Workshop: Active Learning Notebooks for College Mathematics

Hi everyone. We're going to be talking about how to develop an active learning classroom using Wolfram Alpha Notebook Edition. So I want to start off by starting with a question here. So we have goals in each of our lessons that we develop and we also know that these goals may or may not vary in a traditional classroom setting versus an interactive technology driven classroom session. So I just like to get feedback from the audience here on. So in your mind, what is the difference between a traditional classroom and one is technology driven here? Right. So we can have more constant experience for each student. What are some other ways that what are some other ways that a traditional classroom is going to be different than a technology driven classroom? And we're going to talk about all these today, some of the differences and the similarities there and how we can develop classrooms using technology and and also how we can develop a traditional classroom in a technology driven setting versus designing with technology in mind from the ground up. So are there any other thoughts on back here. So we can model possibilities and there is a chance for more risk taking. So what we're going to see today is how we can use technology to create a classroom where there's more chance for exploration and being able to see to see how we can use technology to conceptualize and visualize, et cetera here. So are there any more thoughts on, on. On the differences here between traditional classroom and an interactive classroom? Right. So in a traditional setting we have more lecture driven. So we're going to see how you can develop a class from the ground up using technology, but also how you could combine this with a traditional section as well. So what I'm going to be talking about today is Wolfram Alpha Notebook Edition. So if you're not familiar with this product, it combines the natural language processing of Wolfram Alpha and along the way teaches the Wolfram language. So and I'll get into some examples here of what I mean by that, but just let me bring up an example. So if you want to know what is one plus one, right. We can type in enter here and. That will evaluate and we're going to see some useful information here. So notice that we have this input cell right here and that is the Mathematica equivalent of for this particular case, it's just the same, it's just that one plus one. But we're converting our natural language processing into a Wolfram language equivalent here. And then also I just want to point out some of the things that you get when you have the cell here are you can get things like the step by Step solution, we have related computations and you can even get the full Wolfram Alpha results. And then notice here we have the suggestions bar here. And you can use this to provide further information and explore from there. So for this, for instance, we could get information on is this a perfect number? And so on and so forth there. So if you're not familiar with Alpha Notebook Edition, this is what we're looking at right now. And we're seeing that we can get all this information here and then we can start exploring. So what I was going to talk about today is how we can use this in the calculus classroom, how we can use it in linear algebra, how we're going to be able to use it in differential equations. And then also if you had probability and statistics and data, how could you use that in those classrooms as well? So I won't be going into detail on all these topics. I will start with some calculus. I'm going to go over a section of calculus and I'm going to go over. The focus today will be on linear algebra. But I wanted to provide this notebook to you here so that you can, once you have access to this notebook, then you can start exploring and use this in your own classroom. For instance, so we have various quizzes that we've developed as part of the WolframAlpha math team, which are called interactive plot quizzes. So I'll just show one of these here. And so this would be a review of your pre calculus as you're coming into calculus. But I just want to show that we also can review our material here. And so we've developed these quizzes here so you can have these interactive quizzes to be able to figure out like, okay, if I move this here and I move this here and I move this here and then we'll provide feedback on whether that's correct or not. And of course this one I've. It's not going to be correct, but what you can do is it will tell you the points that you got correct and it'll tell you the points that you got incorrect. And then you can adjust the sliders from there. But the point being that we can develop interactive classrooms now where you can manipulate these various diagrams and we provide the different viewpoints that you'll learn about in a classroom, such as the symbolic viewpoint. You'll learn about numeric driven viewpoint. We also have, we also provide the equation when you get that correct so that you'll be able to see equational formats so you get the various mathematical ways of talking about things Numerics, symbolics, and then we also have verbal descriptions as well, which is part of four different ways of organizing data. There's. So we also have created activities here where you can. Well, in your introductory calculus course you get to talk about functions and derivatives and tangent lines. And so one activity you could think about doing is you could just plot two functions and have the students recognize which one's a