how-to-do-math

How to Do Math

Description of the video:

0:01
Sheila: So what makes math so different from other subjects?
00:06
Charlie: Well, it can feel different.
00:13
Sheila: That feeling is real.
00:14
That struggle is real.
00:16
But, it doesn't mean that you can't do it.
00:18
It's actually part of the learning experience - even for the experts.
00:21
Charlie: Yeah, and math can also look different.
00:24
You know, it can almost be scary to some people.
00:26
Almost like a language.
00:27
And if it is this language, then it's going to be a language of discovery... of finding
00:31
the unknown.
00:32
Sheila: So, you're looking for something.
00:35
Maybe think of it as a journey or a quest.
00:36
Charlie: A quest for what?
00:38
Sheila: The "Why?"
00:39
Charlie: The what?
00:40
Sheila: The "Why?"
00:41
That's the concept behind solving these problems.
00:44
It's more than just finding the solution for one problem.
00:47
It's about how you get there.
00:48
Charlie: Yeah, and even the tricky ones.
00:50
You know, the weird ones.
00:51
The ones that are on a test that don't look like anything that you practiced in your homework.
00:55
Sheila: Right.
00:56
So let's take a look at how to find the "Why?" - in three places.
01:05
Charlie: So when you're taking notes in a lecture, it could be tempting to just write
01:08
down what you see on the board.
01:09
Sheila: Yeah, it's the instructor's thinking.
01:12
It's their explanations, their answers to the questions, "Why did we do it this way?"
01:16
or "How did we know how to do that?"
01:17
And that part might not be on the board.
01:19
It might just be in what they say.
01:20
Charlie: Exactly, so it's the explanations, the thinking.
01:24
It's the finding answers to the how and the why that you want to focus on when you're
01:28
taking notes in class.
01:30
Take a look!
01:32
Emily: So, now our job is to represent B and O inside of this rectangle because inside
01:37
of this rectangle is going to represent all of the students at IU.
01:39
So, we'll start by drawing this set B.
01:42
We'll represent it as a circle.
01:45
So, we need to draw this circle B like this because we need a region outside of B that
01:50
would represent the set of students at IU who don't want to major in business.
01:58
We'll draw the set O like this: the set of IU students who live out-of-state.
02:02
We need to draw it so it has some overlap with B but isn't completely overlapped with
02:06
B. Here's why: this middle region would be a student who can major in business and who
02:12
lives out-of-state, and that can definitely happen, that's possible.
02:15
But this region right here represents out-of-state students who don't want to major in business.
02:20
There can be students who want to major in business and live out-of-state.
02:24
And there can also be out-of-state students who don't want to major in business.
02:31
Sheila: Reading math textbooks is not like other reading.
02:34
The information is tightly packed.
02:36
Every word and every example count.
02:38
Charlie: Exactly.
02:39
So, instead of just simply reading it, like this, you may have to read it like this and
02:43
go back and forth with pencil and paper in hand.
02:46
Defining new concepts and terms through your own words to better check for your understanding.
02:51
Sheila: Yeah, you have to work through the examples yourself to really look for the "Why?".
02:55
It might even help to draw out the concepts and examples to get a clearer picture of the
02:59
thought that goes behind them.
03:01
So reading a math textbook might look a lot less like this and more like that.
03:06
Check it out.
03:09
Emily: OK, so my goal is to try to figure out what's going on in this paragraph.
03:13
OK, so Figure 1.4.
03:14
"Venn diagrams provide us with a geometric way to represent the decomposition of a set
03:19
into subsets."
03:20
OK, so decomposition means breaking things apart.
03:23
"For example, in Figure 1.4 we illustrate how the set A can be decomposed into subsets
03:29
A intersect B."
03:30
OK, so Figure 1.4.
03:35
So, I'm decomposing A, so I'll make A stand out a little bit.
03:39
I know from class, intersection means "and".
03:44
OK.
03:46
