Hey everyone, and welcome to Tutor-Terrific. Today we are going to derive part 2 of the sum and difference formulas, the formulas for the sign of the sum of two angles and the sign of the difference between two angles. These formulas, like the ones in part 1 for cosine u plus v and cosine u minus v, are fundamental in trigonometric analytics, or analytic trigonometry, as it's also known.   Now the proof or the derivation for this set of formulas is very different than part 1, so let's get started. So here we have this, what seems like a very wacky set of right triangles and other geometric shapes, and this is what we're going to use to derive the sign of u plus v formula. Now, notice these two angles here. I've created two right triangles with each of them and the bottom one is angle u and the top one is angle v. It's as if the triangle A, E, B was sat right on top of the hypotenuse of triangle A, F, E, and that's exactly how we've set it up. So I want you to notice something, the angle in question is u plus v and that angle is actually here. This is all u plus v. If I add those two angles together, I hope that makes sense to you. Now, we're going to write out directly using this figure and a way to express the sign of u plus v. Right off the bat, if we use this particular right triangle here, triangle A, B, C, we will see that this angle, the sign of that angle will be the opposite side B, C over the hypotenuse A, B. So we will write it like that, B, C over A, B. Okay, now everything else after this point is going to seem rather unintuitive to you. Some proofs happen to be that way, which makes them that much more remarkable to have solved in the first place. Here's what we're going to do. I'm going to break up B, C into its two segments that it's equal to. B, C, as you can see, is equal to B, D plus D, C. Okay, so let's write this instead like that. I'm not going to change the denominator, just the top. It's going to be B, D plus D, C. Now, we're not finished yet. What I want you to notice is that D, C is actually congruent to E, F, Y, because this is a rectangle, a rectangle, if you know from geometry, which you should, is a parallelogram with all right angles. So that means opposite sides are congruent, because for a parallelogram, all opposite sides are congruent. So I can rewrite this as B, D plus E, F all over AB. And finally, what I'm going to do is I'm going to split this up into two fractions, okay? So now I have B, D over AB plus E, F over AB. Okay, now I'm going to do something extremely unintuitive to you that you wouldn't see coming, but notice how I've used a couple right triangles, but I haven't quite used this little guy right here, triangle DBE. Now, you might say this is very wild and strange, but I'm actually going to multiply this first fraction by something. Here is the unintuitive thing I'm going to multiply by. I'm going to multiply this fraction by BE over BE, which won't change anything because BE over BE equals one. And so BD times BE over AB times BE. This is perfectly rational and reasonable to do. The other fraction I'm going to, unintuitively again, multiply it by this giant hypotenuse AE of the lower triangle, AFE. So I have, oops, not, equals plus EF times AE over AB times AE. Now here's the beauty in all of this. I'm going to, since these fractions are multiplied by each other, I can reorder terms because it's really all one fraction and the commutative property stands for these types of numbers. So another way to write this is BD. I'm going to switch the denominators over BE times BE over AB. Still completely equivalent to the original product. And since the BEs cancel each other, it's equal to my original fraction I had on line two. And I'm going to do the same thing for the other set of fractions. I'm going to reorder those denominators. So I have EF times AE over AE and then over AB. Okay, now the whole point in doing this is because of what these ratios stand for. But first, I need to define a few other angles. Notice how these particular sides, AF and DE must be parallel. These two lines are parallel and you can tell because of the angles. This is a right angle and this is a right angle. Since they add up to 180, these lines must be parallel to each other because they're connected by a single transversal, EF. What this means by the alternate interior angles there, if you look at side AE, that is another transversal for these two parallel lines. Thus, by alternate interior angles theorem, angle U is here but it must also be here. This angle right here is also U. Now, that doesn't matter itself. This little triangle isn't that important but what is important is this triangle I highlighted earlier which seems to be ignored. If that is angle U, look at this angle here, angle BEA, that itself is a right angle because the way I stacked the right triangle on top of another one. What that means is part of it is U, the part I've already marked. The other part that I haven't marked must be the complement of U which means it is 90 minus U. Again, this is not that important except for what it means. Okay, so if that's 90 minus U, what it means is the most important and that is what angle BE, excuse me, angle DBE is.  Okay, so that angle U is the same as this angle U which leads us back to here. Our very important fractions. Let's talk about what BD over BE is. Okay, so with respect to angle U, BD over BE would be adjacent over hypotenuse which makes BD over BE equal to cosine of U. Okay, now we're starting to get somewhere with trig instead of all these crazy letters. Okay, let's look at BE over AB. Okay, now we're looking at a giant triangle like this. Okay, and we have angle V here. So let's look. What would BE over AB be to angle V? It would be the sine of angle V. Check it out. Opposite over hypotenuse. So that represents the sine of V. Okay, next we have EF over AE. So that's a part. Those two lines are a part of this triangle here, this lower right triangle, the larger one. And you can see that according to angle U, EF over AE is opposite over hypotenuse for angle U. So EF over AE is the sine of U. I think you see what's happening here. Why this was set up this way? Lastly, let's just make sure. AE over AB. AE and AB belong to the stacked right triangle that's slanted. If you look at angle V, AE over AB is adjacent over hypotenuse. So cosine of V. Look at what we have here. Remember where we started. We started all the way back with setting up some expression for sine of U plus V. And look where we got. We got all the way down to this, the desired result. So that means we have derived the sine of U plus V formula. Good job guys. Wow, that was a very unintuitive proof, but it works very well.  Okay, so I've made myself some more room here. And I've rewritten the sine of U plus V formula that we just derived. Notice though that I reordered the two product terms so that it looks like what you see when you open your math book before this was in front and this was behind. But I can totally reorder those two because of the commutative property of addition. Three plus two is the same as two plus three. That's why I was able to do this. It's just an example of its use. Now, I need a formula for sine of U minus V. And I'm going to do it by using the formula I just derived. Here's how I'm going to do it. I need a plus in my formula in order to use what I've already derived.  I did something similar in part one when we were working with the cosine addition and subtraction formulas. Now, let's go ahead and write what this equals. Okay, I'll write it down here. So according to my already derived formula, this equals the sine of U cosine negative V plus cosine of U sine negative V. Okay, as we did in part one, we will need to utilize even and odd function behaviors. Cosine is even. So if I plug in the negative of some value, what will happen is that the original value when it wasn't negative will be popped out, okay? So that means that cosine of negative V equals cosine V. So it's almost as if with even functions, you didn't plug in the negative, you get the same result as if you plugged in a positive. So this is just sine U cosine V. Now, let's come over here and look at sine negative V. Sine is odd. Any odd function when you plug in the negative of your original positive input, you get the negative of the output. So sine of negative V is equal to negative sine of V. So we get cosine U negative sine V. Okay, now there's a nicer way to write this so that we don't get confused that this is multiplication. We just bring the minus out in front.  And we have finished. We have proved both formulas for the sum and difference of sine two angles. Now, before I end this video, I just want to make a quick shout out to Khan Academy for the inspiration for this proof. Khan Academy did a similar proof back in 2007 and posted it in the glory days, the beginning of YouTube. All right, guys, thanks for watching. And now this is Falconator signing out.


Good afternoon everybody and welcome to the Berkman Center's Tuesday luncheon series. We're so thrilled to have so many people come out to hear Justin Reich talk to us about the future of online education and lots more and that the response has been as robust as it has been. I'm sure more people will be streaming in so so glad to have you. Right before I get started and before we introduce Justin a couple of logistical notes. If you've not been here before this is a Berkman Center tradition but we announced that we are being webcast live so hello to our webcast listeners out there we are going to post this talk on our website afterwards so please just know that comments etc will be recorded the talk is being recorded and we'll be posted online and for folks watching on Twitter please tweet at us if you have questions or comments and we'll keep an eye on that stream for the Q&A portion and we'll elevate those to Justin during that time and folks in the room as well since there are so many folks here we'll do in person Q&A as well as from Twitter. Did you have a question? A hashtag can be... Berkman. Hashtag Berkman or you could just at the Berkman Center on Twitter and we'll we'll keep an eye out for that. And we're so so pleased to have Justin Reich join us. Justin is a very proud Berkman Center fellow he has been a fellow for the past couple years now. He is the co-founder of the EdTech teacher a professional development firm the EdTech researcher blog for Education Week which is really terrific and you should check that out there's a link from the Berkman Center website to that blog and starting on June 3rd he will be the Richard L. Mentchell Harvard X research fellow so joining the team at Harvard X here so we're excited to have him around Harvard more and to help us think through all these really challenging issues so welcome to all of you and welcome to Justin thanks I appreciate it so thank you all very much for being here I apologize for inviting you to a talk inside on what is the most spectacularly beautiful day in May I hope any of you on the live stream are watching this from rainy cold miserable places where it's not such an imposition on your time I'm really grateful and honored to have your time and to have different people from different parts of my life be here so thank you this is a remarkable time in world history it's a remarkable time in education for the very first time in world history we