I hit a wall in college where math just stopped being something I intuitively "got". I'm sure that given enough time and motivation, I could have continued being "good at math" even at the higher levels, but these were luxuries I did not have, given everything else on my plate at the time.
My biggest problem with math, especially once I got into academia, was how it was taught. So many professors would scribble what seemed like nonsense on the board (symbols that change from professor to professor, or even from lecture to lecture) and then go on to say things like "...and the proof is trivial" or "...it obviously follows that...", and I'd sit there wanting to shout "NO, NO it's not obvious!"
Finally, I'd find a tutor to explain to me what it was I was missing, and it really WAS obvious. If only it had been taught that way in the first place!
Admittedly, not everyone has the same learning style, but the classes I took seemed really tailored towards the students who already had the intuition that I lacked.
I agree, college mathematics education is usually horrible, especially in the start. I remember back when I was attempting to read my calculus 1 textbook and not understanding the thing at all. I realized that about 3 or 4 classes later in discrete math the reason why is because it was using fairly foundational mathematical terms and concepts such as sets, proof by induction and so on that it was small wonder why my calculus textbook was incomprehensible to my grade 12 mathematical education.
I confronted a professor about why they have it backwards and don't teach discrete math course or similar foundational course FIRST so people can actually read their textbooks, or at least let people take that path. They basically said that since it's not relevant to many majors and it's harder for most people since they don't have the 'mathematical mind' they do it in that backwards way. The professors being Math PhDs, don't adjust their any of their classes enough for the lack of foundational knowledge. It frustrated me very much. I don't think it's a big mystery why you probably hit that college wall when most college curriculums are set up that way.
You have hit upon my biggest gripe with mathematicians, their love of 'notation', or more precisely their love of writing in symbology that makes mathematics seem more arcane than it actually is.
When ever I've run up against "impenetrable" math I often ask "So how would you use this?" and connecting it to the real world helped tremendously.
> and I'd sit there wanting to shout "NO, NO it's not obvious!"
Toward the end of my degree program, I became the annoying guy in class who would do exactly that. I remember one time in particular (I think the topic was something on wavelets, which I barely remember now anyway) when I stopped the professor and said "Can you explain all of that over again, from the beginning?"
Worked great for me, but I'm not sure what the rest of the class thought of it. At the time I just assumed they were as lost as me and would appreciate it, but that may well have not been the case.
Usually "obviously", "trivial" and such are used to point out that: "this should be obvious/trivial by now", if not: you are getting behind/need to study more/be better prepared before class/....
I was an art major in a mostly engineering school. I always had trouble with math but loved learning it. I also hit a wall in college like you. The first day of class the teacher asked how many engineering students there were in the class. Just about everyone raised their hand. This was her sign that she could teach fast. I knew I was screwed immediately. I wish she had asked how many art students there were so we could go slower.
I remember the math stopped making sense. The teacher would do exactly as you described, saying things like "...it obviously follows that...", etc. A girl who sat next to me would try to explain but was no less clear than the teacher. All the engineering students just got it.
My grades started high and then rapidly fell each week until I hit a string of zeros for a month. I was too proud to ask for a drop but eventually did but only after skipping a month of classes. The teacher was kind enough to understand that I was trying but my effort was for naught. She gave me the drop.
I've never pursued math any further, having felt defeated.
i got As in almost all my classes in math without really understanding anything. It was like a train I couldn't stop. I couldn't ever slow down for a month and say "I'm going to work on trigonometry this whole month so I can know what I'm doing when I do integration." I just had to memorize the steps to get the right answer, get my A, forget it all, and wait for whatever was coming next.
Years later I go straight back to the beginning and figure everything out, starting with Serge Lang's Basic Mathematics. I didn't even know where the Pythagorean Theorem came from, and when I learned it the second time around, it was damn beautiful.
my advice is to go back all the way to the beginning and get a book written by a real mathematician. I.M. Gelfand's Algebra and Trigonometry were truly enlightening.
> So many professors would [...] say things like "...and the proof is trivial" or "...it obviously follows that...", and I'd sit there wanting to shout "NO, NO it's not obvious!"
Agreed. What makes a great teacher is not the depth of their knowledge of the subject matter, but how well they are able to put themselves in their students' shoes and overcome the "Curse of Knowledge"[1]. Unfortunately for students, university professors are usually hired for the former (knowledge & research) and not the latter (real teaching ability).
