Is there a math concept that never made sense to you? Which one or ones do you wish someone would explain in new ways?
If none of these come close you can mention something as a comment.
Yes I think fractions cause people as many problems as these other "more advanced" ideas. I've seen students in calc 2 who still had messy ideas about fractions. It's not trivial and just because we teach some of it to 5th graders doesn't mean everyone knows how they work.
@futurebird I have managed to squeak by in the US education system without taking calculus: it is a mystery to me. I didn't like the calculus teacher in high school and managed to get a degree without taking it in college.
@madgeface @futurebird saaaaame. Anything that's theoretical and not tangible was always over my head, I took something calculus related in college, introductory calc + trig, something like that, but the teacher kept explaining then going "oh wait I got that wrong" and correcting herself, I managed to pull a B in the class but honestly had no idea how I was gonna get from there to linear algebra.
@madgeface @futurebird
Same here sometimes but in the end I had a very good teacher 2 years bevor the exams and going to the University! Best math teacher I've ever had during my whole 13 year school career!
@stiefel_fan @futurebird Yup - I had a great math teacher in hogh school that taught me geometry, trig, and stats. I also have a good friend who's a math professor. Both of their love of math has leaked into me. I went to Carnegie Mellon and was terrified to have a repeat of my awful experience with Algebra I all over again, so took stats.
@futurebird
when I was in college I tutored many other students, and I had to keep a list of common misconceptions about fractions and how to address them with me. It's especially bad in programming, because you need to understand how fractions are supposed to work in order to write good code, but you also need to understand how divide, modulus, floating point divide, etc, differ from mathematical fractions.
Number theory is fun. I think it should be introduced in grade school. One of my favorite courses. The concept of ordered infinite sets. Fat points. Limits.
Understand Integral vs Rational vs Real numbers and fractions will always make sense.
I have a vivid memory of asking a lot of "why" questions about fractions to my 6th grade teacher, and her (with no malice) being completely unable to provide anything beyond "because, obviously".
@futurebird Proofs. I simply cannot understand how simple an axiom is supposed to be to start with or how to get from one axiom to another; why the first “proves” the second, third, and so forth. This conditional logic makes no sense to me. Ironic, because I don’t have trouble with any other part of math.
If I'm understanding what you are asking correctly I think the reason why you can't understand is because there isn't one correct answer to the question:
"what axioms do we need to include to proceed with this proof?"
It's contextual. And some teachers don't do a good job conveying this. We'll say "oh that's obvious you don't need to explain THAT." then a moment later flip out because someone didn't support their next step with an axiom. There is a logic to all of this, but yeah.
@futurebird @blbc IIRC Peano started proving that 0 + 1 = 1, so there's no bottom I guess.
@futurebird @blbc I mean I'm not arguing that what he did was silly. The man was trying to set a foundation for Arithmetics.
Ok it was a bit silly, but still.
@adriano @futurebird @blbc I remember seeing a proof that there are no integers between 0 and 1 and thinking it was rather silly.
@futurebird @adriano @blbc Lewis Carroll's paradox seems to insist that even when there is a bottom, there is no bottom, and I've never been able to decide whether it is a legitimate problem or not.
@mattmcirvin @futurebird @blbc Look, the important thing is that when you reach the bottom, you do not eat the thing that says EAT ME
@futurebird @blbc The philosopher and logician W.V. Quine once remarked that asking which statements are axioms is like asking which points in Ohio are starting points.
@futurebird @blbc teaching proofs is _so_ hard. In part because the early problems students have to look at are either "stuff they already take for granted" or "simple toy problemsb only a math nerd could love". And as you say, we have to pick and choose which prior axioms or proofs the student is allowed to use, since most of the problems, or all, already have proofs and we don't want them to just bypass the challenging part.
@btuftin @futurebird @blbc I only did proofs in middle school geometry, and I can't relate to them being contextual. The way we were taught was formulaic, and they were boring problems about angles. Given a starting position, how do you arrive at the end position? It had a lot more to do with computer programming: can you be explicit with your language such that a proof-reading machine could follow your work?
Years later I read Lockhart's Lament, and I wish I had learned other proofs too.
@semitones @btuftin @futurebird @blbc my goodness, this is pure Paulo Freire!! I've tried to be a math instructor in a freire-an school and failed miserably. This letter actually makes me want to try again.
@btuftin @futurebird @blbc I'm reading this thread and I'm flabbergasted at how poorly maths is taught wherever you're from. Grade and high school math is established well enough that it can be taught as a natural sequence of proofs from the fundamental axioms starting from grades 5. (There IS the matter of teachers' capability of course. When my sister got out of middle school thinking mathematics wasn't a logical subject I was «WHAT HAVE THEY BEEN TEACHING YOU THESE YEARS?!»)
