Math was rote memorization and regurgitation of algorithms which I had trouble memorizing without explanations of how and why they worked. Later on I found out the answers to these questions are called "proofs," they are very fundamental to mathematics, and I am quite good at them. I did not excel at the math I was given because I was plagued with foundation questions about why any of this added up (sorry) logically and they were blockers to true understanding for me. Logic was completely missing from the curriculum. Digested mathematical trivia were handed to me like I was some kind of child. I was never given the power tools I wanted to build my own knowledge, perhaps for fear of my safety or my innocence.

People who were good at studying and taking people's word for things sailed through and to this day can't prove things rigorously. These kids don't like taking things apart.

It was only during some education electives I took that I realized that a well known problem in STEM education is the group of students who are intelligent but have rather slow, idiosyncratic, and methodical ways of assimilating new information. Imagine the quantum of solace that gave me.

I chose Computer Science at random from an admissions catalog because I liked writing programs even though I was "bad at math," and because I couldn't afford film school. 4 years of discrete math, modern geometry, multi-variable calculus, linear algebra, and algorithm classes (on top of other fun but not as fun-damental courses) later and I realize what a load of boring and disingenuous shit public school math was.

*

Why do so many gifted kids think they don't like "math?" Because "math" sucks and schools don't let kids build things anymore - materially or otherwise.

[Edit] Grammar, new insights, the sort of thorough editing I should have done the first time.

P.S. I remember hitting calculus in college and drawing the "hard" professor. ("Oooh", people exclaimed sympathetically.) I pulled an A+ and enjoyed the course, because the professor knew and cared about what he was teaching. He just insisted that you did, too.

That's usually what it means to be the 'hard' prof. I had a couple of those in CS and I took every class they offered. Most students would drop their classes so that always left me and just a couple others to have nearly a private class with a great professor.

In college, though, it's often times the professors who create the problems.

I had some good, dedicated and effective high school teachers (although they were perhaps the minority). Unfortunately, as I recall and/or experienced it, the math department was not particularly strong.

People who thought they were getting a typical "survey" class -- LOL! This professor wasn't having any of it.

Perhaps a bit unfair, if you were trying to balance an overall course load that was already heavy. But of itself, worth it.

Good point also. My High School math department consisted mostly of a temp with an education in English Lit, a basketball coach who had to fill a teaching position in order to be allowed to coach, a professional motocross rider who taught part-time and gave out a sheet of formula every test and an obviously insane man who randomly selected a student every week to sit out in the hallway for no reason.

Fast forward to the present. I am now a sophomore at a university with an incredibly strong math department. I've finished the core mathematics curriculum, and I am currently taking two grad courses (Algebraic Topology and Grad Linear Algebra). I plan to take two more in the spring, and hopefully participate in a undergraduate research program in the summer.

This is coming from someone who got a 600 on the math section of the SAT (800 on the the other two).

But I wouldn't recommend going into math like I did. It has been an incredibly hard task. I am very, very, VERY jealous of those of you who had an inspired early math education. Oh, the work that could have been saved...

Although, unlike you, I don't think I ever recovered.

Mistakenly, my impression of how to learn maths was based around the ideas of learning how to calculate new idea X and then spending hour upon hour of rote exercises.

One day I had an epiphany, being a computer programmer it dawned on me that I should not be doing all these calculations by hand, only enough to get the basic idea.

Since then I have only bought undergraduate or graduate text books, and used sage for learning math, its just thousands of times better as I now get to learn the real concepts and not just rules (or put another way where axioms do appear I now understand _why_ they are formulated).

Also the other thing that has long lacked in my early math teaching was that in no way were we taught anything about the art of creating and reading proofs, I think this is a shame as it deprives people from the interesting mental rigour that is required to accomplish this.

I think the average high school mathematics syllabus is being devised by the criminally insane. Personally I am a great believer in the work of these folk http://www.computerbasedmath.org/

I'm intrigued. Do you have links to papers/articles about this phenomenon?

