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Hello, everybody. Welcome to another episode of Undercooled, a materials education podcast. As always, I'm your host, Tim Chambers. I'm here with my co-host, Steve Yalasov, and we are going to talk about math today, a topic that is very near and dear to both of our hearts, and hopefully to yours as well. Absolutely, especially in materials programs.
I can't tell you how many of my colleagues here at Michigan and my colleagues all over the country are constantly complaining that our students are just not prepared with their math skills to handle the kinds of math we do in material science. This includes like thermo and kinetics and electronic materials, tensors for mechanical properties. We always seem to have a large number of our students who just don't know what we're talking about, and it's a problem.
So I think this is a great thing to talk about today. Yeah, and we're going to talk about some of the curricular structuring, the sequence of courses, because we want to prepare our students for success. At the end of the day, that's really what it comes down to, is we want them to engage with these difficult high level topics in a way that's meaningful so they actually learn. And the mathematical foundation of that is a critical ingredient for their success. So how do we get them there?
I guess that's our question today. Yeah, and even before that, I think we need to talk about some of the special problems that we're facing, because the students we're getting now all went through COVID. And in COVID days, we let everyone just pass. And so a lot of students just never even learned this material. What they learned were lots of bad habits that they can get away without doing anything and we'll excuse them. So this is all conspired to make it even worse.
To be honest, we had this problem before COVID. Oh, sure. It got a little bit worse, but this is a never ending challenge of engineering education, is that it seems like in some ways the math departments aren't actually teaching the math that we need our students to know. Yeah, and it's even a little deeper than that, because we always get 30% of the students have no problem with the math we throw at them. And they went through those same courses. And so what gifts?
And so I think the math department does teach the material, but they only teach it to a certain kind of student that can receive it. And I don't think they're being sensitive to the other students who really need to learn this material. And they're the ones that get Cs, C pluses, and they pass, but they really haven't learned the minimal amount to succeed in the future. And so part of this, I think, is pedagogy. Part of it is, you know, students are different.
But none of that matters when they get it in our class, because we have to deal with it. That's right. So I think what we're going to be talking about today is how do we deal with it as a materials department without trying to cascade or, you know, complain about our math department? Well, if we're working on a solution, then let's start at the end. What is it that our students need?
You know, as a materials department, what are the key components of understanding and being able to do math that our students have to be able to accomplish? That's a great question. And I really think we know the answer. And it's basically the sequence of calculus that has become Calc 1, Calc 2, Calc 3, none of which says anything about what they learn in their class. But you know, it's very interesting to me, at least at Michigan, and I kind of believe this has happened elsewhere.
They turned four semesters of calculus, differential equations, and linear algebra into five semesters. For whatever reason, the students coming in now, the first calculus class they take is limited just to limits, series, review of trigonometry, and what a differential is. And then they wait for a whole other course to teach integration. And they do integration before they teach multivariate calculus. So they never even learn, you know, integration of volumes because that comes later.
And then they take an ordinary differential equations course, and that's four courses, and they're done. But what happened to linear algebra? Yeah, what happened to statistics? What happened to complex variables? These are all critical topics that we also need. They are.
So our students come to us, and they might have some inkling of those things, but you know, I would have rather they turn this into a five course sequence where they combined differentiation and integration into one course, taught a course on multivariate calculus, taught another course on ordinary differential equations, and a fourth course on linear algebra, and a fifth course on engineering math, where they cover orthogonal series, they cover complex numbers, they cover statistics, and even
discrete math for computational applications. So that would be ideal. But instead of, again, complaining about what would be ideal, why don't we talk about what are the things that students often have trouble with when they get to our higher level courses? And so I'll go with my first one, my favorite one. Our students don't even know how to do trigonometry. They have an idea of what sines and cosines are. None of them know what hyperbolic sines and cosines are.
And more importantly, they don't understand the relationship of series expansions and exponentials to sines and cosines. So when they get to where we're trying to explain structure factor and diffraction, you know, we put e to the ik dot x, and they like look at us like we're from outer space. I just had that experience two weeks ago, I was doing a review of x ray diffraction for the lab class and e to the ik dot, you know, ik vector dot x vector and we were gone.