have technologies that allow us to take the insights of the greatest teachers in the world and transmit them globally for the very first time we have brand new technologies that offer the promise of substantially increasing human learning and achievement for the very first time just in the last couple of years we've developed new computational tools that allow us to use technologies to individually prescribe courses of study to young people to track their learning and to chart their progress for the very first time in just the last couple of years we've combined insights from the media arts and from computer science to develop strategies like gamification that allow us to leverage human motivation in different forms of play in human learning so really it's just been in the last year or two that sort of everything has transformed the only question really left to ask is whether this is a moment of disruptively transformational revolution or of revolutionarily transformational disruption so I'm kidding but I'm only half joking and I think the interesting thing is I can't tell you exactly which half I'm joking about so I'm pretty certain that some of the ideas and some of the technologies that we're developing right now have tremendous potential to expand the human achievement I'm also in and learning and development I'm also absolutely certain that a lot of the things that we're doing with online learning with education technology are taking some very old and in some cases very tired ideas and education and putting them in shiny new form factors I spend every day of my life in this space and I often can't tell which is which so what I want to try to do with you for the next 45 or 50 minutes or so if I can contain my enthusiasm is to try to wander rather capaciously through this terrain one of the things that I love about my job and my work is that education is just the process by which we take young people and help them become full-fledged humans is inexhaustibly fascinating anytime you tug on one little piece inside that process you find it connected by a thousand threads to all different kinds of ideas and human learning and ultimately to all of the different facets of human society I had my undergraduate students at MIT about a month ago I invited them I forced them to have a debate on the following question should students in the Commonwealth of Massachusetts be allowed to independently study mathematics using online learning aids like Khan Academy and then take a standardized test to place out of that course to get course credit for that and to be able to be advanced without actually sitting in a class in that one hour debate these were some of the topics that I managed to jot down that we ended up talking about over the course of that hour we talked about cognitive science we talked about transfer we talked about busing we talked about broadband access and infrastructure we talked about teacher labor markets we talked about teacher professional development we talked about moral development in a conversation which seemingly was about a rather narrow question of independent online study you find very quickly that these ideas are hitched by as if by a thousand threads to all different kinds of ideas in human society and that is for me what makes this work so fun and exciting so what I want to try to do in this talk is pull on a couple of threads that are related to online learning and try to see how they connect to other ideas and education policy and in human society I'm gonna get it's a I'm gonna intentionally try to be wider ranging let me give you three anchors three essential questions three ideas to help you sort of navigate through what follows three kind of principles that organize my thought the first really important principle is that students learn as much from who we are as educators as from anything we teach them students learn as much from the architecture of schools from the grammar of schools from the systems that we create for schooling for the roles that we position students in in those systems as anything else that flows through that system the kids learn as much from who we are then is what we teach them the second really important principle I want you to hang on to is I find it extremely useful in education technology and online learning to develop the discipline of asking the question what's new what's really new here so this is one of my favorite things to pick on this is an iPad app called near pod and what near pod allows you to do is take these portable multimedia super computers that students have in their hands take a slide from a student screen and force it on to the screen of all of the students in their class which accomplishes very similar kinds of functionality as what previously was accomplished with chalkboards with flip charts with overhead projectors with whiteboards with document cameras with interactive whiteboards and with LCD projectors that's eight generations of technology that we've developed to take lecture notes and get them into the notebooks of students it's helpful to ask the question regularly what's new the third anchor that I want to give you which is useful in thinking about all kinds of technologies is what are the second and third order consequences of our decisions what are the unanticipated and unintended consequences of changes that we're making when we make changes to take some kind of perceived initial gain what's the possibility for loss what's the possibility that we have to lose this is a sensitive question for me I know that there were people in the 19th century who went to their graves believing that grammatical parsing the teaching of grammatical parsing was essential to the continuation of Western civilization and most of us have stopped learning grammatical parsing and we seem to still be here there must be other grammatical parsings in our curriculum in the grammar of schooling there must be all kinds of things that we're doing now that we're really afraid to lose and when they're gone we're gonna laugh at ourselves for worrying about them but there must be other facets of education that we've developed over the last thousands of years of turning young humans into adults that would be very very sad to lose it would be very very sad for us to leave by the wayside as we try to make other kinds of changes this is a talk in some ways more than anything else about loss about the fear of loss about the risk of loss it's a talk that's a little bit about policy about how little changes we make with technologies or big changes we make with technologies can influence the design of systems and it's secretly actually a talk about civic education it's secretly a talk about how we prepare young people to be citizens in our society but civic education is a little bit of a less sexy title than backpacks of cash and mook madness the but you but you've I've showed you my cards the thread that I want to pull on first is the thread of personalized learning personalized learning so about a year and a half ago I started hearing everywhere people being interested in personalized learning people who disagreed about everything else in education were agreeing that personalized learning was a good thing people from all over the political spectrum on education we're saying two things about personalized learning first that it was going to be good for kids and second that technology was going to make it possible when people who disagree about everything else agree that personalization is important you can be very sure that they disagree about what personalization means that they're not actually talking about the same thing so what I want to do is sort of unpack a couple of definitions of personalized learning probably the you know the place to start the signature study on personalization was done by a guy named Benjamin Bloom who's most famous for a taxonomy of ideas that he came up with but he did another study called the two sigma study or informally known as he put a bunch of students in a classroom in a control condition and the experimental condition was taking students and assigning them at random to be assigned to work with tutors and the students who worked with tutors perform a typical student working with tutors performs two standard deviations better on assessments afterwards as a typical school student in the classroom condition and thus we sort of took the entire 20th century edifice of educational research to prove something that like medieval lords knew a thousand years ago which is that tutoring works the problem of course is that we don't have the resources of medieval lords to assign each of our little pookies their own individual tutor and so the really interesting question becomes to what extent can we design systems that accrue the benefits of tutoring without incurring the expenses of tutoring and so there are a couple of different models that we're exploring to try to make that possible I'm gonna end up talking of and there's lots of these I'm gonna contrast two of them up I'm gonna oversimplify them in some ways I'm gonna be more critical of them for rhetoric sake than perhaps they deserve and I'll come back to these but I'll talk a little bit about blended learning and a little bit about connected learning them so let's start with this idea of blended learning and make some sense of it so the idea of blended learning is that we're gonna leverage technology to create learning environments where students have some element of control over time place path or pace that we're gonna somehow allow students to have their own navigational pathway through the curriculum to make this possible you have to fix the curriculum you have to have a body of curricular knowledge that you agree in advance that people should be learning and then what you're gonna do is you're gonna develop computational tools that assess students learning and you're gonna use those computational tools to develop a series of algorithms which say okay the student is performing at this level so here's the next logical learning object to send them they're having a problem in this way so this is the hint or this is the video or this is the question that we need to serve up it's virtually impossible logistically for a single teacher in a classroom of 35 students to be able to do this without some kind of technology mediated system a really useful example of a tool that gets used in blended learning environments is Khan Academy how many of you have watched a Khan Academy video from the beginning to the end okay so if you throw in that end thing and a few hands go down they're like about 30 seconds into one how many of you have ever completed one of the practice problems in Khan Academy okay good how many of you have earned a leaf in Khan Academy okay good good so if you some of you didn't raise your hand to practice problems and did raise your hand at leaf which makes sense but so if you go to the Khan Academy website if you go to the learn menu the first thing under the learn menu is the knowledge map and the knowledge map is a visual representation of the entire mathematics curriculum starting with single-digit addition and working its way through things past calculus like linear linear algebra and matrix algebra and things like that and the idea is that this map encapsulates the entirety of the mathematics curriculum and allows students to progress