And they said A intersect B, the football-shaped region.
03:49
So here, they have that pointing to this middle region.
03:52
Looks like a football.
03:56
So these are things that live in A AND live in B.
04:15
Charlie: So now we're looking for the "Why?" in your math problems.
04:17
But you're not going to find it here in the answers.
04:20
Sheila: That's why your instructor might not be crazy excited when you get the answer right
04:24
to, say, problem number eleven.
04:26
It's because the "Why?"
04:27
goes beyond any one problem or one solution.
04:29
Charlie: It's in the way you get there.
04:31
The process of working through a problem and the tools that you learn to use throughout
04:35
the way.
04:36
Sheila: The only thing is, you might run into what feels like a sort of weird paradox when
04:40
you're working on the problems.
04:42
It's that you learn the most at the very edge of your learning ability.
04:45
So that means that learning new information can feel uncomfortable, or frustrating, or
04:50
might even feel impossible.
04:52
Charlie: And yet it's in this same place where the sweet spot for growth is.
04:55
But, how do you get there?
04:57
How do you get past these feelings to where the real learning takes place?
05:00
Sheila: Well, you build a problem-solving system with several key parts.
05:04
First, you take the problem apart.
05:06
You break it down and you analyze it.
05:08
You can ask some questions.
05:09
Charlie: Like, "What kind of problem is this?"
05:11
What tools, strategies, concepts, and equations will you need to use to solve it?
05:15
What are the steps and how do you know?
05:18
Sheila: Second, you follow them and as you go, lean into the struggle.
05:22
It's OK.
05:23
It's part of it.
05:24
Charlie: And finally, you check your work.
05:26
Did you get it right?
05:27
And if not, what happened?
05:29
Analyze your mistakes.
05:31
Did you find the "Why?"
05:33
If there was another problem like it that looked a little different, could you solve
05:36
it?
05:37
Sheila: Now that is what leads to true understanding.
05:39
It changes your brain.
05:41
Because every time you do this, you're creating more and more robust neural connections.
05:46
The ones that you'll need to nail these problems.
05:48
Charlie: In his book The Talent Code, this is what Daniel Coyle refers to as deep practice.
05:53
It's how you get better.
05:54
It's how you get good.
05:57
Let's see this at work.
05:58
Emily: OK, so this says Z lives in (oh gosh) A intersect B complement union C. OK, so I
06:09
have no other way to do this except to break it down.
06:12
So the first thing I'm going to see is I'm going to break this up into the two different
06:16
sides of the union sign.
06:19
I'm going to try to do this side first because for unions you only have to satisfy one of
06:23
the sides to get in.
06:24
So, if Z is in this side then I don't even have to worry about this thing, so I'll try
06:28
that first.
06:30
So, does Z live in C?
06:32
I look over here... it doesn't!
06:35
So I'm not lucky!
06:39
So I do need to check this thing on the other side to see if it's in there or not, so let's
06:42
see what this is.
06:45
A intersect B - this is going to get shaded in.
06:49
So, A intersect B is anything that's in A and in B. But this is the complement of that, so
06:59
it's anything on the outside of that green.
07:01
Well, Z lives in here so it can't live in the complement.
07:07
OK.
07:08
So Z also isn't in this region.
07:12
Z doesn't satisfy either thing on either side so this one is also False.
07:21
Sheila: To find the "Why?" in class, try taking notes on the instructor's thinking and explanations
07:26
- instead of just copying what you see.
07:28
Charlie: To find the "Why?" in the textbook, give this a shot.
07:32
Read it like you need it.
07:33
Like every word counts.
07:35
Work through examples with pencil and paper and seek understanding.
07:38
Sheila: And to find it in the problems - well, you'll need the answer, too - but you'll
07:42
find the "Why?" in the process of how you got there.
07:45
Charlie: And if you remember nothing else from this video, remember this: keep going!
07:51
You can do this.
07:53
And we'll see you on the other side.

