i've always been interested in math is far as i can remember and actually farther back than i can remember the first time i can remember my own interest is when in first grade our teacher asked all of us what we wanted to be and i never went around and i was very proud that i actually managed to say mathematician but i think she didn't expect that far from what she thought a five year old should say s you want to be a magician then i thought well no i wasn't quite what i meant but what i later realized is what i wanted to show people is the magic of math so maybe she wasn't so far wrong after all  so at least since i was five or six i remember math is my home base when ever i thought things were just crazy i remembered oh peace and quiet of a math problem  world was your own pieces in it were there were limited there weren't too many crazy things that could happen and you could just chase your intuitions and ideas where every wanted to take them so i always found great happiness in learning math and doing math problems and then what happened in school was i was always just at least somewhat ahead of the curriculum in the school so for example in first grade you know i already knew how to add and subtract and so i basically didn't spend any time doing what they were teaching us the teachers that whole year you can do something else so i then learned other stuff while everyone else was being really brain damage by the regular teaching method and so i was spared that just by being a bit ahead and so now in second grade third grade i was farther ahead because i hadn't been subjected to the usual teaching system and so then i got to do other stuff and while the everyone else was forced to memorize multiplication tables i got to you know learn about division and square roots and i went farther and farther so just from there happenstance of being curious about math early and then became bit ahead i always was spared the curse of our usual math teacher so i really loved math and still do and so i always wanted to use math in some way and then as i got towards high school i got very interested programming which now i realize oh that's just a species of math and so it wasn't that different but then i thought oh it's different because math is on paper and programming these machines but really computers are mathematical systems we made them that way so you're actually doing math but a different slightly different kind or a way of implementing it so that's what i wanted to do then in high school i got selected to be part of the u s physics who in p had team so there was twenty of us from  round the country who were training to represent the united states in the rest of the world so i didn't end up being one of the five people who went overseas but i learned so much physics in that few weeks in the summer and my teacher there actually now are theorizing craft who is a renowned physics teacher in the country he's now a professor of science education nearby and you must boston and it's nice to still be in touch with my old teachers but i then decided ok in college what i'm going to do is study physics and so i did that but before i'd done that i had only been interested in for computer programming how to make computers smarter their particular way of being smarter was to play games better and so i had written with a graduate student carnegie mellon a world championship a solo program and so it played really good othello far better than i could barely knew the rules but i knew them well enough to program it then the computer did really really well and so that was fascinating to me and it was fascinating to me to see how the computer would learn so we figure out a way of teaching and have the computer learn from its own mistakes which is also what you know we'd like humans to do with the documents learn from their mistake let's see if we can teach a computer to do that so i had that thread going as well as the mathematical traditional mathematical thread and physics and so in college i did both  and then in graduate school i just started out studying neuroscience because my old love for learning and there first was teaching computers but then i was wondering well how did humans learn and how is knowledge represented so i was a graduate student in neuroscience but then my old bug bit me of physics and i decided to return back to physics which meant i lost my fellowship that i had for graduate school so my ad were in the sciences you normally you don't pay your own tuition it's either the research group you're working on or the fellowship not has your back covered so to say so he was ready to do that but i thought well i don't want  cost him so much money so let me be a teaching assistant i've heard of this course called order of magnitude physics which everybody said it was the best course they ever took at celtic so i said oh well i should take it and i thought oh no better that i should be a teaching assistant i should help teach it and i'd learn even more so i did that and at the same time i was studying for my qualifying exams in physics for the ph d qualifier requirement and so in those three months of being a teaching assistant trying to stay basically one week ahead of the students and studying for my qualifying exams i learned more physics than i had in my entire four years of my undergraduate degree so i thought high physics teaching and learning are really broken could be much much better and then i only later realized oh yeah i've had this interest in learning for a long time ever since i was trying to teach computers to play othello and much better and i always had this interest in math and physics so i guess it was inevitable that i'd be interested in the teaching about and physics and engineering and how to prove them but that's how i always loved math and physics and wanted to show that to other people and part of the way of showing that is improving the teaching so that you see the beauty and how it helps explain how our world is put together and how we put it together in different ways and hopefully better ways but with the teaching of math and science than engineering i see so much that could be improved and made better in fact that's just a very politely speaking the more honest me says i am just outraged by how terrible it is and how people's love that they often have from young gets crushed by how we teach it we for example we teach multiplication just as memorization six times nine hundred fifty four then the ninety six is fifty four and then we memorize the community of