so look I’m curious by a quick show of
hands how many of you have taken a
calculus class at some point in your
life all right another show of hands how
so math class in calculus class in
particular tends to get kind of a bad
rap much to the dismay of mathematicians
like myself because calculus is a
subject that happens to contain some
extremely powerful and imaginative ideas
and if you don’t mind I’d like to share
a couple of them with you right now
imagine an object in motion it could be
a car driving down the highway or a drop
of rain falling to earth it feels
obvious to most of us I think that at
any given moment in time that object is
traveling with a certain speed but what
could we possibly mean by such a claim
the Greek philosopher Zeno of Elea
who lived two-and-a-half millennia ago
was so perplexed by motion that he cast
it as paradoxical how is it asked they
know that an object’s motion is possible
in the first place when at any given
instant if we freeze time the object
doesn’t seem to be moving at all that
question might sound a little silly at
first listen but the more you think
about it the more you’ll realize just
how difficult it is to answer one
brilliant way of thinking about this
problem 2,000 years in the making
came to fruition with the work of Isaac
Newton in England and Gottfried Leibniz
in Germany who reasoned about this
problem in the following way so look as
you watch that object move through space
you can imagine that is traveling along
a number line occupying different
numerical positions at different points
in time this in turn allows you to plot
the position of the object as a function
now suppose you want to understand
something about the object’s motion at
just a single instant like this point in
time shown in the picture here’s the
idea take a second point on the position
curve then get out your ruler measure
the change in position between those two
points
that’s this vertical distance right here
and divide it by the chorus
bonding change in time that’s this
horizontal distance here now take that
second point on the position curve move
it a bit closer to the first and measure
again and again and again the remarkable
fact is that it’s possible to describe
what happens at the very limit of this
process as our window of time shrinks
down to just a single moment this in
turn allows us to give a precise
mathematical description of the object’s
motion at just one point in time this
idea which applies not just to moving
objects but to any quantity that exists
in a state of change gave us a
conceptual revolution allowing us to
think about and solve problems not just
in math but in every branch of science
in engineering and in economics in a
totally new way here’s another problem
that fascinated the ancient Greeks
suppose you behold out there in space a
certain region whose area you would like
to know suppose for example you want to
find the area of this swimming pool of
course at the pool had the shape of a
rectangle or the shape of a circle then
finding the area would be
straightforward but what if it has a
weird shape like this well this time the
insight of Newton and lightness is
perhaps even more remarkable take a look
at the upper boundary of that pool and
just imagine that it describes the speed
of an object in motion just like our
object from before as it travels in
between two times a and B and it turns
out that the area underneath the curve
which lightness denoted like this is
nothing other than the distance that the
object travels its position at time B –
its position at time a in one fell swoop
the task of computing a complicated area
is reduced to hitting a simple
difference of two numbers this insight
which likewise has myriad applications
is known as the fundamental theorem of
calculus and it counts in my book as a
really beautiful idea so like many
others I teach calculus to college
freshmen every year in fact in the US
alone over a million high school and
college students study calculus annually
it’s worth asking I think how do they
experience this grant subject well I’m
afraid that a lot of them look a bit
like this or maybe like this turns out
that two students calculus is too often
seen as dry or impenetrable or
terrifying or just some hurdle that they
have to leap over in order to get to
what they really want to study and math
teachers everywhere I must endure the
perennial question what am I ever gonna
have to use this stuff which to be
honest always makes me want to say isn’t
that a little bit like going up to Van
Gogh and saying listen Vincent starry
night is nice but these paintings here
is good for anything like do they have
any practical applications don’t you see
how the work justifies itself and then
contemplating it and riches your life I
claim that math like painting can be an
intrinsically meaningful activity but of
course the story doesn’t end there
because calculus actually is useful
indeed it is the gateway to the STEM
fields it’s a subject whose central
ideas anyone hoping to pursue a career
in science technology engineering math
even economics and finance needs to
understand and yet according to a recent
study conducted by the mathematical
Association of America at least one in
five students don’t succeed in calculus
at the college level at least one in
five students get a D or an F or drop
out of the course entirely so what’s
going on well look I don’t pretend that
the statistic can be boiled down to a
single cause but I do think that a huge
part of the problem is not calculus
itself but the way in which calculus is
typically taught and if you think about
it calculus education has barely changed
in over a hundred years let me describe
for you in a nutshell what I’m gonna