I really felt college level math (abstract algebra, groups, monoids and such) was abstract painting in hieroglyphics, until I made a full turn into programming where you start to speak about abstract patterns that makes absolutely no sense (iterators, monads) at first, then I felt that this way of thinking seemed to be about finding your own solutions by being "mathematical" in the way you model your problem. Then abstract algebra started to feel like 'math + generics' and I went into re-reading my college textbooks. I still don't understand more than 15%, but I feel I have a chance.
I agree about the way its taught, teachers either forgot their own learning process, or they're all very advanced brains aiming at younger advanced brains that can unfold the possible application behind the abstractions.
This resonates with me- particularly the 'it obviously follows', (huge jump in complexity and then --->) 'so we already know that x, so obviously- y'.
I think a lot of teachers aren't cognizant of the fact that what is obvious to then isn't automatically obvious to students. I had what I would say is at best a mediocre HS math teacher and completely tuned out. It was my fault in the end, but the teacher didn't help.
I'm now trying to learn a bunch of things that require math and mathematic theory when you get to higher levels. So - learning math is what I have to do. It's kind of fun, and yes it is hard work. Like, brain-hurting hard work.
I know that my personal experience won't be of much of help to lots of HN-ers who are passed the highschool - early college-age, but I can say that in my case my love and understanding for Maths was inspired by two great teachers that I had, my highschool Maths teacher and my Calculus prof in my first year in college.
Maths is not about arithmetic computations or getting the "exact" answer, is realizing that things like convergent series or Real numbers are extraordinary things, almost magical, as in you somehow get the sense they all come from a different Universe. Sometimes one is lucky enough to have these things revealed to him, like it happened for me.
> I hit a wall in college where math just stopped being something I intuitively "got".
That's the college experience. Everyone that goes to that school is smart. Since the professors go to this level, all that remains is hard work. They understand that everyone there is smart.
One thing that worked well for my wife was a series of games from Nintendo called Fire Emblem.
This isn't a series of educational games: it's actually turn-based strategy. But the mechanics are all derived from very simple arithmetic, and although they don't give you the actual formulas, they give you, up front, every single number that goes into them formulas. Use a FAQ to get the small number of formulas involved, and the randomness all but vanishes: for any given unit on the field, you can always tell exactly which other units can attack it, how many times those units can try to attack, how much damage they'll do if they hit (and the exact odds of them hitting), how much damage you will do in response if your counterattack hits, and so on. The game will do this for you, but only for units that are directly in range at any given moment. With the numbers, you can calculate for any unit on the field, and that lets you start thinking multiple moves ahead.
The end result is that if you work out the math in your head, you can Neo your way through the games, and this is exactly what my wife did. I have never been the math-head in the family, but before she started this, I was still handier with numbers than she was: now it's the other way around. I should look into this myself.
Great suggestion! Just for fun and a share: my parents didn't allow me to have a video game system as a kid, but they did allow me to have computer games for my Commodore 64.(!)
My personal favorite was "Algebra Dragon" in which you slayed dragons by solving equations.To this day I'm convinced it helped my mindset towards math.
I aced all my high school math classes including calc. Because my school was small and rural they didn't have any more classes for me. So senior year I had no math.
Got to college, took a math placement exam and bombed out, so upset. Then as I was leaving knocked a chair down and everyone stared at me.
I ended up with a BFA in Drama.
Fast forward 10 years and like the author I worked my ass off to get into a top MBA program and not only that, major in finance.
So yeah, it can be done. Hard work, and not accepting the bullshit line "oh I'm not good at math." And without attacking my own gender, women tend to be let off the hook more easily with this excuse, as if we accept that girls can't do math.
Fuck that.
This post rocked. Thanks to the HN community for bringing it to my attention.
For whatever reason, I hit a poor grades stretch in math for my first three years of high school. It was just boring, and high school had distractions. My school almost blocked me from taking calculus. When I hit Calculus, I had my "Aha, I get it now" moment, and had great grades in math ever since.
Some students don't need to be motivated to work hard. Others do. Some in the latter camp are led to believe that they're not good at math, when the reality is that they're just not motivated for it. I'm glad that I ultimately found my motivation.
