Math education: It's not about numbers, it's about learning how to think - <AUTHOR> http://www.nwaonline.com/news/2017/jul/03/slaying-the-math-monster-20170703/ <comment><USER4> Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late.

I never had any problems with math until I went to university, so I was merely a passive observer of everyday struggle for some people. I honestly believe that foundations are the key. Either you're taught to think critically, see patterns and focus on the train of thought, or you focus on numbers and memorization.

The latter obviously fails at some point, in many cases sufficiently late to make it really hard to go back and relearn everything.

Math is extremely hierarchical and I believe schools do not do enough to make sure students are on the same page. If we want to fix teaching math, I would start there, instead of working on motivation and general attitude. Those are consequences, not the reasons.

<comment><USER5> > Maybe I'm wrong, but I have always believed that if you want people to be > good at math, it's their first years of education which are important, not > the last ones. In other worlds, push for STEM should be present in > kindergartens and elementary schools. By the time people go to high school > it is to late.

I indeed believe you are wrong. I started getting interested in STEM when I was 16 and now have a masters in math on the one hand and work for one of the most prestigious Tech firms in existence on the other. Meanwhile, when I _was_ young, I was encouraged to enter creative fields (mostly writing in my case) which I did have enthusiasm for but then dropped as a teenager. So I switched at least twice, what I wanted to do with my life and still turned out fine.

Honestly, the whole "you need to get them when they are young" idea never convinced me very much. It's never too late to learn a thing and often you need a certain age to actually appreciate something. You don't have to be a childhood prodigy to be good at something.

I rather find the idea pretty harming; I've met a lot of people who did not pursue the career they wanted, because they bought into the myth, that you need to start programming at 6 to get a job at a company like Google or Facebook.

Let kids be kids for a while. Don't worry, they can always figure out what they love later and it's never too late to reconsider.

<comment><USER6> Anecdote time: I was a late bloomer in math too. From about 6th to 10th grade, I was terrible at math. I think it was because of inapplication problems.

Yes, you get a word problem about Jack and Kwanzeeah trying to fill up a pail or something, it was nonsense to me. Why are they filling up a pail? Why are they using a bucket that is 173/9th's the size of the hose? Why do they have to not let it overflow? Why do I have to decipher this inanity, why can't they just ask me to do this stupid problem of stupid fractions? Why the cloak and dagger?

Then you get to 'real' algebra and you start to find roots of quadratic equations and other useless information. Dear Lord! When has anyone ever done this for any reason? Yes, something something eigenvalues something something computational modeling something runge-kutte. But for real though, quadratic equations are just nonsense gibberish to weed out the poor kids from the rich ones. Trig, yes, it's useful when you are building a shed that has to be totally perfect and crisp otherwise the rich white lady will not pay you. Geometry was a bit 'fun' actually, but in the same way that learning that Custer's Men took a poop at this road-side stop on the way to Little-Big Horn.

It wasn't until Chem in my 10th grade year that I actually 'got' algebra. We were doing moles to molarity calculations to get the solution to the right pH and make it all turn pink or something. It was then that I realized that all this mathy stuff was actually useful to me, that I could use it to make things easier and better for myself and my family. Like, I could 'know' what to do then. Before that it was just drill and kill and bad grades and shame. I remember staying after class and into the next one in the room, just sitting in the back doing these calculations over and over. I got kicked out for doing math! It was such a relief! I finally 'got it' when I needed it for something real to me personally.

For me, it was the issue of 'math' being this blunt-shame-thing that made no sense to use ever. But once I _needed_ to use math, it was trivially easy. Learning math for each of us is individual, we each have our own motivations that are unique that need to be met. I know that is not easy to institutionalize, but if you want to do it, you have to find the 'correct motivation for each kid.

<comment><USER7> I'd agree with this completely. It's less about teaching what, how and when than it is teaching WHY you need to know this.

In high school, if you can convince a single person that they need to know calculus, more power to you. 99% of people don't and it's literally a waste of time. Teach people finance though...suddenly we have math that people will know is important.

<comment><USER8> I was so pissed in college when I found out that all of those equations we had to memorize in physics were just the integrals.

Why did I have to memorize all that shit if there's a simple way to derive the formula? Jesus.

<comment><USER9> At my college, there were "calculus based" and "non calculus based" physics classes. It seemed like the calculus based class was a lot easier for just the reason that you state.

</comment></comment></comment></comment></comment></comment><comment><USER10> I studied Mechanical Engineering, and it was my experience that several professors are only interested in having the students learn how to solve problems (which in the end boil down to math and applying equations), instead of actually learning the interesting and important concepts behind them.

My wife went to school for Architecture, where she learned "basic" structural mechanics, and some Calculus, but still cannot explain to me in simple words what an integral or a derivative is. Not her fault at all: her Calculus professor had them calculate polynomial derivatives for 3 months, without ever making them understand the concept of "rate or change", or what "infinitesimal" means.

For me that's a big failure of our current "science" education system: too much focus on stupid application of equations and formulas, and too little focus on actually comprehending the abstract concepts behind them.

<comment><USER11> Nailed it.

I remember "hitting the wall" in Honors Algebra II, when I just couldn't keep up with all the new things I was expected to compute: radicals, synthetic division, logarithms, etc. It was a major dent in my math confidence, which up until that point had be reinforced by being told I was "good at math".

I didn't start regaining that confidence until Chemistry, when I learned what pH actually meant. That sparked the epiphany that _scale_ was deeply important in nature, and with that motivation, logarithms seemed much simpler.

We so quickly forget that most math wasn't discovered as some abstract exercise. But we still teach it that way, with the concepts before the applications.

<comment><USER12> This is compounded in the weird American class called "precalculus," which is just a random grab bag of math crap that might maybe needed before calculus but might not. In most precalc classes I've encountered, not only is there no explanation of why the concepts will be used in calculus, but there is little attempt to relate the concepts covered in the class. Let's go from linear equations to complex numbers to polar coordinates! And let's not mention that complex numbers and polar coordinates are related!

To top it all off, a lot of young college students taking precalculus are taking it as a terminal math class -- they'll never take calculus! It's just for a credit! So you've got some poor kid who hit that wall in high school and is _repeating_ precalc and going through some unit on polar coordinates who will never take another math class again and will somewhat justifiably think this whole thing is stupid.

In theory, I strongly support "math for liberal arts"-type classes instead in college. When done well they might cover voting systems and how you can prove there isn't a perfect one, gerrymandering and how you can use math to get several solutions, perspective drawing and its relation to projective geometry and Renaissance art, a bit of number theory and basic cryptography....

<comment><USER13> >"math for liberal arts"-type classes

I took a class that met in a room used by a class like that the period prior and got to listen to the better part of it. In the one lecture the prof coaxed them into creating an algorithm for the number of moves to solve a rubix cube based on the number of cells who's configuration is known and where those cells are located on the cube. In a different lecture they were doing the math necessary to cut hypoid gears on a manual milling machine. They didn't even know what they had did until he tied it back to the underlying concept.

The quality of those courses depends on the prof.

<comment><USER12> > The quality of those courses depends on the prof.

Entirely. In theory they are great :)

The one you listened in on sounds interesting!

</comment></comment></comment></comment></comment><comment><USER14> Learning "how to think" is just one part of it. The other part - the one that makes it _much_ more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math _specifically_. The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day that learning them takes a lot of training, _years_ of intensive experience dealing with mathematical structures of one kind or another, so it should come as no surprise that people have difficulty learning math.

<comment><USER15> Is that really true before calculus? In my recollection, even in pre-calculus it was pretty easy to accurately tie constructs to physical representations. Limits is the first time I remember learning something that could not be generalized to a physical form, because it specifically dealt with what happened when a function "broke".

<comment><USER16> I find this sentiment somewhat surprising.

Limits is incredibly easy to link to real physical things: movement, speed, acceleration.

If speed is distance over time, what's the speed of a moving object at a very specific instant?

<comment><USER15> Are you thinking of derivatives? I agree that those are relatively easy -- it's the rate of change for a function. Or, in geometric terms, it's the slope of the tangent line of the function's curve.

Like I said, limits are used when a function "breaks", so they're naturally hard to visualize. I'll use my typical example for when I discuss teaching math by solving problems: Approximating the area under a curve using rectangles. The thinner your rectangles are, the more accurate your approximation is. So you narrow the rectangles, getting more and more accurate... Until you hit zero width, where the value of the function suddenly drops to zero... A discontinuity. A limit lets you find what the value _would have been_ had that discontinuity not happened. So, there's a clear visualization of what the limit is letting you do _in this particular case_ \-- pretend like the rectangles of zero width don't have zero area -- but there's no clear generalization on how to visualize a limit.

<comment><USER16> Will, speed breaks when you consider a specific instant in thing because there is no time span nor a distance the be traveled.

<comment><USER15> Ah, good point. I forgot about that problem with instantaneous speed.

