How to Develop a Mindset for Math

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 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?


Posted November 26, 2007, under Guides, Math
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    Comments

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

      Joel Dietz — November 27, 2007 @ 3:58 am

    2. 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 — November 27, 2007 @ 4:14 am

    3. Kalid: actually even positive numbers are not that real. You see three cows, three lines, but not three as a concept :-)
      What 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!

      .mau. — November 27, 2007 @ 6:17 am

    4. 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.

      larry — November 27, 2007 @ 6:35 am

    5. 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.

      Bob — November 27, 2007 @ 6:48 am

    6. “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)

      she — November 27, 2007 @ 7:35 am

    7. 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.

      Jonathan — November 27, 2007 @ 8:35 am

    8. Great stuff. Looking forward to your next post.

      Gilbert — November 27, 2007 @ 8:37 am

    9. Hi
      Really looking forward to your next post about Imaginary numbers.

      wow — November 27, 2007 @ 9:11 am

    10. > 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.

      Bill Mill — November 27, 2007 @ 9:33 am

    11. 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.

      Joe — November 27, 2007 @ 9:35 am

    12. 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

      Doug — November 27, 2007 @ 9:41 am

    13. 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.

      Tim — November 27, 2007 @ 9:45 am

    14. 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.

      Alex U. — November 27, 2007 @ 10:22 am

    15. 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.

      Eddie — November 27, 2007 @ 10:27 am

    16. @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.

      Kalid — November 27, 2007 @ 10:32 am

    17. 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.

      Kalid — November 27, 2007 @ 10:43 am

    18. 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.

      PeeJay — November 27, 2007 @ 11:13 am

    19. 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!

      Erik E — November 27, 2007 @ 11:22 am

    20. You probably find “Does Mathematics Reflect Reality?” interesting

      anh — November 27, 2007 @ 11:59 am

    21. @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.

      Kalid — November 27, 2007 @ 12:14 pm

    22. 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.

      Alicow — November 27, 2007 @ 12:35 pm

    23. 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.

      Brian — November 27, 2007 @ 1:20 pm

    24. 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 :)

      Me — November 27, 2007 @ 1:24 pm

    25. 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.

      Curly — November 27, 2007 @ 1:39 pm

    26. The cow example is wrong. -3 cows is having none, but owing three to someone, not someone owes you.

      Jeremy Stein — November 27, 2007 @ 2:19 pm

    27. @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.

      Kalid — November 27, 2007 @ 2:32 pm

    28. @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 :) .

      Kalid — November 27, 2007 @ 2:33 pm

    29. 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.

      jeff — November 27, 2007 @ 3:08 pm

    30. Thanks Jeff — I’ve been having some brain-bending thoughts about imaginary numbers that I’m excited to get down onto “paper”.

      Kalid — November 27, 2007 @ 3:15 pm

    31. 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’!

      Marie — November 27, 2007 @ 4:10 pm

    32. @ 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.]

      Z — November 27, 2007 @ 4:13 pm

    33. @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.

      Kalid — November 27, 2007 @ 4:24 pm

    34. 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.

      Dorai — November 27, 2007 @ 5:20 pm

    35. Elevation is a better example for negative numbers. Sea level is zero. Denver would have a positive elevation. Death Valley a negative elevation.

      auferstehung — November 27, 2007 @ 5:29 pm

    36. 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.

      Mark — November 27, 2007 @ 6:11 pm

    37. 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.

      cariaso — November 27, 2007 @ 6:36 pm

    38. 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.

      Ken — November 27, 2007 @ 6:55 pm

    39. 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.

      Pablo — November 27, 2007 @ 7:40 pm

    40. @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.

      Kalid — November 27, 2007 @ 8:18 pm

    41. 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.

      John — November 27, 2007 @ 9:13 pm

    42. Even if it is not directly related to this article, I’d suggest also to have a look at Mathematics: A Very Short Introduction by 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.

      .mau. — November 28, 2007 @ 1:33 am

    43. @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 :) .

      Kalid — November 28, 2007 @ 11:54 am

    44. 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.

      PeeJay — November 28, 2007 @ 3:58 pm

    45. 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.

      wookie — December 1, 2007 @ 2:30 pm

    46. 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 :)

      Kalid — December 1, 2007 @ 3:17 pm

    47. 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.

      Mohammad Ali — December 9, 2007 @ 7:39 am

    48. 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.

      Kalid — December 11, 2007 @ 9:36 am

    49. 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?

      Zac — February 29, 2008 @ 7:13 pm

    50. 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.

      Kalid — April 24, 2008 @ 7:51 am

    51. 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.

      John — April 30, 2008 @ 8:29 pm

    52. 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.

      John — April 30, 2008 @ 8:31 pm

    53. 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.

      Kalid — April 30, 2008 @ 8:43 pm

    54. 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.

      Prashanth Ellina — May 9, 2008 @ 7:09 am

    55. 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.

      Kalid — May 9, 2008 @ 11:34 am

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