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.
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.
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.
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.
This helped me to “trust” it in more advanced contexts where geometric intuition becomes exceedingly difficult.
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.
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.
That is what word problems try to instill.
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!
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.
"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.
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.
https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...
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.
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).
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 :)
"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
We use these skills a lot more than we may realize automatically and without really realizing we are utilize the skills.
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?
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.