(The answer to both is "all the time", but that's not the point.)

I was a math major who did independent research in Elementary Number Theory and decided to switch paths and become a computer programmer (many years ago). It's true, I haven't encountered many sigmas, deltas, and derivative signs in all those years, but there's absolutely no doubt in my mind that I'm exercising the same mental muscles built with math.

OP mentions “wondering, playing, amusing yourself with your imagination.” Hell, build something.

Assembling ones and zeros to help real people solve real problems is every bit as wonderful, playful, and amusing as anything I ever did with pure mathematics. And probably even more rewarding...I've discovered that it's much more likely to see a customer do a happy dance with software that never existed before I built it than proving Fermat's Last Theorem or figuring out what Galois was up to on that long last night.

Mathematics is like AI. As soon as a company/whoever builds upon one of the advances in the field and people en masse start using a practical result that comes out of it, it stops being called mathematics/AI. When we hear the word "mathematician" we almost right away imagine an absent-minded professor not contributing anything immediately useful to the real world.

People are basically arguing about language, not realizing that they are pawns of a subtle phenomenon that the name of a discipline taught in school refers not so much to knowledge based on and derived from that discipline in the world at large, but to the actual act of teaching that discipline to children and young adults, and sometimes to performing academic-model-funded (meaning charity-like funded) research.

It's a bit sad because this trick the language plays on us probably devalues in the eyes of young people something that is actually a useful/practical skill. Kids may say to themselves, "Why do I need this? I don't want to be a schoolteacher," and they will be both right and wrong at the same time, not realizing how many high-paying jobs from engineering to banks to Wall Street require the mental skill they chose not to spend time developing.

Do I use any of the math I learned even in high school today? Almost never. And certainly a lot of the math I learned in college I have not used at all since then.

That said, I would still do the same thing again if I had the choice. The mental tools I acquired through having an extensive education in mathematics have served me immensely well in my career and elsewhere. So much of math is about following or evaluating the conclusions to assumptions, which is the very foundation of science and of computer science as well, but also critically important in engineering, craftsmanship, and especially in troubleshooting. So much of software development comes down to research, investigation, logic, and the scientific method, and the skills I built up in studying math have given me a very sharp edge in that regard even over folks who have CS degrees from quite good schools.

I found my studies of theoretical math in college enormous fruitful just as I found jogging beneficial. But I thought doing numerous arithmetic problems in middle school to be of minimal value and the rote memorization of things like multiplication tables to be worthless.

In short, much like exercise, studying math should be done in a way that is pushing your personal limits at least a little.

I seem to have learnt checkpoints within the times tables; e.g. when asked 76, I do 75+7. I know 10 and 5 times tables instantly, and adding 7 is trivial, so I clearly never bothered to actually learn the 6 times table.

Rote learning is generally inferior. Give a man a fish versus teach how to fish.

Memorizing multiplication table in English is a lot less intuitive, which I attribute to many people's disdain (or pure hatred, depending on whom you ask) of it.

I'm all for teaching a man to fish, but if he's starving at this moment, save him now by giving him the fish fish first before teaching. A dead man doesn't do much learning.

As for my son, he hasn't learned the multiplication table yet, but he's always interested to have us repeat it to him. (Ah, the curious minds of 3-year-olds.) In a few years, once he has learned the concept of multiplication, it'll be easier for him to apply it for the rest of his life if he has the table memorized. Before that time comes, the table is nothing but nursery rhyme to him. And that's fine with me.

Good to know I'm not alone.

Same with addition, 1324 + 1872 is just 2000 + 324 + 872 which is just 3100 + 24 + 74 which is 3196.

Math is about creativity, somewhat ironically. ;)

It's a technique taught in schools now.

(http://www.mad4maths.com/math_help_multiplication/)

See also "Chunking" for division.

(http://www.mad4maths.com/math_help_division/)

There is a distinction to be made when discussing 'rote learning'. It's always better to understand than to memorize, but there is a case to be made for memorizing some (more mundane) information as an optimization.

Here's what Russel Ackoff professor of systems sciences had to say about it

http://www.youtube.com/watch?v=UdBiXbuD1h4 (3 parts)

Analytical thinking allows you to analyze the system at hand in a way that let you describe its behavior, its structure; it allows you to answer the “how?” question. Yet, it never helps you answering the “why?” question.