</comment></comment></comment></comment></comment></comment><comment><USER17> > _it 's about learning how to think_

It's about learning a set of thinking skills, not _how to think_. Many people who know no math can think and function very well in their domains and many people who know lots of math function and think poorly outside of math.

<comment><USER18> Math skills (esp proofs) shape the way you approach problems, giving the problem structure and providing you the tools to reduce it into components, relations, and dependencies. To make mysteries unmysterious, such tools are indispensable.

I'd go so far to disagree with you that people do well without learning math skill (those tools of thinking). If you can deconstruct a problem, then you certainly learned that skill somewhere, just not in a math class.

Abstraction and reductionism are unusual capabilities to acquire on your own. As I recall, they're almost nonexistent in pre-linguistic societies like hunter-gatherers. If you have no words for abstract concepts, your thinking will be strictly concrete.

<comment> FLUX-YOU I feel like there's at least three ways of thinking about problems that are practical:

\- systematic (math, rule-based)

\- social (history, emotions, people-based)

\- creative (art, invention)

These all overlap and feed into each other because we're a complex species, but we really only ever hear about the necessity of math to "learn to think".

</comment></comment></comment><comment><USER19> My problem with Math education was always that speed was an enormous factor in testing. You can methodically go through each question aiming for 100% accuracy and not finish the test paper, while other students can comfortably breeze through all the questions and get 80% accuracy but ultimately score higher on the test. This kind of penalizing for a lack of speed can lead to younger kids who are maximizing for grades to move away from Math for the wrong reasons.

Source: I'm slow but good at Math and ended up dropping it as soon as I could because it would not get me the grades I needed to enter a top tier university.

<comment><USER20> You couldn't finish your tests in high school, but you're good at math? Have you ever been evaluated for test anxiety?

[https://www.adaa.org/living-with-anxiety/children/test- anxie...](https://www.adaa.org/living-with-anxiety/children/test-anxiety)

There's no shame in it (although high schoolers can be assholes), and it can help your school accommodate your needs eg extra time to take your exams.

<comment><USER19> I've never been evaluated but I don't think test anxiety fits in my case - <USER2> speed issue was only ever in math / physics.

My accuracy on the questions I got to was very high, I just couldn't go fast enough to complete enough questions. Same deal on SAT type math papers too.

<comment><USER21> Where I did my undergraduate degree they squared the marks scored for each question before adding them up! Slow but accurate was an advantage.

<comment><USER22> But, most exams at that level gives enough time to make them accurate.

</comment></comment></comment></comment></comment><comment><USER23> Disclaimer: I'm CTO of [https://www.amy.ac](https://www.amy.ac) an online math tutor.

From our experience most people struggle with math since they forgot/missed a curtain math skill they might have learned a year or two before. But most teaching methods only tell the students to practise more of the same. When looking at good tutors, we could see that a tutor observes a student and then teaches them the missing skill before they actually go to the problem the student wanted help with. That seems to be a usefull/working approach.

</comment><comment><USER24> This is something I've been pondering quite a bit recently. It is my firm belief that mathematical skill and general numeracy are actually a small subset of abstract thought. Am I wrong in thinking that school math is the closest to deliberate training in abstract reasoning that one would find in public education?

Abstract reasoning, intuition, and creativity, to me, represent the underpinnings of software engineering, and really, most engineering and science, but are taught more by osmosis along side the unintuitive often boring mechanics of subjects. The difference between a good engineer of any sort and one that 'just knows the formulas' is the ability to fluently manipulate and reason with symbols and effects that don't necessarily have any relation or simple metaphor in the tangible world. And taking it further, creativity and intuition beyond dull calculation are the crucial art behind choosing the right hypothesis to investigate. Essentially, learning to 'see' in this non-spacial space of relations. When I'm doing system engineering work, I don't think in terms of X Gb/s throughput and Y FLOPS... (until later at least) but in my mind I have a model of the information and data structures clicking and buzzing, like watching the gears of a clock, and I sort of visualize working with this, playing with changes. It wouldn't surprise me of most knowledge workers arrive have similar mental models of their own. But what I have observed is that people who have trouble with mathematics or coding aren't primed at all to 'see' abstractions in their minds eye. This skill takes years to cultivate, but, it seems that its cultivation is left entirely to chance by orthodox STEM education.

I was just thinking that this sort of thing could be approached a lot more deliberately and could yield very broad positive results in STEM teaching.

</comment><comment><USER25> My theory is that math anxiety is really anxiety about a cold assessment.

In other subjects you can rationalize to yourself in various ways: the teacher doesn't like me, or I got unlucky and they only asked the history questions I didn't know.

But with math, no rationalization is possible. There's no hope the teacher will go easy on you, or be happy that you got the gist of the solution.

Failure in math is often (but not always) a sign that education has failed in general. Teachers can be lazy or too nice and give good grades in art or history or reading to any student. But when the standardized math test comes around, there's no hiding from it (teacher or student).

<comment><USER26> Marking a math test is only objective when an answer is entirely correct. What mark do you give a student who makes a minor arithmetic error on a single step of a multi-step problem?

<comment><USER25> Objective tests for math ability are pretty good. Arguing that they are only 99% objective doesn't really change my point.

People aren't afraid of making tiny mistakes. They are afraid of looking at a problem similar to a dozen they saw in lecture; and having no idea what to do, and then being told under no incertain terms that they did not succeed.

</comment></comment></comment><comment><USER27> I was a math teacher for 10 years. I had to give it up when I came to realize that "how to think" is about 90% biological and strongly correlated to what we measure with IQ tests.

This may be grave heresy in the Temple of Tabula Rasa where most education policy is concocted, but nonetheless every teacher I ever knew was ultimately forced to chose between teaching real math class with a ~30% pass rate or a watered-down math Kabuki show with a pass rate just high enough to keep their admins' complaints to a low grumble.

In the end we teachers would all go about loudly professing to each other that "It's not about numbers, it's about learning how to think" in a desperate bid to quash our private suspicions that there's actually precious little that can be done to teach "how to think."

<comment><USER28> Unfortunately your realization is empirically unsound, and the problems of "every teacher you know" have as much to do with lack of available expert teacher attention per student and standard expected teaching styles and school structure with substantial friction/latency in feedback and huge amounts of wasted time as with inherent student ability.

Start here [http://web.mit.edu/5.95/readings/bloom-two- sigma.pdf](http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf) but then there is a vast literature (thousands of studies and other research papers) exploring the general topic of math pedagogy. Some approaches and some teachers are radically more successful than others, given comparable students.

If you hand me one of your 20th percentile students who is struggling but willing to learn, and give us the same amount of time which would otherwise be spent in a classroom for a year of one-on-one face-to-face time, with the student also spending a typical amount of time working independently, I can have them outperforming your 80th percentile students by the end of the year, without issue. The difficulty of course is that direct mentoring by an expert tutor is too expensive for society to be willing to pay for at scale.

The main reasons that your teacher friends are stuck is because (a) many if not most of their students are unprepared before they arrive in any particular course, which is largely down to structural social factors and school scheduling inflexibility, and (b) the combination of lectures and independent work with inadequate feedback are for the average student a terribly inefficient and ineffective way to learn. Neither of those has all too much to do with biological predestination or whatever.

<comment><USER27> Heh. You can cite all the old wives' tales, superstitious folklore, and education research papers you want. None of that makes it more likely that a 95 IQ student will ever understand precalculus.

<comment><USER28> I actually have direct experience getting a (I would guess) 95 IQ student to understand precalculus, tutoring for a few months back when I was a high school student ~15 years ago. She got an A, after having had a string of Cs in every prior high school math course, and not being properly fluent with 9th grade algebra at the beginning of the year. [I don't say this to brag; I expect any competent tutor can achieve the same thing with a typical below- average student if the student is motivated and they get lots of 1-on-1 time.]

However, I can well believe that a 95 IQ student who was dramatically unprepared at the beginning of the year and believes himself to be an irredeemable failure will have a lot of trouble in a standard lecture + homework drills format high school precalculus course taught by a teacher with no time to help him catch up on remedial material at his current level of understanding, give him any special attention, or make the class material engaging enough to give him reason to care. In that context, failure (or scraping by with an undeserved C) is the obvious outcome, but can't fairly be blamed on the student's effort during that year or some kind of mental deficiency.

(Personally, I skipped half of precalculus and somewhat wish I had skipped the other half; I found it to be an uninspired grab-bag of disconnected topics, with much too rote a focus.)

</comment></comment></comment></comment><comment><USER29> Late to the party but wanted to share my experience.

I was an Applied Math major at Berkely. Why?

When I was in 7th grade, I had an old school Russian math teacher. She was tough, not one for niceties, but extremely fair.

One day, being the typical smart ass that I was, I said, why the hell do I need to do this, I have 0 interest in Geometry.

Her answer completely changed my outlook and eventually was the reason why I took extensive math in HS and majored in math in college.

Instead of dismissing me, instead of just telling me to shut up and sit down, she explained things to me very calmly.

