I can remember failing math class hard throughout my entire academic career. I never passed a math class past primary school, and ended up dropping out of high school. The concepts simply never made sense to me, and as they were presented were just meaningless rote exercises of memorization. Math was something I had completely written off as ever being able to understand.
Then I took calculus and trig classes in college. Since I was also learning graphics programming at the time, I was able to actually make sense of math with a real world connection. Playing with little graph equations on shadertoy.com made me realize graphics was all just math. Everything immediately clicked. I was able to intuitively visualize the trig functions and graph equations and do transformations of them because I could picture what the resulting image would look like. Ultimately the problem for students I think is just creating this connection to reality.
It would be incredible if trigonometry classes had a lab component where students work on a 2D game, such as tower defense.
I ended up taking a fun programming elective in college where my team did exactly that -- we created a 2D game. That experience made me appreciate trigonometry more than my high school teachers and Khan Academy ever did.
My experience too - I wanted to make games so when we had trigonometry and analytic geometry in class I was like "yes please, I can use this, more". Same with vector and matrix algebra later.
Meanwhile I had little use for calculus so I only learnt it enough to get a good grade.
Having a real world problem to apply it to makes learning math far easier. I spent a lot of time in 2020-1 tutoring kids in math and the first thing I did was find out what they were into and then worked math into helping them do that thing.
It is like this for all subjects. Beyond the 5th grade, applicability drops a lot. Knowing how to read simple sentences, perform elementary math is enough to get by with most tasks, but personal finance will be a struggle.
My school rarely ever directed mathematics instruction in relation to another skill or problem. Sure math might be used in industrial arts, or in home economics, but the instruction phase of the math is always 100% abstract. If people knew all the interesting things you can do with math they would be more likely to use math more often and remember what they learned. This was certainly the case for me and may have been the only thing that made me even care about math at all. I am referring to only one math class I took in high school which did this, and that was geometry. I loved my geometry teacher. He was the only math teacher I ever found likable.
I loved the geometry class because it helped me a lot when thinking of practical considerations to 3D graphic arts I was creating on Amiga computers at the time not to mention the value of the discussion of area and volume. Did you know Pythagoras was a kind of cult leader?
Geometry wasn't just abstract, it was practical. How much carpet do I need for a room? How much water will fit in this cylinder? If I have a disc that has to be one square foot, what would the diameter be?
I had some enthusiasm for algebra one level math for solving an unknown variable using what is already known. I didn't learn the value in this at school. My dad was an electronic technician (Worked at Argonne Labs) and I was interested in what he did there. He taught me all there is to know about ohms law, finding resistance, conductivity, current voltage, using simple procedures. We did parallel circuits, tank circuits, ac theory. In less than a years time he gave me a complete 2 year associates degree understanding of dc and ac circuits.
Most of us found our math teachers some of the most arrogant people in the school. They demanded that we show our work and show it going down the way they personally approached solving problems. If the result was correct but if they didn't like your preferred procedure they would lower your scores. They never could offer a sound reason for their demands. Perhaps they did have a sound reason. They just didn't bother to sell us on it.
I suppose it isn't right of me to condemn all math teachers this way as my experience is kind of limited. This is just what I felt about it while being a student in high school and middle school 30 years ago.
My math teacher was not like the ones you describe, he really let me go off the reservation if I ever wanted to approach the problem in some wierd zany way. IMO, that's how it should be.
On the other hand, mastery of the tools and methods requires repeating it and spending a lot of time with them untill they "make sense" or rather, you get used to them. So that is useful as well. Most people can apply mechanical methods well, but the part where I see a disconnect is applying them to new problems and new contexts. It's like... Most people don't even realize you can do that. Would love to find a scalable way that teaching can approach that as well. It is a crucial mental tool.
I'm sorry you had so many less than awesome teachers. I had an Algebra 1 teacher in middle school (United States) that really made a huge difference. The way she taught was so different than every other teach I had up until that point. She approached everything in a meaningful and useful way. It really set me up for success then and now.
> If the result was correct but if they didn't like your preferred procedure they would lower your scores. They never could offer a sound reason for their demands. Perhaps they did have a sound reason. They just didn't bother to sell us on it.
The reason was always "because the goal is to see if you've learned this technique, not if you can solve this specific problem". Which is a perfectly fine reason that I sympathized with, though I agree some were poor at articulating it. Where I had issues was when I didn't know what technique they wanted me to use, and then lost points for not guessing it correctly. That part is what always seemed unfair - if the goal is to test a technique, it should always be clear what the intended method is, and they need to explicitly specify it in the problem when it's unclear. The exercise is supposed to be in math, not ESP.
but isn't that the case with all instruction, not just maths?
As a counterpoint to your example, I, too, learnt the basics of electronics when I was young - Ohm' laws, Kirchhoff's laws, etc. At the time I found it extremely uninteresting, and I certainly didn't apply it to my everyday life and still mostly don't.
In phys ed. I had to play handball a lot. I never played handball as an adult. Running around a courtyard for 20 minutes? I don't do that, ever -- heck, why don't we teach kids to run behind a bus to catch it, or in a crowded airport to catch a connection? Now that would be usable skills...
History, geography, literature (how often do you find yourself finding all the metaphors in a poem?), most of what we learn at school is not "used", nor is it instructed in relation to another skill or problem. And yet, when the issue comes up, it's always maths that is under scrutiny.
There is a lot to review in the way we educate children, all the way to university. Not everything we teach has to be for utilitarian purpose, and yet we need to find a way to interest the kids into it, which we largely fail, in maths in particular, but in other disciplines too. Some of it is about helping them relate it to their every day lives, and some of it to expand their horizon. It's not easy, we don't do it well, and most suggestions I've seen are mostly cosmetic or superficial, and I'm not smart enough to have come up with anything clever myself either, sadly.
Yeah. One underappreciated thing about education is exposure. It's not that I've been walking around every day for the past decade being capable of matrix multiplication. But having been exposed to it once, it's amazing how quickly the knowledge comes back when you need it. I did advent of code over Christmas and when the 3D transformations came up I thought about it for a bit, then remembered... something about matrices... linear algebra..? And then searched for the resource I needed to refresh myself on it. Just the fact I was exposed once a decade ago means I was able to refresh myself in a few minutes.
Given the relative innumeracy of the public, some argue we should teach stats and probability instead of calculus, if there's a choice. Those things are more applicable for everyday choices like "should I buy a lottery ticket or invest in the market?" "How likely am I to die from disease X versus side effects of its vaccine?" "Should I buy comp insurance or pay out of pocket if I break my car?" etc.
Using probability and statistics and understanding their foundations are two distinct things. Prob/stat can be taught (to the point of utility) without calculus and often is. More advanced applications are inaccessible or only learned by rote without calculus, but that doesn't mean that you need calculus first.
Of course, this is what commonly comes up in topics like this, a confusion over the two notions: Understanding the basis of something and understanding it well enough to utilize. You don't need a graduate level physics course in electromagnetism to be a competent electrician.
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.
> 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.
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
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".
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
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.
> [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.
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.
>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.
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.
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.
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.
I doubt even the "real world" ones will make much impression or sense for kids of that age.
The farmer Bob example for the fence? Definitely real world. Minimum wage farm hand might not be the one doing the buying of fencing material that's true. 10 year old kid learning that math? Doesn't care one bit about fences, even if you make it "Your family has a dog and you just moved into a new house and there's no fence ..."