So, we [Ackoff and his colleagues] had to develop a new way of thinking; the “systemic thinking” [,or sometimes it’s called “synthesis”]

Synthesis, or systems thinking, allows you to truly understand the system and answer the “why?” question. Actually, it also has 3 steps (the same numbers of step for the traditional analysis), and each single step is interestingly quite the “opposite” to the corresponding step in analysis!

Traditional Analysis

1. You take the thing you want to understand apart.

2. You explain the behavior of each part taken separately.

3. You explain the behavior of the containing whole. So if we’re trying to explain a university, we have to first explain the education system of which the university is a part.

Synthesis (or Systems Thinking)

1. You take the thing you want to understand as a part of a larger whole.

2. You aggregate your explanation of the parts into an understanding of the whole.

3. You dis-aggregate the understanding of the whole into an understanding of the parts, so that we explain by identifying their role or function of a system in the larger system which it’s a part. And the explanation of a behavior of a system lies in its role or function in the larger system of which it’s a part.

Example - You go to a business school and you look at the course structure. You study marketing as a separate subject, finance as a seprate subject, production as a seperate subject. The net result is at the end of the business school you have no understanding of what a business is and not even the understanding of its parts. You can't study production independently without understanding how it interacts with marketing, finance, personnel, product and so on

I was a math major in college. I don't use the actual math I learned, but 4 years of solving really hard math problems at Cal honed my problem solving skills and helped me develop a type of stamina to try my hand at them for long periods of time.

My grades weren't the best, and having math Olympiads in your classes sometimes makes things harder than they should be. That said, I treasure the experience. I learned to push myself in a way I hadn't before. I learned the value of being rigorous, and even the value of getting creative to solve problems. I learned that even after 4 hours of staring at a problem, I should keep at it because I may get that ah-ha moment at the fifth or sixth hour.

So do I use math everyday? Not really. Do I use the ability to solve problems, to abstract my knowledge in one area and apply it another, to carry through long logical thoughts everyday? You bet your ass I do.

This is a defense of pure mathematics. Pure mathematics can certainly be a thing of beauty and that elegance is the draw which pulls in most mathematicians.

But applied mathematics are also a powerful tool. One needn't love the elegance of pure mathematics to appreciate the utility of applied mathematics.

I mostly like applied mathematics, though I appreciate pure mathematics. I like physics and applied computer science. Exploration into them wouldn't be possible were it not for advanced mathematics, but I usually only learn new mathematical concepts when it's demanded by the application that I'm studying. That's not the "wrong" way to appreciate mathematics and the implication is that I've almost always already answered, "Where am I ever going to use this?" before I learn add new mathematics to my toolkit.

The need to optimize and the associated knowledge is not quite as much of a problem as it was in the recent past.

This leads to a hidden form of bloat, programs written in a terribly inefficient way that still perform adequately in isolation but that fail to compose in a useful way because that quickly slows things down to the point where it no longer produces an answer in acceptable time.

That probably falls under ignorance in your enumeration but it may be a sub-category labelled 'appears unnecessary'.

There ought to be a an energy label for algorithms :)

Isn't that what we use asymptotic analysis for?

There should be carbon footprint metric for their implementation as well.

;)

Because they are playing the flute and they are playing football. They enjoy it now and that's what counts, they don't even think about the future.

And yet they come into my math class and raise their hand half-way through my demonstration of the mean value theorem to ask me when they will ever use this.

Because the "mean value theorem" sounds like boring shit and their only hope is that it may be useful in the future.

If you teach them programming to solve their physics homework and make it a game using math, they won't ask.

Never.

True. That's actually how I ended up as a programmer - math in school was boring, trying to make a video game was not. I finally learned trigonometry only when I was trying to figure out how to rotate an object in 2D space for my game. Also, there was a point in time when I used a "formula for distance/vector length in 2D space", but I had absolutely no idea what that "Pythagoras' theorem" is useful for.

For the first problem you could use anything from tight to algebra to calculus to solve it. For the second, you might easily find yourself knee deep in a PDF giving you a "remedial treatment of jacobian matrices"

Similarly, I'd prefer students leave high school fluent in basic algebra, vs. trying to force-march people through algebra 2, trig, calculus, etc. when it's clear they'll just hate the subject afterwards. (Some students will read ahead, great. The others, since they actually enjoyed math, will pick up the other stuff later).