through that curriculum at their own individualized pace because the entire curriculum is there available for them the entire assessment mechanism is available for them in any moment to be able to pass through let me give you a short video here of some teachers talking about their experience with Khan Academy and I'll cut it in the middle but it's a tremendous impact actually that I'm seeing through Khan Academy the most impact I see are for those kids who really need the challenge and I also see the spark in the kids who struggled who in a whole group math lesson can appear to be lost and then shy and then try to hide and now they can hide behind their computer screen so to speak and continue to work at their own pace a lot of students are very cognizant of where they are in relation to their peers some of them won't want to appear smarter some won't want to appear slower those students will hesitate to ask questions they just want to be right there in the middle teaching to the middle only does the middle good and you see it seems like you're leaving out two-thirds of the class that way and so there has to be a better way to do it Khan Academy offers an assessment piece that's incredible it's so right there and immediate it's a quick picture of where students are at where they are having issues where their struggles are so I know who I need to visit with and where I can help them where I can come alongside and remediate and support they love those aha moments where they can be the teacher and they can tell me about something that they think I don't know how to do my students select weekly goals for themselves based on their own data so I have them looking at their own data now in a graphic way they're able to visualize in a very concrete way their progress you know usually you just move through a math classroom it's like okay yeah we finished this chapter we finished that book but the kids don't really see everything that they've accomplished and this lets them see it and it makes it real for them and then at that point I was excited for my kids because I saw that again as being a tool for them that ultimately would help them become responsible for their own learning and deciding hey this is where I need work good so that gives you a little bit of a sense of what some of this looks like in practice and again the ideas we're gonna take an individual student and we're gonna figure out how we can have an individual student map her own pathway through this material using a wide variety of online resources and using these computational tools to accelerate each individual student through this pathway as quickly as you know and efficaciously as we can get them through it and I you know I'll be critical of some of these things we go on but I definitely want to honor those teacher voices who are coming and talking about how this is raking a real difference in student learning in their classrooms. One of the most common critiques of education that is built on a kind of factory model where we take students in batches and we sort of define what we want to stamp on them and then we put them in these little eggshell crate-shaped schools and move them along as batches through these assembly lines one of the ways of thinking about blended learning that I find helpful is to think about maintaining a lot of the features of the factory model but giving each kid their own assembly line how can we how can we still stamp the same stuff on students but how can we move them through that assembly line as as as efficaciously as possible for each individual student. I'm gonna take a couple of detours here to a couple of important features of the system here's me sort of pulling on some other threads so one of the detours that you get into as you start getting into this is the idea of unbundling the functions of teachers so right now we ask teachers to be good at a lot of different things they have to be good at direct instruction they have to be good at providing pastoral care for students for designing lessons for assessing student understanding for translating the assessment data into actionable follow-up for individual students for developing curriculum for answering questions from tutoring students from modifying parents for providing the physical security of the classroom and for the students they're very few teachers who actually can be good at all of these different kinds of things and so one of the ideas that's embodied within some of these notions of blended learning is trying to unbundle some of these functions to try to take apart the functions of the teacher and say maybe there are only certain teachers that are actually really good at developing reading fluency in fourth and fifth graders it's actually pretty difficult to find people who can move the needle as we say in educational jargon on reading scores especially for low performing fourth graders and fifth graders what if we had those teachers only do that and we supported those teachers with a set of folks who are performing sort of simpler lower grade kind of tasks in the same way that we no longer in medicine have large groups of general practitioners but instead what we have is a you know sort of pyramid of staff from orderlies to surgical specialists who provide different kinds of functions one of the fun things to imagine with this is what some of the more sort of orderly like positions would look like so I was on a plane recently with one of the authors of the Gates funded measures of effective teaching study and one of the things that the measures of effective teaching study found was that to have reliable observations of educators they needed to have the educators rated on four separate occasions like classroom observations on four separate occasions by four separate raiders if any of you have any experience in the administration of schools when you hear that like four separate occasions for each individual teacher there's like a little voice in the back of your head that goes impossible so I was sitting on the plane with one of the authors of the study and I like so politely sort of turned him was like impossible and he said no no no companies are going to emerge and the companies are going to take fish eye 360 degree cameras and put them in the middle of rooms and they're going to stream video from classrooms to these cubicle farms where people are going to be sitting in these cubicle farms watching teachers and grading them on rubrics all day long for eight hours a day for 40 hours a week for 50 weeks a year that's what unbundling of education that's what sort of specialization in education looks like and then they're going to take this rubric data and they're going to feed it back to principals and schools who remediate on it it's an interesting question like who is the right person for that job and is there anyone willing to accept that job that you would actually want to evaluate your teaching so so that's one side of unbundling I mean the part of unbundling that we talk about much more are who's going to be at the pinnacle of these pyramids who are going to be the rock star teachers who are going to be the ones who are creating these online resources who are teaching the MOOC classes people like Sebastian Throon on the right who taught introduction to AI and people like Sol Khan who's actually talking tomorrow at both Harvard and MIT so if you want more of this go find him there's some so what's going to happen is that the people who are the best at providing direct instruction on certain materials are going to be so much better or at least incrementally better than all of the other folks doing this that will use their direct instruction tools in classrooms around the country there's a whole conversation that's happening right now around Harvard as a bunch of San Jose State philosophy professors have been objecting to how Michael Sandals justice lectures might be used you know and and I mean Michael Sandals with the classic rock star teacher I mean he fills stadiums of like 70,000 people in South Korea to lecture on justice and moral authority which like says something really great about South Korea but there's two there's two ironies of rock star teachers the the first irony is that the way that we achieve personalization is by giving people access to the exact same set of direct instruction that we all watch the same kind of Academy lectures in order to move at our own individualized pace again it sort of raises that question like so what do we mean by individualization by personalization the second irony is that these people who presumably would be at the kind of pin at the top of this education pinnacle would actually not have any direct responsibility for any particular student the people with sort of the most celebrity the most the most cachet wouldn't wouldn't actually interact with young people in anything more than token ways so that's a little bit about unbundling and I should say that unbundling that that rock star teachers probably aren't an inherent feature of blended learning environments but unbundling probably is you can't really have a blended learning environment without unbundling some of the functions of educators so again blended learning was this idea that we're gonna let students move independently through the curriculum by put right defining the curriculum in advance putting the curriculum in some online space and using computational tools to help students rock their way through connected learning addresses the same set of challenges around personal learning from a very very different perspective so the heart of connected learning is the idea that connected learning is realized when a young person is able to pursue a personal interest or passion with the support of friends and caring adults and is in turn able to link this learning in interest the academic achievement career success or civic engagement let me give you another sort of little short video from one of the architects of connected learning and again I'll sort of let it run for a minute and then pause we really think that that part of what's wrong with the current educational system and why people talk about it is broken is because it's fundamentally starting with the wrong questions the educational system often now starts with a question of outcomes it starts with what do we want kids to learn what are the goals and what's the content what's the material that they need to cover and then everything is defined by that it doesn't almost matter who the kid is so long as we're going on pace through the material and through the content and reaching those educational standards or those outcomes because that's our starting point our core question is what's the experience we want kids to have so the core question is around engagement and as soon as you start with is the kid engaged what is the learning experience we want the kid to have then you have to pay attention to the kid you have to start with in the design world you have to start with the user you have to start with the experience of the young person of a learner so instead of starting with the outcome which is for in most educational systems a math problem or a math fact or a literacy fact which is not particularly it's decontextualized it has no relevance to the learner we instead start with what is the experience really what do we want them discovering so connected