welcome to the science for the public lecture series science for the public is an organization committed to bringing science information and issues to the general public visit our website for our program massachusetts institute of technology and an associate professor of applied science and engineering at olin college of engineering dr mahajan took his undergraduate degree in mathematics from oxford and physics from stanford and he received his ph d in theoretical physics from cal tech in one nine hundred ninety eight he became a founding member and the first curriculum director of the african institute for mathematical sciences in capetown south africa later he joined the faculty of old in college of engineering and he was also associate director of the teaching and learning laboratory at mit dr mahajan is very much committed to improving mathematical training of science and engineering students especially and beyond that too including intuitive techniques in mathematical education in general he would be the first to say this is not some version of the new math rather he wants people to learn the importance of guessing and trusting a hunch in the process of solving a problem that's what scientists often start with too much rigor he says mortis we urge you to see his very popular ted x talk at cal tech about all of this the links are on our then page for tonight on the science for the public website mahajan has produced two textbooks both published by mit press demonstrating the intuitive approach to mathematics i'd like to point out that both are available for course in print but also online under creative commons noncommercial share alike license we're very honored to welcome sanjoy welcome so i'd like to get your input about mathematics generally to start with most people think it's way too hard at least in this country and that also it's for engineering and science but not for the rest of the world what's your take on that please i think mathematics is as important as language it is a language and it's the language in which we live so much of our modern society and it's one reason historians have actually looked into this why is it that mathematics is so important numbers in particular and strangely enough it's because we have democracy and we have a large country and so it turns out that as societies grow in size and in democratic nature they need numbers more and more which seems a bit strange why would that be you know what it seems to be is that in a small elite govern society you don't need numbers you just have the elites in a back room cutting all their deals you know so you think of england in the seven hundred one thousand shares or even a lot of the  hundred but as a society becomes more democratic you need some way of having a universal trust that people can look to because they say oh well just talk well that's your opinion and that's just my opinion how can we find a way of building consensus across a whole bunch of different opinions and it seems like numbers are one of those ways that people trust across a wide spectrum of ideas and opinions and so numbers become more and more important as we become more democratic to other developed nations that there's there's this problem here but with it we can come back to it but you have this great mathematics background yourself what does mathematics bring to life apart from this need in the practical world today but a lot of people especially from mathematics say no this is a window to reality what's your take yeah you mathematics is like that it's connective tissue that link together the world it makes no sense without mathematics the patterns in the world why do things fall how do they fall how do they move the social work that's the physical world the social world the flows in the social world money moving around  resources all of that there's mathematical constraints on it if you put something if it moves at some speed and it moves or certain time it gets only so far you can get ten times as far so mathematics in some way yes that's a rather interesting but do you think that's true of other cultures of their own i think other developed societies actually live with reality more yeah oh yeah for example we don't have an infinite amount of oil we have public right yes whereas we say oh let's just tear up all the trade i ingest so more cars yes and the american way of living exactly exactly so you could say that we have a very irrational we're not reality focused person festive perhaps but on the other hand you can't operate a modern economy or anything else anymore without without the mathematics but i was wondering or yeah i was just wondering if you see nature for instance differently because of your background oh yeah i don't think i  actually little mini springs but they're the bonds between the electrons and protons in every year molecule of nitrogen and so why is that a spring well mathematically it behaves like a spring so when the sunlight comes in and we goes the electron wiggles just like it was attached to the proton by a spring and so in the atom you have the spring vibrating up and down and then that spring itself gives off light and so that light we're seeing is the light from all these little springs that are being shaken up and down each one the little radio so you get insight immediately just because of knowing this and see the connection exactly exactly that is one of the things that i think scientists say right away you see patterns you couldn't do it otherwise you wouldn't see patterns you would see connections and you'd also overflow yourself your mental registers so you only have you know seven or eight say slots in your short term memory to think with and then after that it just gets overflowed and then you can actually think anymore and so if you look around the world and you see thousands of different little anybody things of no connection you can't make any sense you have to find ways of connecting it and one of the ways is through mathematics that's interesting because it seems as though the brain is toward that actually that may be a very fundamental part of human brain because it's making patterns is so important but that's only way to survive yes right if you to books you give a lot of attention to problem solving because they're designed to help engineers and scientists think a little differently when you say you know intuition and gesine and going with your hunch es this is very different from the way engineers and scientists are trained and so how does that approach benefit scientists and engineers to start with well benefits them because so much math education is to put it mildly a big fat lie so what you see in most math courses is here's a formula or here's a proof and no one ever tells you what problem are people trying to solve why did they think