tangent line, which one is the, the actual function there and, and vice versa. And so if you want to plot a derivative, you want to plot the function, you can have your students do that. So that's one of the topics I'm going to talk on a little bit here. So I'm just going to quickly go over this section here. So a common activity is again trying to recognize a function versus a derivative. And so one thing that you could do in your interactive classroom though is, is there's this function called a canvas. And you could have your students, they could sketch here what they think the function will be, so they could start sketching a function. So there's various drawing tools you can use here. So you can do things like this. There's also free form input. So the point being that they could draw a function here, they can save that. There's various ways to take the function that they've drawn now and we could email that to the instructor. Wolfram also has a technology called data bins. I won't get in that today, but there's data bin technology where you could have the students data automatically sent to the cloud and it would be saved and then you'd valid grade and to know whether their information there is correct. You can combine that with input form so that the student could submit reasoning for why they drew a given picture. But the point being that you can create these interactive activities and combine all the different representations that you'll encounter in mathematics. So what I want to talk about now is tangent lines. So I'm going to go over this section and we'll get into the linear algebra and I also have follow up questions as we're going along here. So with a tangent line different about viewing that. But one of the new things that we've been working on is what are called calculus results. These aren't available just yet, they're still being reviewed. But in the future we're going to have the ability to, in your classroom to be able to access these various definitions so that you can get started with a classroom fairly quickly and have those definitions there available for use. So for instance, for the tangent line. We have, we've spent a great deal of time on collecting the information here so that we can provide that information in a format that a user can, so that the teacher and the students can quickly access that information directly from a notebook and go with it from there. Okay, so we have this information available on the tangent line. So we spent time collecting the various definitions that you'll encounter in a calculus classroom. And so you'll better use these to develop your own lesson plans here. So with a tangent line, we usually have, we have the limit definition there that we could use. We can use the Wolfram language, compute that out. So we can also use these, we provided various palettes here and ways of this, what we call 2D input here. So we can just use a structure in a cell. So that's what I've done here. So this is using this limit icon here. So this 2D input will allow you to input the text as you would more of a traditional math, so book formatting essentially. And it's also available on our, on our website as well. So there's the 2D input. So once you add that to the input, then you can go about evaluating things here to be able to, for instance, compute out a limit. So this will give us information on that tangent line. We've also, as we talked about in an earlier talk, if you were here for that, for our resource function talk, we've also developed resource functions to be able to compute all these things as well. So in the classroom you could directly apply this tangent line function that we developed to be able to compute all these out. So once we have this information again, we can see our various forms of equation. We have our standard form equation, we have our point slope equation, we have our slope intercept equation. So that's going to lead me to my next question that I wanted to ask everyone here. So in designing our course, we can format it as we would a traditional course where you're competing out the limits by hand there. And we can also be able to develop a course from a more conceptual viewpoint. So I just wanted to, before we get into that conceptual viewpoint here, I just wanted to get your feedback on what do you think the benefits and drawbacks of developing an interactive course in the same way as a traditional course would be back here? Right. It'll be a lot more work in the front end. Large, Right? There's, there's a large volume of resources available, right? That's correct. Using a technology driven classroom, we can more personalize the, the classroom to the, to the individual Student, we can provide the different representations that a student might be familiar with as opposed to. So with a traditional classroom you're essentially having usually a lecture based classroom or you can also have group work. But the point being that in this new setting of a tradition, sorry, of a technology driven classroom, you can individualize the lesson to the student there. And so you can provide various viewpoints that in a traditional classroom that you wouldn't build experience. Right. So we can, in a traditional classroom there may be one or do two different styles of learning, but as we're going to see today, there's different viewpoints that you can get that the student will be able to, to have. In this new technology driven classroom, we can make it a lot more conceptual, we can add various plotting and graphics and. Right. So in this technology driven