0:01

Hey fellow

0:01

mathematicians. Welcome to the

0:04

podcast where Math is

0:04

Figure-Out-Able. I'm Pam.

0:09

And I'm Kim.

0:10

And we make the case

0:10

that mathematizing is not about

0:14

mimicking steps or rote

0:14

memorizing facts, but it's about

0:17

thinking and reasoning; about

0:17

creating and using mental

0:21

relationships. That math class

0:21

can be less like it has been for

0:25

so many of us and more like

0:25

mathematicians working together.

0:30

We take the strong stance that

0:30

not only are algorithms not

0:35

particularly helpful in

0:35

teaching, but that mimicking

0:39

algorithms actually keeps

0:39

students from being the

0:42

mathematicians they can be. We

0:42

answer the question, if not

0:47

algorithms and step by step

0:47

procedures, then what?

0:51

So in the last two

0:51

episodes, we began a

0:53

conversation around models and

0:53

modeling, right, and how it's

0:56

important to consider the

0:56

progression of modeling First,

0:59

the model the situation, then

0:59

the model of student thinking,

1:03

and then using that model as a

1:03

tool.

1:06

And in this

1:06

progression of modeling, notice

1:09

that it's not about any model.

1:09

It's about models that can

1:13

become tools, important tools.

1:13

So in this episode, we're gonna

1:19

get specific about a couple of

1:19

recently popular models that we

1:23

think should be de-emphasized

1:23

and what we can do instead. So

1:29

y'all, I often get asked about

1:29

what I think about number bonds

1:34

and tape diagrams, or sometimes

1:34

called strip diagrams, and how

1:40

they play out in some current

1:40

textbooks. People wonder about

1:43

that. People are really curious

1:43

about how they fit. I think

1:48

people might be noticing that we

1:48

don't use those two models... at

1:54

all.

1:55

At all. So Pam,

1:55

let's start with number bonds.

1:58

What is a number bond? And will

1:58

you describe a number bond for

2:02

the listeners?

2:03

Okay, so I think a

2:03

number bond is a well

2:06

intentioned idea of helping kids

2:06

focus on the relationships at

2:10

hand, focus on what's sort of

2:10

happening in the problem. And so

2:15

Often, it

2:15

looks like circles and there's

2:20

two circles that are sort of the

2:20

parts. And those circles are

2:24

kind of connected to this

2:24

larger, the whole. And so

2:28

there's like a big circle,

2:28

that's the whole, and two

2:30

circles that are the parts. And

2:30

so - like explaining this over a

2:35

podcast is tricky. So Kim, pick

2:35

a number.

2:38

65.

2:39

Okay, so a number

2:39

bond for 65, could be I put 65

2:42

in sort of the total place. And

2:42

then I kind of draw two lines

2:46

that come off of that circle,

2:46

and then I draw two smaller

2:48

circles. And I could say 60 in

2:48

one and five in the other. Or I

2:53

could do a different number bond

2:53

where 65 in the total. And I can

2:56

have 50 in one and 15 in the

2:56

other. Or I can even have one

3:00

where I have 65 in the total and

3:00

I could have 70 in one and five

3:05

in the other because 70 minus

3:05

five is 65. And then I might

3:08

have to play around with which

3:08

one's bigger and all the things.

3:11

Okay, so it's like sort of like,

3:11

how are these numbers related to

3:15

each other? Alright, so let's

3:15

also talk about the other one.

3:17

So we talked about number bonds

3:17

and what they look like. Kim,

3:20

what's a tape diagram? Or strip

3:20

diagram?

3:23

So a tape diagram

3:23

is a rectangular box, maybe,

3:26

with a couple of empty spaces

3:26

where students would take

3:31

numbers from a problem they're

3:31

given. And they are supposed to

3:35

put the numbers in the correct

3:35

spot. So like, if it was a

3:38

missing add-in problem, they

3:38

might fill in one part and the

3:42

total from the numbers in the

3:42

problem. And then they need to

3:46

find the other part. It's a way

3:46

to organize information from the

3:50

problem.

