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Math education: It’s not about numbers, it’s about learning how to think (nwaonline.com)
565 points by CarolineW on July 11, 2017 | hide | past | web | favorite | 330 comments

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.

> 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.

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.

But once I needed to use math, it was trivially easy.

This is close to my experience as well. Math, for me, wasn't super difficult, but it didn't necessarily come naturally either, and I got really bored with the tedious, mechanical aspects of it in algebra, college algebra, trig, etc. Geometry and Calc I were mildly interesting, but I was handicapped a bit in Calc I because my basic algebra skills weren't as strong as they needed to be... because I found basic algebra to be mind-numbingly boring.

Fast forward 20+ years and I'm not working on re-learning some lower level math + leveling up on higher level calculus and linear algebra. This time, it seems easier (although I might not go so far as to say "trivial") and I attribute that largely to the fact that I'm much more motivated now. Now I have specific reasons for wanting to learn this math (mostly to do with machine learning and AI) and that seems to make all the difference in the world.

I am in kind of a similar situation. Was never really good or into math during school, been a software developer for 10+ years (mostly backend and web stuff) and at 34 am now working on my calculus and linear algebra after spending some months brushing up my basic algebra skills (which i still do on the side to keep "fit"). I am actually toying with the idea of going back to college to do a Masters in Data Science because i got so fascinated with Machine Learning and AI as well. Soo, you rock, keep it up!

Seems like both of you are doing what I plan to do once I finish my Udacity SDC Nanodegree course. I've known for a while (ever since AI Class and ML Class in 2011) that I need a refresher and more on math, because ML. This Udacity course just drove it home.

I'm tired of hearing about "take the derivative" and "perform the integral" (or whatever), and not quite understanding it. Heck, just what I wrote probably indicates I don't quite get it!

Also, my intuition on probabilities and statistics isn't there either, so I want to do something to fix that as well. So, when I finally get done - that's my next goal (after a bit of a break).

It isn't that I'm terrible at math, I just never used much beyond basic linear algebra after leaving high school. I never went much beyond high school: I got an associates from a tech school (worthless, since the school is long defunct), and I've taken a few community college courses, plus my online MOOC stuff that's more recent.

I'm not hurting career-wise; in fact, that's never been an issue as a software engineer (started when I was 18, and just kept going, earning more over time and having fun). I'm not doing bad right now, living in Phoenix. It ain't SV - but then again, it doesn't have the downsides of it, either (not that there aren't certain downsides).

Now - after having done MOOCs in ML/AI - I am finding that I need more understanding and intuition about calculus and other mathematical subjects, so I can really apply what I have learned, and understand it at a deeper level, and perhaps do more with it. I've given thought to going back for a BS and maybe an MS (not that I need it - I just want to do it).

Good luck to both of you; I hope it all works out great!

Best of luck to you too! Try taking some community college (CC) courses too. Calc. is not fun or easy, and having a professor to help 'keep you honest' may be very useful in the long term. CCs are very useful for these things, cheap, good schedule for working people, and an attitude of 'here to learn, not to degree'.

Pro-tip: Make sure to talk to current student outside of the lab space and without the PI around to get the real deal on how useful the classes are.

I'm not working

should read

I'm now working...

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.

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.

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.

I remember the day in my last year in high school, when my physics teacher looked at us with a frustrated face and asked:

"So, you still didn't have derivation in math class?"

We explained, that we will have it in the final month of the school year, if we are lucky. Her reaction was:

"But we kinda need it already. But I should be able to teach you that in 20 minutes."

And she did. And then she spent the rest of the lesson going through some old equations that she might have spent hours explaining: "Remember how long it took us to get to i.e: equation relating gass pressure and temperature? We always used that finicky helper variable. I think I even named it delta. In reality, you just derive this base form, and do a simple substitution."

Ok, my memory is a bit hazy. But she went on and on, through stuff we have been ehm, deriving without calculus for hours, that took with calculus around 3-4 steps.

Coming up with actual uses would have been enormously helpful to me when I was younger. All I was literally ever told was "you need this to pass the exam"

The one thing I really wished I'd have learned more of, younger, is stats. Knowing how basic probability works would have been fantastic - it took a lot of poker theory to really understand it in my bones.

I think motivation is key. That said, I think that the engagement in the 'personal finance' section of a HS class would also be low, maybe not for graduating seniors, but definitely for freshmen. Teaching is tricky, my SO is in education and it's not easy. Kids don't tell you things like their hunger issues, romantic issues, home life, etc. They may not even know they are having an issue, young as they are. Hormones are never going to stop making them nuts. Top that all off with budget problems (most US districts have not increased funding since 2008, pop. growth and inflation be damned) and other bureaucratic stuff, and you get a heck of a mess. Staying motivated as an educator is pretty tough, let along inspiring them to be motivated. At the end of the day, the kids are responsible for themselves largely, just like most humans have always been.

All I ever learned in high school math was isolated "number tricks" that had to be memorized and then performed on que.

Parrots have deeper understanding than was required of us.

I didn't even get a cracker.

Word problems: Designed to see if you can recognize numbers in a contrived short story and plug them into the rote-learned algorithm that the teacher just covered. Fun. It seems like if we taught the application alongside the math, it'd be easier to "get" what the math is supposed to represent.

I've been playing with computer graphics recently; I've understood trig for a long time, but this is the first time I've actually felt a need to understand matrices. Exploring applications on my own is the only way that I've really felt connected to math since maybe late grade school or early high school.

Doesn't Randall xkcd assert that the paramount math problem that most people need to do is how to split the bill fairly when you take your friend out for their birthday?

If you're splitting the tip and the tax, and everyone is pitching in $3 to cover the Birthday Girl, that's more complicated than a lot of those problems they use that nobody cares about.

(In fact, I think if you put all of those math problems in terms of fairness, you'd get more kids to pay attention. Nobody wants to get cheated, and that's what happens when you're bad at math and/or finance).

> I think if you put all of those math problems in terms of fairness, you'd get more kids to pay attention. Nobody wants to get cheated

That's quite insightful! I can think of a few related approaches that could round it out, like frugality (eg. only buy the paint/bugspray/lumber/etc. you actually need for your project), and laziness (do the least amount of work possible that still gets the job done).

Possibly more important is why logically you're not wise to gamble, work out odds and compare them to every day occurrences.

Trig and Simultaneous equations are probably the two parts of high school maths that I actually use on a regular basis.

I don't think it matters what ethnicity or race you are, it's good to be able to actually get things the right length. And it's a lot easier to do it on paper than run around for half a day with a tape measure.

Where do you use simultaneous equations on a regular basis?

I'm an engineer at an industrial plant simultaneous equations are very common in engineering problems. Just about any "optimization" problem can be represented as a system of simultaneous equations even very complex non-linear dependencies are modeled like this.

i.e a common problem such as calculating the optimal blend of fuel in a reactor is done based on solving a system of linear equations with constraints such as Loss on ignition, maximum allowable ash content, minimum specific energy etc.

Industrial design as well uses simultaneous equations extensively. Do you want to build your aircraft wing out of CFRP or a titanium alloy? One way is to use a constraint based selection method something like strength to weight ratio vs cost.

edit: I should add 99% of the time these are solved programmatically with the aid of a computer. You do not solve these problems "high school math style".

Last I used one was for mixing paint. I will admit it's not super regular, but I use them once or twice a year.

I often think that kid psychology is probably extremely well suited to kinetic problems. Things that involve their own natural desires, all kids want to move, jump, interact with the world. Connect their eagered mind with a way to denote the world and let them abuse math to communicate and reason their own desires.

A kinetic/kinesthetic approach probably would have worked well for quite a few classmates of mine, but definitely not for me. In fact, the only approach that would have appealed to me even less would be musical.

Hi Balgair, your comment made my day, i was feeling the same as you at that age.

Glad to have done so!

First, 16 is still pretty young. Second, just because you didn't have STEM hobbies when you were young doesn't mean you weren't given the framework to excel at STEM. It's not "you learn math or you don't", rather the argument is "you develop the framework for mathematical thinking or you don't".

For what it's worth I also did better with literature and history than math when I was young. Also, I struggled to pass Calculus 2 but breezed through upper level CS courses. It's a very fuzzy thing.

> First, 16 is still pretty young.

Depends on what we are talking about. The post I was responding to was talking about "the first years of education" and with 16, we are way past that.

Nice to hear some sense, I'm surrounded by tiger parents sending their kids to STEM summer camps & after school classes because it'll help them get into MIT or whatever. Its pretty sad as most of them would be much better off if they had some fun and had some friends.

So basically you're saying that it is a preferred solution to teach kids math poorly, let them develop insecurity and aversion towards it and then spent a lot of money to let them overcome it than just do it right in the first place?

Yeah, I know that's not what you mean, but you're missing a point. The premise of the article is that US has a STEM problem. I merely suggested that they're allocating resources wrong to fix it. I didn't mean they should introduce rigor and discipline or increase hours - I said that the best way to achieve results in teaching math is to fix it early, when it's most commonly broken (in my experience).

It actually helps you if you decide to switch interests, because you got the basics right, which in this case is the ability to think in an abstract way.

If you don't agree with me, you should have provided arguments that math problems do not arise early in education system, but they are introduced later.

One last thought - timing _does_ matter. That's why almost every professional athlete starts as a child. The question is how much you gain for starting early and I agree, that it is usually unreasonable to put too much pressure on kids. Pressure is not necessary to fix the system as far as I see it, though.

Went to college with a guy. Freshman year calculus. He was really slow with it. I helped him, and we did homework together. Now he's a math major. Totally surprised me, but he's now majoring in a subject that he wasn't super good at.