classroom setting, as you mentioned, you're going to get different viewpoints here, you'll get different representations, you'll be able to, we can really, we'll be able to really see the conceptual here that you'll be able to interactively to see how if you're algebra class, see how the various planes intersect, for instance, and so on There other thoughts on this. So I would say Wolfram Notebook Edition allows the students to start much earlier on the process because we developed this natural language processing so that the student will just type in, for instance, say, integrate x squared from 1 to 2 as you would in Wolfram Alpha. And so that's one of the benefits of Notebook Edition is that you'll be able to do the input much as you would into Wolfram Alpha. But then once you have that, you'll be able to start going further along and develop more information based on that initial data that you had there. So one of the benefits of Wolfram Alpha Notebook Edition is that you'll be able to start with something such as a tangent line and then you might want to plot that tangent line and, and so you can start sequencing together these various operations so that the student will be able to see, you'll be able to go beyond traditional, okay, here's the integral, here's the derivative and you'll be able to actually visualize it. You'll be able to use the information there to continue onward there. The point being we're going to see that you can use this at any level of the classroom. So this could be used at high school, this could be used at community college level and beyond. So we've also developed, there are a lot of information on linear algebra in a system as well that we've developed, so I'll start talking about that next. But those were very good points there and very good responses. So let me go ahead and we'll start talking about linear algebra now. And as I'm scrolling through this notebook, I want to point out we make use of a lot of the calculus definitions here. And we're also making use of, there's a lot of activities you could incorporate into your classroom where you can both compute things by hand, right? You can compute that integral by hand. But if we're goal is to compute the area between two curves, we're also going to provide our method here of just specify error between those two curves and the given endpoints there. So let me go ahead and start talking about linear algebra here. So when you're in a linear algebra classroom, you're going to be talking about plotting various curves here. So in a linear algebra classroom, you're usually going to be seeing a couple of equations. You're wanting to solve that for a given intersection point, for instance. So one of the benefits of using a technology driven classroom is that you can start making plots and visualizing the intersection of those two lines. And then we also can start solving our systems here, right? We can solve the system for the given intersection point and then you might also want to write that as a matrix. So you could gather the coefficients and rewrite your system as an augmented matrix. But the point being is that now we have, now that we have technology, we can start doing things like varying parameters. So you could assign activities where they're going to be varying a parameter and then they have to start describing the reasoning. So in a traditional setting, you're most likely going to have been familiar with seeing the matrix equivalent. So we, we have matrix equivalent. We can work with that, we can solve the equations. But the point being, and now you can develop an interactive classroom where you can visualize how, how is changing a parameter going to change things such as is one solution, is there many solutions, et cetera here? So we have these interactive manipulates here that will allow you to vary a parameter. So I said here, plot X plus Y equals C, X minus Y equals 1 and then we're going to vary C. So C is now a parameter, right? And so what does that look like as we're varying that parameter here? So we see that we can vary the parameter C here and the student can then determine whether or not there's one solution, implementing solutions, zero solutions, et cetera here. In this particular case, we see that there's one solution here on this given range. And so the student can interact with this activity here to. To get an idea behind what is actually happening, instead of approaching everything from the point of view of either a matrix or from the point of view of manipulating equations. So you get this visual feedback here. So you can make that correspondence between the algebraic and the conceptual. You can visualize what is actually happening there. I provide another activity here. So if we verify does the following set of equations have a unique solution? So we can adjust our slider here, and we can see that we can then provide information on does this have a unique solution? We can complicate this further. So we have now we have these two equations. I moved the parameter C to in front of this Y term here. So we have X plus C times Y equals 1 and X minus Y equals 1. And we have from X between negative 2 to 2 Y from negative 2 to 2. And we're varying C. So in this case, does this always have a unique solution? Does it have more than one solution there? So the student can start moving a slider here. And okay, for some values of C, it looks like there's just that one solution, right? We just get this one point here. But what happened if I adjust the slider here? So now we can. We can see that there's not