not be the ideal solution to the problem and it'd be a great deal of work to set up (but like Wiki it could come together with the help of thousands of volunteers). Unless we tackle the problem in a tangential manner - in a different and hopefully more successful approach from the past then I don't think we'll ever solve the problem. BTW, I found parametric equations very interesting from the outset. It was obvious to me how they could overcome an impasse and thus solve some otherwise awkward problems - although I can't say I had such insight uniformly across other branches of mathematics.
huetius 5y ago
I love this post, I would only add that “useful” is not the inspiring criteria it is made out to be, as demonstrated by the example of Farmer Bob’s fence. What needs to be resuscitated is the transcendent, aesthetic dimension.

by rland 5y ago

Shared404 5y ago

> it's drivel, no better than "find the perimeter given a=5, b=6." Not only no better, but actually worse - if you add a word problem around the problem that doesn't fundamentally add anything or make the problem easier to understand [0] all you've added is another layer of text to parse through. [0] Sometimes it can be helpful. Adding a relevant layer of physicality to help reason about the problem can be nice, especially in calculus.

netizen-936824 5y ago
Adding the extra physical descriptions to calculus actually helped me better understand what the equations meant and represented. Definitely beneficial in some scenarios to some people
- Shared404 5y ago
  Same here - that experience is why I aded that exception. Until Calculus/Physics though, I don't think there's much added benefit - except maybe an occasional "here's how algebra helps while grocery shopping".
  - netizen-936824 5y ago
    One other thing I find super helpful is understanding exponential growth and decay. That gets used in quite a few places besides just the sciences
  - jacobr1 5y ago
    When I tutored elementary kids, physical examples really helped with fractions.
    - Shared404 5y ago
      That makes sense. Seems reasonable that as you introduce new concepts having a physical equivalent would make it easier.
Jarwain 5y ago
I think that act of translation is pretty helpful; it's not often irl math is presented nicely, but rather as a scenario that one has to solve. Translating a scenario into a math problem is practice for that, I think. To try and inspire people, though, maybe word problems could be written to be more relevant
- after_care 5y ago
  I always found the context in which I’m solving a word problem to be the most critical piece to accurately solving it —- and the reason they aren’t that great. Oh, I’m taking a test it over polynomial expressions —- I already know how I’m going to solve whatever word problems I run across.
fatnoah 5y ago
> [0] Sometimes it can be helpful. Adding a relevant layer of physicality to help reason about the problem can be nice, especially in calculus. I think you've hit the nail on the head here. I had tremendous difficulty with Calculus in High School. Everything was introduced via mathematical derivation an proof. I'd gone over halfway through the year with Cs and Ds and, even though it was almost 30 years ago, I remember the moment it clicked. I was sitting the library working on some problem about pools draining and "OMG, it's just about how fast things are changing" popped into my head. That made derivatives, area under a curve, and all of those things make sense to me. Ironically, I ended up being the only person in my class to get a 5 on the AP test and then ended up as a EE major, which was very calculus heavy.
huetius 5y ago
This rings true to me. I was only able to get an intuition for calculus when I was able to conceive of it in geometric terms. I later learned that this was how Newton originally derived it. This helped me to “trust” it in more advanced contexts where geometric intuition becomes exceedingly difficult.

paulpauper 5y ago

>What we really need is for every student to realize why math is useful -- so that if they want to, they can go learn it so that they can make some kind of technological contribution to the world. That is what word problems try to instill.

Broken_Hippo 5y ago
Poorly. Many word problems I had were't actually reflective of real world, everyday living. You know, the sort of things folks working a minimum wage job might encounter. I find it absolutely amazing that we haven't done this better since it has been an issue for literally decades.
- arvinsim 5y ago
  I enjoyed math in my Physics classes because of the real life context it is used in.
  - jacobr1 5y ago
    I read about some schools that offer hybrid physics+calculus classes for exactly that reason. Why even bother to separate the subjects. Other schools have experimented with this like project based learning - build a robot or whatever - and then add in tutoring to help close the knowledge gaps that aren't getting picked up. There are plenty of ways to structure learning beyond the Prussian model.
- unethical_ban 5y ago
  I had an eastern European physics teacher in the 2000s. Our classes had word problems about baseballs thrown from the top of buildings to calculate acceleration/speed. I remember him saying "back home all our kinetic physics problems like this were about artillery". Seemed like some good real world application. Heh.
  - inter_netuser 5y ago
    It is. MIT is the public research arm of the DoD. Most of fundamental physics research is military connected apnea way or another.