And I, you know, I backed up, we worked through it, but I had to do a lot of unpacking there that I was surprised had to be taught that far into the sequence. That's right. And of course, this is where series expansions really help.
Because the only way to prove that any of this stuff is correct is by doing this series expansions and, you know, taking science and cosines and, and doing their series expansions and multiplying one of them by I, and you add it up and woohoo, it's the same as a series expansion of e to the ik dot x. And so that gives students, you know, it's not that that proof is such an important thing for the practical use, but it gives students a lot of confidence that they understand why
that equation works. And so they were supposed to learn that a long time ago, and they didn't. And probably because they learned a little bits of it across many courses, and nobody ever coupled it together. That's the exact experience I've been having, especially in the last year or two for students who did get a lot of this core math during COVID. A lot of the puzzle pieces are there, but the puzzle has never been built. Right. So that was my first one. What's, what's your next one?
Well, I have some personal feelings about this because I, once upon a time as an undergraduate student at the University of Michigan, had an amazing linear algebra for scientists and engineers course. And it was a very practical, essentially semester and how to set up and solve eigenvalue problems was really the punchline of it. And that completely transformed the way I thought about math, the way I thought about how math is used to solve scientific problems.
Really my understanding of quantum mechanics, a lot of it stemmed from that course. And so when I'm looking at what our students here in MSC need to know, I am often finding ways in which students don't think about functions, about series, about vector spaces in the way that I do. And that can be a challenge in communicating with them when I want to teach something mathematical effectively, but we're just using a completely different framework.
And I feel like if they had that understanding of vector spaces, it would enable so many more discussions and so much more understanding of other topics. Like for example, discrete math, as you brought up earlier. It's funny you say that, you know, I, I was a math major, so I had a lot of this in a different way than I think a lot of engineers get it. But I probably got it more like a physicist got it because physicists need this too.
And while I've got to say even three space, you know, just XYZ vectors, I wish our students knew more about that. I teach our introductory course and I'm blown away by students who don't know how to take a dot product in vector notation. Like what could be easier? And they just don't understand that. And they don't understand its connection again to trigonometry. That's at least they have some inkling of what a vector space is.
Now, of course, the big thing I wish our students would learn, and no one teaches this in the first few courses of math. I never learned it in the first few courses of math. I learned it in applied math. Well, I actually learned a real analysis. But the simple idea, and it really is a simple idea, that the space of all functions is a vector space. That blew my mind when I learned that.
And once you realize that, then you can define a basis set where the basis elements are mutually orthogonal, meaning their dot products are zero, and you normalize it so they're unity. That's all you have to do. But then to know that by a linear combination of these, you can map out any function. Well, not any function. You can't do real functions.
But for any countable space, and of course, that's another thing, how hard would it be to just explain the different sizes of infinity to all of our students? I love having that conversation with students. They feel so enlightened, so philosophical, like I'm thinking about infinity. But it's interesting as well. Yeah. And it goes back to the 1850s, where they figured all this stuff. You know, Cantor wrote his book, which was called The Laws of Thought, because we're stuck here as humans.
This is another one of my pet peeves. Psychologists get all flipped out when they say things like, "Don't anthropomorphize." And of course, what does that word mean? It means don't ascribe human characteristics to inanimate objects or animals, because they're not humans. I'm sorry. When you take physics, you learn about how the electron feels the proton. Right. And how the system wants to reach its lowest free energy state. And yeah, it's just how people think. Well, we're human.
And it's how logic is. Logic is all about how we think as humans. And we're kind of stuck, because we're humans. We're not dogs. We're not rocks. And so there's only one way for us to think, and that's the anthropomorphize everything. But you should be aware of what you're doing. That's very important, because we know that just because we think that way doesn't mean that other things are going to think that way. But that said, The Laws of Thought was the first book on logic.
And Cantor introduced his ordinal numbers and his cardinal numbers and all that, the basis for number theory. But it's such an important thing, because ordinary differential equations are easy to solve, because they, the solutions are countable. Whereas partial differential equations are really difficult to solve, because the solution space is uncountable. And it goes right back to what you said, the eigenvalue problem. What are those? They're really just the weights on the basis functions.