Minimum wage framing crew helper. You forgot your square. Definitely real world. But you got your tape measure. Use Pythagoras to figure out if your wall is going to be square. Does the minimum wage framing guy care when he's an 11 year old kid? Nope!
Figure out how high your school building is with math and without going to the roof and a loooong "tape measure"? That was cool! Especially since we actually went outside and did it.
Also, and this isn't math but at least science related: gasoline fumes are flammable. Very. Real world. Minimum wage lawn mowing crew guy might want to remember. Loved the presentation our chemistry teacher did. Put some gasoline into a small can and tried to light it. Nothing happens. Had to pour it back out. Next step: let us calculate exactly how much gasoline we'd have to put in for the fumes to be ignitable. Then he did just that, lit it and man that was a nice explosion and flying can!
To be a little nitpicky about these sort of problem "How many fence posts should Farmer Bob buy"?
"Farmer Bob would be pretty stupid to drive an hour to the hardware store to buy 20 fence posts and 100m of fence, because Murphy's Law and what if one happens to be bad or someone cuts wrong and we're short 2m of fence?"
yes, growing up on construction sites does that to you, the problem is badly phrased if it's saying "should buy" instead of "needs" ;)
>To be a little nitpicky about these sort of problem "How many fence posts should Farmer Bob buy"?
Sure. I'll go with that. Farmer Bob can be a part of a case study. We start off simple.
Farmer Bob buys exactly as many posts as necessary... then he discovers shrinkage/theft/errors
The next project, Farmer Bob uses linear extrapolation from the past 5 projects and buys X% extra posts... but then he discovers new employees break things
The next project Farmer Bob uses Linear Regression, broken down into key variables (like # of new employees)
Hehe I like this! I don't even think it's nitpicky from your parent at all and your "progression" I would say works very well with how school should work.
At first kids just learn how many fence posts and fence is gonna be there in the end. I.e. how much do you need to buy ignoring some things. Like you ignore friction losses in physics in many cases, especially in school.
Extrapolation and Linear regression you learn later on to solve the same problem better! Love it!
Similar thing happens (or at least happened when I was in school) in chemistry. We learned the Bohr model and it could be used to explain everything that we learned about. Towards the end we learned that there are some things that model can't explain and that there's another model that can explain those things and so we learned about that and also that it's totally fine to keep using the Bohr model for a lot of stuff as it's easier. Like ignoring friction losses in physics and the predictions/calculations come close enough in most cases that it's still useful but much easier to comprehend and calculate than if you had to take them into account.
> Figure out how high your school building is with math and without going to the roof and a loooong "tape measure"? That was cool! Especially since we actually went outside and did it.
I still find it incredible that "Daumensprung" [0] is apparently not an international concept (I learned about this in Math class growing up in Germany).
Unfortunately the Wikipedia link is only in German, but the short explanation is:
* Extend your arm horizontally in front of you, and raise the thumb to point upwards.
* Close one eye and align the thumb with the edge of a large object in the distance (e.g. a building).
* Close the other eye and note how far the thumb virtually "jumped" across the large object.
Now, knowing the distance between your eyes and the length of your arm, you can use trigonometry to either
* knowing the distance to the object, deduce the width of it
or
* knowing the width of the object (e.g. using a window, doorframe, or car as a reference size), deduce the distance to it.
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.
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 :)
I find math to be incredibly useful in my day-to-day life, and am constantly quantifying things and making computations based on those quantifications. I specifically find double booking accounting, geometry, probability, and set theory to be useful in making daily decisions.
highschool and to an extent middleschool math do spend a lot of time on stuff that doesn't matter to the average person. but it covers some things that matter a lot.
what will the sales tax be on this item at the store?
can I afford the items in my grocery cart?
if I buy a TV on my credit card and pay it off in three months, how much extra am I paying?
if I keep doing what I'm doing, when will I be able to retire?
Most people generally do not use these math skills. Most use the very basics. Never once have I used or seen another developer use Calculus, for example. Most devs will have no idea what a "polynomial" is, either (grade 8). Let alone factoring a quadratic polynomial!
I agree with you that math is pretty much useless in most people's lives. However I disagree that teaching it to everyone is useless. The whole point of making kids learn a whole bunch of different things is that it gives them a chance to figure out if they are, in fact, interested. If you don't provide lots of different topics of learning to children in school, where are they going to find out that one of those topics might be just the thing that fascinates them? Most parents aren't going to expose their kids to every topic under the sun. I think it's great that schools expose kids to math, physics, chemistry, biology, languages, music, physical education, social studies, history, geography, cooking, sewing, painting, carpentry, religion, you name it. I hated doing at least half of those things when I was in school, but I still appreciate in retrospect that I did it, because now I know much better what I like and don't like.
Exactly. Most moth is not useful at all, even for senior software engineers. I've researched this topic a lot. I'd say the useful math peaks at grade 9/10. You know: Algebra, Geometry, Trig, Graphs. Obviously, for a very small fraction of the population, higher math is useful.
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.
"It is a great profession. There is the satisfaction of watching a figment of the imagination emerge through the aid of science to a plan on paper. Then it moves to realization in stone or metal or energy. Then it brings jobs and homes to men. Then it elevates the standards of living and adds to the comforts of life. That is the engineer's high privilege."
- Herbert Hoover
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).
> 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.
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.
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.
I’m just curious, but what do you think is the difference between “critical thinking” and “common sense”? Ostensibly, they both seem to mean something like effective thinking done outside the boundaries of predefined dogmas. On the other hand, “critical thinking” often appears to be Orwellian-Newspeak for “accept the dogma I am telling you”. Granted, maybe “common sense” plays the same role, but for a different audience, and the principle distinction is a class divide.
I struggle to see how “critical thinking” can mean anything other than the application of tools to ascertain the true state of things. Certainly far from “accept the dogma I am telling you”
>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.
> "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.
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.
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.
You don't - I do (including some math only seen in uni).
It all depends on your job, and since we can hardly know in advance, school tries to cover all bases. Drill and practice or the other proposed solutions won't help with that.
Because most school is forced knowledge work on problems that don't fit the person's problem situation.
When people are free, they learn the math they need for the problems they are trying to solve.
Traditional, compulsory school forces children to solve problems they don't have. See Karl Popper's idea of the bucket theory of mind, or David Deutsch and Taking Children Seriously.
> most school is forced knowledge work on problems that don't fit the person's problem situation.
Have you never been in a situation where you are about to take train going at 80 mph; wondering when you'll meet up with your friends going in the opposite direction at 90 mph?
I recently saw a great photo [1] by Ben Cooper of a Falcon 9 on ascent crossing the Moon from the photographer’s perspective. I wonder how much math went into finding that vantage point.
It's not really that hard. I set up several systems to do this for the previous total solar eclipse. The ephemerides for the moon are easy to download and calculate the position in the sky (IE, altitude and azimuth at time t) with a python script.
I believe also the launch vehicle has a launch window (the moon moves 15 degrees per hour) and launch trajectory. I'm lazy so I'd compute the extends of the launch vehicle's motion in the sky (from earliest possible launch to latest possible launch), and then intersect that geometry with the moon position geometry without explictly trying to solve the equations simultaneously. That should back-project to shapes on the ground at which point you could reasonably expect to be able to get a good shot, and then you'd do some adjustment in your pointing in real time.