She said doing math beyond improving your math skills improves your reasoning ability. It's a workout for your brain and helps develop your logical thinking. Studying it now at a young age will help it become part of your intuition so that in the future you can reason about complex topics that require more than a moment's thoughts.

She really reached me on that day, took me a while to realize it. Wish I could have said thank you.

Wherever you are Ms. Zavesova, thank you.

Other beneits: doing hard math really builds up your tolerance for building hard problems. Reasoning through long problems, trying and failing, really requires a certain kind of stamina. My major definitely gave me this. I am a product manager now and while I don't code, I have an extremely easy time working with engineers to get stuff done.

</comment><comment><USER30> When people talk about the failure of mathematics education, we often talk about it in terms of the students inability to "think mathematically".

It's impossible to tell if students are capable of thinking mathematically, however, because I have not met a single (non-mathlete) student who could give me the mathematical definition of... anything. How can we evaluate student's mathematical reasoning ability if they have zero mathematical objects about which to reason?

</comment><comment><USER31> This part really resonates with me as well:

 _" You read all the time, right? We constantly have to read. If you're not someone who picks up a book, you have to read menus, you've got to read traffic signs, you've got to read instructions, you've got to read subtitles -- all sorts of things. But how often do you have to do any sort of complicated problem-solving with mathematics? The average person, not too often."_

 _From this, two deductions:_

 _• Having trouble remembering the quadratic equation formula doesn 't mean you're not a "numbers-person."_

 _• To remember your math skills, use them more often._

What I remember from high-school and college was this: I'd take a given math class (say, Algebra I) and learn it reasonably well. Then, summer vacation hits. Next term, taking Algebra II, all the Algebra I stuff is forgotten because, well, who uses Algebra I over their summer vacation? Now, Algebra II is harder than it should be because it builds on the previous stuff. Lather, rinse, repeat.

This is one reason I love Khan Academy so much. You can just pop over there anytime and spend a few minutes going back over stuff at any level, from basic freaking fractions, up through Calculus and Linear Algebra.

<comment><USER32> I wanted to quote this passage for another reason -- it seems like you're agreeing with it, but I completely disagree with it, especially this:

> But how often do you have to do any sort of complicated problem-solving with > mathematics?

There are a LOT of chances to do that kind of complicated problem solving, especially if you're shopping or comparison shopping on your own. It's not that people don't have the chances, it's that people avoid the work involved in doing those kinds of comparisons.

<comment><USER31> I mean, in a certain sense I agree with you. There are _chances_ to do that stuff, yes. To a point. Basic algebra could come up in a shopping scenario for example. But even then, people aren't going to be doing that stuff every day, or even close to it. And when you get even slightly more esoteric, like, say, exponents... how often are people (especially kids) really going to be using exponents, or logarithms, or the quadratic formula, etc. in their daily lives?

Sure, if you go out of your way to actively _look_ for reasons to find that stuff, you can find them. But my point is that when you're a kid learning math, it's summer vacation and you're busy playing with your friends, you _aren 't_ out actively looking for reasons to apply the quadratic formula, etc. At least not for most people, from what I've seen.

</comment></comment></comment><comment><USER33> You say "it's not about numbers, it's about learning how to think," but the truth is it's about both. Without the number skills and the memorization of all those number facts and formulas, a person is handicapped both in learning other subjects and skills and in succeeding and progressing in their work and daily life. The two concepts -- number skills and thinking skills -- go hand in hand. Thinking skills can't grow if the number skills aren't there as a foundation. That's what's wrong with the Common Core and all the other fads that are driving math education these days. They push thinking skills and shove a calculator at you for the number skills -- and you stall, crash and burn.

The article brings out a good point about math anxiety. I have had to deal with it a lot in my years of teaching math. Sometimes my classroom has seemed so full of math anxiety that you could cut it with a butter knife. I read one comment that advocated starting our children out even earlier on learning these skills, but the truth is the root of math anxiety in most people lies in being forced to try to learn it at too early an age. Most children's brains are not cognitively developed enough in the early grades to learn the concepts we are pushing at them, so when a child finds failure at being asked to do something he/she is not capable of doing, anxiety results and eventually becomes habit, a part of their basic self-concept and personality. What we should instead do is delay starting school until age 8 or even 9. Some people don't develop cognitively until 12. Sweden recently raised their mandatory school age to 7 because of what the research has been telling us about this.

</comment><comment><USER34> Is this a US thing? Why would people still think that math is about numbers? Math is about patterns, which got drilled into us by our teachers in primary school. I really don't understand how US education system can fuck up so badly on fundamental subject like math.

<comment><USER35> It is a US thing. And it's because until middle school (age 11-13) your teacher is likely not that familiar with math (they have taken math courses, but likely little formal math beyond geometry and trigonometry, perhaps some courses on math pedagogy). So for them, math is all about the numbers, they barely got to the more abstract concepts.

In middle school in the US you start getting teachers specialized in their teaching subject matter, but often tangentially. Any one with enough college math credits can teach middle school math (with some other certifications, usually). High school you may, finally, get a math teacher who is a mathematician. If you're lucky.

<comment><USER36> What kind of education is required from teachers in US? In Poland it is Bachelor's degree in relevant topic (for elementary and middle school) or Master's degree (for high schools) plus a course in pedagogy. Most teachers however get Master's degree even if not required (they get salary increase for that). It is hard for me to imagine math teacher not good in match themselves.

<comment><USER12> The sibling commenter is right re: bachelors and certifications, but due to shortages in the US often teachers with no certification in a subject are teaching. See Table 2 in [1] and find that 70% of high school math teachers have a math major and 81% are certified, while 10% of public high school math teachers have neither a math major nor a certification in math.

[1] [https://nces.ed.gov/pubs2015/2015814.pdf](https://nces.ed.gov/pubs2015/2015814.pdf)

</comment></comment></comment></comment><comment><USER37> Wow ... this blows me away ... in a few short hours, so many people chimed in sharing thoughts ... It is great ... Would like to share mine as well. Fundamentally, math to me is like a language. It's meant to help us to describe things a bit more quantitatively and to reason a bit more abstractly and consistently ... if it can be made mechanical and reduce the burden on one's brain, it would be ideal. Since it's like a language, as long as one knows the basics, such as some basic things of set theory, function, etc., one should be ready to explore the world with it. Math is often perceived as a set of concepts, theorems, rules, etc. But if one gets behind the scene to get to know some of the original stories of the things, it would become very nature. At some point, one would have one's mind liberated and start to use math or create math like we usually do with day to day languages such as English.

</comment><comment><USER38> For me, math has always been a source of unplugging. I'd sit at my kitchen table, put in some headphones, and just get lost in endless math problems.

Interestingly, now as a masters student in a statistics graduate program, I've learned that I don't like "doing" math but get enjoyment from teaching it. I really like it when students challenge me when I'm at the chalkboard and I'll do anything for those "ah-ha!" moments. The best is at the end of the semester hearing students say "I thought this class was going to suck but I worked hard and am proud of the work I did." I'm hoping that on some small scale I'm shaping their views on math. Or at least give them the confidence to say, "I don't get this, but I'm not afraid to learn it."

</comment><comment><USER39> I think a major issue with math problems in school is that they're obvious.

By that I don't mean it's easy. But when you're grappling with some problem, whatever it is, eg find some angle or integrate some function, if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?"

And you won't have any excuses for why you didn't think of it. Because math is a bunch of little logical steps. If you'd followed them, you'd have gotten everything right.

Which is a good reason to feel stupid.

But don't worry. There are things that mathematicians, real ones with PhDs, will discover in the future. By taking a number of little logical steps that haven't been taken yet. They could have gone that way towards the next big theorem, but they haven't done it yet for whatever reason (eg there's a LOT of connections to be made).

<comment><USER40> Reminds me of this quote:

"The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles."

Alan Turing

<comment><AUTHOR> You might want to edit that to fix the name. It's "Turing", not "Turning".

<comment><USER40> Thanks. That's what I get for posting on my phone :)

</comment></comment></comment></comment><comment><USER41> I wish school curricula would embrace that "learning how to think" bit.

With the sole exception of Geometry, every single math class I took in middle and high school was an absolutely miserable time of rote memorization and soul-crushing "do this same problem 100 times" busy work. Geometry, meanwhile, taught me about proofs and theorems v. postulates and actually using logical reasoning. Unsurprisingly, Geometry was the one and only math class I ever actually enjoyed.

</comment><comment><USER42> As my old boss once said, "never confuse _mathematics_ with mere _arithmetic_."

<comment><USER43> I've always liked that line, but it's clearly being cheeky and there are times it's applicable and times it's not.

All arithmetic is math, but not all math is arithmetic.

Except that if you dig deep enough, all math is arithmetic under the hood. Algebra is doing the arithmetic on variables before they're numbers. Calculus is just useful shorthand for compound arithmetic; the derivative is a divide and the integral is a sum. Exponentiation is repeated multiplication, which is repeated addition. Linear algebra is arithmetic on blocks of numbers.