Asking "When will I use this?" reveals that we don't understand a major goal of math education: fluency with powerful mental models, not just factoids you can directly apply.

I think everyone would be much better off if they left high school with a mastery of basic algebra and left trig and calculus for College (for most kids).

There's no doubt that our math education destroys curiousity, and if we could teach math in a non-utilitarian way, as a sort of artistic endeavor, that would be amazing, but... is it even possible? Has it ever been tried? Whenever I think about this, I think about season 4 of The Wire.

I have a slightly silly book about birdwatching. It has a section which talks about birding competitions - "Big Days", they call them - and the final two sentences stick in my head, always:

Don't go away from this thinking that birding is an endless succession of fabbydoo games or fun, fun, fun without a moment's rest. Quite the opposite. Birding is mostly about looking at birds. It is no use to pretend otherwise.

A useful motto.

Birding is very popular, of course, even in the go-go all-action United States. Other popular activities include knitting, gardening, hiking, sudoku, crosswords, reading, fantasy football, Angry Birds, writing comments on message boards, and, ahem, hacking. All of them solitary, cerebral, and socially unglorified. (Though I guess fantasy football is arguable: You can probably spend an entire evening over beer talking with your friends about the performance of their fantasy football teams. For all I know it's even popular in high school. I knew several people in high school who were reliable authorities on every active major-league baseball player...)

People have a warped view of other people's lives in general, and particularly the lives of high school students. I suspect that high school kids enjoy quiet, solitary activities as much as anyone else: In other words, more than you notice. The quiet hobbies are just unobtrusive, so they don't get a lot of press and they don't show up in movies or TV except around the edges.

Speaking for myself, I think I stayed away from band in school because it was full of people and attention. I gravitated toward the quiet sources of pleasure and meaning. Of course, I was blessed with a math club mentor who knew what "recreational mathematics" was...

See this twelve minute video: http://www.ted.com/talks/lang/en/dan_meyer_math_curriculum_m...

In high school, I actively participated--and later led--my school's math league team. Every Friday, we were either home or away for a match with another school. My school wasn't particularly known as a math hothouse (unlike other schools with gifted programs), but we managed to get pretty far in the playoffs for the years I was there, finishing 3rd in the district once even.

Back then, I also actively participated in math and science contests, which eventually led me to be invited to a week-long math seminar. During that week, we had daily team competitions and other activities that were fun, social, and mind boggling at the same time.

And it doesn't hurt on that one faithful June morning too many years ago, I also met the person whom eventually became my husband. wink

http://pauli.uni-muenster.de/~munsteg/arnold.html

He said everything I could possibly want to say as a reply here but probably better then I ever will be able to put it:

It is only possible to understand the commutativity of multiplication by counting and re-counting soldiers by ranks and files or by calculating the area of a rectangle in the two ways. Any attempt to do without this interference by physics and reality into mathematics is sectarianism and isolationism which destroy the image of mathematics as a useful human activity in the eyes of all sensible people.

tl;dr: If you force someone to do something, and you don't want to make them resent the thing you're forcing them to do, you should explain to them why you're forcing them to do it, i.e., why it's in their best interest, whether it's math, music, learning a language, etc.

The first day of the year has 7 seven choices. It's either a leap year or it isn't. Also 823 is a prime number. Can you now figure out why this isn't true? (You can also prove it by looking at the calendar when you double click on the time in Windows.)

The person did not want to be enlightened and I was deemed a buzzkill.

The lesson I learned is to be careful when educating someone when it may take away that person's 'awe and wonder'. That is why atheists should avoid trying to reason away the faith of the religious.

If we are all predisposed to take unconsidered facts as an insult to our intelligence and not as a prelude to a discussion, then that is a whole other educational issue.

Also please accept my apologies if you think the word enlighten confers a condescending tone. I was going for 'greater knowledge/understanding'.

Something like "91 is the only prime number that isn't blue", perhaps.

It's expensive to not know math.

Good old Pythagoras: 24^2 + 32^2 = 40^2.

Because the sides were long he had a nice big right angle to work with.

That said: your advice sucked. The 3:4:5 triangle is a better choice for obvious reasons (smaller numbers) and not so obvious ones (less sensitivity to measurement error due to larger internal angles).