I've got 2 really smart boys in elementary school. My wife and I are amazed at how bad the math education is. Our observations, based on a few kid-years of school across both a public and a private school:

- In an attempt to make math easier to learn, they're teaching math really abstractly. "skip counting" and a bunch of other stuff. It's basically all the tricks you figure out after you understand what you're doing, but they're teaching the tricks and not the math itself. - Further to the last one, they don't teach things like times tables. Rather than just getting kids to memorize the times tables, they make them go through all these hoops to get them further along. I realize that times tables are no fun, but neither is struggling through every problem. - Most teachers are not good at math. One teacher would send problems that didn't make any sense. I sat through a presentation by the teachers on the "new math" and it was clear they really don't know what they're doing, they're just teaching an algorithm. - Most teachers don't like math. How do you instill a sense of love for something that you hate?

Many of these things go for other subjects too. It's just that it's more obvious in math.

She was an English major in college. She didn't take a math class her senior year of college. She took a CS Elective to fulfil her math requirement in college (that's where we met. The class met once a week and a different Grad student or Professor would give a talk about whatever project they were working on. Over the course of the semester we would have to write a page "report" about 3 of the talks we heard)

So, she graduated, then got a Masters in Education and then started teaching (among other subjects) Math to 9 year olds but she hadn't taken a math class in 6 years.

She not only didn't like math, but she actively avoided it for over half a decade and then she was tasked with teaching the basics to kids...

Part of the problem could be that there are lots of teachers like her, that don't even know how to do the math themselves.

What they all had in common was the ability to see my aptitude and find materials for me that I could benefit from. These teachers may not have been math whizzes, but they definitely weren't afraid to give their students material they weren't completely comfortable with themselves. They also encouraged me to do more difficult math and were very supportive in general. Their attitude carried me very far, even if they couldn't answer some of the more difficult questions I would ask. My calculus teacher was very good, but he was a gigantic asshole.

The majority of the comment was praising my female math teachers for caring enough to go outside their comfort zone to foster a love of mathematics. If more teachers in general had the temerity to venture into the unknown when educating their students, especially in math and science, we might not have serious gender gaps in most professional fields.

Part of the problem I think is that we assume that people who don't get math early on are just "bad at it" as opposed to people who don't get reading. If a student doesn't learn to read, we think "how can we better teach them? Do they have dyslexia? What are ways we can teach dyslexic children to read? We don't ask the same questions when a student doesn't get math, we just say "oh, well, they suck at that then."

Case in point: Took AP Computer Science my sophomore year of high school. I got C's on most of the tests (mostly code tracing) for the first two months or so. After getting a particularly bad grade on a test (near failing, and I got all A's in everything else), I went home and thought "fuck this" and pulled out some Java book and read it for about 5 hours. After this, I moved from being about 40th percentile to best student in my class. I got high A's the rest of the year without trying particularly hard, and I actually enjoyed the class, and this lead to me starting to program on my own for fun, and eventually becoming a pretty good programmer.

I was exceptionally fortunate to have a math teacher in high school who is probably one of the best high school teachers in my state. She loved math. She was excited as hell to get to teach it every day. I had her for 7 of my 8 HS semisters, and I credit much of my appreciation for math to her. I was still never a good math student - B+ at best due to lack of discipline in cross-checking answers and the like - but I loved (and still love) math. Calculus was such a blast for me that I took it twice in HS (the second time around for college credit).

I think the other half of the equation is that math is hard. It requires a level of mental discipline and precision that most people don't possess, and have no interest in possessing. I think that for most of us here, who have at least some interest in programming, this isn't obvious. We think in discrete terms and proof-like concepts, but most people don't. Math takes more brainpower and more self-discipline than most any other primary education subject, and people naturally follow the course of least resistance.

A couple of weeks ago we started multiplying. To do 5 * 3, we make 5 circles, draw three X's in each circle, and then count them up.

Last week, we started dividing. To do 16 / 3, he'll draw 16 X's, then circle 3, then 3 more, etc. At the end there is one left over (poor little guy). Then, he counts up the circles and writes: 5 R 1.