People need good teachers at all ages. I'm glad his college professors helped him out enough to lead him to major in math.

Mind and persons are complex, they can shift entirely. I've seen a girl that sucked at math she had to take lessons in HS. In college she flipped around and got a Master in Math.

On my side I was good at math but crashed in college, so much I avoided it, it's the reason I went into programming. Until 2 or 3 years ago I clicked again, I started to see through abstract math and now I'm back to math/phys because it's beautiful again.

If your buddy finds pleasure and can walk the path I'm not surprised he's still walking :)

>That's why almost every professional athlete starts as a child.

Professional athletes are highly competitive. People don't hire basketball players to throw balls into hoops; they hire them to beat other basketball players. Unless you are one of the best humans at basketball, you are worthless as a basketball player.

In almost any profession, you do not need to be the best to have value.

> So basically you're saying that it is a preferred solution to teach kids math poorly, let them develop insecurity and aversion towards it and then spent a lot of money to let them overcome it than just do it right in the first place? Yeah, I know that's not what you mean, but you're missing a point.

If you knew it's not what they meant then why did you start off saying that's what they meant? If you thought they were missing the point why didn't you just say that?

To better illustrate the point. I actually started with that, decided that it sounds too aggressive and added that paragraph to make that clear.

> 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.

While starting early is neither necessary nor sufficient for success, the question should be: do the benefits of starting early outweigh risks? Like the parent comment, I think they do, if done correctly and gently, but it should be decided case-by-case for each child. The only risk I can think of is: if you push too hard, it can have the opposite effect of making the child hate it. So, one has to do it with utmost sensitivity and by responding to cues and feedback from the pupil.

I find that a pretty strange approach to parenting. IMO you should optimize for happiness of your child, not for carefully manipulating them into the path you laid out for them. My perception of a good parent is to encourage your child to discover and pursue their own passions, even if they contradict yours. You can open doors, but they need to walk through it by themselves. YMMV.

Huh? Learning can be fun, last I heard. In any case, that's what I meant when I said 'feedback'. Further, there are some skills that are absolutely necessary to learn to function in society, basic math, basic language skills, etc. More important are skills like perseverance, managing one's own emotional well-being, self-control, delaying gratification, etc. It is one's responsibility to at least attempt to teach these to their children.

I was referring to this sentence:

> The only risk I can think of is: if you push too hard, it can have the opposite effect of making the child hate it.

That seems like a weird "risk".

You say you weren't interested in maths when you were young, but were you bad at maths when you were young?

Not being particularly interested and not being taught math is not the same. You probably had normal math just like anybody else and was not behind other people all that much.

But I agree that the tendency of programmers and public to assume that if you already don't know a lot about computers, you are lost case and stand no chance is harmful. There is strong tendency to overstate how difficult things are and that makes people look elsewhere.

Are you also suggesting that we shouldn't teach kids to read until they are in high school and decide they are interested?

No, I'm suggesting that we shouldn't push for kindergarten kids to read and analyze Shakespeare, because reading is so important (and in the process de-emphasize other subjects). I'm saying that the level of math education happening early (though I can only speak for my home country) is fine, just like the level of reading education happening in the early years is fine. It is completely okay to only reach Calculus and math as a structural science in your late teens and believing that you can teach that widely to kindergarten kids just doesn't work out particularly well for anyone.

For me, at least, understanding math was all about understanding how to translate symbols to concepts.

Seeing an integral sign and understanding what it means, for example, can help simplify seemingly complicated mathematical expressions and make them easy to understand.

way I see it, the comment is in writing and at least a bit creative. Its hard to separate language from math and higher learning. Now, psychology is even mor important with regards to learning. In that sense, keep a positive attitude ;)

To this point, I have been doing a plenty of 1on1 lessons with kids trying to catch up with Math in high school/college/matura exams and one this I have noticed is that things which are problematic to these people are things which were covered quite well already in elementary school.

Later the curriculum only expands on these problems so it's even harder and harder to catch up.

I have been working with college freshmen struggling with basic linear equations and such. While the profession is marginally unrelated to Math (e.g. Graphical Designer), I still wonder to this day how people can have 20-30-40 years and not know how they could (with a calculator in their hand!!) calculate the price after the discount. How can someone feel ok with this?

Schools (at least around here) do not hold kids back anymore. No matter what.

Some of them get behind in one or more subjects (math, say) as early as 1st or 2nd grade and all their later teachers are stuck trying to give them the remedial math help the desperately need, while teaching their other students on-grade-level material, and also trying to teach the remedial kids enough of the new stuff that their scores on that year's standardized test don't land the teacher in hot water (though the kids won't get it at all). The result is that the kids fall farther behind every year but keep failing up to the next grade. They'll enter e.g. 6th grade with ~Dec. of 3rd grade math skills (more often than not their other subjects aren't much farther along, though not always). If they get an LD diagnosis they'll get some extra help but by then it's too late.

It sucks for all concerned.

Tying students' progress through the curriculum to their age is silly. If you think about it for a single second, the whole thing is totally absurd. Vague pre-puberty/tween/teenager distinctions make sense, but within each grouping/building? totally unmotivated.

Yearly age-to-competency distinctions continue by sheer force of tradition, and are harmful to all but the totally mythological "average" student.

But it works at-scale, which is where the focus is. Bang-for-the-taxpayer-buck.

Further, socially-speaking, being with your actual peers is extremely valuable, obviously.

But yes, the ideal would be a tailored-to-each-child education. Personally, having seen the power of a good Montessori education, I think it's frickin' genius and should be the template for all education (specifically the grouping of 3 yrs together -- e.g. 3,4,5-year-olds together), so kids are perpetually working through the cycle of "look up to someone, mimic someone, mentor someone".

... but I digress.

Fact is, grouping by age works socially and it's much, much cheaper than a tailored education, which is all we an afford (get taxpayers to pay for).

At some point, I expect a software company to make headway in this space and you'd see a bunch of kids staring at iPads all day and a bored teacher playing Minecraft at the front of the room, only engaging when someone gets stuck.

My kids are homeschooled, and I'm in charge of teaching Math, because my wife is one of those "I suck at math" people. Her parents focused exclusively on memorization when they helped her with homework, and then she got a few bad teachers, so her mental models are completely off. She's getting better, but that's not really my point.

The math program I use with my kids is this one: http://www.defimath.ca/ecole-maison/ (in French)

It was developed by two mathematicians who actually studied in classrooms what worked on kids when teaching math. They've come up with a method where kids learn negative numbers and multiplication _before_ learning positional numeration (numbers greater than 10, with units, tens, hundreds, thousands, etc.). I've seen first-hand how these concepts just click in my daughter's mind, and how she often comes up with the new concept herself when you introduce the prerequisites in an order that makes sense to her.

They learn about negative numbers and multiplication before they learn to count past ten? Or they just learn counting without learning how to write the numbers?

The latter. My daughter actually learned it because we mixed different learning methods early on (we don't anymore, it confused her), but you have to admit that writing numbers greater than 10 kinda involves multiplication (by powers of ten, but still), so it makes sense to learn multiplication first. Negative numbers were seen while learning addition / subtraction (the concept was basically adding a negative number), which made a lot of sense too.

That sounds wonderful. My kids are about the right age to do this, and I'm curious, are there any English-language versions of this text?

Cheaper? I also don't know if that's necessarily true. If it's ineffective for many, is it a cheaper solution or just simply cheaper?

Personally, I think Khan Academy is one of the few forward-thinking organizations doing something right in education. It tries to skew learning towards a 'mastery-based' model which keeps kids from falling behind by not advancing them too early. Of course, teachers and parents still need to implement the usage of Khan Academy in this way for it to be effective.

>Further, socially-speaking, being with your actual peers is extremely valuable, obviously.

I would disagree... I think that there is huge value to being around a variety of people, and a strong negative value to only being around people in the same phase of life as yourself, especially when you are in the nasty phase. I think I made it through that part of my life because I had an after-school job where I fixed computers for a local office. I even enrolled in the vocational program at my highschool that let me out of school early to go to work (except every second Monday, when we learned how to spot shoplifters, short-change scams, and other hazards of the retail life.)

I think the number of people who felt good about their social lives, the things they did socially and the things, socially that were done to them in high school is... small. As far as I can tell, for most people, college is important because it is a kind of recovery from this, and prepares you for a workplace where conflict is muted, where yes, if you are good enough, you can still be an asshole, but where being an asshole has a pretty heavy cost that must be made up in other ways.

For me? Being expected to behave like an adult while being treated like an adult around a bunch of adults was amazing. It gave me a reason to keep going in high school, and when I came of age, I was all set to get a really nice job.

Not being argumentative -- is the current system actually cheaper? I'm not enormously familiar with Montessori, but is the student/teacher ratio vastly different, or are the various learning materials any more expensive over time than the books, computers, and reams of paper that traditional schools use?

I went to a public elementary school which had only mixed-grade classes (2–3 grades at a time), and it was just fine, and didn’t cost any more than having one grade per class with kids all precisely the same age.

>Tying students' progress through the curriculum to their age is silly. If you think about it for a single second, the whole thing is totally absurd.

Sounds quite logical to me. If we assume a gaussian distribution (which tests seem to verify), most kids of the same age will have the same skills/level.

So at worse you mismatch what's taught to some kids towards the edges, whose level there are ways to accommodate anyway in most school systems (skipping a class or two for extra smart kids, or staying behind/supplementary teaching for less than average smarts).

And it's not just about learning and who can cram more into one's head (akin to e.g. preparation for the Olympics), but also about sharing the same teenager and adult-making experiences as other kids of your age, which is probably even more important that what's actually taught (the majority of which most people will forget anyway).