And the way that you solve a partial differential equation is you map out the characteristic lines. And then every point of the characteristic line contains a new set of solutions that lie on a cone called a mange cone. And of course, there's a uncountable number of those points, and the cones sweep out. So that's how you do it. And that's how people solve these problems. But it all goes back to very fundamental things, and you need linear algebra for that. So how can we ignore linear algebra?
We deal with tensors. What are tensors? They're matrices. You've got to diagonalize stuff. You've got to look at all the off-axis elements. Right. Think of it as a transformation acting on a space. Precisely. So a little bit of understanding, not with proofs. You don't have to sit there and prove it all. Just explain that functions represent a vector space, that you have these basis sets of functions that you use linear sums of.
And you tell them, we can prove this if you really want, but you can approximate any real function arbitrarily closely if you take enough terms in your series. So brings it back to you. You've got to understand what series are. You've got to understand vector space is more than just three space and how functions can be a vector space. And then you need the mechanisms of linear algebra to actually be able to do computations in all of these things.
So with that little bit of information, it can go so long. So I totally agree with you. Second thing. And my next thing is kind of related to that. You tell students that the solutions to these are Bessel functions and you write them down and they freak because they're really ugly. And unless you understand that a Bessel function is just a basis set that makes life easier. It's a convenient collection of functions. That work in circular boundary condition areas.
And the same is true for any orthogonal series, any orthogonal series, Fourier series, Legendre polynomials, you know, Bessel functions, Hermit functions, they're all same. They're all exactly the same from a simplistic view of vector spaces and bases. Why don't we teach it like that? I wish we did. Well, if that's our finish line, then the question becomes, how do we get students here? What can we do from a curricular point of view?
What can we do in the classroom to achieve that goal so that we can help our students think in this more sophisticated, but also simpler and more effective way about the mathematical problems they're dealing with in their MSE courses? So I think there's two approaches. One is the simple, but really stupid idea. Just make them take more math courses. Again, why academics think the solution to any problem is... Is classes and lectures.
And, you know, yeah, that might work, but it would take forever. And we don't really have room in the curriculum to do that. So we need to be more thoughtful and more innovative, I think, than that. And so we've talked a lot about this, but Tim, why don't you talk about what you think might be a solution to this? Well, first I should unpack the genius idea of the more courses, because even though throwing classes at a problem is exactly, as you said, Steve, the academic way.
In our case, the way this manifested a couple years ago at Michigan was having this conversation that you, dear audience, just heard, what are our students missing? And then finding the courses where those topics are taught around campus. Many of them math, but some of them in stats, some even in science departments.
And we used one of our elective course slots to require a fifth math class out of this list so that students would have a little bit of agency and have an opportunity to specialize more in an area that they found valuable to themselves, be it statistics, linear algebra, or whatever. And this idea was great on paper. The factor that we neglected to account for is that our students are smart and they understand how the school works very well.
And so they all took their extra math class senior year. Long after the point at which it would have actually done them some good, I'm very happy for their future employers that they're getting a more complete product, but it did not do any good for solving the preparation issue of students being ready to succeed in advanced MSC coursework. So we're going to chalk that up as a noble experiment that didn't quite pan out. That's right.
In addition to this, I think the student in the median of the distribution needs more math to be able to do our courses in mechanics, our courses in electronic materials, in thermo, in our, you know, partial differential equations course, which is transport. We also need to think about the students who are even more poorly prepared. How do we help meet them where they are and bring them forward so that they can succeed?
And there's a lot of pedagogical tricks to do that, like team-based teaching, but you need to do it at the right time and you need to make sure that you're not trying to overcome a massive gap. So instead of waiting till their senior year, this is something we need to do as soon as they declare as students in our department. So we catch them early. We can certainly add more complex math topics when we get to thermo, when we get to kinetics, but we really need to bring them up to speed.
So I think it speaks to the point that when we design a system to help our students all get to a certain level of competency for skills, ready to move on to the next step, we need to do that early. So I think we're going to get rid of our fifth math course and instead require a sophomore level math course.