Like the OP, I also went to journalism school and can confirm that, while Statistics 101 is sometimes a highly recommended elective, math is seen as not needed, especially for those who want to tell "stories with impact."
I would've agreed back then as a student, but I also happened to be studying computer engineering so I took math for granted. In the professional world, it's astonishing how hard it is to explain ratios and basic enumeration to those who didn't try math, and how that greatly affects the range of story ideas you can conceive of.
And I say that as someone who still has to look up the quadratic equation...something virtually all college grads learned at least in high school. But there's a huge chasm between knowing that the quadratic equation exists and is applicable and not remembering that it exists at all.
"The difference between good at math and bad at math is hard work. It’s trying. It’s trying hard. It’s trying harder than you’ve ever tried before."
-while that's certainly true, when people say they are "bad at maths" they usually mean exactly that they have to put in a lot more effort (=trying harder) to reach the same level of math skills as the "gifted" guys.
Jim Fowler's calc II course on Coursera[1] is wrapping up right now, and I think he's the most engaging math professor I've ever had to please of learning from. Highly recommend.
He's also a maintainer of an OSS MOOC platform, MOOCulus[2], built with Rails.
Even if you're not interested in doing the course work (though really, it's only six homeworks) everyone should take a second to watch some of his videos... definitely some of the most entertaining, engaging, and clear math lectures I've ever seen. I also highly recommend it.
At first, I was terrible at maths at University and indeed had to retake a year due to my awful attitude. Oddly enough after almost flunking out completely I came to really enjoy the more abstract mathematical aspects of CS (e.g. lambda calculus) and even in the mainstream maths classes I ended up getting rather splendid marks which eventually led to me getting a 1st.
At a post-grad level I then ended up working in an Electrical Engineering department with Control Engineers - rather ironic for someone who started off almost failing because of maths...
Maths went from something I had no interest in, and therefore did terribly, to something I loved.... maybe it was age/maturity or just plain getting a scare!
[NB I'm very happy we had the UK style degree grading system rather than GPAs]
I think the author is making an excellent point about needing better math in journalism. An article in the NY Times yesterday about online shopping China had this line (pulled from a press release) "Tmall.com, one of Alibaba’s shopping sites, said Chinese bought...two million pairs of underpants, which if linked together would stretch 1,800 miles..." Why should I believe the rest of the article when the author is quoting someone saying that underpants are 57 inches wide?
To cross check your claim about the author, I did:
1800 miles ~ 2000 miles
5280 ft/mile ~ 5000 ft/mile
So 10M ft / 2M underpants = 5 ft/underpant. 12*5 = 60 in/underpant, which is close to 57, so I trust your statement :)
This is great and needs to be said more often. I feel as though I am the opposite. I absolutely love math, but I am terrible at it. I have taken tons of math, both formally in undergrad (up through diff eq) and on the side on my own. I struggled like hell through Calc 2 because I just couldn't visualize the transformations. I still feel like I want to take two years off and just start over from scratch.
I have the exact opposite problem in math courses. I can visualize the transformations and what the math means, I can know the exact approach to take to solve a problem, but I can't seem to get all the way through without making stupid mistakes. "Partial Credit" for showing work never really helped me much.
"The difference between good at math and bad at math is hard work. It’s trying. It’s trying hard. It’s trying harder than you’ve ever tried before. That’s it."
I love this quote. I think that nails it. When you see someone run a marathon, you don't think they are just naturally good at exercise, you recognize that they've put in a lot of hard work.
I think the value of hard work is under appreciated. Thanks for writing this.
I think the main problem is kids aren't taught abstract problem solving, and don't learn how to process the deep concepts that you find in math. This is largely due to our rigid, rote memorization math teaching methods (here is formula x, here are the type of problems formula x can be applied to, now do this pracitce set with 100 of them, and so on).
The majority of math currently taught k-12 is also largely useless to most students who don't go into hard sciences. This time would be better spent learning math as an art. They could struggle with problems without knowing the formulas ahead of time. They could modify the axioms and see where that leads. And so on, working their way through the different areas of math, and actually internalizing the concepts. In high school, the curriculim would include a life skills: arithmetic/statistics class, to ensure that day to day practical math skills are learned.
There was a really good paper on HN a few months about math as an art form which influenced my thinking in this regard.