De-mystify the Numbers

How best to take notes, use the textbook, and work the problems in a wide variety of math-related courses. Our goal: that you'll look at this and think, "I could do that."

Download a transcript

Full-Length Demos

Venn Diagrams: The Basics (Complete In-Class Demo) Class notes

Description of the video:

00:40
Anthony: We'd like to invite you to imagine yourself right now as immersed in a grand
00:44
story. This is a story full of rewarding careers. It's shaped by a broad and deep education
00:51
and lifelong learning. And it's marked by engaged participation in a truly global 21st
00:57
century world. We are asking you to invest in the idea that a college education at its
01:05
maximum potential is an absolutely essential ingredient in making sure this life happens.
01:10
But it won't happen automatically. You must make it happen. It's up to you to put the
01:17
pieces of your college classes and experiences together into some meaningful whole. You are
01:22
the main character in the story, but you're also the author and so that means that how
01:28
it turns out is entirely up to you. The end hasn't been written yet but you've already
01:35
started on the beginning. Fortunately you have some help, and that's where we come in.
01:41
My name is Anthony Guest-Scott and I'm the Academic Coordinator here at the Student Academic
01:45
Center on the IU Bloomington campus. This online workshop series is all about how to
01:52
make the most of college, but it's a "most" that's unapologetically ambitious. It's a
01:57
most that includes, but goes well beyond, the idea that a degree is simply a job ticket.
02:04
We're trying to raise the bar high, as high as we can put it.
02:08
I want to talk a little bit more about the main character in this story now: you. And
02:14
I'm going to describe the kind of learner you'll want to be if you want to to wring
02:17
every last drop of potential out of your college degree, on both a personal and professional
02:21
level. First, you want to be a reflexive learner. Reflexive learners are self-aware. They set
02:28
goals. They monitor their motivation and comprehension. They track their progress and performance.
02:35
They think about their own thinking and they connect this with those goals.
02:40
Second, you want to be a self-regulated learner. Self-regulated learners understand that learning
02:47
and studying are active processes that are largely under their control. They know it's
02:53
up to them to get what they need and what they want out of their college degree. And
02:59
so they must make it happen and they'll find the resources they need to do that, or if
03:04
they don't exist, they'll create them themselves. And when they do encounter obstacles, they'll
03:08
go and they'll seek support. And we have a ton of support on campus.
03:13
Third, you want to be a deep learner. Deep learners, like everyone else, care about grades,
03:18
they care about jobs, but they're most motivated by the love of learning itself. And so they
03:24
want to do well on all of their college work because they see it as an opportunity. An
03:29
opportunity for personal growth that will lead them somewhere exciting, to a life full
03:34
of curiosity, wonder, understanding, and compassion. They know their college degree will provide
03:42
them the critical thinking they need to deconstruct the world as it is now and the creativity
03:47
they need to reimagine it as it might be.
03:51
If you can get behind these ideas, behind this kind of learner (reflexive, self-regulated,
03:58
and deep), then that will lead you on a path where you'll learn to mine your college courses
04:03
for ideas that will permanently transform the way you think, act, feel, and experience
04:08
the world. This is what we hope for you and this is what we want you to hope for yourself.
04:15
So how do we get there? There are two roadblocks to student success that come up over and over
04:21
again. They are attitudes and application. Sometimes what's standing primarily in a student's
04:28
way are their mental models, their whole way of thinking about a certain topic, say, studying,
04:34
or taking notes. That's the "attitudes" part. Other times students have a pretty good handle
04:39
on that, they know what they need to do, but they're not quite sure how to actually do
04:43
it. That's the "application" part. So we're going to be dealing with both things. We're
04:48
going to be talking a lot about big ideas and philosophies, approaches to ways of thinking
04:53
about the topics we address, but we're also going to be talking a lot about very practical,
04:58
everyday strategies you can use right now to start making a real difference in your
05:03
education. What we hope is that we'll help you create a positive habit. Something that
05:09
eventually you won't even need to think about.
05:10
All right, now a word about the format. Each episode in this series comes in a video and
05:16
audio only version and they're about 20 to 25 minutes apiece. Our hope is that you can
05:21
get through two or three topics that you need help with in about an hour. And they're mobile
05:25
and accessible so you can watch or listen to these while you walk around campus, wait
05:30
between classes, while you're working out, or washing the dishes. If you visit us online
05:36
at our blog you'll find that each episode comes linked to a number of supporting documents
05:40
that you can download, online resources that are connected, etc. Also on the blog you can
05:47
talk back to us, so please do come and post questions, even very personal ones, and we'll
05:53
answer them. Suggest other ideas for episodes and we'll get to them.
05:58
All right, that's all for now. I want to you to remember that story I talked about at the
06:03
beginning, the one you're writing. This series is about that journey. So enjoy, good luck,
06:13
and please let us know what else you need.
Venn Diagrams: The Basics (Complete Textbook Demo)