A smart college senior could do it directly (IE, not lazily compute a bunch of points and manually intersect them on a screen).
Having done a lot of long road trips, I have often entertained my tired brain by trying to calculate how long it will take me to get to mile marker x, or city y (SPRINGFIELD 400 says the sign) based on how fast I'm going, then how long it would take if I was going 1mph faster or slower, or 5mph, etc. It's relatively simple but to do so entirely mentally seems to take a lot longer for me than I if I could just jot a few notes down. It gets more fun if you try to account for how long it takes for you to slow down for an exit, take a piss and get a snack, then get back up to speed.
I didn't know, for the longest time, what a derivative was. I still don't, not really, though I can bluff my way to a reasonably correct answer using the above analogy and going from dry "rate of rate of change" sort of language into more practical concepts.
Yeah really, when I finally found a reason to get interested in higher level math, I then realized it built on much knowledge from my 12 years of school that I neglected to care about to learn, all further down the chain of abstraction and further away from my problem to the point where it’s no longer interesting again.
Uh, there is a carpenter with a Web site that uses the "3, 4, 5 rule", of course, the Pythagorean theorem.
The first order ordinary differential equation initial value problem
d/dt y(t) = k y(t) ( b - y(t) )
can be use to model viral growth. Once it kept two crucial FedEx BoD members from walking out and saved the company.
Similarly for the law of cosines for spherical triangles for finding great circle distances.
Linear programming (LP) gets used right along for actual, genuine LP problems and also as a means of approximation for problems that are not linear. The problem of minimum cost flows is a special case of linear programming.
If want to sort keys, e.g., for some positive integer n, if want to sort n numbers into ascending order by comparing numbers two at a time, then heap sort does that in worst and average case in O(n log(n)) and, thus is the fastest possible -- this is from a cute counting argument, the Gleason bound, A. Gleason.
Statistics is important and parts of it are awash in math, not all of it trivial.
The design and operation of the Webb telescope is awash in math.
> Knowing a triplet ain't employing Pythagorean theorem.
3^2 + 4^2 = 5^2 = 25 guarantees that the angle between the side length 3 and the side of length 4 is a right angle which is what the carpenter very much wanted to know. E.g., the sides might be 30 feet, 40 feet and 50 feet, e.g., for the foundation of a house, and presto, bingo the carpenter knows that the 30 foot side and the 40 foot side form a right angle, are square, which can be very important to a carpenter and not so easy to know otherwise.
Yup, and knowing that having triangle sides multiples of 3,4,5 guarantees that it's a right triangle is not the same as employing pythagorean theorem. You know one triplet, and not why it works, or that e.g. it works for 5, 12, and 13 and 11,60, 61 as well.
A whole lot of people know that measuring a bookcase corner to corner is a good way to square it up, too. But they may not be masters of geometry beyond knowing a rule of thumb.
So much of modern math education, especially once you get into high school, is centered around _computation_ instead of _reasoning_, so you get a lot of kids who spend an entire year doing random integration problems or charting a bunch of useless functions/conic sections instead of really understanding fundamental structures, reasoning, problem solving, etc. It's just rote computation, but we have computers for that now.
I can understand the average person not using the math we learn in school, but it’s crazy to me that even the average engineer doesn’t beyond basic arithmetic. I don’t mean this in an elitist sense but I feel like I was the same way a couple years ago. Until one of my mentors kinda guided me through the thought process and understanding the math is extremely powerful and useful.
In my opinion serving up 2/3 cups of a semi-solid cheese and reserving a quarter of that demonstrates a quite excellent intuition about the problem of 3/4 of 2/3. Probably 3rd-grade children would be congratulated upon discovering this strategy.
The article mentions "More drill and practice" ...
I don't think this is the right way. I believe something happens with math to many/most students which leaves them less than fluent in mathematics.
Somewhere along the way, they stumble for a few weeks and fall behind. Maybe their family took a vacation. Maybe the teacher's explanations don't make sense to them. And then there's a deficit.
Pretty soon, supporting that student through math becomes a whole lot more about drilling and memorizing strategies and understanding is deprioritized.
This becomes insurmountable, especially with the "layer cake" model of how math is taught in US schools. Pretty soon you'll never catch up in understanding. But you probably just attribute it to "not being good at math".
I'm working with a student right now, who is in precalc and "bad at math." He is actually really bright. Some missing knowledge and intuition about fractions has made everything since much harder. And the problem's never been fixed because he's constantly been in a survival mode in his math classes.
These students never end up doing the kinds of things that kids are ahead in math are taught to do to crack free-form problems:
- Make a good guess of the answer
- Form a mental picture and use it to select strategy
- Work a simpler problem and see how this can be extended
- Organize your givens ; use dimensional analysis for a hint as to what operations may be required or what picture should be drawn
- Look for symmetry and transformations that result in simpler cases
- Does positing an incorrect answer reveal any information about the solution?
IMO, teaching these things -- strategies-- should be more the goal than doing lots of kinds of applied math work by rote in a way that deeply depends upon previous work.
There was another “I’m bad at math” thread a while back I stumbled across and drilling was considered basically the way to get good at math by most people posting.
… which is all the more reason I’ll never become good at math. I’d rather drill a hole in my head than subject myself to drilling back to back, contextless, boring problems I don’t care about.
I think there's a few points where drill makes sense. In each case, after you have a good understanding, and you're seeking to become fluid and avoid dumb mistakes. (Even if you have a good understanding, if you're consciously thinking about each step you'll error a fair bit, and on a 20 step problem you can't afford this error rate and slowness)
Arithmetic facts, basic algebraic manipulation, big polynomials, derivatives and integrals.
But we do much more drill than that. And, worse, it's often as homework, where you can be guaranteed a decent slice of the class is practicing it wrong.
I think I've used almost everything I learnt in maths at school at one point in my adult life. And not just because I'm a programmer. The other day I was using trigonometry to work out how to install a projector into a room.
From what I can tell, most people don't do things like install projectors into rooms. They either get someone else to do it, or just figure out some placement that is good enough via intuition or trial and error. If people have jobs that require it, they usually remember the bare minimum for their job and aren't remembering general concepts to help them solve brand new problems.
If there is causality here, I'm not sure which way around it is. Either they don't need maths because they don't stuff that needs it, or they don't do stuff that needs it because they can't do maths.
This is a good question, why is it that seemingly nobody uses the math they've been taught?
When I was in school, my math teacher used to write AMBS on the board before calculus lessons: Algebra Must Be Strong. What happens when you study calculus is you end up rehashing algebra over and over again until it's second nature. Consequently if you find someone who's done well at calculus, you can be sure they can do algebra.
Someone mentioned this in relation to RenTech. Why do they hire mathematicians who have studied incredibly complex things with seemingly no relation to trading? One reason might be that people who have done that know the fundamentals very, very well.
Similarly accountants don't use vectors or geometry at work either, but they certainly are comfortable with arithmetic and its application in spreadsheets.
When I look at my own education, it is also the case that the things I'm least sure of are the deepest, latest courses that build on everything else. Probably if I built on top of those I'd be more comfortable with them. Taking this line it would appear that you can't just teach people the math they'll use, because they won't really get it until they try to do the next thing.