@oblomov @futurebird @blbc I'm curious how much experience you have in instructing a full classroom of students, with varying interest in math and current skill level. But hey, if where you are from has solved math instruction, I'd love to see some literatureb on the approaches. Though apparently the approach failed your sister, or was she in a different school system?
@btuftin @futurebird @blbc I do have experience teaching. The USA have a notoriously lacking track record at teaching maths.
https://www.nationmaster.com/country-info/stats/Education/Mathematical-literacy
Low quality teachers will affect negatively the students' performance regardless of the school system guidelines.
@btuftin @futurebird @blbc This is why I like proof writing in algorithms-focused classes. There, you can focus on really straightforward problem. For example, prove this function has no unhandled input combinations. Static analysis handles that for programmers *by creating a proof*, so it even comes with an easy answer to the “when will I ever use this in real life” question.
On a related note, I think we really need to introduce the concept of algorithms *much* earlier in math education. They’re the key to understanding huge amounts of math.
@futurebird @blbc "Then a miracle occurs."
Simply stating that "it follows that ..." can work, but it should be done with caution.
A proof is a lot like writing an essay in that you don't just need to know your subject matter and be correct about how the math works, you need to know your audience. You need to show that what you are proving follows from the theorems or axioms you and your reader *already* agree are perfectly supported.
Some students will read what they need to prove and reason for themselves why it MUST be true, but they struggle to express this reasoning clearly.
Wow, I have the exact opposite experience. I think visually and kind of “see” proofs. I roll things around mentally until they fit. 95% intuition.
I think that's a terminology problem. Anywhere outside of mathematics, a proof is a thing that you do to be able to claim that something is true.
In mathematics, a proof is something that you do to demonstrate that things are self consistent. A mathematical proof doesn't say 'X is true' it says only that 'X is true if all of the axioms are true'.
Outside of mathematics, 'true' is never an absolute because you always have some measurement errors, and so any notion of a thing being proven is probabilistic. Courts talk about 'beyond reasonable doubt' and 'in the balance of probability' as the requirements for evidence-based proofs of guilt for precisely this reason.
This is a really important distinction for applied proof techniques. We use formal verification a lot, but I always remind people that formal verification doesn't prove that things are correct it proves that all of your bugs are present in your specification. You can then step further back and prove that some properties are present in the specification, but that doesn't mean that the properties that you're proving are the ones you really want.
@david_chisnall @blbc @futurebird > A mathematical proof doesn't say 'X is true' it says only that 'X is true if all of the axioms are true'.
Uh, an axiom is something that assumed without proof to be true in all circumstances. A proof applies the agreed axioms (or, as a shortcut, already established theorems) to show that if CONDITIONS are true, then RESULT must follow.
@mansr @blbc @futurebird I'm not sure what you think you're saying that contradicts anything I said.
An axiom, as you say, is something that is assumed to be true. You can start with the axiom that 1+1=3
and build a structure of proofs. The validity of the proofs within the framework of the axioms is independent of the fact that the axiom is something that most people who are not GPU designers would view as nonsense.
@david_chisnall @blbc @futurebird You said "if the axioms are true." Axioms are by definition true. Everything else, I agree with.
@mansr @david_chisnall @blbc @futurebird no, axioms are not ‘by definition’ true, axioms are ‘assumed’ true. The canonical example is Euclid’s parallel postulate https://en.wikipedia.org/wiki/Parallel_postulate
If you assume it false, you get non-Euclidean geometry of various types.
@Tom_frog @david_chisnall @blbc @futurebird That's a level of pedantry that makes any discussion impossible. If you want to allow axioms to be false, the truth value of each statement must be included in the axiom, and then we're back to them all being true by definition.
If you still want to argue, I suggest a visit to Monty Python's Argument Clinic.
@mansr @david_chisnall @blbc @futurebird
Not trying to be pedantic, but you are actually right by definition when you say the truth value is included in the axiom. So if we say “assume (ie take as an axiom) for a given straight line and a point not on the line, there is exactly one line through that point parallel to the first” , you get Euclidean geometry.
You can also assume there are many such lines, or zero, and you get useful and interesting non-Euclidean geometries.
@david_chisnall @blbc @futurebird
" ... it proves that all of your bugs are present in your specification."
I love this statement!
@futurebird @blbc @david_chisnall
Makes me think of Bayesian statistics.
@david_chisnall @blbc @futurebird > Courts talk about 'beyond reasonable doubt' and 'in the balance of probability' as the requirements for evidence-based proofs of guilt for precisely this reason.
I suppose this could be expressed mathematically using conditional probabilities, but juries might struggle with that.
@mansr
@david_chisnall @blbc @futurebird
I was on a jury in a court case to determine whether the defendant was a "sexually dangerous person" who should be diverted to a special program.