We go up and down the stairs for positive and negative numbers. He really enjoys all this. Kids enjoy almost anything you do together. He'll even sit down and write his own problems, and problems for me to do.

I'm not expecting too much from the schools on math when he starts in a couple years, so I'm hoping to cultivate that interest at home, and maybe he can share it with his classmates.

For example, a trick to multiplying by 9: hold up all 10 fingers, and then put down the one that matches what you are multiplying. The answer is the number of fingers still up (appending the count of those to the left of the down finger to the count of those to the right).

Example in ASCII Art: 9 x 4:

    ! ! ! . !      ! ! ! ! !
    left hand      right hand

Edit: Here's a link: http://listverse.com/2007/09/17/10-easy-arithmetic-tricks/

The "tricks" he likes right now are the ones he can understand like n * 1 = n for any value of n or n / 1 = n (because you have one circle around everything).

I've tried showing him some tricks, but they aren't surprising and cool to him yet.

I really came to enjoy math / calculus when I was taking a physics class and started to understand the world better through the lens of my new math knowledge. Einstein's theories made so much more sense when you could think about the physical limits imposed by the math. Otherwise I couldn't care less about the Disc method and rotely calculating the volume of objects around an axis...

So to many children, learning math without practical application is like teaching someone a successively harder alphabet/vocab every year without ever writing an essay or delivering a speech... Or like learning how to read music and play harder scales without ever performing a piece.

Effective learning needs a good balance between application (the fun) and mastery of technique (the sweat and tears).

Although a "gifted" child is perhaps more likely to enjoy math than children less likely to receive encouragement and praise for their work in math; there is nothing wrong when a bright child's interest is in history, art, writing or even sports.

To put it another way - as the WPT shows, smart kids often prefer Texas Hold'em to Chess. The idea that intelligent people should desire abstractions rather than concrete engagement with the world is as old as Plato's redirection of Socrates' project from the actual corruption of youth to tomes of political theory.

Likewise, the article is based on a theory that smart kids like math rather than acknowledging that the data show that many of them don't.

That's not really true; it just assumes that some gifted kids would, a priori, be expected to like math.

If Math is a language, then the first 10 years of instruction are essentially spelling tests. There is very little flexibility nor room for creative thought in those years of instruction. Most kids never get to a proof or other areas without a defined path from start to finish. That is like only studying English grammar and never taking a literature test. Sure it can be interesting, but it is hardly creative.

I see mathematics, especially the more abstract areas, as much more similar to the arts than to the sciences.

The answer is that it gives you intuition about how these things behave, even if you always use a calculator to come up with the correct result.

And mental back-of-the-envelope calculation is unbelievably useful in every day life; if you can't do it, I can guarantee that you're either making worse decisions for it, or wasting a lot of time optimizing your every purchase by typing it on a calculator.

Similarly, you need to practice proving and reading proofs so you actually have the right intuition about what constitutes "proof". Unfortunately, all of western culture, and american culture in particular, is so far divorced from the concept of sound logic, that it is very likely that most people don't even realize they are missing something.

It's not any of the specific proofs that matter. It's the exercise. You go to the gym to exercise your muscle; You do proofs to exercise your logic.

And this is speaking as someone who is absolutely terrible at mental math.

Given the current state of math, very few people would ever be able to contribute anything. Others can skip to the plan B immediately.

Doing a proof from scratch is akin to being given a compass and told that somewhere past the wilderness is the promised land where others have blazed trails to but otherwise given no other help.

You know you can create the proof (since its a book problem) but how to is entirely up to you.

Programming is instantly addictive because you create from the day two. Math isn't.

It isn't, because students aren't given problems to be solved. They're given the solutions first.

The profound feeling of figuring out a solution to a problem (a real one, unlike the ones given at school) is hard to explain to someone who did not experience it. That's why, having choice to do Masters degree in Math or CS, I choose Math.

Programming is addictive, learned long before you decide to specialize in it, and you make up problems for yourself creating unique software products from day one.

With math, you only reiterate the same thing over and over.

That's why, having choice, I choose to work as a programmed early and do as little as possible without being kicked off at my higher education facility.

