[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
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.