call the standard model of calculus
education amazed will be called a
standard model of math education the way
in which the subject is typically taught
go something like this step one by a
thick heavy expensive calculus textbook
price somewhere at $200 and in which the
entirety of the subject is laid out in
excruciating detail with key theorems
and results
neatly boxed on every page like they’ve
been handed to us from the gods rather
than produced through human imagination
and struggle step to attend lectures
where you’re never called to action or
made to consider an idea on your own but
rather where you’re treated as a totally
passive recipient of knowledge sitting
quietly at a desk while a professor at
the front of the room lectures out of
that expensive book step 3 solve
homework problems that emphasize
formulas and rule following but they
tend to obscure the very ideas that
motivate the entire subject in the first
place and finally more of the same
except this time in the form of
nerve-wracking high stakes exams that
several years ago our math department
was teaching calculus according to the
standard model and several years ago we
faced a crisis our DFW rate that’s the
percentage of students who are getting a
D or an F or withdrawing from class with
an unacceptable 35% and unsurprisingly
student satisfaction as measured by
course surveys was extremely low in the
face of this crisis my colleagues and I
realized that it wouldn’t do to make
superficial changes to our curriculum we
needed to fundamentally change the way
we teach now having been along for much
of this journey I can tell you that the
kind of pedagogical change we had in
mind is not easy took years of
experimentation and effort it’s
something that requires close
collaboration among our faculty and it
also crucially required communication
with students who often weren’t used to
thinking of math class as anything other
than what you get with a standard model
I can also tell you that this kind of
change is in my opinion not only worth
it but essential so let me tell you what
some of the things we did are to change
the way we teach calculus for starters
we decided to get rid of that expensive
textbook my colleagues and I realized
that we could simply write up our
lecture notes polish them and make them
available to students online for free we
retooled our lectures so that they
contain an active component so that
students would be asked to solve
problems in class and not only that but
discuss them both among them cell
and with the instructor we redesigned
almost all of our course materials so
that they place an explicit focus on
concepts and we made it clear to
students that not only would we be
asking them to focus and be tested on
routine problem-solving skills we would
also be asking them to understand the
big ideas of the subject and finally
being an institute of technology and
recognizing that today’s students have
digital literacies the previous
generation simply didn’t have we got rid
of written homework assignments as well
we designed a calculus learning platform
that allows students to solve problems
in a totally freeform manner using the
software students can enter every step
of the solution in whatever way they
like and if they happen to be on the
right track the software is capable of
riding instant positive feedback on the
other hand if students make a mistake in
the course of problem solving then the
software can recognize that mistake in
real time and in many cases offer
detailed feedback to help the student
understand exactly where they went wrong
and improve the software can grade
homework assignments automatically and
in case a student has the wrong final
answer but has strong supporting work
will automatically assign them partial
credit
and since the software is recording and
analyzing all the mathematics that
Student Center is able to provide the
instructor with detailed analytics that
help them understand both where the
class as a whole
and where individual students are
struggling and succeeding since
implementing these changes we’ve seen
our DFW rate drop from 35% all the way
down to 10% we’ve seen an increase in
student satisfaction and in freshman
retention rates and crucially by
rethinking the way that we can teach
calculus we’ve generated ideas for
teaching other subjects differently both
in the math department and beyond if
you’re an instructor teaching a subject
in accordance with some standard model
that’s outdated and that isn’t engaging
your students then I encourage you to
think about how you can teach
differently as well you can make a
change that occurs at the level of your
institution or a change that occurs at
the level of your classroom like making
your lectures more active whatever it is
if it helps make the subject come to
life in the minds of your student
in my opinion it’s worth it and much
needed when students first see the
fundamental theorem of calculus they
sometimes see this and only this a
formula you know it’s like reading
Shakespeare but only paying attention to
the spelling and the grammar failing to
understand that the words on the page
actually mean something and it’s human
nature it’s the path of least resistance
the student says look professor just
give me the rules for succeeding in your
class and I’ll follow them but that’s
not how learning works
the great irony as I see it is that
instructors can fall into the exact same
trap teaching their subjects according
to the same old methods in the same old
formula year after year the instructor
says look just give you the rules for
teaching this class and I’ll follow them
but that’s not how it works I believe
that justice students must be aware of formulas so to most educators thank you