always just going to be one solution, that you could have infinitely many solutions there. So the point being that with this visual format, you can see you can make that connection between the algebraic and the verbal and the various descriptors we have for a set of equations here. Here's an example we have of an equation which has no solution because we have these parallel lines here. We can verify this using the Wolfram language. So as I was saying earlier, we have this natural language processing, right, where the student can just type in, solve X plus Y equals one, X plus Y equals free. So that if you're coming into the Wolfram language without experience from being developing Wolfram code, not only can you still input the equation as you would in WolframAlpha, you also get back the Wolfram language equivalent. So we're teaching Wolfram language along the way here. And so you can see that we have this conceptual view and we also have our more traditional, okay, what is our system of equation solutions going to be? If you just want to verify your solution here, we can again rewrite this as an augmented matrix. You could provide the various row operations. So we could say row reduce that. So, and it's going to be leading into my next question here. So we can row reduce this we can see what the form of that would look like. So in a traditional classroom, you're going to be doing the various steps that will lead to that, which we also have that as well. And you can directly access that from the notebook here. But that leads me to my next question here. So we have these different viewpoints. We had a conceptual, the mechanical, the data and the visual. What should our balance be between those? Do you have any thoughts on how should we balance our different viewpoints? Should we focus on the conceptual? Should we focus on the mechanical, the data, the visual? Right. We can, we can provide all of the various forms here. We can vary mixture of conceptual, mechanical, data and visual, and it can be responsive to the individual student. Was there a hand raised over here? Okay, any further thoughts on this? On conceptual, mechanical? Well, Right, we don't want to. Right, we don't want to make it too mechanical. So you want to provide those various viewpoints there. And then it would just be about. If we focus on the mechanical, as you're saying, it would just be about the tools we're developing. Right. We need student centered content. Okay. There are no further comments on that. So that leads to my next question here. Does a student still need to learn the standard algorithms to individually do the mechanical calculations? Why and why not? And if the students should learn this standard algorithms, in what way does technology help the process? So do we have any thoughts on, on this here? Mm, Right. We could compute this once by hand to appreciate the technology. Right. We could see that, for instance, you could be looking at a four by four matrix, right. And you want to provide the row reduction operations on that. And that's going to be quite a lot of material to go over. And then once you see that you can compute that by hand once, then. And then compare that to what it will look like once you have technology to do that calculation, then that will provide a striking viewpoint of. Okay, so now you understand why we develop algorithms to be able to compute these things especially. So what if I had a 10 by 10 system of equations or 20 by 20? Once you start reaching larger and larger levels of data, as you were mentioning, we'll come to appreciate the technology there. So are there any further thoughts on what about the standard algorithms? Does technology help in this process? So you can use that as a jumping point for further exploration. Right. So you can create activities based on. So you could since envision creating two regions and sliders between those two regions and developing insight into how the area between curves would change as I adjusted the function for instance. Right. So you can combine both information where we're just taking that natural language processing and getting straight to the answer versus using that information to further develop and launch it as an activity for further exploration. I saw a hand up here. Right. We can use it for error analysis. We can figure out where our mistakes are. We could provide data analysis on that individual activity. Right. So we could collect information on the individual threads of where students are making similar mistakes, for instance. That's right. Right. So we can implement algorithms without knowing about all the various details. Right. So students, so you're mentioning that we can complicate and explore. We can, that we don't have to start with just knowing that algorithm. We can then use that algorithm to expand our knowledge. That's a very good point. So essentially. So students can start thinking outside the box essentially here. Any further thoughts on this? Right, so these are all very good points that are being made here. So we can use technology to both explore and discover new areas, but we can also use technology to help us part of that development process. So there are actually abilities in the Wolfram language itself to say, if we were in an algebra classroom, we wanted to add something to both sides, we can actually do that as well. So we could say, we could say we have an equation x equals 2 and we want to add 2 to both sides. So we're using technology. You could explore how is that equation adjust as we add something to both sides. And