The critically important distinction there is precisely that between, say, Haskell and Ruby. :)

Absolutely. Why bother with all that unnecessary complexity in Ruby?

Where a functional nut would sit and worry about getting a design right that correctly abstracts the properties of any right triangle, a web developer is happy to hard code the values for 3:4:5 because that's all that's required. And he isn't wrong to do so.

If you want to do something, you don't need much math. If you want to do it WELL, you do.

The second you want to optimize something, you need math. Whether you're figuring out the analytics of an A/B test (stats!), or finding the best inputs for optimal outputs (set a derivative equal to 0 anyone?), or making an algorithm more efficient by lowering its asymptotic complexity.

You can do stuff. Math just makes it better. Like salt on bland food.

Full disclosure: I'm defending my PhD in math in 2 months! My opinions may be biased.

Sorry, off topic, and not personally directed at you, but I'm getting so tired of everyone pretending that A/B testing is something fancy as opposed to a willful ignorance of the past hundred years of work in experimental design. Will you all please go look up orthogonal arrays and linear models?

Now, mind you, all of them know that, in order to get the right overall tip, every person can just calculate the tip on what they themselves ordered and pay that -- and it will magically sum to the right value. They might also know that if you had to tip self-consistently -- that is, if you also had to pay a 15% tip on your tip, and a 15% tip on your tip on your tip, and so on, that this process converges and is essentially the same as just paying an effective tip. (They might even be able to tell you that the effective tip is 15%/0.85, if they're still reasonably young.)

Those are the sorts of things they're good at. I know some who will talk your ears off about the convergence of an involution series in a noncommutative algebra and the problems of just finding a nice notation which makes the problem not look ugly, because the idea is really just so simple once you skip past the messy derivation to the intuitive result.

Computing the actual tip? Not so much. Heck, that effective tip? That's 3/17ths, if you really churn the mental gears! Who has time to divide 3 by 17?! That's what I have a calculator for. Except, well, I didn't bring a calculator because we never use them -- but that's what the computer in my office is for.

It's not clear in what you're writing if you mean "when presented with a bill, how much to tip" or "given the final payment, how much of it was tip", or something else.

(You're actually doing it a little more complicated than it has to be because it sounds like you're spending a little memory remembering the number you took 10% of. Take half, add it to the original, which is presumably still right in front of you, then drop the digit after.)

I learned a lot of mental calculation because my background is in applied physics, where you're expected to reason about the size of things etc. (A good example question might be, "how far do you think an air molecule can go before it bumps into another air molecule?")

So, just for example, to convert Celsius to Fahrenheit you multiply by 2, subtract 10%, and add 32. It is exact. So 25°C → 50 → 45 → 77°F. To convert Fahrenheit to Celsius you subtract 32, divide by 2, add 10%, then add 1% if you really feel like it; it is approximate. So 52°F → 20 → 10 → 11 → 11.11°C. Those sorts of things I will do in my head when people give me a number in one or the other temperature. Tipping is especially nice when compared to the fact that in New York State there is an 8% sales tax which is not added to any of the prices on stuff at the store -- calculate a tip and halve it.

[Naive reasonings on air: small upper bounds can be given by the fact that it has to travel easily through the small passageways in your inner ear, but my favorite calculation is to take (kinematic viscosity) / (speed of sound), both of which are things that an applied physics geek should roughly know.]

People seem to associate "mathematicians" with "wizardry in arithmetics," but even before I finish undergrad, I understand "1 + 1" doesn't necessarily have to be 2. That's how mathematicians think (at least those who are interested in rings, fields, etc).

That stupid 15% tip? It's child's play and has long since been delegated to calculators.

No kid asks when band or football is going to be useful in their life because they already have a premise around it: start from the bottom, and you have four years to become a leader. Math doesn't provide this opportunity, because the moment you get a passing grade in a math course, you move on to the next class. There's no leadership here. No opportunity to lead that class you passed with what you now know. Nothing shifts. On another note, you do gain group work skills and peer bonding, like in other said activities.

The calendar thing pisses me off, like him, but for other reasons.

Primarily, this isn't because people don't know math. It's because they don't want to do it. We're currently in an era where it's easier to share information--good and bad--than for us to figure it out on our own. The problem here isn't that no one knows math, it's that "share" and "be a sheep" is much higher on everyone's bucket list than "do something by myself and learn from it." This is especially with kids. Instead of frustration, however, he's provided with a the perfect warmup problem. Ask kids to figure out if the answer is true or not using what they know about math, five minutes pass, and you show them how to handle it. You make two points now: math is useful, and don't always believe everything that gets shared around the internet.