> Sounds quite logical to me. If we assume a gaussian distribution

In this context, Gaussian is a pretty useless assumption without fixing a variance. Proposed alternatives range from "already implemented" to "totally infeasible" depending on variance.

> which tests seem to verify

Not really. For each individual subject area, maybe, and again, Gaussian is pretty uninformative.

But the odds of a student being "average" in every subject area != the odds of a student being "average" in a given subject area.

> whose level there are ways to accommodate anyway in most school systems

Except the whole point is that there are not currently ways to accommodate this in most school systems! From GP:

>> Schools (at least around here) do not hold kids back anymore. No matter what.

Also notice that holding back a student in math is possibly net detrimental if the student is not also behind in English and Science.

> but also about sharing the same teenager and adult-making experiences as other kids of your age, which is probably even more important that what's actually taught

Again, the prepubescent/tween/teen division is much less granular/restrictive than the competency-by-age-X division.

Sounds a bit circular: when we test kids taught in single age groupings they have "narrowly" grouped grades. Therefore we should teach then in single age groupings because they have narrowly grouped grades.

That doesn't really help us to know if the top achievers are being held back, for example. Perhaps we can speculate that lower achievers are being pulled up?

Outside school you're almost always going to be in age diverse groups; I think a larger element of that in public schools would be better. IMO it helps to emphasise that children are there to make their own education and not simply to be part of an age defined peer group where it appears you're doing something just because of your age, not because of the educational opportunity.

Yeah. To my mind, the problem is that learning is very individualized (not even taking into account learning disabilities!), and yet mainstream U.S. pedagogical theories really only target the middle of the bell curve. What's needed -- at least for a huge chunk of students -- is something more like Montessori, where a learner who's struggling can receive individualized attention from a teacher or a peer, and where the learner has free access to tactile/visual/whatever aids which may be overkill for most other learners, AND -- perhaps most importantly -- where the natural human tendency to sit and focus for hours at a time on a single task is fostered and encouraged.

Montessori is awesome when done right. It's also easy to get wrong. Moving an entire educational system over to it is a formidable task.

There is plenty of research that holding kids back isn't actually an effective intervention, for example: http://www.sciencedirect.com/science/article/pii/S0022440506...

Retained students (the current terminology for "held back") are much more likely to stop attending school early, and on average do not see larger academic gains in their year retained than their equivalent peers who are not retained.

The ONLY effective intervention is dramatically increased services for that student, delivered rapidly once the deficiency is detected. Many high performing charter school networks actually use this systematically, but only by overworking their teachers and generally burning them out within 5 years (retention rates at 5 years are frequently <10%, compared to 50% in the broader profession). Most teachers in mainstream district schools are covered by union contracts that limit their required working hours, so for the vast majority of schools even this isn't an option. Actually delivering these services in a sustainable manner is a financial burden American taxpayers appear largely unwilling to shoulder.

>calculate the price after the discount

It is the discount they care about, not the price. These have no calculations, only gut feelings, which are perfectly exploited.

Btw, how many world-ruling decisions are made this way?

Oh reminds me of something peculiar. I gave math lessons to a 16yo guy. Mainly quadratic root formula. He understood the "high level" bits, but he failed so often at basic algebra after plugging the coefficient into the formula that he felt completely demotivated. It was odd to try to explain to him that he got the idea right, but failed at the basics. I told him that to mean "you can do the hardest, you will be able to get the easiest" but he interpreted it at "whats the point if you cant do the low level bits" ...

Not all are created equal. Our children are unique.

One of my children showed natural talent in language at 9 months with no prompting. This was brought to our attention by childcare staff. Another of our children showed a natural talent with mathematical concepts at about a year old.

Even as they grew, our linguist struggled with math (for years) and our procedurally oriented child struggled with language (for years).

To this day, these two children maintain these core differences. It took at least 6 years for our linguist to crack basic arithmetic (even basic addition) which was at least several years behind our proceduralist.

I found out later that some leading child psychologists recognise different brain types in very young children (exactly as I found). An Internet search on brain types of children will show some high profile child psychologists who talk about this in depth, despite some strong "opinions" (ie. devoid of evidence) that oppose these studies.

Our linguist, with minimal pressure, has developed into a strong mathematician (at least grades wise) but to this day has never demonstrated anywhere near the natural ability of our proceduralist.

I have drawn the same conclusions with my own siblings and my wife's siblings. At a very young age, our own strengths become apparent without intervention. I am very glad I never pushed my kids to be equal (or even close to equal) in all skills. I consider most uses of the word "equal" worrisome (except for equal opportunity, a concept frequently downplayed in the last decade or so. Even Zuckerberg's famous open letter was unclear on such a fundamental concept).

OBS: when I asked our linguist to step through basic math, they understood the concept but could not do the work independently. Someone in this thread described a similar story for their child and attributed this to a lack of "confidence". For my child, I wholly reject that it was confidence related. When things clicked for our linguist, they clicked. If anything, our linguist's ability to crack the basics took patience on my part. I wanted my child to succeed quickly but I restrained myself (thankfully).

People need to realise that not all kids are the same. We have innate strengths. We have different learning styles, different learning rates, and different interests and motivations. I strongly reject the modern populist theory that we are all equal in ability and I believe we do significant harm because of this factoid. The motives behind this factoid concern me deeply.

If I could offer one piece of advice, your child(ren) are unique. Don't ever let anybody tell you that your child's strength or weakness comes from social conditioning. The only social conditioning cones from extreme behaviour (eg. Heavy handed forcing of "equality" under the banner of political correctness is extremely harmful, rather than focussing on potential and opportunity. This heavy handedness is also driving some extremely destructive social engineering under the banner of "equality". If you are watching academic trends you should be horrified as a patent).

I always encourage(d) play at a young age (physical activity, math games, language games). However, if you make this more than games (if you call this teaching and you start to measure), you set kids up for failure, especially when many children need patience and time.

According to PISA rankings, most western countries (especially English speaking) are not the top performers. I have hinted my beliefs of the root cause of this in this post. I predict most western countries will slip in ranking even further (especially English speaking countries). If things play as I expect, the slip will be significant in the next 10 years.

I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables.

Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi without first explaining how it's calculated? Where does the "four-thirds" part come from in calculating the volume of a sphere?

Not to mention that there's never enough history to go along with the mechanics - who discovered the quadratic formula? What was their life like? Why were they playing with quadratic equations in the first place? This make math seem less like magic incantations and more like something that was sort of cobbled together by flawed weirdos in order to solve real-life problems, and evolved over time.

that sounds utterly dreadful. computer science is often taught like this; to its folly.

imagine a kid learning to speak 'stop just memorizing words, you need to understand how language was derived before you learn how to speak it'.

its completely counter intuitive to how creatures learn. learn easy things, especially those that relate to problems we deal with, and then get deeper into the subject if necessary.

This would be true if calculating the volume of various shapes was something that kids did for 16 hours a day, every day.

Formulae without context are meaningless to most.

the pillars of science are theory and experiment.

memorization is a side-effect of these, not a foundation.

> Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi without first explaining how it's calculated?

If we're waiting until children are capable to derive those equations before we get them to use those a bit, then we're waiting for very long. And if you just keep telling them to add numbers and multiply numbers together for many years in a row without giving them any interesting problems to go with it, there'll be no one interested in math by the time they'd usually have the abstract maturity to deal with the more foundational modern math problems.

We don't ask Computer Science students to write an OS before using one, we don't ask carpentry students to build a hammer and cast nails before using them.

the beginning of the end of my math studies was eighth-grade algebra. i GOT algebra, it was easy, and i enjoyed it. ..until the quadratic equation was introduced.

"hey, that's really neat! how does it work?"

"oh, you'll learn that in calculus, which we won't allow you to take for another four years."

It's common not to explain where the quadratic formula comes from (which is silly, it's straightforward enough to show, but standard math education curricula are shot through with this blind formula memorization nonsense), but… do they really tell you you'll learn it in calculus? It's got nothing to do with calculus.

Not to mention, the quadratic formula per se is needlessly complicated. If you break it into two or three steps it makes a lot more sense:

Step 1: move pieces around and divide by the leading coefficient to put equation into the form x² + b = 2ax (or if you like, x² – 2ax + b = 0; or feel free to swap the sign of b if you prefer). The equation for the parabola is then x² + b = 2ax + y.

Step 2 (optional): rearrange that equation to get (xa)² = a² – b

Step 3: x = a ± √(a² – b)

In this form, the “discriminant” is just a² – b, the x coordinate of the vertex is a and the y coordinate is ba², Viète’s formulas tell us that the two roots satisfy ½(x₁ + x₂) = a and xx₂ = b. If the coefficients are real but the roots are complex, then we know each root has amplitude √b and phase arccos(a/√b). Etc.

Or even better in some contexts is the form x² + b² = 2ax, in which case we have a = ½(x₁ + x₂) [the arithmetic mean of the roots] and b = √xx₂ [the geometric mean of the roots], and the discriminant is a² – b², which has a nice symmetry.

> I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables.

This was tried. It was called "New Math". Spectacular failure. Do you want to know what worked? Memorizing times tables.

There's just no way around the fact that drilling is key to early mathematical learning.

I was a proponent of "New Math" types of philosophies before I had a kid. It was only when I tried to teach concepts did I realize how important it was to be able to be able to arithmetic quickly off the top of your head. Its hard to explain, or even to grasp, how fundamental basic arithmetic is to almost everything in math. For us arithmetic is as simple as thinking a thought -- but only when I tried to actually explain concepts to a young kid did I realize how difficult it is for them to understand these concepts when arithmetic isn't ingrained in their brain.