So we abstract over the individual arithmetic operations for our own convenience, there are too many of them. But it's all still fancy arithmetic.

Maybe we should call math "fancy arithmetic" to help all the people who are intimidated by math...

<comment><USER44> Actually, the opposite is true. If you dig deep enough, it turns out that arithmetic is far more complicated than most other areas.

<comment><USER43> Then maybe instead of calling math fancy arithmetic, we should call arithmetic "hard math"? :P

Seems like you agree more than disagree; math and arithmetic aren't easily separable, if you're being serious instead of cheeky?

<comment><USER44> Well, I was trying to make two points.

1) Arithmetic (number theory) is far harder than most people realise, to the point that its considered one of the most complicated mathematical disciplines.

2) Most subject areas in math bear little to no resemblance to arithmetic, so it doesn't make sense to call all math 'Fancy Arithmetic'.

<comment><USER43> Sure, understood and I agree. I was also trying to make one point: all math is built on arithmetic operations if you dig down to first principles, so it does make some sense to call all math 'fancy arithmetic'. There is no math at all without the concept of addition sitting somewhere under it.

But I'm not actually proposing that, just sharing a cheeky point of view to contrast @taneq's boss, so don't take me _too_ seriously. ;)

<comment><USER45> > I was also trying to make one point: all math is built on arithmetic > operations if you dig down to first principles,

I am a newb at this stuff, but I believe (basic) number theory actually builds on recursive structures, and then uses induction to make statements about behavior.

[https://en.wikipedia.org/wiki/Peano_axioms#Addition](https://en.wikipedia.org/wiki/Peano_axioms#Addition)

What is interesting is how the same approach works for analogous structures like lists - because of the monoid abstraction.

<comment><USER43> This is interesting! I'm not sure, but I think this may reinforce what I said, not contradict it. I'm suggesting that math operations we know all build on top of addition, not that addition is necessarily the atom. Addition can build on top of something else, and still sit under all higher level math.

But that said, I didn't learn addition in terms of monoids, and I don't know anyone else who did. This might be a grand unified theory of math under which addition fits, but physical addition has no recursive monoid analogue. You can add two weights together to get a measurable sum, and it does not depend on a recursive structure that uses induction. You just put two separate things side by side on the scale.

Just because we can explain addition using monoids and induction does not mean that addition _is_ made of monoids and induction.

Also, I probably wouldn't use Peano theory as the starting place to teach addition. As a pedagogical tool, this is advanced math, and I guess would be a bigger impediment to learning than Greek symbols, for beginners. Right?

</comment></comment></comment></comment></comment></comment></comment></comment><comment><USER46> Enough "I" statements already. It's ironic how many people seem to think their personal experience is somehow relevant on a post about "critical thinking."

The _ONLY_ sane way to answer these questions: \- Does math increase critical thinking? \- Does critical thinking lead to more career earnings/happiness/etc? \- When does math education increase critical thinking most? \- What kind of math education increases critical thinking?

Is with a large-scale research study that defines an objective way to measure critical thinking and controls for relevant variables.

Meaning you don't get an anecdotal opinion on the matter on your study-of-1 no-control-group no-objective-measure personal experience.

</comment><comment><USER47> My mind was blown when I came across this Github repo that demonstrates mathematical notation by showing comparisons with JavaScript code [https://github.com/Jam3/math-as-code](https://github.com/Jam3/math-as-code)

I think I often struggled or was intimidated by the syntax of math. I started web development after years of thinking I just wasn't a math person. When looking at this repo, I was surprised at how much more easily and naturally I was able to grasp concepts in code compared to being introduced to them in math classes.

<comment><USER48> I found programming (started my first year in college) to be easily accessible, and quite a natural next step due to my success in math at the time. It is interesting how folks develop a mental model of a problem space and scaffold knowledge onto previous models to understand problems at hand. What do you feel made programming accessible when you started?

</comment></comment><comment><USER49> I often worry that mathematics education is strongly supported on the grounds that it is about "learning how to think", yet the way it is executed rarely prioritizes this goal. What would it look like if math curriculum were redesigned to be super focused on "learning how to think"? Different, for sure.

</comment><comment><USER50> I can't agree more. Math is about intuition of what the symbols are doing. In the case of functions, intuition about how the symbols are transforming the input. I've always thought I was "bad at math." It wasn't until my late 20's when I took it upon myself to get better at calculus and I used "Calculus Success in 20 Minute a Day[0]" did I finally realize why I was "bad" at it; I never understood what I was doing.

That series of book really put intuition at the forefront. I began to realize that the crazy symbols and formulas were stand-in for living, breathing dynamic systems: number transformers. Each formula and symbol represented an action. Once I understood Math as a way to encode useful number transformation, it all clicked. Those rules and functions were encoded after a person came up with something they wanted to do. The formula or function is merely a compact way of describing this dynamic system to other people.

The irony was I always thought math was boring. In retrospect it was because it was taught as if it had no purpose other than to provide useless mental exercise. Once I started realizing that derivatives are used all around me to do cool shit, I was inspired to learn how they worked because I wanted to use them to do cool shit too. I went through several years of math courses and none of them even attempted to tell me that math was just a way to represent cool real world things. It took a $10 used book from amazon to do that. Ain't life grand?

[0]:[https://www.amazon.com/Calculus-Success-20-Minutes- Day/dp/15...](https://www.amazon.com/Calculus-Success-20-Minutes- Day/dp/1576858898)

<comment><USER51> But math _isn 't_ about intuition, much to the contrary, it's about rigorous reasoning. In math "intuition" and "common sense" are worth nothing. Only objective proof matters. It's a valuable lesson too, learning not to take anything on a hunch and only take as true what what you carefully and logically prove.

<comment><USER52> you use intuition to come up with the argument. then you use a proof to convince others.

<comment><USER51> No, you use intuition to come up with an idea the use that idea to write a proof _for yourself_. Intuition is invaluable as a starting point, but it's often misleading.

<comment><USER52> you're very naive. i think you're probably still an undergrad or maybe have never done serious math.

[https://terrytao.wordpress.com/career- advice/there%E2%80%99s...](https://terrytao.wordpress.com/career- advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/)

<comment><USER51> But the article agrees with me:

> The point of rigour is not to destroy all intuition; instead, it should be > used to destroy bad intuition while clarifying and elevating good intuition.

In mathematics, intuition is the starting point and guiding principle, but never worth anything by itself without rigour.

</comment></comment></comment></comment></comment></comment><comment><USER53> Permit me to make a tangentially related comment of interest to parents reading this thread: This camp for 11-14 y/o kids: [http://www.mathpath.org/](http://www.mathpath.org/) is absolutely excellent. My kid loved it so much they attended three years. Great faculty... John Conway, Francis Su, many others. If you have a math-loving kid of middle- school age, I encourage you to check it out.

</comment><comment><USER54> I completely agree. I think we start all wrong too, the first memories I have of maths at school was learning how to compute an addition, a subtraction and later a multiplication and division. Then we had to memorize by heart the multiplication tables.

That can be useful of course (especially back then when we didn't carry computers in our pockets at all times) but I think it sends some pupils on a bad path with regards to mathematics.

Maths shouldn't be mainly about memorizing tables and "dumbly" applying algorithms without understanding what they _mean_. That's how you end up with kids who can answer "what's 36 divided by 4" but not "you have 36 candies that you want to split equally with 3 other people, how many candies do you end up with?"

And that goes beyond pure maths too. In physics if you pay attention to the relationship between the various units you probably won't have to memorize many equations, it'll just make sense. You'll also be much more likely to spot errors. "Wait, I want to compute a speed and I'm multiplying amperes and moles, does that really make sense?".

<comment><USER55> A friend's kid did not know how to do subtraction with just numbers but when you said it like he had some currency in a shop then he understood it. That was peculiar although the kid was very young (5-6 I guess) so it's interesting how kids develop ability to do abstract thinking as they age

<comment><USER56> That's the way my kids were able to grasp basic math. For fractions we used pizzas. Being able to hang the abstract concept on something that they were used to dealing with made it work for them.

</comment></comment></comment><comment><USER43> I wonder if a large part of our math problem is our legacy fixation on Greek letters. Would math be more approachable to English speakers if we just used English?

I like to think about math as language, rather than thought or logic or formulas or numbers. The Greek letters are part of that language, and part of why learning math is learning a completely foreign language, even though so many people who say they can't do math practice mathematical concepts without Greek letters. All of the math we do on computers, symbolic and numeric, analytic and approximations, can be done using a Turing machine that starts with only symbols and no built-in concept of a number.

<comment><USER57> If the use of greek letters is a major hurdle to learning maths, I'm guessing that the learner has misunderstood something fundamental about symbols in maths. It shouldn't make any significant difference if the symbols are Latin, Greek or even made up by the lecturer (as long as they are sane).

