"what he really liked was solving problems. The text of each chapter was just some advice about solving them. He said that as soon as he got a new textbook he'd immediately work out all the problems—to the slight annoyance of his teacher, since the class was supposed to work through the book gradually."
is literally me. I did that. Every year at my school I did exactly that. Once I actually turned in my solutions and my math teacher was quite upset because she didn't know what I'd do for the rest of the year in her class. She thought I was being arrogant and I should take in the material slowly, not swallow it all like a whale. But I wasn't arrogant or anything, because unfortunately this skill didn't transfer to the rest of my classes. I wasn't particularly good at history or physics or anything else, only math. Even now, I have tons of Schaums at my home. Like this one - http://www.amazon.com/Schaums-000-Solved-Problems-Calculus/d...
I work problems in it just because it is a craving - I simply have to solve it. Sadly, society doesn't pay for this sort of addiction. I have been a professional programmer for the past 2 decades to pay the bills, but I secretly hate programming, debugging, programmers, git, the whole enterprise - just seems so stupid & futile. But hey, atleast I can spend my salary on Schaums.
>Every year at my school I did exactly that. Once I actually turned in my solutions and my math teacher was quite upset because she didn't know what I'd do for the rest of the year in her class. She thought I was being arrogant and I should take in the material slowly, not swallow it all like a whale. But I wasn't arrogant or anything, because unfortunately this skill didn't transfer to the rest of my classes
Accusing people of being "arrogant" is a cheap way feel righteously indignant at the expense of someone smarter than you. I was fortunate enough to have some very nice teachers in grades 11-12, who complimented me on my intelligence and didn't try to take me down a notch for the sake of their own egos (nothing wrong with the teachers below grade 11, these issues just didn't come up as much at that time).
I studied mathematics as an undergrad, and later got into programming. Now I do machine learning as my job, and study dependently typed programming languages for fun. If you like mathematics I highly recommend Haskell and Idris (or Coq or Agda, but I found Idris them most approachable, as a programmer). In 50 years I think everyone will be using something dependently typed languages (or some other kind of language that is also fundamentally different to existing languages).
This is a bit of tangent, but I looked up Idris since I'd never heard of it. I'm currently learning Haskell and they look really similar. What differentiates the two? What can you do in one that you can't do in the other?
As practical matter, Idris is a new experimental language while Haskell is a mature, stable language with a useful standard library.
But fundamentally Idris is theoretically more advanced than Haskell. The core differences are
1. Idris functions can be proven to terminate (if you choose).
2. In Idris, types are first class values, and you can have dependent functions: functions whose return type depends on their input value.
An example of something you can do in Idris and not* in Haskell, in Idris you can define a vector type Vect n a, which is the type of vectors of length n with values in type a. You can also define Fin n, the set of integers less than n. Then you can define a function index : Fin n -> (Vect n a) -> a which takes an integer less than n, a vector of length n with element of type a, and returns an element of type a. This function is guaranteed to return a value, because the index is guaranteed to be in the correct range.
*For some meaning of "not": you can probably do this in some way in Haskell.
Yeah, you can definitely do that example in Haskell – if by Haskell you mean GHC. GHC has been slowly but surely adding dependently-typed features for quite some time now, which will culminate in a proper DependentTypes pragma sometime soon (for some meaning of soon, anyway).
There are a lot of other differences between Haskell and Idris, though. Totality is a pretty big thing, and Idris is also strict by default. It also has support for proof tactics, which I don't see Haskell getting anytime soon.
I'm not an expert, but I wouldn't think so. To me, full dependent types just means you have pi and sigma types. I don't see anything fundamental about pi and sigma types that would prevent you from having bottom in your language, or from allowing users to define partial functions.
Yes, it happens at compile time. To understand how proofs work here, you want to look at the Curry-Howard correspondence. Basically, there's a correspondence between types and propositions, and between the values inhabiting those types and proofs of the corresponding propositions. So a proof looks like a value of a certain type.
> The text of each chapter was just some advice about solving them. He said that as soon as he got a new textbook he'd immediately work out all the problems
I literally did the opposite of this. I went so far as to make deals with my teachers that if I got an A on every test, I wouldn't have to do any work outside of class. Maybe I figured it to be a challenge, maybe I was plain lazy. Whatever it was, that didn't prepare me for college.