I hate to admit it, but I am now a believer that strong arithmetic skills are important, and drills get you there. I don't like to call it memorization, since I'm not sure it necessarily is simply memorization. But you do need the answers at a moments thought. It should be as natural as saying a word.

And its not to say that you don't teach concepts concurrently... but that the arithmetic is fundamental.

That said, I still believe that the long division algorithm taught isn't so useful. :-)

I generally agree with your comment but have specific disagreements with two parts:

Its hard to explain, or even to grasp, how fundamental basic arithmetic is to almost everything in math.

I disagree. Not hard to explain at all. Point the person at a page of geometry problems and say, "Imagine struggling with 90 + 90 at the same time as you struggle with the concepts here." Point the person at a page of polynomials to factor and say, "Imagine trying to do these if you hadn't memorized basic multiplication facts." And so on.

I still believe that the long division algorithm taught isn't so useful.

At the risk of not knowing precisely which algorithm you're talking about, I can't imagine one that doesn't work by taking a large/hard division problem and breaking it down into small/easy division problems. And that's useful because it's a great example of what math does for your thinking.

To be specific, I think the important principle math teaches is that, when faced with a big, hard problem, break it down into smaller, easier problems. I would rather describe math as a "learn how to break down problems" discipline rather than use the vague and pretentious "learn how to think" description. All areas of education help your thinking.

I never memorized my times tables and do just fine with graduate level mathematics.

If you ask me 9 * 7 I still do 10 * 7 = 70 - 7 = 63 in my head; but, I seem to handle spectral graph theory just fine.

It's way too early to judge "New Math" to be a failure. I think the idea of focusing on concepts over computation 100% in the right spot. The biggest problem to its acceptance is cultural. Parents don't feel comfortable with math concepts, and in my opinion, most of the negativity is coming from that insecurity. So they demand for things to be taught the "old way", even though that's produced a generation or two of mathphobic Americans.

EDIT: As pointed out, I mistook OPs invocation of New Math to be talking about the much maligned Common Core rethinking of math education. I was briefly a high school math teacher, but before the roll out of these changes, so I can't comment first-hand on what the new curriculum looks like in the actual teaching. But I do know how poorly prepared my students were for math beyond arithmetic. They were trained with similar curriculum that I had experienced growing up in the 90s, which I think is poorly thought out.

Apologies for the confusion caused by me not recognizing the term New Math.

>I think the idea of focusing on concepts over computation 100% in the right spot.

Concepts over computation (or well, before computation) is probably right. But New math was about learning the abstract before the concrete. This was predictably an abject failure.

I think the best way to teach math is to follow the trajectory that humanity took when discovering it. The key that's missing is that math doesn't just come out of thin air, its just a systematization of precise quantitative thinking. If we motivate the concepts using real world examples, then explain how to abstract away the particulars into a general procedure, then these connections will get made that make math "real" and relevant.

Not even wrong. You think New Math refers to the current modifications of common core. New Math actually refers to a series of cold war changes made in response to perceived russian scientific dominance.


Oops! Forgot about that. I assumed the OP was griping about Common Core.

BTW, "not even wrong" is a rather rude way to point out a misunderstanding.

yeah it is a bit rude..sorry! I just get very few opportunities to say it- and I was like yo- this is a good chance. Are you familiar with the history of the term? with pauli?

Yeah, I'm familiar. To be fair, it was a good opportunity. Well played :D

I disagree. I am not uncomfortable with math in general or the "new math". What I am uncomfortable with is the fact that my kids, and everyone else's kids, are essentially lab rats in a massive experiment. The result of which is 1+ generation of kids who can't calculate tax in their head or split a restaurant bill . They can't manage more than a few number without a calculator and are missing many of the basic blocks of math that are learned through rote.

I find it disheartening that I have to teach my kids basic fractions, ratios, and transformation cause the teachers don't or barely touch on it. Kids are supposed to "discover" and "explore" math, whatever that means. In my opinion it's all bullshit.

Math, in many ways, is like an engine, either it works meaning the answer is correct, or it doesn't.

You're confusing "New Math" with common core.

Disagree. If you read the common core standards, they are very sensible and make no curricular or pedagogical recommendations. "Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph." CCSS.MATH.CONTENT.HSF.IF.B.6.

The way people are teaching math in the US is baffling and weird, but Pearson's textbook design has little to do with the common core per se. Most common core-labelled material is from something else with a new shiny CC cover slapped on.

I think you misunderstand, "New Math"[1] was a thing from the 60's, he clearly took it as an adjective talking about common core, the current "new math", but not the same as the "New Math" bitwize was saying had been tried and was a failure.

[1] https://en.wikipedia.org/wiki/New_Math

I know New Math from the 1960s; the sibling comment to my post indicates that ghettoCoder is talking about Common Core today and I disagree with his characterization of Common Core. New Math as practiced in the 1960s is a fascinating and very different failure case, although it probably had similar problems in implementation (poor teacher training and shoddy repackaged curricula).

I know. I was replying to the comment and using the his/her terminology. It's actually a blended common core/discovery math hybrid.

idk, I studied higher level math very seriously, and even there, practicing computations is pretty key to understanding the underlying concepts. Concepts and abstractions are important, but they don't stick without practice.

I agree that it's an important part of the process, but I think it works best when it's connected with actual application, and not just a worksheet full of disembodied computations. I saw way too much of that as a student and as a teacher.

> I think the idea of focusing on concepts over computation 100% in the right spot

Concepts of what? Without intuitive understanding of basic computation procedures, what concepts can anyone build out of nothing?

Math is concrete. It's from observation of actual quantitative phenomenon.

And yet it's completely abstract as sets and numbers don't need a universe like ours to exists, they only need axioms.

Modeling the world is a job that math the tool was created for, but it certainly isn't concrete just because the world that needed it was or is.

You forcefully disintegrate concepts and physical world. All concepts have certain degree of abstractness for human understanding. It's not unique to Math. And it's not necessary that abstractions automatically alienate something from concreteness.

To split them, and think that, just because there is abstraction, and it's OK to develop the concept without the concrete substrates of actual experience is trying to make dream come true.

And I argue that no one should live in a dream. And it's obvious that only those dreams that have a strong connection with the real world have realistic chance of being made real.

Is it way too early, or has it produced a generation or two (or three!) of mathphobic Americans? How many generations of avoiding rote memorization would it take to judge an alternative, in your view?

Many problems that people have with math seem to stem from not having internalized the most basic facts about addition and multiplication. If you don't know at least to 10 by 10, each tiny step of working through a problem will tend to be interrupted by counting. Fluency requires memorization.

I knew a guy with a Physics PhD who would agree completely with bitwize, so I don't think it is just parental insecurity.

PhD physicist here. Struggled through the 'normal' public education math program, and didn't get good traction on stuff until college when my instructors starting focusing on more 'evidence based' math and physics.

Also I never completely learned my multiplication tables to the point where they were reflexive, such was my loathing for rote memorization. To this day I sometimes need to pause to mentally crunch something. This sucks for small numbers but it means I have the mental tools to grind out bigger ones that other people would need calculators for.

Memorizing time tables and similar rote learnign means that precisely kids who could be good at math hate it. Meanwhile, those who have good memory and sux at problem solving think they are good at it.

People who were taught by memorising go into outrage a new type of exercise is introduced. Suddenly thinking is needed and that is bad in their eyes. Not exactly success. Meanwhile, you can memorize time tables in later age if you decide it is useful (people rarely do).

I think this is a very astute observation. I know a lot of people who develop problem solving skills later in life after realizing that their mathematics education in high school was woefully inadequate. I don't know anyone who memorized a multiplication table by choice, and I know several professional mathematicians/engineers who haven't memorized their 10x10 or 12x12 multiplication table.

Once people see what they need out of mathematics for success in life, they never choose to memorize multiplication tables. But they do often learn new problem solving techniques.

That should tell us something about which is more important.

(But of course, we don't have to choose between the two either.)

I never managed to memorize my times tables. I never managed to memorize other formulae either, which meant that in an exam without a cheatsheet I'd have to do things like draw a bunch of triangles, measure them (graph paper) then re-derive the Pythagorean formula that I vaguely remembered involving a square root.

It sucked.

Blind memorizing is a problem. Memorizing things that you have learned concepts behind makes a student faster. As an example, the way I've taught my kids multiplication starts with: let's count by 2s! Great, now you know that, let's do 3's! 4's, ... 12's. We got to the point where we could count fluently (speed + accuracy), and incorporated fingers to help. Then we started saying things like, "if we count by fours five times, what do we get?" and then we changed our words to "four times five." All my kids have done well so far with this method. There is some memorizing to become fluent, but it is fully backed by understanding.

The next steps are to start doing things backwards, "how many times do we count by four to get to twenty?" And we start to introduce notation. Bingo, simple division. This leads directly to simple fractions. This opens up conversations on adding and subtracting fractions, then multiplying and dividing them.

My oldest kid, now in college, could add, subtract, multiply, and divide fractions by second grade and understood them. Oddly, she had a horrendous time learning decimals. Her mental model of numbers was fractions and decimals were "weird." She would have to change things like 5.045 to 5 45/1000 to understand it, and wanted to work with it as a fraction. It took a long time for her to get comfortable working with decimals.

