Conrad Wolfram: Teaching kids real math with computers
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Conrad Wolfram: Teaching kids real math with computers


We’ve got a real problem with math education right now. Basically, no one’s very happy. Those learning it think it’s disconnected, uninteresting and hard. Those trying to employ them think they don’t know enough. Governments realize that it’s a big deal for our economies, but don’t know how to fix it. And teachers are also frustrated. Yet math is more important to the world than at any point in human history. So at one end we’ve got falling interest in education in math, and at the other end we’ve got a more mathematical world, a more quantitative world than we ever have had. So what’s the problem, why has this chasm opened up, and what can we do to fix it? Well actually, I think the answer is staring us right in the face: Use computers. I believe that correctly using computers is the silver bullet for making math education work. So to explain that, let me first talk a bit about what math looks like in the real world and what it looks like in education. See, in the real world math isn’t necessarily done by mathematicians. It’s done by geologists, engineers, biologists, all sorts of different people — modeling and simulation. It’s actually very popular. But in education it looks very different — dumbed-down problems, lots of calculating, mostly by hand. Lots of things that seem simple and not difficult like in the real world, except if you’re learning it. And another thing about math: math sometimes looks like math — like in this example here — and sometimes it doesn’t — like “Am I drunk?” And then you get an answer that’s quantitative in the modern world. You wouldn’t have expected that a few years back. But now you can find out all about — unfortunately, my weight is a little higher than that, but — all about what happens. So let’s zoom out a bit and ask, why are we teaching people math? What’s the point of teaching people math? And in particular, why are we teaching them math in general? Why is it such an important part of education as a sort of compulsory subject? Well, I think there are about three reasons: technical jobs so critical to the development of our economies, what I call “everyday living” — to function in the world today, you’ve got to be pretty quantitative, much more so than a few years ago: figure out your mortgages, being skeptical of government statistics, those kinds of things — and thirdly, what I would call something like logical mind training, logical thinking. Over the years we’ve put so much in society into being able to process and think logically. It’s part of human society. It’s very important to learn that math is a great way to do that. So let’s ask another question. What is math? What do we mean when we say we’re doing math, or educating people to do math? Well, I think it’s about four steps, roughly speaking, starting with posing the right question. What is it that we want to ask? What is it we’re trying to find out here? And this is the thing most screwed up in the outside world, beyond virtually any other part of doing math. People ask the wrong question, and surprisingly enough, they get the wrong answer, for that reason, if not for others. So the next thing is take that problem and turn it from a real world problem into a math problem. That’s stage two. Once you’ve done that, then there’s the computation step. Turn it from that into some answer in a mathematical form. And of course, math is very powerful at doing that. And then finally, turn it back to the real world. Did it answer the question? And also verify it — crucial step. Now here’s the crazy thing right now. In math education, we’re spending about perhaps 80 percent of the time teaching people to do step three by hand. Yet, that’s the one step computers can do better than any human after years of practice. Instead, we ought to be using computers to do step three and using the students to spend much more effort on learning how to do steps one, two and four — conceptualizing problems, applying them, getting the teacher to run them through how to do that. See, crucial point here: math is not equal to calculating. Math is a much broader subject than calculating. Now it’s understandable that this has all got intertwined over hundreds of years. There was only one way to do calculating and that was by hand. But in the last few decades that has totally changed. We’ve had the biggest transformation of any ancient subject that I could ever imagine with computers. Calculating was typically the limiting step, and now often it isn’t. So I think in terms of the fact that math has been liberated from calculating. But that math liberation didn’t get into education yet. See, I think of calculating, in a sense, as the machinery of math. It’s the chore. It’s the thing you’d like to avoid if you can, like to get a machine to do. It’s a means to an end, not an end in itself, and automation allows us to have that machinery. Computers allow us to do that — and this is not a small problem by any means. I estimated that, just today, across the world, we spent about 106 average world lifetimes teaching people how to calculate by hand. That’s an amazing amount of human endeavor. So we better be damn sure — and by the way, they didn’t even have fun doing it, most of them — so we better be damn sure that we know