For me it always was about getting how the system works not the actual lesson. From that angle it is hard to be bad at anything at high-school level. Either you got it how school works and were good or you did not and were bad. For me there was no in-between and all the "people have different talents"-stuff. For me it was about a combination of people skills, short-term memory and keen perception.
I took it to the extreme though and optimized for the amount of free-time, which forced me to change schools.
> Maybe I figured it to be a challenge, maybe I was plain lazy. Whatever it was, that didn't prepare me for college.
Exactly this. I study CS, in the end I lack the discipline to force me to do stuff I am not interested in. Taking tests without visiting the classes and learning for 3 days does still work for smaller conceptual classes, for math or practical ones not so much. It is kind of childish, but I still need the "beating-the-system"-incentive to learn complex stuff. Math always sounds mildly interesting to me and I get the concepts quickly, but i lack the discipline to really internalize it for a few months, especially bottom-up. For me it is easier to come from the other side, for example digging through scikit-learn and learning the math after I already got the big picture.
> It is kind of childish, but I still need the "beating-the-system"-incentive to learn complex stuff.
I can relate. I had a few experiences like that in college.
I once got a bit depressed and skipped a lot of classes; when I finally dragged myself to a numerical methods class I was told that I might be unlikely to pass it at all given all my absence. For some reason this made me so interested in the topic itself that I spent ton of time learning and internalizing the concepts, aced all the assignments and in the end I put the PhD that taught our lab classes in a very awkward position - he wanted to give me the best possible mark but he couldn't given my initial absence and the established rules (he actually did stick to the rules he set and gave me a reduced grade, for which I highly respected him and later choose him as my BSc advisor). Funnily, the momentum I gained actually transferred to other classes so I pretty much aced everything that year.
I had a lot of other situations of the kind of "what do you mean this language doesn't even have functions? I'll hack it until it gains them." leading to the most crazy final project submitted; or "what do you mean I can't ace this class? I still have 24 hours left to do a project!". As long as I was feeling that I'm beating the system in a most overkill way possible, taking the doomsday scenario and sticking it back to the faces of naysayers, no task seemed like a chore. I was in perfect state of flow.
Sadly nowdays it's very rare that I find myself in such scenarios. But when I do, I literally don't need to sleep at night.
>Math always sounds mildly interesting to me and I get the concepts quickly, but i lack the discipline to really internalize it for a few months, especially bottom-up. For me it is easier to come from the other side, for example digging through scikit-learn and learning the math after I already got the big picture.
You probably already know this, but that just means you're an inductive learner, not a deductive learner. I'm also an inductive learner, but unfortunately the majority of math instruction is based on deductive learning.
i hear you, but clever deal anyways. my laziness in high school definitely didn't prepare me for college. my high school geometry teacher started grading our homework after she found out i wasn't doing it, even though i was acing the tests. i still didn't do the homework, banking on the idea i'd get an A anyway. she changed the grading to make homework worth just enough to give me an A-. =\ short term thinking...
I owe my career to not doing math homeworks. School math wasn't something I cared about, but I wanted to make games so I learned programming instead of doing math problems. Ended up catching up with the math through gamedev anyway - it was very easy to learn once I cared about it.
In the end I had mediocre math grades throughout school, but learned a ton of skills I now use to create stuff.
I thought I was absolutely the worst person in the world at math. Turns out, crappy teachers and a teaching methodology that is diametrically opposed to one's optimal learning style count for a lot in school.
However what was amazing to me was how I went from zero confidence in my math skills to actually being excited about math when I took geometry. I never studied once in that class--just absorbed and immediately internalized what the teacher said. I could look at a proof and it just sort of visually made sense to me and clicked and I could step through it because of the pattern recognition. To this date I've never experienced anything intuitive in that fundamentally primal manner.
What has been great lately is Khan Academy and a growing interest in teaching myself software development has rekindled my interest in math. I also owe a huge debt of gratitude to Kalid @ BetterExplained.com (he frequents HN) for getting me past my fear that higher-level maths were beyond my capabilities. They weren't--I just needed to find a way of applying them in an intuitive manner that I could easily internalize vs. staring at equations and their definitions.