A thing that needs addressing is the notion that we should only teach people things that will be useful, which seems to underlie this article. We should teach people how to explore, and math is the premier example of a large space that can be explored with very little access to resources and evidence. You can do it just in your mind, and you don't need any apparatus or conclusions from people who had such apparatus, like you do in science.
When I started learning woodwork I was really excited to apply geometry which I hadn't used much since highschool (even working with geospatial data vis). However as I've progressed I find myself using math less and less and it's often faster and more accurate not to measure or calculate.
To give an example, making a three legged stool I needed 3 equally spaced spaced points around a circle, I could find the center, draw the first point then use a protractor to find the other 3 points at 120 deg. But it was far far quicker to just use a divider, step around the circle a few times adjusting until 3 steps takes you back to the first point.
Of course this is still some form of math, but it doesn't involve any calculation. I feel there's a whole world of very useful techniques that work on a different abstraction, though I'm struggling to name or describe to properly.
A similar trick is the 3:4:5 rule[1] which I have not used for woodworking but that's where I first heard of it. I have used it for general site planning for tents and ensuring fire breaks and similar, though.
> Many students are only taught math as symbol manipulation. Less instruction is focused on identifying situations where it might be useful. We need to give students more training in noticing and converting everyday situations into the math problems they know how to solve.
No, please don't do this. When I was learning math as a kid I found word problems incredibly irritating and stupid. It's just a way to waste a few seconds of my time when I just want to solve the problem and get to the next one. Word problems are just disrespectful
I think the reason most word problems suck is that they're manufactured abstractions on the same problems elsewhere on the worksheet--which seems to be your critique.
In real life, a lot of those situations are hidden, but there are often multiple correct ways to approach the problem and hopefully some intuition will help you if you're too far off track. 2/3rds of 3/4ths of a cup of cottage cheese shouldn't be more than one cup and shouldn't be a tiny amount, either. (I also don't see a problem with people who scooped it onto the table). Similarly to the woodworker (elsewhere in this thread) trying to evenly measure out where 3 legs on a stool would go.
To me this is coming at the issue from the wrong angle. Sure, mathematics classes in elementary through high school have the veneer of teaching mathematics, but they also teach logical thinking and problem solving.
Identifying a problem (in math class the problems usually come pre-identfied until you tackle the dreaded story problems) and breaking down the problem into logical steps, are useful skills to have even if someone never does another math problem in their life.
My answer: because most of the math we learn in school is useless.
I see a lot of math homework questions on Quora, because students copypaste their homework all the time:
What is log (3x +2)- log2 =1?
How do you prove that X^2-3 is irrational?
What are the roots of the equation x4−4x3+6x2−4x+4=0?
When on earth are you ever going to use any of that?
The word problems are no better.
In a group of 20 adults, 4 out of 7 women and 2 out of 13 men wear glasses. What is the probability that a person chosen at random from the group is a woman or wears glasses?
Mr. Tay had 20 female fish and 5 male fish in a pond. Mr. Tay’s father brought some male fish and put them in the pond. Then he found 1/5 of the fish are female. How many male fish did his father put into the pond?
In a school interhouse competition, 80% of the students turned up for the athletic competition, 60% turned up at the football match. What percentage of the students attended both functions?
These just aren't the kinds of questions anybody ever actually asks. The math required to solve them is pragmatic; are there no pragmatic examples they can come up with?
The fact that students are copypasting their homework doesn't thrill me for their ethics or their intelligence. But the fact that the problems themselves are so stultifyingly dull almost makes it seem reasonable.
Maybe they should get engineers to teach math courses the way they wish they had been taught them, rather than math people who sometimes still seem to think we will be doing this without a calculator?
Engineers have specialty interests. I'd just as soon have it taught by a home economics department, or at worst an MBA. Things everyone should be able to do: read a bill, allocate a budget, estimate how much food to buy for 2 people versus 10 people, plan for retirement, maybe convert a recipe from metric to imperial units. Perhaps get a shop teacher.
Let the future STEM majors learn the rest in their classes. It takes a long time for a lot of that stuff to pay off, and much of it remains irrelevant.
Giving STEM majors a head start is helpful, and a lot of those things aren't primarily mathematical.
I'm not sure what math you need to read bills in the age of specialty apps. I can't remember the last time I needed to do any arithmetic without my phone.
Estimating how much food to buy for 2 people requires understanding nutrition science, whether all your guests ate at all today, how much space you have to store leftovers, etc.
The only math I can think of is jus t multiplying by number of guests, But then you might have kids or people with dietary restrictions and now half your planning is just individual person by person stuff.
Math won't really help you plan a budget either unless you know all the finance specific stuff, or enough fairly advanced general math that it's just all obvious to you.
There's way too much estimation of real world stuff and most of the controllable expenses(for an average poor person) are impulse fast food orders and such that they already know are costing a ton.
I'm not sure trust an MBA to teach... anything about economics. They are way too high on their positive mindset business optimism attitude to really understand that the cost of getting Starbucks actually is important.
A lot of the "Useful real life math" is stuff that people won't use anyway, either because an app does it, or because retirement is later and doordash is right now. Is math actually the most effective tool to make someone want to stop that?
I think this question stems from a confusion as to what school is actually for. It seems to me that school is at least as much about orderly management of the social hierarchy as it is about teaching useful skills.
The reason why math is taught in school is not because math is useful (it is useful in certain situations, but this is only tangentially related). We teach it because it keeps students busy and out of trouble, and lends itself well to unambiguous testing and is therefore an effective tool for sorting students into the various social classes.
This all became clear to me when I was in the middle of my math PhD and wondering why all of the math courses I had taken did next to nothing to prepare me for actual mathematical research.
Viewed from this perspective, it’s obvious why we don’t use the math we learn in school.
Why don't we use the [History|English Lit.|Chemistry|etc.] that we learn in school?
As a generality, isn't the math taught to most teens in public schools aimed pretty squarely at preparing them to take the standard calculus courses in their first couple years of college? If so, hardly surprising that most ex-students don't notice much use for that stuff in the real world.
And since the social function of that is usually to demonstrate that you're determined / focused / upper-class enough to successfully slog your way through it - gee, what would be the point to remembering or using it after getting your "made it through" bragging rights?
English is critically important to help you understand complex written works and to write in a way that is clear and doesn't make you look dumb. Advanced chemistry might be useless, but basic chemistry/physics is useful. And I feel that people would understand what they are doing a lot more if they understood physics a little better. For example a lot of people think a cone makes amplifies noise rather than understanding that it simply directs them in to a smaller space. Chemistry helps you understand why oils stick to your hands and don't wash off until you use soap.
Basic math is very useful but highschools go so far off basic math that they teach massive amounts of seemingly useless stuff.
People don't use math because... it's only useful if you actually really know it.
"Useful" to me means "Lets you do something you couldn't before with a phone".
Aside from imperial inch fractions(Not a fan of those!), most things require fairly little understanding of fundamentals.
Because most people aren't in tech jobs where they can use it.
I don't have a checkbook or do my own investing. I'm single, and very little math is needed to pick whatever is cheapest, especially when only certain sizes of some things are practical for one person.
CAS systems can do basic algebra, FreeCAD does geometry...
FreeTaxUSA does taxes, I don't have a car to optimize insurance costs for...
It's always pretty cool to see a chance to use math IRL just because it's such an uncommon novelty, but I don't see it very often.
To unlock new capabilities you have to learn all those basics, then move on to learn the advanced stuff that computers can't already do.