why we’re doing that and it has a real purpose. I think we should be assuming computers for doing the calculating and only doing hand calculations where it really makes sense to teach people that. And I think there are some cases. For example: mental arithmetic. I still do a lot of that, mainly for estimating. People say, “Is such and such true?” And I’ll say, “Hmm, not sure.” I’ll think about it roughly. It’s still quicker to do that and more practical. So I think practicality is one case where it’s worth teaching people by hand. And then there are certain conceptual things that can also benefit from hand calculating, but I think they’re relatively small in number. One thing I often ask about is ancient Greek and how this relates. See, the thing we’re doing right now is we’re forcing people to learn mathematics. It’s a major subject. I’m not for one minute suggesting that, if people are interested in hand calculating or in following their own interests in any subject however bizarre — they should do that. That’s absolutely the right thing, for people to follow their self-interest. I was somewhat interested in ancient Greek, but I don’t think that we should force the entire population to learn a subject like ancient Greek. I don’t think it’s warranted. So I have this distinction between what we’re making people do and the subject that’s sort of mainstream and the subject that, in a sense, people might follow with their own interest and perhaps even be spiked into doing that. So what are the issues people bring up with this? Well one of them is, they say, you need to get the basics first. You shouldn’t use the machine until you get the basics of the subject. So my usual question is, what do you mean by “basics?” Basics of what? Are the basics of driving a car learning how to service it, or design it for that matter? Are the basics of writing learning how to sharpen a quill? I don’t think so. I think you need to separate the basics of what you’re trying to do from how it gets done and the machinery of how it gets done and automation allows you to make that separation. A hundred years ago, it’s certainly true that to drive a car you kind of needed to know a lot about the mechanics of the car and how the ignition timing worked and all sorts of things. But automation in cars allowed that to separate, so driving is now a quite separate subject, so to speak, from engineering of the car or learning how to service it. So automation allows this separation and also allows — in the case of driving, and I believe also in the future case of maths — a democratized way of doing that. It can be spread across a much larger number of people who can really work with that. So there’s another thing that comes up with basics. People confuse, in my view, the order of the invention of the tools with the order in which they should use them for teaching. So just because paper was invented before computers, it doesn’t necessarily mean you get more to the basics of the subject by using paper instead of a computer to teach mathematics. My daughter gave me a rather nice anecdote on this. She enjoys making what she calls “paper laptops.” (Laughter) So I asked her one day, “You know, when I was your age, I didn’t make these. Why do you think that was?” And after a second or two, carefully reflecting, she said, “No paper?” (Laughter) If you were born after computers and paper, it doesn’t really matter which order you’re taught with them in, you just want to have the best tool. So another one that comes up is “Computers dumb math down.” That somehow, if you use a computer, it’s all mindless button-pushing, but if you do it by hand, it’s all intellectual. This one kind of annoys me, I must say. Do we really believe that the math that most people are doing in school practically today is more than applying procedures to problems they don’t really understand, for reasons they don’t get? I don’t think so. And what’s worse, what they’re learning there isn’t even practically useful anymore. Might have been 50 years ago, but it isn’t anymore. When they’re out of education, they do it on a computer. Just to be clear, I think computers can really help with this problem, actually make it more conceptual. Now, of course, like any great tool, they can be used completely mindlessly, like turning everything into a multimedia show, like the example I was shown of solving an equation by hand, where the computer was the teacher — show the student how to manipulate and solve it by hand. This is just nuts. Why are we using computers to show a student how to solve a problem by hand that the computer should be doing anyway? All backwards. Let me show you that you can also make problems harder to calculate. See, normally in school, you do things like solve quadratic equations. But you see, when you’re using a computer, you can just substitute. You can make it a quartic equation. Make it kind of harder, calculating-wise. Same principles applied — calculations, harder. And problems in the real world look nutty and horrible like this. They’ve got hair all over them. They’re not just simple, dumbed-down things that we see in school math. And think of the outside world. Do we really believe that engineering and biology and all of these other things that have so benefited from computers and maths have somehow conceptually gotten reduced by using computers? I don’t think so — quite the opposite. So the problem