There's a better math joke in there somewhere...

Music on the other hand can be seen, at its most elegant, as taking the complexity of the world and reducing it or rather tying it together, coalescing it into patterns, rhythms, and other things that approach the simplicity and fundamental nature of what one starts out with math, or at least the earliest lemmas of a mathematical system.

I think both are intensely pleasurable activities in their own way because they let you trace this path from fundamentals to complexity and back. Math and football as challenges that have you climb up the mountain of complexity, but that still remain close enough to the base to be simple and absolute and beautiful in these ways; and music as an initially complex and tumultuous thing that gets rid of just enough entropy to be simpler but not fake.

Re-learning it will be much easier though. Kind of like remembering how to ride a bicycle.

Anyway, what are the times when you'll need math? When you're trying to solve interesting problems. You will never use math if you're doing routine programming work. However, if you need to create an algorithm that detects clients within a radius of 1 mile of your location, you'll need math. If you need to rate those clients on a curved basis, you'll need math. Not skull-fracturing math, but still math. Many, many other examples.

So if you're pulling data from a database and displaying it, no math. If you're actually creating that data or heavily modifying it, math.

I don't think anything past calculus is necessary unless you're doing some very specialized work, but up to that point is an absolute must for any computer scientist.

This isn't exclusive to mathematics. I'm sure there are thousands of blog posts lamenting the lack of philosophy, comparative literature, computer science, etc. instruction.

Math teaching is being carried along with a thousand year momentum and while the current curriculum made sense when many jobs involved building catapults, bridges, ships or cathedrals and jobs as a carpenter or mason were more numerous, they make little sense now.

Notice also that most of these subjects were set at a time before education was necessary so that if you went into say a gymnasium; odds were you wanted to be a teacher, academic or engineer. So topics like trigonometry, geometry, calculus and higher algebra (matrices, analytic geometry) made sense.

But these days, the vast majority of people do not need these subjects. On top of that there is a great crime. The most pertinent topics to modern living are given short thrift. Subjects like understanding basic statistics (including mean, mode, median, stdev, and variance), probability (including expected value), basic decision theory and estimation. All of these would have far more use to every day life and could be fully taught in the context of how they would help in real life (media, gamblers fallacy, money management etc).

Advanced topics would be things like distinguishing conditional and joint probabilities, counting (combinatorics), graphs and networks, exponentiation and logarithms, common plots (logistic, exponential, parabolic), rate of change, and the relationship between a circle and triangle (must be taught with animations). These things actually still have use to many people.

There is also the question of should math be taught at early years at all? A 12 year old can probably compress their previous 7 year math eduaction to 1. Maybe just numbers, (Z,+) or (Z,*)? I don't know but I think the question of if math education starts too early deserves to be looked at. The current hatred is in part due to a cycle of teachers who hated the subject having to force learn it in college and then having to teach it by force in addition to other subjects they may be stronger in.

Instead let the child's curiosity guide them. So they would arrive not by force and a bunch of unmotivated subjects but by curiosity. To augment the lack of math classes there will be game classes to teach reasoning. Not just video games but also card, dice and board games - with the caveat that the game must be PSPACE complete. Just let the kids play and compete. Maybe they will learn the same type of reasoning that will be useful to learning about chains, posets, groups, first order logic or probability. The more determined may even go read about the topics.

For video games the game must allow scripting. I think such a policy would just about eliminate thoughts of the pointlessnsess of math.

Kids are not dumb, they will do impressive things if it interests them. Adults make this mistake of underestimating kids all the time e.g. whenever they say things like "whaoh that was done by a 12 year old?!"

Personally I think learning to program is far more useful than any of those skills and yet it is usually presented as an 'elective' course in most schools.

Not to say that I don't appreciate math, but if we look at cost vs benefit, I think our current curriculum could be updated somewhat. Maybe some math classes could be offered as 'electives'. Of course, there were other less useful classes I was required to take as well.

Actually, quadratic equation might be one of the most important things in high school maths - lots of practical problems, especially involving optimization, tend to be reduced to (or approximated by) quadratic equation.