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

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

Usually great stuff here but this one was trivial. Maybe you should have a separate blog for elementary school students 😛

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 conceptWhat 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 “hammers drive nails” 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

@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.

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

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:

http://science.slashdot.org/comments.pl?sid=327729&cid=20978989

———-

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) [amazon.com] by George Polya

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes [amazon.com] by Daniel Solow

Mathematics 6 [perpendicularpress.com] by Enn R. Nurk and Aksel E. Telgmaa translated and adapted by Will Harte

Algebra [amazon.com] by I.M. Gelfand, Alexander Shen

The Method of Coordinates [amazon.com] by I.M. Gelfand, E.G. Glagoleva, A.A. Kirilov

Functions and Graphs [amazon.com] by I. M. Gelfand, E. G. Glagoleva, A. A. Kirillov

Trigonometry [amazon.com] by I.M. Gelfand, Mark Saul

Basic Mathematics [amazon.com] by Serge Lang

Kiselev’s Geometry / Book I. Planimetry [amazon.com] by A. P. Kiselev (Author), Adapted from Russian by Alexander Givental (Editor)

Euclidean Geometry: A first course [solomonovich.com] by Mark Solomonovich

Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra [amazon.com] by Tom M. Apostol

Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications [amazon.com] by Tom M. Apostol

———–

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.

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.

There are several intermediate steps which connect Urnary and Roman.

http://en.wikipedia.org/wiki/Tally_marks

visually highlight precounted intervals. This creates a new symbol out of 5. This system is well suited for incremental counting of potentially large numbers – such as counting days on a prison wall.

As the numbers grow large, a similar concept can be applied to group blocks of 50 or 500. Mayan, Egyptian, and other historical solutions exist.

http://en.wikipedia.org/wiki/Maya_numerals

We’re not limited to groups of 5, that’s just ( pun) handy. When playing Cricket (Darts) you’re using a base 3 symbology

http://en.wikipedia.org/wiki/Cricket_%28darts%29

While the Babylonians used a base 60. This number is usually chosen since it divides evenly in so many ways. This allows even partitioning among 2 to 6 people.

As the numbers grow larger we need groups of groups, and a new symbol for each. Roman numerals assigned a unique symbol for 1, 5, 10, 50, 100, 500, 1000.

This system can compactly represent very large numbers, in a very small space. But making the numbers follow a more uniform progression simplifies math. By standardizing on a 1,10,100,1000,… pattern all sorts of nice mathematical regularities pop out. Suddenly a mathematical operation such as multiplication that was difficult, could instead be calculated using a 10×10 lookup table. By choosing a binary system that table becomes 2×2.

I recall sitting through 3/4th of a semester of linear algebra. While studying for a test I finally had all the pieces in front of me, having that eureka moment. I’d wondered ‘why didn’t he just say that?!’ Having since done some teaching, I can appreciate that even when you know it, it can be very hard to convey. But I think too often we do fail to mention the forest while teaching the trees.

If I had ever previously encountered the Mobius Transformation, I suspect this youtube video might have been similarly insightful

http://www.youtube.com/watch?v=JX3VmDgiFnY

Visual understanding, while not the only path, is often a key insight.

ps. loving the blog.

I have a hard time understanding why you would suggest negative numbers are not “real” (I understand the connotation). They are just as “real” a positive numbers (which are also contrivances of our imagination). I use the example of elevation for my math students as suggested by auferstehung (only I use the idea of a hole in the ground to illustrate the idea of negatives). And, to continue the analogy negative numbers are just as “real” as complex and imaginary numbers as well, with real-world applications and examples.

I think the best mindset is like you said think concretely, with concrete physical examples that you can sense and relate to. Everything in mathematics becomes abstract too soon in school mainly; when I’m doing trigonometry for real-world applications I’m not thinking what tangent really is which is a ratio of two sides but simply as a formular which I use and a button on my calculator. I think that you can’t make an mathematical concept abstract until you’ve grasped it’s physical real sense extremely well and then can progress to what if situations that aren’t present in the real world.