that allows for time for the student to go beyond just always knowing what to do next. With technology you can show, okay, so a lot of times a student doesn't have time to discover. So and you'll just start with, oh, we can add negative 2 to both sides to solve that equation. Well, in a technology driven classroom, there is a way to let the technology do the exploration for you. So you could say, okay, I'm going to add, instead of adding negative two to both sides, I could add two to both sides. And then so now I've complicated my equation more. And then, so you can start pinpointing how am I going to go about solving this? Instead of being given that you just add negative 2 to both sides and that's because of an additive inverse, and so on and so forth. You can start exploring why. Why did I choose negative 2, for instance, to simplify that equation. So we also have various activities on calcus, but I'm going to close this section for now. So what I was wanting to talk about next is so now that we've seen this conceptual format there's also three dimensional formats. So I'm going to talk a little bit about that and then we're going to get into more of the standard linear algebra. But I wanted to show that in a linear algebra setting, usually you don't get to experience what that actually looks like. Right. So you have the equation. Usually you start with an equation, you may start with substitution, you may start with elimination. But what technology allows you to do is you get to visualize what does that look like? So you have these two planes, X plus Y plus Z equals 1, you have X minus Y equals 0 and they intersect and where are they intersecting along here? So you can see where are they intersecting along? You can add another plane. So now we can look at free planes and how many solutions are going to be how many different ways can those planes intersect? So you don't always have exactly one solution. Sometimes you have infinitely many solutions, sometimes you have exactly one, sometimes you have no solutions. And you can see why that would be true. You can see where all the planes would be intersecting, for instance, so that you can have a visual. So in a traditional classroom, a lot of times in mathematics you'll get into this algebraic viewpoint. And so now if you're a visual learner, you can find that connection with the visual. Mm, Somewhat. That's a good question. And. We, we do have natural language and that is available and is expanding further as we encounter different entries here. That might be a good question for. So Michael's here as well. So I'll let Michael speak for a second on this. So you can vary the parameters again in 3D so you can see how this interactively works here. So I can vary this plane here and we'll notice how it's intersecting. So you can see if it's intersecting along a line, if it intersects at a single point. Sort of. Not only the visual, but you can also start looking at the algebraic conditions again. So if you have, if you want to plot a parametric equation here, for instance, well, we can solve the equation so that you can natural link use natural language processing to solve the equation for an individual part still. So here we could solve this equation Z for K minus X minus Y. You can collect all that information that we just formed here so that you could then look at the parametric plot and you could actually see that line of solutions there. And then you can adjust the sliders here to see how that varies in three dimensional space. You can start asking more open ended questions and allow the technology to explore these more open ended questions. So you could ask is there ever a value of k for which there is a solution? Is that solution then going to be unique? And you could specify what parameters, set of parameters you're looking at then. So we varied this parameter k here. So and then also point out you can adjust the, the view we're looking at here and we can adjust this plane and you can see if there's ever a point at which there's actually a solution here. Is there examples where there's not a solution? So are all these planes intersecting? So you can start asking these open ended questions compared to a traditional classroom where you might just be looking at the equations. So for instance, if you're developing a biology classroom or you're in a setting where you want to use technology and aren't looking at the algorithm itself, then you can start doing things like well, okay, I'm going to row reduce this and then we'll row reduce it and you'll be able to read off the what or not there's a solution there. So this is connected to that last problem. So if the parameter was k equals 1.16, then we read off this last equation, we get 0x plus 0y plus 0z equals 1. So there is no solution to that equation. So we'll see that it corresponded to when we adjust the slider to 1.16, that it wasn't intersecting, that not all three planes are intersecting there. Then we could adjust this as well. So you can make that connection, right? You can make the connection between a matrix, a set of equations, and the conceptual idea of moving around, having a parameter and then moving that parameter and seeing how those planes adjust to that. So you can actually see that there, that not all three planes are intersecting at one unique point there. Which I have another question for everyone here. So now that we've seen some examples of this, I want to ask you to spend say three to five minutes here thinking about what challenges that you think you would encounter in developing