Khalid Azad has a passion for learning concepts at a deep, intuitive level, he writes at BetterExplained.com, which serves millions of readers every year and is used in dozens of university courses. He's also authored a popular math book, It's a pleasure to welcome Khalid. Hi Khalid, thanks for joining us I, I have to ask you first off, what got you interested in this wonderful intuitive approach to learning that you use? >> You know it actually started with a really frustrating math class I remember I was in my freshman year at Princeton, you know, I had always enjoyed math in high school, middle school and I was in my first class and I was just getting destroyed. It was around the midterm, and I wasn't happy, things weren't going well. The subject that I'd always liked just wasn't making sense I remember the finals had come around and I was cramming and cramming and cramming, and suddenly some analogies started coming to me about twists and rotations and turns and that really applied to the class, so I was able to visualize the concepts instead of just looking at the symbols and equations. And because of that, I didn't really need to memorize anymore, I could just think through it, and I could walk myself through my mental diagrams and the equations kind of snapped into place, and so after that I realized that a lot of learning could become so much more pleasant and more fun after we found the right mental metaphors, diagrams, and analogies Instead of trying to, you know, bash our heads against the, the raw definitions. So from that I started trying to write and share my, my insights that seemed to help me. >> It sounds like we share the same approach I'm, that's, that's the kind of thing that I was trying to do, too I finally sort of cottoned on to that. So you can, can you talk a little bit more about what approach you actually do use? >> Sure just to keep things straight in my head, I have this little, mnemonic I use, and I call it the ADEPT Method. And basically what I'm looking for in an explanation of something, I try to find an analogy, a Diagram, an Example, a Plain English description and then a Technical Description. And the, the general idea is actually kind of a C.D.M, diffused thinking Is that you, when you're first learning, you want to start with the sort of fuzzy overview and then gradually sharpen it up, until you get to the technical details at the edge. So you start with the analogy, the diagram, you get some examples, you have a plain, plain English description, and at the very end, that's when you go to the technical description. And you want to save that really focused thinking, for me, for last because that's when I already have the framework in place to think about it. >> Oh I like that, can you, can you share a, a specific example of the ADEPT method? >> Sure I think one of my favorite topics is actually the the imaginary number, so, most people myself included for a long time, it is confusing, sound imaginary, real safe you know, how can we use in real life and after breaking it down with the method, I was able to find some metaphors that seemed to help me. >> You just use an analogy, then, to start with this? >> Yeah, so, in this case, imaginary numbers, a lot of the the time we'll start the definition, the technical definition, and that doesn't really make things click one thing that helped me was the analogy about a 90 degree turn. So if you imagine that negative numbers are sort of a full 180 so you're going forward, then you turn all the way 180 to backward. A imaginary number is basically when you start forward you actually just go up, so instead of going all the way backward you're actually just going up. So, imagine maybe you're going east and west negatives are sort of flipping between east and west, but imaginary number is the idea that hey maybe I should take a 90 degree turn north. So now you're going along and woop you go north, and suddenly you're in a different direction, It's not just duress it's a new direction it's a second dimension really and that's kind of the key analogy behind it. >> So in, in some sense then this can actually become like a diagram or a map. >> Exactly, this topic is really easy to visualize because, we already have the idea of second dimensions and going different directions, so in my head, I kind of imagine, first a number going along, and it gets a rotation, which turns and then it turns again. So I have this mental diagram in my head. And that actually leads right into an example that well, you know, a lot of kids wonder, well what happens when you take four left turns? Well if you keep turning, you'll face back the same direction after four turns, and with imaginary numbers if you turn four times, you'll face the positive numbers again. And so the idea is that your diagram can really, you know, almost lead to an example that you can figure out intuitively without having to do a lot of math it kind of worked through the results I got. >> I used to do something very similar, but I had, what I had was a visual with a toilet paper roll that was unrolling because of scrolling across the screen It was a lot of fun but, but moving on so then you, actually just use a plain English description. >> yeah, so once you've, wrapped your head around it an analogy, a diagram, read about the example, start sharpening up what you've understood and you want to describe it in your own words. So rather than going right to the technical definitions, lets think about what happens. I might say that we were moving in one direction and we moved 90 degrees or we moved into a second dimension or there's kind of a different type of number, there is one number that goes this way and one type that goes this way. In your head, you want to get your own understanding, put it in your own words and eventually you sharpen it up and can I think a lot of times we try to understand things just by someone else's words instead of really internalizing it ourselves. >> Well, I, I think this is the perfect example of the real imaginative world of mathematics and science and engineering, I mean, we imagine things in ways that people don't ordinarily imagine and, and I love how you're approaching this, because it really brings this to life. But, you, what you do then you move next on to an actual technical description of what you were describing? >> Yeah, that's right, so the last step is follow me to use the formal notation that came up with my little analogy here, its kind of like you have a song in your head you might be humming it, doo ta dit da doo, but, humming in your head but, other people don't really know what you're talking about. You need to find the sheet music, you know, explain, you know the scale that you're using, and the notation, the timing, that way somebody else can really have a handle for the hymn, instead of this little song that's going in your head. So you want to take, you know, the rough words and kind of break them down in a formal notation. In the math world, the notation is "i", lower case "I" and that actually represents, the imaginary dimension and it's kind of like the number one but in just that imaginary dimension, It is in there, now we can start talking about other ideas in a more technical sense. So when I say, oh I want to face forward and to take two terms, and I'll be facing negative, what that means is you start with one, multiply by i, multiply by i again, and you should be facing negative. Which should be is negative, so a mathematician might write one times i times i is negative one. Now you can simplify this you could say, oh, one times i squared equals negative one, and you can simplify it further by just saying i squared equals -1. And then the last step, is if you really want to simplify it, you just take the square root of both sides, and now you can say that i equal the square root of -1. Now, this makes sense to us hopefully now because we've been able to walk through it step by step, and turn our kind of intuitive fuzzy description into the really technical one, and it's really painful to start with a technical description I equals the square root of negative one, and try to guess what that could've meant, and I promise you don't see all the steps ahead of time. So I realized to myself I really needed to start at the fuzzy level, and work my way down to technical. >> Yes I, I think that's the thing about imaginary numbers people often don't realize that they can dance, that they can twist and rotate and that's kind of what they do It's quite beautiful actually, do you have final learning tips for our viewers?" . . >> Yea, I think the biggest tip that I might say is just, have a sense for yourself when something's not clicking. And realize that it's probably just because of a missing piece. So you know, in the past they might have just reread the same passage again and again because it wasn't working, but maybe if you want to defer a diagram, Google Images or maybe a talk or a slideshow, or something that might have a different metaphor that can help you. I realize, you know, with the ADEPT you know, analogy diagram example plain-English technical, I need all the parts It's kind of a part of a complete breakfast I need all the different pieces, and then when I have them, I know the idea is very quick in my mind. So, if I'm working on something and it's just a technical, maybe an example, it's usually not enough for me to really internalize it, so I start searching for the elements there, I didn't even know that there were other pieces that I should be looking for so for me, it's kind of a, checklist that helps me kind of zone in on what to do with it. >> Great well, thank you so much I know our viewers will benefit greatly from these ideas, which I myself have found really valuable thanks again.