It's a pretty long term investment of time. Maybe worth it, but still a lot of work.
FreeCAD can do geometry, but if you can’t reason your way through the constraints you need to set for it to do that then you’ll be flailing around. FreeCAD definitely saves me from doing arithmetic, but not from doing maths.
I usually find the constraints are a pretty direct translation of how I would describe a part in non-tech nical writing.
Math definitely is helpful, but unless you are a full time MechE, it's not like you need to be able to pass a high school geometry class to be able to print a replacement part for something.
When you say “imperial system” you almost certainly mean US Customary Units. These are not the same system. The United States has never used the Imperial System. Also, US Customary Units have been defined in terms of SI units since 1893.
Personally I don’t find multiplying by two any harder than multiplying by 10.
Multiplying by all numbers is equally easy or hard with calculators, the problem is fractions break UIs.
Try adding 3/16 plus 1/2. On a calculator that is three separate operations, and if the output is decimal, you have to convert back to a fraction.
And when automated systems convert to fractions, the amount of error is not obvious. With decimals, the computer just adds more digits unless you limit it to a certain number of places. It is safe to add a few extra because you can ignore them.
With fractions the whole point is smallish integers. If a computer says 3/8 inch I have no idea if it's really 3/8 or if its 11/32 and the computer rounded.
It can't just always show the value in 128ths of an inch because... nobody knows what those are and rulers aren't marked that way. So to have a computer display fractions means understand how it does so, it's not obvious(Unless you displayed explicit error values, which would be pretty great)
The whole thing came from a time when very skilled humans with no tech did everything themselves and seems optimized for that at the expense of everything else.
Decimals way easier for applications where all the math is 100% done by the CAD program.
Nobody is using 128ths of an inch. That’s 7 thou. Realistically anything smaller than a 32nd is going to be in thousandths of an inch and using specialized tooling.
3/16 + 1/2 is 3/16 + 8/16 which is 3+8 or 11/16. That’s simple arithmetic. No calculator needed.
No calculator needed assuming you can remember multiple intermediate variables or have the conversions memorized.
It's a great "power user" system for skilled professionals in the pre computer age who practiced with it regularly, and it works well in things like woodworking where you still have a lot of things done on paper or just in someone's head.
It's not the best for CAD first workflows, when you're trying to model everything down to where to clamp the edge guide blocks for the saw, and changing something without changing the model and confirming it will all fit isn't a thing.
In 3D printing you get down to 0.4mm or occasionally even 0.2mm gaps between things, although none of it is really an exact science, and you assume some sanding may be needed to make things fit unless you have an amazing tuned printer.
But I suppose woodworking is less of an issue, because you're probably not going to get even 1/16th precision by hand unless you're really skilled, in which case you probably have fractions memorized.
I feel like there could be an Explanation #4 which is something like: for many people lots of math is used so infrequently, it's hard to retain after you're done with school.
Maybe more drills would help that?
I did pretty well in high school math and got through the most advanced math classes my small town school district had to offer. Then I opted out of driving to the next town over to do a more advanced math class with something like the 5 other kids who qualified. I think it was pre-calculus or something.
So I ended up not doing math for a year, then I really struggled in my first college math classes, and had to retake one of them.
Since then I've basically just forgotten everything because it comes up so rarely. How often do I need to do fractions? almost never, and when I do, I can just look up a factions calculator. Never mind trig or calc.
Day do day, I think I need basic addition, subtraction, multiplication, and division. And I figure out percentages a lot. But I'm also doing all of that with a calculator.
I think financial cautions, like compounding interest are the most advanced things I run into regularly.
I have a different take on this situation (thought it bears some resemblance to "Explanation #2" in the article). My years of experience in maths education has led me to the conclusion that the role of maths, particularly in child education, has, or should have, little-to-nothing to do with maths itself: it is a means unto a different end, and should not be used as the end unto itself.
For example, why do we make (US) high-school students study elementary algebra? You would be hard-pressed to find an application of the quadratic formula in everyday life, and yet I maintain that it, and the course at large, has an important pedagogical role. All pre-collegiate levels of maths education (arguably including elementary calculus, but certainly everything before that) presents the student with increasingly complex situations, the tackling of which requires increasingly complex problem-solving strategies. From this perspective, all levels of maths education are effectively drilling structured logical reasoning, problem decomposition, pattern matching, rule application, abstract thinking, abstract/concrete translation, precise/"technical" communication, etc. Maths simply provides a rather nice vehicle for presenting these concepts in an ordered fashion with natural increases in complexity. I even believe that the introduction of integers to pre-schoolers is a valuable vehicle for introducing causality and linearity of time.
Along the way, there are certainly valuable real-life skills that are clear applications of maths concepts, and I would agree with the article that most maths programs are woefully deficient in really underscoring the value of these applications. The article opens with a comic about needing "two-thirds of three-quarters", but anyone who has ever scaled a recipe, or worked with so-called "bakers percentages" knows that this is far from being simply a theoretical word problem.
As an aside, I spent a number of years teaching elementary arithmetic to adult students (usually in their 50s-60s, and, without a hint of hyperbole, commonly unable to solve "2 + 2" without memorization). My role was to find creative ways of communicating arithmetic concepts to people for whom the traditional approach often failed. This strengthened my belief in the above perspective by providing an interesting corollary: many of these people had come to learn the "maths-adjacent" skills, traditionally taught via maths instruction, in fascinatingly creative ways. If the ordering and equal-spacing of the integers was not something they were able to truly grasp as children, sometimes they would surprise me with their approach to other parts of their lives for which perhaps you and I would have a more "normal" solution to.
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[ 3.3 ms ] story [ 317 ms ] threadThen I took calculus and trig classes in college. Since I was also learning graphics programming at the time, I was able to actually make sense of math with a real world connection. Playing with little graph equations on shadertoy.com made me realize graphics was all just math. Everything immediately clicked. I was able to intuitively visualize the trig functions and graph equations and do transformations of them because I could picture what the resulting image would look like. Ultimately the problem for students I think is just creating this connection to reality.
I ended up taking a fun programming elective in college where my team did exactly that -- we created a 2D game. That experience made me appreciate trigonometry more than my high school teachers and Khan Academy ever did.
Meanwhile I had little use for calculus so I only learnt it enough to get a good grade.
I loved the geometry class because it helped me a lot when thinking of practical considerations to 3D graphic arts I was creating on Amiga computers at the time not to mention the value of the discussion of area and volume. Did you know Pythagoras was a kind of cult leader?
Geometry wasn't just abstract, it was practical. How much carpet do I need for a room? How much water will fit in this cylinder? If I have a disc that has to be one square foot, what would the diameter be?
I had some enthusiasm for algebra one level math for solving an unknown variable using what is already known. I didn't learn the value in this at school. My dad was an electronic technician (Worked at Argonne Labs) and I was interested in what he did there. He taught me all there is to know about ohms law, finding resistance, conductivity, current voltage, using simple procedures. We did parallel circuits, tank circuits, ac theory. In less than a years time he gave me a complete 2 year associates degree understanding of dc and ac circuits.
Most of us found our math teachers some of the most arrogant people in the school. They demanded that we show our work and show it going down the way they personally approached solving problems. If the result was correct but if they didn't like your preferred procedure they would lower your scores. They never could offer a sound reason for their demands. Perhaps they did have a sound reason. They just didn't bother to sell us on it.