I wouldn't be surprised if I would have ended up as an engineer if I hadn't had such a poor experience with math when I was younger. I am pretty resentful of it. Fortunately I can take steps to change that, and I am.
I encountered many students with such misconceptions of their math ability when I taught as a grad student for two years. With many of these, it turned out they had a strong talent for it, they just didn't have a good pre-college math education. I would tell those students that they possessed a good ability at it and encourage them to try pursuing a math-related career.
I don't know if I made a difference in that regard, but it appeared to help bolster their confidence in themselves a little. Educators should do more to reinforce their students and help improve their passion. The state of our K-12 education system is horrid overall here in the US.
> Educators should do more to reinforce their students and help improve their passion.
Absolutely. I studied CS, failed at Algebra and Analysis and decided to drop out. Perhaps I lack the talent/intelligence but I've definitely lacked a good math experience in school and tutors at university haven't been helpful at all. Now I am doing my masters degree in business (with little passion for it and mediocre grades) wondering what I am going to do with that degree. I still dedicate my spare time improving my programming skills but only occasionally find time to make significant progress.
As (failed) physics major, it sounds so strange to hear someone claim to be good at math but not physics. To me physics is 98% math. It's a few empirical facts from experiments and then bucketloads of math.
There are some skills in physics which are sort of foreign to the math method of problem solving. (Speaking as a physics major)
For example, picking a nice reference frame in simple mechanics problems is something that a physical intuition is good for. Same with spotting symmetries in an EM problem. Also, in physics you need to have a good grasp of what to ignore because they have only a small effect on the solution or because it operates on a different scale (fringing effects, transient solutions in ODEs), which often relies on a very hand-wavey type of reasoning.
Essentially, physical intuition often does not map to mathematical intuition
I have not taken enough math to actually speak for them; this is mostly gleaned from talking with math major friends and my own speculations.
I majored in math and physics. The same kind of intuition that you use for physics you also use for math. In physics you may call it choosing a reference frame, in math you call it choosing a coordinate system. At least for me it's exactly the same kind of thinking. You hit the nail on the head here though:
> Also, in physics you need to have a good grasp of what to ignore because they have only a small effect on the solution or because it operates on a different scale (fringing effects, transient solutions in ODEs), which often relies on a very hand-wavey type of reasoning.
Physics is often taught in an exceedingly vague and hand-wavey way. Physics students have to learn to ignore that nagging feeling that something is not quite right. This is impossible for a mathematician. A mathematician wants to cleanly separate the math from the problem that is being solved using math. The problem specification consists of a list of assumptions, and the solution of the problem consists of 100% rock solid math. Physicists weave the two together, so that in the end it's often not clear what is actually being assumed. Furthermore, it's usually not explained based on which experiments those assumptions are justified. A counterexample is special relativity. There it's clearly assumed that the speed of light is constant, and the experiments on which that assumption is based are explained, and from there it's mostly logical deduction. In other topics that is sadly not the case. I would love a physics education where you start with the experiments and work from there, instead of saying "Bam! Here are Maxwell's differential equations. Now deduce things from that based on hand-wavey arguments". I don't mean having the students perform the experiments themselves, just describe what somebody else did and what the results were, and why that led people to believe that the laws of physics are as they are. To make time for that, we should remove the endless by hand solving of special cases of special cases. We live in the 21st century. Instead use numerical methods everywhere, which easily tackle the general case. Got n electrons with initial positions and initial velocities, and you want to see what happens? No problem.
Thank you so much for this; you just described my undergraduate experience in physics perfectly, especially the endless nagging feeling that something isn't quite right -- I never managed to make that feeling go away. I ended up taking a bunch of maths courses towards the end of my undergraduate and can still remember one particular functional analysis lecture on dual spaces where I finally figured out what the hell those dual spaces in quantum were about. Sadly, this was after I had struggled through and finished the full quantum sequence.
Ironically though, I ended up drifting into the EECS department to do applied physics and I've found an environment much more similar to mathematics (and to the experiment-based approach you advocate) than to the physics department -- when you're trying to build systems rather than just solve problems, you can't just wave your hands. Instead, you have to pick apart your assumptions and figure out why you can ignore certain things and not others.
