Math uses made-up rules to create models and derive 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’s a selfish goal too — I want to convince you to share your 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 -3 means “somebody owes me?” and forces them to repay you? That’s pretty neat how a number can change behavior — I should use that trick on the kid who borrowed my xbox.
Me: Sigh. It’s not like that. When he gives you the cows back, you go from -3 to 3.
You: Cool, he gives you 3 cows and you jump 6, from -3 to 3? Amazing arithmetic you’ve got there. Care to show me sqrt(-17) cows?
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 give up 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 cover complex numbers, calculus and other elusive topics by focusing on relationships, not proofs and mechanics.

When you think about future, do you see where you are going to be? Are you certain that you will have what you want in a certain period of time? Do you have goals? Is your life the way you want it to be? If you go back to where you were three or five years ago, would you be happy to see the status you are having right now? Do you believe that if you lead your life like this, you will wind up not much different from where you are now? Will you happy? These many questions relates to what we call “time perspective”.

Do you know that successful people have longer time perspective than the ones who are not so successful? Successful people think of the impact of the actions to their lives in three or five years late. They ask themselves if the situation helps or have some impact to their lives long term. On the contrary, unsuccessful people think about their short term gratification. They hardly have plan for future and their care less on their time planning if they have one. We can easily notice if we are a good time planner by seeing if we are often impulse buyers. If we buy things without plan or just by them because we accidentally bump into it. There is a tendency that we are the short time perspective people. We need to hurriedly correct that.Why does the long term time perspective matter? Because it relates to how we plan our lives for the good future not only just for being happy for a short while and then bear the negative consequences. Whether a person is successful is measured long term. Some even says that everything in life is measured by what you have on the last day of your life. I think it is logical and well spoken.

For some investment like buying a house, we understand that people need to prepare well in advance before we go into it. We know that we need to prepare financially since the almost all of the house payment is for long term. We need to know the location and understand how we and our families can commute within the area. We need to know how the utilities are provided. Will there be any new construction nearby? How about the pollution problem and people nearby? We need to think a lot about these before we buy a house right? Why is that? Because buying a house is an expensive process for most people and it has big impact to the lives of us.

Planning a life is like buying a house. Unfortunately, most people take their lives for grant and ignore the process. They are not so careful on leading their lives and a lot of them end up in trouble in the latter years of their lives. Tragically, a lot realize this when it is too late for them to do anything. Don’t let this happen to you.

Start to honor your time perspective. See things in the longer term view. Ask yourself what will happen to you five years from now if you keep doing what you are doing. Will you be better off or worse off? Once you create a habit of asking this question, you will amazingly improve your time utilization which in the end will help you in the future.

Although there are a lot of concepts teaching that you need to focus and do things for the moment of now, there is the other angle of the time usage for the future results. This article talks about long into long term perspective and how to use your time effectively.