Undercooled: A Materials Education Podcast | Materials and math: teaching MSE students the mathematical tools they need to succeed

Tim and Steve talk about math education at the college level, what engineering students are and are not getting from their math courses, and the math skills MSE students need to have to be successful. We also tease the creation of a new sophomore course at Michigan - Math Methods for MSE! Also a little nerding out about math topics from different levels of infinity to functions as bases of vector spaces.

This episode is sponsored by the University of Michigan Materials Science and Engineering department (https://mse.engin.umich.edu).

Creators & Guests

Host

Steve Yalisove

Host

Tim Chambers

What is Undercooled: A Materials Education Podcast?

A look into active learning, flipped teaching, team based/project based learning and much more. Everything related to teaching materials science and engineering will be covered. Kindly sponsored by the University of Michigan Materials Science and Engineering Department

[MUSIC]

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