> "Bam! Here are Maxwell's differential equations. Now deduce things from that based on hand-wavey arguments". I don't mean having the students perform the experiments themselves, just describe what somebody else did and what the results were, and why that led people to believe that the laws of physics are as they are.
It's odd that you chose that example, because there is a rich history leading up to Maxwell doing his work, then Hertz verifying it, then Heaviside refactoring the notation.
Perhaps the deeper issue is that there are only so many lecture hours in a semester for adding these details, and you gotta start somewhere.
Physicists seem very hand-wavey to mathematicians, and mathematicians seem very hand-wavey to logicians. If you want to learn what real formal rigor is like, study logic.
True, but a lot of chemistry is mostly math too, and I flunked organic chemistry the first time. I think its because there is some domain knowledge involved in these subjects, which you have to actually care about. I don't particularly care which element has how many isotopes and what the valency is and how many other elements it can bond with, though I did memorize the atomic weights of all the elements in the periodic table and Avogardo number and few such stats. Those things helped a little, because I was ultimately able to pass my chemistry exams just by solving the numerical stuff and ignoring the chemical stuff. I handled physics in same fashion . There's something called the distance equation, where you can express the distance travelled by particle in terms of velocity and acceleration. We had to simply memorize that and use it in problems. I forgot what it was in the exam. So I just derived it from definitions. The teacher was quite surprised, because at that stage we hadn't been taught calculus, so how did I derive it ? Well, I had solved calculus problems by myself, so I just did this - http://pastebin.com/T6PrePh8
Organic chemistry and biochemistry are very different from physics, but chemistry is a very large field and these are just a small part of it.
You should have been exposed to physical chemistry or computational chemistry or maybe even quantum chemistry or analytical chemistry, if your are mathematically inclined.
...but unfortunately they are considered very advanced topics in most learning institutions, even if one could start with them from the very beginning, instead of "classical chemistry". And more unfortunately, after the tedium of "classical chemistry", what you are presented with next are very boring aspects of "organic chemistry" or "biochemistry".
Aiieee, physical chemistry. That thing approaches pure evil.
The chemistry department in the university I studied at required 4 courses of physical chemistry. The introductory course was bad enough - the course had mandatory practice sessions, where assistants were at hand to aid with the supposedly trickier bits. Each session was 1h45m straight.
5 weeks in, there was a supposedly simple exercise. When nobody at the class got even past the initial hurdles in the first 15 minutes, the assistant decided to show how it's done. He failed to finish the calculations in the remaining 90 minutes ... and he knew how the steps went.
Eventually I changed my major from chemistry to CS.
EDIT: btw, the assistant in question was a post-grad so lack of domain knowledge was not the reason.
>> I forgot what it was in the exam. So I just derived it from definitions.
Hah, I did the same :-) ... unfortunately I also forgot the conventional names, and just used 's' (for speed) instead of 'v', and 'd' (for distance) instead of 's'. My teacher did not object the derivation at all, but told me that the results are wrong, because of the letters used ...
Well, point taken - now, I try to pay extra attention and memorize idiosyncrasies like naming conventions ... it saves time when communicating with others.
That reminds me of a sessional (mid semester exam in India) when I got confused on speed of light being 3x10^8 m/s or 3x10^9 m/s. Strange are the ways of mind. I knew the relation between c and electric constant and magnetic constant. So derived it from that.
I was a math + physics double major in college, plus I was also deeply into programming, electronics, and music.
In my view, the starting point for physics is a deep curiosity about how things work. But I'm not sure that aspect of it is ever taught. Rather, it's assumed that good physics students arrive at college, having developed that instinct on their own as kids -- taking things apart, breaking things, asking questions, maybe having curious parents.
Instead, the emphasis in teaching physics is almost purely on the math. Certainly, what defines physics as a unique discipline is the interest in studying problems that lend themselves to mathematical analysis. Solving the textbook problems involves identifying the equations corresponding to the wording of the problem, then solving the equations. That's a skill, but it's not really physics.