hnmm23 5y ago

Any good resources that help instill the "why"?

eternityforest 5y ago
I'm working on a reference guide here for exactly that: https://github.com/EternityForest/AnyoneCanDoIt/blob/master/MathELI5.md https://github.com/EternityForest/AnyoneCanDoIt/blob/master/... It's basically "Here's some cool stuff math can do, and interesting historical anecdotes, and how to make a computer do it without actually understanding any of it, and here's what you could do if you actually understood it" There may be errors because... I don't actually understand it and can't derive it myself, so I'm relying entirely on sources. But I think there's value in a text on math by someone with no talent for it, since most books even for beginners assume people have a certain level of ability and treat things i would call very hard as obvious. I don't have any excercises or material on how to actually DO the math, just how to use a CAS solver, I leave the actual teaching of math to those who understand it and focus on showing how it's relevant at all to those of us who don't want to go back and study for two years.
cknakal 5y ago
This website has helped me out a lot with understanding the ideas that math represents: https://betterexplained.com/ https://betterexplained.com/
munk-a 5y ago
I'd suggest the youtube channel 3Blue1Brown - the teacher that runs the channel likes to pull problems into physical space and applications often.

alexashka 5y ago

> Math is incredibly useful For what? It isn't useful for teenagers who are trying to get laid and be cool, at all. It also isn't useful for what 99% of humans end up doing, unless by math you mean basic arithmetic, enough to add and subtract on your 'smart' phone. Math isn't useful. People who are amazing at math and are interested in it - they can be useful, but mostly not. What they don't teach in school is how the real world works, because it is fucking outrageous. What they teach is how to go be a cog in a machine, run by idiots. They don't teach it by telling you, they teach it by making you sit still, listen and do arbitrary, idiotic tasks for 12 years. Sure they teach you a thing or two along the way, but anyone would learn more from doing anything else they were interested in for 12 years. That's the dirty little secret. If you let people do what they are interested in, they'll learn math, programming, exercise, health, whatever, because it'll be necessary to achieve their goals. But you can't make people who've had a taste of achieving their goals slave away their whole life to help some idiot achieve theirs. Plus most parents are fearful idiots who never treat children like human beings, they treat them like their personal pet project whose goal is to line up with society's idea of success. Pathetic. Oh well, that's people for you :)

foxes 5y ago
The way the world and universe actually works seems to be described by maths you probably have no hope of understanding with that attitude.
waynesonfire 5y ago
You're an idiot. Math is everywhere. You can be ignorant of it, it won't kill you. But you're going to have a hard time in just about every endeavour I can think of.
- alexashka 5y ago
  Yup, humans have been struggling for millions of years. Then math finally came to the rescue.
  - waynesonfire 5y ago

_y5hn 5y ago

"Why" won't meet the KPIs and metrics though.

jrumbut 5y ago
What I find interesting is that so many schools don't meet the KPIs and metrics anyway despite stripping away most of the educational experience that isn't test prep. If that was yielding great test scores it would be one thing, but even that isn't happening in many states (or any state that I've looked at but I only pay attention to a few).
- _y5hn 5y ago
  The tests are made to fit a distribution. But test prep is so demotivating and inhumane, much less people are able to cope with it.

technothrasher 5y ago

> What we really need is for every student to realize why math is useful We need to do this for science education as well. The foundation of science that we need the population at large to be educated in is not the steps of the scientific method, how to dissect a pig, or the difference between a covalent and an ionic bond. It is instead logic, critical thinking, rationalism and empiricism, and why science is superior to "common sense". Yet for some reason our primary educational systems tend to focus on the what way more than this why.

