Why teach calculus?: Daniel Ashlock at TEDxGuelphU

[Music] so I'm going to talk about a sort of an abstract topic why do we teach people calculus how many people in the room actually have taken or have to take a calculus course at some point at 12 yeah how many of you are sure you're going to need it okay we had a 90 % drop in the number of hands that went up and that's part of what I want to talk about you may notice that um while they look sort of like snowflakes the last one is horribly wrong no natural snowflake has a five-fold symmetry and this is actually the result of my trying to de-stress at some point so here is something many of you may have been horrified by in the past this is the actual key to calculus The Dread limit so what we want to do is find that purple line there that's going along and just brush the parabolic black curve at one point and the way you do that at least in most calculus classes is is you start with two points the one you're interested in and one much higher up and you find that line then you sneak up a little and you find that line then you sneak up a little and you find that line then you sneak up a little and you find that line now as long as you're sneaking up each of those lines has a point the one you're interested in and it also has a rise over a run so you can find the line but then at the end you've just got the one point so the rise is zero and the run is zero what's Z divided by by zero actually it can take on almost any value and that's sort of the point of limits so limits this horrible calculation where we start out dividing by something that's going to be zero and then do a bunch of algebra and at the end we're not dividing by it anymore so we can just plug it in um it's a way of figuring out what answer you would get if you divided by zero if you could divide by zero not a fun process and it's sort of the first thing you have to learn in calculus so um here's a little bit of the truth calculus is built on infimal differentials DX if you've had a class is probably the first infimal differential you've seen and they were invented a long time ago by lib nits and then Newton invented them well no actually Newton invented them first but he refused to tell anybody so liess gets credit we use libnet notation for them um and there was like a 200-year argument about what they were what they meant if they existed if they were rigorous and eventually we managed to put them on a footing it turns out that they are reciprocals of infinite numbers how many people in the room didn't know there were a lot of different kinds of infinite numbers okay yeah so the rest of you did know it cool um and in fact this is this was one of the sort of horrifying things that grew out of this we managed to demonstrate that there was no Infinity large enough to tell us how many different types of infinity there were if there were it would lead to a horrible logical contradiction and that's what happens when you think about math too much and it sort of arises naturally out of calculus which is something that for some reason we think we should teach to people who are in their first year of University I am I conveying to you some sense of the insanity of this okay so what's the result almost all calculus classes are taught as if calculus is a religion rituals for solving specific problems are memorized and very little if any mathematics is actually taught the tech needed to the techniques you need to actually understand calculus are taught to people somewhere between the fourth year of a math major and the second year of graduate school at that point you've had enough practice that it won't actually put brains on the ceiling when your head explodes but you we can't teach you calculus so what we teach you are the results and useful techniques that arise from calculus so how did we get here well first of all calculus is incredibly useful for physics and Engineering it's an important field of mathematics it's used some in chemistry and biology and economists both use and misuse it quite a lot um the original reason calculus was taught so broadly in University was because the Russian Communists have more intercontinental ballistic missiles than we do and so for a very long time Russia which was about 150th the size of Western Europe and the United States working together terrified the world and one of the side effects of that was we decided we needed a lot more aerospace engineers and Rocket scientists and so calculus was made the default math we had to teach everybody because as I said it's incredibly useful in engineering okay so there's also an appalling ignorance to the alternatives to teaching calculus as the first advanced math course I'll go into that in a minute and finally calculus was when calculus was adopted as the default advanced math high schools taught a really solid course of algebra trigonometry and geometry a high school would have eight math courses that everybody would take and you had to pass them all to graduate I believe we're down to needing to take one and a half math courses to graduate from high school now and if you're bound for University that goes all the way up to like 2 and a half and they've deleted 3/4 of the math curriculum from Ontario high schools worse they've stopped teaching math they've started teaching how to get a good score on a standardized test which is actually you know very