hands how many of you have taken a
calculus class at some point in your
life all right another show of hands how
so math class in calculus class in
particular tends to get kind of a bad
rap much to the dismay of mathematicians
like myself because calculus is a
subject that happens to contain some
extremely powerful and imaginative ideas
and if you don’t mind I’d like to share
a couple of them with you right now
imagine an object in motion it could be
a car driving down the highway or a drop
of rain falling to earth it feels
obvious to most of us I think that at
any given moment in time that object is
traveling with a certain speed but what
could we possibly mean by such a claim
the Greek philosopher Zeno of Elea
who lived two-and-a-half millennia ago
was so perplexed by motion that he cast
it as paradoxical how is it asked they
know that an object’s motion is possible
in the first place when at any given
instant if we freeze time the object
doesn’t seem to be moving at all that
question might sound a little silly at
first listen but the more you think
about it the more you’ll realize just
how difficult it is to answer one
brilliant way of thinking about this
problem 2,000 years in the making
came to fruition with the work of Isaac
Newton in England and Gottfried Leibniz
in Germany who reasoned about this
problem in the following way so look as
you watch that object move through space
you can imagine that is traveling along
a number line occupying different
numerical positions at different points
in time this in turn allows you to plot
the position of the object as a function
now suppose you want to understand
something about the object’s motion at
just a single instant like this point in
time shown in the picture here’s the
idea take a second point on the position
curve then get out your ruler measure
the change in position between those two
points
that’s this vertical distance right here
and divide it by the chorus
bonding change in time that’s this
horizontal distance here now take that
second point on the position curve move
it a bit closer to the first and measure
again and again and again the remarkable
fact is that it’s possible to describe
what happens at the very limit of this
process as our window of time shrinks
down to just a single moment this in
turn allows us to give a precise
mathematical description of the object’s
motion at just one point in time this
idea which applies not just to moving
objects but to any quantity that exists
in a state of change gave us a
conceptual revolution allowing us to
think about and solve problems not just
in math but in every branch of science
in engineering and in economics in a
totally new way here’s another problem
that fascinated the ancient Greeks
suppose you behold out there in space a
certain region whose area you would like
to know suppose for example you want to
find the area of this swimming pool of
course at the pool had the shape of a
rectangle or the shape of a circle then
finding the area would be
straightforward but what if it has a
weird shape like this well this time the
insight of Newton and lightness is
perhaps even more remarkable take a look
at the upper boundary of that pool and
just imagine that it describes the speed
of an object in motion just like our
object from before as it travels in
between two times a and B and it turns
out that the area underneath the curve
which lightness denoted like this is
nothing other than the distance that the
object travels its position at time B –
its position at time a in one fell swoop
the task of computing a complicated area
is reduced to hitting a simple
difference of two numbers this insight
which likewise has myriad applications
is known as the fundamental theorem of
calculus and it counts in my book as a
really beautiful idea so like many
others I teach calculus to college
freshmen every year in fact in the US
alone over a million high school and
college students study calculus annually
it’s worth asking I think how do they
experience this grant subject well I’m
afraid that a lot of them look a bit
like this or maybe like this turns out
that two students calculus is too often
seen as dry or impenetrable or
terrifying or just some hurdle that they
have to leap over in order to get to
what they really want to study and math
teachers everywhere I must endure the
perennial question what am I ever gonna
have to use this stuff which to be
honest always makes me want to say isn’t
that a little bit like going up to Van
Gogh and saying listen Vincent starry
night is nice but these paintings here
is good for anything like do they have
any practical applications don’t you see
how the work justifies itself and then
contemplating it and riches your life I
claim that math like painting can be an
intrinsically meaningful activity but of
course the story doesn’t end there
because calculus actually is useful
indeed it is the gateway to the STEM
fields it’s a subject whose central
ideas anyone hoping to pursue a career
in science technology engineering math
even economics and finance needs to
understand and yet according to a recent
study conducted by the mathematical
Association of America at least one in
five students don’t succeed in calculus
at the college level at least one in
five students get a D or an F or drop
out of the course entirely so what’s
going on well look I don’t pretend that
the statistic can be boiled down to a
single cause but I do think that a huge
part of the problem is not calculus
itself but the way in which calculus is
typically taught and if you think about
it calculus education has barely changed
in over a hundred years let me describe
for you in a nutshell what I’m gonna
call the standard model of calculus
education amazed will be called a