One time that I think it paid off. I told her that 0.999... is equal to 1. She said false. I said, no, it is true. Can you tell me why? At this time, she was in algebra, and I was expecting to show her how to prove it using algebra. She had a much better way of looking at it. In about a couple of seconds, she said, "well, 1/9 is 0.111... and 9/9 would be 0.999... and that is also 1." Her answer was much better than mine. :)

An example of when memorizing is bad (ie, when the underlying knowledge is skipped) was her 7th grade algebra teacher. In teaching the laws of exponents, he said "anything to the 1st power it itself and anything to the 0th power is 1. We don't know why, it is just one of those math things." Teaching like this is why we have students who, later in high school, can't do x^0 or x^1 because they think, "it is either 1 or 0 or itself, I don't remember." As opposed to applying mental models and patters to see that 3^3 -> 3^2 -> 3^1 -> 3^0 is just dividing by 3 each time. These students know 3^2 is 9. So they should know that the next is 9/3 = 3 and that the next is 3/3 => 1.

1) Did New Math actually try to do those things, or did you just bring it up to poison the well against any alternative pedagogies?

2) Did New Math fail because of poor results or because of popular revolt? (Would New Coke have failed if there was never any such thing as the original Coke?)

2a) If New Math did actually have poor results, was it because it hewed too closely to the goals I brought up, or because of other issues?

3) Memorizing times tables clearly didn't work, or we wouldn't be having this discussion

4) I never said that drilling times tables isn't important, so I'm not sure what your last sentence is in response to

> drilling is key to early mathematical learning.

I was told to memorize the times table, and tried but never managed to succeed. Instead, I found that I got along just as well by memoizing them instead; that is, I would compute the parts I needed on the fly in the margins of the paper. (Example: Say I need to find 37. I happen to know that 33=9, which I can double to get 36=18, plus 3 to get 37=21. These figures would be written down, so when I later needed 47 I could easily add another 7 to get 28. I had similar tricks for various other numbers, and could generally get the figure I needed--if it wasn't already written down--in a few hops.)

These contortions don't seem to have significantly affected my mathematical development, but they did* improve my logic and reasoning skills (or possibly merely showed that I had them). Particularly now that nearly everyone I know carries a calculator in their pocket, I don't see why we would continue to focus on rote learning over actually understanding how the underlying principles work.

Hah... glad to see I'm not the only one who does that. I never did truly memorize the times table, but I can work out most small multiplications easily enough.

I still feel a certain tinge of guilt though, over not memorizing that stuff.

> This was tried. It was called "New Math".

No it wasn't, at least writ large in the USA.

And that's not what New Math was. Explaining what multiplication is doesn't demand an introduction to set theory.

If we read Feynman's CRITICISM of New Math, we actually find that he ADVOCATES for exactly what your parent is suggesting ("cobbled together.. in order to solve real-life problems"). So clearly, your parent isn't describing "New Math". Or perhaps Feynman is just a raving lunatic.

So I'm no advocate for "New Math", but I do oppose the argument you're making here, in which "New Math" is taken to mean "anything other than memorizing times tables" and is then denigrated on face. Without regard to the fact that the most vocal opponents of "New Math" were in fact advocating for exactly what your parent post is suggesting.

> Spectacular failure

So brief was the new math intervention that, to this day and despite all of the hoopla, we don't have a good empirical basis for claiming new math worked or did not work.

New Math was barely attempted, and its primary opponents were mathematically illiterate parents and teachers. This is just true, even if there were mathematically literate opponents to New Math, e.g., Kline or Feynman.

(But also note Meder’s reading of Kline. It's also worth noting that Mathematicians are maybe not the ultimate authority when discussing secondary pedagogy, especially in the mid 20th century. I have no basis for this belief, but IMO lots of mathematicians who weighed in on New Math were very possibly waging a sort of proxy battle as part of a larger war over the future of their own field -- pure vs applied.)

> Do you want to know what worked? Memorizing times tables.

Is this satire (honest question)? For all the things we don't know about math ed, we know that this doesn't work. Students who memorize times tables are routinely incapable of multiplying 12 by 13 or 55 by 55.

> There's just no way around the fact that drilling is key to early mathematical learning.

No, there isn't. But there's also no way around the fact that drilling without understanding is why a whole bunch of students who are "good at math" can't get through even the most dumbed-down versions of proof-based courses in college, or in some cases can't even get through a full calculus sequence. But they're "good at math" because they can rattle off 12*7 real fast!

New Math advocates (and their opponents!) were all at least correct about one thing: we REALLY SHOULD seriously ask what good is learning "math" if the student does not become a better problem solver. It's not 1417 anymore -- problem solving is important, but human calculators don't pull down living wages.

Probably the answer is that we should all be equal opportunity critics: memorization without understanding is intellectually lazy and limits growth potential, while understanding without practice is for most learners a contradiction in terms.

Also Kline spends a good part of his book shredding the standard math curriculum (especially rote memorization and mindless application of standard algorithms) and agreeing that it needs reform, at the end advocating constructivist alternatives, with word problems, use of physical manipulatives, and motivation via applications to other fields.

His beef with the New Math is with an emphasis on axioms, deductive reasoning, rigorous abstract logic, linguistic purity, and symbol manipulation, rather than with teaching conceptually or letting students think for themselves. He also doesn’t like the specific content of the New Math (set theory, inequalities, alternate number bases, boolean algebra, modular arithmetic). [I haven’t studied the New Math curriculum enough for myself to know how fair these arguments are.]

His key criticism: “Psychologically the teaching of abstractions first is all wrong. Indeed, a thorough understanding of the concrete must precede the abstract. Abstract concepts are meaningless unless one has many and diverse concrete interpretations well in mind. Premature abstractions fall on deaf ears.”

> Do you want to know what worked? Memorizing times tables.

That obviously hasn't worked as most people are quite awful at math and society at large hates it. Memorizing times tables has been a spectacular failure as has memorizing formulas.

Math is about problem solving, not memorizing answers to common things; it's the focus on memorization that's made so many people bad at math to begin with. Common core is an attempt to address this by focusing on how the problem is solved rather than what the answer is, it's freaking parents out, but it is a better approach if you're actually trying to teach math.

> I never had any problems with math until I went to university

So, here's the thing. Did I have no problems with math until uni because I'm smart? Because I have a specific type of brain? Because of my upbringing? Because I had very good education in my early years? Some combination of the above?

I struggle to believe that people can't do algebra. I am convinced that with some thought and help they'd find it trivial. However, is that my own experience overriding my ability to take their perspective as true? I dismiss "I can't" as "I don't think I can", but who am I to judge that better than them?

I saw what looked like this sort of thing with my daughter. She struggled in her high-school algebra 1 class, but whenever I would sit down with her and she would work out problems, she had no trouble (but she also had no confidence that she could do it).. She ended up having to repeat the class, which ruined her confidence even more, and she hated it even more the second time through.

At the start of middle school, her teachers were recommending accelerated math classes, but she never ended up doing that, because she stopped liking math and stopped trying. Why? Maybe because it was unpopular or she didn't like her teacher... I don't know. In any case, she went from kinda-liking math to completely hating it over the couple of years, even though when I was there, she seemed to understand it just fine.

Teachers are important. I had consistently bad grades in science until I my 8th grade science teacher that liked to blow stuff up.

I really like him and I wanted to impress him with how smart I was and I remember putting in a lot of time doing all my homework. I got my only "A" in science for the first two quarters. Then he got fired and became a harley mechanic. Back the C's and D's

Accelerated math could have been a solution -- I know that when I was bored in math I did worse, and when the problems got more interesting I did better. Now I have a math PhD, even though I got so bored in one class earlier on that I had to drop it in order to avoid failing.

There are a lot of social factors in US schools, too. I notice a lot of young women doing poorly in algebra and precalc even though they're perfectly capable if you sit down with them and ask them to simply write down the steps of solving the problem. The social benefits of being "bad at math" can be seductive in the short terms and the long-term benefits of being good at math are entirely invisible to many young women. Guys at least see some male nerds on TV making good money; girls who want to be like nerdy tv stars know that they have to go into forensics and don't know that math is very handy for figuring out the rate of cooling of bodies and decay of tissue.

Maybe she found easier math boring, therefore did low effort, therefore had bad results. It is a cycle, happen to kids that have aptitude but are in easy class. The teacher might have been right.

I used to have bad grades from classes I considered easy.

Encourage her to give it another try when she gets to late high school or college. I know a lot of math phds who fell out of love with math around middle school and then fell back in love during their first proof-based course in college.

Yeah, this.

I hated math in middle-school and early high-school. It was boring and rote.

But proofs based math, and learning how to think of math as a language, instead of a collection of overly specific "solutions" crammed into my head by teachers, was the key.

I never got proofs. I wish I did.

The thing that made it click for me was getting a "first principles" book for something relatively simple like Linear Algebra (or some basic CS-flavored discrete math book). These books prove everything, and reading the proofs the author wrote made me recognize what you needed to do to prove something.

Would you recommend one?

There are people who have dyscalculia, essentially math dyslexia, and it causes problems beyond simply being able to do basic algebra. https://en.wikipedia.org/wiki/Dyscalculia

I don't really think the situation in the article is directly about people not being skilled in math. It's about math-phobia with it's after-affect of people not having skills in math.

It seems like an especially American thing - being good at math makes someone a freak and so it's something kids actively avoid at the ages in which being a freak is not tolerable.

That kind of attitude would need to be actively confronted because what happens is that kids who previously were developing skills tend to drop out given the peer-pressure.

Edit: best documenting link I could find but Google book won't let me copy a quote:


I blame TV in the 80's and 90's..and to today. The smart kid is the dorky, beat up kid. The dad is usually a guy who thinks he can figure stuff out and repair stuff, but hes wrong and just breaks everything.. better just buy it at the depot.

Our media praises stupidity

"School is about the advanced stuff, not the basics" is the call of good students, often ones who learnt the basics from their parents. It's also kind of popular with bad students, who think that they might suck at the basics but have untapped powers if they were doing something else.