Working hard at math might be necessary, but it doesn't have to be painful! I have found learning about the history of math has me puzzling over difficult concepts on my own time because it is fun.
The book I am reading (Journey Through Genius) is really good, but not very advanced math. I would love to find some self-study math courses that approach math not as a bunch of symbol-pushing but as an art with a history.
I've thought a lot about this and haven't found many good resources. I finally gave up and started reading history of math books. Unfortunately, most of them are aimed at grad students in history or grad students in math.
It has frustrated me enough that I've started developing my own versions. I taught a history of ancient mathematics class to homeschoolers locally and now I'm writing a book that works through Euclid while placing everything in historical context and focusing on the story of its development.
All of these are great. I'd like to add, maybe not for everyone, Calculus by Spivak. For me, it was the calculus book for which I had been looking for a long time.
I always thought of myself as good at math, but I had an unexpectedly hard time in required freshman calculus courses en route to a CS degree. I think the source of all of these paradoxes is that math is too broad a thing to be simply good or bad at. The objective in first year calculus seemed to be to practice basic techniques until they became rote. If you were an engineer, you'd have much more calculus to do and the early stuff was like learning to tie your shoes. I didn't encounter any really mind-bending concepts (I actually had to retake some calculus I originally took in high school and placed out of); the problem was that I couldn't do the problems fast enough on the tests, and I had to devote way more energy than expected to practicing doing them fast, which felt sort of like learning to play a musical instrument -- training muscle memory so the mind didn't have to be involved. I figured that calculus wasn't for me and I stopped after I met the degree requirements.
My point is that different scenarios involve different "math." For engineering, math is a hill to climb on the way to the interesting part. For CS, math is not about learning math per se, it's about learning to prove things and think rigorously. For math majors, it's about deep concepts in math itself. You could be good at proofs and bad at adding numbers in your head, and be good and bad at "math" at the same time.
Dividing people into good at math and bad at math is pretty meaningless. If you were truly bad at all math, you could probably not function in any job, because everyone needs to do something that could be called math. On the other hand, hardly anyone is just plain good at math, because you would have to be good at pretty much every hard subject that exists (statistics, economics, physics, and so on). If anyone is that smart, there can't be more than a couple of them.
Part of what makes (re)learning basis math difficult is the lack of math textbooks written for adults. Most math books are either too advanced (assume prior knowledge) or too basic (assume reader is retarded). In both cases, an otherwise intelligent adult with some gaps in their knowledge will get discouraged.
I wrote a math textbook which starts off from the very basics (numbers, equations, functions) and proceeds all the way to university level topics like calculus and mechanics.
I also like the approach of the Art of Problem Solving curriculum, though it doesn't currently have elementary school material. I haven't used it on my own kids yet because they're still too young, but I did buy the set and I think it looks good. If you read "Good Questions for Math Teaching" you will probably modify the way the problems are presented to make them more open-ended, but I like that topics are introduced with a problem that is later explained, instead of explaining then drilling.
Finally, I thought that this online course was a nice introduction to the approach. It's a pretty short course, but I just put the audio on an mp3 player so I could listen while working on other things and there were only a couple of places where I couldn't tell what was happening in the video.
Sadly, most of these books are pretty pricey for the page count, but the material I thought was quite good. If you were to get only two, I'd say go with "Young Mathematicians at Work" and "Good Questions for Math Teaching" because they will probably give you the quickest jump start, especially if you can get the "How to Learn Math" course.
If you're interested in where my evidence for the approach outlined here comes from, my main sources are the following books:
I have a 4yo and 6yo. The 4yo can add 2 to some numbers up to about 20. The 6yo can add any single digit to any number below 100. Most of the time. They both can read pretty well. We practice both math and reading every 'school' night for about 15 minutes.
The only goal is to familiarize them with doing stuff with numbers. The 6yo's kindergarten class has heavy emphasis on math, so I try not to get in the way of what they are doing. I avoid confusion by just staying away from the same stuff (at first, I thought I'd reinforce, but that turned out to be a bad idea. The 6yo needs a disconnect from what's happening in school, not a reminder. It's tiring).