Oh no, wait - I did as little as possible even before I began working. I just amn't good at that learning thing perhaps.

If you do that, yes, it's dull and pointless. The thing is, you don't need to do it. It's like complaining that programming sucks because writing hello world over and over again is boring.

...but then, as they try to write out the steps of the "obvious" proof, or explain them to me verbally, we discover that they don't have nearly as good a handle on all the definitions and the details as they thought they did. Having to wrangle a proof, even a simple one, does an unparalleled job at developing a deeper understanding of the theory.

(For example, in my database theory class: Prove that any two-column table must be in BCNF. On the way through to the "trivial" two- or three-line proof, one has to understand what it means to be a key of a relation, what a functional dependency really is, and the notion of transitive dependency. Even if you never decompose a table into BCNF, the other concepts help you build some important mental models.)

What I am referring to is the level where math is a set of tools. Rather than doing algebra, I frequently finding myself create new algebras. Every time I create a new algebra (in order to model data, for example), I informally prove that it meets the required axioms to be the mathematical ring or group I mean it to be.

It's a whole different type of math. It's like reading shakespeare (where there are a bunch of difficult words) vs writing shakespeare (where a master of the English language invents new words to suit his purpose, while still conveying the message clearly).

I find this particularly troubling since I remember as a student that my peers would constantly complain about math courses using the rationale "Am I ever going to really use this?". Both sides are arguable for certain subjects (I can't remember the last time I really used trigonometry, but I do think that learning it helped me to reason better in other domains). However, I find it troubling that statistics is not mandatory/highly encouraged, yet it is something that applies to an incredible amount of everyday activities.

Absolutely. Teaching HS students trig but not statistics makes zero sense.

That said, the way statistics is taught is absolutely awful (probably even worse than most math education), so I'm not sure it would make much difference.

One of things I noticed when I was in HS (~10 years ago) was that since it was such a new phenomenon to teach stats in public HS, the teacher was essentially learning alongside us. This was no fault of his own, but simply that when he was training to teach, the value of teaching everyone basic stats was not appreciated.

I can only hope that trend will change.

She's one of the worst teachers I've ever had, and it's a shame. She definitely does know what she's doing, but she has absolutely no ability to pass that knowledge on effectively. In some ways it's no different than the argument this article makes about more "traditional" high school math classes--except for the great abundance of story problems, with which most students have very little prior experience.

From what I've seen from tutoring 3 younger siblings in math throughout their high school days, I think they should spend the first 3 years of high school drilling Algebra into their heads. Then in the last year they can spend half a year on geometry and half a year on trig. Or maybe keep kids in Algebra until they can demonstrate an absolute mastery.

The biggest problems my sibs had with higher math wasn't the higher math, it was the algebra underlying it. They get one year of Algebra in 8th grade and the move on to Geometry assuming they have mastered it, but they havent.

When I was in high school many of my classmates told me the 'did not like word problems.' I now realize that kids who do not like word problems are kids who do not understand what they are being taught.

Ant that is most of the kids.

This, adding the fact that it's "socially acceptable" to claim that a person might not be smart enough and give up entirely, is what makes children fail in the end and deemed to be "stupid".

Edit: If you're further interested in Salman Khan's point of view you can check his TED talk here http://www.youtube.com/watch?v=gM95HHI4gLk

Math often isn't intuitively easy (lots of rules to learn!), lots of boring grindwork to get the grade, and can't be faked.

Because it didn't come "automagically" like the other subject (or at least how I knew the other subject could if I had applied myself), I figured I just wasn't good at it. Reinforce this with lots of people who also thought they just weren't good at it and leveraging all that into a pile of excuses, I did famously bad in math in K-12.

In college I decided to start over and finally tackle it, I had to learn how to accept that some subjects are hard, that grindwork has value, and how to actually build competence in something rather than just having it. I figure if I was so smart, I should be able to figure out how to figure out math.

And it worked! I ended up picking up a math degree as a side product of learning how to learn math while getting my C.S. degree. Got great grades up through some reasonably upper-level math courses.