It hinged partly on a 10 point dangerousness test the state psychologist administered, and one guy on the jury was a data scientist who asked about the P-value of evidence supporting the test. The rest of the court and jury, including the expert witness were like, wha?
@futurebird I was in a "math/physics/informatics" profiled class, and our math teacher was an absolute legend. Friends who went on to study math easily coasted on what they learned in high school for a year or two.
That said, conditional probability remains Black Magic to me.
@rysiek
Not exactly math, but I could never wrap my brain around monads.
@futurebird
@Mux the main thing to remember is that the Monads have no windows.
Oh wait, you weren't talking about not the Leibnizian ones. Never mind!
@davefischer @rysiek @Mux @futurebird
I have injected Trurl and Klaupacius into several use cases I've written for my clients.
1/ Conditional probability explained to my kids:
Deck of cards. Shuffle. Now draw cards until you get an ace:
P(Ace) = (Number of favorable outcomes) / (Total number of possible outcomes)
Let's break it down:
Favorable outcomes: These are the outcomes we're interested in, which is drawing an Ace. There are 4 Aces in a standard deck.
2/ Total possible outcomes: This is the total number of cards in the deck, which is 52.
So, plugging in the numbers:
P(Ace) = 4 / 52
This can be simplified to:
P(Ace) = 1 / 13
Therefore, the probability of pulling an ace from a standard deck is 1/13, or approximately 7.69%.
This equation represents the basic foundation of probability.
Also card counting, which will get your picture in The Book and you'll never get in a casino again
@tuban_muzuru @rysiek @futurebird Invaluable in a game of bridge, though.
@benfulton @rysiek @futurebird
A friend of mine in high school ( with an Erdős Number of 1 ) went on to become a very serious quant. Also a phenomenal poker player.
But he and I both hated bridge.
@futurebird haha I still don't get it.
Draw until you get an ace. Probability is 1. You will get an ace if you keep drawing. Probability of drawing an ace before only 4 cards remain?
Is it
4/52
plus 48/52x4/51
plus 47/51x4/50
plus... Down to
Plus 2/6x4/5?
No.
What is conditional probability?
It's the probability of an event happening given that another event has already occurred. Think of it as narrowing down the possibilities. We're not looking at the whole universe of events anymore, just a specific subset.
@tuban_muzuru @japonica I'd like to not be mentioned in this thread, thanks
@rysiek @futurebird I kind of understood that in my math courses, but never got any good at a lot of the probability stuff. Abstract algebra's the one that I could never get, probably because I had trouble with the language. That and differential equations made me switch my major to computer engineering.
@rysiek
"Conditional probabilities are difficult. All probabilities are conditional."
John Cook
Part of the reason conditional probability and probability more generally are confusing is that it insists on living on the margin of human language expressed in words and sentences and mathematical representations of that language.
And our language simply IS NOT precise when talking about cause, effect, probability, dependence and a whole host of topics. It's a big mess.
@futurebird @rysiek There are a bunch of "rhetorical fallacies" that are false when taken as statements about logical implication but that I've always thought appeal to us because they resemble correct statements about conditional probability. You can take Bayes' Theorem as a statement of how much you can really affirm the consequent.
@futurebird yeah, that makes a lot of sense.
@futurebird I get fractions because I worked doing lots of carpentry and construction for years
@futurebird I guess fractions always felt a little bit weird until much later in maths when you feel more at ease not calculating everything.
Like, when I was comfy leaving things as an expression, I had less anxiety around fractions.
@futurebird I get fractions now, but I changed schools mid-year that year. The old school hadn't gotten to them yet, the new school had already finished them, and my teacher assigned another student in the class to help me catch up. It went ... poorly. It took years for me to bounce back from that.
@robotistry @futurebird A similar thing happened to me with (1) cursive handwriting, particularly the horrible capital letters, and (2) the multiplication table. I switched from a second grade class that never got through them to a third grade class that assumed we had them down already or would catch up in our spare time. I am weak with both of them to this day.
@robotistry @futurebird I never had to do fractions because when I was in primary school decimalisation came along just in time so fractions, being old-fashioned, were thrown out along with everything else old-fashioned, so I avoided that
I avoided most of maths because I went to a lot of different schools consecutively and similarly they hadn’t done a thing at the previous school that they’d already done at the next school, so most stuff I got away with legitimately not having the slightest clue how to do
Eg, late in life I encounter the word ‘calculus’ and people talking about it as though they did it at school, yet my entire time at school I don’t think that word ever popped up at all, whatever it means
Of course there’s always the chance that a thing in maths did appear in the lessons and I simply wasn’t aware of it ever because I was always messing about distracting the other kids instead of paying attention