Description of the video:

00:04
Emily: OK, so my goal is to try to figure out what's going on in this paragraph.
00:08
OK, so Figure 1.4.
00:10
"Venn diagrams provide us with a geometric way to represent the decomposition of a set
00:15
into subsets."
00:16
OK, so decomposition means breaking things apart.
00:19
"For example, in Figure 1.4 we illustrate how the set A can be decomposed into subsets
00:25
A intersect B."
00:26
OK, so Figure 1.4.
00:40
OK, so I'm decomposing A, so I'll make A stand out a little bit.
00:48
"...will be decomposed into the subsets A intersect B." OK, I know from class, intersection
00:57
means "and".
00:59
OK.
01:00
And they said A intersect B, the football-shaped region.
01:04
So here, they have that pointing to this middle region.
01:07
Looks like a football.
01:11
So these are things that live in A AND (because that's what that is) live in B.
01:27
OK, so they say A can be decomposed into subsets A intersect B. Got it.
01:31
And A intersect B complement (the crescent-shaped region).
01:36
So they point to this that looks like a crescent, so this thing must be A intersect B complement.
01:41
Let's try to figure out why [and] say what that means in English.
01:45
OK. A intersect B complement (keep it color-coded).
01:52
So, this is almost exactly the same as this except I have this complement thing.
01:58
And from class, I know that complement means "not".
02:01
So things that live in A AND DON'T live in B. That makes sense with this picture because
02:09
out of the stuff that's in A, this is the part of the stuff that's in A that's not in B. OK.
02:14
But it seems like they aren't going to label these regions.
02:17
Let's try to figure out what they are.
02:19
Out of these two other regions that are left, this guy right here looks an awful lot like
02:27
the blue region.
02:28
So this guy looks an awful lot like that.
02:30
So whenever I try to write this out I'm expecting the English part to be very similar
02:35
to the English part in this.
02:37
Let's see what happens.
02:38
This seems to be the stuff that's in B AND NOT in A. So in B AND NOT in A. Cool, that
02:50
looks like this, that makes sense.
02:52
I'll write that in English.
02:56
So things that live in B AND (because it's still intersection this whole way through)
03:09
NOT in A. And this literally looks like the mirror image of this, just with As and Bs
03:15
switched around, so I buy that.
03:18
OK, but the last region I have to do is this outside region.
03:23
So there's four regions.
03:27
So what could that be?
03:30
Seems to be the set of things that AREN'T in A AND AREN'T in B, so that makes sense.
03:35
So I'll try A complement intersect B complement.
03:40
And this seems to fit in because it seem to be the only piece of these pairings of A and
03:44
B that I've missed.
03:47
OK, makes sense.
03:51
Things that are NOT in A AND NOT in B.
Venn Diagrams: Three Circles! (Complete Problem Demo) Problem solved