standard model of math education the way
in which the subject is typically taught
go something like this step one by a
thick heavy expensive calculus textbook
price somewhere at $200 and in which the
entirety of the subject is laid out in
excruciating detail with key theorems
and results
neatly boxed on every page like they’ve
been handed to us from the gods rather
than produced through human imagination
and struggle step to attend lectures
where you’re never called to action or
made to consider an idea on your own but
rather where you’re treated as a totally
passive recipient of knowledge sitting
quietly at a desk while a professor at
the front of the room lectures out of
that expensive book step 3 solve
homework problems that emphasize
formulas and rule following but they
tend to obscure the very ideas that
motivate the entire subject in the first
place and finally more of the same
except this time in the form of
nerve-wracking high stakes exams that
several years ago our math department
was teaching calculus according to the
standard model and several years ago we
faced a crisis our DFW rate that’s the
percentage of students who are getting a
D or an F or withdrawing from class with
an unacceptable 35% and unsurprisingly
student satisfaction as measured by
course surveys was extremely low in the
face of this crisis my colleagues and I
realized that it wouldn’t do to make
superficial changes to our curriculum we
needed to fundamentally change the way
we teach now having been along for much
of this journey I can tell you that the
kind of pedagogical change we had in
mind is not easy took years of
experimentation and effort it’s
something that requires close
collaboration among our faculty and it
also crucially required communication
with students who often weren’t used to
thinking of math class as anything other
than what you get with a standard model
I can also tell you that this kind of
change is in my opinion not only worth
it but essential so let me tell you what
some of the things we did are to change
the way we teach calculus for starters
we decided to get rid of that expensive
textbook my colleagues and I realized
that we could simply write up our
lecture notes polish them and make them
available to students online for free we
retooled our lectures so that they
contain an active component so that
students would be asked to solve
problems in class and not only that but
discuss them both among them cell
and with the instructor we redesigned
almost all of our course materials so
that they place an explicit focus on
concepts and we made it clear to
students that not only would we be
asking them to focus and be tested on
routine problem-solving skills we would
also be asking them to understand the
big ideas of the subject and finally
being an institute of technology and
recognizing that today’s students have
digital literacies the previous
generation simply didn’t have we got rid
of written homework assignments as well
we designed a calculus learning platform
that allows students to solve problems
in a totally freeform manner using the
software students can enter every step
of the solution in whatever way they
like and if they happen to be on the
right track the software is capable of
riding instant positive feedback on the
other hand if students make a mistake in
the course of problem solving then the
software can recognize that mistake in
real time and in many cases offer
detailed feedback to help the student
understand exactly where they went wrong
and improve the software can grade
homework assignments automatically and
in case a student has the wrong final
answer but has strong supporting work
will automatically assign them partial
credit
and since the software is recording and
analyzing all the mathematics that
Student Center is able to provide the
instructor with detailed analytics that
help them understand both where the
class as a whole
and where individual students are
struggling and succeeding since
implementing these changes we’ve seen
our DFW rate drop from 35% all the way
down to 10% we’ve seen an increase in
student satisfaction and in freshman
retention rates and crucially by
rethinking the way that we can teach
calculus we’ve generated ideas for
teaching other subjects differently both
in the math department and beyond if
you’re an instructor teaching a subject
in accordance with some standard model
that’s outdated and that isn’t engaging
your students then I encourage you to
think about how you can teach
differently as well you can make a
change that occurs at the level of your
institution or a change that occurs at
the level of your classroom like making
your lectures more active whatever it is
if it helps make the subject come to
life in the minds of your student
in my opinion it’s worth it and much
needed when students first see the
fundamental theorem of calculus they
sometimes see this and only this a
formula you know it’s like reading
Shakespeare but only paying attention to
the spelling and the grammar failing to
understand that the words on the page
actually mean something and it’s human
nature it’s the path of least resistance
the student says look professor just
give me the rules for succeeding in your
class and I’ll follow them but that’s
not how learning works
the great irony as I see it is that
instructors can fall into the exact same
trap teaching their subjects according
to the same old methods in the same old
formula year after year the instructor
says look just give you the rules for
teaching this class and I’ll follow them
but that’s not how it works I believe
that justice students must be aware of formulas so to most educators thank you
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