different from math okay now I tried to put these quotes up here to sort of move my agenda along and the one at the top is the that annoys me the most I've never heard somebody say I'm a baker and had somebody reply oh I'm terrible at baking I've never had somebody say I'm an auto mechanic and you know the answer is usually something about well I'm glad there's somebody that can do that or you know oh I do some of that myself not I suck at that but people are proud of being mathematical nitwits they're happy they can't do math mind you it means they have a lower income it means they're easier to cheat it means they're easier to lie Li to and yet they're proud of it and this is largely the result of the way we teach math and I think this focus on trying to teach people in possibly weird math for their first math is at least part of the problems so Alternatives combinatorics and probability Theory this leads into statistics and gives you a solid foundation for it how many of you here have to take a statistics course at some point yeah most of the hands went up okay um and the other thing is it gives you problem solving skills and makes you as Citizens harder to cheat it's also far easier to teach than calculus there's linear algebra this is used extensively in computer Graphics if you have a game console there are several chips in it that sit there and do nothing but linear algebra to manipulate the point of view and render the shapes you know when you're looking at Lara Croft you're actually looking at about 600 linked matrices it just looks like a cool woman okay so it's all so linear algebra is used in business it's used in planning Logistics ecology um for instance one of the things that happens is plants that have just fallen off a um seeds that have just fallen off a plant aren't ready to germinate and there's an internal biochemical process by which they become ready to germinate and so you build this model where there are new seeds seeds a year old seeds that are 2 years old seeds that are three years old you have probabilities of either sprouting or going on to the next year and when you fit the parameters to that model what you get is an understanding of when you need to put down the herbicide assuming we haven't learned not to yet um to kill the weed there's a wonderful plant called a fox tail that puts hemoglobin into its seeds and the amount of hemoglobin that goes into the seed gets smaller the further out from the core of the plant the seed is and what happens there is the hemoglobin absorbs oxygen starving the embryonic plant of oxygen but since it's a different amount in different seeds they come up at different times that's why Fox tailes keep sprouting All Summer in low numbers and the compartmental model for that one has a complicated term for how much hemoglobin is in the seed but that's linear algebra you need not calculus the other thing is linear algebra actually is necessary to do both Advanced calculus and differential equations it's easier to teach than calculus too so that one would be good even for engineers as a course before they take calculus algorithmic mathematics used in computer science extensively useful and underused in biology so I said I have an endowed chair in bioinformatics what guf did was they hired me to dig biologists out from enormous piles of data it used to be 20 years ago that when you found a gene you had about two years to characterize the gene find mutants and write a couple of papers on the gene now it's the sequencer found 200 genes this afternoon quick do something with them and so what they have me do is I do what used to take a biologist two years in 80 Seconds I don't do it as well and I have to use a computer and I have to talk to biologists a lot to make sure it's doing it right but this algorithmic mathematics is taking over a lot of fields of biology um it's also critical for many types of modeling how hard it is depends on how you approach it you can actually do non-trivial algorithmic mathematics with Excel so graph Theory this is you've got some you've got some locations you know the routes between them you know how long they are and you build a network structure graph theory is enormously easier to teach than calculus it's used in City Planning Logistics epidemiology one of my students won a best paper at the conference award at a conference with 800 plus papers working on epidemiology using graph Theory um so social network analysis you can bet Facebook has 50 or 60 phds that know graph Theory working on how to well essentially make you look at Facebook more but it's still interesting mathematics um if you haven't seen vard's videos and I actually looked up the spelling so I'm sure that's right they are wonderful they cover many mathematical topics she speaks about twice as fast as I do and she has wonderful pictures um the one on hexaflexagons is beautiful as well and in fact um she's an also I think one of her lines is so suppose you're sitting there in a boring Calculus class and you can't stand because the guy's going so slow and he's not making any sense at all and it's not really math anyway but then you think about some graph Theory Well turns out that and then she does these beautiful pictures so it's worth looking up her videos as well what's that well okay so you can read the word but okay so this is one this is my second favorite Einstein quote I'm I'm