It's nice, vague, and populist. Everything thinks they know how to do critical thinking. Everyone thinks they know how it could be taught. I suspect most people think that they have above average critical thinking skills. As long as they don't have to agree on how or what critical thinking is, everyone can agree it's a good thing.

The other one that's dumb is "you learn best when you teach yourself". Sure, if you are really super-interested in something then you learn better, and tend to teach yourself a bit, but correlation is not causation.

Not sure it's exactly the early years. But I think there's zero point to moving past a topic until you have mastery & can think WITH it (not just, about it, or hack through it on paper). And since the early years are often where that fails, they need to be rethought.

I remember listening to an episode of The Science Show podcast where the guest was arguing (based on their experience) that chemistry and physics concepts should be taught to young children as it's actually easier to teach it to them. With the stated reason being that if they're taught it at a later age, as they are now, they've already built a view of the world where those concepts are somewhat unintuitive. If they learn the concepts earlier they are more easily absorbed.

I found their point fairly compelling, and I wonder if it applies to mathematics as well.

The episode was http://www.abc.net.au/radionational/programs/scienceshow/you...

One of the most fun things I ever did with my five year old son (now six) is build a cloud chamber with dry ice and a fish tank and talk about it. Strongly recommend.

I wouldn't make math/STEM a special emphasis when they're young.

It's more important to foster an positive learning environment and encourage/draw out curiosity and creativity at this age. Help them become curious about the world, fascinated with what they're learning and find enjoyment in learning.

In terms of subjects, aim for a strong reading focus at an early age as it will pay dividends in all other subjects (even math) if they can read/comprehend well.

Salman Khan's One World Schoolhouse does a good job exploring this topic I think. On the one hand many students struggle with what he calls a "swiss cheese" understanding of math: lacking mastery of fundamental concepts that leads them to struggle as they grow older. On the other hand he advocates for a more holistic approach to learning overall from the beginning, exposing children to various subjects and exploring how everything is connected.

Anecdotally, I am also not in agreement with you on this. Another poster mentioned not 'getting' algebra until 10th grade--this didn't happen to me until roughly my second year of college. Anyway, I eventually became very motivated to excel at math. I ended up changing majors, transferring to a school with a more serious math program, and doing so well under a very heavy courseload that I won some department honors and scholarships. I basically went from not knowing high school (middle school?) algebra to demolishing my Rudin course in about 3 years. Honestly I think most people could do this and probably a lot more if they had the desire and access to a university/books

> memorization

I clearly recall memorizing the times tables when I was young. It suddenly dawned on me that "times" wasn't just a word, it meant adding the number so many times. I no longer needed to memorize the table, I just applied the rule.

Some university engineering classes present formulae and tell the students to just apply them. This is disastrous in my not-so-humble opinion. Where the formula comes from should always be taught, so students understand the formula and the assumptions it was derived from.

I disagree that you have to get them young... but I do think that math classes need to start involving proof-based work rather than calculation-based work by the time students hit middle school because it's such an explicit exercise of critical thinking muscles.

Recalling my own experiences with middle school and high school, I saw my peers frequently fall into the trap of trying to memorize pipelines of steps while we were learning algebra, geometry, trig, pre-calc, and then calculus. The ones who "didn't get it" would fall apart to varying degrees when confronted with problems where you had to think in terms of combining toolboxes of strategies to unfamiliar but analogous patterns, but I don't necessarily think it was their fault, because they didn't know better and rote memorization had worked consistently until this point.

I often feel a bit sad about it because I did a ton of proof based stuff outside of class (as a mathlete), and that was a HUGE part of my development in logical reasoning, intellectual rigor, and critical thinking.

> By the time people go to high school it is to late

Just anecdotal evidence, I know, but for me it was the exact contrary that happened. As a 5-year old kid I had learn to read all by myself (my dad had taught me to recognize the letters and how to pronounce them), but I had huge conceptual difficulties in teaching myself things like subtraction (what do you mean you can take/subtract a thing from another thing? what happens when the thing which you're taking/subtracting is bigger than the thing from which you're subtracting? you cannot do that, it would result in a non-thing, a thing that is below 0, and all things below 0 technically do not exist).

In elementary and middle-school I was passable in maths, while very good in humanities, but all this changed in high-school, mostly because of my teacher, who taught me (and my colleagues) how to think about math. He's one of the reasons why I decided to pursue a STEM education and why I'm currently a programmer.

Absolutely. See Toom (2010) Word Problems in Russia and America, http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd...

I was always relatively at ease with math even though throughout lower education it was mainly about memorizing rules or formulas. In college it became about proofs, which was a completely different ball game and not something I'd been prepared for from preschool.

To me the real important thing is the attitudes imparted by your parents and peers. As noted in the article, it's acceptable to fail at math. Parents who have their own history of math anxiety probably don't really push their kids to excel in that field. Just the notion that I was expected to "get it" and keep trying until I did (perhaps helped along by some natural stubbornness) may have made the difference in my case.

school teachers, administrators, and legislators most likely did not do STEM, so that pattern will most likely continue.

My experience differs. My brain was intuitively ok until high school. The abstraction shift around cross references between log / exp / S 1/x went right above my head; same for discrete math basis such as prime relationships.

My conclusion is that every one will have strong and weak spots, and it's very important to stay alert to this and not beat around the bush thinking it's not important.

Took me more than a decade to get back into intuitive thinking, where you can play mentally with symbols and ideas just like LEGOs (as I was around high school).

>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.

This is the opposite of my experience, I did horribly in math and hated it until I took calculus (which, I guess I should have taken in highschool but I tried to avoid it since the rest of math sucked.)

EDIT: just want to clarify: I did have a reasonably intuitive understanding the whole time, the earlier stuff just felt like busy work.

I couldn't develop internal understanding of calculus, so while solving the equations wasn't that hard, I never felt really comfortable with it. Discrete mathematics - total opposite. Proofs were very intuitive to me.

If I was to review (CS-related) math after a break of many years, what would be a good starting point? High school math? Discrete math? What are the foundations and the path like for the stuff more closely related to algorithms and core CS stuff?

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.

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.

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....

> In theory, I strongly support "math for liberal arts"-type classes instead in college.

I love it. They could call it "math"! ;)

Seriously, I think that captures an important concept: for the most part, math was developed to serve the needs of humanity.

I used to wonder, how do we know our numbers are the "true" numbers? Did we just happen upon one way of computing, and there are countless alternatives that could have been explored?

Of course, the answer to that is actually "kinda, yeah". Abstract algebra leads to all sorts of systems that behave more or less like our familiar numbers and operations. And besides the whole numbers, which represent physical quantities, every extension to things like integers, reals and complex numbers are used not because of their tangible reality, but because they do a good job of modeling human concerns. Because my brain has a limited ability to pursue abstraction for its own sake, its this aspect of math that really captivates me. I think that's true for most people.

>"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.

> The quality of those courses depends on the prof.

Entirely. In theory they are great :)

The one you listened in on sounds interesting!

> when I learned what pH actually meant

potential (for attracting) Hydrogen. I had to look it up.

Reminds me of an anecdote in Richard Feynman's book in which the Brazilian education system revolved around memorizing definitions without understanding the underlying concepts: http://v.cx/2010/04/feynman-brazil-education

That was such an interesting read. I live in the US, but am actually Brazilian, born, raised, and educated there (BS and MS). I did have some GREAT professors in college who you could tell loved their subjects so much, and would give amazing, passionate lectures, full of real-life examples, etc. Those shaped my career. But some others were just like the ones in the article.

Funny enough, all the good ones were much younger professors (like, in their 30's, 40's), with very few exceptions. Since the original article is from the 80's, maybe Feynman's feedback had some effect? I did go to one of the best universities in the country, so maybe my experience is also different from other people's. I hope it isn't, and I hope it continues to get better over there.

It's actually a little scary how much of what we end up doing in life depends on the kind of teachers we get at a younger age. I can trace my academic pathway almost entirely by a few key teachers that had a huge impact on my interests.

A really good art teacher or a really good math teacher can really change the kind of future you'll have.

> 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"

I've taught Calculus close to a dozen times now. I talked about integrals. I talked about derivatives. I tried to get into the intuitive nature of rates of change and the limiting process of Riemann sums. I used analogies and real-world examples and illustrations and interactive computer demonstrations.

Students don't listen to that part of the lecture. They just don't. They listen to the part where I start doing problems on the board, because to them math is about doing things, not concepts. That's only reinforced when they're asked on tests to just do things rather than demonstrate understanding of concepts. If, in a college class, you try to change that up by asking concept questions, you get bad end-of-semester evaluations because you've challenged them in a way they didn't expect and their parents aren't paying good money for them to be challenged.

(And realistically it's hard to ask concept questions at that level of mathematical maturity anyway. As answers you get mostly word salad with a smattering of misused jargon. I've tried.)

If you try to mix it up and use techniques from inquiry-based learning (Moore-style, flipped classroom, etc.) then you get a stern talking to from administration.

> , or what "infinitesimal" means.

Probably because even defining infinitesimals with anything resembling rigor is tricky, let alone using them productively. I wouldn't even try it in less than a junior or senior class. There is no such real number that is smaller (in absolute value) than every other real number yet bigger than zero. Infinitesimals, at the calculus level, are a convenient hand-wavy fiction to mask limits and a notational artifact from simpler times.

> 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.