law that six times on is the same as nine times six but students don't learn it as the meaning of multiplication they just learned as pieces of linguistic symbols so a string of sound six times nine is fifty four and it could be anything and there are all kinds of experiments now that show that the way people mostly learn they haven't understood what we were hoping they would be learnt at a much more surface level and much more simple manipulation level or parent learning to a famous example of that was a study which asked students about pitchers of water so you have pitcher of water here that's forty degrees and a pitcher of water here that's a one hundred twenty degrees so that's quite hot and that's you know fairly cold a bit above freezing and then you mix the two pitchers together what's the temperature of the water and what his students say is they say oh do they mix together and think about what happened no they say oh there will be hundred sixty degrees so they just realize ok when you're adding things together you add you know it's that surface level so we have two numbers are forty and one hundred twenty and so we add them together because that's this it's this type of problem it's an addition problem so there's no connection to the world there's no meaning in what they were doing so it's just meaningless simple manipulation so of course after years and years of it people start to resent it start to hate it and as soon as they can be free they quit and they stop doing those subjects so we wonder well why do so few students want to study math and science well there's your answer because it doesn't seem to have anything to do with reality so what i would like to do my teaching and how i'd like  teaching to improve is i'd like you to connect to reality and so that you build deep structures of knowledge that help you see how the world is put together and it's not a separate compartment of your mind called school which is separate from the world and you do school manipulations here and then you go out and do things in the world know these two are actually one in the same and what you learn in school and in college are waves of understanding and changing the world  o first way to improve that is to change our focus our teaching right now we focus almost entirely on content you have to learn this one topic in this next topic the next topic in this next topic we transmit one topic after the other that we don't think well how are these ideas stitched together and how could students teach these ideas together so should they stay in their minds for the long term so what i eventually realized is that the topics are important and nice and important but they're not the fundamental thing we're trying to teach we're trying to teach ways of thinking that the itch can cut across all the topics now of course you have to have actual content that you're teaching otherwise you're just talking about here but you need ways of thinking otherwise each topic is this isolated piece of knowledge which soon gets blown away by the winds of new knowledge and information that stream through us so how do you connect things together well first you have to find out what are the ways of thinking that are really valuable and generally valuable and so that's what i ended up spending a lot of time many years working on and so my book street fighting mathematics was a temp to do that for mathematical problem solving so there are six ways of thinking and modes of reasoning in there that we can use for applying math problems across the world of science and engineering and then my next book which is just about published in november the art of insight in science and engineering tries to do the same for science and engineering for developing modes of reasoning ways of thinking that help us master the complexity of the world around us and so both of you these books they're freely available to everyone one that's already published is already out there street fighting mathematics in the next one soon as it's published will be freely available to everyone so that this knowledge can be widely used and people are don't have any barriers to accessing it incorporating into their way of thinking so what are these ways of thinking so that in the art of insight in science and engineering i found that you could divide the ways of thinking into two broad groups because the whole goal is to master the complexity of the world because that's what happens if you and that's what we'd like students to do and what they usually can't they come to a new problem and say oh my god i've never learnt that particular problem that seems so impossible how do i even get started it's as if they're confronted by this giant overgrown jungle of mess and they need a few machetes so they can hack their way and start making a path and clear a path so the first thing you do with complexity is you have to organize for example your desk if it's just his pile of papers with no rhyme or reason to it you can never find anything if you never do anything you spend all your time just chasing your tail looking for what you want you organize your papers into piles you for example clean up one part of your desk the next part of your desk or your kitchen that's what i try to do one end of the kitchen to the next in the kitchen not the entire thing at once which feels too overwhelming so there w papers that deal with teaching science these are all my papers that i need to read the next day these are all my papers that deal with teaching engineering so you may now have higher level ways of looking at what was once a big mess or for a kitchen ok here is my spoons here my forks here mice nice if you don't put them all in one big jumble of a drawer then in fact we even have drawers the purpose being to make obstructions so these form the first group of tools organizing complexity or if you get if organizing is not enough you have to discard complexity so that's the other choice the other route you can take and there's two ways to discard the complexity you can either throw away complexity that isn't really there it was just on the surface kind of like barnacles incrusted on a rock and you can scrub it away and you know lost yourself or to the information in the problem or you can throw away information and so example of throwing away a spurious complexity not really there is using symmetry so you look at the problem from the right way and the right orientation the right point of view and all of a sudden things are simple you haven't changed the problem you just