going to end with my favorite Einstein quote but that is a picture I did in an attempt to relax I often do fractal pictures and this one it isn't just that it's a fractal it's one that I sort of um I made an error writing the program and it did something I'd never seen before and you know much as with though much less significant much as with Alexander Fleming's discovery of penicillin I went wait that's odd oh cool and it turned out to be a new type of fractal and it's actually quite ornate that one I did for somebody for a Christmas card it's got vaguely christmy colors the thing is the image is generated by a very short computer program 36 lines only eight of which have anything to do with the math the rest are things like hi this is a program please give me some place to store the picture and so on so eight lines of computer code generate that and most people that have finished four semesters of calculus couldn't understand it and they couldn't write the eight lines of code I think that's a little sad if we're going to teach people math why don't we teach them math that does stuff like that I mean it's not even hard to teach it to people once you've got the computer so what's stopping us from solving these problems first of all most K through2 teachers came up through the calculus based system and were convinced that math is hard ritualistic and has few real applications this is like child abuse it's the gift that keeps on giving we teach incomprehensible religious math to the people who will be teachers and so they teach incomprehensible religious math to their students and it goes on and on we've been treating calculus and differential equations as the final ultimate math since the late 1940s and so there's incredible institutional inertia expect we've never done this before type objections I um we recently developed here at g a one credit your freshman physics and calculus at the same time course integrated physical science 15510 and what I did was I tripled the amount of math I was covering and covered it much faster and the result was one person fled two people dropped normally you get 10 to 12 people flunking and 20 people dropping from a course the size I had so teaching the math much faster I think worked a lot better partly because people didn't get bored and stop coming to class we also wo it together with the physics demonstrating actual applications I talked with Martin who was the physics instructor and we collaborated and made sure that stuff meshed at least fairly well okay and there were a lot of we've never done that before objections when we were setting it up universities currently being treated by the government as optional extras that can be used to balance the budget the government officials would never say this but if you look at how the money is allocated it's obviously true this means big revisions to the way we teach math are unwelcome budgetary disasters then some faculty focused on teaching and that word some is there for a reason would need to be retrained or replaced we've got some wonderful faculty here who are teaching focused they can teach any course you throw at them they know what they're doing we've also got some that know how to teach a course and if we wanted to change the course they would be dead and so having faculty like that since you don't have to pay them much because they don't know how to do much is another part of keeping the budget under control but people that are hired because they're cheap tend to be less versatile and then the final barrier is the textbook industry how many people in the room have paid too much for a textbook wow okay more hands than any other question for the record and one of them just stayed up a while maybe you paid too much for more than one textbook um so essentially I have two published textbooks um one of them is on how to use Darwin's theory of evolution as an engineering optimization tool it's fairly obscure um the other one is a freshman calculus text that bizarrely I translated from American to Canadian so now um I also developed math 10:30 here the business math class if any of you have taken it yeah I apologize I know but remember the business professors were the ones that told me what I had to put in the course um and we've decided that we're Andrew the TA who helped me write the book have decided we're going to give the book away we're going to put it on the web we're going to keep updating it everything will be completely free I'm also working on an update of my Evolution as an engineering design tool book it will be given away free I have had it with the textbook industry right now their current plan to keep sucking money out of you guys is that they will come up with these websites that you have to pay to subscribe to that will write my homework problems and test for me and grade them now I have a number of problems with that so does the university the university has a rule against you being charged one cent beyond your tuition for anything you have to have to get your grade um chemistry was already using these tools and when that rule got passed they had a real problem and had to somehow compensate but the thing that really worries me is um they're not really all that good at writing questions and tests and if I give a test that contains impossible horrible wrong problems I'm going to have real trouble blaming the Textbook Company so I'd