Math uses made-up rules to create models and relationships. When learning, I ask:

- What
**relationship**does this model represent? - What real-world items
**share this relationship**? - Does that relationship
**make sense to me**?

They're simple questions, but they help me understand new topics. If you liked my math posts, this article covers my approach to this oft-maligned subject. Many people have left insightful comments about their struggles with math and resources that helped them.

## Math Education

Textbooks **rarely** focus on understanding; it's mostly solving problems with "plug and chug" formulas. It saddens me that beautiful ideas get such a rote treatment:

**The Pythagorean Theorem is not just about triangles**. It is about the relationship between similar shapes, the distance between any set of numbers, and much more.**e is not just a number**. It is about the fundamental relationships between all growth rates.**The natural log is not just an inverse function**. It is about the amount of time things need to grow.

Elegant, "a ha!" insights should be our focus, but we leave that for students to randomly stumble upon themselves. I hit an "a ha" moment after a hellish cram session in college; since then, I've wanted to find and share those epiphanies to spare others the same pain.

But it works both ways -- I want you to share insights with me, too. There's more understanding, less pain, and everyone wins.

## Math Evolves Over Time

I consider math as a way of thinking, and it's important to see **how** that thinking developed rather than only showing the result. Let's try an example.

Imagine you're a caveman doing math. One of the first problems will be **how to count things**. Several systems have developed over time:

No system is right, and each has advantages:

**Unary system:**Draw lines in the sand -- as simple as it gets. Great for keeping score in games; you can add to a number without erasing and rewriting.**Roman Numerals:**More advanced unary, with shortcuts for large numbers.**Decimals**: Huge realization that numbers can use a "positional" system with place and zero.**Binary:**Simplest positional system (two digits, on vs off) so it's great for mechanical devices.**Scientific Notation:**Extremely compact, can easily gauge a number's size and precision (1E3 vs 1.000E3).

Think we're done? No way. In 1000 years we'll have a system that makes decimal numbers look as quaint as Roman Numerals (*"By George, how did they manage with such clumsy tools?"*).

## Negative Numbers Aren't That Real

Let's think about numbers a bit more. The example above shows **our number system is one of many ways to solve the "counting" problem.**

The Romans would consider zero and fractions strange, but it doesn't mean "nothingness" and "part to whole" aren't useful concepts. But see how each system incorporated new ideas.

Fractions (1/3), decimals (.234), and complex numbers (3 + 4i) are ways to express new relationships. They may not make sense right now, just like zero didn't "make sense" to the Romans. We need new real-world relationships (like debt) for them to click.

Even then, negative numbers may not exist in the way we think, as you convince me here:

You: Negative numbers are a great idea, but don't inherently exist. It's a label we apply to a concept.

Me: Sure they do.

You: Ok, show me -3 cows.

Me: Well, um... assume you're a farmer, and you lost 3 cows.

You: Ok, you have zero cows.

Me: No, I mean, you gave 3 cows to a friend.

You: Ok, he has 3 cows and you have zero.

Me: No, I mean, he's going to give them back someday. He owes you.

You: Ah. So the actual number I have (-3 or 0) depends on whether I think he'll pay me back. I didn't realize my opinion changed how counting worked. In my world, I had zero the whole time.

Me: Sigh. It's not like that. When he gives you the cows back, you go from -3 to 3.

You: Ok, so he returns 3 cows and we jump 6, from -3 to 3? Any other new arithmetic I should be aware of? What does

sqrt(-17)cows look like?Me: Get out.

Negative numbers can **express a relationship:**

- Positive numbers represent a surplus of cows
- Zero represents no cows
- Negative numbers represent a deficit of cows that are assumed to be paid back

But the negative number "isn't really there" -- there's only the relationship they represent (a surplus/deficit of cows). We've created a "negative number" model to help with bookkeeping, even though you can't hold -3 cows in your hand. (I purposefully used a different interpretation of what "negative" means: it's a different counting system, just like Roman numerals and decimals are different counting systems.)

By the way, negative numbers weren't accepted by many people, including Western mathematicians, until the 1700s. The idea of a negative was considered "absurd". Negative numbers **do** seem strange unless you can see how they represent complex real-world relationships, like debt.

## Why All the Philosophy?

I realized that my **mindset is key to learning. **It helped me arrive at deep insights, specifically:

**Factual knowledge is not understanding.**Knowing "hammers drive nails" is not the same as the insight that any hard object (a rock, a wrench) can drive a nail.**Keep an open mind.**Develop your intuition by allowing yourself to be a beginner again.

A university professor went to visit a famous Zen master. While the master quietly served tea, the professor talked about Zen. The master poured the visitor's cup to the brim, and then kept pouring. The professor watched the overflowing cup until he could no longer restrain himself. "It's overfull! No more will go in!" the professor blurted. "You are like this cup," the master replied, "How can I show you Zen unless you first empty your cup."