@Dorai: I agree that it takes a bit of time to view math as models and relationships. The why and how have a yin-yang relationship; each one feeds the other. Unfortunately, schools tend to teach the “how” and leave “why” as an exercise for the reader :). As you say, we learn a lot of math without realizing why it works and where else it can be used.

@auferstehung: Elevation is another example. I wanted to choose one where a negative number didn’t have a nice clean meaning.

@Mark: Thanks for the info! I’m glad you liked the article, I have quite the reading list now

@Ken: I agree. Negative numbers are just as “real” (or better said, just as *fake*) as any kind of number (positive, fractions, imaginary). The goal was to explain that negative numbers are a figment of our imagination, which is easier to initially grasp than “counting numbers are a figment of our imagination”. I’ll be speaking about this more in upcoming articles, I hope to make it more clear.

@Pablo: Yes, I think there is a cycle of learning “how” and learning “why”. Unfortunately, many times we only learn the “how” (i.e., tangent just becomes a button on the calculator, as you say) vs. knowing “why” it has the properties it does. Thanks for the comment.

Interesting article, I’m glad you’ve pointed out some things here and that I’m not the only one feeling as you do. Here comes a rant. I am a college student in Calculus II (for the second time) and I often wonder “Who murdered math?”. It was my favorite and strongest subject throughout most of grade school, but in college I have developed a powerful hate for it. It has become “this is how you do it so now go home and do enough problems until you memorize it, then regurgitate it onto the exam.” There’s no soul to it anymore. Some of us aren’t interested in finding the area of the surface of a cylinder and don’t see ever needing to do so. Relate it to us, make it interesting. To me, if something needs to be vigorously memorized, it’s not being presented in a proper, meaningful way. Relationships should be drawn between concepts, like the relationship between Summation and Integration. I may be wrong, but it seems there is too much of an emphasis on breadth, not enough on depth. Please continue writing articles like this one. I doubt you’ll be able to make me a lover again, but maybe, just maybe, I’ll become less adverse to the subject.

Even if it is not directly related to this article, I’d suggest also to have a look at

Mathematics: A Very Short Introductionby Timothy Gowers,http://www.amazon.com/gp/product/0192853619 .

Gowers has also a blog, http://gowers.wordpress.com/ , but he did not update it for a while.

@Me: my understsnding is that this post – at least for the time being – is more a “call to arms” than an essay. Kalid will correct me, but I have the impression that he is trying to find the best (ok, a quite good) approach to teaching math, but he does not have the Answer right now.

@cariaso: Thanks for the history, that’s some great background info. Yes, even when insights are in your brain it can be tough to get them out in writing. I try to use diagrams, analogies, dialogues, anything that can help convey the topic. Different approaches click for different people.

@John: Thanks for dropping by, you aren’t the only one who feels that way. “Who murdered math?” is a great quote, I think that phrase epitomizes what happened to many people along the way. Lots of people I know liked math as a kid but hated it later in life. I’ll keep writing, and maybe we can relight that spark :).

@Mau: Thanks for the info! I have quite a reading list now, I’m excited to start into it. Yes, this was a “call to arms” of sorts (nice description) — I want to do an assault on the insight-free math we’ve come to expect, and I’m looking for battle strategies :).

John @ 41: (Hoping you check in here again)

I understand you perfectly. I *hated* calculus and differential equations was a nightmare.

My major (CS) required, in addition to the usual calc, diff-eq and other maths, a Discrete Mathematics class. Within the first week, I was seriously hooked. Followed up with a couple abstract algebra classes and mathematical logic. After those, I enjoyed the hell out of math again. Because I understood the model – those classes are all about the ‘why’, not so much about the ‘how.’