looked at it in a different way and now it seems like oh how could i not have known that but you thrown away some complexity when you do that certain example of the using symmetry is suppose you have two and one and put them underneath then you have one in three that's four two into that's four and one and in three and one that's four so now we combine those two ways of doing one two and three we just get a bunch of fours and there are three of them so that's twelve  symmetry that this problem is the same when you turn it around and then you've got rid of the complexity of it that all the numbers are different now all the numbers become the same just by this one symmetry trick but as po said method is a trick i used twice so once you use something like that twice you realize oh there's a method there and you give it a name you make an abstraction and collect all those uses together like you do for a folder of all your papers on one topic so that's an example of using symmetry in getting rid of complexity but no loss of information example of getting rid of information but actually throwing away information given the complexity is rounding numbers so when i go to the grocery store i always have a game with my daughters we carry thing in the cart and then we have to do is estimate how much the bill is going to be before it gets totalled up or before we see it even being totaled up now the hard way of doing that if you want to exact answers you have to look at what the price of every single thing is and add it all up now computers are great at that which is why the cash register was invented and it does that for us but if you want to get a feel for the numbers you have to throw away the complexity so now one of my one daughter who's four she always says it's eight dollars no matter how much and i say ok i'll be great but unfortunately that's not right but that's the number she understands for my older daughter and i we have more of a discussion about it the way i usually do it is i look at the shopping cart and say wow roughly how many items are there eighty items and not how much is each thing just as i was picking it off the shelf ok maybe three or four bucks so that's going to be something like three hundred dollars for the bill and so i've thrown away a whole bunch of information point no item was exactly three dollars or four dollars and therefore not exactly eighty items in the cart but it's rough round things you throw away complexity but then you get some sense of what things should be in the world and you can apply this principle to all kinds of calculation in the world and ways of understanding the world to reduce the complexity such that we can fit it in our minds so that we people can master the complexity of the world and not just say well you know my computer do that for me so these in some made up made nine modes of reasoning and ways of thinking too for organizing complexity  three for throwing in the complexity with no loss of information and for for throwing away complexity with loss of information and so these ways of thinking nine together cut across science and engineering and what i did is i realized finally this came to me i'm very late in life after working on this book and we teaching for maybe fifteen years then i finally i saw oh really the ways of reasoning are more important than the individual pieces of content which you may not use again because the ways of reasoning are more general and more transferable to students first students later in life whatever they may do now as i was saying before you need the content or the ways of reasoning are just air so they're taught in some context of some actual problem but the ways of reasoning these nine ways of reason are what i want students and all readers to take away so they can simplify the large complicated wonderful world that we live in and feel they have some mastery over it and some connection to it so i'd like to show a couple examples of so they're the same except for one thing which is that this is two centimeters thick in this is one centimeter thick but they're both thirty centimeters long about one foot and five centimeters or about two inches wide and they're made out of the same piece of pie and they were cut for me by the machine shop of the cambridge university physics department and i've used them ever since to demonstrate sound and blog can you hear that so that was the thin block now i'm going to tap the thick block in exactly the same way which was just holding it here and tapping it right in the middle you think that will have a higher pitch lower pitch sure the same pitch as the note i made with the thin block so when i ask groups of physics students or often physics professors and mechanical engineering professors and students to think about that i usually give people a few minutes to argue back and forth about what they think will happen and then we try it and then before we try it though people make their arguments so here are some of the arguments that come up the one hand thi s block the thick block because it's thicker has a longer distance for the sine waves to move so it takes more time for the sun we have to go back and forth from this thick block and then block so the  note made here in the thick block should be lower the other hand some people say well but the thick block is stiffer than the thin block because of that maybe this should be higher frequency than other people say well yeah that's true but look at this length this length is the same as this length and so the sound going to take the same time to go across this length so the frequency should be the same now people say yeah and it's also made of the same wood so the sound speeds the same should be the same so take your guess would you think it will be you know give you five seconds to just make your gut intuition and then we'll try it ok so here is the thing block again just to refresh our memory can you hear how it's quite a bit higher from the thick block so this is by far the minority guess usually no one guesses that it will be thicker maybe one person out of twenty and even then they're not sure they're sort of apologetic for guessing something that most people don't think but why might that be well one of the ways of thinking about materials is as springs is just a big collection of springs so this block is just whole bunch of springs connecting a whole bunch of balls and those balls are what atoms now would isn't really an element so there no such thing as an atom of wood but just pretend that's part of