we’ve really got in math education is not that computers might dumb it down, but that we have dumbed-down problems right now. Well, another issue people bring up is somehow that hand calculating procedures teach understanding. So if you go through lots of examples, you can get the answer, you can understand how the basics of the system work better. I think there is one thing that I think very valid here, which is that I think understanding procedures and processes is important. But there’s a fantastic way to do that in the modern world. It’s called programming. Programming is how most procedures and processes get written down these days, and it’s also a great way to engage students much more and to check they really understand. If you really want to check you understand math then write a program to do it. So programming is the way I think we should be doing that. So to be clear, what I really am suggesting here is we have a unique opportunity to make maths both more practical and more conceptual, simultaneously. I can’t think of any other subject where that’s recently been possible. It’s usually some kind of choice between the vocational and the intellectual. But I think we can do both at the same time here. And we open up so many more possibilities. You can do so many more problems. What I really think we gain from this is students getting intuition and experience in far greater quantities than they’ve ever got before. And experience of harder problems — being able to play with the math, interact with it, feel it. We want people who can feel the math instinctively. That’s what computers allow us to do. Another thing it allows us to do is reorder the curriculum. Traditionally it’s been by how difficult it is to calculate, but now we can reorder it by how difficult it is to understand the concepts, however hard the calculating. So calculus has traditionally been taught very late. Why is this? Well, it’s damn hard doing the calculations, that’s the problem. But actually many of the concepts are amenable to a much younger age group. This was an example I built for my daughter. And very, very simple. We were talking about what happens when you increase the number of sides of a polygon to a very large number. And of course, it turns into a circle. And by the way, she was also very insistent on being able to change the color, an important feature for this demonstration. You can see that this is a very early step into limits and differential calculus and what happens when you take things to an extreme — and very small sides and a very large number of sides. Very simple example. That’s a view of the world that we don’t usually give people for many, many years after this. And yet, that’s a really important practical view of the world. So one of the roadblocks we have in moving this agenda forward is exams. In the end, if we test everyone by hand in exams, it’s kind of hard to get the curricula changed to a point where they can use computers during the semesters. And one of the reasons it’s so important — so it’s very important to get computers in exams. And then we can ask questions, real questions, questions like, what’s the best life insurance policy to get? — real questions that people have in their everyday lives. And you see, this isn’t some dumbed-down model here. This is an actual model where we can be asked to optimize what happens. How many years of protection do I need? What does that do to the payments and to the interest rates and so forth? Now I’m not for one minute suggesting it’s the only kind of question that should be asked in exams, but I think it’s a very important type that right now just gets completely ignored and is critical for people’s real understanding. So I believe [there is] critical reform we have to do in computer-based math. We have got to make sure that we can move our economies forward, and also our societies, based on the idea that people can really feel mathematics. This isn’t some optional extra. And the country that does this first will, in my view, leapfrog others in achieving a new economy even, an improved economy, an improved outlook. In fact, I even talk about us moving from what we often call now the “knowledge economy” to what we might call a “computational knowledge economy,” where high-level math is integral to what everyone does in the way that knowledge currently is. We can engage so many more students with this, and they can have a better time doing it. And let’s understand: this is not an incremental sort of change. We’re trying to cross the chasm here between school math and the real-world math. And you know if you walk across a chasm, you end up making it worse than if you didn’t start at all — bigger disaster. No, what I’m suggesting is that we should leap off, we should increase our velocity so it’s high, and we should leap off one side and go the other — of course, having calculated our differential equation very carefully. (Laughter) So I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being there, computers that are now ubiquitous almost. Calculating machines are everywhere and will be completely everywhere in a small number of years. Now I’m not even sure if we should brand the subject as math, but what I am sure is it’s the mainstream subject of the future. Let’s go for it, and while we’re about it, let’s have a bit of fun, for us, for the students and for TED here. Thanks. (Applause)