<comment><USER58> In theory it shouldn't matter but it does. We brainwash students to think in terms of y being a function of x.

y = f(x)

If you give a problem with x = f(y) it really confuses people. If I label the horizontal axis y and the vertical axis x in the cartesian plane then confusion ensues.

t = b(a) where t is the velocity and a is time

That's confusing. I suppose it's analogous to me saying

"The male, fluffy, white, large sheep."

There is a certain amount of brainwashing that occurs with the symbols that we use. Using symbols other than the ones that have been standardized on is very difficult cognitively. What you say is true in theory but I think not in practice.

<comment><USER59> If this is enough to "really confuse" someone than there's a serious problem with understanding VS rota learning.

<comment><USER58> I'm certain you would be confused by non-standard usage of notation. It's not just symbols and that any symbol can be used. Try reading or speaking English with the adjectives in an order that is not expected. It's confusing as hell but grammatically correct. Our brains get trained to see things in a certain way and when the symbols get jumbled up so do we.

(x1(x2+x3) - x1(x2))/x3

is simply not as understandable to a first year calculus student as

(f(x+h) - f(x))/h

I've been teaching university level mathematics for 25 years and I know mixing up symbols confuses people. Try reading Newton's Principia. It's really hard to know what he's talking about.

EDIT: Try this equation. What famous one does it represent?

(sex(sxe) - sex(xes))/(sxe - xes) = xse sex/xse sxe

</comment></comment></comment></comment></comment><comment><USER60> True, Math is ultimately about how to think but students need to memorize and grasp the basics in addition to making sure that new material is truly understood. That's where things fall apart. We are bombarded with new concepts before we ultimately know how to use what we learned. How many people use imaginary numbers in their daily life? Need I say more?

We don't communicate in Math jargon every day so it's ultimate a losing battle. We learn new concepts but we lose them since we don't use them. Additionally a large number of students get lost and frustrated and finally give up. Which ultimately makes math a poor method to teach thinking since only a few students can attain the ultimate benefits.

Yes, Math is important, and needs to be taught, but if we want to use it as away to learn how to think there are better methods. Programming is a great way. Students can learn it in one semester and can use it for life and can also expand on what they already know.

Also, exploring literature and discussing what the author tries to convey is a great way to learn how to think. All those hours in English class trying to interpret what the author meant was more about exploring your mind and your peer's thoughts than what the author actually meant. The author lost his sphere of influence once the book was publish. It's up to the readers of every generation to interpret the work. So literature is a very strong way to teach students how to think.

</comment><comment><USER61> Why aren't people taught how to think explicitly? The Greeks and the Romans thought it was a good idea.

</comment><comment><USER62> Is there any other subject that is given as much leeway for its abysmal pedagogical failures?

"Economics, it's not about learning how money and markets work, it's about learning how to think."

"Art, it's not about learning about aesthetics, style, or technique, it's about learning how to think."

"French, it's not about learning how to speak another language, it's..."

Math has a problem, and it's because the math curriculum is a pile of dull, abstract cart-before-the-horse idiocy posing as discipline.

</comment><comment><USER63> The true eye-opener for me was reading Number - The Language of Science by Tobias Dantzig. The philosophy part of math as an abstraction layer for what is observed or deducted was a nice touch.

</comment><comment><USER58> I teach math at a community college. I've tried many times to teach my courses in such a way that understanding the concepts and thinking were the goals. Perhaps I'm jaded by the failures I encountered but students do not want to think. They want to see a set of problem types that need to be mimicked.

In our lowest level course we teach beginning algebra. Almost everyone has an intuition that 2x + 3x should be 5x. It's very difficult to get them to understand that there is a rule for this that makes sense. And that it is the application of this rule that allows you to conclude that 2x + 3x is 5x. Furthermore, and here is the difficulty, that same rule is why 3x + a x is (3+a)x.

I believe that for most people mathematics is just brainwashing via familiarity. Most people end up understanding math by collecting knowledge about problem types, tricks, and becoming situationally aware. Very few people actually discover a problem type on their own. Very few people are willing, or have been trained to be willing, to really contemplate a new problem type or situation.

Math education in its practice has nothing to do with learning how to think. At least in my experience and as I understand what it means to learn how to think.

<comment><USER64> I've had the same experience - teaching math for a short time at a State University. I think the basic situation is that by the time students have reached the university level, the habit of looking for a set algorithm for completing a class is completely ingrained and moreover, the average student has little-to-no free time so they feel a panic concerning anything that will take an _unpredictable_ amount of time.

</comment></comment><comment><USER65> Exactly, as ordinary hacker i was always afraid of math. But after taking mathematical Analysis I realized how wonderful math is. These day i am in love with pure mathematics. It literally corrected my brain pipeline in so many ways and it continues to do it further and further.

I have thought about changing my major to pure mathematics too.

</comment><comment><USER66> You know what helps kids (and adults) learn math? The abacus/soroban. Yes, automaticity with math facts/basic math is important but what's really important is being able to represent the base-10 system mentally.

The abacus is an amazing tool that's been successful in creating math savants - here's the world champion adding 10 four-digit numbers in 1.7 seconds using mental math [https://www.theguardian.com/science/alexs-adventures-in- numb...](https://www.theguardian.com/science/alexs-adventures-in- numberland/2012/oct/29/mathematics)

Students are actually taught how to think of numbers in groups of tens, fives, ones in Common Core math -- however, most are not given the abacus as a tool/manipulative.

</comment><comment><USER67> I'm studying math right now and I have that problem. We're just being vomited theorems and propositions in class instead of making us think. There's not a single subject dedicated to learning the process of thinking in maths. So I think we're learning the wrong (the hard) way.

<comment><USER68> Many undergrad programs (in the U.S.) now have a course in math proof, which is intended to introduce students to proof techniques. It seems helpful to learn how to write proofs without the pressure of learning the content of a subject (like linear algebra, abstract algebra, and so on) at the same time.

</comment></comment><comment><USER69> Well, I actually liked math and took kind of a lot of it in K-12. I was in my 30s before I knew there were actual applications for some of the things I memorized my way through without really understanding.

When I homeschooled my sons, I knew this approach would not work. My oldest has trouble with numbers, but he got a solid education in the concepts. He has a better grasp of things like GIGO than most folks. We also pursued a stats track (at their choice) rather than an algebra-geometry-trig track.

Stats is much more relevant to life for most people most of the time and there are very user-friendly books on the topic, like "How to lie with statistics." If you are struggling with this stuff, I highly recommend pursuing something like that.

</comment><comment><USER70> i always found munging numbers and memorizing formulas discouraging. i think physics classes teach kids more math than math classes and in more interesting ways (or at least have potential to).

<comment><USER40> Interestingly, my experience was that your two sentences are totally contradictory. Physics and engineering was all about using memorized formulas to solve word problems, and mathematics was where you learned how those equations systematically fall out of some obvious assumptions.

I think the real take-away is that both physics and mathematics can be taught poorly, and both can be taught well.

<comment><USER70> i get a feeling you're not talking about elementary school. but i agree - <USER3> can be taught good or bad. bad is dumb memorization, good is understanding real world phenomena.

</comment></comment></comment><comment><USER71> Programming needs to be taught alongside Algebra I. Especially in a language like Haskell or Scheme where algebraic refactoring of type signatures looks like normal algebra notation.

<comment><USER72> Learning multiple different highly abstract concepts at the same time does not seem like a solution to the difficulty of learning a single highly abstract concept.

<comment><USER35> My experience with this comes from college, but I disagree. I added a math major late into my education (I was one course from a minor, then 4 more from a major so why not?). I ended up in several math courses at once that were often taken separately, but specifically abstract algebra and a higher level (4xxx) linear algebra course.

I pretty much aced both, though I probably would have gotten an A in them taken separately. The reason I did better, though, was that they were both algebra. They were both expressing the same concepts, but on different _types_ of objects. The moment that that clicked in my head, my Monday lesson in linear became the clarified by my Tuesday lesson in abstract, which was further clarified by my Wednesday linear.

If you structure it correctly (particularly, with programming, by selecting appropriate problem domains for programming exercises), I'm quite confident you could create a better math and programming education experience by properly pairing the two.

<comment><USER72> Maybe! I just don't think programming (even in Haskell!) is similar enough to algebra for this to be effective, no matter how much we all want it to be. Programming still requires a mental model of how machines compute, which is a good thing to learn about but not (in my opinion!) a strong complement to learning math.

Don't get me wrong though, I learned programming and calculus at the same time, I'm not saying you can't learn both at once, I just don't think one helps much in learning the other, for a novice in both.

</comment></comment></comment></comment><comment><USER73> One of the most critical skills I see differentiating people around me (co- workers and otherwise) who succeed and those who don't is an analytical, pattern-recognizing and pattern-applying mindset. Math itself is quite useful, but I really like the way this particular article highlights the mental blocks and misconceptions that seem to particularly crop up around mathematics; those same blocks and misconceptions tend to get applied to other topics as well, just less overtly.

</comment><comment><USER74> A couple of years ago I did the _Introduction to Mathematical Thinking_ course on Coursera [1]. Even though I found it hard, I enjoyed it and learned a lot, and I feel I got some insight into mathematical though processes. Recommended.