inter_netuser 5y ago
common sense is just a form of implicit learning. many do a great job at it.
dekhn 5y ago
it's funny when you mention the difference between covalent and ionic bonds. my teacher in high school chemistry or physics told us cutting a piece of paper was a physical, not a chemical change (the textbook spent a lot of time differentiating those). It wasn't until many years later that I learned that cutting paper breaks hydrogen bonds. H-bonds are neither covalent nor ionic, but more like in-between (partial sharing of electron density by two different nuclei) but can be modelling (approximately) using ionic models (partial charges separated by a distance). Is breaking an h-bond physical, not chemical? I want to talk to my teacher.
- hickimsedenolan 5y ago
  Not an expert, but it is a physical chance. H-Bonds are considered as a weak interaction, and making or breaking them (via phase transitions, or in this case breaking H-Bonds) do not change the chemical structure of the compounds or molecules, making them "weak" and "physical," because it is not that hard to do so.
  - hickimsedenolan 5y ago
    change*, not chance

m4thisfunaway 5y ago

The classic "Lockhart's Lament" paper makes a similar point, that math education would benefit from a major shift in pedagogy (PDF): https://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...

geoduck14 5y ago

>Educators say that they're trying to do this, but I'm sorry "how many meters of fence does Farmer Bob need to keep the sheep inside the pen" is not inspiration -- it's drivel As a counter point, I'm planning on adding drainage to my back yard. I actually am measuring how much pipe I need and how much water I can hope to drain out.

albrewer 5y ago

> "how many meters of fence does Farmer Bob need to keep the sheep inside the pen" is not inspiration -- it's drivel I recently had to do a line integral to figure out how much coiled pipe I could fit inside a trench of a given width (trying to lay out geothermal loops in my yard to see if they fit (they don't)). I came up with a parametric formula that approximated the coiled pipe shape and integrated to get a length formula. Advanced math comes up in the weirdest ways.

hilbert42 5y ago
"I came up with a parametric formula..." Parametric forumae and equations can be incredibly useful. I remember learning about them at school but we never applied them to any realworld examples. Even at the time I recognized that without practical examples our understanding of them would be problematic. Unfortunately, the problem with most mathematics training - especially so at high school - is the overly theoretical approach used in teaching it. It seems to me that at least in high school we should place much more emphasis on the practical application of mathematics. The theoretical approach is all very well for the 'gifted' and those with a mathematical bent but it's often of little help to many other students. I blame this mainly on those who write the textbooks and set the syllabuses. After all, they're the ones who are good at math so they project their theoretical worldview of mathematics onto everyone else. A classic example of the problem is the dozens of trigonometrical identities that we had to learn at school many of which were of no known practical use to us students at that stage of our training. Sure, at university they came into their own but by then one's already more adept at mathematics so they then start to make sense. The same can happen at university too, I recall the tedium of learning vector algebra and matrixes without them being of any seeming use. At the time, no one bothered to explain how incredibly useful they are when it comes to manipulating Maxwell's equations, etc. It's occurred to me it would be very useful if we had a website (and perhaps also in reference book form) devoted to providing mathematical solutions to thousands of realworld problems of a practical nature. For instance, problems could be grouped alphabetically so they're easy to find. Once located, one would find various solutions to the problems (each solution using a different mathematical technique, etc.). For instance typical entries could be: Interest, Compound Interest, Simple Moreover, the website could be made interesting if we also included additional mathematical information such as short biographies on famous mathematicians together with how they arrived at their discoveries, etc. Of course, the site would be graded so as not to scare off the mathematically timid, that way it would also useful to those who are more advanced in mathematics. I envision it as a short of Wiki for math solutions. This may not be the ideal solution to the problem and it'd be a great deal of work to set up (but like Wiki it could come together with the help of thousands of volunteers). Unless we tackle the problem in a tangential manner - in a different and hopefully more successful approach from the past then I don't think we'll ever solve the problem. BTW, I found parametric equations very interesting from the outset. It was obvious to me how they could overcome an impasse and thus solve some otherwise awkward problems - although I can't say I had such insight uniformly across other branches of mathematics.

huetius 5y ago

I love this post, I would only add that “useful” is not the inspiring criteria it is made out to be, as demonstrated by the example of Farmer Bob’s fence. What needs to be resuscitated is the transcendent, aesthetic dimension.