I tried. Every single time I taught the class I tried to teach concepts. It's really easy to fall to the temptation of just working problems on the board every day, because trying anything else will just give you grief. Grief from the students, grief from your colleagues, grief from the administration. Grief from people like you who constantly chastise us for not being able to explain to teenagers that they should pay attention for the whole class. (I love my adult students though. Non-traditional students are the best.)

It's not worth rocking the educational boat if you don't already have tenure. If you do rock the boat you get a reputation for being hard or weird and students avoid your class anyway.

You hit a nerve. I've left academia for this and other reasons, but as you can tell I'm bitter about it.

> You hit a nerve. I've left academia for this and other reasons, but as you can tell I'm bitter about it.

Looks like I did! Sorry about that. I fully understand where you are coming from, as I've had the same experience when trying to help college colleagues with course work, tutoring for a few courses in college, or even giving private lessons to HS kids. Some people are 100% not interested in learning -- they want the bare minimum to pass the course and move on.

Not sure how we can change that, though.

I had one specific professor in college would would abuse those inquiry-based learning techniques that you have mentioned, and it was absolutely the best course I had during my 7 years in the university. Anecdotal, but students loved him. He ended up being my MSc supervisor :)

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.

"all these notions are so different from what we have to deal with every day that learning them takes a lot of training"

I disagree in the general sense. I think most of mathematics with which "normal people" (by that I mean people who don't get a masters or phd in mathematics) have problems is directly correlated to problems and situations in the real world. The problem is that we teach mathematics as if we were computers - which we are not. Its about formulas, recognizing patterns, precise answers and proofs instead of general understanding and intuitions about things - which is where humans excel and how we work and remember things.

For example, being able to solve quadratic equations is nice but IMHO what's much more important is knowing how the coefficients and their signs shape the plot of the lines and how that might effect the trajectory and hang-time of a projectile. Also being able to look at a projectile and give an approximate quadratic equation is again IMO more important than knowing all the steps in the quadratic equation.

But this is just my opinion.

Henri Poincaré once said: "Mathematics is the art of giving the same name to different things"

I know the temptation is great to give lots of examples while teaching math, but you get the risk of only teaching little examples to students and not the abstract vision that math education demands.

Take your quadratic equations, it's all nice to demonstrate how coefficients shape the plot. But when you start talking about projectiles, you end up ignoring that quadratic equations are used... heavily... everywhere.

You may see them in chemistry, crypto or nuclear physics. But if your brain only remembers projectiles, it doesn't have the ability to abstract new problems the right way.

As a counterpoint to that, I flat-out despised abstract Math until I got to college precisely because I found it useless. At best I knew I was probably going into engineering in college and I'd been told "engineers use calculus". That was my sole motivation to learn anything about abstract math, and it was pretty bad. It didn't help that the "examples" given in class were mundane toy problems (ex: "how long will this arbitrary tank take to empty?" Well gee I don't know and I don't care. Is this the last tank of water on the island or something?)

Now physics on the other hand, chemistry, computer science, any time I could apply math to a real-world situation got me motivated. I liked Math that could solve actual existent problems, and I learned said math better as a result. The first time I learned about projectile motion I literally started picturing vectors everywhere for the rest of the day.

It also didn't stop me from learning abstraction later in college. In basic signal analysis we learned about the Fourier transform, and then we learned about the Laplace transform and how it was simply a more general Fourier transform (or the reverse if you prefer). Mastering Fourier first gave the Laplace a relatable mental context, it made the abstractions more real.

By contrast I had a Math professor once who tried to teach us theoretical induction and work his way down to examples. I had no idea what the fuck he was talking about until he hit the examples, then everything clicked all the way back up.

Maybe that's just how I'm wired, but I would argue for more and more relevant examples. There are tons of historical uses of math that people would find interesting. Talk about the math used to build the Hoover dam, or telegraphs, fly satellites, or make medicines, or decode Nazi transmissions. And talk about it in real terms with character, if you must simplify for the sake of time then say that's what you're doing. Make people realize they're learning something that has uses beyond Math class or the rarified ivory towers of Math PhDs. That by learning the material they're learning how to do meaningful things. The original, wildly popular SimCity games were basically raw spreadsheets and formulae, but people learned those spreadsheets and formulae because the games had character.

I know to a lot of Math people the abstractions are the "real" part, but most people I know simply don't process the world that way.

This is especially a problem in college. Few math profs seem to enjoy teaching and seem reluctant to ground their subject using concrete illustrations. Without these, there are no tangible points of reference, aside from derivations from known equations. But mere transformations rarely offer insight without adding some sort of context.

I think the crux of the problem in teaching math is the need to illustrate each concept from MULTIPLE perspectives: theoretical, tangible, incremental, graphical, dynamic, etc. It's only by connecting the concepts to manifestations that most of us appreciate the meaning of math. Few are satisfied by disembodied equations or proofs. Most of us need some form of grounding using what we already know to make sense of new concepts. Otherwise math is just castles in the air.

If you have 45 minutes, watch this talk: http://blog.mrmeyer.com/2016/nctm16-beyond-relevance-real-wo...

There's also a hurdle in having to work through the wildly varying notational and structural views/tastes of the individuals who created their specific categories/theories (biggest problem at least for me - while programming languages are nice and structured and standardized, traditional math is highly impure/unstandardized and relies too much on how individuals feel like when they write things down)

Learning math is like learning a language. It simply takes a lot of time and effort. If you read the rules of German grammar cover to cover, that will not prepare you to speak German; in order to be fluent, you MUST practice, over and over again, using and interacting with German.

Likewise for math. There is no "new book" to learn math that is going to blow the cover open on everything. It's a problem of motivating people to become fluent in math. I would like to see discussion in those terms -- fluency, because that's what it is.

"The reason is that mathematics creates its own universe of concepts and ideas"

This also applies to reading music and understanding music theory. Written music is a map of events/time - which is totally different than any other math I studied. I've noticed that a lot of musicians pick up programming pretty easily. I've often wondered if learning more symbolic systems that are different from natural language makes it easier to pick up additional systems like programming languages.

Many other commenters have pointed out how great examples and ties to the real world are; the love of applications is well-represented in this discussion. But I'll throw out beauty as an underrated motivator!

I love math because it's beautiful, primarily, and this isn't represented often in elementary or high school education. The idea of "function" or the idea of "constructive proof of bijection" are beautiful concepts. The patterns of math are beautiful (so beautiful I created an adult coloring book called Math with Crayons that focuses on Aztec diamonds, fully packed loops, lozenge tilings of hexagons known as totally symmetric self-complementary plane partitions -- you know, the stuff people are actually doing research on today!). Numbers to me are kind of boring. But how many kids get to draw fractals in class and then talk about the concept of dimension, and then argue about how one could rigorously define dimension, with the example of trying to define dimension for fractals as an edge case? How many kids get to draw non-crossing partitions and Dyck (lattice) paths and then try to prove that they are both enumerated by the Catalan numbers? How many kids get to think about the mathematical structures you allude to, and play with the fundamentals of category theory? Well, very few. And why?

Their teachers don't know any of that stuff.

If teachers only know computation, that's all they can teach.

If we had teachers who really had breadth and depth in math teaching math, I think we could start teaching concepts, structures, and beauty very early. I've had conversations about the basic ideas of category theory (relations not objects!) with many kinds of people. This is not out of the grasp of a kid. Set theory -- not out of the grasp of kids. The very idea of what a number is: in high school I got to take a seminar with other high school students on this, looking at John Stuart Mills and Frege and Russell and Whitehead and all this thinking about what a number is. But this is extraordinarily unusual material for a high school or even college student. Of course it's difficult to deal with these notions later, then -- people have been trained out of their natural curiosity and into a view of math as computation at worst and manipulation to find x at best, rather than seeing it as a way of dealing with beautiful structures.

> Learning "how to think" is just one part of it. The other part [...] is learning to think about math specifically.

Do you feel like you were taught how to think about math specifically, or how to think at all? Like probably most people here on HN, I love math and programming, but I'm struggling to recall whether and how I was taught to think in school. We went through a series of problems, and talked about the syntax and mechanics of solving those problems, using various frameworks (geometry, calculus, diff. eqs) So it feels like we learn math by example. Is learning how to think directly part of education, or only indirectly something that happens by seeing what others did?

I think that people are rarely taught how to think; this basic skill usually comes (rather quickly) in the process of "learning by example" - part of which is struggling with concrete problems - whether it is in mathematics, in programming, or in life in general.

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".

Even simple objects, like vectors, for example, can very quickly become so abstract (higher dimensions; complex vector spaces) that the physical intuition stops being useful and can even lead to mistakes. This is the point when new kinds of mathematical intuition becomes necessary, and building them is very difficult and takes time.

I remember doing only simple (two-dimensional Euclidean) vectors before calculus -- length, scaling, addition, subtraction... Agreed that they obviously can get pretty hairy, but I don't think that side is exposed to people before calculus? Complex vector math was calculus III for myself.

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?

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.

I think these things have already been discussed to death some time around the year 1700, when they were called something like "mathematical analysis of infinitely small magnitudes" or some such; the difference between an "infinitely small magnitude" and zero was already clearly understood. The notion of a "limit" came only later, in an attempt to put this stuff on a firm logical foundation and was (and still is) seen more of a formality than something aiding our intuition in this area.

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

Ah, good point. I forgot about that problem with instantaneous speed.

lol sorry for the typos! Wrote that from my mobile phone at night.

"Ma'am, you were driving at ninety miles an hour."

"But, sir, I have only traveled twenty miles!"

I think the most important part, especially for people who do not intend to be mathematicians, is how to apply math to solve problems. The math courses I took in high school and college were all about theory and nothing about application. These courses were taught as if it were still the 19th century and computers don't exist.