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63 thoughts on “Conrad Wolfram: Teaching kids real math with computers

  1. The problem with math is that no companies is hiring people with degrees in math.
    Believing that is too theoretical and not applicable to real life situations which I think is a stereotype what we need to change.
    Lots students are afraid that after spending 4 years studying math they find themselves unemployed and their only way out is spending 2 years doing a master and 3 more years doing a PHD.
    If only companies take the chance of employing a math student.

  2. Could not agree with the message of this video more. to quote a man of highest respects "Education is what remains after one has forgotten everything he learned in school". Emphasis on other aspects of mathematics needs to be of focus when we learn the subject. Leave the routine work to the machines that do routine things.

  3. the math ability that older people have, doing arithmetic in their head, is fantastic and there's no reason why that shouldn't be taught. I doubt it'd take a huge amount of time to teach. One may not have an excel sheet in front of them, and it may be quicker sometimes to do something in your head.

  4. Why is a written javascript exam a bad thing? it can show you really know it. You're not going to have to write how to use 3rd party tools – knowing all their procedures. But the fundamental parts of the javascript language, you should know enough to do them pen on paper. And doing them on pen and paper tests that.

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  6. Brilliant. He's exactly right. Let the computers do the calculating and let's start asking the kids some tough conceptual questions about the material. Have them assemble the pieces of the puzzle we put in front of them. The way we teach math is changing, and we should embrace that change.

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  8. I think Wolfram covered this though. Much of the basic math should still be tought in the same way. Learn your addition rules, your times tables, long division, etc by hand. It's when you move into calculus, trigenometry, and all the other maths that are hard to say and harder to do that we need to move away from the traditional model and teach students to see them as logical problems rather than squiggling numbers.

  9. I used to call the alumni for my school. 90% of the math majors I called were employed as programmers – a lot of high level computing is nothing more than converting a huge forumla into machine instructions. The other 10% were PHD's doing research. This is in the US though, it may be different around the globe.

  10. Coding is logic, not memorized libraries. As long as you know the fundamentals – you know that something CAN be done and have an idea about how to do it – does it really matter if you have the documentation up in a separate window to make sure you're passing the right values?

  11. When I said fundamentals, I meant not involving libraries. Just knowing things like iteration, invoking functions, As well as being able to write the algorithms in that language. Though I suppose knowing the libraries to the extent you mention, is fundamental, and indeed, for knowing libraries, using documentation is fine and even normal! I seriously doubt anybody would be given an exam on pen and paper where they had to have memorized libraries.

  12. nobody that learns math and does well in a quality exam, is going to think it's just "squiggling numbers". If somebody managed to learn the process without understanding what they're doing, then maybe the exam should have picked that up.. What are "your addition rules"? I'm talking about doing math in your head. We actually had exams in calculus and trig, so I know them.We didn't have drills or final exams on mental arithmetic, neither would teachers of today have had.

  13. Isn't this unethical? Mr. Wolfram has a huge stake in the computerized algebra systems market. He has 0 experience in teaching at the collegiate level, much less high school or elementary school. I do think that Mathematica can help speed things up, but there is a lot gained and much missed by skipping the hand calculations. I think Mr. Wolfram means well, but that is how the highway to hell gets paved.

  14. I would like to see how a student can prove the rationals are dense in the reals, using Mathematica®. Does mathematica have a theorem prover that could actually be used to teach elementary kids rather than graduate students? When that is done, then there is a hope to teach mathematical reasoning using computers.

  15. You're saying that there's a conflict of interest in the guy encouraging children to learn, design and implement, on their own, the software that he currently sells?

  16. When did the alumni graduate?
    They are not employed in math related jobs, they changed careers.
    Companies now prefer computer science rather math majors for programming work.
    And back in the days there weren't computer science majors they were are math majors, bill gates was in applied math.
    Right now there is a distinction between math major and computer science major.

  17. that computer program is a nice toy for children, not suitable for teaching math
    making the mind rely on computers for every calculation, also what would you do if computer broke down or anything happened to it?
    We need hand calculations in many situations were computers are not available or it would take time to operate one, also computers are dumb anyway. By learning hand calculations we are learning how to create computers, and then use computers only when they are needed.
    I disagree with you completely
    And I believe that computers dumb math down

  18. As a programmer I agree that programming is an excellent tool to really force yourself to learn concepts. The logical thinking skills I have gotten from programming and computer science are simply unmatchable from anything else. If your school offers computer science, I urge you to try it.