learning starts from the position of the individual student and if blended learning takes the curriculum is fixed then most at most instantiations of blended learning take the curriculum is fixed most instantiations of connected learning assume that the curriculum has to evolve that assume that the curriculum has to be assaulted in some ways has to be transformed because there are a whole bunch of practices that kids find incredibly engaging that are really useful in the world that provide pathways to meaningful adulthood that have no treatment at all in school setting so things about media arts and gaming and coding and hacking and a lot of places engineering and computer science really don't have a lot of room there so what we want to do in connected learning is take this individual student and we want to surround her with a wide variety of online learning resources online communities mentors learning experiences experiences in schools experiences at home experiences in all kinds of third spaces in libraries and maker spaces and hacker spaces and try to try to use technology and the connections of technology to allow this individual student in a sense to shape her own curriculum to shape her own pathways towards learning and to define for ourselves starting with this question of engagement what what's going to be meaningful to her and how can we translate what's meaningful to her into a set of ideas around academic engagement and practices around academic engagement and meaningful work and civic learning so I've positioned these two theories of blended learning and connected learning as sort of you know evolutions and pedagogical thought over the last year or two but in fact in many ways they tie very closely to some of the oldest debates that we've had in education in some ways these are sort of new form factors new expressions of a debate that's been sort of kicking around between these two guys Edward Thorndike and John Dewey for about a century or so so Edward Thorndike sort of conceived of education as a science that there was a certain amount of stuff that we were trying to get into kids and we could measure the rate at which different teachers delivered that stuff into kids head we could figure out what's the sort of right titration or dosage of interventions to get stuff in kids head what kinds of leakage happens in kids head and all of this can be sort of scientifically evaluated and measured and tracked and then there was a guy named John Dewey whose philosophies are pretty complex to articulate in a simple sentence but the ones that are most commonly used or that he thought of education much more like an apprenticeship than as a sort of science science of delivery he thought of education much more like life itself than as a separate thing you do to prepare for life and in a sense sorry these are flipping through without my permission and in a sense what we're seeing now is sort of a shadow boxing of these ideas continuing presented in kind of new language in new ways the Dean the former Dean of the Harvard Education School and logman who is a historian said helpfully I used to tell my students only in part to be perverse that one cannot understand the history of education in the United States during the 20th century unless one realizes that Edward Thorndike one and John Dewey lost and if you look at the application of blended learning ideas and connected learning ideas sort of in the contemporary milieu you'll find much more conversation much more interest much more engagement around the ideas of blended learning around the ideas of a sort of accelerating individual student pathway through fixed curriculum then through creating students own pathways through learning I sometimes say that educators dream and Dewey and live in Thorndike I want to take a minute a detour to to bring up what I think are some really helpful ideas that Mako Hill who's in the corner here has discussed that I think help frame some of this as well so again this is another like pulling on a thread that hopefully will get us back Mako studied Wikipedia and he studied failed attempts at Wikipedia he studied seven other Wikipedia like projects that appeared around the same time as Wikipedia was started I mean he interviewed all the founders and he was trying to figure out why some of the why I went why did Wikipedia persist in these other seven initiatives not persist and he ends up creating this two by two which I think is really helpful where he defines each project is having as being either an established product or an innovative product and a statin innovative process or established process so some of these efforts at knowledge management were established products with established processes so they weren't doing anything better than Britannica was doing at the time so they didn't make any progress there were a number of these knowledge management initiatives that had really innovative products that said we're going to change the shape of what knowledge management looks like we're going to redefine the encyclopedia for the 21st century and people said I don't know what that is and I don't know how I can participate in that and I don't know how I can contribute to that and so those products were not successful as well so the one that succeeds has this pair of characteristics which is that it's an established product anyone looks at a Wikipedia article and goes that's an encyclopedia article I know what that is and I know how I can collaboratively participate in its creation because I understand what this form is I think these insights may have a lot to suggest to us about how education technologies are going to evolve anyone who looks at a Khan Academy video for two seconds has a light that crashes in their brain that goes oh this is a lecture at a chalkboard I know exactly what this is I'm totally familiar with this and it's in a kind of different environment but the product is established anyone who sends two seconds looking at a practice problem on Khan Academy will go oh this is a worksheet problem I know what this is I know what this form looks like and in fact it may be that that kind of familiarity you know without some kind of clever intervention is essential to scaling and growth it may be that we're that that that we're extremely attracted to the forms that we recognize in part because almost all of us have 12 or 16 years of edge of experience in education we're all experts in education at a principal who once told me the reason that we still have textbooks in our classrooms is because when parents walk into classrooms they expect to see textbooks and it's really cheaper to buy a bunch of textbooks and line the walls with them that it is to pay staff to sit around and modify parents who are wondering where the textbooks are that conservatism I think is something to keep in mind as we think about what's growing what's taking and what's not it sort of raises some questions about whether or not success whether or not scale is a really good indicator of success or whether or not scale is a really good indicator of doing what we've been doing before zing let me talk about two thing let me talk about the thing and it's sort of obvious that blended learning and connected learning have in common which is that they both start from the position of the individual they both in different ways start from the position of the individual student in one of them we're trying to get the individual student to move at an optimal rate through a fixed classroom and then the other we take the individual student and we try to position them in the center end of an ecology of learning but in both cases what we're trying to do is think about how we can change the primary unit of analysis in schools from something to individuals one of the things that you might say we're doing if you're really concerned about the ways in which students are harmed by being treated as cohorts by being treated as groups you might say that we're thinking about students as individuals rather than as batches if you are really interested in the ways in which students can collaboratively co-author their learning experiences and form ties and relationships with one another then you might say that what we're doing is changing the focus of education from communities of students to individuals I'm going to use the word group which is sort of a neutral word here but I'll give you both of these words as options I'm a I'm a formal social studies teacher as I'll talk about later and so it's my obligation to give you the tools to undermine my arguments but one of the things about making the individual the locus of education is it finds a very natural fit having the individual be the locus of pedagogy it finds a very natural fit with having the individual be the locus of school funding okay topic shift having the individual be the locus of school funding this has been recognized for at least 40 years so there's a guy named Ivan Illich who wrote this book called the schooling society about 40 years ago and he's one of these figures in American political thought that reminds you that the that the educational the political spectrum is probably best conceptualized as a horseshoe rather than as a line where the two ends of a political spectrum tend to bend towards one another if any of you are watching like the stop common poor hashtag on Twitter right now it's a perfect example of this we're like Glenn Beck and Maria Montessori have found a common cause together. Ivan Illich is the only person I know but I'm excited to hear if there are others who favorably cites the work of Paolo Freire who's this sort of Marxist postcolonial educator and Milton Friedman and he does so in the following way so following Freire Illich argues that schooling systems are sort of postcolonial totalitarian and oppressive systems which are designed to sort of prepare young people to serve their oligarchic overlords and they should be destroyed and they should be replaced with individual learning experiences and he has this chapter in the book called learning webs which is incredible to read because what he does is he imagines that using mainframe computers with punch cards what learners are going to do is identify publicly their learning interests on these punch cards and there's going to be a series of mentors and teachers who are also going to encode their talents on these punch cards and they're going to be able to connect with one another and form these webs form these sort of small M marketplaces where learners and educators are going to be able to come together he basically in 1970 in his mind invents the world wide web and then invents an educational system on top of it it's one of these sort of books of prognostication which is it's a little bit like reading snow crash if you're into that sort of from the from the future so so then he starts thinking about all right we've got rid of all the schools so what are we going to do with all this money that we were sending to school boards and he says this guy Milton Friedman has a great idea Milton Friedman's problem is that school boards are local monopolies they have no reason to get better they have no competition so rather than sending our education earmarks tax dollars to school boards let's give them to students and the students can spend them on whatever schools they want and Ilich says this is a great idea