this even might be true what ideas would you used to check whether it's really true so the whole construction of this house you never see all you see is a completed house and then you tell students all right well now go make your own house oh you know you don't think you can do that oh you can't make a perfect house in the first trial you just not you must not be home go here not inside and so very so what we do is we just have the students instead of actually showing them how we don't tell them any of that we just shown the finished product and then we look like wizards and so that it's very good for the egos about teachers but it's very bad for the egos of the students i can't resist asking you know the great  accomplishments in both engineering and structure architecture from ancient times like the peruvian walls the incan walls with you know cement or anything and people figuring this stuff out and building norma's buildings that are architecturally very complex and so on but they didn't have this kind of education well some of them had some of this education actually so in the temples  is the you know in language there's many ways to express yourself yeah ok well write your essay in the way you think best captures we want to say and convince people so that's a creative process and there's lots of freedom there but with matheson we tell him ok you want to get here is your problem the one answer and we teach people there is one way to get from the problem to the answer and that kind of teaching actually produces terror and people they think ok i'm here on this one mountain peak and i get this other man ph and there's this you know valley  so they've done they've actually developed that in more areas of mathematics so they've taken subject x and y and z and each of those they've learnt that we're thinking and so then what happens is something new comes along like an actual problem and they say oh i never took a course in that and you say well of course not i didn't take a course in every single problem that you might have used to find ways of thinking that you're ready to use for any problem and that requires a flexibility and so if you're still here for you know just in your own personal relations if you're scared or worried you become less flexible may become more rigid exactly that's what happens so it becomes defensive and intellectual defensiveness in this case though it's it's almost wonderful to hear that they're terrified they've done really well with the regular system right and so we filtered for the people who have done so well with it you know you want to change that yes right and how do they react because you you you work with these students and turned them sort of upside down in the process and thinking of a couple of reviews of your book by mathematicians and engineers who said oh this just changed my thinking you know i had to work at it but it changed my thinking and i wonder how the students react well i find what i have to do is i have to be very unsubtle about it so i give them problems that they think they should be able to solve for ok why is the sky blue yeah i know they take in the physics of spring they're taking all the physics to bed with that kind of question but they never put those things together and in with ways of thinking they were allowed an answer and then they realized oh wait a minute even though i know all these little pieces there are some big picture that i'm missing and you know in some ways i'm a little i'm kind and i rub their noses in the dirt just to show them that hey it's not your fault you're right and you're curious but you just learnt the wrong  thing you learned a lot wrong way of thinking and so you need to learn other ways of thinking so that you can solve new problems and not just solve problems that you've already seen before or that are similar to what you've already seen before so that's the way i try to get them to change their thinking do you see a big result after this kind of progression once they get through this barrier that they have a different mindset yeah i've got nice letters from students after  they graduate go on to other courses or other jobs and they say oh yeah that was so useful i was you know on this project and it seemed like it was going to take months and months and then i said wait a minute let me just use my street fighting approach  here and figure out what's the one main problem in this program and just solve that and then oh the thing ran ten times faster and my whole problem was solved and then i had two months where the bosses let me do whatever i want because they were ready for to spend three months on that project right but i think maybe my favorite of all was one time this was in cambridge and we invited the england we did by did all the physics students to dinner with physics professors and so there was two or three of us with about twenty of our students but i was happy to see them taking these ideas of ok we don't have to have the exact springing subject it's not even clear how you'd better that but we can just use in our everyday life right if speaking of everyday life not quite but it seems as though  certainly in science i'm not sure whether it's true in engineering but in science people do have to make guesses simply because you don't have the facts at hand so the theoretical level what came out of this  it seems unnecessary is is is there less fear with people with that kind or people in science like in theoretical areas of science so that's the interesting part is that people who do these professionally they've found one way or another even if their education didn't teach it to them they found some way to make guesses so either they didn't find that way and they left doing science or they found some way and they stayed in it and even maybe even more true  engineering and so all the people who are doing these fields say oh yeah this is exactly what we should be teaching our students when i ask people what is it first what frustrates you about students you know about what we what they learn after march they are degree course they say well the thing that frustrates me most is the students will do a calculation no come up with absolute nonsense and they'll just write it down because it came out of the calculator yes you know without thinking design number make any sense is it even actually measure the world yeah and so you know my colleagues and scientists engineers around the world they say you know this is so important to teach people but somehow they just haven't managed to teach it to them you know engineering students probably especially apparently gripe a lot that a lot of them not the courses they did take don't so let's help them they don't and that's sort of your main point there what why is this happening is because so the there's people