Taking them made the all rest (relatively) easy; linear programming, coding theory, complexity theory, differential geometry, you name it, it was fun. AND, not nearly as challenging as basic calculus had been for me. So much fun in fact, I spent a litle extra time to get a second BS in math! (I will admit to struggling a bit with the Stat series though, probably because it bored me)

An intro course into discrete math could be just the ticket to resurrect math for you. Sure worked for me. I recommend looking into it.

I’m looking forward to reading more.

I am a former physics student with a form of dyslexia who “hit the wall” in university math… the point at which no amount of effort, practice or explanation could reproduce the concepts that I enjoyed so much on paper.

Like I said, I’m looking forward to more… I miss enjoying math as a concept as opposed to the terror and fustration it brought to my university years.

Hi wookie, thanks for the comment. Yes, unfortunately math lives up to the stereotype of being painful, not because it actually is, but often because of our approach.

Teachers often forget that many math “discoveries” were stumbled upon when tinkering. Newton used calculus without a formal definition of derivatives and did just fine. The constant e was discovered when tinkering with interest rates, not someone declaring “we must know the value of lim n->inf (1 + 1/n)^n”! But people forget this, or don’t know.

I felt a similar frustration and will do my best to take a different approach to learning

I think it was a wonderful article and very thought provoking too. I guess the writer is right when he says we need to be a but tougher mentally because we stop thinking very soon and give up. And definitely I agree that math should not be treated as a ‘plug and clug’ kind of a thing but rather learning the way it came about to be. I think this article should be read by more people here in my home country in Pakistan.

Thanks Mohammad! I’m glad you enjoyed the article, it was a very precarious situation for me as I was almost discouraged after enjoying math for so long.

One “Eureka!” moment I had came recently when studying matrices in school (for the second time (I hate the fact that Analysis is a repeat of advanced algebra)). I was wondering why the determinant of a matrix was ad-bc and not some other combination of the elements of the matrix. It took about 10 minutes of thinking about Cramer’s Rule before I understood exactly what the point of it was.

I happen to believe that teaching the history of mathematics along with the subject itself is probably the best approach to the subject. You really have an appreciation of why things are the way they are when you see how they came to be that way.

Seeing relationships really helps you understand math; it’s great being able to see the derivative as an “instant change” and the integral as a “cumulative change” – might make a lot of people less scared of calculus, and who’d oppose that?

Hi Zac, thanks for the comment! That note about Cramer’s rule and determinants is really interesting, I hadn’t thought about the reason for that format either.

Yes, having math & history go side-by-side really helps show *why* certain techniques developed, not just how. For me, it makes understanding much clearer and more interesting.

And on your calculus topic, that’s exactly it — I want to show that “scary” topics like calculus really aren’t that bad :). Thanks for the note.

I think you’ve got your representation of negative numbers all wrong. In your example, you said that if I gave my friend three cows, I had -3 cows. I disagree. I would explain things this way: I *promised* my friend 3 cows. Then when I get three cows, I will give them to my friend, bringing my total to 0 cows.

Representing “owed assets” is a more difficult problem. The best way is to separate your assets according to whether they are concrete or whether they are owed to you by someone else. Then you can add them together to get your total theoretical net worth.

Note: To get total net worth, you would also have to subtract the amount of money you owed to other people. I kinda forgot that part.

Hi John, thanks for the comment.

Yeah, the example I used may be more confusing than helpful. The idea was to show that negative numbers aren’t “real” — they have a certain relationship, and in our world, debt seems to model a similar relationship. So we use them to represent the bookkeeping of debt.

Hi Kalid, Thank you very much for this wonderful resource. I admire the way you explain things, especially math. Thanks again.

I am looking for a math book which can teach me things in a way similar to your posts. Can you think of any?

Thanks in advance.

Hi Prashanth, thanks for the comment, glad you are finding it useful :). Comment #33 (above) has a list of some books people have found useful.