there's only one problem with it which is that it doesn't go far enough kids shouldn't have to spend these dollars on schools they should be able to spend it on any learning experience they want to they should be able to send it any way they want to so Ilich in 1970 ish puts together these four ideas that one of the central problems of schools is that the batch processing of students limits the development of their intellect and maybe destroys their humanity that the proper unit of analysis of education is not the class or group but the individual that technology enables the optimal allocation of learners to learning resources and that the proper mechanism for funding this kind of marketplace of learning is redistributing education earmark tax dollars from school boards to individual students the connected learning folks are sort of more keen on the top three the blended learning folks really are pretty excited about all four that people have found that there's a kind of natural alliance between the there's a kind of yoking of people who are interested in free market economics and people who are interested in education technology that these two sets of ideas seem to fit well together with one another in you know in any kind of policy process this kind of constituency building is complicated right so there's some people who are totally jazzed about blended learning who are like whoa those free market ideas might be a good lever to get more blended learning happening in schools and there are other people who have been working on this free market stuff in schools to the last 50 years are like wow this blended learning stuff might be a wedge might be a lever to advance this thing that we've been working on for a long time and there's probably some number of people out there who are like it legitimately jazzed about both ideas and our advocates of them but increasingly advocates of individual online blended personalized learning are finding common cause with people who are interested in free market ideas about funding education probably the ur text around this right now is a set of essays published by the Fordham Institute called education reform for the digital era which is definitely worth a read it's it's fewer technologists it's mostly people have been working on this free market stuff for a long time they've gotten really good at some of the language around this so one of the problems that Milton Friedman has he kept calling the funding mechanism vouchers and i don't know about you but like when i think of vouchers i think of the cab voucher that you get at Logan airport and you like walk down the row of taxi cabs trying to get a cab home and all the drivers are like no that thing is worthless so people have been trying to find new words for vouchers so the healthcare folks came up with a pretty good one which is premium support premium support that sounds pretty good the education folks have done much better they've described vouchers as weighted student backpack funding giving children backpacks full of cash how many of you in this room object to taking backpacks filling them with money and giving them to children like how many of you hate children and so you and so you'd have to say to yourself um at least there are no organizations advocating these policies that are so crass as to put images online depicting children wearing backpacks with money flowing out of them um but it turns out that you would be disappointed um as i would um i want to talk for a minute about the weighted piece because this is important both to be fair to the idea and to critique it so what do they mean by weighted backpack funding well obviously if we give every kid the exact same amount of money then you're just piling on benefits to affluent kids so one of the things that you have to do is you have to weight the backpacks you have to give low-income students more money in their backpacks than other kids and if you can give more money in the backpacks to poor kids than wealthy kids then schools who are trying to attract the services who are sort of trying to attract clientele in this marketplace of education are more likely to reach out um to those low-income students so if this would be possible what you have to believe um is that sort of conservative free market folks um are going to dismantle school board faced funding put in this weighted redistributionist policy and then vigorously defend this redistributionist policy when middle-class people find out that in their neighborhood some child got a check for $16,000 and your precious little poop you get a check for $12,000 um you have to believe that sort of Americans will tolerate that um we have we have all kinds of redistributionist policies in this country right now we engage them through these totally Byzantine layers of bureaucracy so that none of us can actually tell how much money the federal government or state government is sending to somebody's little pookie versus your own little pookie they they play a kind of function um but to be fair to the folks who are advocating for this they do have plans to to think through this um so whereas for 50 years freedman's advocates have tried to voucherize schools they now have new options made possible by technology which is they can voucherize at the class level um you could buy your education a little bit like buying sushi plates off a conveyor sushi belt restaurant um so you could buy algebra two from Khan Academy you could buy biology from the big valley creation science museum you could buy PE from Reebok you could buy english from the national writing project you could buy sex ed from CVS you could buy Spanish one from Rosetta Stone um you could rent a cubicle from your local charter school or from a local warehouse which has the cubicle farms patrolled by security guards who are attending to the computer equipment and you know if you're more affluent then you can have some tutors on site to deal with remediation um we start positioning education not as the process of taking young citizens and having a shared responsibility for helping them become young adults but position students as consumers who are using their scarce reap sources to optimize their own learning trajectories to their own individual benefit um and hoping that um what happens is that markets do what we hope that markets do best which is to efficiently distribute these resources um you know Ivan Illich when he imagines this process sort of imagines like small m marketplaces the connected learning folks imagine small m marketplaces like little local village bazaars um the blended learning folks do not imagine small m marketplaces um they imagine opportunities for now hundreds and millions and approaching billion dollars of venture capital to go into different kinds of initiatives which would be able to serve students at massive kinds of scales um so this conveyor belt sushi um is conceivably a kind of extreme or radical vision of the future but what interests me most about a bunch of these ideas is I think it's not hard to imagine what the beginnings of these changes look like what the thin edge of the wedge of these changes look like I think if any of us started to talk about say small rural high schools we'd start to say to ourselves man it's really not fair that instead of these small rural high schools kids don't have access to an ap physics class or an ap statistics class they should be able to take that online we've done that seem fair that they would be able to sort of buy an additional class on top of their schooling if it's your student if it's your little pookie who's in some terrible math class it's extremely attractive to imagine that you could keep the rest of your school and yank your little pookie out of this terrible math teacher's class and be able to buy a sort of additional math instruction um to allow them to continue to advance their education um without necessarily being stuck in this classroom that they don't want the thin the sort of the thin end of these ideas is very very sharp it's very very compelling but I think it wouldn't take too much of this kind of change for the community functions of schools to be substantially disrupted there are middle schools in this country that use processes like teaming where they have cohorts of students or communities of students who have the same set of teachers and math and english and science and history and that group of teachers can kind of follow and track the progress of those students over the course of their school day and over the course of their school it wouldn't take much more than 10 or 15 percent of kids to start opting out of this piece or that piece of their education for those kinds of exercises not to seem so useful anymore there's all kinds of interdisciplinary activities which happens in the best schools which you would imagine wouldn't make sense anymore if kids are sort of opting out individually of some of these different kinds of things there are science fairs there are art shows there are sports events there are pep rallies there are performances of understanding we're in the very best of circumstances communities of students come together and they learn from one another and they develop norms and they violate those norms and they collaboratively author their learning experiences and to me it doesn't seem like you would need to do too much to introduce some of these ideas about students as individual learners to make those community efforts seem much less worthwhile to seem much less possible to seem much less valuable the wedge of these ideas is a very very sharp tip and there is a whole lot of broader change potentially coming behind the wedge we could ultimately literally turn children from students to educational consumers changing their relationships to their schools their towns their neighborhoods their communities their peers rather than being part of a community institution where learners have reciprocal obligations towards one another students are consumers who are using their scarce resources to buy a set of educational products that are designed to optimize their individual learning opportunities so i want to reveal one further thing here which is that i'm a history teacher what history instruction means to me the best history instruction is when you bring groups of students together ideally as ideologically diverse groups of students together as possible and have them look deeply and closely at a set of shared texts and to discuss with one another to debate with one another challenge one another ask questions of one another there's nothing to race through there's no way to optimize the process of that kind of engagement you can't do it alone you also i would say can't do it without me there's also a variety of topics that show up in history which every student should learn i've been working for an organization called facing history in ourselves which develops a series of units around the holocaust and around the civil rights movement the holocaust is not something that kids get fired up about initially it's not something that they stumble across on their own but studying the darkest chapters of human history reveals the fragility of our democracy the fragility of our civil society the dangers of totalitarianism the dangers inherent in human beings it's something that all students should learn when i taught in an independent