who write just purely for writing not necessary to express to other people what they're saying ok and i think that's find but that of those aren't shouldn't be the people who teach writing or teach art or teach math to other people who need to use math yeah because they have a completely different way of looking at it and so when you take a math course say as an engineer if you take it from out department usually they'll teach you how mathematician things which is ok here is there's a difference and there's the and then the mathematicians have this culture of kicking away the scaffolding which is ok here's a proof here's a theorem here's a proof but no idea of the guesswork that when intel why do we think this theorem yeah why would we think this might even be remotely true why would we even care about it none of that happens right and so i remember when i said undergrad i used to torment the math grad students by just asked their i'd asked them well so what are you working on for your ph d because i was curious and they would tell me something and i usually wouldn't understand it very well i would understand some of the words but then it say can you give me an example and they would look at me i think we haven't gotten there yet we have got is that and so are not many got into it and we're not even they're not even thinking in that way but if you're going to use mathematics to understand the world the world is one big example yes you know if you can't find an example you're not looking at the world right so once you introduce this to students they sort of change their their mindset and they can move on but you brought up an interesting point is that that there seems to be a divide between the way mathematicians think in the way engineers think but still even studious you say students in engineering take their courses for mathematicians and that is a they might get misled that way but even so they sometimes feel frustrated that whatever it is they're learning in this package is not somehow apply couple to the kinds of things they need to solve this do you agree yeah and  one i find is after they come out of most of the math courses they can actually solve problems that require any kind of creative use of mathematics you know they're really good at doing long chains of algebra yes that does i mean they are they exactly are not exactly spite of this deficiency in the in the in the curriculum they have accomplished tremendous things some of the accomplishments in the last say fifty years are amazing but you only need a small fraction there to be really understand what's going on to do that and what i'd really like to see is that fraction become not so small yes so that most of the society understands how to use mathematics right and that's really the next thing this is that this does seem to be and they lead all the time when we are i think rather consciously were raised to believe that in this country at least i have discovered elsewhere that's not the case right and so here in this country we actually believe man it is an inborn test that's another know it's a gift either you have it or you don't exactly know i mean i remember when i was a kid in school you know obviously i was good at math then too and people say  that's what you want to be fraid of it right and so if you think of it as a skill yes you know no one thinks oh you're just born being able to play basketball yeah room realize you need to have a passable coach you have to your swimming and you have to be taught how to swim or you're just born knowing how to ride a bike well no you have to be taught this but we think crazily for mathematics it's inborn ability either you have it or you don't which is very convenient for teaching because it means you want to teach but you know it means we should actually think have all this research now on expert performance what makes expertise we should just use that for math teaching is going right you talk about these various kinds of tools you know in your in the one book street fighting book but that can you give us some ideas about things that people ideas you would employ for everyday use for young people ordinary mortals the same for most people i think what goes wrong in their mathematics education is that math becomes meaningless so numbers don't have any meaning so in fact you can almost see it if people are reading an article which has numbers and you can just replace every number with very grunting pretty much how people hear the article so and that's terrible because they what they means is politicians in the charlatans of various stripes can produce all kinds of schemes and tell people lies and just basically bamboozle into doing things that are against that that's right so what you have the first step is you have to start giving meaning to numbers and the only way to make meaning of things is you have to connect them to other things you understand so you first have to start connecting numbers to other numbers you know this is just you know almost basic knowledge of friendship how do you get people together they have to have connections and become friends with each other well same thing with numbers you have to get numbers to be friends with each other and you have to make friends with numbers so the first most important tool is comparison every time you see a number compared to something else so i'll give you an example so i read in one of the boston papers maybe a couple years ago there was stanley cup was happening or the stanley cup was wanted something that and this was a d in boston so the newspaper said and in a million cheering fans filled the streets you know near north station or wherever they stadium was to celebrate the stanley cup coming home and now ok so now when you compare it you realize har that number is just bogus and so that's the fundamental tool this is stark comparing numbers and always do things by comparison oh that's two times bigger that's one half as small that's three times bigger well that's one hundred times bigger but it's thout for example murder rates traffic accidents so do you always see them given as just simply a straight number there were x thousand say thirty thousand traffic deaths in the u and there's maybe three thousand traffic deaths in the u k that's a i think that's roughly right the question is let's give meaning to these numbers you can't just compare them just like that because there's a different number of people in the u s and u k so the comparison is well in the u s there's ten times as many people five times as many people in the u s as the u k let's say ten times the traffic this so the u s is twice as dangerous on the roads as you case and then you have some meaning to these numbers you started to and so when you first start doing it you know just like when you first start making