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Hi. It's Paul Andersen and this is Science and Engineering Practice 5: Using
Mathematics and Computational Thinking. There's no established definition for what mathematics
is. But it's essentially an abstract study of quantities, structure, shape and change.
And it's very important in science. What is computation then and what is computational
thinking? Computation is essentially using computers to do calculations for us and a
way of thinking about doing science. Because mathematics is important in science. It allows
us to represent variables in our studies with actual mathematical variables. In engineering
it's important to improve design. And if you look at the root of any science you're eventually
going to find mathematics. I love this quote from John Louis von Neumann, who's one of
the most famous modern day mathematicians. And he said, "If people do not believe that
mathematics is simple, it is only because they do not realize how complicated life is."
What that means is if you're going to study engineering or if you're going to study science,
you're going to have to become good at mathematics. It's at the core of everything. Because science
is like an onion. So if we were to look at biology as a science and you were to look
at it at just the outer level, biology, you're just going to find biology, things like cells.
Things like ecosystems. But if you dig a little bit deeper you'll find that biology is built
on biochemistry, these micromolecules. And that in turn is based in chemistry. And chemistry
is based in physics. And if we look far enough at the core we're going to find mathematics.
So mathematics is really important. It's intimately tied with science. And so let me give you
a couple of examples of that. One formula almost everybody's familiar with is E=mc2.
A lot of people have been looking at this equivalence between mass and energy. But Einstein
was the first one to come up with the formula. The formula that simply explains how mass
can be related to energy and vice versa. And so when we convert for example in a nuclear
explosion we're converting a little bit of that mass to a massive amount of energy. Or
if we were to look at James Maxwell's study of electromagnetism, so he came up with these
equations, Maxwell's equations which explain electromagnetism but they also explain optics
and electrical circuits. And so you can find these simple core abstract ideas that govern
the way science behaves. What is computation then? Computation is doing calculations using
a computer. And computers and calculations and math have really revolutionized all of
science over the last fifty years. And Seymour Papert, who's one of my heroes, did some early
studies on computer programming and how students learn. Developed a programming language called
logo. Was the first one to coin this phrase, computational thinking. It's a way that we're
using computers to help us model and help us understand the world. Let me give you a
concrete example of that. You've maybe seen this demonstration before. The idea is that
if we put a bunch of mouse traps around, let's say a gymnasium, and we were to balance on
each of them one ping pong ball. Or let's put two ping pong balls here and then we were
to trigger that, those ping pong balls will fly up into the air and they'll land on other
mouse traps which will trigger more mouse traps and so you have what's called a chain
reaction going on. Now if you were to set that up in a gymnasium you could imagine this
is going to take a long time, but we can use simulation software to kind of do that for
us. And so this is NetLogo. It's a software program that you can download and it's an
extension of that Seymour Papert's first logo program. But basically what you do is here
you can set up a number of different agents that do different jobs. And so in this one
the red is going to represent what mouse trap has been triggered and then these whites are
going to represent ping pong balls that fly off. And so when you start it you can just
step through the process one after another. But when you really want to run the simulation
you just click on this go button up here and it will run the simulation really really quickly.
So you can see a quick simulation on this mouse trap problem. It gives us data over
on the the side and we can run it again and you'll find that it doesn't look exactly the
same but it looks very similar. But we can run it again and it doesn't look exactly the
same but it has a similar behavior. And so we can get data from that. And so we can use
computers to gather a huge amount of data and they allow us to make better decisions.
And so for example, Mathematica is a computer algebra system that can allow us to model
really complex mathematical problems and solve those. And we really live in the age now of
big data. In other words we're getting so much data that we now have to figure out a
way that we can deal with that data. Examples could be genomics, these are gene sequencers
that are sequencing DNA and species after species, we're figuring out what are the letters
within their DNA and we have to make sense of that. Or connectomics is looking at how
the neurons, for example, in our brain are connected together and there's a huge amount
data. Or meteorology. So it's big data. All this data is being collected and we have to
figure out the science that goes behind that data. And so it's important that our students
are able to work with data. And they should be working with data from day one. And in
engineering we use computational thinking to do simulations. And so this could be a
simulation on the space shuttle and the forces that are put on it. Or this would be an actual
driving simulator used by the military. And so in engineering we're using computation
to test designs that we have. And so what do we want our students to be able to do.
We want them to be able to use mathematics in the science classroom. And that starts
with looking at quantities and proper units and then starting to establish mathematical
relationships. We also want them to start doing some computational thinking. And we
can do that in two ways. We have them start building models and then we can actually run
simulations. And so what's a nice progression for that? In other words, in the elementary
school we want them throwing darts at this board. We want them doing mathematics and
computational thinking. And we just want to get better and better and better over the
years. Get closer and closer and closer to that bull's eye. And so according to the framework,
a good way to actually start is to start them with quantities and units from day one. So the
moment that they can count they should be using rulers and thermometers and protractors
to actually be making measurements. Those measurements should have proper units that
go along with them. They should also be collecting data as soon as they can and organizing that
data into some kind of a chart. As soon as they can they should be using spreadsheets
if that's available. We also want them to look at mathematical relationships. And so
we want them to start using words and then those words eventually become symbols. So
if we're studying motion in a car for example, we could have them come up with sentences
like this, "Distance equals velocity multiply it by time". So we want them to start by looking
at words and explaining what they're seeing or their observations, but we want to transition
that to symbols. So math, that we can then manipulate and that we can use to refine our theories.
When it comes to mathematical modeling like I mentioned, spreadsheets are really really
important. There's so much data available online right now and we want kids to start
looking through that. Becoming comfortable with it. I am always surprised when they get
to my high school class how many students have never ever used Microsoft Excel before.
Never used a spreadsheet. You might go out and not be a scientist when you graduate high
school but almost everybody is going to use spreadsheets in their job at some point. And
so we want our students to start doing that as well. We want to start doing, as they move
into middle school and high school, using probes, so that we can gather data. This would
be a motion sensor or a thermometer hooked up to a computer so they get this idea that
we can gather a huge amount of data and then we have to go through that. And then don't
neglect simulations. Mathematical models or computational models. This again is NetLogo
one on climate change. And you can see that they can do things like add clouds, remove
or add carbon dioxide, and they can see how that is going to manipulate the environment.
Because theories are important but we want to do experimentation and computers allow
us to do a huge amount of experiments in a short period of time. And it's pretty easy.
NetLogo is a great example of one. But there's a lot of other modeling softwares out there.
And so again, mathematics and science are really intimately tied together and we want
students to not have a fear of mathematics. We want to show them why mathematics is important.
Because that is something that science teachers really can do. I hope that was helpful.