school i taught the history of race in american african american history the school was probably 90 white my entire class was made up of african american and latino students when students have choices they don't always pick in the ways that we think would be most valuable for them they're things that are worth having students learn i think we should make students study these things and i think we should make them study them together none of this i should say dampens my excitement for the potential of new technology mediated models i want my daughter to learn math if she gets really excited about math i want her to have pathways to individually follow her pursuits and passions i want my student my daughter to explore her interests to explore her ideas to have opportunities in the curriculum to find what she's passionate about and pursue them whether or not that's what everyone else in the curriculum thinks that she'd be pursuing and in the end these opportunities that place her as an individual in the center of the educational process may be so compelling that it's worth surrendering it's worth giving up some of the community learning experiences that are embedded in the dusty old buildings that we call schools i know that there are all kinds of ways that students are harmed are harmed when we treat them as batches rather than treating them as individuals i also think there are all kinds of things that we might find that we lose when we stop treating people as communities of learners and instead treat them as individuals i think we should consider very carefully what it would mean for our society to define students as consumers rather than as future citizens for which all of us share in the obligations of their development i think the changes that network technology has enabled could transform schooling very very quickly i think the thin edge of the web is very sharp and very compelling the thin end of the wedge is seductive i'm seduced by it but i think we should not let those changes happen without a very serious consideration of what would be the civic consequences of personalized learning so and rant let's do this for just a couple of minutes because we can go to 145 why don't you turn to the person next to you and chat for a minute about what some of the ideas were there that you want to ask questions about or push back on or celebrate or contribute to last time i gave a talk at berkman harry louis the former dean of harvard college totally tore me apart in very helpful ways it was like getting beat up by mike tyson so i hope some of you will i hope some of you will contribute to that process i'll take a minute to sort of flip through the twitter feed and see if there are folks who aren't here who want to ask some questions or contribute so we'll do that from two minutes and then i'll sort of surface some questions ready go hey let's come back let's come back the reason that i became a teacher is that i get bored very quickly the great thing about being a teacher is that whenever you don't like what's happening in a room you can just tell everyone to stop and do whatever you want to do with them that's the that's the whole point really that and sort of what is the small miracle of watching people develop and become better than they were before um i'm terrible at this but i'm really going to try i'd love to surface sort of i would love to not respond to people initially i'd love to surface like five or six or seven questions that people have and try to contain myself from responding to them sort of see what the vibe is in the room and then see if i can give a kind of synthetic response there are a couple of good questions from shane landrum wherever he is asking about places where students are already used to thinking themselves as higher ed consumers thinking about you know what some of these online learning environments like mook will do for under-prepared students so those are some good questions that are showing up there what are some other sort of questions or comments that are in the room over to mit so i thank you so i think that many people in education feel like they're being kind of dragged towards mooks and i'm wondering if you could talk a little bit about who or what is doing the dragging so i was struck by your video especially from covington elementary which is the health was kind of wrapped in the gates brand so covington elementary has 455 students three of whom are on free lunch and you can compare that to morse elementary here in cambridge which has 427 students 202 of which are on free lunch and i was wanting to talk a little bit about the political economy of mooks not just the mixing of the blended learning and the free market folks and the people who are kind of pushing these software platforms but the broader alliance of who is really behind this who's really buying into this to whom does the to these models innovations intuitively make sense because of the world that they're coming from that's great if you can address the high tech high touch divide it seems with the rock star um professors as you put it that it's almost a winner-take-all market if they were in this marketplace everyone would go to them which would reduce the quality of education on an individual basis hi i had two questions one is what do you think the blowback on this would be i mean i come from new york city we have a new york city public teachers union you know the school system i mean i don't think the teachers would be so keen on a system that might you know undermine their ability to somewhat control their future as part of the system etc and secondly you mentioned you're part of teaching history we're just an organization i know something about and you know they're fairly progressive and etc so i'm curious okay how have they or for that matter any other school that you can or organization that you can pick uh taking your ideas and them into practice and are they not putting them into practice what was you know the blowback from them to give us a sense of okay well what does this potentially really look like good good right here just briefly um you touched you touched on the fact that not all students do seek out what's what society may think would be helpful we see a lot of instances where students for in their spare time are looking for entertainment rather than rather than education and one of the major functions of schools is to provide society with the homogenized and some would say some some sort of standards of quality rather than what certain communities might might say is as appropriate or desirable as wondering if you could address one trying to get a a desirable and somewhat standard basis of knowledge and two how do we protect students or children from on one side businesses that are trying to take advantage that might be taking advantage of the process for purely for their bottom line and two how do we and possibly from from their parent society or whoever who might find personal political social religious agendas more compelling than what others may think would be important for education maybe two more this woman here in the yeah yeah in our group um there was concern about the disadvantaged um children and uh from families where they don't support education and how they could possibly use this system with benefit without guidance and it's part of the thing of not choosing on their own good let me let me pause there I appreciate everyone can contributing let me let me respond to a few of those and if we have some more time we'll we'll we'll get it and I probably won't get it all of these but let's let's start with a disadvantaged one because that's something that I care about um an awful lot so there's two kind of core findings from education technology over the last 30 years the first core finding is that teachers leverage new technologies to extend existing practices the vast majority of what happens when you put new technology in the teacher's hands and they go oh like how can I lecture and give worksheets with this doesn't matter what the technology is there there's always people who are on the innovative edge trying to think about new ways of doing things but that's sort of one core finding the other core finding is that the bulk of that innovative practice tends to get distributed to more affluent students um even when we make gains you know there's a sort of phrase the digital divide even when we make gains closing access gaps and there's been some remarkable changes in access gaps over time you know Pew just did a study that says that 93 percent of families um with high school age students have a computer and an internet connection in their house that's incredible it's considerably higher than sort of the baseline percentage of households that have access to a computer with internet connection and to me part of that says that there must be some things that we're doing in society to make that possible it also sounds like families are doing some sort of incredible work to be able to provide students with the access that they need but the access digital divide probably isn't as important as the digital divide of usage in the ways in which people have access to different kinds of technologies the ways in which people use those technologies and certainly what we see in school settings is that the technology is primarily used in schools serving disadvantaged students for sort of drill and kill practice credit remediation kinds of things and it's really affluent students who use technology for you know complex performances of understanding different kinds of media things like that so you could imagine that some of these sort of online learning environments as they're taken into schools you know in low-income settings they might be used you know there are some schools that say what we're going to do is we're going to get rid of all of our art all of our PE all of our music all of our electives we're going to hire the very best teachers we can but to pay these teachers a hundred and twenty thousand a hundred and thirty thousand dollars a year two hours a day students are going to go into these you know I'll call them cubicle farms which is pejorative but that's what they look like and they'll sit in front of a cubicle for a couple of hours a day supervised by tutors and they'll sort of click through learning games and exercises and things like that that's not what's going to happen in affluent settings if these tools are used in affluent settings it's going to be to sort of strategically remediate or let students advance to do what kind of you hear people talk about with ideas of flip classroom that will take all the lecture and direct instruction in practice and have it happen in students own time so that what happens in classrooms is that we have these really rich project-based problem-based curriculum having spent a couple of years at MIT I sort of feel like I understand some of the vision sometimes when Sal Khan talks I feel like having been at MIT helps sort of articulate it helps me understand what he's articulating where basically says what students should be able to do is race through their you know the sort of curriculum part of their learning as quickly as possible so they can get to do really cool projects and practices and other things like that which is if you talk to MIT students what they do is they optimize their pathway through