throwing away information so here's a wood atom with other wood at another wood and they're all connected by springs and when you bend this block which is what happens when you hit it it gets bent you're bending all the spring you're stretching all the springs and you're putting a whole bunch of energy into those springs and because this block is thicker the spring here at the top he compared to the top of this block get stretched far more twice as much so they have much more energy in them and they're twice as many springs so this block takes so much more energy to bend so it's much much much stiffer so the spring model when you work out all the factors of two shows you that this block is much stiffer than this one and far stiffer by a factor of eight in turns out than this one which is far more than its weight is only a factor of two heavier so that slows it down but the fact of stiffness speeds it up so much that this has a much higher frequency almost a factor of two so that spring model that was the clean walk the spring model is an approximation it's a model it's a way of throwing away information about the world because it's not exactly a spring it doesn't behave just like a spring but it's close enough and it's that willingness to use approximate ways of reasoning to throw information that overwhelms us that allows us to make simple models that help us understand the world so that's one example of ways of thinking and how they fit into a problem and a way of looking at the world here's another which is these two combs so these two cones they're also cut out from the same sheet of paper but this one has twice the radius as this one so it is four times the area of the paper that it was made of and the way they were made is you put a circle out of the paper and cut one corner of the circle out and throw it away and then just tape it together and here's the tape line so there's your comb now the question is when i drop these cones which i'll do when i'm standing on the chair and second when i drop them how fast will they fall relative to each other so here are the two cones which i'll drop in just a second and you'll see which one gets there first or are they about the same so over three choices are that this big cone goes about twice as fast or about the same or that the small cone goes about twice as fast now why would the big cone go twice as fast well because it's heavier so it pushes through the air faster why might the small cone go faster oh because it's smaller so it doesn't have so much air to fight through or maybe they're the same because there's two effects balance not clear which one is right so let's try it one two three so they hit almost exactly at the same time so those two effects almost exactly cancel almost exactly the same time and what that tells us is that the two effects exactly balance and cancel each other out so on the one hand the big cone has more air resistance because it's bigger the other hand it has more drive because it's heavier but because those two effects cancel out it didn't matter whether it was big or small they landed at the same time what we know is that er proportional to the area of the cone and then you add one more way of thinking which is part of the whole collection of ways of thinking for throwing away information without losing any information to it throwing away complexity without losing any information that method of reasoning within this group is called dimensional analysis which says that you can't add apples to oranges so you can add feet to seconds or meters to square metres so with that one idea plus the result of this experiment that they land at roughly the same time you can work out the way  on speeds areas and density of air you can work out the gas mileage of a seven forty seven over a car you can work out how much peanut butter a bicycle it's needs to bicycle one thousand miles so you can work out the fuel efficiency of a bicyclist and you can do that all from the simple paper experiment and you can check the conclusion with this one follow up experiment so the follow up experiment is to stack four little cones together so here is one two three four so now i've made a heavy small cone four times every in this one and now the conclusion of the other because it's bigger because it's the same size as this one can't because the air density is different because it is the same for both of them so it has to be because it's going to fall faster how much faster well it turns out it should be a factor of two to compensate for the factor of irresistible so let's check that so we could actually drop them both the same time and measure the speeds and say oh yeah that looks like about a factor of two but instead what i can do is i can drop them with this one at half the height of this one looks about half the height and then i drop them at the same time and hopefully they land together exactly at the same time with almost exactly half the height for this single code so that's confirmation that the other experiment can lead to nonsense and gives us more confidence to use it to understand the physics of fuel consumption in the world around us the purpose of these ways of thinking in these demonstrations is to show you a different way of thinking about math science and engineering that it's not just a series of formulas one after the other which need to be memorized and been reproduced on a test in the numbers plugged into these pieces to get some other numbers but it's a way of understanding and making sense of the world it's a way of also getting away from that too often happens from just teaching through formulas as we teach people riggers important riggers important riggers important and the importance of rigor but then the result is people get rigor mortis they get so stiff they are so afraid that if they don't have the formula they can't do anything that they sit there paralyzed and there's no joy in that there's no enjoyment in looking at the world and understanding it with that kind of paralysis so these ways of thinking in these are ways of unblocking unsticking and freeing your vision in your sight your ears in your soul senses to understand and see the connection in the world around you and the demonstrations or to show that these apply to real objects that the world is actually in some way built of these ways of thinking and it's a beautiful way of looking at the world and that science math and engineering have a lot to say to us in that everyone should keep studying those for many years and make our world better