Truth is, I don't think I'll ever really take an interest in math. I haven't really done any looking into it in a decade, and probably couldn't solve and integral to save my life. I still can't get over the notion of not being naturally "good" at it. But I learned tons going through the process and am satisfied that I could learn the subject now even if I've forgotten all the details these days.

I'm pretty convinced after going through it all that most people could eventually learn to handle most of the maths through at least single variable calculus if they can learn how to learn it -- and I think that that process is highly personal and highly specific to the individual, but it's at least doable.

Most of the time anyone who really learns math ends up teaching it to themselves. It can be tough to find self motivation if you are struggling at all so the only kids who like math are usually the ones that instantly grasp concepts, they can do the rote busy work in a few minutes and spend the rest of class day dreaming about math ideas or reading ahead in the textbook to sections they find interesting. This is where the love of math comes from.

Edit: this was 27 years ago BTW: you would hope (gifted) kids now come into contact with game creation faster now.

What can I do, as a parent, if I see that my children are not being taught math at the proper pace? I could tutor my children at home to a certain extent, but that just raises all sorts of other questions:

I've got a solid math background, but no education background, so what kinds of resources are available to me to establish an effective home tutoring program?

How can I tell if the pace I am setting is too fast, too slow, or just right?

In the unlikely event that one of my children is a "math outlier," my knowledge of math, although in the 90th (95th? 99th?) percentile, would prove woefully inadequate: where would I find an (affordable) math tutor with comprehensive knowledge of math?

This last question is the only one I think I have a decent answer for: find a mathematics graduate student looking to earn some money on the side.

The books are not very difficult for parents to understand and give you a baseline that you can follow very closely. Also, I hope that since it is unlikely that my child will do the same book in school I do not run the risk that the she will refuse to do math in school since she has already done the book.

Since you have the book you can set the pace based on how difficult the lesson of the day seems for the child. You can do one page per week or 10 pages per day (both these things have happened to me). Of course there can be several levels of understandings of the same lesson and in my case I usually am happy with the lowest level. To correct for that I do sequentially 2 different books that have the same material (Singapore Math provides multiple books for the same level). I skipped some chapters about weights and volumes since these seemed too involved for my daughter (3 at the time), but I have done everything else that is on these books.

I must say that until now this has been a wonderful experience for me. I have never needed to ask my daughter to do math, anytime she sees me free she asks for it herself. And almost always I am the one who tries to cut the lesson short, making sure that next day she will want to come back wanting more.

http://www.khanacademy.org/#browse

I bookmarked it because I plan on going through all of the lessons so I can refresh myself. I've only watched a few clip but they seem helpful.

Here is the NYTimes article

https://www.nytimes.com/2011/12/05/technology/khan-academy-b...

History became my least favorite subject after that, because suddenly math was about solving problems, not being a human calculator, and history was still just memorizing names and dates and facts and regurgitating them back out on demand, which I am terrible at.

People often hate ego-damaging objective evaluations of their performance, and math is full of them. Learning math is, almost by necessity, a humbling experience (it's always possible to come up with more difficult math questions).

So a first step to the successful teaching of math is to teach patience, persistence, humility, a sense of what it "feels like" to learn a difficult concept, and a certain comfort with not-yet-understanding.

It gets even worse because you spent six years in grade school grinding long division (oh god) then finding out it's almost useless. The introduction of more interesting math times out well with the rebellious phase where you stop trusting adults.

Only reason I ever got into math was programming. I'm just so happy none of the highschool computer teachers knew how to program and had us memorizing MS Word instead, because I'm sure they would've ruined that to.

As a tangent I was talking to a friend who studies astronomy. She does math for fun, but haaates programming and sees it as dull busywork. Her first introduction to it was through school and it's all "punch in these numbers and see what it does".

If kids see how math is needed to trade in Chicago, to gamble in Vegas, to optimize at Google or to solve the German tank problem kids will want to know more.

I remember every guest speaker in K-12 math classes -- only about one per year. And I'm probably exposed to more math from HN links than I was as a non-technical undergrad.