[1] [https://www.coursera.org/learn/mathematical- thinking](https://www.coursera.org/learn/mathematical-thinking)

</comment><comment><USER75> As a someone with a degree in applied mathematics, I feel the problem with learning mathematics is more often than not a problem or a fault of the instructor of mathematics.

Many instructors approach the subject with a very broad understanding of the subject, and it's very difficult (more difficult than math) to shake that understanding and abstract it to understandable chunks of knowledge or reasoning.

</comment><comment><USER76> If we want to teach people how to think, I propose that math isn't the best way to do it. I can't tell you how many times I complained about how senseless math was. The real-world application is very limited, for the most part.

Contrast that to if I had learned programming instead. Programming definitely teaches you how to think, but it also has immense value and definite real- world application.

</comment><comment><USER77> Anecdotally, I was a pretty average math student growing up and a pretty good math student in university. One of the reasons I studied math in college was to improve what was objectively my weakest area intellectually, but I found that once we were working with much more abstract models and theories, I was more competent.

</comment><comment><USER78> I always had the feeling I failed to grasp math because I never got good at mid level things.

It took me reeeally long to grasp things like linear algebra and calculus and I never was any good at it.

It was a struggle to get my CS degree.

Funny thing is, I'm really good at the low level elementary school stuff so most people think I'm good at math...

</comment><comment><USER79> There just needs to be faster feedback than _once a test_.

[https://opinionator.blogs.nytimes.com/2011/04/21/teaching- ma...](https://opinionator.blogs.nytimes.com/2011/04/21/teaching-math- advanced-discussion/)

</comment><comment><USER80> When I first saw it I thought the sign in the mentioned tweet may have been because the deli was next to a mathematics department and the professors/students would stand around and hold up the line while discussing math.

Overactive imagination I guess.

</comment><comment><USER81> Dammit, I glanced at the domain and expected a statement from the gangsta rap group... School is cool, kids!

</comment><comment><USER82> School is not about learning but learning how to think. The way it is now it's more about showing off than it is about anything actually useful. They don't reward effort, they reward talent.

</comment><comment><USER83> tl;dr A couple of numbers-nerds are sad and offended that math is not as recognized as reading and literature, where there are great works that speaks of the human condition and illustrates life.

Also they have the mandatory "everything is really math! ™". "LeGrand notes that dancing and music are mathematics in motion. So ... dance, play an instrument."

Just because i can describe history through the perspective of capitalism or Marx theories, does not make history the same thing as either of those.

</comment><comment><USER84> On a somewhat related tangent, I think about programming the same way.

I always tell people programming and syntax are easy - it's learning to think in a systems and design mindset that is the hard part.

</comment><comment><USER85> I agree, but have a small caveat: math does typically strongly involve numbers, so in a way, it is about numbers, though it's definitely not about just memorizing things or blindly applying formulas.

It just bugs me sometimes when people make hyperbolic statements like that. I remember coworkers saying things like "software consulting isn't about programming". Yes it is! The primary skill involved is programming, even programming is not the ONLY required skill.

</comment><comment><USER86> Maths problems are cool too, like counting apples and oranges :) (Or gold and rubies)

</comment><comment><USER87> How do you "learn to think" _without_ numbers?

Depressing.

</comment><comment><USER88> Only really a problem in the USA. In civilized countries, there's no particular aversion to math or to disciplined thinking in general.</comment>


Stress That Doesn't Pay: The Commuting Paradox (2004) [pdf] - <AUTHOR> https://www.econstor.eu/bitstream/10419/20544/1/dp1278.pdf <comment><USER3> > _Contrary to the prediction of equilibrium location theory, we find a large > negative effect of commuting time on people's satisfaction with life. People > who commute 23 minutes (one way), which is the average commuting time in > Germany, would have to earn 19 percent more per month on average in order to > be fully compensated._

When I worked in Manhattan, I spent the extra $1,000/mo. to live within a couple blocks of work, rather than a 45-min commute away off in Brooklyn like most people I knew. It's _expensive_ to live next to work.

But here's the thing: I _always felt guilty about it_. That it was "unnecessary" money to spend, even though I had it. That it was irresponsible.

However, I knew from (painful) experience it was still the rational thing to do. Once you subtract time spent sleeping, at work, and "maintenance" (shower, tidying, etc.) you really only have maybe 5 hours a day of personal time max. If you commute 45 min each way... that's only 3.5 hours. And when you factor in that you need a couple hours to relax, eat dinner, etc. -- that's the difference between having time left over for a rewarding hobby or not, which qualitatively changes your life in a fundamental way, and keeps you sane at work. Whether it's going to the gym regularly, reading books, doing yoga or dance, whatever it is.

But it's still _so hard_ to mentally justify paying such a premium for your own leisure time. It's _really_ easy to tell yourself, no -- I'll save money, commute, and I'll still manage to find the time, it'll be enough... because you're optimistic you can have it all. But you can't. You'll lose your hobby or sleep or your happiness.

<comment><USER4> This is the justification for remote. You get the benefits of a full salary while avoiding close-to-work living costs _and_ commuting time.

I spent a hellish couple of years in London living in a pleasant area but commuting to and from a startup near the centre, and since then I've been remote-only.

There are other trade-offs, but it's hard to beat for low stress levels and free time.

<comment><USER5> You avoid the close-to-work living costs, and also the close-to-work living benefits.

I'd save time, but then I'd spend it driving into the city, anyway, to do most of the things I want to do.

<comment><USER6> clarification: I didn't downvote you. dunno why people would.

but if you feel like that, consider if you're living in the "right" place for this. "remote" could mean instead of living in san francisco, living in a small town far away. or heck, somewhere in europe, where I live in the very city center on a student budget.

<comment><USER7> This probably won't be a popular opinion here, but some of us actually want to live in San Francisco, and not just for the job market.

</comment></comment></comment></comment></comment><comment><AUTHOR> From the paper: An individual with a 60 minute commute has to earn 40 per cent more money to be equally satisfied with life as an individual who can walk to the office.

I found this very interesting, given the fact that I know a lot of people who work in major cities but live outside the city for financial purposes. Might make more sense to bite the bullet and pay high rent to live next to your workplace.

This is complicated for couples though, because it is hard for you and your partner to find employment in the same part of a city. So one of you will probably have to make a sacrifice.

I would be very interested in how remote work figures into all of this. Does the interaction with humans outweigh the benefits of being able to roll out of bed and be at the office?

<comment><USER8> > Does the interaction with humans outweigh the benefits of being able to roll > out of bed and be at the office?

I wonder if this harrowing 911 call sheds some light on this question:

[https://www.newyorker.com/humor/daily-shouts/i-work-from- hom...](https://www.newyorker.com/humor/daily-shouts/i-work-from-home)

<comment><USER9> I don't consider working hours proper human interaction. It's professional interaction. Working from home is not recommended for every personality type.

</comment></comment></comment><comment><USER10> I commuted for almost 20 years in the NY Metro area. My journey, on one of the least reliable mass transit lines, took 4 hours from my day... if it was on time. I was well-paid and lived in a nice home, in a great neighborhood. The real problem with this commute is it takes over your life. During the work week, there is no me time, no time for fun or hobbies. I would sandwich in workouts and some downtime to unwind daily, but I was deeply sleep-deprived. Sure, I slept on the train and meditated, but the lack of reliable transportation on a daily basis really retarded any progress on those fronts. And that impaired my ability to fully enjoy my family, actively participate in my kids' upbringing and to genuinely enjoy life. I kind of had it all, yet, that darn commute raised my stress levels so high, it was ruining every aspect of life. I drank more than I wanted, and endured marital discord.

Moral of the story is look at your commute as the top factor determining your quality of life. Think well and hard about that extra 30-60-90 minutes because it means a bigger house and backyard.

<comment><USER11> I've been doing the same for only 4 months and I can feel a lot of similar feelings beginning to set in. I have no idea how you managed it for so long!

</comment></comment><comment><USER12> I'm always surprised seeing so many people accepting to commute.

The time waste, the money waste, the environmental waste.

I guess they got sold on the idea they can listen to a podcast on the way to the office...

It's important to think about what we do. Even if we are taken by the storm of life. Pause, and think about what you are accepting to put yourself through.

<comment><USER13> I guess I'm in the minority here, but I genuinely enjoy my commute to work. It's the sole reason I turn up.

With my new place of work I pass through rolling hills, farms, vineyards, rivers, and by the time I get to the office I am delightfully refreshed and my heart is pumping.

When I was in London I would traverse the same distance, but loved weaving through the narrow traffic, sprinting up the small hills, passing over the river and then racing through central, fighting through the markets, and feeling absolutely elated by the time I got into the office.

For this reason alone I always place myself at least 15km from my work place. Its a nice transition between working mode and home mode, and its a good way to separate the two.

Plus it's easier to justify home office to the boss on those super treacherous days.