The one thing I can confidently say is that what works changes as they pick up more concepts, and have more influences on learning (ie, other parent, school, other kids). My kids are not especially smart or gifted. The trick is that we stay just at the border of what they can do. We try new things, which fuels their confidence when they get it. We practice old things, which fuels their confidence with little cost. We stay focused and brief.
I try to get as many information and pedagogical tricks as I can. But they only work, if they work, for brief points in time. Kid thinking changes much faster than adult thinking.
This is just my opinion and intuition (so no experience with it), but I have the idea that establishing that numbers are an abstraction is probably more important than adding them together. I have the notion that rote practice of arithmetic is mostly just an easy and transmissible way to help establish this concept (it sort of follows from this that people who have trouble with things after arithmetic are often missing this abstraction. Plenty of people who think they are bad at math can tell you how long it will take to drive somewhere (basic algebra), they just don't like multiplying).
A concrete way of demonstrating this would be to point out that several same size groups of objects have something in common.
I wouldn't focus on "math" so much as logic at that age. This link may be useful to you - http://www.scholastic.com/teachers/article/ages-stages-helpi... (my source: my wife, who is a Reading Corps member that also assists in teaching math to 4 year olds)
[+] [-] Smudge|12 years ago|reply
My biggest problem with math, especially once I got into academia, was how it was taught. So many professors would scribble what seemed like nonsense on the board (symbols that change from professor to professor, or even from lecture to lecture) and then go on to say things like "...and the proof is trivial" or "...it obviously follows that...", and I'd sit there wanting to shout "NO, NO it's not obvious!"
Finally, I'd find a tutor to explain to me what it was I was missing, and it really WAS obvious. If only it had been taught that way in the first place!
Admittedly, not everyone has the same learning style, but the classes I took seemed really tailored towards the students who already had the intuition that I lacked.
[+] [-] mahyarm|12 years ago|reply
I confronted a professor about why they have it backwards and don't teach discrete math course or similar foundational course FIRST so people can actually read their textbooks, or at least let people take that path. They basically said that since it's not relevant to many majors and it's harder for most people since they don't have the 'mathematical mind' they do it in that backwards way. The professors being Math PhDs, don't adjust their any of their classes enough for the lack of foundational knowledge. It frustrated me very much. I don't think it's a big mystery why you probably hit that college wall when most college curriculums are set up that way.
[+] [-] ChuckMcM|12 years ago|reply
When ever I've run up against "impenetrable" math I often ask "So how would you use this?" and connecting it to the real world helped tremendously.
[+] [-] aethertap|12 years ago|reply
Toward the end of my degree program, I became the annoying guy in class who would do exactly that. I remember one time in particular (I think the topic was something on wavelets, which I barely remember now anyway) when I stopped the professor and said "Can you explain all of that over again, from the beginning?"
Worked great for me, but I'm not sure what the rest of the class thought of it. At the time I just assumed they were as lost as me and would appreciate it, but that may well have not been the case.
[+] [-] a1a|12 years ago|reply
Usually "obviously", "trivial" and such are used to point out that: "this should be obvious/trivial by now", if not: you are getting behind/need to study more/be better prepared before class/....
[+] [-] vinbreau|12 years ago|reply
I remember the math stopped making sense. The teacher would do exactly as you described, saying things like "...it obviously follows that...", etc. A girl who sat next to me would try to explain but was no less clear than the teacher. All the engineering students just got it.
My grades started high and then rapidly fell each week until I hit a string of zeros for a month. I was too proud to ask for a drop but eventually did but only after skipping a month of classes. The teacher was kind enough to understand that I was trying but my effort was for naught. She gave me the drop.
I've never pursued math any further, having felt defeated.
[+] [-] jrs99|12 years ago|reply
Years later I go straight back to the beginning and figure everything out, starting with Serge Lang's Basic Mathematics. I didn't even know where the Pythagorean Theorem came from, and when I learned it the second time around, it was damn beautiful.
my advice is to go back all the way to the beginning and get a book written by a real mathematician. I.M. Gelfand's Algebra and Trigonometry were truly enlightening.
[+] [-] pshin45|12 years ago|reply
Agreed. What makes a great teacher is not the depth of their knowledge of the subject matter, but how well they are able to put themselves in their students' shoes and overcome the "Curse of Knowledge"[1]. Unfortunately for students, university professors are usually hired for the former (knowledge & research) and not the latter (real teaching ability).
