Every maths misconception is actually secondary school maths teachers' fault, because they don't think like formalists.
They talk as if mathematical objects really exist, and then wonder why students object to certain mathematical objects existing.
>"i" is defined as the square root of -1! What does this even mean to a student? Nothing! There never was a general concept of "square root" given to them. And there's no reason to believe that just saying something at random won't break all of the features of the real numbers. Students intuit this, and feel like they're being conned.
>0.9 recurring = 1 What does this mean to students? Nothing! They don't even know the definition of an infinite decimal expansion. If they did, this would be trivial to understand. But in reality, it is impossible for students to "understand" except with goofy sophist arguments.
>"i" is defined as the square root of -1! That's not how it is defined in highschool
Joseph Thomas
The issue is that even accomplished logicians will defend a Platonist position (in fact, instead of ever defining a Platonist position, they declare the opposite as absurd and speak negatively of those who hold it).
But looking back at math reforms (pic related), you can't teach the general kid interesting conceptions, appearently. Probably, there will always be just plug and chuck and crying about it
Formalism was completely btfo by the time of Gödel and Tarski you goober. Most mathematicians and logicians subscribe to some version of structuralism nowadays. Some are reservedly platonist, I'll give you that, but even they are not even close to 'muh forms'
Nathan Brown
>'muh forms' ? For what do you use the term forms?
I view structuralsim mostly as a way of setting up mathematical notion to speak about and work with them. So I also don't know how you really can put it opposed to Platnoism/formalism. By "structuralsim", you either speak of a system where the structure of e.g. the natural numbers exists in the real world or that it's still providing symbolic frameworks for things hopefully exemplified in the world.
Describe your structuralism idea and how it's not either of the above.
Andrew Gutierrez
>Every maths misconception stems from a lack of rigor
I think that goes without saying. We used to have 'school maths'-threads on here, where we discussed at length everything wrong with school maths and how to fix it. Mathematicians tried it already, in the 60s, and it was a total failure:
I used to get mad at school maths at times, but I don't anymore. I just accepted it's shitty, and when I see kids I tell them that university maths is nothing like school maths plug-n-chug. I think that's all you can do pal.
I haven't read pic related, but apparently it deals with this exact problem.
Juan Ortiz
>I haven't read pic related, but apparently it deals with this exact problem. Dammit, I wanted to write that book.
Grayson Bennett
But the problem is that maths educators of any type have no business teaching maths topics in such a way that they literally cannot be understood.
Not only is that dishonest, it's not even practical. If a student gets the idea that complex numbers are a bunch of bullshit, obviously he will never use complex numbers for anything in his life.
I'd actually prefer if practical manipulation of numbers was the emphasis, and all of the abstract stuff was thrown out or became a different subject. It would be better than butchering it.
Evan Sullivan
What are you smoking? No concept of a square root?
That's how I learned it. What other way would there be to teach it?
Liam Smith
>What other way would there be to teach it? >teach It's not "teaching" anything to define i as the square root of -1. It's just creating the impression that maths is a game where you can say any combination of words you like.
The honest way to get from real numbers to complex numbers is to define complex numbers as ordered pairs of real numbers, and then note that the complex numbers contain a substructure that has the same properties as the real numbers.
Lincoln Ward
Structuralism in general is the view that mathematics is the study of structures, which has a range of 'realist' vs 'anti-realist' debates internally. Essentially they hold that the structures are real but have varying positions about what the structures are structures of. Hence {{}} and {1} (or any other notational differences) can be thought of as the same thing without always appealing to the existence of abstract objects. Sure, some mathematicians say they're structures of abstract objects, but the others, which would be gratified by your viewpoint, argue that the subject matter of mathematics is something else. Am example is mathematical fictionalists who say that the subject matter of math is the same as that of fiction, where propositions can be 'true in mathematics' in accordance with the structures of math, but false in reality because numbers and such aren't actually real. Another example is Kantians, who argue that mathematics studies structures hard-baked into the human mind.
Blake Evans
Where do you draw the conclusion that you can suddenly say any combination of words you like? Are you trying to suggest a student would apply that same logic erroneously to some other problem?
Caleb Phillips
>university maths is nothing like school maths plug-n-chug Im going to college next year
math always seemed cool (logic and proof stuff, not really what Ive done so far) but Im not good at math
the only math class excelled in High School was honors Geometry
in algebra and precalc i did pretty poorly
should I even bother pursuit math in college?
Im terrified of taking Calc next year too lol, but I need it for premed
send help
Levi Myers
>Are you trying to suggest a student would apply that same logic erroneously to some other problem? It's not a hypothetical, this is actually what happens.
If you can take the concept of "Square root" (which was only defined for nonnegative reals) and randomly apply it to -1, and give it a name...
Then why can't you give a name to 0/0? This fundamental misunderstanding is where concepts like "nullity" come from.
If you define something at random using words that shouldn't apply, then 99/100 times you'll lead yourself into nonsense.
The only reason that giving 1 a "square root" works in because of deep connections in mathematics.