  19. I totally agree to this speech and I am a huge fan of Wolfram Research.

    I recently signed up for the free pro trial of Wolfram Alpha and tried the Problem Generetor. What I found there were exactly that kind of problems Conrad was pledging against in this speech. Why? I'm kind of disappointed right now.

  20. If I follow his idea to its extreme, why would I learn anything at all ? Why not have a chip with an unlimited access to internet grafted in my brain and just vegetate all day watching cat videos ?
    I agree on 1 point: it's more important to learn the why than the how of a problem. If I know the laws of physics, I'll know what to do to resolve a problem. If I know economics, I'll know how to calculate investments, depreciation and whatnot. 
    But what he suggests is to become intellectually lazy and dependant on machines. 

  21. Does anyone find it strange that we only teach students how to program in Pre-Calculus? In my grade school years math classes had zero programming involved, until my Pre-Calculus course(which was on a graphing calculator). 

  22. Math is a beautiful world of numbers. Numbers are the keys of our everyday life. By teaching calculating, we make people think faster and better. we tech 'em to get out of different unexpected situations by finding the better way of solving 'em out. Math makes people better. In fact that in almost every school people are divided in two huge groups: the ones who are better at math and physics, and the others who are better at learning history or literature. I'm not saying that the way they teach us in school is completely good, We surely need to be taught more about why we all need to get all that information. But I hope us to never get rid of basic calculations by hand in order to make all of us to love math. Pardon my english

  23. Wolfram talks a good game, but his talk is vacuous because he doesn't really focus on the formulation and analysis of a mathematical problem before you can use a computer to do the "calculations." Moreover, he make no mention of the fundamental axiom of computing, which I mastered so well when I was interacting with the Operations research and DP Systems (now called IT) Departments of a large corporation: "GARBAGE IN, GARBAGE OUT."

  24. A very good video indeed, and I agree, this is the way math should be taught. But I don't agree we should discard teaching the calculation process. We'll never understand how the computers are doing it if we do so. We'll never doubt what the computers tell us is the answer. Computers can make mistakes, because humans can make mistakes, and computers were built by humans.

    I do however think that calculating by hand is an absurd way to do math, we should be teaching kids to do it mentally. And there are two ways I know we could approach this, one is doing what some schools in Japan, China and Korea among others are doing, teach them to use an abacus (and by abacus I mean chinese or japanese abacus; not the western crap), when you're good with an abacus I hear it is very easy to do math with an imaginary abacus instead of a real one, and that it can be done on average in about 1/10 of the time it would take on a real abacus. Meaning not just mental math, lightning fast mental math. For everyone. The calculating process is important because it is literally teaching kids how to solve problems or puzzles on their own. They need to understand the process if they're ever caught in a situation where they do not have access to a computer, or if they are trying to solve a problem they can't properly express to a computer.

    The second approach I can imagine for tackling mental math is looking to mental math geniuses like Scott Flansburg for ideas on how to properly teach mental calculation. Our brains are more powerful than computers, and we have a good chunky part of our brain dedicated to calculation only, it would be an absolute waste of resources not to use it.

    As for bringing computers to exams, this is not a problem when we have things like the raspberry pi (easily affordable to schools) and linux (highly customizable), it wouldn't be hard to set things up, computers wouldn't be hooked to the internet, no USB ports to thwart cheating, just offline computers with all the applications the students have learned to rely on (like a set of programming languages, and perhaps libreoffice or something to write answers to the questions on the exams)

  25. Don't agree much. There's oddly an acquired satisfaction from the gained ability to successfully perform tortuous computation. This developed satisfaction is extends then to making useful things.
    It's actually more than hand calculating, it's an exercise in ability to concentrate, to keep a line of though, to pay extreme attention to detail, to apply and employ creatively a handy set of tools. This whole process then leads to people that can make computers, program and come up with new technologies.