I think another issue is that we expect parents to be able to introduce kids to advanced topics. What happens when you get a brilliant kid with no access to education outside of the classroom? More often than not at least a few years of missed accelerated learning.

In three months I learned more about the logic behind mathematics, and by extension more about math period, than I learned in the previous 14 years in public school. I took one math class from here years and years ago and I can still do calculus. The Japanese method for teaching math is just simply amazing (it helped that she was ranked number one in the nation (Japan, not the US) when she was in high school, this woman was seriously brilliant, she just got knocked up by an American and ended up having to quit Todai).

It's no wonder most kids get turned off by that.

I do educational research now (engineering/science/history education - but yeah I still shy away from research on math education).

The reason I believe is because kids aren't taught why math is relevant or useful to them.

For example I still remember in first grade, we got handed a big book of math problems and had to go through them all over the course of the year. It was so boring, I raced to finish it as soon as I could. I remember racing with other students in fourth grade, too, to see who could finish tests the fastest. That probably did have a role in my math abilities improving. There have been studies of having kids 'race' through math problems really fast so that they learn to do it more automatically, and this is apparently common practice in China if I recall.

This may help learning math, but it doesn't help and possibly hurts interest & motivation to learn math. And studies have shown that interest and motivation are what correlate the most with our career choices, not test scores or abilities.

Finally, yes, there are solutions already out there that teach math in a way that makes it relevant and more interesting to students - they just haven't spread all over yet. The Realistic Mathematics Education (RME) project out of the Netherlands is very old, and there have been similar efforts since then. They are basically theoretically grounded in what is known as situated cognition. All cognition/learning is tied to the context. Jean Lave for example showed how some Brazilian street children had developed very sophisticated math skills on the streets. Some less depressing contexts for learning math skills might be in the grocery store, or in creating a game or other software app (which is how I came to finally see the uses of differential equations and matrices and trigonometry and the like after college - creating educational software applications).

John Dewey knew about this 100 years ago. He said we shouldn't educate students for the future (which is uncertain and unimportant to kids), but instead educate them for today. How is what you are teaching them useful to them right now, in their own lives, not the lives of adults or professionals.

"Education is not preparation for life; education is life itself."

But let's be honest: It is BORING. I don't sit around doing math things in my head when I could be doing anything else.

Okay, occasionally I'll find some interesting math thing and play with it, or find a (real life) word problem and decide to solve it. But other than that, it's so amazingly boring. Learning math is even more boring.

So why don't kids like math? Because it's boring! I don't blame them.

Some people are blaming the teachers, or the education system, or blah blah blah. Okay, maybe they -could- be doing things better, but has anyone ever done their job perfectly? I've yet to meet that person. Instead, they're doing their best, just like everyone else.

I started learning Japanese a few years ago. What I didn't expect to learn was how many different ways there are to learn a new language. And the best way isn't any single one of them... It's to combine a bunch of them together. And not a particular set, either. You should combine all the ones that work best -for you-.

Having learned just exactly how complicated it is to make the perfect set of lessons for a single person, I looked around and saw how differently everyone learned. It's not only impossible to create a perfect set for 1 person, it's impossible to create a good set that matches everyone. The best you can do is catch the people who don't learn well on their own and hope the rest will teach themselves.

So then I look at our system, and I'm not surprised that I see that's exactly what they're doing. They're trying to catch the stragglers and leaving the brightest to fend for themselves. And they can. I did. But had I -known- that was happening, I'd have forced my education to go differently.

My girlfriend is homeschooling her son for various reasons, but chief among them was that he hated school. He was bored and picked on, and yet still getting bad grades. She made him a promise that if he brought his grades up to a certain level, she'd homeschool him the next year. Unsurprisingly, he easily hit that level.

It's been going great for them. She has accelerated everything to the point that he is constantly learning, and he gets top marks on everything. Then, because there's free time, he gets a little vacation to have fun for a while, then back to work the next semester.

They also go beyond the required instruction and do projects based on the material. I can't think of anything better to create lasting memories than that.

tl;dr - Our system is fundamentally flawed.