This reminds me of Feynman’s Six Easy Pieces. Math and physics aren’t arcane formulas and ethereal reasoning: they relate to the real world. Understanding what’s really going on behind the math is surely a key to really doing math well (and discovering that math is actually fun).
Pi is more than circumference divided by diameter. It’s a measurement of the curvature of space. Cool!
@Peelay: Thanks for the great examples! I love hearing about people’s a-ha moments, it helps remind me I’m not the only one who enjoys them. And I agree that any subject can benefit from this approach.
@Erik: Thanks for the info! I’m reading Feynman now and I love his approach - I wish I had a chance to see his lectures. That note on pi is really interesting.
I just came across your site and I really like it! I’m one of those people who somehow (sadly) managed to escape high school and college with the math of a 6th grader. Now at 28, I’m trying to learn what either wasn’t explained well or what I just didn’t get. I really like your approach and will continue reading.
Any learning can also be validated/strengthened by attempting to teach someone else the concept you think you’ve conquered.
I think it was Feynman who felt he never truly understood something if he couldn’t explain it to a fifth grader.
I’ve always been better than average at math, but struggled with higher mathematical concepts, so I thought this would be a helpful article for me to read. However I found it instead to be confusing, muddling, and rather pointless (as in missing a unifying point).
It seems that instead of attempting to explain how to develop a mindset for math, you instead cover several scarcely related mathematical concepts, leaving it up to the reader to try and figure out what the heck any of this has to do with having a ‘Mindset for Math’.
I seem to be in the vast minority amongst the commenters though, so feel free to disregard me 
Thank you for an interesting read. Being an engineering major (and thus taking many math classes) I have thought a lot about what math “is” and how to learn it best. Below I will share my current view on the matter, which I perhaps will adjust after re-reading and thinking through your post. Please feel free to comment (or ignore!).
I think of math as a thousand little “tricks” you can use to solve a problem - 1 + 1 = 2 is one trick, the Pythagorean theorem is another trick, binomial coefficients yet another, and in order to truly master math, you need to have seen most of those tricks, for example by reading about them in a math textbook or having someone (ie a teacher) teach you them.
Solving an unsolved problem - even a really hard one - just involves finding a new trick (cos^2(x) + sin^2(x) = 1, for example), and that process I view as pretty iterative - throw a thousand ideas at the problem and eventually something works (which is why the really hard problems take so long to solve - they require as of yet unseen tricks, and these tricks are, I think, discovered mostly by accident - then again, I am not a mathematician, so I may be wrong about this) - this “something” becomes yet another trick which can be used again and again.
Please note that my view need not be contrary to your view - sure, it may seem like plug’n’chug, but to “learn” a trick can (and should!) also involve actually understanding why it works.
The cow example is wrong. -3 cows is having none, but owing three to someone, not someone owes you.
@Alicow: Glad you enjoyed it! Don’t worry, I admire your courage in coming back to learn. Most people give up on math (science, history, etc.) and never return. Good luck!
@Brian: I completely agree, part of the reason I write for this site :). Teaching forces you to really simplify your thinking, and be prepared for “simple” questions that really make you wonder.
@Me: No problem, not every article will gel with everyone :). The point was that I’ve understood math better by considering it as a series of models with relationships, rather than mechanical calculations. By focusing on relationships I get a deeper, a-ha understanding.
The counting example shows how our models can evolve over time – no single one is perfect.
@Curly: I think our views are compatible. Sometimes the tricks lead to new insights, sometimes the insights lead to new tricks. I’m not sure which one comes first (or if the order changes sometimes).
@Jeremy: I’ve updated the article to be a bit more clear. The point was to show that negative numbers aren’t “real” in the way we normally think. We humans have complex relationships (borrowing and debt) and use negative numbers to represent them. But there’s really no such thing as a negative cow – it’s all in our mind.
@anh: Thanks for the link, I’ll check it out.
@Eddie: I’ll probably have to update the article to be more clear on this point. I chose a non-standard interpretation to be “clever” but I think it’s coming back to bite me :).
another good article kalid. i like your approach and i look forward to seeing some more articles that teach the meaning behind the formula instead of expecting people to learn through rote memorization.
Thanks Jeff – I’ve been having some brain-bending thoughts about imaginary numbers that I’m excited to get down onto “paper”.
Interesting post! I used to tutor a middle school girl who struggled even with elementary math (like negative numbers). I think your article articulated the reasoning behind math rather well–as Jeff said, nice approach, and keep the math posts comin’!
@ Joe #11:
“It took a few days, but I finally tracked down some good books that explain the theory behind the equations and it’s been a much more rewarding experience.”
If you happen to read this, can you post the titles for a couple of these books? [If others have recommendations, those are welcome, too.]
@marie: Thanks for the encouragement, I’ll try to keep cranking them out 
@Z: Joe was kind enough to send me a list from this post:
Joe wrote: “Anyway, I’ve bought almost the whole list. I’ve read Polya’s book which was
very dry but enlightening. Now I’m reading Lang’s Basic Mathematics and
Gelfand’s Algebra which are amazing. Lang’s book is so wonderful. I don’t know
why it’s not used more in school.”
Thanks for the help Joe! I look forward to checking these out, I really enjoy books with a focus on understanding.
Nice article Khalid. I think a few early insights make a subject interesting. But to think of Math as models and relationships takes a bit of maturity. I don’t think I would have thought of it when I was learning it in school.
Another aspect that may improve math based thinking is understanding how the concepts are applied. We learn a lot of Math without ever understanding the application.
The Why of Math is as important as how. And conceptualizing this as a set of inter-related models is a bit meta, but certainly enjoyable way to retain the essence in your mind.
Elevation is a better example for negative numbers. Sea level is zero. Denver would have a positive elevation. Death Valley a negative elevation.
[…] How to Develop a Mindset for Math | BetterExplained - An interesting approach to understanding how math works and why it works. […]
[…] Khalid has a nice post on How to Develop the Math Mindset. Math uses made-up rules to create models and derive relationships. When learning, I ask: […]
Excellent post, looking forward to reading more… I totally agree with your mindset in regard to maths, ie used for modelling and showing relationships.
@auferstehung - excellent example, I also like the use of time (GMT) offsets as well.
A great book i’m reading now is titled ‘Mathematics and the Physical World’ by Kline avaliable at Amazon.