their lecture classes most of which are really of not particularly high quality and then they get to do these totally awesome you're up undergraduate research opportunity experiences you know where they're firing laser you know one kid was late to my class the other day he said I'm sorry I'm late Mr. Reich I lost track of time in the wind tunnel you know like who says stuff like that you know okay I guess I guess if you're in the wind tunnel that's fine so I mean I think it's absolutely possible to imagine a dystopian future perhaps not that different from the dystopian present that we have now where these kinds of tools are used to replace teachers sort of in low-income settings and to create cubicle farms and you know in affluent settings are used to do what we really hope they can do which is that there you know there's a lot of stuff that's great about these systems you know computers are infinitely patient tutors you can ask them the same question over and over and over again and they will never get bored they will never tire of you hearing that you want to do it one more time that's not the case with me or with other educators like I'll get fed up and send you away I'll probably send you a computer which is infinitely patient I do want to say sort of responding to some of the questions that have come up about different circumstances and I think pretty I personally think pretty differently about k-12 and higher ed so one of the things is that somewhere you know we have some parts of educational system which we believe we have a social contract that students should get an equitable start in life that it's not fair for instance if some kindergartners get much better access to education than other kindergartners I mean that seems a little unfair at some point somewhere around the age of 18 as a society we stop believing that we start believing that we should start sort of allowing you know certain students who excel in certain areas to race ahead in those areas and certain students who are not who want to take other pathways to take those different kinds of pathways in a sense at some point in the educational system we want to treat students as consumers we don't want every single person who goes into higher ed to have the exact same position in higher ed and we don't want a state controlled system to allocate which students should go to which systems like the market breaks in lots of places but parts of it you know work better than we imagine for kindergarten there are places where our system is really broken there are tens of thousands of students who would like to enroll in community college classes in california who can't because there aren't seats for them so this leads to a really difficult problem if you're an educator who's concerned about inequity urie triceman who's a professor at the university of texas gave a beautiful talk at the last national council of teachers of mathematics conference where he said poverty is an issue that we have to wrestle with as citizens opportunities for learning are issues that we have to wrestle with as educators so there are structural inequalities the fact that there are not enough investment in community colleges in california which we have to deal with as citizens but there are some of us who are in education who have to say okay society is going to provide whatever society is going to provide and whatever we have that's what we have to use to make the best possible opportunities to learn and if there are a whole bunch of students in california who don't have access to classes because we're not making seats for them and we could create new seats cheaply by having them take place in online spaces then that's what we should do then that's what we should do now the problem with doing that is you conceivably while sort of optimizing the opportunities for that particular generation of students create a structural system where policy elites can say hey this is working pretty well like if we cut a little bit more out of the community colleges over here we could fill up the rest of it with these online courses and what makes it further complicated is that some of the online courses are objectively as good as courses taught in face-to-face settings there's a really interesting study done by Matthew Chingos at the Birking Institution who did a study about an entirely computer mediated statistics class students who took this entirely computer mediated statistics class did as well on an assessment of statistics as kids from state colleges and community colleges who took a typical class of introductory statistics with 40 students and a teacher in it and the automated class took them half the time it took them half as many weeks as the full class so there seems to be pretty robust evidence in that particular case that you could conceivably serve a considerable portion of students better if you got rid of those classes and just had students take online classes you know like if you believe the findings of the study which i think are pretty rigorous and pretty robust then there might be circumstances in which you would say this really is a better option for kids not all you know i mean one of the other thing which complicates all of this is that you know one of the things we're learning is that there's an extraordinary diversity of brains any study that purports to tell you about what's better for kids on average doesn't tell you anything about what's better for your little pookie because your little pookie is somewhere on that distribution that's different from everyone else so those are i think some of the issues that kind of surface when you think about disadvantage i think there are tremendous risks there but i also think we have to look at what are the exciting and real opportunities we need to provide much more post-secondary education for students than ever before you know people come up with different statistics for this but to meet the demand based on population growth and the needs of the labor market you have to build something like 350,000 person universities on the planet every day to keep up there's not you know what was the last time we built a giant new state university in the united states like it hasn't been recently so there are ways in which despite all of these concerns and i think part of what that points to too is a little bit of some of the questions about like what are the motivations and the ideologies of the people who are advancing this my general sense that most of the people who work in the educational space care about students as deeply as i do you know i work with people at big publishing companies and things like that sometimes who do things that i think is appalling and harmful to children but i know that they do it really believing that they're serving the best interests of children i think a lot of the folks who embrace some of these free market ideas would tell you yes you know we want people to make a profit because human society is improved when people are motivated by profit when you know one of the problems we have right now is that schools don't get better because there's no incentive for schools to get better i think those are ideas that we should not entirely ignore i mean we wouldn't have all these technologies to talk about if we didn't have a profit motive to inspire and explore and motivate folks so that's sort of one suite of things there you know around who these people are and some of these issues around educational inequality you know and it's part of why i you know as i said at the very beginning you know half of this is a joke and i don't know which half is a joke and which half is really exciting one of the questions that people asked was about the appropriate balance between individual learning experiences for people and standardized learning experience of a few people so one of the things that we say is to make good citizens it seems like there's some learning experiences that we ought to have in common it seems like we ought to impose certain kinds of things on people but it's also you know i think most of us struggle with the idea that you know there's a group of central bureaucrats who could decide this is the correct curriculum for all students to follow so how do we navigate that tension as a society one of the things that you could say is that for the most part we've been extremely conservative in answering those questions you know this gets a little bit to the question about how are teachers going to react how are how's the established education system going to react to this they're going to in general probably history would say that they're going to react with extreme conservatism they're going to change very very slowly i mean this is an institution that was not changed by the internet like everything else around you the definition of the word friend was changed by the internet and schools are like nope we're going to take a pass on that one i mean that's pretty remarkable what i find compelling though about these issues is that i can foresee policy pathways by which educators would be compelled to change policies looking at individual instances looking at what's best for one individual student for one individual set of circumstances i was talking to someone who runs the independent school down in Florida who said we don't let any of our kids take online classes except they can take two in the summer and you get credit for those yeah you get credit for those but we're not going to let them take any others and we're not going to let them take any that we offer now that's how they're negotiating some of these things you know when that becomes normal what's going to happen in the following year well anybody can take two classes during the year as long as we're not offering them unless you really don't like the one that we're offering now and then you can that's what i mean by the sort of thin point of the wedge i think there are a set of compelling circumstances where you'd say oh this is really good for this specific circumstance for this particular set of ideas these issues are so complicated that we think we can get some leverage by drawing these things in what i could then imagine is us not recognizing all of the potential changes that come behind that wedge all the ways in which that when we make accommodations for individual circumstances we say this is good for this kid in this circumstance that you create a set of policies that make it possible in all kinds of ways um Tony Wagner who i don't agree with on everything said something like i'm really glad that schools are so conservative because everything they're doing and it's in the wrong direction anyway so if they're moving at least they're moving slowly i don't think that's exactly true but i do think the notion of schools is really conservative places is one that's helpful but again what's so compelling to me about this set of issues is that i can see ways in which these issues are sufficiently compelling to to overcome some of these things and in fact there are lots of other people who see the same kinds of things ideas that have been stuck for a long time might be accelerated one of the great things about the federal system we have in the united states um that i get excited about is that we should be able to have experiments around these things we should be able to say like all right new orleans like you know if