Before we get to that, I had an idea for something that could be its own episode, but briefly here as you're talking about interventions and what can we do with the students pedagogically socially to help those who need some extra support in the math area. I know you've been working on some tutoring programs lately and being very intentional about how those peer tutoring groups are set up.
Is that something that you could talk about for a couple of minutes and how that might help in the math area? Sure. This is something that's come to my attention as the program advisor. Many of our, right now we tell our students, "Get in a study group. It's really valuable." But we don't do anything to create the study groups. We leave that up to the students. And what happens is something that's not very inclusive.
So a lot of our students, like our transfer students, students with disabilities, underrepresented minority students, they're not in the club. They're not in this group of students that naturally wants to get together to work out problems and help each other. So they get excluded. And I found that out when I was talking to a student and I said, "Why don't you join a study group?" And I was told no one will let me in their study group. And that just broke my heart.
And so then I started looking around at what other people did. And I found out in our literature science and arts college, this is where physics and math and chemistry, biology are, they have a science learning center. And they've created facilitated study groups. You just go to a website and you choose a time you can make it. You have no idea who's going to be in that group, but it's all groups focused on a particular course and sometimes even a particular section of that course.
And you just sign up. And then, you know, on Wednesdays from eight to nine o'clock at night, you get on a Zoom meeting. But what they do, which is awesome, they hire a student who got a B plus or better in the course. Eighteen bucks an hour to be on every single Zoom call to facilitate the conversation. It's not quite tutoring, but these people help all the students do their homework, help them understand things they don't understand.
But what I love about it is how inclusive it is because anyone who wants to join can join. There's no barrier to joining. There's no social capital involved about who you know and who you don't and what you look like and whether you're the cool kid. All it is is a common need and schedules that line up. And then what they do, which I think is also genius, you worry that, well, you're the people who still want to form their own study groups will just ignore this.
They might, but once they hear that there's a facilitator there who's going to help them, I think it's going to draw everybody into this process. So anyone can really start this. You don't need fancy software. You can use signup genius for free and cap the number of people for a group. And so we're going to actually try to pilot it this term with our kinetics course and see how it goes. But it's a way to help students find study groups, at least for the upper level courses.
And hopefully the college will see this experiment. It's funny, I just saw our associate dean told him about this and he loves this idea. So they're going to be watching what we do and we'd probably do it through our academic, I forgot the name, we have something like a science learning center. So they're actually very interested in maybe rolling this out for our lower level courses for the whole college. So I think it's going to be a good experiment.
And obviously this math course is a perfect place to institute something like this. Yeah, I think that could translate really well to the sophomore level, especially imagine a student has just declared their major, I'm brand new to MSE. I don't know anyone. I don't know anything. Where do I even start? And to create this guided experience of just tell us when you're free and we'll handle the rest.
I think that can be a really great method to reduce that barrier to entry and get them meeting their peers in an effective and helpful social environment right away. That's right. And of course, this speaks to something I know you believe and I believe the idea that the best people to learn from are other students who just learned it or learned it within a year because nothing is obvious to them. Whereas we're like some of the worst people to teach. Right.
Because to us, it's like, what do you mean I don't understand that? It's obvious. Yeah. I've known that for 30 years. Everyone know that at this point. Yeah. Right. And, um, but it also is really valuable even to the better students who are going to be helping the students who aren't as advanced because they'll get to teach it. And we all know we learn much better when we teach the material. That's how you really learn something.
So it's just a win-win and, uh, can be done in parallel with a regular course and should ultimately save the student time because it's a very efficient process to figure out what you're doing wrong. So I think we've come upon this idea that need for this course. And, and I guess, uh, we should come clean and say, uh, uh, Tim here has actually been charged with, uh, teaching a course just like this next fall. For better or worse, it's going to happen. Yes. And I've been thinking about this a lot.
So, um, Tim, why don't you talk about your process of how you're approaching this and how you would, what the constraints and what the opportunities might be. Yeah. My approach to a curriculum design problem is that word design. This is an engineering problem. So you have to use an engineering toolkit to solve an engineering problem. Step one, define the problem. Step two. And then everything goes from there. Who are the stakeholders who's involved in this?
