How I faced my fears and learned to be good at math
niemanlab.org
niemanlab.org
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 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.
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.
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.
I quit after the first year. Best thing I ever did.
The questions that bother me are the type that stroke the ego of the person asking because they already know the answer.
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/....
But even then, in many cases, the professor was simply expecting us to have made an intuitive leap. And those of us who hadn't made it were left behind, with no explanation as to why or what it was we needed to understand.
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.
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).
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.
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.
With a basic course in linear algebra (such as Gilbert Strang's on MIT OpenCourseWare) and potentially some intro calculus you should fly through that course.
Computer science is also essentially about notation and vocabulary, but we have to make our notation understandable to the computer, which is a much higher standard than what mathematicians have to adhere to.
We are in a field that demands a much higher level of rigor than mathematicians are accustomed to, as much as they'd hate to hear it.
Coq notation, lisp-style notation, even python-style notation- anything would be better.
That said, it does bring me back to a math class where a professor, after realizing he needed to introduce a subscript, to a subscript, to a subscript of something that had both a subscript and superscript already, made a comment along the lines of "please excuse my poor notation."
My largest college regret was blowing off linear algebra- it was an annoying class taught in an annoying way (handwritten homework showing your work for each step of matrix multiplication, no proofs, ect). I blew it off because there were no applications of it in anything I cared about.
A semester later, it showed up somewhere in every single advanced computer science class. Really wish there had been a proof-based linear algebra class that showed up later in the curriculum so by the time we reached it we knew it had value.
When given two numbers, if one wishes to find the quantities that give 0 when the second of the numbers is added to the product of the quantity and the first number and the quantity multiplied by itself, one should negate the first number and then either add or subtract the square root of the sum of the square of the first number minus four times the second number, and divide this summation by two.
That's just the quadratic formula in disguise:
Let b and c be real [or complex] numbers, then
x^2 + b x + c = 0
x = (-b +- sqrt(b^2 - 4c)) / 2
(Saying "let's do maths without the symbols" is a little like saying "let's do programming without special languages"... it is very very hard to make it work.)
As wonderful as it is, mathematics needs notation, and lots of it. You can express incredibly complex ideas in mathematics, totally unambiguously, through a collection of symbols. Not to mention that they're universally recognised.
The reality is that mathematics is 100% about thought. You'll struggle to put together the concepts in your mind long before the notation is the real issue. Once you have a clear picture of the abstract space you can use the notation you've learned to communicate the world you've created to others. What could be more wonderful?
[1] http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI...
Suppose we wish to make a rectangle with a given area and perimeter. This is an interesting problem! Does the number of possible answers depend on the specific area and perimeter? Certainly! It all comes down to thinking about squares, since squares maximize the area given a fixed perimeter. If the area of a square with the given perimeter is LESS than the desired area, then there's no way we can make such a rectangle. If the area of the square is equal to the desired area, then that's our only answer! Now, how about if our square's area is larger than the desired area? We'll get two possible different lengths of a given side of the rectangle - one representing the rectangle's width, and the other representing it's height. Or we could also think of them as two different rectangles - a tall one, and its rotation by a quarter turn (which makes it wide). By symmetry, we know that the difference between the square's side length and the shorter side will be the same as the difference between the square's side length and the longer side. How large is that difference? Exactly enough to diminish our shape's area from the square's area to the desired area. And that difference in length is simply the square root of the difference between the square's area and the desired area!
It would have been better with pictures :). Anyway, the quadratic formula is probably the greatest mistake in all of mathematics education. Somehow we use the word "quadratic" and even the phrase "complete the square," but never have I ever seen someone draw the said square!
While I think notation is often great for expressing ideas concisely and precisely, I think an excess of notation not a good way to communicate concepts. Nobody should memorize the quadratic formula! We should understand instead how to think about areas and lengths, and then we solve the problems in quadrature that we want.
FWIW, an animation of the quadratic formula/completing the square: http://en.wikipedia.org/wiki/File:Completing_the_square.gif
Let x, b, c \epsilon C:
x^2 + b x + c = 0 => x = (-b +- sqrt(b^2 - 4c)) / 2
Example of the difference: http://i.imgur.com/gwAqirx.png
generated by code:
\textbackslash{}epsilon: $\epsilon$ \\
Example: $x \epsilon \mathbb{R}$
\textbackslash{}in: $\in$ \\
Example: $x \in \mathbb{R}$
The problem isn't the notation per se, it's that teachers don't spend nearly enough time explaining the notation itself. It's a foreign language that they are so skilled at that they don't understand how unfamiliar it is to their students.
