People just want to know why it's x and not something else or how a letter can have value. They might even think how can 24 + 2 = 5? They just want something to grab onto and nobody is really teaching the concept of a symbol in a math class.
Yeah, lots of existing math texts will forever exist with greek alphabet soup, but we don't have to rely on those as our be-all-end-all teaching tools.
To operate at a high level in mathematics I would agree that having the skill of easily abstracting complex things into compact symbols is a necessary skill, just as I would agree to the same concept applied to software engineering or really any complex engineering system; by the same token, we don't have to START on hard mode with all of our students. Math is infamously difficult for some, largely (I think) because we make it unnecessarily opaque out of some misguided sense of traditionalism.
If we want to have lots of people who are good at math we should embrace whatever pedagogy is effective.
But I've got to say, the short names are not the problem. If you rewrote F=ma as "force is equal to mass multiplied by acceleration" this wouldn't suddenly make it more accessible to swathes of the population. People who are good at maths anyway have no problem with this.
are you really saying that "let function(argument has type RealNumber) has type RealNumber be a function from a real number argument to a real number" is somehow superior to "let f(x) : R->R"
I don't know why this is a startling take. If you encode your ideas in an unfamilar symbology, of course that's going to make it more difficult for someone who isn't familiar with the space.
I'm not arguing that we should teach real analysis this way, or any other high level math class. I'm only contesting GP's comment that there is NO value to be had in using more familiar language to explain a new concept to an unfamiliar audience.
This is the entire reasoning behind "word problems" at the elementary level; they're meant to ground the abstract modeling of a math problem (193 - 3 * 12 = ?) into something more intuitive for a child to understand (If you start with 193 eggs, and you take three dozen away to bake a cake, how many are left?)
> are you really saying that "let function(argument has type RealNumber) has type RealNumber be a function from a real number argument to a real number" is somehow superior to "let f(x) : R->R"
No, I'm saying the there's tradeoffs on either side, and our educators ought to be aware of this.
> math notation is by and for professional mathematicians.
I agree, but we teach math to plenty of people who aren't professional mathematicians. I wouldn't want to do formal abstract algebra proofs in a more verbose form, I'm perfectly happy using the domain notation, but my friends from biological sciences who have to take a calculus course now have to learn both a new symbology alongside the problem domain. I've watched enough of my (clearly intelligent) biology friends slam face first into calculus and spin out. They're not dumb, they can do circles around me when it comes to chemistry, yet they Just Can't wrap their heads around calculus-style math, which leads me to wonder what the difference is between how we teach complex chemistry vs complex math. Questioning the pedagogy is a fairly logical extension of that.
It's hard for me to imagine it without any special notation. It makes me think of the general relativity in words of four characters or less that was posted recently. Sure, it might be possible, but does it really make it easier to understand? Understanding is normally built up in layers. We learn things using big words because that makes it easier than learning with small words. And we learn maths with funny symbols because it makes it easier than learning it with words (or colours or mime or other things you already know).
In fact the analogy to music notation is I think a fairly strong one. People's complaints always sound to me like asking why we don't write "C3 sixteenth note" for music instead of using dots and lines. After all, how are we meant to know what the dots mean and remember the difference between an eighth and sixteenth, or what flats/sharps do? And then the key signature can modify all of it!
The notation just isn't a barrier. Once you learn to read it, it's there because it's a clearer way to write the ideas. The hard part for people is they don't understand the ideas, and don't have the frameworks like key signatures, chord progressions, and meter to place them within. Longer words for variables won't help people understand e.g. inner and outer regular measures, or the open cover definition of compactness. That comes from a lot of work to understand what you're trying to say, the pitfalls of saying it wrong, and precisely how your slightly different way of saying it avoids those pitfalls (or selects the best set of pitfalls if you must pick some kind of degenerate behavior).
Broadly I agree; the semantic density of domain-specific language is often required to operate well in that domain. I disagree some with the "Math doesn't start on hard mode," but I think that's just bikeshedding at some level.
The endemic "I just don't understand math" that my (American) peers have espoused, to me, points to a failure in our (American, public school) instruction practices around it.
We did these sorts of problems for a long time, with addition/multiplication/fractions, and even when we started doing actual algebra the problems were introduced the same way “let’s look at a problem we’ve solved already, and write it in a different way”.
Interestingly mathematical symbols in the past also regularly evolved. Then at some point we just stopped doing that and get stuck in a time which is arguably no longer especially appropriate. So we're left with rather inconsistent symbols, oft reused in different contexts, and optimized for written communication.
Seems like a pretty easy example to make practically, for map have a collection of things, say balls or black. Pick up each one and do a thing to them, paint them blue for example.
For filter do the same except have two different colour balls, if they are yellow they get thrown away, of they are blue they get put in a bucket.
A for loop doing exactly the same you would need to explain the topic at hand, as well as explain iterating an index etc...
Maps and filters also require understanding of higher order functions and the very idea of passing function around as a value. I would argue that implementing map/filter with a loop and then demonstrating how this pattern is generalized as .map()/.filter() functions is better and more accessible
Nobody in school ever tells you that there are glossaries on Wikipedia that tell you the meaning of the symbols. You're supposed to figure it out yourself using vibes.