you go to a new cocktail party and you feel a bit awkward and i don't really know people that well because start to make friends with people and start to chat and the only connection well the same thing these number you'll start to make friends you say oh that's what a million you did that you would not be so easily misled by media nonsense so think of you know all these demonstrators or this or that or troops or anything you like but you're given a good bit of is hard to look at demonstrations and see the total number of demonstrators so there's all these claims are one hundred yes and they all vary yet all vary and so what i find is that when the demonstration supports whatever the government is doing sort of  have to make sure they're too stupid to use it yeah so let's just make sure they don't understand anything about numbers yes but so then once you have the tool of comparison now what you'll find is that all these numbers out there and they start having meaning to you have meanings he want to grab them you get overwhelmed so to solve that problem you need to actually throw away information you need to approximate so there's basically two things you need to do you need to give meaning to stuff and they need to throw away the extra information that you don't need so i can give an example of yet what happens if you don't do that you get lost in the details was i asked students to estimate the health budget of the united states on which to spend every year on health care and so the students she did are actually quite a good estimate she said well you know there's like you know insurance premium per person is something five or ten thousand and they're say three hundred million people so that's you know something like a couple trillion dollars or one and a half trillion dollars let's say i think she got t cluttering your head up with noise you know so make your approximation move on this is almost counter to everything we've all learned at least in my job aeration certainly you'd be so careful and rather detail oriented and you're saying no no gotta get past this you go into it your idea to replace the completely the way we teach math or is your idea complimentary that is you're not throwing out accuracy or whatever what i want to do is turn it on its head so what happens is now we do detail detail we seek figgs is the mantra everyone learns as significant figures and you want all these rules for if you have three significant figures you know after the decimal point or two yes point seven to two and you come and with this other number some people get the number ten of about all and so you learn that and that and then you drive out most of the students in the ones who survive into graduate school maybe as post docs and become faculty there they learn how to approximate and thin way it was too late and i saw only point one percent of the population so i want to turn it upside down and say ok let's actually do things roughly first yeah so and then as we get more more and more need for accuracy we build in more and more accuracy as you get more  mathematical tools right differential equation you do more and more things and you get more and more accurate as you need but the first things in how you approach any problem first should be rough approximate what you feel is it if you can't tell whether it's millions or billions why you shouldn't actually be make general things which is very much a part of a creative process whether it's scientific or artistic or anything else so in i need to close up it looks like so that you can have talk to the audience here a bit but i  i think we can say at this point you think most people could really learn a lot higher level mathematics than they do learn i think well i think they could have a much deeper understanding of how numbers feel in underlie our world that you think everybody's complaining anymore because we're global and we need to change everybody needs to change do you see anything as exemplary or even close in terms of this kind of education yet there was a an example in manchester new hampshire the superdome so manchester new hampshire had the best mathematics curriculum i think the world ever had so the superintendent of schools said the math teacher best math curriculums the world has ever seen and he was driven out by the column the standards to test the testing meaning yeah even then even then there were yeah but i was and i example remodel and that's your approach would be much more along those lines which is a real discovery method as well everything makes sense why didn't your docs yeah i'll write letters to their students are and giving them an environment where now it's number rich and showing them how they connect i needed right i really appreciate that got to go look that up now but before we close i'd like to know something about your own educational experience you were a child you to tell that story again when you were a we child when i was on their body i was just two i think and i was all sitting on the potty and my uncle was sitting with me and then i looked up at the ceiling the rough no i see i what happened and i think all you need is to be one year ahead and then this feedback loop happened so what happened is i was at least one year ahead so when i got to first grade they said oh you already know all this stuff when you do you know everyone with our regular teacher you can go do this or that you were sort of like on your own so i kept doing so i was always protected from the standard math teacher just because i was when you're ahead and so then that just increased and increased because i wasn't crippled by the it's money i did theoretical physics and i am an engineering professor and yes and i turns out i teach all the courses that involve math yeah so all it turns out that all the things that i find very easy to teach they all are connected by mathematical i am yours and then i realized oh they're actually in this department as department they're actually the same course i say same mathematics yeah just give it a different name yes we call it signals the systems here we call it dynamics here we call it circuit cleary there but it's actually one in the same thing and if you can see that well then you've actually advanced because right instead of learning three separate courses you can learn one thing so the students are beginning to understand this thanks to your intervention in this regard i need to write so there's lots to do so i think it's i have a sociological explanation for that which is that until recently mathematics and physics were one subject so mathematicians if you look say in the seven makes some of his there was no separation and it's only basically in the early twentieth century late nineteenth century that they started to split and that's actually when mathematics