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 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!

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.

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.

> 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.

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.

@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.

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.