I really admire the way Richard Feynman explains physics and other topics (videos here: http://vega.org.uk/video/subseries/8). I’m not sure of many mathematicians who explain things in his style.

Great article. Hope the site gets converted to a print version.

My personal favorite for learning about ‘e’ is e:The Story of a Number- Eli Maor. For a good of general mathematics my current favorite is Mathematics: From the Birth of Numbers.

You can also savour this – 50 Mathematical Ideas You Really Need to Know.

Short and snappy.

Thanks Sameer — I’d love to turn this into a book one day. Thanks for the references, I’ll add them to my (ever growing) list :).

Thanks or information

Thanks Kalid. Your posts are excellent and contain some beautiful revelations.

Only one thing (there is always a but). Could you swap the es in the article above. Generally and on calculators in particular e represents 2.71828.. and E x10^n.

That is “e is not a number” rather than “E is not just a number”.

And in the table and below. 1E3 vs 1.000E3 rather than 1e3 vs 1.000e3.

It may seem a trivial point but surely will confuse my students.

Thanks again, Peter

Hi Peter, that’s a great point about the “e”s, I hadn’t realized their potential for confusion. I’ll update the article & diagram.

I am really impressed. A link to your site is going straight onto the Aberdeen College VLE in the hope that it will help and inspire my engineering students.

A good site that has also inspired me:

http://jedidiah.stuff.gen.nz/wp/?page_id=10

@Peter: Thank you for the kind words! I hope your students enjoy it also. I’ll check out the article you linked.

@Peter: Thank you for that link, it was an excellent discussion similar to Lockhart’s Lament. I agree with the way English/Math is taught, with a focus on “grammar” vs. ideas.

Wow! Love this site! I actually have been dreaming of doing something like this for a while, as I noticed in the comments for this and a few others many have found these in interesting moments of synchronicity!

Anyway, I completely agree with you. Our education system should be showing people how to apply what it teaches, thus the student gains interest in the subject and desires to learn, thus learning. Spitting brick wall rhetoric only annoys and encourages the student to tune out the teacher.

Keep up the excellent work and thank you very much for clarifying my understanding of many, many topics!

@Jasmine: Thanks for the kind words! I completely agree, right now much of math education consists of memorizing and plugging values into formulas. The mechanics of math has its place, but it isn’t real understanding (and can be boring/demoralizing).

Thanks again for the note.

Greetings,

Absolutely ignore comment 1! I can’t believe someone would start with that. Getting the right mindset is vital – a process I’ve been learning as part of my PhD, returning to maths after an extended break.

I’m in the process of trying to learn some (for me) hefty, complex stuff from Paul Krugman’s papers. once I’ve nailed it, I hope to follow your example and make it easier for others. E.g. I’m doing interactive visualisations to help –

http://www.personal.leeds.ac.uk/~gy06do/processing/ces_function/

(Though it needs more writing up!)

This stuff is fantastic – and really, really, not just for elementary school! Thanks. One line in particular I’ll stick on my wall:

“Don’t stop until it makes sense, or that mathematical gap will haunt you.”

I’ll do my best…!

Dan

Maybe I’m missing the point here, but shouldn’t we say that the person who takes the cows has -3 cows? In that case, negative numbers certainly do make sense.

This an eye opener, loads of a-ah moments. I wanted to purchase yet another maths book because of math’s abstract nature the ideas aren’t discuss only the equations are presented in text books. I decided not to purchase yet another math’s or novel until I came across this website by accident. Keep up the good job!!!

@Mike: Yes, that’s the way we’d have to look at it in our accounting system: I have 0 cows, and my friend has -3 cows. So nobody has any positive number of cows? The total # of cows in the world is 0 + (-3) = -3?

[The point is that negative numbers are a mental construct that we try to apply to scenarios we encounter in the world].

@Anthony: Thanks for the kind words!

@Dan: Thanks for the comment! Really glad you are pursuing more learning, and I’m even more delighted that you want to share your knowledge with others.