you want to totally rewrite all the rules like you know go for it and if you can dramatically improve student achievement in new orleans then the rest of us should do what you're doing um and if you can't you know leave us alone um you know utah you know Idaho like Idaho if you want to give all of your kids um access to these kinds of online learning environments and let them take online courses let's find out if kids in Idaho are smarter than kids in massachusetts you know 10 or 15 years from now um and if they are let's make changes so so there are ways in which i think the federal system that we have in this country sort of helps us experiment with some of these different ideas um and so i guess if there if there are a final set of thoughts um that i would leave you with um it's that you know hopefully one of the things that emerges from this is that when you look at pieces of education technology um our instinct is to look at them and say um embodied with these are new pedagogies um i think often in new technologies there are not new pedagogies embedded within them sometimes there are and sometimes emerging pedagogies that are evolving based on online tools kind of you know in my world like what Yvonne Illich imagined 40 years ago are incredibly exciting to imagine but even ones that don't have new pedagogies um can have alliances or natural affinities with new policies with new ways of organizing schooling and these new ways of organizing schooling have as much to teach people young people as the content that flows through them so that question of what's new that question of how are we going to change the architectures around students that question of what are the possible second order and third order consequences of these decisions um are questions that i remain very interested in and hope that we pursue rigorously as we think about all of these exciting opportunities that that are ahead um so i really appreciate you spending some of your time with me today on this beautiful afternoon which is just right out through those doors and windows i'll hang around for about 15 minutes and hang out with whoever wants to hang out um and uh and i look forward to staying in touch with all of you thank you


In this video, I wanna show us how we can graph a rational function without any technology whatsoever. We're just gonna use information about the asymptotes and about the intercepts and how the function behaves near those intercepts and asymptotes to graph this thing. Consider the rational function x to the fourth plus one over x squared. If at all possible, I would suggest we factor this, the numerator and denominator. The denominator is already factored because it's x squared plus, or it's just x squared. The numerator, we also really can't factor x to the fourth plus one. If we factor, then we have to use a difference of squares, which because there's a plus right there, there's really no sum of squares factorization with real numbers. We can factor with imaginary numbers, but we won't see imaginary numbers on our graph, so we only focus on real numbers. So the domain of this function is gonna be everything except for zero, because the denominator being zero, x squared equals zero, that implies that x equals zero. So that forbids x equals zero being in the domain of the function. Now, notice this function's already in lowest terms, and so when your function's in lowest terms, if there's something that makes the denominator go to zero, that means it's a vertical asymptote. So this problem with the domain x equals zero suggests there's a vertical asymptote at x equals zero, so the y-axis will be a vertical asymptote. Now, if there's a vertical asymptote at x equals zero, in other words, if x equals zero is outside the domain, that means there's no y-intercept. The y-intercept would be r of zero here, which that would look like one over zero, which does not exist. This function doesn't have any y-intercept, great. But what about x-intercepts? When you take x to the fourth plus one, like I mentioned earlier, the x-intercepts are what will make the top of the fraction go to zero, but you get x to the fourth equals negative one, which would actually say that x equals plus or minus the fourth root of negative one, which is actually plus or minus the square root of i, for which I mean, I can tell you what that is. I mean, that's gonna be, we don't need to worry about all that. I mean, we get these, we're gonna get some, I mean, there's rx-intercepts per se. You get like one over the square root of two, plus or minus, plus or minus one over the square root of two, i, but the thing is these are non-real numbers. These are non-real numbers, complex numbers, yes. But when we're graphing, it needs to be a real number. And so as such, we don't have any x-intercepts. The numerator is always positive. That's important thing there to notice here. So coming back to x squared plus one, it can never equal zero. It's actually strictly greater than zero. It's always positive. So we have a function with no y-intercepts or x-intercepts. That might seem depressing, but that actually is very useful information. We know that there's a vertical asymptote at the y-axis, but we also know that the function never, never crosses the x-axis because there's no x-intercepts there whatsoever. Another thing to consider is that if you wanna do symmetry, notice if you plug in negative x here, you're gonna get negative x to the fourth plus one. The even power will eat the negative, but also the even power of the bottom eats the negative. All the negatives disappear. We end up with the original function r of x. That suggests our function has even symmetry. That is, it's symmetric with respect to the y-axis. We can utilize that at the very end. All this check to see if we have, if we have the correct symmetry when we're done. In terms of the end behavior, this function, you'll notice that the top is x to the fourth, the bottom is x squared. So as x goes to infinity, this will be approximately the same thing as x to the fourth over x squared, that is x squared. And so asymptotically, this will be a parabola. We can do a little bit better than that because after all, since the denominator is a monomial, this looks like x to the fourth over x squared plus one over x squared. So this actually gives us x squared plus one over x squared. And so as x approaches plus or minus infinity, we see that this part right here will vanish towards zero. And so this graph has a parabolic parabola. And so essentially it's gonna be a parabola going forward from there. And so we do have an x-intercept, we don't have an x-intercept, excuse me, we do have a vertical asymptote. It came from the x squared of the denominator, but the power there is two. So the vertical asymptote has an even multiplicity. This tells us that at the vertical asymptote, we're gonna touch infinity. We won't cross, we'll actually touch it. And so I wanna show you, without actually looking at the picture yet, we'll compare that in just a second, what type of picture can we construct here? Okay? So let's graph, let's, let me emphasize where the x-axis and the y-axis is gonna be. So what we know about this graph already is that this function has a vertical asymptote at the y-axis. This is outside the domain of our function. We also have this parabolic asymptote at y equals x squared. So if you think about that, so y equals x squared goes from zero, zero, one, one, it will go through two, four, something like that. So our parabola roughly will look like this, right? And then we go through the other side. Let me fix that. One, two, three, four. Trying to draw this reasonably to scale. So we get this like parabolic parabola, y equals x squared, y equals x, like so. And so these are things, well, the vertical line is something we have to avoid. Does our function cross its oblique asymptote? Well, if there's a crossing there, do we cross, and I shouldn't say oblique asymptote. Does it, is there a crossing of y equals x squared? Well, that means we take the function r of x and set it equal to x squared and we solve that. So we take x to the fourth plus one over x squared. Can this ever equal x squared? We don't know. Clear the denominator in times both sides of x squared. We get x to the fourth plus one is equal to x squared times x squared, that's x to the fourth. Notice the x to the fourth is canceling at one equals zero, which is false. So this tells us that in fact, there will be no crossings of this oblique asymptote. So our function's either gonna, if we're looking at like what's beyond zero, right? We're either gonna be in this sector right here or we're gonna be in this sector right here. Now because we don't have any x intercepts or y intercepts, we don't have any points. We gotta get started somewhere. We need an initial point, a seed, to grow this picture from. So I'm gonna actually consider what happens at r of one. I need a point to get started here. And so if I take r of one, one seems like a good point to use if I can't use the y intercept. You're gonna get one to the fourth plus one over one squared that ends up being a two. So you get the point one comma two, which we can put on the graph. That would be right here, one comma two. By symmetry, I also know that since this graph is reflective across the y axis, I'm also gonna get a point right here negative one comma two, I also computed that. So what's gonna happen if we start at one, two and we move to the right, we have to get close to this parabola. So we either gonna approach it from like above or we're gonna approach this parabola from below. But we can't cross the parabola. Remember, that's what we determined a moment ago. So it turns out it's gotta be this first picture.  Well, what happens moving from this point to the right? Oh, or to the left, excuse me. Do we approach, we have to approach the asymptote. Do we go up towards infinity or do we go down towards negative infinity? Well, if you went down towards negative infinity, there's two problems. You would need an x-intercept which we don't have. You'd have to also cross the parabola which we can't do. So that second option's not possible. It has to be the first option that our function will approach, our function's gonna approach infinity as x approaches zero. So we could say something like as x approaches zero from the right, y will approach infinity, positive infinity. Now, like we discovered before, that as we get close to our vertical asymptote, we're gonna touch infinity. So that means we won't cross to the other side, like over here. We actually have to come back from the side we came from, like so. And then at this point, one comma two, well, we have to go towards the asymptote but we can't cross this. So we're gonna have to do the mirror image, which remember our function's supposed to be symmetric with respect to the y-axis. And so this is my picture, which I claim is in play right here. Let's look at, here's a skeleton that you could construct using the information from above, but if we compare this to the computer-generated image for which we can see that one right here, you can see that, yeah, that's pretty good. That's the picture. I didn't quite extend mine as far up as the computer did, but we see that these two pictures are in fact one and the same thing, that we were able to very accurately graph this function right here.