I was well into a physics major before I really stopped and carefully considered the many different notations used to represent derivatives (dx/dy, f'(x), y', y-dot, Dsub-yX, etc.) I realized that I had developed separate context-specific bodies of calculus knowledge/skill from different fields with different notations and approaches, and that these were all the same thing. Different techniques in different contexts were an artifact of different styles of notation, not differences in the actual math.
I can understand Italian to some extent as a side-effect of my study of Spanish. If I took an electronics class in Italian, I would understand some of the concepts and misunderstand others. Would my troubles to understand certain electronics concepts be due to trouble with electronics or trouble with Italian? Who knows? Both types of misunderstanding would compound each other.
If teachers spent more time carefully teaching this foreign math language before (and while, and after) using it to teach math, I'd guess a lot of students' "math problems" would magically disappear.
The first problem is that the notation is usually "the first symbol that popped into some random genius's head 200 years ago". And once the notation is set, it's set, no matter how poor it is, or how many other places it's already in use etc etc. Then, as you note, sometimes there are multiple notations. Ugh.
The second problem is closely related to the first. Mathematical notation is write-optimised. This makes sense because of the long history. But that doesn't change that write-optimised languages are harder to read and understand, even for experts, than read-optimised languages.
In a programming context if you today reduce all your variable names to single latin letters and all your function names to single greek letters, you will be widely mocked and reviled. In maths it's Just How Things Are Done.
I guess what I'm saying is: the curse of mathematical notation is pen and paper. The boundaries of QWERTY liberated (almost all) programming languages from the curse.
Serious question. What do you think is the general impression of APL programmers? =)
I already compromised by adding "today" and "almost all" as qualifiers.
I actually really like mathematical notation. Once you get used to it, the terseness can make things a lot clearer than natural language.
(If you've ever tried to read old mathematical articles / books that are a few centuries old, you'll understand the power of mathematical notation. Check out "God Created The Integers" or "On the Shoulders Of Giants" by Stephen Hawking if you're curious: these are two books that provide excerpts of highly influential works from earlier mathematicians and physicists, respectively)
Dr. Vesley blew them all away. There was notation, but the real clincher was his absolute clarity of communication. He covered a lot of material, but the pace never felt rushed. In fact, it was so calm and so clear it was refreshing, more like meditation by a babbling brook. I wasn't the only one to feel that way -- the whole class seemed to have a similar experience.
Related to the story in TFA, a friend of mine with a towering math background said a few years ago, "I remember when math was easy -- back when I had time." Math that we've learned and mastered is "easy", but new areas of math can require a LOT of mental energy to gain traction in.
see this video, starting at 8mins: http://www.infoq.com/presentations/Expression-of-Ideas
But following up on your comment, the difficulty of math is that it is build on foundations. If you miss something because you were distracted, that hole is going to be an impediment over and over and it will create more holes until it is very difficult to make progress.
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.
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 run into this a lot when people talk to me about programming and I get something faster than they do. I've spent a lot of time reading book after book, listening to podcasts, learning new languages, and studying new concepts, so it can be really easy for me to fit new information or ideas into some context and get them. I don't think that I'm necessarily smarter for it. I think I've just had a passion for it, so I never get tired of reading the articles and absorbing the material.
Now I'm personally running into the place where there are a lot of things I've wanted to learn for awhile, but my weak background in math hinders me (machine learning, more advanced algorithm analysis, signal processing, machine vision, etc).
I'm working my way through a calc book. I don't think I could have done it in your position, either though. I'd have psyched myself out. I've got to learn it on my on, with my own rhythm.
Kind of rambly, but you should find something related to what you like and maybe jump back in. Just find a learning mechanism that's suited to your background :)
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.
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.
"One thing that worked well for my wife was a series of games from Nintendo called Fire Emblem."
So are you trying to make a scientific claim? This is starting to seem like an anecdote supporting brain training games.
What happened with his wife was that she became motivated to learn number crunching. The game provided or increased a motivation to become comfortable working with numbers. That's all that happened.
Brain training has nothing to do with learning math. It just uses arithmetic the same way cross training uses a mountain trail.
And, you know, one would have to show that there is some "mathematical thought" that can be trained to begin with. I'm not entirely sure there is.
I don't disagree that "practice makes perfect". I disagree with the statement "practice in fire emblem makes perfect in math (not number crunching)".