I got through my physics courses on the strength of my math skills, but was extremely fortunate to have picked up the empirical half of physics on my own, through my hobbies, and from the curiosity about nature that my parents encouraged. But if someone lacks that background, I could see them being good at math, and maybe getting a good way through school physics, but never really getting physics as an end unto itself.
i was a physics major, too. the student in our year who was best at math had trouble with the "hand wavy" stuff we'd do. "well, this term is basically zero, so we'll just remove it" and then she'd pull her hair in frustration, like "how can you just do that? how is it still valid?"
it wasn't enough to tell her "well it still gives us the valid results" - she had to be able to interpret every portion of an expression and then make intuitive sense of it.
at first i was annoyed with this habit of hers, but eventually i started doing it to. for a lot of really math-minded people, hand waviness is anathema, and physics was full of "2 + 2 = 5 for sufficiently large values of 2, so we'll just assume that to make things easier"
"well, this term is basically zero, so we'll just remove it" and then she'd pull her hair in frustration, like "how can you just do that? how is it still valid?"
I've tried to teach friends with that hangup and found it difficult. Those people weren't good at math, though. I feel like with mathematicians, you should just be able to explain with limits.
> I feel like with mathematicians, you should just be able to explain with limits.
Assuming the limit exists without actually proving the sequence converges is going to produce total junk. Jaynes spent a good portion of Probability Theory illustrating this.
I feel that the right approach (at least it worked for me) is to tell the matematician - "see, this term is very close to zero for all practical inputs we can see in real life (proof of which will be left as an exercise for you), so - as you can see from the structure of the equation - we can safely drop it without introducing any significant error to the end result. Yes, the result will be a little bit less accurate; no, no one will notice. And if you happen to encounter a high-energy (or whatever) problem, you can always go back and use the full equation". Explicitly accepting that each handwave introduces errors and imprecision but that sometimes those errors are below what we can measure (or care about) should be acceptable enough for a math-minded person.
^ This. I ended up majoring in pure mathematics as an undergraduate after finding out that applied mathematics courses had too much material to memorize after finding out that physics courses had too much material to memorize. Pure math was so much easier.
You and the person who started this thread seem to think that way. I find there is just as much to memorize in math. fields, rings, groups, and all sorts of things. Oh and those greek letters. I like math and physics and stuff but I never thought there was significantly less memorization in math. Calc 2 techniques of integration and then DiffEq seemed like a lot of special cases that had unique solutions we had to remember.
Each of those is just a definition and a handful (literally like 5 or 6) of axioms. If I memorized the axioms and a few key theorems I could usually derive everything else in the homework and on the exams. Most mathematical objects have very similar structure and the rest is just maps (morphisms) between them.
> Oh and those greek letters.
That's actually a very valid point. It took me a few years after graduating to realize how much myself (and I suspect many other students) are hampered by not knowing the notation well enough. Most people just assume you (and they) know what it means and never think about the actual definition and ambiguities. Even not being able to pronounce Greek letters definitely makes you less able (or at least less confident) to reason with them.
> Calc 2 techniques of integration and then DiffEq seemed like a lot of special cases that had unique solutions we had to remember.
That's the applied math part of calculus. The pure side is just "here's a compact/open/whatever set, prove for any point in it this property holds." Then you find out no one cares about either and it's all just numerical algorithms.
Note that he says "I wasn't particularly good at history or physics" (emphasis added), not "I was not any good at physics." As other comments point out, there is enough to physics besides math that someone (who was good at math) could easily do well in physics without being considered exceptional.
Funny you should say that. During my undergraduate, I got almost fired from my University for being one of the top students at maths and one of the worse in physics - it was in France, a bit different than in the US.
I recall that a physics chapter would start by "we take the Maxwell equations as [complex formulae]..." and for the life of me I wasn't able to understand them, or see where the teacher took that from.
On the other hand, maths seemed more logical, and even now - almost 15 years later - I can correctly recall my undergraduate maths classes, because I have them so well ingrained in me, because I could understand how various ideas connected to each other.
a lot of it is just cultural approach. math people tend to like to rigorously define everything they're doing, and work from there. physics is more like "what can we make up and fudge and twist and bend and then get the right answer." there's a lot of people who aren't as comfortable working in that less-defined context.
some of this is maybe inherent to the aim of physics, some of it is just physics machismo culture, and could be better, imo.