prefer to get myself in my own trouble my own way okay working toward change what can you do about this mess well you can exploit the fact that the internet makes it much easier to get content out I'll say it again look at vard's videos KH Academy which hosts viart and helps support her work um is another attempt to let people educate themselves they put a whole lot of course materials on the web for free where possible push alternate content when I was at the second year I was at g a couple of students and I developed a course a math course for art majors in which we emphasized all the mathematics that was developed to help the Renaissance artists actually learn to make realistic looking pictures perspective and so on there's actually quite a lot of math buried in art and I have um I have three Publications on using mathematics to generate art one of them in the Journal of mathematics and art and so you know it's a going concern um there's math and computer Graphics you can teach mathematical biology without an obsessive calculus Focus I did this back at Iowa State where I used to work you can teach algorithmic math in spreadsheets um even simple things like you know is what the guy said about your car payments true or not and a spreadsheet makes that easy to calculate and then I really want to push the idea about probability Theory as the first Math course um when I was an undergraduate I invented this game you get four six-sided dice you roll them if there are no sixes you lost if there's a six you put or one or more sixes you put them aside roll again and you keep doing this but if you roll and don't get any sixes you're allowed to pick up a six and go again so the game can take a while and the thing is most people think the odds of um actually getting four sixes are about one and five I don't know why but that's where people arrive intuitively and the actual odds are one in 8.2 which I can calculate using linear algebra and that meant that I would sit there and let people roll dice and if they want I'd pay them $5 and if they lost they' pay me $1 and I never had to buy lunch and it's because nobody knew any probability Theory except me in the whole room okay try to move Public Schools away from standardized testing preparation if you prepare your students for standardized tests you just made the results of the test meaningless the test you're supposed to be assessing whether you've learned certain skills if you teach exactly what the test is looking for there's no evidence one way or the other that you taught them the other skills but ask any of my colleagues in the the math department the people coming out of high school have had monotone decreasing uh pardon me they've been getting worse at doing math monotone decreasing is actually a technical mathematical term which I should avoid around people that came out of the current high schools the other thing is it damages the students one of the reasons the math is so repetitive is is they just go over and over the material that will be on the standardized tests in an attempt to get a good score because the school's budget depends on it this is a truly diseased and corrupt system that is making the time and money spent on the testing wasted and it's also really damaging the students okay be aware there are a whole lot of other sorts of math besides calculus set theory group Theory difference equations graph Theory combinatorics agent-based modeling linear algebra and actually I was going to put the whole list on but it doesn't fit on a slide even if I stick to topics you don't need an advanced gree to understand we've worked out an incredible amount of mathematics and almost none of it is taught at University and G teaches less than many others that's my favorite Einstein quote and it speaks to the fact that nobody can try and make you learn all the math because it's infinite Kurt gal proved a truly horrifying theorem called the incompleteness theorem the formal statement of it is any sufficiently complex formal system contains true facts that cannot be proven inside the system in math we worked out something called the axioms of set theory there are a basic set of things that we have to assume before we can do math and it turns out that if you keep going the number of vines you need to explain everything that's true are infinite and so math is quite literally a bottomless well now since my job is discovering new math for me this is good news other people may not be quite as happy about it as I am um but it means there's essentially endless Wonder out there in mathematics and we've carefully designed a math education system that avoids almost all of it um here are two more stress relief pictures that one's a fractal and that one I can't prove it's a fractal but I think it is and it looks cool okay so essentially math goes on forever most of it is unrelated to the real world probably most of it is unrelated to any world that ever existed which means there's infinite room to make your mind stronger and more powerful okay that's probably enough for now

Channel: TEDx Talks

Views: 117,602

Keywords: tedx, TEDxGuelphU Inside Out, ted talks, TEDx, TEDxGuelphU 2013, tedx talk, tedx talks, Why Do People Teach Calculus?, ted x, TEDxGuelphU, Education, Calculus, University of Guelph, ted talk, ted, Daniel Ashlock

Id: 7LUkglCRCVY

Length: 20min 36sec (1236 seconds)

Published: Mon Jun 16 2014