<comment><USER12> Right, it's not like you could make your own path of walking, riding a train, a bus or your own car wherever you want, on your own terms by not _having to_ commute every single day.

But hey, good thing you have a beautiful scenery on the way to the office. It's rarely the case for most commuters.

</comment></comment></comment><comment><USER14> Right now, in the US, in many cities, we have minimum parking laws, that say you need X number of parking slots per commercial space or per work area, within some maximum distance of a couple thousand feet or less. And, yet there's absolutely no minimum living space/apartments/residentials to speak of.

This is completely backwards. We should have minimum living space requirements within every work place. If you want to regulate things, regulate it well: No places to live, then no places to work. If there were 1 place to live within 500 ft for every 1 job, then we wouldn't have this situation where most of the population has to commute 2 hours every day polluting the planet with overly long commutes that could be 10 or 20 times closer. Imagine the reduction in c02 if people were traveling 20 times fewer miles every day.

<comment><USER15> Which was some of the thinking behind the classic attempts at "company towns". On the one hand, yes reducing commutes is a public good, but directly or indirectly connecting your housing to the company you work for has it's own problems (including just the basic problem that your whole life shouldn't have to revolve around where you work). As with most things, you have to be careful what you incentivize.

<comment><USER16> "company town" is a small town phenomenon. Superstar metros are the ones that need regulation that promotes building of housing units.

<comment><USER15> Just because we most associate the "company town" era most with (relatively) small towns (and that wasn't always true, both Detroit and Chicago, as two wildly different examples, have been accused of being company towns in different ways and at different times), doesn't mean it isn't an applicable comparison even to "superstar metros".

The "Google bus" issue in SF certainly didn't make SF a company town, but it did raise the specter of how willing a large enough company can be to routing around a city's infrastructure rather than work with it, stratifying public transit services in complicated ways that caused a lot of real grief for non- Google employees, just as non-employees in many a company town would have had mobility issues.

<comment><USER17> >The "Google bus" issue in SF certainly didn't make SF a company town, but it did raise the specter of how willing a large enough company can be to routing around a city's infrastructure rather than work with it,

What exactly are they supposed to do? Stage a violent revolt of the local government? They're a big company with a lot of money, but that doesn't give them control over the local politics, so there's only so much they can do about the transit situation. Public transit in America is notoriously horrible, and no single corporation could possibly fix it, so it makes perfect sense for them to hire some buses to help their employees commute faster. They're helping keep many, many cars off the road by putting people into buses, and yeah, it's unfortunate that non-Googlers can't use the buses, but if people want better public transit, then they should start voting for local politicians who will improve public transit. People don't want to do that just about anywhere in America, so this is what we get.

<comment><USER15> As one easy counter example, Microsoft spends a lot of money directly into King County transit options (and has done so regularly for a long term relationship), and "money talks" to politicians. It's a lot easier for politicians to be "pro-public transit" if they don't have to fight so much for how to pay for it.

(That has its own issues in the entangling of corporate and political interests, of course, as there are no perfect solutions when capitalism is involved.)

<comment><USER17> >It's a lot easier for politicians to be "pro-public transit" if they don't have to fight so much for how to pay for it.

They don't have to "fight" to pay for it, they just need to use tax money to do so. The problem is that voters don't want that. Voters are perfectly happy to _massively_ subsidize infrastructure for cars (those roads aren't free), but they don't want to subsidize public transit, and claim that public transit should somehow survive on its own (even though roads don't).

>as there are no perfect solutions when capitalism is involved.

There's nothing "capitalistic" or "free market" when it comes to the road system in this country. It's purely socialism. Charge use-based tolls for every road driven on and things will change quickly.

</comment></comment></comment></comment></comment></comment></comment><comment><USER18> I've been working remote for 8 years and so my commute is 0 mins. But in my experience companies that allow remote work have caught on to this commute time==money dynamic and have adjusted their offered compensation accordingly. More than once have I had or heard the conversation that "well, you have the perk of no commute + living in a lower cost area, so based on that the compensation target for you is X - Y", where Y is $10-20k less.

The question I wrestle with: is it fair?

1\. For living in a lower cost area, no. If I live in SF I benefit/pay a premium for living in a nicer area. That should have nothing to do with compensation (unless they are located in SF and need me close by).

2\. For not having a commute, maybe. I think it is fair for a company to pay a premium to have someone onsite. But that premium should be the same whether I walk 5 mins to work or drive 50 miles.

<comment><USER19> I think it's a mistake to frame this with respect to 'fairness'.

If you were another company instead of an individual you would not be having this discussion with your employer.

Would they try to bring the price of IT equipment down, with the argument of a shorter delivery route, if they were doing business with a retail business close by their office?

For HR the best outcome is to get the best people possible for the lowest price possible.

The best outcome for you as an employee is to get the maximum compensation + benefits + bonuses for the least amount of your time/effort.

For the archetype factory employee, their value grows linearly with time spent working. For a knowledge worker the output of a small team of 10/20 people could generate orders of magnitude more, the time spent is not relevant what matters is the value adding output.

So if you want to think in terms of fairness don't frame it in terms of the 40 hours a week, frame it in terms of the profit generated or money saved as a direct result of your output.

<comment><USER20> > frame it in terms of the profit generated or money saved as a direct result > of your output.

As I pointed out here[1] this is hard to do and no company does this. If you _can_ find a way to quantify your value to the company, that's great. 90+% of the cases I've seen where people do it, they're making wild assumptions that no employer would buy, though.

> Would they try to bring the price of IT equipment down, with the argument of > a shorter delivery route, if they were doing business with a retail business > close by their office?

No, but they would bring the price down if they could buy it cheaper from a competitor. If you live in a low COL area, they could try to argue that they can find other employees in your area for whom they could pay less than what you demand, and those people would still get paid more than the average in that area.

There's a reason outsourced employees in other countries don't get paid as much as they do in the US. Do not expect that you can change that dynamic easily.

Of course, if you can demonstrate that it'll be hard for them to find a remote worker with your (perhaps rare) skills, then your argument would be more appealing to them.

[1] [https://news.ycombinator.com/item?id=21685035](https://news.ycombinator.com/item?id=21685035)

</comment></comment></comment><comment><USER21> I live in NYC so perhaps I can give some perspective on why we put up with this. My commute is ~40 minutes from Brooklyn central, and relatively painfree. I work in Midtown. A lot of my friends commute 1 - 2 hours from South Brooklyn, and one co-worker is commuting from outside of Queens(!).

A lot of jobs that pay decently aren't located in the outer boroughs. We could move closer to the city but we simply don't make enough and/or don't have the skills to make enough. Rents, especially near train stops are reaching 2 - <USER2> easily. Also, many jobs are still on the fence about letting more staff work remotely, although tech is changing this. Thus, we put up with crowds, delays and smelly train cars.

I hope this brings a bit more light into things.

<comment><USER22> I also live in NYC (Manhattan, specifically), and I read through this with my own lens: married, with a kid, another on the way. I live 13 blocks from my office.

Some things that cause me grief: 1\. NYC's city income tax on top of federal and state tax 2\. The competitive public school system that disadvantages many and doesn't appeal to my parental administration efforts either 3\. The pay- or-wait-for everything kid-related -- 2s or 3s programs that ask for astronomical amounts of money 4\. Swampy summer urban heat island 5\. AIR quality that probably isn't as good as it could be. If you ever walk on Madison Ave. during morning or evening rush hour, you've inhaled the avenue- wide plume of diesel smoke that saturates the air

In some suburbs things flip in the other direction: 1\. Public schools that are supported by high property taxes 2\. Rather large fees for relatively shorter commutes --hundreds of dollars per month to commute 40 minutes to/from Grand Central or Penn 3\. Lower income tax burden

I'm looking at houses in the suburbs now, and what's difficult to quantify is the level of happiness associated with gaining things like fresh air, extra space, a yard for kids to play and keep themselves busy, closer proximity to family (both sides), and etc.

I think there's lots of folks who positively and negatively value similar things and happily eat the commute time up (as me time, or personal time) in exchange for the some of the benefits I mentioned.

</comment></comment><comment><USER23> It has been noted that most people consider a commute about 30 minutes or least each way every day normal. Transport mode doesn't matter: primitive villages walk that long to their various fields or whatever. As we add faster modes of transport people move farther out to maintain that time.

<comment><USER17> >As we add faster modes of transport people move farther out to maintain that time.

Exactly, which is why widening highways to alleviate congestion is a waste of time and money. More people will just move farther out, canceling out all the savings had by adding the extra lanes.

<comment><USER24> Let's do a thought experiment. Today, Highway A that links suburban region A to city B has 2 lanes and there are Y number of people that can make a certain commute in a certain amount of time.

We widen it to 4 lanes. We get 7-10 years of reduced congestion that steadily gets worse until we are at the same level of congestion.

Casting aside that temporary reprieve, which in itself adds value, the carrying capacity even at current levels of congestion has doubled, do we've now opened up another Y number of people with the option to make that commute.