I suppose it isn't right of me to condemn all math teachers this way as my experience is kind of limited. This is just what I felt about it while being a student in high school and middle school 30 years ago.
On the other hand, mastery of the tools and methods requires repeating it and spending a lot of time with them untill they "make sense" or rather, you get used to them. So that is useful as well. Most people can apply mechanical methods well, but the part where I see a disconnect is applying them to new problems and new contexts. It's like... Most people don't even realize you can do that. Would love to find a scalable way that teaching can approach that as well. It is a crucial mental tool.
The reason was always "because the goal is to see if you've learned this technique, not if you can solve this specific problem". Which is a perfectly fine reason that I sympathized with, though I agree some were poor at articulating it. Where I had issues was when I didn't know what technique they wanted me to use, and then lost points for not guessing it correctly. That part is what always seemed unfair - if the goal is to test a technique, it should always be clear what the intended method is, and they need to explicitly specify it in the problem when it's unclear. The exercise is supposed to be in math, not ESP.
As a counterpoint to your example, I, too, learnt the basics of electronics when I was young - Ohm' laws, Kirchhoff's laws, etc. At the time I found it extremely uninteresting, and I certainly didn't apply it to my everyday life and still mostly don't.
In phys ed. I had to play handball a lot. I never played handball as an adult. Running around a courtyard for 20 minutes? I don't do that, ever -- heck, why don't we teach kids to run behind a bus to catch it, or in a crowded airport to catch a connection? Now that would be usable skills...
History, geography, literature (how often do you find yourself finding all the metaphors in a poem?), most of what we learn at school is not "used", nor is it instructed in relation to another skill or problem. And yet, when the issue comes up, it's always maths that is under scrutiny.
There is a lot to review in the way we educate children, all the way to university. Not everything we teach has to be for utilitarian purpose, and yet we need to find a way to interest the kids into it, which we largely fail, in maths in particular, but in other disciplines too. Some of it is about helping them relate it to their every day lives, and some of it to expand their horizon. It's not easy, we don't do it well, and most suggestions I've seen are mostly cosmetic or superficial, and I'm not smart enough to have come up with anything clever myself either, sadly.
Of course, this is what commonly comes up in topics like this, a confusion over the two notions: Understanding the basis of something and understanding it well enough to utilize. You don't need a graduate level physics course in electromagnetism to be a competent electrician.
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.
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.
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".
Seems reasonable that as you introduce new concepts having a physical equivalent would make it easier.
To try and inspire people, though, maybe word problems could be written to be more relevant
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.
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.
This helped me to “trust” it in more advanced contexts where geometric intuition becomes exceedingly difficult.
That is what word problems try to instill.
Seemed like some good real world application. Heh.
MIT is the public research arm of the DoD.
Most of fundamental physics research is military connected apnea way or another.
The farmer Bob example for the fence? Definitely real world. Minimum wage farm hand might not be the one doing the buying of fencing material that's true. 10 year old kid learning that math? Doesn't care one bit about fences, even if you make it "Your family has a dog and you just moved into a new house and there's no fence ..."
Minimum wage framing crew helper. You forgot your square. Definitely real world. But you got your tape measure. Use Pythagoras to figure out if your wall is going to be square. Does the minimum wage framing guy care when he's an 11 year old kid? Nope!
Figure out how high your school building is with math and without going to the roof and a loooong "tape measure"? That was cool! Especially since we actually went outside and did it.
Also, and this isn't math but at least science related: gasoline fumes are flammable. Very. Real world. Minimum wage lawn mowing crew guy might want to remember. Loved the presentation our chemistry teacher did. Put some gasoline into a small can and tried to light it. Nothing happens. Had to pour it back out. Next step: let us calculate exactly how much gasoline we'd have to put in for the fumes to be ignitable. Then he did just that, lit it and man that was a nice explosion and flying can!
"Farmer Bob would be pretty stupid to drive an hour to the hardware store to buy 20 fence posts and 100m of fence, because Murphy's Law and what if one happens to be bad or someone cuts wrong and we're short 2m of fence?"
yes, growing up on construction sites does that to you, the problem is badly phrased if it's saying "should buy" instead of "needs" ;)
Sure. I'll go with that. Farmer Bob can be a part of a case study. We start off simple.
Farmer Bob buys exactly as many posts as necessary... then he discovers shrinkage/theft/errors
The next project, Farmer Bob uses linear extrapolation from the past 5 projects and buys X% extra posts... but then he discovers new employees break things
The next project Farmer Bob uses Linear Regression, broken down into key variables (like # of new employees)
I think we can get a book deal with this!
At first kids just learn how many fence posts and fence is gonna be there in the end. I.e. how much do you need to buy ignoring some things. Like you ignore friction losses in physics in many cases, especially in school.
Extrapolation and Linear regression you learn later on to solve the same problem better! Love it!
Similar thing happens (or at least happened when I was in school) in chemistry. We learned the Bohr model and it could be used to explain everything that we learned about. Towards the end we learned that there are some things that model can't explain and that there's another model that can explain those things and so we learned about that and also that it's totally fine to keep using the Bohr model for a lot of stuff as it's easier. Like ignoring friction losses in physics and the predictions/calculations come close enough in most cases that it's still useful but much easier to comprehend and calculate than if you had to take them into account.
I still find it incredible that "Daumensprung" [0] is apparently not an international concept (I learned about this in Math class growing up in Germany).
Unfortunately the Wikipedia link is only in German, but the short explanation is:
* Extend your arm horizontally in front of you, and raise the thumb to point upwards.
* Close one eye and align the thumb with the edge of a large object in the distance (e.g. a building).
* Close the other eye and note how far the thumb virtually "jumped" across the large object.
Now, knowing the distance between your eyes and the length of your arm, you can use trigonometry to either
* knowing the distance to the object, deduce the width of it
or
* knowing the width of the object (e.g. using a window, doorframe, or car as a reference size), deduce the distance to it.
[0] https://de.wikipedia.org/wiki/Daumensprung
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.
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 :)
what will the sales tax be on this item at the store?
can I afford the items in my grocery cart?
if I buy a TV on my credit card and pay it off in three months, how much extra am I paying?
if I keep doing what I'm doing, when will I be able to retire?
We use these skills a lot more than we may realize automatically and without really realizing we are utilize the skills.
"It is a great profession. There is the satisfaction of watching a figment of the imagination emerge through the aid of science to a plan on paper. Then it moves to realization in stone or metal or energy. Then it brings jobs and homes to men. Then it elevates the standards of living and adds to the comforts of life. That is the engineer's high privilege." - Herbert Hoover
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).
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.
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.
https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...
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.
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.
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.
It all depends on your job, and since we can hardly know in advance, school tries to cover all bases. Drill and practice or the other proposed solutions won't help with that.
When people are free, they learn the math they need for the problems they are trying to solve.
Traditional, compulsory school forces children to solve problems they don't have. See Karl Popper's idea of the bucket theory of mind, or David Deutsch and Taking Children Seriously.
Have you never been in a situation where you are about to take train going at 80 mph; wondering when you'll meet up with your friends going in the opposite direction at 90 mph?