I think that quote might be a little too cut-and-dry in hindsight, more that you *can* understand anything simply. There’s always a period of “haunting confusion” before understanding a topic — but we can always push through! Thanks again for writing.

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

Me: Numbers are a great idea, but don’t inherently exist. It’s a label we apply to a concept.

All numbers have the same amount of “inherent” existence or non-existence.

@Peter: Great point. Yes, all numbers can be considered an abstraction or a creation of the human mind.

There’s a lot of philosophy of mathematics that deals with this dilemma — one view is “”God created the integers, all the rest is the work of Man. (Leopold Kronecker)” [substitute Nature, Flying Spaghetti Monster, etc.].

Another view is that the universe is the result of a mathematical structure (http://discovermagazine.com/2008/jul/16-is-the-universe-actually-made-of-math), so in that case it’s math creating man :). It’s fun to think about.

hey kalid,

I have gone through the post and comments, and also read views of proponents of formalism, and I must say that i could pick up most of the points correctly but somehow I could not think in a free and intuitive way when i tried to learn maths on my own (using standard books though).

When i try to think of concepts, find relationships as in real world, just nothing to my mind. Can you guess me (assume that i have average IQ) whats wrong have I been doing and suggest me an intuitive way of conceptualizing that is faster and can also be applyied in real world

@hitendra: Finding the right mental model/analogy can be tricky. I usually try to look at the history / context of an idea to see where it was used & what it was useful for (calculus, for example, was primarily used around rates of motion, so intuitions can involve speed, acceleration, etc.). I try to find a few ways of looking at a concept and “circle around” to see if I can connect them. You might find this post useful:

http://betterexplained.com/articles/developing-your-intuition-for-math/

This is probably silly to all of you, and off topic. But I am in 5th grade and our teacher gave us a challenge question: What is the relationship with the numbers 10 and 1,000. He wants us to write/draw 3 relationships. I may be overthinking this, I can think of the obvious, but then the draw part is confusing. Any ideas?

Maths, its always a dream for me…….. Not to learn it but to get passing marks. My college teachers of Anna University so much spoiled the interest tat i almost run from the subject. Perhaps they themselves dont know the subject.

Im so much frustrated with maths that i have left my mechanical engineering degree since i was not able to clear the maths paper alone. Now i finally say or have to say, Maths Sucks and the teachers who teach them sucks more…

I opine that now we really need essay writing just about this good topic, because this is our business to cater all people with the supreme issue close to to about us.

go head keep on you are doing good

Perfect method!

Thank you!

@wisdomlover: Glad it helped!

am a linguist, was never good at maths. Probably THIS i was really looking for .. the real concept behind Maths. This is simply superb.

THAAAANKs

i thank you for the brief explanation and it has really given more ideals for teach my student. i think you are point about each topic is very good and i will always visit your site.

reading dr robert h lewiss paper on math, had me searching for a deeper understanding this lead me to this page. after reading your article i picked up my copy of ‘trachtenberg speed system in math’ (which i bought a year ago without reading)

then i tried to find out why the system for finding 11 x n works. with this deeper understanding i was able to find my own (quick and simple) way of multipling any number by 12,6 and 7 in my head i assume i can find out (on my own with out the book) the rest of the formulas for multiplication..im hopeful and excited!

thanks for your article

@Ashley: Awesome! One of the best feelings in math is getting an idea to click for yourself :).

Hey, cool article.

I found your site tonight and it is great.

One thing though, about the cows. The abstract version of negative cows you use makes no sense to me. I agree with the imaginary person arguing with you. You can’t gain three and jump six. Rather, I imagine it as this: Negative three cows means that YOU owe someone three cows, but have none. I have ten cows on my lot. I sell seven. I have three. 10-7=3. I then meet a man in the market. I promise to give him five cows, and he pays me for five cows. Uhoh. I only have three on my lot. I give him those. I have zero cows on my lot, but I owe the guy two more cows. I have negative two cows. When I recieve my next shipment of cows, the first two I get off the cow truck go to him. 3-5=-2. The shipment has only two cows. -2+2=0. I have zero cows.