**Be creative.**Look for strange relationships. Use diagrams. Use humor. Use analogies. Use mnemonics. Use anything that makes the ideas more vivid. Analogies aren't perfect but help when struggling with the general idea.**Realize you can learn.**We expect kids to learn algebra, trigonometry and calculus that would astound the ancient Greeks. And we should: we're capable of learning so much, if explained correctly. Don't stop until it makes sense, or that mathematical gap will haunt you. Mental toughness is critical -- we often give up too easily.

## So What's the Point?

I want to share what I've discovered, hoping it helps you learn math:

- Math creates
**models**that have certain**relationships** - We try to find
**real-world phenomena**that have the same relationship - Our models are
**always improving**. A new model may come along that better explains that relationship (roman numerals to decimal system).

Sure, some models *appear* to have no use: **"What good are imaginary numbers?"**, many students ask. It's a valid question, with an intuitive answer.

The use of imaginary numbers is limited by our imagination and understanding -- just like negative numbers are "useless" unless you have the idea of debt, imaginary numbers can be confusing because we don't truly understand the relationship they represent.

**Math provides models; understand their relationships and apply them to real-world objects.**

Developing intuition makes learning fun -- even accounting isn't bad when you understand the problems it solves. I want to cover complex numbers, calculus and other elusive topics by focusing on relationships, not proofs and mechanics.

But this is my experience -- how do you learn best?

## Other Posts In This Series

- Developing Your Intuition For Math
- Why Do We Learn Math?
- How to Develop a Mindset for Math
- Learning math? Think like a cartoonist.
- Math As Language: Understanding the Equals Sign
- Avoiding The Adjective Fallacy
- Finding Unity in the Math Wars
- Brevity Is Beautiful
- Learn Difficult Concepts with the ADEPT Method
- Intuition, Details and the Bow/Arrow Metaphor
- Learning To Learn: Intuition Isn't Optional
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Honest and Realistic Guides for Learning

## Leave a Reply

159 Comments on "How to Develop a Mindset for Math"

Thanks for the comment, though I think this works for adults too :). I’ve seen far too many people approach math from the plug-and-chug angle, I want to encourage a more intuitive approach, especially when teaching kids.

This post is a lead-in to some of the more advanced stuff I’ll be covering (complex numbers, calculus of e) where intuition is usually left in the dust.

Kalid: actually even

positivenumbers are not that real. You see threecows, threelines, but notthreeas a concept πWhat I want to say is that positive integers are so deeply inside us that we have forgotten that they are a creation of our minds too! (A Platonist may freely change this with “an idea residing in the Hyperuranus”)

I agree with you that learning math through models would be better than the usual approach, but I also believe that you have to find the “right” model not only for the observed data, but also for the person who is learning. I would not talk however about “imperfect and incomplete” models; it gives an impression of something wrong going on. Wouldn’t it be better if you say “we choose what we are interested in, and what we may discard; then we find a way to deal with the former in a way useful for us”. It’s the same thing, but it sounds different!

Thank you for your explanation.

I’m 50 years old and it’s been almost 30 years since someone has helped me so well with getting math. I have hope again. Thank you, Kalid.

I didn’t find it trivial at all. It’s a philosophical foundation for future exploration. I think all endeavors have one though most are unstated. By stating the thing you’re able to review your work against it; when you deviate, do you change your work or your foundation?

An unstated philosophy denies self-reflection.

You can choose which is better.

By the way, I presented a very convincing argument about negative numbers, didn’t you? I surprise yourself sometimes.

“Maybe you should have a separate blog for elementary school students”

I disagree ENTIRELY with your post and the assumption.

I found the blog great, because HE REASONS.

You know what is needed? To teach people. Whether this is in math, or in school, or on Linux …

How can people learn AND understand if they do not grasp something?

This blog is in fact one of the best I have read lately (coming close to “how to do startups from paul graham” lately… reddit isnt that bad after all)

Thanks for all the great math posts, this is what i’ve been looking for, writing to help me understand the bigger picture not just, as you say, plug and chug formulas and rules.

Great stuff. Looking forward to your next post.

Hi

Really looking forward to your next post about Imaginary numbers.

> Factual knowledge is not understanding. Knowing βhammers drive nailsβ is not the same as the insight that any hard object (a rock, a wrench) can drive a nail.

This is a point that cannot be stressed enough. We must always be vigilant against believing that we know things which we merely know the names of. There’s a great blog at overcomingbias.com that frequently drives this point home in many interesting ways.

I think articles like that teach concepts are important. There’s too much of the “plug and chug” in all fields nowadays — even IT. The number of HowTos that simply list each step drastically outnumber the amount of works that attempt to explain how things work. And it’s a wonder why most people nowadays can’t troubleshoot a simple PC or Server when they don’t have the steps listed out for them.

I used to love mathematics and have started to refresh myself on it in my spare time. I picked up a few simple books on Algebra and was totally discouraged by their methods of teaching — simply use whatever shortcut possible to solve an equation. 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.