[1] http://en.wikipedia.org/wiki/Curse_of_knowledge
[+] [-] agumonkey|12 years ago|reply
I agree about the way its taught, teachers either forgot their own learning process, or they're all very advanced brains aiming at younger advanced brains that can unfold the possible application behind the abstractions.
[+] [-] SamBoogie|12 years ago|reply
I think a lot of teachers aren't cognizant of the fact that what is obvious to then isn't automatically obvious to students. I had what I would say is at best a mediocre HS math teacher and completely tuned out. It was my fault in the end, but the teacher didn't help.
I'm now trying to learn a bunch of things that require math and mathematic theory when you get to higher levels. So - learning math is what I have to do. It's kind of fun, and yes it is hard work. Like, brain-hurting hard work.
[+] [-] paganel|12 years ago|reply
Maths is not about arithmetic computations or getting the "exact" answer, is realizing that things like convergent series or Real numbers are extraordinary things, almost magical, as in you somehow get the sense they all come from a different Universe. Sometimes one is lucky enough to have these things revealed to him, like it happened for me.
[+] [-] trentmb|12 years ago|reply
[+] [-] skierscott|12 years ago|reply
That's the college experience. Everyone that goes to that school is smart. Since the professors go to this level, all that remains is hard work. They understand that everyone there is smart.
[+] [-] Millennium|12 years ago|reply
This isn't a series of educational games: it's actually turn-based strategy. But the mechanics are all derived from very simple arithmetic, and although they don't give you the actual formulas, they give you, up front, every single number that goes into them formulas. Use a FAQ to get the small number of formulas involved, and the randomness all but vanishes: for any given unit on the field, you can always tell exactly which other units can attack it, how many times those units can try to attack, how much damage they'll do if they hit (and the exact odds of them hitting), how much damage you will do in response if your counterattack hits, and so on. The game will do this for you, but only for units that are directly in range at any given moment. With the numbers, you can calculate for any unit on the field, and that lets you start thinking multiple moves ahead.
The end result is that if you work out the math in your head, you can Neo your way through the games, and this is exactly what my wife did. I have never been the math-head in the family, but before she started this, I was still handier with numbers than she was: now it's the other way around. I should look into this myself.
[+] [-] melindajb|12 years ago|reply
My personal favorite was "Algebra Dragon" in which you slayed dragons by solving equations.To this day I'm convinced it helped my mindset towards math.
[+] [-] melindajb|12 years ago|reply
Got to college, took a math placement exam and bombed out, so upset. Then as I was leaving knocked a chair down and everyone stared at me.
I ended up with a BFA in Drama.
Fast forward 10 years and like the author I worked my ass off to get into a top MBA program and not only that, major in finance.
So yeah, it can be done. Hard work, and not accepting the bullshit line "oh I'm not good at math." And without attacking my own gender, women tend to be let off the hook more easily with this excuse, as if we accept that girls can't do math.
Fuck that.
This post rocked. Thanks to the HN community for bringing it to my attention.
[+] [-] mathattack|12 years ago|reply
Some students don't need to be motivated to work hard. Others do. Some in the latter camp are led to believe that they're not good at math, when the reality is that they're just not motivated for it. I'm glad that I ultimately found my motivation.
The OP seems to have found this moment too.
[+] [-] danso|12 years ago|reply
I would've agreed back then as a student, but I also happened to be studying computer engineering so I took math for granted. In the professional world, it's astonishing how hard it is to explain ratios and basic enumeration to those who didn't try math, and how that greatly affects the range of story ideas you can conceive of.
And I say that as someone who still has to look up the quadratic equation...something virtually all college grads learned at least in high school. But there's a huge chasm between knowing that the quadratic equation exists and is applicable and not remembering that it exists at all.
[+] [-] MatthiasP|12 years ago|reply
-while that's certainly true, when people say they are "bad at maths" they usually mean exactly that they have to put in a lot more effort (=trying harder) to reach the same level of math skills as the "gifted" guys.
[+] [-] eloisius|12 years ago|reply
He's also a maintainer of an OSS MOOC platform, MOOCulus[2], built with Rails.