stop being so useful for physics so it really was people were trying to see what's the structure in the world and how can we explain it and mathematics was the explanation for but also astronomy so the greeks were fascinated by astronomy and so partly they were looking at the structures in the astronomical world to their patterns of periods in the  and string theory and all so that's actually i think string theories are further illustration of my point which is i think string theory is going to collapse at some point it's big ocean liner of thousands of different theories and it hasn't put them in sort of dial that number and we have no idea why that number is what it is like how have you electron is string theory you would like it to explain some of these numbers but it hasn't explain anything in that department at all and people say oh well the math is very beautiful maybe so but it doesn't seem to have anything to do with the physical world so i think string theory is also illustrates the problem it's the flip side of it once the mathematicians in the physicists are separate then you start doing mathematics that has no relevance to the physical world example his special theory of relativity started i mean it's hard to know when anything fully started because these ideas start in this sort of back here mind but he described it as starting from wondering what it would be like to ride on a beam of light you know and that was just wondering about it and then so that became the special theory of relativity and then he started to wonder well how do i connect this theory of motion with gravitation for by then you know two hundred years because for example the special theory says that no signal can move faster than the speed of light for it is related to that moving on the beam of light but in newtonian gravity for example suppose the sun has some mass here and we have a feel the gravitational field one turning gravity of the mass of the sun changes instantly responds to field a change in the gravitational field of the sun and the earth would change its orbit instantly but i said well that can't be true that doesn't work with special relativity how can i put those two together so yeah there's a bit of mathematics in there but it really was a physical problem he was wondering about an intuitive problem and he figured out ways of melding with together and so he yeah he was a great example of intuitive reasoning in fact all to the end of his life that was his objection to quantum mechanics that it makes no sense you know he never believed in any of the math he said ok fine you can have your math but it's physical nonsense there must be something underlying it underneath it that gives you these probabilities and i refuse to give that up then i think he's he's been he was right and we're going to see more and more people pointing in that direction and saying oh yeah einstein's intuition was right yeah yeah entanglement i think of is actually what's really common there is that there's two different observers who happen to share some knowledge and then we we are mistakenly putting that into the system but it's really states of knowledge in the observer and people are starting to work out that way of thinking but it's a new way of thinking it's related to what's now called quantum basie and basie in ism as well but the old ways of looking at quantum mechanics are starting to crumble and so i would even put too much money on entanglement so i'm not necessarily so here's a way of getting things to seem like they travel to speed of light which is suppose we both you know we exchange a message so we both know something and then we go very far apart from each other and then people ask me a question do you know what's the answer to x and i give an answer and they ask you at the same time what's the nature of x and we both give the same answer it would seem like that knowledge went faster than the speed of light because really far apart but actually where it came from is that we both knew it together and there was some common kind of knowledge we had we share for whatever reason so i think we'll see more ways of looking at quantum mechanics like that and not actually information travelling at the speed of light sure so what happens is the sunlight comes in what sunlight is is it's basically it's the representation of wiggling electrons in the sun so the sun is really hard the electrons are going up and down and so one way we represent that is to say there is an electric field which is moving up and down and so there is an electric field which puts a force on electrons and also in protons but they're so heavy they hardly move so let's just take the electrons so the electron in the nitrogen atom moves up and down like this now the sunlight has several colors in it let's say just to let's just take red and blue right it's a mix of the whole spectrum of the stick those two so there's a red light which is a  fast jiggle and each of those jiggles radiates so the slow jiggle radiates red light and the fast jiggle radiates blue light from this atom now the the thing about the way is that the faster they accelerate the faster the more radiation they get and it's actually a very steep dependence so if you double the acceleration you quadruple the amount of radiation so the because the blue light gives  faster jiggle exhilaration is higher and them for their slow reggie go so there's a lot more blue light coming off of these one thousand atoms than red light just because of the way radiation works so the result is as the b moves through all the air molecules it's coming along there's this nice straight beam so when you look at the blue sky you're not looking at the sun directly you definitely should look at the sun directly you're looking to the side and the sun is coming that way through the air molecules you're looking at them this way and they're radiating out of the beam so what's coming out of the beam is more blue light then red light because the blue light jiggles the electrons so much more and they radiate so much more so you see much more blue than re even though blue and red are stay calm roughly in the same amount in sunlight so sunlight looks white but what comes off it looks blue and then you can check that this isn't total nonsense which is there is a time when it's sort of ok to look directly at the sun once that yeah a sunrise or sunset so what's happening then is that you know you're looking directly at that beam which is going through a very long distance of the atmosphere because it's just barely grazing the atmosphere here is the earth and here is the sun going through and here is you so what's happening is all the blue light is being kicked out of the sun beam and what's left is the red light so the blue sky and the red sunset are actually the same thing