[1]: http://www.launchphotography.com/Starlink_4-6.html
I believe also the launch vehicle has a launch window (the moon moves 15 degrees per hour) and launch trajectory. I'm lazy so I'd compute the extends of the launch vehicle's motion in the sky (from earliest possible launch to latest possible launch), and then intersect that geometry with the moon position geometry without explictly trying to solve the equations simultaneously. That should back-project to shapes on the ground at which point you could reasonably expect to be able to get a good shot, and then you'd do some adjustment in your pointing in real time.
A smart college senior could do it directly (IE, not lazily compute a bunch of points and manually intersect them on a screen).
I didn't know, for the longest time, what a derivative was. I still don't, not really, though I can bluff my way to a reasonably correct answer using the above analogy and going from dry "rate of rate of change" sort of language into more practical concepts.
Citation needed.
The first order ordinary differential equation initial value problem
d/dt y(t) = k y(t) ( b - y(t) )
can be use to model viral growth. Once it kept two crucial FedEx BoD members from walking out and saved the company.
Similarly for the law of cosines for spherical triangles for finding great circle distances.
Linear programming (LP) gets used right along for actual, genuine LP problems and also as a means of approximation for problems that are not linear. The problem of minimum cost flows is a special case of linear programming.
If want to sort keys, e.g., for some positive integer n, if want to sort n numbers into ascending order by comparing numbers two at a time, then heap sort does that in worst and average case in O(n log(n)) and, thus is the fastest possible -- this is from a cute counting argument, the Gleason bound, A. Gleason.
Statistics is important and parts of it are awash in math, not all of it trivial.
The design and operation of the Webb telescope is awash in math.
Knowing a triplet ain't employing Pythagorean theorem.
> [lots of following examples that are moderately deep applied math used in lots of skilled professions]
Most people don't do this kind of work or analysis.
3^2 + 4^2 = 5^2 = 25 guarantees that the angle between the side length 3 and the side of length 4 is a right angle which is what the carpenter very much wanted to know. E.g., the sides might be 30 feet, 40 feet and 50 feet, e.g., for the foundation of a house, and presto, bingo the carpenter knows that the 30 foot side and the 40 foot side form a right angle, are square, which can be very important to a carpenter and not so easy to know otherwise.
A whole lot of people know that measuring a bookcase corner to corner is a good way to square it up, too. But they may not be masters of geometry beyond knowing a rule of thumb.
I don't think this is the right way. I believe something happens with math to many/most students which leaves them less than fluent in mathematics.
Somewhere along the way, they stumble for a few weeks and fall behind. Maybe their family took a vacation. Maybe the teacher's explanations don't make sense to them. And then there's a deficit.
Pretty soon, supporting that student through math becomes a whole lot more about drilling and memorizing strategies and understanding is deprioritized.
This becomes insurmountable, especially with the "layer cake" model of how math is taught in US schools. Pretty soon you'll never catch up in understanding. But you probably just attribute it to "not being good at math".
I'm working with a student right now, who is in precalc and "bad at math." He is actually really bright. Some missing knowledge and intuition about fractions has made everything since much harder. And the problem's never been fixed because he's constantly been in a survival mode in his math classes.
These students never end up doing the kinds of things that kids are ahead in math are taught to do to crack free-form problems:
- Make a good guess of the answer
- Form a mental picture and use it to select strategy
- Work a simpler problem and see how this can be extended
- Organize your givens ; use dimensional analysis for a hint as to what operations may be required or what picture should be drawn
- Look for symmetry and transformations that result in simpler cases
- Does positing an incorrect answer reveal any information about the solution?
IMO, teaching these things -- strategies-- should be more the goal than doing lots of kinds of applied math work by rote in a way that deeply depends upon previous work.
… which is all the more reason I’ll never become good at math. I’d rather drill a hole in my head than subject myself to drilling back to back, contextless, boring problems I don’t care about.
Arithmetic facts, basic algebraic manipulation, big polynomials, derivatives and integrals.
But we do much more drill than that. And, worse, it's often as homework, where you can be guaranteed a decent slice of the class is practicing it wrong.
From what I can tell, most people don't do things like install projectors into rooms. They either get someone else to do it, or just figure out some placement that is good enough via intuition or trial and error. If people have jobs that require it, they usually remember the bare minimum for their job and aren't remembering general concepts to help them solve brand new problems.
If there is causality here, I'm not sure which way around it is. Either they don't need maths because they don't stuff that needs it, or they don't do stuff that needs it because they can't do maths.
When I was in school, my math teacher used to write AMBS on the board before calculus lessons: Algebra Must Be Strong. What happens when you study calculus is you end up rehashing algebra over and over again until it's second nature. Consequently if you find someone who's done well at calculus, you can be sure they can do algebra.
Someone mentioned this in relation to RenTech. Why do they hire mathematicians who have studied incredibly complex things with seemingly no relation to trading? One reason might be that people who have done that know the fundamentals very, very well.
Similarly accountants don't use vectors or geometry at work either, but they certainly are comfortable with arithmetic and its application in spreadsheets.
When I look at my own education, it is also the case that the things I'm least sure of are the deepest, latest courses that build on everything else. Probably if I built on top of those I'd be more comfortable with them. Taking this line it would appear that you can't just teach people the math they'll use, because they won't really get it until they try to do the next thing.
A thing that needs addressing is the notion that we should only teach people things that will be useful, which seems to underlie this article. We should teach people how to explore, and math is the premier example of a large space that can be explored with very little access to resources and evidence. You can do it just in your mind, and you don't need any apparatus or conclusions from people who had such apparatus, like you do in science.
To give an example, making a three legged stool I needed 3 equally spaced spaced points around a circle, I could find the center, draw the first point then use a protractor to find the other 3 points at 120 deg. But it was far far quicker to just use a divider, step around the circle a few times adjusting until 3 steps takes you back to the first point.
Of course this is still some form of math, but it doesn't involve any calculation. I feel there's a whole world of very useful techniques that work on a different abstraction, though I'm struggling to name or describe to properly.
[1]: https://www.ezwoodshop.com/345-triangle.html
No, please don't do this. When I was learning math as a kid I found word problems incredibly irritating and stupid. It's just a way to waste a few seconds of my time when I just want to solve the problem and get to the next one. Word problems are just disrespectful
In real life, a lot of those situations are hidden, but there are often multiple correct ways to approach the problem and hopefully some intuition will help you if you're too far off track. 2/3rds of 3/4ths of a cup of cottage cheese shouldn't be more than one cup and shouldn't be a tiny amount, either. (I also don't see a problem with people who scooped it onto the table). Similarly to the woodworker (elsewhere in this thread) trying to evenly measure out where 3 legs on a stool would go.
Identifying a problem (in math class the problems usually come pre-identfied until you tackle the dreaded story problems) and breaking down the problem into logical steps, are useful skills to have even if someone never does another math problem in their life.
I see a lot of math homework questions on Quora, because students copypaste their homework all the time:
What is log (3x +2)- log2 =1? How do you prove that X^2-3 is irrational? What are the roots of the equation x4−4x3+6x2−4x+4=0?
When on earth are you ever going to use any of that?
The word problems are no better.
These just aren't the kinds of questions anybody ever actually asks. The math required to solve them is pragmatic; are there no pragmatic examples they can come up with?The fact that students are copypasting their homework doesn't thrill me for their ethics or their intelligence. But the fact that the problems themselves are so stultifyingly dull almost makes it seem reasonable.
Let the future STEM majors learn the rest in their classes. It takes a long time for a lot of that stuff to pay off, and much of it remains irrelevant.