People want to share stories. I get it. But if we're not being rigorous about it, then we're just fooling ourselves. And then the conversation devolves into a circlejerk where everyone thinks they're brilliant.
Yes, there is more to math than number-crunching, but to claim that number-crunching is not math is to forget the very roots of the discipline. This is where it begins, and it's where people who have been out of practice for a while return. Give it some respect.
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.
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.
You don't jump right into advanced topics, most streams give you an intro sequence first assuming you know little to nothing about the topic.
That said coursework is like < 50% of the value of an MBA, it really isn't the point of the degree.
Plenty of smart but somewhat wayward humanities/soc sci majors end up graduating into a random office job, find out that surprise! they perform well and enjoy business, and proceed to do an MBA.
Disclaimer: I'm not an MBA, this is second-hand from MBA'd friends and acquaintances.
agree about coursework not being the point, networking is; but coursework to a point is a factor for many things: career changing, or in my case, burrowing into math and finance so deeply you can hold your own with the best and brightest in the world. (for example you learn when to recognize when you are not the smartest person in the room, so it's time to shut up and let other people teach you)
1) spent 2 years in an MFA program in Stage management at Rutgers. I have dozens of blog posts in me about what theatre teaches you that is directly applicable to business especially the collaborative nature of software development. 2) I burned out. I saw how Tony award winning lighting designers lived and realized that my desire to make art wasn't greater than my desire for health care, vacation and a nice place to live. 3) I took a year off to be an admin assistant. this was at the dawn of computers at every desk. Because I'd built my first computer at the age of 10, and taught myself to program in basic in junior high, I was very comfortable with computers and troubleshooting. I quickly morphed into the office computer guru. installed network cards, taught the CEO of the mergers and acquisitions firm how to use a mouse (!), etc. As a fun aside one of my tasks was to print out my boss'e mail for him. :P 4) Fortunately this same boss took a shine to me (in a not creepy way) and taught me a lot about finance, balance sheets, and how deals got done. He said to me one day, you're smart enough to be on the other side of the desk, go do something with your life. 5) I took a job working for a company that sold equipment to the entertainment industry, turned around my first division of the company at the age of 25 (just as the internet rose), then took over a sales territory and tripled it. (that, along with my sales experience got me fantastic references) 6) About this time I got encouragement to go to business school since it was obvious I had a knack for it. I got good advice to not bother unless it was one of the best. 7) I took 3 classes to prove my status and A+ed all them: microeconomics, business math, and accounting. University of Connecticut night school, while working full time. 8) I studied long and hard for the GMAT and got a great score.(again I was just before the online practice questions. I used to sneak the test booklets out of kaplan, and photocopy them so I could work problem sets on the plane.)I scored high enough on the GMAT that when I graduated in the horrific post 9/11 economy I got a job at Kaplan teaching gmat classes here in SF. 9) I wrote good essays about the future of Tech and its impact on the entertainment industry--which have all since come to pass (i.e. content proliferation, the rising value of top talent). 10) Accepted to Cornell Johnson School at the age of 30 (top 10 school). 11) if I May humblebrag, in spite of the fact that I majored in finance so that I could improve my math skills, which cost me GPA, I still finished in the top third of my class.
Was luck a factor? for sure. But so was my persistence, drive, and commitment. Going through this process taught me a lot about those things. About never giving up. About working hard. I came from a dirt poor family and the internet didn't exist, so there was no one to really mentor me early on and I had to fix the mistakes of doubt and lack of self confidence and ignorance I'd made as a young woman.
People ask me how I got the courage to start a company. After what I just described, after pulling myself out of the hole of poverty and debt, this is a cakewalk.
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.
Approaching it without other academic pressures and plenty of confidence from being successful in other areas also helps. Plus the article gave no hint that he was taking a full schedule. Personally, I would likely get bad math grades if I had to go back and study math and other things.
Perhaps for these students that "didn't get" or were "bad at" math, maybe they just need to be given some time to only focus on math to raise their confidence and abilities. I had similar results as the author when all I was doing was working and doing some math during the day. I also barely graduated high school.
I'm hoping to move on to statistics, which I slept through because I was an immature shit-head my sophomore year.
Then a review multivariable and diff-eq, and the end goal is to do the full Stanford Machine Learning class, taught by Andrew Ng [2]. I also really want to tackle "Underactuated Robotics", which is another MIT OCW course.[3]
[1] http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebr...
[2] http://see.stanford.edu/see/courseinfo.aspx?coll=348ca38a-3a...