Makes sense?

@Patrick: Thanks for the comment! I think I need to reword the example, as its purpose isn’t clear :-).

My key idea is that numbers don’t inherently exist, they’re ways to express relationships. Getting stuck on whether negatives exist or not means we’re missing the point: what type of relationship does it represent?

As you’ve mentioned, negatives can represent the idea of “debt” — and the cool thing is it exists “alongside” the positive numbers!

Here’s an example. Negatives were only invented (in Europe) in the 1700s. But clearly debts existed before then… so how did they track them? Credits and debits!

If you have 3 cows, then great, you have a credit of 3 cows. If you promise to pay someone 5, then you have a debit of 5 cows (and maybe you equalize it, so your Credit is 0, and your Debit is 2).

Negatives are a new type of relationship where the credits and debits exist on the same number line — pretty neat stuff! But it’s a bit of a mind-bender to see the numbers as one continuous line, instead of two buckets (what I have, what I owe).

The key idea for understanding new math (new types of numbers, etc.) is to focus on what type of new scenario it’s describing.

The deference between success and failure is success never stopped trying.

@Older man: Math is about not giving up :).

Very nice article. Having a separate site for school maths would be good for kids.

Please keep writing. God bless you!

Really fantastic article, a big help in my confidence to practicing math for college. Thank you so much!

@Hash: Thanks! Yep, some articles focused on kids topics would be a good idea.

@Jason: Awesome! One of the best part of really getting math is it increases your desire to learn more:).

Math support

see:

http://itunes.apple.com/us/app/iformulas-for-iphone-ipod/id492412908?mt=8

and try iFormulas

Wow, a fantastic article and an awesome website! Keep on the good work!

The “negative cows” with the “jumps 6″ problem works much better, if you stop trying to sum cows on the same sheet for both persons.

If each of you tracks your present cows on his own sheet. Then you always go one cow step for what you currently own.

If, additionally, each of you tracks the cows “which you could have (back)” on a different sheet (for each debitor) , then you always go one hypothetical cow step there as well.

Only the second sheet can go both positive and negative. Depending on you will to “share cows “.

The first sheet cannot, because it tracks the cows which currently can give you milk.

Kalid, I love your work and I’m trying to soak it all in. May I suggest an improvement here? The farmer owning cows is positive and the farmer owing cows is negative (being owed cows by another person is positive). This way if he owns 3 and owes 3, intuitively he has zero even though you can see three in his field. (You also avoid the problem of jumping six when he’s paid three that someone else owes him and suddenly has three.) My credit is positive and my debit is the equivalent negative.

Keep it up!

Thanks for the motivation. I realised it is always the relationship between mathematical concept and its application to the real world that make mathematics an eye opener to the common senses.

its pleasure to read this article because i worried many times by seeing the mysterious formulas and equations without any physical meaning and visualization of what going inside it.

maths formulas acts like a calculator if we give the inputs it will show the output but we cant able to see what is going inside it

in order to see what is going inside it we want to figure it out how the formula is extracted, but it is not simple in order to know the starting point of one mysterious formula we want to extract 100s of formulas from their starting point that is incredible

but every thing is possible that is the faith

i request all the guys posted here to share their resources such as best maths videos and wonderful books for the overall development of the similar thinking people like you!!!!!!!

Excellent and inspiring Article. Thank you. I have come to appreciate and love math recently so I can totally relate to everything you are saying here. I think its all about using unconventional methods to try to understand difficult concepts. For me, its rewriting things down in my own words, and building diagrams/models.

Thanks Mashhood, glad you enjoyed it.

Talk about maxima minima, concavity or other ideas and I may not get it but show me a graph and it starts to click.