What are the different needs of the different populations involved? What does success look like? What's a minimum viable product look like? So in my mind, this is completely an engineering problem of creating this solution to a need that's been identified. And so a lot of this at this point is having the conversations with all the people involved, talking to faculty and saying, what do you wish your students were better prepared for when they get to your course?
And some of it is talking to students and saying, what do you feel like you're struggling with this semester that you wish you had additional help or additional instruction in? Some of it is even consulting friends that I have an industry and saying, what's the math that you wish you actually knew when you graduated that you never got? And so we start to define these sets of topics.
And I'm hoping that as these conversations continue, that it'll turn into a Venn diagram and there will be some sweet spot, some overlap of what different people want. And then that can really start to define course topics. Beyond that, there is also just the personal aspect of what do I, the instructor believe in? What can I teach effectively? What can I teach in a way that I think is meaningful and useful? And then trying it, throw the pasta at the wall, see what sticks.
That's always the last step in a new course is just giving it your best and then finding out what you were right about and what you were wrong about. And then you iterate because it's an engineering problem and that requires iteration as well. The other thing I think that's a real opportunity is that you're very aware and understand a lot of literature of evidence-based teaching.
So I presume you're going to be putting in a massive amount of active learning, lots of feedback loops so that students get their reps in and actually do the work themselves, without fear of failing, so that they can actually learn the material. Yeah, absolutely. Something that I know a lot of students don't appreciate and I find also quite a few faculty don't appreciate is that you need to have the conceptual understanding of the math, but it's also just a skill.
It's a practice that you develop through repetition, but that the goal for our students should be to achieve a level of automaticity with this skill, not to have to sit down and think for three hours about how to do this integral, but to sit down and have the integral solve itself while their pen just walks across the paper because they've done it before and to say, "Oh, I guess I didn't have to think so hard about that after all," because that frees up their cognitive resources for the
stuff that we really care about, which is what's the physical meaning of this math? How do you use this to describe or predict or do something in the real world? If they're spending all of their brainpower saying, "Where do I put the x squared?" they don't have the bandwidth left over to think about these more scientific topics that we hope to get them into and how that transforms or how that translates into the pedagogical structure of the course.
Steve's still having some ideas here, but my initial vision is something that's a lot more like a studio course where the students come in and it's more like these guided practice sessions. Here are some problems we're going to work through together, get in those reps, get in that practice, but also get real-time feedback exactly as you said, Steve. There has to be that, "I attempted something. Where did I succeed?
Where did I fail?" and have someone there who can keep pushing the learners forward and slowly climb that tall, tall mountain of someday looking down and saying, "You know what? I'm actually pretty okay at math after all." That's the dream. That's fantastic. Obviously, we're going to be able to talk about this in about maybe 10, 12 months from now and see how it actually went, probably 10 months from now. I'm very hopeful that this will be a good thing. I know that I just read a book.
I think Tim's probably going to take a look at it too, "Greeting for Growth," which was written- I just started reading it today. Written by two math professors at Grand Valley State. I think there's a lot of really good material in that book that meshes very nicely with what Tim and I have just been talking about for the pedagogy to choose for teaching this class. I think I'm extremely excited about what Tim is about to undertake. Of course, I'll help you any way I can, Tim. I appreciate that.
I'm only mildly terrified. It's not going to be just a bunch of lectures. It could be on YouTube. It's got to be a whole process. That's already out there, and it's been done better by someone who knows it better. It doesn't work. Our students try to learn with video, and it just doesn't work. I think it's because they really need these social engagement with each other. I think that's the secret sauce.
Yeah, I'm sure most of our students don't think of math as a social activity, but it can be, and it should be. That's something that I hope we can help them discover through this new attempt at teaching the math that our students need at the time and place where they need it in our curriculum. All right. With that, I think we'll say goodbye. Was there anything you want to say at the end? No, I think that's a great place to wrap up. To the audience out there, thanks for coming.
We'll see you next time.