[3] http://ocw.mit.edu/courses/electrical-engineering-and-comput...
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.
-- Albert Einstein
Source: http://en.wikiquote.org/wiki/Albert_Einstein
The above quote seems to be the genesis of the folk trope, "Never commit to memory that which can be looked up in a book." To me, what you describe above is why people think they are "bad at math," because they cannot commit to memory long equations with foreign-looking symbols. Even the function sheets provided during some math exams are needlessly obtuse, for the sake of proving whether or not somebody can remember exactly what the different constants or inputs into the functions are. It would be like encountering a function in somebody's code that had cryptically-named parameters without comments or documentation.
-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.
The ease of doing it in class or breezing through homework assignments was backed by hours a night of looking at high level concepts and talking about math with people more advanced than me, including teachers during lunch breaks and after school - a habit I'd had since late elementary school.
At the end of the day, they weren't "bad at math", I just put a lot more effort in to my practice - and it showed on game day, as it were.
Just like the guy in my school who was "gifted" at basketball spent hours a night practicing since early elementary school.
edit: I just realized you said high school.
I picked high school though because most of the comparisons about being "bad at math" start around then. The reality is that I just put in more work than many people and was a class or two ahead of the average student up through the start of university, where I went on to study math (and was at a similar starting point with most other math majors at my particular school).
At the end of the day, I'm just an average fish in the math pond, but it's easy to think you're not as good if you don't see the work backing my talent in high school and base your comparisons for life on that (or even the first year of undergrad).
In college, I had to put in a lot more work, but the math still came somewhat naturally to me. I'm pretty sure I still had to put in less work than some of the other college students to reach the same level of understanding, though.
I neither put in as much work, nor reached the same level of understanding, as the 'gifted' college students, though.
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
The Calculus One videos are available at http://youtube.com/kisonecat and the new Calculus Two: Sequences and Series videos will be posted soon.
I don't agree with the author's claim "The difference between good at math and bad at math is hard work."
I have a M. Sc in math, but never worked hard at it. I went to the classes, paid attention to what was told and visited most of the exercise classes, but, except for the first few weeks, never prepared for either of them. That left plenty of time to spend on other things, such as sport and reading the art of computer programming, Compute! Magazine, Byte, etc. (partly because of that, I am not that good at math)
IMO, it is the same with about everything else. For example, there are people who just throw a baseball well, and there are those who can learn to throw a decent one through hard work.
Two reasons: 1. The desire to create something or solve a real problem
2. The interpreter
I disagree. Math ability is highly genetic: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2913421
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]
It's one of my favorite MOOCs of all time, a fantastic intro to mathematical thinking.
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...
"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.
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.
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.
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.
[1] http://www.hup.harvard.edu/catalog.php?isbn=9780674057555
* 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)
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.
I think this article is targeting a much larger audience than the small percentage of the population that has genuine problems with math.
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
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?
I basically skipped (invented) the modelling step. In practice you would observe the download size f as a function of time t, and then ask yourself "which function f(t) correctly describes what I see." I wanted to describe a download that gets faster and faster with time, so I picked a quadratic function f(t)=At^2 as a template, then chose A=0.002 so the function matches the problem statement.
Suppose instead I wanted the download rate to be uniform (constant download speed), then the function which describes the file size would be of the form f(t) = mt+b. More specifically, the function would be f(t)=720/600t+0=1.2t. (You can check that f(0)=0 and f(600)=720, as in the problem setting).
In general, once you become familiar with the main function families f(t)=mx+b, f(t)=ax^2+bx+c, f(t)=Aexp(kx), f(t)=ln(x), f(t)=Asin(kx-ϕ), etc, you will be able to describe real-world phenomena by using one of these equations. This ability to "model" the real world through mathematical equations is one of the super-powers that comes with math knowledge, but my download example doesn't quite manage to communicate that. I will have to rework it...
For a slightly better example of "modelling" using equations, check out page 26 in the PDF preview: http://cnd.mcgill.ca/~ivan/miniref/miniref_v4_preview.pdf#pa...
My son is 3, so at the moment we're just counting everything and making little groups to add them together.
* 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...
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.
A concrete way of demonstrating this would be to point out that several same size groups of objects have something in common.
It is alarming (for our country) that this needs to be offered at the university level, and even more alarming that most students are (according to the OP) failing it.
I recently was able to apply a few math techniques in a project to solve a problem elegantly[1][2]. It is useful in my career as a software developer.