  26. Meanwhile in the US, the math curriculum is written without a Math teacher and simple arithmetic now takes 133 steps instead of 6.

  27. Sorry but teaching math fundamentals / basics by practicing programming is pointless. Why? For programming in a high computer language you don't need math at all. What you need is logical thinking, systematical thinking. But even after 20 years of practicing programming — even "professional" — you won't understand any equation better than before, given you never solved one before. I started programming when I was 11 years old. My brother was teaching me Turbo Pascal and Basic programming in 1996. First on a very old C64. Later on the first Pentium machines, 90MHZ, 4 MB RAM if you know what I mean. I started "real" programming in C by 1999 when I was 14 years old. Beyond my logical thinking and the most basic math skills (add, sub, mul, div, etc.), I never ever needed any further math. And it would have been a blocker for me, if it would be like that. The schools I visited were terribly low-level. I learned only the very basics in school. And even after 18 years of programming, I never needed to solve an equation. Why? Because all mathematical problems are already solved in high level programming languages. What you need is logical thinking. And yes, for training logical thinking, practicing programming helps. But NOT for understanding math. Im 29 years old now. I'm programming since 18 years. And I can't solve any equation because I never learned the basics. Now I want to study psychology and for this I need decent skills in statistics. That's why I'm motivated to learn math now. And it's not as easy as described in the video. A lot of things are even more irritating to me. A function is programming has a different concept than a function in math. When you do programming, you can write down the logic in a specific, well-defined and formalized way. In math, you have a formula composed out of "elegant" math terms. Math formulas are like a recursive summary of summaries compared to computer programs. If you come from programming this looks like a limitation or like totally abstract logic and you don't know how to apply that for a specific problem. You can't learn such things using a computer. After a lot of years I'm using paper again: To practice writing down those equations. Solve them. MANUALLY. To understand the basic concepts. BUT AFTER I've understood the basics of a specific math problem, I use Wolfram Alpha to proof my calculation results. So yes: Computers are GOOD for COMPUTING 😉 For proving my results while I'm learning math on paper. But they are worthless for learning the fundamental concepts of math. Today, as a professional programmer with lots of years of experience, I wish I would have had a good math teacher and the motivation to learn it back when I was a kid in school. With the lack of basic understanding I'm using my computer to consume A LOT of YouTube videos, teaching me how to solve equations by hand — like a teacher would do in school. This is what helps me to understanding math. For my understanding, most of the difficulty in learning something new relies on MOTIVATION. I mean, I want to study psychology — I have specific questions in my mind. Questions, I can answer by using math only. So I have the motivation to learn it. I don't need a computer to have fun learning math. I have fun while understanding more and more because it makes sense to me and I'm seeing a progress in my self-development.

  28. I know what modern High school math classes look like. I see them several times a week. They use computers to look at what is effectively YouTube videos, though they are forbidden to go to actual YouTube. To copy the notes that the teacher and the books should have given them. They assume everyone has internet, and the teacher explains how to do the simple things that they already know, and the things they do not understand despite reading the book, the teacher reads the book word for word. Don't you think they already tried that. Only those in engineering and or programming clubs begin to understand the processes. They learn it from each other and their couches not their math teachers. I know states think just because it is on a computer it is automatically easier, it is not. Math on computers is easier if put in the right contexts, like building a program to solve an equation or a calculator. It is far more complex when solving problems in a way that is meant to be done by hand.

  29. I also learned a lot of mathematics by programming. However, as soon as you make something a compulsory subject it will be dead.

  30. While I do believe in the immeasurable value of using computers in math education, I do think there is the danger of its over-dependence.
    For instance, just because a computer can calculate faster than a human, that does not mean it is necessarily correct.
    Someone had to know the math and the most efficient algorithm to program the solution into the computer.
    The problem is how do you know that programmer did their own math correctly or chose the correct algorithm?
    You don't unless someone else has verified it.

    I also disagree with Wolfram's statement that calculations are more suited to computers than humans.

    The purpose in learning math and calculations is not just to get an answer.

    Learning to do calculations as much in your head as possible develops the cognitive skills that you would not get in just relying on a computer.