your own technology driven classroom. And so feel free to discuss with someone next to you. But we'll spend say three minutes here, I want you to think about what challenges you might face and then some ideas that you have for a technology driven classroom. Okay. All right, so now that we've had a chance to look into this, so you've spent five minutes talking about challenges in the classroom. So I'd like to hear, to hear back from everyone what challenges you think that you're going to face and trying to implement a technology driven classroom. Right? So do you have any thoughts on, on some challenges you may face or what material that you have in mind for developing your own classroom here? M. Right, that's, that's a good point. So one of the challenges you may face is providing good feedback. There are ways to, to work on that. For instance, you can develop. So as I was talking about earlier with the interactive plot quizzes, you can develop technology where you provide feedback at different stages. So, so for instance, you may have gotten, you may have been looking at the mean value for him and you may have adjusted the tangent line to the, to the incorrect place. For instance, you can provide feedback at that step. But that is, that is a very good point that one of the things you have to keep in mind when you're developing a course like this is you need to provide appropriate places for feedback. So as you're going about designing a course, that's one thing to keep in mind is how can we provide good feedback to the student and how can we adjust this information that we're developing here? I also want to point out there is question interface technology that's developed by Wolfram as well on the question selector. So you can start thinking about those concepts as well when you're trying to provide feedback. Were there any other thoughts on challenges that you may face when you're developing your own classroom? Right, that's another challenge that could be faced. Any further thoughts? That's a good question. On the Raspberry PI. I'm not sure, but I can, we can talk to someone and we can also Michael Trott is here as well. So you may be able to speak to Michael Trott as well. But I also want to point out that we have. So as you're mentioning with the phone, there is Wolfram Cloud and Notebook Edition is built into that as well. So you'll be able to access Wolfram Alpha Notebook in the cloud as well. So all these activities I presented, except up to the point of the, of those 2D inputs I was showing, that's still being worked on in the cloud, but the notebook will be able to run in the cloud as well. Right, That's a great point. So when you're developing an activity, one challenge you could meet is the problem could be so open ended that you're going to have 100 different answers. So as it's being pointed out, we can narrow down when we're setting up the problem to a specific definition. So for instance, and as you were mentioning, with a basic reproduction number there for a given set of equations, if they're looking at for instance, the susceptible infected recovered model. There's a well defined equation there that you can look into. There's also a lot of demonstrations and let me point this out here. I didn't get a chance to show this here, but there's all these demonstrations here. And so there are ones on the life sciences, there's ones in particular on biology, high school biology for instance. And you can start using these in your classroom as well. So the activities are already here and the question you were just talking about would be something such as herd immunity, right? And so you can, this is already built in essentially, and you can start adjusting the sliders and you can look at how those two numbers, they such as your final time and your initial fraction immune, affect outcome there. And there are also ones in the sir as well. So that's a good point that you need to have your problem be well defined. And it's also a good point in that there are already some of these functionalities built in so that you can start exploring the concept from the very beginning of the class. So you could start, okay, what happens as I make this initial fraction immune increase? What happens to my two curves here? And then you could start getting into susceptible infected recovered, and you can start exploring activities along the lines there. So let me get to summarizing here. And those were all very good questions and points there. So to summarize here, wolframarapha Notebook Edition allows you to develop a course from the ground up or to use it in a course you already have in place. So you could take your traditional lecture and you could supplement this with more interactive activities so that the students, for instance, when they're seeing those equations initially of the linear system, they actually can conceptualize and visualize what does that actually mean? And a lot of times you don't get to experience that. And so I feel like that's one of the key things of your technology driven classroom is that you can actually not only see the equations, but get to hear about what you're talking about there. So I just like to take the time to thank everyone for attending and there are some remaining talks by our team. So tomorrow, Thursday, what's new and stuff by step. So we'll be talking about a lot of our new technology. There's so seeing those algorithms in action. And then we also have an office hour. So if you had any further questions from today, you're wanting to talk to us and we'll have this officer.