You might be interested in the book Where Mathematics Comes From, on the embodied basis of mathematical understanding.

I think it would be useful to create an animated, controllable (directly manipulable) visual model to represent different mathematical transformations and relationships. We all imagine a number line for example. You can use bars to represent numbers. I tend to think of them flipping over to the right when multiplying (by a positive number), for example.

See also the virtual manipulatives site here:

http://nlvm.usu.edu/en/nav/vlibrary.html

I don’t know about anyone else, but when I was in elementary school (late 80s early 90s) in Austin, TX we had this sort of idea being pushed. It was called “Math is Real” or something like that, and they made the teachers teach math using all real world examples. I don’t think anyone ever said the word “Relationship” at that point, and it didn’t continue into harder math like geometry, algebra, calculus… etc. But personally, as a visual imaginer I do like to learn about relationships in math to truly understand them. Something about looking at a graph of a real phenomenon and then seeing an equation that approximates the data gives me an intuition that simply memorizing equations does not.

I always learned better when I had some application for the math I learned. Basic math, like algebra, is so extremely useful. I love to learn how math can explain the behavior of real world things. One of my teachers went on a rant one day about how all numbers are imaginary, none of them really exist, they are just concepts. I love how the human imagination can be so accurate and useful in that way. I think that in the future, counting systems might be far more complex than our current ones. Those Eureka moments are what makes math so interesting to me. I had one when playing with prime numbers, but I won’t explain it here, it’s too complicated. I agree that rote memorization of math is horrible, because I forget things learned by rote so quickly, it hurts me in the long run.

Love the concept, and I think you’ve hit the nail on the head about why our schools are failing to teach math to our kids.

One correction, though. If I have -3 cows, it does not mean someone owes me three cows. Rather, it means that not only do I have zero cows, but I owe 3 cows to someone else.

@mau: Excellent points! Yes, I agree regular, positive numbers aren’t real either — though the story wouldn’t work as well as people generally accepted them (unlike negatives which have a struggle). And a rephrasing might help — “incorrect” isn’t quite right, it’s more the model isn’t the most elegant or compact way to represent the problem. Thanks for the comment!

@Larry: I’m so happy you found it useful! I think anything can be understood by anyone, so I hope you enjoy the future posts.

@Bob: Thanks for the clarifying thoughts. Yes, I wanted to get my approach to learning out on paper — and the nice thing is it helped clarify it for me as well π

@She: Thanks for the support, I’ve enjoyed writing this blog. Yes, everyone starts at different levels, and even the “experts” have something to learn.

@Jonathan: Appreciate that — yes, I detest plug and chug too.

@Gilbert, wow: Thanks!

@Bill: Thanks for dropping by — I’ll have to check that site out. Rote memorization and “labeling things” is the bane of true learning.

@Joe: I totally agree. I did an article on version control, and was shocked by how many tutorials just throw command-line arguments at you instead of explaining the high-level concepts.

Especially in IT — facts become obsolete, understanding stays current. I’d love to check out those books you found if they take a better approach to learning.

@Tim: Thanks for the info. I’m happy that your school had that approach, I wish more did! Unfortunately it was fairly rare in my education. I’m a visual learner too, which is why I enjoy creating diagrams for things — it’s just another way to look at it.

[…] test 11/27/2007 How to Develop a Mindset for Math | BetterExplained […]

Hi Doug, thanks for the info! I like that idea, as we have so many pre-conceived notions about what a number “is” — there’s many ways to look at it.

I’m on board; the above is not a triviality.

I’ve had several a-ha moments, one in chemistry and two in math come to mind.

First math a-ha moment: Coming up with what was previously a bizarre thing for me, the quadratic equation. This while studying algebra (Galois theory, to be precise). This came after I completed the calculus and diff-eq series but without ever having a real feeeling for it. I really had to work at them. But then I finally understood WHY all those equations and methods worked. At last! I understand the model! Much of what I had previously struggled with, all that calculus and stuff, suddenly became very much clearer.

Second moment (Hey, YOU made the pun necessary!) came in my Mathematical Logic class. The a-ha? Negative numbers, imaginary numbers, infinity, all abstractions. Some parts do not necessarily have “real world” instantiations. Maybe it would be better to say “exist without verbally anthropomorphic counterparts.” What is infinity? It’s a symbol I say. A symbol that works. Yes, but what does it MEAN, you ask. It doesn’t MEAN anything, I reply, other than the role it plays in the formal system that is mathematics. It’s ony a symbol. I suppose one could say I finally understood the model of mathematics.

Thinking back on my entire formal education, I believe it’s ALWAYS been the case that true understanding – in ANY field of study; math and chemistry yes, but also social sciences, literary theory, you name it – comes only after understanding the respective underlying model.

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!

You probably find “Does Mathematics Reflect Reality?” interesting

@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 π

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.