    Part of the reason we learn math is NOT just to get the right answer, but it is to develop the mental muscle.

    Body builders do not get their physical muscles developed by getting a machine to lift their weights for them.

    The analogy of developing a mental brain builder can only be accomplished by doing as much as possible in your head.

    Machines can help build bridges and buildings, but someone has to have the knowledge and skill to design those machines.

    Computers are just toolsto help you do calculations,
    but they can not do the thinking for you.

    Wolfram is an misguided idealist that believes that he can build a thinking machine that can completely replace a human.

  31. Can he provide us some sort of guidelines or a "syllabus" kind of thing? I think the idea is great! Not only for math, but programming as well. I am a programmer and I still struggle with some math concepts, anyone up for designing such a syllabus?

  32. This lecture in beginning confuses math with analytics and statistics. The 4 steps are fundamental to how to solve a problem analytically. Makes some good points about teaching math early. I am not a great mathematician like wolfram however as much I know about math, I respectfully disagree that we teach computer based math. This would not allow individuals to understand mechanics of math conceptually. By reading once we shall retain less and lose math specialists. If we have not done matrices by hand can not explain what is going on or how to feed computer with right form of vectors. It will weaken foundation….

  33. I pay for YouTube red. This video, by embedding a blackberry ad towards the end, violates the contract I have between YouTube and myself to not play ads in YouTube videos. Despite this video being a high quality ted talk, I have reported the video to customer support for spam.

  34. this is amazing, ive been thinking about this since I was in high school, I always hated not knowing the formules and the use for them, hence why no one ever remember them or cares for them

  35. I use Mathematica myself, as a degree holder and math hobbyist, and a lot of the time I just use it to help me clarify and visualise things. I actually ENJOY figuring out real word problems WITHOUT using a computer, then use the technology to verify the answer, as it's so much more satisfying to do it yourself.

  36. I'm using Mathematica to study Calc 2 and Multivariable Calc, and ODEs…I now resent all my teachers that insisted doing computations by hand. I am learning more concepts per day than I learned in an entire week in High School math. Any curriculum that focuses on computation rather than emphasizes concepts is a weeder class and against your interests as a student.

  37. At 10:35 his quick substitution of x^2 with x^4 had me ROTFL. It's even funnier because he stopped at 4 for the deepest and most abstract reason, completely at cross purposes with his own narrative, and yet his original point stands.

  38. really compelling. I think it is great future in the math educational domain that math teaching would be divided into two sides. first it is the compulsory course which teach students  about step 12and 4 and the elective course to teach students step 3

  39. i went to the wolframalpha and it said become a pro member to see the equation's solution. This fuckin why people learn to calculate, to not to be the slaves of the likes you!!

  40. He is not saying to do away with the calculating step completely, but rather to give more emphasis to other steps.

  41. The man speaking is a great mathematician. So many people in this comment section are misunderstanding what he is saying, his core arguments. Math is something beautiful, but most students in schools today, will never see that beauty. Their curiosity will get bogged down in repetetive, time-consuming calculations, that they do not understand the ultimate purpose for. Teach the children firstly creative maths, how to use it like an artist uses language, make it fun, interesting, challenging. Then learn them how to use the maths for solving different problems, that has a real-world potential. Teach them how to ask mathematical questions first, how to play with the math, how to visualise it, how to sense it.

  42. To be honest i think it's time to stop saying what's better for other people. It's impossible to predict that, so i think the only solution for education is the free market.

  43. I decided I was going to re-teach myself integral calculus; that my "A" grade in the 3d tier university I attended wasn't deserved. I got good at doing really difficult integrals. But I found that for any integral I could solve (evaluate), some truly obsessed math geek could solve an even more difficult one that I had no clue how to approach. So feeling "adequate" was always moved one more step away. It finally hit me that not only wasn't this making me any better at math, but I was forgetting the basic structure of what integration does. I was forgetting the art of "setting up problems." I was solving puzzles, like a retired man in a coffee shop solving a crossword puzzle. Ultimately it had no value besides self-satisfaction.

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