Description of the video:

00:04
Emily: OK, so I'm about to attempt number ten on the homework. And this one seems tough
00:10
because all I've dealt with so far have been two Venn diagrams, and this seems like a three-circle
00:16
Venn diagram. So they have a circle A, circle B, and circle C. So which of the following
00:30
is a true statement about the Venn diagram shown in figure 1.8? Whoops, I forgot to label
00:34
my points. They said I have an X, a Z, and a Y.
00:38
OK, so statement 1 is that X is an element (or "lives in" - that's what they told me
00:46
to think of that as). X lives in A intersect C. OK, so let's see if it satisfies both
00:53
of the conditions because this is intersection. I'm looking over here - X lives in A, so
00:58
that part's OK. I guess I'll put a little check mark. X lives in A, but X doesn't live
01:06
in C. And if this is an intersection and has to be in both, it's only in one, so that's
01:11
not true. So I will put a false.
01:14
OK, so part B looks horrendous. I'm going to write it down, but I'll wait. I will wait
01:21
to do it.
01:23
OK, so c.) says Y lives in B complement. Well I know that that complement symbol means NOT
01:32
in B. But Y DOES live in B. It doesn't live outside of B, so that's also false. That makes
01:41
sense.
01:42
Good. Good, OK let's do d.) next. D.) says X lives in C complement, so X lives outside
01:53
of C. X DOES live outside of C. OK, true. That's good because I was getting too many
02:01
falses.
02:02
E.) says Y lives in A complement union C. OK, so this is the one where I just have to
02:13
satisfy one of the two things on either side of that symbol. So let's see what it satisfies.
02:18
So does Y live outside of A? It does because this circle A - Y lives outside of it. So
02:27
Y satisfies this condition on the left. Cool. Y DOESN'T live in C. So Y satisfies one of
02:35
the conditions. It doesn't satisfy both. But that's OK because it's union, and union
02:40
means you have to satisfy at least one of them. This is a true statement, that's true.
02:47
OK, so let's go to the big one. OK, so this says Z lives in (oh gosh) A intersect B complement
02:59
union C. OK, so I have no other way to do this except to break it down. So the first
03:05
thing I'm going to see is I�m going to break this up into the two different sides of the
03:08
union sign. I'm going to try to do this side first because for unions you only have to
03:15
satisfy one of the sides to get in.
03:17
So, if Z is in this side then I don't even have to worry about this thing, so I'll try
03:21
that first. So, does Z live in C? I look over here... it doesn't! So I'm not lucky! So I
03:31
do need to check this thing on the other side to see if it's in there or not, so let's see
03:35
what this is. A intersect B - this is going to get shaded in. So, A intersect B is anything
03:45
that's in A and in B. But this is the COMPLEMENT of that, so it's anything on the outside of
03:52
that green. Well, Z lives in here so it can't live in the complement. OK. So Z also isn't
04:02
in this region. Z doesn't satisfy either thing on either side so this one is also False.
04:10
OK, so the question said, "Which of the following is a true statement?" So there's two true
04:16
statements, my answer would be "D" and "E."

Exercises

  1. This video uses one word to explain what to look for when doing math. What is that word, and why do you think we chose it? How might this approach to math be different from what you are doing now?

  2. Next time you are in your math class, try taking notes not just on what you see, but on how the instructor thinks through the process of solving the problem.

  3. Try reading your math textbook for understanding, with paper and a pencil nearby to work through the examples.

    • You could try using the Frayer Model to enhance your understanding of each new concept.
  4. When you are working your math problems, what kinds of questions can you ask yourself to talk through them, “translate” them, break them down, and work through them step-by-step?

  5. Instead of working the same type of math problem over and over in a row, try mixing them up.  Read more about why here (look under “Mistake 2. Repetition”).

  6. How could you adapt some of these strategies for other kinds of STEM courses?

Download the exercises