[1] https://github.com/SGrondin/map-reporting/blob/master/src/ge... [2] https://github.com/SGrondin/map-reporting
________
As with learning to program or play piano/violin, you can:
- take private lessons
- take group classes
- join some kind of study group or peer teaching. I've seen meetups, google groups for that, and here's a subreddit for Bishop's ML text, which takes a decnet amount of concentration: http://www.reddit.com/r/mlstudy
It was kind of by accident. I knew I needed to memorize the formulas, but I couldn't find a situation where I could sit down and do them without becoming anxious and distracting myself.
On a run, there is no alternative. The eventual runner's high helped me memorize things even more, and I found I'd associate the mental high that comes with working on a math problem for a long time with a runner's high.
I'm super exited to share this wisdom with my kids. I wish I knew this secret when I was in high school.
I have a feeling that in a century people will look back at formal education today and view as barbaric and ineffective.
Now i find really enjoying differential equations.
Journalism has many problems, but make no mistake, this certainly is the biggest. Journalism hasn't been a respectable profession for at least a hundred years. Sure, there are examples of good journalism, but sadly they are the exception. The overwhelming majority of journalism today is either meaningless pop-culture drivel, or hyper-politicized, sensationalist propaganda.
Its not even the journalism industry's fault. Its just a sad fact that unbiased, factual articles are incredibly hard to write, and nearly impossible to make interesting to the average person.
The National Press Club, founded in 1908, is seen as the beginning of professionalism in journalism. They were built upon promulgating a set of ethics and standards for journalism, including a focus on the facts, accuracy, and bias-control.
Before that there was very little focus on what we consider bias-control. In the early years of the United States up through the 1890s, newspapers were most often owned by political parties and were explicitly for slinging innuendo, suggestion, rumor, and scandal at the political opponents of that party. So if you think it's a new problem or that it's bad now, history has shown that it has been worse.
I think the biggest problem in my approach to learning was my refusal to practice. From other classes, I was used to reading the textbook and understanding immediately. That meant to study for math exams, I would read the chapters, try to learn principles, and memorize some formulas.
I later realized that I would never get any better without doing practice problems - and lots of them. The best advice I've gotten is to start with the problem, and then learn.
I just wanted to share to you guys that this morning I really feel bad on my way to work (I'm an iOS developer). I told myself if only I'm really good at math maybe I could have use a lot more algorithms for my apps that I was making. All I can think of until I got to the office was math math math math. And I admit I was envious of other people who are really good at math.
Now I'm not envious anymore! Thank you Mr. Matt Waite!
There is: http://en.wikipedia.org/wiki/Introduction_to_Algorithms which is pretty readable.
And there are a lot of MOOCs:
https://www.coursera.org/course/algs4partI
https://www.coursera.org/course/algs4partII
https://www.coursera.org/course/algo
https://www.coursera.org/course/algo2
If you work yourself up from there and solve one problem after another you'll be quite good at these things in a few months time.
John von Neumann
The only way to get used to things is to work with them regularly!
C is the average grade. B is above average. A is excellent. You think he's not "good" because he's above average while learning calculus at 37?
When you (or I) hear "I'm not good at math", you presumably take that to mean "it doesn't come easily, and I certainly wouldn't want to study it full time". But I think that when a lot of people hear the words "good at math", they take it as a statement about just basic competence. People saying "I'm not good at math" all too often mean "numbers scare me" or "I would never be capable of understanding that at all". And I think that's what this article is responding to.
Alternatively -- and this is much more common -- self-coach. But then you're in the hard work territory that the author speaks of, since you'll be banging your way through a number of walls headfirst.
Oh no, you're THAT person.
I was 'good at math' but 'bad at spelling' and that was normal and natural. Except I found out it is just as much a lie as the opposite thought the author had.
Even in 2nd and 3rd grade (American elementary school system), any time we had a "spelling test" I got 100%. The first time I looked at any word, I remembered how to spell it perfectly. To this day I still have the ability. I can read a 5,000 page essay and immediately notice every typo and grammar misusage; and when writing anything myself, the only reason I ever misspell a word is due to an accidental typo, not a result of forgetting the correct spelling.
I saw many classmates struggle with trying to remember the correct spelling of various words, and I never had to do any of that.
However, in contrast, I've always had to put in much, much more effort than some of my peers to properly grasp math concepts. Some people could basically be introduced to a concept once. I wonder if this may be due to dyscalculia on my part, becuase the math I struggle with the most is basic arithmetic and algebra; I'm ok with the more advanced concepts, generally, except when I have to do the real number crunching.