I don't think it's just cultural. Ultimately in physics you are often faced with a pile of observations from nature that you have to make sense of. Vs math, where you usually have a mass perfectly defined err.. well math that you're trying to build forward upon.
I think the best advances in both math and physics are made with people with an excellent grasp of when to move forward with existing constructs, and when to start shaping new ones...
Math in physics (at least most of the physics I have done, which covers classical 19th century stuff mostly) is just a tool, if you can take a shortcut, take it! If you can approximate and cheat, do it! What matters in physics is the path from observation to modeling. Math is but a tool.
Now I agree that the math gets rather solid, but it's nothing compared to real math at an equivalent level.
I think my experience is particular because in France where I studied, you study both in parallel very intensively. So the math in physics always kind of seems trivial to you... But still my best physics teacher taught me that it was way more about the "feel and model" than the "exactly prove" that mathematics consists in.
Math is certainly required to understand the theory and to synthesize experimental observations into a working model, but I wouldn't put it at more than 50-60%. Most of the math is fairly basic relatively speaking, even at the highest levels of physics.
I didn't get that far in math, but in physics classes after Junior Physics Lab , I once or twice questioned the assumptions. There's always a theorem you don't know about.
I would say math is necessary but not sufficient to be successful in physics. Also, if somebody has the aptitude but not the drive, failure can certainly be an option.
I think that I have a reasonable aptitude for maths and I like solving problems, but I always felt like textbook problems more-or-less consisted of plugging the numbers into a standard template. Maybe my education was uniquely terrible but I'm surprised that there are people who can blast through textbooks and enjoy doing it.
I'll agree with you on that. I enjoyed math problems, but only until I grasped the concept. Then it became tedious. I then preferred to move on to more complicated problems. Probably why I like programming, because there is always a more complex program to be made or worked on.
If you know SICP, I think it's a good example of a textbook with nice problems. My math textbook in French education system were to a similar level of quality. The first two problems were usually to make sure you understood the basic of the chapter, but the others were much harder or opened new perspectives, or used very different context. It was a real pleasure to read the textbook and the problems.
I see you and the OP liked that feature of math problems in the textbooks. I, however, disliked them completely. Here's why:
The format is usually on of: you have a chapter with explanation, samples, and rules. Fine and dandy, and then you get to the end of the chapter with a bunch of questions. The questions, usually, start off with simple, menial ones that test your knowledge of the chapter. These are not challenging in any way, they're just gatekeepers to make sure you memorized and can apply what was taught in the chapter.
And then, they completely flip it around and make the questions completely different and non-standard. They throw you into the deep end for no reason, without any progression. I wouldn't mind them twisting and slowly warping the questions with more complicated constructs that add new/unique elements, but they hardly ever did that. They instead just threw them all, haphazardly, into the "difficult" questions.
Again, I wouldn't mind that if I had some place to look for the answers in that book. Perhaps in an earlier chapter, which would mean that the "hard" problems in subsequent chapters would try combine the concept in the recent chapter with things learnt in older chapters.
Sometimes, I really think they don't always spend as much time on those practice/bonus questions as much as they should. We'd very easily get out of the whole "memorize + apply" rut of education, and into "learn, apply and extrapolate", which is where real intelligence/knowledge is.
I would say truly _revolutionary_. A spectacularly useful tool.
And yet, there is something to Devops Borat's quip that in 1990 entire Internet fits in head, but in 2012, just git no longer fits in head. (paraphrased)
Revolutionary, sure. I just think it solves a problem that shouldn't exist in the first place. Like if we could prevent cancer, we wouldn't need revolutionary cancer medication.
If I go into Google Docs, I can watch different people edit a document at the same time and nobody is really thinking about version control. I think software development should be the same way.
Maybe. (I say maybe because I could see it turning into pair-programming-like chaos.)
But mostly, git solves a different problem. I find it incredibly useful on my own one-person projects, because it lets me explore forward in numerous directions, branching as I go, and easily roll back and move forward again.
I was just thinking yesterday, it would be cool if I could select a block of code and then pull a slider and walk back and forth just that one block of code while leaving everything else untouched. Anyway, we're way off on a tangent.