[1]: https://class.coursera.org/sequence-001/class [2]: https://github.com/ASCTech/mooculus
[+] [-] ics|12 years ago|reply
[+] [-] scott_s|12 years ago|reply
[+] [-] verteu|12 years ago|reply
I disagree. Math ability is highly genetic: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2913421
[+] [-] arethuza|12 years ago|reply
At a post-grad level I then ended up working in an Electrical Engineering department with Control Engineers - rather ironic for someone who started off almost failing because of maths...
Maths went from something I had no interest in, and therefore did terribly, to something I loved.... maybe it was age/maturity or just plain getting a scare!
[NB I'm very happy we had the UK style degree grading system rather than GPAs]
[+] [-] columbo|12 years ago|reply
It's one of my favorite MOOCs of all time, a fantastic intro to mathematical thinking.
[+] [-] smithbits|12 years ago|reply
[+] [-] BlackJack|12 years ago|reply
1800 miles ~ 2000 miles 5280 ft/mile ~ 5000 ft/mile So 10M ft / 2M underpants = 5 ft/underpant. 12*5 = 60 in/underpant, which is close to 57, so I trust your statement :)
I think this sort of "back of the envelope" calculation is something that just comes by practice/habit and is not some innate ability. I love this article by Jon Bentley on the topic: http://www.csie.fju.edu.tw/~yeh/research/papers/os-reading-l...
[+] [-] AndrewKemendo|12 years ago|reply
[+] [-] blktiger|12 years ago|reply
[+] [-] dominotw|12 years ago|reply
[+] [-] SamuelMulder|12 years ago|reply
"The difference between good at math and bad at math is hard work. It’s trying. It’s trying hard. It’s trying harder than you’ve ever tried before. That’s it."
I love this quote. I think that nails it. When you see someone run a marathon, you don't think they are just naturally good at exercise, you recognize that they've put in a lot of hard work.
I think the value of hard work is under appreciated. Thanks for writing this.
[+] [-] dwaltrip|12 years ago|reply
The majority of math currently taught k-12 is also largely useless to most students who don't go into hard sciences. This time would be better spent learning math as an art. They could struggle with problems without knowing the formulas ahead of time. They could modify the axioms and see where that leads. And so on, working their way through the different areas of math, and actually internalizing the concepts. In high school, the curriculim would include a life skills: arithmetic/statistics class, to ensure that day to day practical math skills are learned.
There was a really good paper on HN a few months about math as an art form which influenced my thinking in this regard.
[+] [-] bridger|12 years ago|reply
The book I am reading (Journey Through Genius) is really good, but not very advanced math. I would love to find some self-study math courses that approach math not as a bunch of symbol-pushing but as an art with a history.
[+] [-] SamuelMulder|12 years ago|reply
It has frustrated me enough that I've started developing my own versions. I taught a history of ancient mathematics class to homeschoolers locally and now I'm writing a book that works through Euclid while placing everything in historical context and focusing on the story of its development.
[+] [-] thetwiceler|12 years ago|reply
[1] http://www.hup.harvard.edu/catalog.php?isbn=9780674057555
[+] [-] nextos|12 years ago|reply
* How to Prove It (Velleman)
* Algebra; Trigonometry; F&G; The Method of Coordinates (Gelfand)
* Geometry (Kiselev)
* Calculus Made Easy (Thompson)
* How to Count without Counting (Niven)
* Introduction to Probability Theory (Hoel)
* The Little Schemer (Friedman)
The proceed to more advanced texts like:
* Naive Set Theory (Halmos)
* Linear Algebra Done Right (Axler)
* Geometry Revisited (Coxeter)
* Infinitesimal Calculus (Keisler)
* Concrete Mathematics (Graham)
* Information Theory, Inference and Learning (MacKay)
* SICP (Abelson)
[+] [-] tokenrove|12 years ago|reply
[+] [-] mtdewcmu|12 years ago|reply
My point is that different scenarios involve different "math." For engineering, math is a hill to climb on the way to the interesting part. For CS, math is not about learning math per se, it's about learning to prove things and think rigorously. For math majors, it's about deep concepts in math itself. You could be good at proofs and bad at adding numbers in your head, and be good and bad at "math" at the same time.
Dividing people into good at math and bad at math is pretty meaningless. If you were truly bad at all math, you could probably not function in any job, because everyone needs to do something that could be called math. On the other hand, hardly anyone is just plain good at math, because you would have to be good at pretty much every hard subject that exists (statistics, economics, physics, and so on). If anyone is that smart, there can't be more than a couple of them.