So when it's all said and done, the number of people that can make that commute has doubled, even if the time it takes remains the same.

Still a win in terms of opening up options and mobility. More trucks can make deliveries. More people can commute to work.

<comment><USER25> The temporary reprieve's value is probably a minute or two each direction saved tops. The cost for that is also not paid for by the people using it.

<comment><USER24> >The temporary reprieve's value is probably a minute or two each direction saved tops.

Do you have a source for that? How could you possibly know without knowing current distance and average speed?

</comment></comment></comment></comment></comment><comment><USER26> The transportation mode is critical for a commute. The time spent commuting can be mitigated somewhat if you are active in the process.

As an example, I live relatively close to my office in the suburbs, and most days I walk ~50 minutes each way for my commute. I could drive it in 10-15, but my family has reduced from 2 to 1 cars this way (owning a car is expensive and a massive headache). I also spend far less time at the gym because I walk many miles per day just commuting, plus I feel fresher when I get to work. I can definitely say I am much more satisfied with this arrangement than if I was driving for the same duration. The major negative is less time at home with the family, which I admit is a challenge but one I've been willing to put up with for the other advantages.

</comment><comment><USER27> I wonder how the equation would change if flex/remote options were available, eg. telecommuting 1-2 days/week. Imo the 5 days a week commuting grind is much more debilitating that 3-4 times a week.

Going meta, the inertial insistence on 'butts in the seat 5 days a week' needs to be disrupted. I know there are more startups and young companies these days that understand flex and telecommuting, however, this needs to flow into the large mass of old school companies somehow.

</comment><comment><USER28> Didn't fully understand the statistical analyses tables in the paper because college statistics was a long time ago, but to me it seems like other variables had way more effect on life satisfaction than commuting time. To me those charts imply that having a partner (or being married) and having an education improve your satisfaction way more than adding subtracting 60 minutes to your commute.

If we're able to weight correlations of different life parameters with life satisfaction, where is the "how cost-effectively to maximize your satisfaction according to statistics" guide?

All my coworkers commute 40+ minutes to have a larger house for their family and a slower pace of life than in the city.

This seems like pretty biased against commuting. Is it because everyone here is salty about having to commute to SF or Silicon Valley because rent is prohibitively expensive?

</comment><comment><USER29> Abstract:

>People spend a lot of time commuting and often find it a burden. According to economics, the burden of commuting is chosen when compensated either on the labor or on the housing market so that individuals' utility is equalized. However, in a direct test of this strong notion of equilibrium, we find that people with longer commuting time report systematically lower subjective well- being. Additional empirical analyses do not find institutional explanations of the empirical results that commuters systematically incur losses. We discuss several possibilities of an extended model of human behavior able to explain this 'commuting paradox'.

</comment><comment><USER30> I suppose my mindset could change later; but I live a rather hectic and noisy life with small children (as I'm sure a lot can relate).

Having a 40 minute commute entirely to myself to listen to anything I want to (or silence) is almost a sanity check for me.

</comment><comment><USER31> I'm definitely in no commute camp. My office is less than a 5 minute walk away from my apartment. It's literally a distance of two blocks. I'm in apartment right now for lunch cooking myself a meal.

</comment><comment><USER32> Another data point: I know somebody who has a long commute (at least 2 hours a day), but commute by train and is allowed by his employer to work during his commute. He also works from home one day per week. Which sounds like an excellent compromise: he's not socially isolated by remote working; his commute time is included in his working hours; he can afford a nice home in a less expensive place.

</comment><comment><USER33> Just the number of minutes is a very coarse measure.

Important for me are seating and no changes in public transport and by bike the route is important.

</comment><comment><USER34> Looks like once air taxi or commutes proliferates, down town residential house prices will plummet

</comment><comment><USER35> I just returned to a two hour commute instead of a 20 minute commute because of stress induced by the 20 minute commute job.

Is it ideal? No, is it worth it? Yes.

</comment><comment><USER36> It's not a paradox, they aren't paying the full price for energy. Soon they will!

</comment><comment><USER37> Not all commutes are equal. Self driving will change many aspects. You will be able to do recreational stuff while in the car.

<comment><USER38> That is absolutely wrong. I take a train for ~50mins and walk ~10mins. I used to just do the ~10min walk. Work/Life was so much better when I just did the walk. Sure I can read and listen to podcasts and play phone games, but what I'd really like to be doing is be with my wife and son, or really just not in transit.

<comment><USER39> I'll offer a contrast. For me 10 minutes wasn't enough because it made me feel like I lived at work. Now that I travel 45 mins the increased physical distance results in mental distance as well and feels incredibly liberating.</comment></comment></comment>


Micro HTML and CSS Only Framework for Slides/Presentations - <AUTHOR> https://twitter.com/giuseppegurgone/status/1096051462407405568 <comment><USER2> Not sure what to make about this idea of going off piste with your own elements.

Next there will be people wanting to roll their own 'bus ticket' elements, 'parking-ticket' elements, 'food-label' elements and so on. Div soup will start to look hygienic again.

Can't people give the elements in the HTML5 a fair go?

Besides 'css-slide' is actually quite a thing to type, might as well use 'section' with it autocompleting in the editor.

<comment><USER3> Look upon [https://www.webcomponents.org/](https://www.webcomponents.org/), ye mighty, and despair...

</comment></comment><comment><USER4> The "inspired by" article is where the beef is: [https://www.chenhuijing.com/blog/html-slides-without- framewo...](https://www.chenhuijing.com/blog/html-slides-without-frameworks/)

</comment><comment><USER5> And not a bit of Javascript in sight! love it.

<comment><AUTHOR> I imagine you could add keyboard navigation with a bunch of radio inputs with the same name (and hidden to screen readers)

<comment><USER6> You can add the tabindex attribute to non-form elements

<comment><AUTHOR> yup that'd enable tabbing. With radio inputs you could use left and right keys to go to the prev and next slide

<comment><AUTHOR> something like this [https://twitter.com/giuseppegurgone/status/10961238298925916...](https://twitter.com/giuseppegurgone/status/1096123829892591616)</comment></comment></comment></comment></comment>


LLVM 3.7 Release - <AUTHOR> http://lists.llvm.org/pipermail/llvm-dev/2015-September/089935.html <comment><USER2> Exciting to finally see the release! But the feature I was most interested in appears to have been pushed to 3.8. Namely, the addition of -fsanitize- trap=undefined (which will allow for some parts of UBSan to run at compile- time rather than adding run-time overhead and bloat).

(Also, I know many people are not terribly worried about the overhead and bloat—and, to be fair, it is pretty minimal—but, I'm a little obsessed with keeping things about as light and small as I can make them.) :P

Either way, great to see the new stuff (especially the additions to the the static analyzer and formatter)! Here's looking forward to 3.8!

</comment><comment><USER3> Very exciting. Since the current rustc is based on a fork of llvm, when can we expect it to be part of the mainline llvm?

<comment><USER4> I believe what you're asking is when rustc will be using a stable, unpatched version of LLVM? Rust is always working to get its custom patches upstreamed, but as soon as it does it typically comes up with more. :) That said, most of Rust's custom patches are to do with performance improvements due to compile- time information that C and C++ have no equivalent to, so theoretically Rust could probably exist on a stable branch of LLVM with nothing but a performance hit (and this may actually be the case if there exist OS package managers which don't want to package a custom version of LLVM with their Rust distribution, though I don't know if any package repos actually go to this length).

<comment><USER5> Debian rustc package is built against Debian LLVM 3.6 package. All Rust tests do pass.

<comment><USER6> On FreeBSD, I've seen bugs in release builds using that approach. The compiler works, but the compiling doesn't.

</comment></comment></comment></comment><comment><USER4> Excellent. While we're on the topic, does anyone happen to know the status of LLD?

<comment><AUTHOR> LLD has been redesigned. LLD as released in 3.7 can link itself faster than MSVC's LINK.exe at a factor of around 2x. It can correctly link some large projects like Google Chrome and LLVM itself but I'd still call it rather experimental.

Work is ongoing to support features like COMDAT folding.

<comment><USER4> Thanks for the update. :) I know that the folks in the Rust project are curious about LLD's potential to provide a linker that Just Works on any of the platforms they target.

</comment></comment></comment><comment><USER7> > Clang 3.7 includes an implementation of control flow integrity, a security > hardening mechanism.

I wonder which operating system/Linux distribution will be the first to activate this by default when building binaries. It's extremely effective in preventing memory unsafety bugs to get exploited without a big performance hit.

</comment><comment><USER8> I don't really understand why important targets like x86/x86-64 and ARM don't appear at all in the release notes. It seems unlikely that no work at all was done for them...

<comment><USER9> The MIPS and PowerPC folks were simply much better about writing release notes.

</comment></comment><comment><USER10> The BPF target looks promising for writing seccomp filters. Anyone tried it?

</comment><comment><USER11> Nice to see references to LDC and LLVMSharp in the release notes.

</comment><comment><USER12> Finally, official OpenMP support!</comment>