I like the way you explain things Kalid. It’s like I’m sitting down over coffee with a math professor who will show me how math really works rather than just plug and chug, which is how I got through college. Thank you.

Thanks Mike, that’s exactly the vibe I hoped for so it means a lot (the cover for the Amazon ebook actually has a cup of coffee on it :)). I think math should be a conversation about ideas that can then adapt into more technical details if needed, just like we can enjoy music and get into the actual notes/scales afterwards.

I really love your explanations. hopefully mathematics is a loving subject that needed to be cared for. I love the subject too much .

Thanks for making math more understandable for many more people

I love that story about the professor and the zen master. It reminds me of something my grandmother once told me. ‘A man who knows it all will never know anything else.’ I took that lesson to heart all my life.

Hey! I am in class 12 and my exams are around the corner. I just gave my pre board examination of maths. Although i had practiced my ncert maths book thoroughly but still i wasnt able to solve questions. I had gone blank. The questions werent difficult but i dnt know what happened that day. Any tip that u can provide for final examination in march 2015

HI i am haris , am math teacher, i glad find this site it’s make me more thinking about math again, change mind set for me and my student, thanks

Dude, what are you on? Like seriously?

There comes a time when you do not question the concept, you just accept it, because there is no understandable explanation. Any attempt to explain these concepts will simply cause more headaches than the original problem.

Sincerely,

Calculus Students

Hi, I haven’t read every part of this article yet, but I have a comment on the example given about a person lending his neighbor 3 cows and creating a negative amount. At first I didn’t really believe that this would create a negative number example but I read on. I had a “Brain Tickle” when I read the that the number shifts 6 places from -3 to 3 when the neighbor returns the cows back.

But I do see a way for the negative to exist without this happening.

If I owe a person 5 dollars and then someone gives me 5 dollars I don’t think I go from -5 to 5. I believe I go from -5 to zero because the money doesn’t belong to me. Perhaps this is in a perfect world thinking but I have always assumed this to be true.

When I was young and I “lent” money to my older sister and then my mom would give her money I would tell her that it was mine. Therefore she didn’t really have money to spend but if she used it she would be using my money.

She always looked at me like I was crazy because she never wanted to repay me. I should try to factor in that the Bible says that we are to forgive debts after 7 years… 😉

maths is are had subject we all need to no maths

Maths is better learnt when people comment their ideas. This is comment section is proven to be useful. With a little motivation we can get our ideas across the board easily. Thanks to all of you. I leave with you my little maths quote.

The ideas of reality has a better effect on the mind which makes mathematical concepts tranformational objects.

– Kenny Hall

I encourage you to like this page on facebook. Page name: Taste these quotes. Thanks to better explained, this helps me a lot. Let’s not forget we learn better through cooperation.

Am from Nigeria and um…… l am good in maths. but I find it difficult to understand why am solving a question, what’s it use and application and so on. I love your blog, it encouraged me to learn more but differently using some of the tips you provided. thanks a lot . your blog is awesome.

It’s with every student. Most just want to pass and the good ones want to get a 100 with the formulas they memorized. Some book though do provide some help learning things in depth.

I’m sorry, but I really don’t like your “debt” analogy of negative numbers, and here’s why.

From what I understand from the article, if I have one cow and then lend it to a friend, I have -1 cows. OK, that seems fair enough, but what if I start with 11 cows? I have 11 cows and lend 1 to a friend. By the analogy given, I would have 9 cows remaining. But any fool can look at my field and see 10 cows grazing, not 9.

Here’s an alternate analogy that I think makes more sense, still using cows and debt. Let’s say my friend has 10 cows and I have none, poor me. So I ask to borrow one of his cows, and I’ll give it back at the end of the week. I still have a net 0 cows in my field, because there’s one there, and a negative cow, that I need to give back. Everything is fine and dandy, until the cow dies. Now I have nothing but still owe someone something. The total balance is -1 cows.

this is a good opportunity,help to me for my studies