The way mathematic notation is taught is inherently unstructured. You're expected to just get it.
I often see people make the mistake of trying to teach inappropriately abstract things to small children, because that's what the pros do, and we want the little kids become pros as soon as possible. Problem is, trying to skip the fundamentals is only harmful in long term.
First kids need to learn what all that stuff means, and then we can proceed to teach them the shortcuts.
What I should really do is create myself a cheat sheet of symbols to code...
That said, as a person who moderately enjoyed math in high school and university, this functional notation would make me hate math infinitely more. It's would look like Lisp, which, at high level, looks just as cryptic as algebra. The sheer amount of braces and mistakes that would be made when reading and writing them is nauseating.
Infix notation, for all its flaws, provides important visual aid for understanding the structure of the expression (the sum of two fractions looks very different from fraction of two sums for example). Whereas with functional notation it's like working on linear textual representation of abstract syntax tree. Trust me, nobody wants to read, write or transform one by hand
https://aplwiki.com/wiki/Comparison_with_traditional_mathema...
;-)
Also it should be sum(x+3, x, 1, 4) since you need to encode what the iterator variable is as well.
Edit: Oh wait, someone else mentions map/filter, did they mean this as a combination of range->map->sum and the latter two numbers are the range portion, like sum(map(x+3, 1..4)) ?
Edit2: And now I'm remembering sigma. I think it would have been more obvious to me if the order was flipped and your issue handled the way it is in that notation: sum(x=1, 4, x+3), though I'd still prefer the range notation: sum(x=1..4, x+3)
Granted I always found sigma a bit quirky for separating the range ends like that. Either x below and 1..4 above, or x=1..4 below/above would have been more intuitive.
But it's just a notation you learn once and then you know it.
The way I was taught it and the way that worked now for my now 3 year old is just to say pirates buried a number under the X, and that we need to guess what they buried. If the concept of a number being hidden is a barrier to understanding for anyone they have seriously bad teachers.
This was what? 5th grade?
What kind of crap teachers never taught that
It's rarely the fault of the teachers.
The problem is, in many MANY MANY schools, teachers are more like social workers that have to compensate for utter horrifics outside of school. You got a ton of children so poor they didn't have breakfast which means their first (and all too often: only) meal will be the school-provided lunch (Covid showed that - a bunch of schools were open at least for lunches). You got children that are literally homeless and living with their parents in some car on a Walmart parking lot. You got children whose parents are in and out of jail. You got children living with their siblings in way too small, pest and mold ridden "apartments". You got children whose parents don't have money to pay for basic school supplies. You got children who are dealing with mental, physical and sexual abuse. You got children where the parents are constantly on drugs or seeking for drugs. You got children with a drug dependency on their own - if they're lucky it's just tobacco or weed, if not it's opioids. You got children with parents or siblings with serious mental or physical health issues. Or you got children with their own mental and physical health issues, or if you want it worse, children with these issues but without access to any kind of treatment. You got children that are being weaponized in nasty divorces. You got children that are being weaponized by street gangs. You got children committing crimes from petty theft to dealing drugs just to survive. You got children that have to literally work (and states like FL pushing to have more working children). You got children having their own children already (either from sexual abuse, from under-education about their own bodies, or intentionally because they fell for some stupid challenge/dare). You got children dealing with bullying, you got some who actually are bullies because they have no other way of dealing with their emotions or getting lunch money. You got children with parents with about zero interest in them. You got children who worry that they'll come home and find out their parents got snatched and disappeared by ICE. You got children who worry that ICE will storm their classroom and deport them. You got children who worry they might not survive the school day because someone will shoot at them. You got children who are constantly on the move because their parents' employment/deployment requires absolute mobility. You got children who are LGBT and have to deal with ever increasing hate against them (and LGBT youth already had significantly higher suicide rates than before the GQP made it a culture war issue).
The US doesn't have any kind of system to help these children but schools and libraries, both are horribly underfunded (there's some school districts where teachers gotta take up second jobs because the government can only afford paying them for 4 days a week), and all too often teachers have to pay with their own money for students' school supplies.
And on top of dealing with these kind of nightmares, they actually have to try and teach these children something - even if the children in question aren't anywhere near a headspace where they can actually learn.
Education has a problem with scaling, especially at the elementary level. Sometimes people figure out a nice solution, but when you tell them "great, and now do this in every village" the problem becomes obvious. But there are kids in that village, too, and you want them to know reading and math and hopefully also something more.
Not if you actually provide the money. Europe gets this down decently well - although I'll admit, in rural areas in Germany we got some serious consolidation issues thanks to urban flight.
But at least our teachers are well paid government jobs and the job is decently attractive.
(Yes, this is a political opinion. No, do not blame me for that. Politics does not come wrapped up neatly with a bow tie in a box. If you want to debate the veracity of my claim, go do that instead.)
I'd do no such thing because you are completely correct - the only thing I'd add is that poverty, while being very dominant, isn't the only issue that desperately needs to be fixed.
10•_ + _ = 73