At my elementary school this was actually encouraged for students excelling in maths[1]. It was offered - but not stressed as important whatsoever - that if you completed the official book of problems ahead of time, you could begin working on the extra book(s) named "Mathimagination". If my memory serves me, there was no extra credit afforded for taking this route; rather, it was purely to keep students interested in and progressing with a subject they enjoyed and in which they excelled. I credit my teacher that year for getting me excited about mathematics, which carried on for many years.
On a related note, my high school had an Advanced Placement English class offered to students selected by previous semesters' English teachers. This option came with an unfortunate snag for all students enrolled in the French immersion track: both the AP English class and a required French class occupied the same period. We were frankly offered the choice to stick with the French immersion we'd been part of for more than 10 years (having started in kindergarten), or to convert to the English track by dropping all French-language classes entirely.
Yeah, the AP class that year was quite small with not a single person dropping the French track. It turns out that people enrolled in the French track are much more likely to land placement in AP English, as being fluent in more than one language steers a person into understanding and appreciating languages more than someone immersed in a single language. Such a ridiculous scheduling blunder by the administration; just the memory of not being part of that class more than 10 years ago makes me sad.
[1] Having grown up in North America where it is common to refer to mathematics as the singular "math", it is still weird to type "maths" even after having picked up the habit a few years ago.
>[1] Having grown up in North America where it is common to refer to mathematics as the singular "math", it is still weird to type "maths" even after having picked up the habit a few years ago.
Maths is also singular, it just ends with an S. (People who say "maths" say "maths is my favourite/worst subject", not "maths are")
Actually, many dictionaries specifically label "mathematics" as plural, but concede that the singular form is far more popular in practice.
I tend to find only one use case where I find a plural form simply appears more natural: prefixing it with "the", as in "The mathematics necessary to explain the universe are complex." Replacing the "are" with "is" just does not sound right. Funnily enough, the natural pattern with the North American "math" becomes "The math necessary to explain the universe is complex.".
If you're that into math, I'm going to second the recommendation of the guy who recommended Haskell. Haskell is VERY math-y, in a way that most programming languages... aren't. ;)
> the whole enterprise - just seems so stupid & futile
Why do you think that? I'm a programmer (who enjoys the challenges of learning new languages (currently Elixir, which is sweeeeeet), coding maintainable code, as well as debugging) and don't think that. Sure, most if not all of my code "out there" is going to get thrown out before another decade passes, but what other job lets you create vast information machines using just your mind and fingertips?
Would you say that working out those problems has given you a solid fluency in math, in that you spend little to no time now on approaching a new problem? I want to develop a strong intuitional base in math, and from what I've read solving problems is the way to go.
> Once I actually turned in my solutions and my math teacher was quite upset because she didn't know what I'd do for the rest of the year in her class.
Obvious answer: move to the next class. Repeat as desired.
What's with all the bad teachers in these anecdotes? I can't imagine any teacher I've ever met getting upset because of something like that. Quite on the contrary.
fascinating. You say society doesn't pay for the addiction but if you could translate a verbal problem into a solution, I think society would pay for that. I mean being addicted to solving problems laid out for you to knock out one by one.. but if somebody was trying to build something and didn't really know how to lay out the elements or variables into an equation, you could solve that and get paid.
"what he really liked was solving problems. The text of each chapter was just some advice about solving them. He said that as soon as he got a new textbook he'd immediately work out all the problems—to the slight annoyance of his teacher, since the class was supposed to work through the book gradually."
is literally me. I did that. Every year at my school I did exactly that. Once I actually turned in my solutions and my math teacher was quite upset because she didn't know what I'd do for the rest of the year in her class. She thought I was being arrogant and I should take in the material slowly, not swallow it all like a whale. But I wasn't arrogant or anything, because unfortunately this skill didn't transfer to the rest of my classes. I wasn't particularly good at history or physics or anything else, only math. Even now, I have tons of Schaums at my home. Like this one - http://www.amazon.com/Schaums-000-Solved-Problems-Calculus/d... I work problems in it just because it is a craving - I simply have to solve it. Sadly, society doesn't pay for this sort of addiction. I have been a professional programmer for the past 2 decades to pay the bills, but I secretly hate programming, debugging, programmers, git, the whole enterprise - just seems so stupid & futile. But hey, atleast I can spend my salary on Schaums.