I would rather see teachers accept that computers exist, and teach students how to solve problems, even very complicated ones, using software (e.g. Mathematica, R, or whatever). Keep the theory to an absolute minimum (leave the theory courses to people who are interested.)Really, so what if you don't learn how to do calculus manually?

I suppose one can learn how to think by doing math, or they can just learn how to follow rules...and perhaps some may want to think about match specifically. However, I think most people would be encouraged to learn math if they are shown that math is just a way to model the world they live in and it can tell them specifically about some of the assumptions they have about it. For instance, I might assume if you put some fuel in a rocket and point it straight up that it will go into space and be somewhere. Math can tell me where it will be and the path it will take to get there...people identify more with those concrete things.

> 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.

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.

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".

Math is a subset of logical thinking. There are deconstructions of thought that math cannot handle, for example in domains where the axioms are too complex, or perhaps even unknown.

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.

In my view, speed plays a role, though I can't say that it should play such a central role.

I think that in a lot of disciplines, you have to become fluent at manipulations, and at seeing and thinking in higher level patterns. Being able to think your way through a more complex problem would seem to benefit from, if not require, seeing multiple steps ahead in a progression. At least this is my perception.

My experience in school math was that it wasn't enough to satisfy myself that I knew how to solve a problem. I then had to work my way through a whole bunch of similar problems until I could perform the manipulations quickly. This is also how I managed to commit the definitions, axioms, and theorems to memory. If I didn't do that stuff, then I got my arse handed to me on the exam. I gave my kids pretty much the same advice.

You couldn't finish your tests in high school, but you're good at math? Have you ever been evaluated for 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.

I've never been evaluated but I don't think test anxiety fits in my case - the 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.

Different tests challenge people differently. It may be that you only needed an accommodation in computationally heavy exams.*

Truthfully, my mention of accommodations isn't for you--I'm assuming you're out of high school already and you've found a professional niche that works for you. I mention test anxiety and the professional workarounds for high schoolers or their parents.

*) This is actually moving well beyond my expertise. I know about the existence of test anxiety, and I've accommodated students with it.

A brief consult with Dr. Google surfaced some clinically recommended accommodations that probably would have helped you. Sorry I don't have a time machine!


This one's my favorite: "Reducing the number of tasks required to demonstrate competence (such as 5 math problems instead of 25)"

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

But, most exams at that level gives enough time to make them accurate.

Disclaimer: I'm CTO of 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.

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.

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).

Funnily enough, this goes away to some as you move along. Say I'm taking an exam and don't know how to do a proof. I pray and think that some equation is important, so I write it down with a little context. If I've done a great job up to that point, my professor will be more likely to think that I hit upon the key idea and give me 3/4 credit. If not, I'm probably more likely to just be marked wrong. You can also lose points for stylistic reasons if your professor is opinionated about them (-1 with "it is clear that" underlined and "don't write this ever again" in red ink :) ). Some professors will fail you if you get too technically correct with logical symbols everywhere instead of writing mathematical English. If you show too many steps you're wasting everyone's time; if you don't show enough or you don't seem confident you might get marked down. Etc

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?

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.

0 pts - space shuttle exploded.

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."

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 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.

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.

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.)

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.

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?

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.

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.

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.

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.

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.

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.

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.

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

For all grade levels a BS and certifications is all that's required in most of the US. Sometimes a specialized degree is required or helpful (this is regional) for certain areas. Like a degree in primary education may get you a job as a 1st grade teacher in some state where a degree in English wouldn't (because that degree, ostensibly, conveys the knowledge and ability to specifically help young students like identifying learning disabilities and modalities, or some roles bordering on social services like identifying psychological trauma). At the higher grades (starting at 6th grade or so in some areas, but more often just high school) a degree in the subject matter is preferred or required. But this can often be worked around by demonstrating capability (certification exams and classes) or having enough course work in the subject (like most STEM degree holders have enough math to qualify as a math educator).

I was taught multiplication by rote memorization of the tables. It didn't go well as I'm bad at rote memorization.

People ask me if they see the beauty in math when I'm complaining. I do, but I also see the other kids outside playing while I'm held back from recess until I can tell you what 7*8 equals. Fuck that so much.

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.

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."

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).

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

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

Thanks. That's what I get for posting on my phone :)

> if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?"

Really? There are many times where I thought "How am I supposed to think of that?" In fact, that's probably what would happen most of the time.

The problem with high school math is that it has no context. It's just a bunch of seemingly arbitrary rules.

It's as if when learning programming, you would learn fragments of various languages here and there but never connect the various pieces into anything meaningful.

Lesson 1: boolean variables in Java

Lesson 2: If statements in python

Lesson 3: Three-part for-loop in C

Mid-term exam:

Q1: In the C language, how do you separate the components of a for loop? (correct answer: semicolons)

Q2: What can you put in the right hand side of the '=' symbol in a boolean variable declaration in Java? (correct answer: an expression that evaluates to either true or false).

Q3: You are supplied a function that can take any number and determine if it's a Crastomarian number. We have 20 foot ball players each wearing a T-shirt with its own number. The numbers go from 5 to 24. We want to find how many players have Crastomarian numbers. The program to find the correct answer will use a for loop. Write the header part of the for loop. (correct answer: `for(i = 5; i < 25; i++)`)


And this is supposed to be how you learn programming.

Also note: there's almost no way to know the answers without rote memorization. Of course you will never absorb these concept by osmosis because your day to day activity will not involve writing programs that do things. Instead, you're just memorizing the rules given to you and doing exercises that help you recall which rule to use for which situation.

If you haven't memorized the for syntax and written 25 different variations of it, you will not be able to know it.

And when I say that you write 25 different variations, I mean you write them as part of pointless exercises of course because these are not in the context of a program.

It's more like just repeating these lines until the pattern somehow hopefully sinks into your mind.

for(x = y; x < z; x++)

for(y = 10; y < j; j--, y++)

.. etc

My university education experience seems to contradict this notion. Many classes were heavily focused on Java to allow students to learn data structures and object-oriented programming, in general. Other classes, like Operating Systems Programming were focused on C. In the end, you learn that there is the right tool for the right job.

I think you misunderstood my post (or perhaps I wasn't able to articulate the point clearly).

My University computer science was also broken up as you describe. The introductory class was in Pascal, the next course was Java, and then each course used its own language.

That's not what I'm talking about at all.

I think the more pervasive problem in schools is a lack of mastery in basic foundational skills before moving on to more advanced topics.

Too many students lack the ability to perform basic algebra. I mean simple stuff like simplification, factoring, working with exponents and even fractions. Faced with a typical problem in calculus, of course they're going to flounder and make errors when each calculus problem involves perhaps a dozen simple algebraic manipulations.

These basic things have to be drilled and practiced A LOT before moving on to more advanced topics. There is no way around that. No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.

> No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.

Conversely, no amount of memorization and drilling will provide students with the intuition necessary to solve problems just beyond the grasp of naive symbolically-driven proof search algorithms.

In a typical non-proof-based calculus sequence, tricky u substitution and integration by parts problems are the two places where you really see "I was good at math in high school" students start to struggle because they lack a good conceptual foundation in trig and logarithms. Up until that point the students with trig identities etched into their skulls by practice and drilling do just fine. But from that point onward, they really start to struggle because they need a bit of geometric intuition and some practice with more effective problem solving techniques than "blindly apply equations until something works".

Really, both are needed -- drilling and practicing, as well as a solid (and preferably proof-backed) intuition about elementary properties of trig funcations and logarithms.

I agree, you need both. I am not optimistic about the state of mathematics education. So many things have to "go right" for a student to master mathematics.

The issue is that being able to see that little path of logical steps is a skill on it's own, and most students don't know even how to begin approaching a new mathematical problem.

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.

As my old boss once said, "never confuse mathematics with mere arithmetic."

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...

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

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?

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'.

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. ;)

> 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.


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

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?

Exactly. That's why there are three kinds of mathematicians, those who can count, and those who can't.

I don't understand why this point seems so important to people. Seems like a semantic point, if we're talking about math education in schools.

I mean, there is theory & practice in every field, technique and discovery. A medical scientist studying the development of aberrant cells in a petry dish has a pretty different job to a pediatrician or radiologist. They generally seem to coexist happily under the "medicine" label. For first year undergraduates, they would probably need to take most of the same classes.

Arithmetic is theory too. Graph theory has lots of practical applications. I don't think it's that kind of divide. Arithmetic is just a very small part of mathematics.

As someone who went to university for mathematics (and computer science) I can tell you this is the most common misconception. People think that I spent years studying how to become a calculator.

I love talking about both subjects I went to university for actually and both can be difficult to get people interested in. There are massive misconceptions about both.

Yes, this is one of the most annoying things about people who don't know anything about math.

- They see a shopping bill or a phone number and ask you if you can add them.

- They see random equations, often in the background, and ask you if you understand what it means.

It's like when you study mechanical engineering and suddenly everyone wants you to fix their car.

A professor of mine who was quite bad at arithmetic used to say, "I'm a mathematician, not an arithmetician."


- arithmetics (a.k.a. Number Theory) is a vast branch of modern pure mathematics;

- numeric calculations, i.e. how to do them efficiently, is a very important part of the (applied) mathematics.

The parent commenter's old boss certainly did not mean Number Theory by "arithmetic".

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.

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

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.

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?

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.

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?


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.

For me, before you get a level where you can understand proofs or even how to read them, you need to understand why lemmas, axioms, and other components are true. My take is more about how to learn to understand the calculatory side of math, which I think is a prerequisite for taking on the reasoning and logical side of it.

you use intuition to come up with the argument. then you use a proof to convince others.

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.

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


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.

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