[+] [-] Spoom|12 years ago|reply
1. http://en.wikipedia.org/wiki/Dyscalculia
[+] [-] jebus989|12 years ago|reply
[+] [-] thatswrong0|12 years ago|reply
I think this article is targeting a much larger audience than the small percentage of the population that has genuine problems with math.
[+] [-] saraid216|12 years ago|reply
[+] [-] ivansavz|12 years ago|reply
I wrote a math textbook which starts off from the very basics (numbers, equations, functions) and proceeds all the way to university level topics like calculus and mechanics.
Check out the "No bullshit guide to math and physics:" http://minireference.com
[+] [-] bobbles|12 years ago|reply
Example from your page:
Suppose that you monitor the file size during the entire download and observe that it is described by the function:
f(t)=0.002t2[MB].
- 'Supposed that it is described by the function' - Where does this function come from? Is it derived from something else?
[+] [-] DanBC|12 years ago|reply
My son is 3, so at the moment we're just counting everything and making little groups to add them together.
[+] [-] aethertap|12 years ago|reply
* Good questions for math teaching (http://amzn.com/0941355519)
* Young Mathematicians at Work (http://amzn.com/032500353X)
* Number Sense Routines (http://amzn.com/1571107908)
* Dr Wright's Kitchen Table Math book (there are three) - http://amzn.com/0982921128
I haven't finished these next two, but they look promising:
* Fostering Geometric Thinking (http://amzn.com/0325011486)
* Fostering Algebraic Thinking (http://amzn.com/0325001545)
I also like the approach of the Art of Problem Solving curriculum, though it doesn't currently have elementary school material. I haven't used it on my own kids yet because they're still too young, but I did buy the set and I think it looks good. If you read "Good Questions for Math Teaching" you will probably modify the way the problems are presented to make them more open-ended, but I like that topics are introduced with a problem that is later explained, instead of explaining then drilling.
http://www.artofproblemsolving.com/Store/curriculum.php
Finally, I thought that this online course was a nice introduction to the approach. It's a pretty short course, but I just put the audio on an mp3 player so I could listen while working on other things and there were only a couple of places where I couldn't tell what was happening in the video.
* https://class.stanford.edu/courses/Education/EDUC115N/How_to...
For a more general book about early childhood education, I really liked
* "Engaging Children's Minds - the Project Approach" (http://amzn.com/1567505015)
* Making Thinking Visible (http://amzn.com/047091551X)
* Learning Intelligence: Cognitive acceleration...(http://amzn.com/0335211364)
Sadly, most of these books are pretty pricey for the page count, but the material I thought was quite good. If you were to get only two, I'd say go with "Young Mathematicians at Work" and "Good Questions for Math Teaching" because they will probably give you the quickest jump start, especially if you can get the "How to Learn Math" course.
If you're interested in where my evidence for the approach outlined here comes from, my main sources are the following books:
* Effectiveness In Learning (cognitive load theory) http://amzn.com/0787977284
* Visible Learning (synthesis of 800 meta-analyses) - http://amzn.com/0415476186
Edited because I put some of them in the wrong spots and forgot links...
[+] [-] ericssmith|12 years ago|reply
The only goal is to familiarize them with doing stuff with numbers. The 6yo's kindergarten class has heavy emphasis on math, so I try not to get in the way of what they are doing. I avoid confusion by just staying away from the same stuff (at first, I thought I'd reinforce, but that turned out to be a bad idea. The 6yo needs a disconnect from what's happening in school, not a reminder. It's tiring).
The one thing I can confidently say is that what works changes as they pick up more concepts, and have more influences on learning (ie, other parent, school, other kids). My kids are not especially smart or gifted. The trick is that we stay just at the border of what they can do. We try new things, which fuels their confidence when they get it. We practice old things, which fuels their confidence with little cost. We stay focused and brief.
I try to get as many information and pedagogical tricks as I can. But they only work, if they work, for brief points in time. Kid thinking changes much faster than adult thinking.
[+] [-] maxerickson|12 years ago|reply
A concrete way of demonstrating this would be to point out that several same size groups of objects have something in common.
[+] [-] bovermyer|12 years ago|reply