I'm not sure what math you need to read bills in the age of specialty apps. I can't remember the last time I needed to do any arithmetic without my phone.
Estimating how much food to buy for 2 people requires understanding nutrition science, whether all your guests ate at all today, how much space you have to store leftovers, etc.
The only math I can think of is jus t multiplying by number of guests, But then you might have kids or people with dietary restrictions and now half your planning is just individual person by person stuff.
Math won't really help you plan a budget either unless you know all the finance specific stuff, or enough fairly advanced general math that it's just all obvious to you.
There's way too much estimation of real world stuff and most of the controllable expenses(for an average poor person) are impulse fast food orders and such that they already know are costing a ton.
I'm not sure trust an MBA to teach... anything about economics. They are way too high on their positive mindset business optimism attitude to really understand that the cost of getting Starbucks actually is important.
A lot of the "Useful real life math" is stuff that people won't use anyway, either because an app does it, or because retirement is later and doordash is right now. Is math actually the most effective tool to make someone want to stop that?
The reason why math is taught in school is not because math is useful (it is useful in certain situations, but this is only tangentially related). We teach it because it keeps students busy and out of trouble, and lends itself well to unambiguous testing and is therefore an effective tool for sorting students into the various social classes.
This all became clear to me when I was in the middle of my math PhD and wondering why all of the math courses I had taken did next to nothing to prepare me for actual mathematical research.
Viewed from this perspective, it’s obvious why we don’t use the math we learn in school.
As a generality, isn't the math taught to most teens in public schools aimed pretty squarely at preparing them to take the standard calculus courses in their first couple years of college? If so, hardly surprising that most ex-students don't notice much use for that stuff in the real world.
And since the social function of that is usually to demonstrate that you're determined / focused / upper-class enough to successfully slog your way through it - gee, what would be the point to remembering or using it after getting your "made it through" bragging rights?
Basic math is very useful but highschools go so far off basic math that they teach massive amounts of seemingly useless stuff.
"Useful" to me means "Lets you do something you couldn't before with a phone".
Aside from imperial inch fractions(Not a fan of those!), most things require fairly little understanding of fundamentals.
Because most people aren't in tech jobs where they can use it.
I don't have a checkbook or do my own investing. I'm single, and very little math is needed to pick whatever is cheapest, especially when only certain sizes of some things are practical for one person.
CAS systems can do basic algebra, FreeCAD does geometry...
FreeTaxUSA does taxes, I don't have a car to optimize insurance costs for...
It's always pretty cool to see a chance to use math IRL just because it's such an uncommon novelty, but I don't see it very often.
To unlock new capabilities you have to learn all those basics, then move on to learn the advanced stuff that computers can't already do.
It's a pretty long term investment of time. Maybe worth it, but still a lot of work.
FreeCAD can do geometry, but if you can’t reason your way through the constraints you need to set for it to do that then you’ll be flailing around. FreeCAD definitely saves me from doing arithmetic, but not from doing maths.
Math definitely is helpful, but unless you are a full time MechE, it's not like you need to be able to pass a high school geometry class to be able to print a replacement part for something.
Personally I don’t find multiplying by two any harder than multiplying by 10.
https://en.m.wikipedia.org/wiki/United_States_customary_unit...
https://en.m.wikipedia.org/wiki/Mendenhall_Order
Try adding 3/16 plus 1/2. On a calculator that is three separate operations, and if the output is decimal, you have to convert back to a fraction.
And when automated systems convert to fractions, the amount of error is not obvious. With decimals, the computer just adds more digits unless you limit it to a certain number of places. It is safe to add a few extra because you can ignore them.
With fractions the whole point is smallish integers. If a computer says 3/8 inch I have no idea if it's really 3/8 or if its 11/32 and the computer rounded.
It can't just always show the value in 128ths of an inch because... nobody knows what those are and rulers aren't marked that way. So to have a computer display fractions means understand how it does so, it's not obvious(Unless you displayed explicit error values, which would be pretty great)
The whole thing came from a time when very skilled humans with no tech did everything themselves and seems optimized for that at the expense of everything else.
Decimals way easier for applications where all the math is 100% done by the CAD program.
3/16 + 1/2 is 3/16 + 8/16 which is 3+8 or 11/16. That’s simple arithmetic. No calculator needed.
It's a great "power user" system for skilled professionals in the pre computer age who practiced with it regularly, and it works well in things like woodworking where you still have a lot of things done on paper or just in someone's head.
It's not the best for CAD first workflows, when you're trying to model everything down to where to clamp the edge guide blocks for the saw, and changing something without changing the model and confirming it will all fit isn't a thing.
In 3D printing you get down to 0.4mm or occasionally even 0.2mm gaps between things, although none of it is really an exact science, and you assume some sanding may be needed to make things fit unless you have an amazing tuned printer.
But I suppose woodworking is less of an issue, because you're probably not going to get even 1/16th precision by hand unless you're really skilled, in which case you probably have fractions memorized.
Maybe more drills would help that?
I did pretty well in high school math and got through the most advanced math classes my small town school district had to offer. Then I opted out of driving to the next town over to do a more advanced math class with something like the 5 other kids who qualified. I think it was pre-calculus or something.
So I ended up not doing math for a year, then I really struggled in my first college math classes, and had to retake one of them.
Since then I've basically just forgotten everything because it comes up so rarely. How often do I need to do fractions? almost never, and when I do, I can just look up a factions calculator. Never mind trig or calc.
Day do day, I think I need basic addition, subtraction, multiplication, and division. And I figure out percentages a lot. But I'm also doing all of that with a calculator.
I think financial cautions, like compounding interest are the most advanced things I run into regularly.
For example, why do we make (US) high-school students study elementary algebra? You would be hard-pressed to find an application of the quadratic formula in everyday life, and yet I maintain that it, and the course at large, has an important pedagogical role. All pre-collegiate levels of maths education (arguably including elementary calculus, but certainly everything before that) presents the student with increasingly complex situations, the tackling of which requires increasingly complex problem-solving strategies. From this perspective, all levels of maths education are effectively drilling structured logical reasoning, problem decomposition, pattern matching, rule application, abstract thinking, abstract/concrete translation, precise/"technical" communication, etc. Maths simply provides a rather nice vehicle for presenting these concepts in an ordered fashion with natural increases in complexity. I even believe that the introduction of integers to pre-schoolers is a valuable vehicle for introducing causality and linearity of time.
Along the way, there are certainly valuable real-life skills that are clear applications of maths concepts, and I would agree with the article that most maths programs are woefully deficient in really underscoring the value of these applications. The article opens with a comic about needing "two-thirds of three-quarters", but anyone who has ever scaled a recipe, or worked with so-called "bakers percentages" knows that this is far from being simply a theoretical word problem.
As an aside, I spent a number of years teaching elementary arithmetic to adult students (usually in their 50s-60s, and, without a hint of hyperbole, commonly unable to solve "2 + 2" without memorization). My role was to find creative ways of communicating arithmetic concepts to people for whom the traditional approach often failed. This strengthened my belief in the above perspective by providing an interesting corollary: many of these people had come to learn the "maths-adjacent" skills, traditionally taught via maths instruction, in fascinatingly creative ways. If the ordering and equal-spacing of the integers was not something they were able to truly grasp as children, sometimes they would surprise me with their approach to other parts of their lives for which perhaps you and I would have a more "normal" solution to.