John Anderson
should be >The only reason that giving -1 a "square root" works is because of deep connections in mathematics.
Joseph Hill
lol I wrote this as if Im drunk and English is my second language. Sorry, I'm just exhausted and my mind is elsewhere right now
Adam Thompson
It's all logic in university. Give it a try, I did poorly at school maths too, and I do great at uni maths.
You probably never had a proper introduction into algebra and calculus, that's why you can't possibly understand it.
Michael Reed
A lot of the people I took Calc I and Precalc with said that Precalc was actually harder than Calc I if that means anything to you, and I maintain to this day that Calc I was the easiest math class I ever took, but I had an amazing teacher for it so that was probably more to her credit than anything.
Nolan Scott
>Im terrified of taking Calc next year too lol How did calculus get such a reputation for being such a hard subject?
Austin Baker
Ive heard this myself
We started doing "calc" according to my teacher in precalc right now. I have an A because of how easy it is but obviously it isn't real calc, its simple limits -- an introduction at best.
Hopefully I have the experience you did.
>It's all logic in university. Give it a try what classes in particular? does this start in calculus?
For me? A lot of my smart asian friends have D's in Calc BC this year
Dominic Clark
>>The honest way to get from real numbers to complex numbers is to define complex numbers as ordered pairs of real numbers, and then note that the complex numbers contain a substructure that has the same properties as the real numbers. The honest way is to adjoin to the reals a root of the polynomial X^2+1. Your suggestion to just call them ordered pairs of real numbers makes them completely opaque.
Carter Robinson
I thought that I really disliked maths, when I was doing precalc. Once I started calculus, even though it was watered down high school calculus, I realized that math was actually fun. Calc I is a lot easier than precalc, especially if you have done any study on logic and proofs.
Matthew Gonzalez
Math and logic student here. In my experience many non-honors first year courses are still very industrial and plug-n-chug with no interest in actually teaching you what's going on. Calc and linear algebra especially. Once you get past that and start getting into analysis and algebra in second year you'll be much better. Lots of people get pushed through first year courses are prerequisites for other fields, so they have similar methodology to school math.
So TL;DR either tough it through first year or take honors if you think you're ready.
Kevin Garcia
>Your suggestion to just call them ordered pairs of real numbers makes them completely opaque. With defined addition and multiplication, of course. The whole thing is rigorous, and not even hard to understand.
>The honest way is to adjoin to the reals a root of the polynomial X^2+1. That's the opposite of honesty. Why do you know that such a root exists? You're using terms from advanced studies of polynomials to found the advanced study of polynomials.
Isaiah Watson
>what classes in particular? does this start in calculus?
It starts with Analysis I and Linear Algebra I at my uni in Austria. It shortly introduces you to logic, proofs, sets, and then goes through the material with logic and rigor. It's really easy actually, since you don't need any prerequisite, just logical thinking.
Ryan Howard
>It starts with Analysis I and Linear Algebra I at my uni You're lucky
>Austria aaaaand that's why
Dominic Gomez
>Why do you know that such a root exists? Because we made up a number that is a root of the equation. Just like we did with negative numbers, fractions, etc. for other equations.
Liam Walker
But you can't just make up a number that has Property P! I mean you can, but why should anyone go along with you? Why would you think that applying all the normal operations on your new number won't lead to nonsense? There's no reason to think so.
If I define a number which gives you 7 when you multiply it by 0, nobody should listen to me.
Sebastian Howard
Negatives, fractions, etc. can and should also be constructed. You don't have to do it at school level, but you shouldn't lie about it.
Nolan Nelson
You keep using words like "honest" and "lying" and I honestly don't know what you're talking about.
Alexander Gutierrez
>I mean you can, but why should anyone go along with you? Why would you think that applying all the normal operations on your new number won't lead to nonsense? There's no reason to think so. Hmm, it's almost as if this applies to every mathematical construction before you do anything with it.
Liam Sanders
No it doesn't. By definition, a construction is built out of parts you already had. So it can't introduce further contradiction into the system.
What you're suggesting is just adding an axiom out of nowhere. The axiom that a certain polynomial has a root. The natural reaction is "wait what?"
It turns out that the axiom is fine because it matches a construction, but it's no wonder people mistrust maths when people talk like this.
Grayson Sullivan
We need to start from kindergarten, instead of pulling the axiom that every number has a successor out of thin air, we should show how to construct them from sets like {}, {{}}, {{}, {{}}}, ... That will be much more intuitive.
Mason Roberts
They should teach and use imaginary numbers in trigonometry. Less abstract is better for most people.
Easton James
>have 4.0 in highschool >immediately make C in cal I My only consolation is that I might have had an A if I did my homework.
Jaxon Butler
Honestly, what do you even do if you want to master advanced math but feel like you never learned any of it well?
Leo Lopez
>What you're suggesting is just adding an axiom out of nowhere. You show that it works out after you suggest it. I didn't realize I only got one sentence to explain complex numbers.