3:51

Yeah, exactly. So I might have the total kind of outside the rectangle. I've seen

3:53

where I might have one

3:57

rectangle, and like you just

3:57

said, the parts are inside and

4:00

the total is kind of outside the

4:00

rectangle. I've also seen it

4:02

where the total is its own

4:02

rectangle and then the parts are

4:06

kind of a rectangle below that

4:06

cut in pieces, and you sort of

4:10

put the parts in that second

4:10

rectangle that is the same size

4:14

as the total. Okay, so let's

4:14

talk about how those two models

4:21

are, in a way, a model of the

4:21

situation. So we've talked about

4:27

our modeling context that we

4:27

like, where kind of the

4:30

beginning step is a model of the

4:30

situation. And so in a big way,

4:34

we feel like number bonds and

4:34

strip diagrams are models of the

4:39

situation. They are an attempt

4:39

to get the relationships out

4:43

there to make them visible. So

4:43

which one's the part? Which

4:47

one's the whole? Oh, do you have

4:47

two parts and you need to find

4:50

the whole? And so both of those

4:50

models can sort of help kind of

4:55

get that understood. If you

4:55

think about a problem solving

5:00

step-by-step procedure -which we

5:00

don't advocate - but often that

5:03

first thing is understand the

5:03

problem, right?

5:06

Yep.

5:06

It's like important.

5:06

And why is that the first thing?

5:09

Well, because it's important to

5:09

understand the problem, or

5:12

you're probably not going to

5:12

solve it correctly. So of

5:15

course, we need to understand

5:15

the problem. And so this is a

5:18

well, my number bonds and strip

5:18

diagrams, are well intentioned

5:21

attempts to say to kids,

5:21

"Understand the problem". Where

5:24

it can go awry is when we demand

5:24

that kids draw it every time. If

5:28

I'm a kid who already understand

5:28

what's happening in the problem,

5:32

I don't think it's very helpful

5:32

to demand that I draw the number

5:35

bond or the strip diagram. It's

5:35

just like an extra step that

5:38

doesn't help me; it's not

5:38

improving my understanding. If

5:41

I'm the kid who doesn't

5:41

understand what's going on in

5:44

the problem, it might help to

5:44

draw the number diagram or the

5:48

strip diagram, it might help to

5:48

do that. But often, it also

5:51

becomes like sort of this extra

5:51

thing to do. It's like, how big

5:54

do I draw the rectangle? And

5:54

does it need to be tall?Kids

5:57

like to pay attention to all

5:57

this weird stuff that may not

6:01

have to do anything with the

6:01

problem itself. So it kind of

6:04

becomes like this extraneous

6:04

thing that may or may not be

6:07

helpful. So do we want kids to

6:07

understand what's happening?

6:10

Absolutely. So also another

6:10

thing that can kind of get in

6:13

the way. So think CGI, have we

6:13

ever done a podcast episode on CGI?

6:18

Surely we've mentioned it.

6:19

If we haven't, we'll

6:19

do one soon. But if you're not

6:21

familiar with Cognitively Guided

6:21

Instruction, and we really

6:24

appreciate Cognitively Guided

6:24

Instruction - which is referred

6:27

to as CGI - because they did a

6:27

lot of really groundbreaking

6:30

research that then has allowed

6:30

us to build on that research.

6:34

But sometimes we might have the

6:34

opportunity to get a little bit

6:37

mixed up here. In CGI, there's a

6:37

strategy called direct modeling.

6:43

Direct modeling is all about

6:43

acting out what's actually

6:46

happening in the problem, a good

6:46

thing, right? Another well

6:49

meaning like, let's model what's

6:49

going on in the problem. That

6:53

can be a very necessary thing

6:53

for kids to do when they are

6:57

beginning problem solvers. They

6:57

need to like actually act out

7:00

what's happening. And then they

7:00

use that acting out to help them

7:03

solve the problem. But that can

7:03

be different than a model of the

7:07

situation. Because it can be...

7:07

how do I say this? Direct

7:14

modeling is where you're

7:14

actually like acting out the

7:17

problem, what's actually

7:17

happening in the problem. Strip

7:20

diagrams and number bonds are

7:20

less of acting out the problem

7:25

and more of a model of the

7:25

relationships at hand, or a

7:30

model of the relationships that

7:30

are happening in the problem.

7:33

But what's interesting is, all

7:33

of those are a model of the

7:38

situation. In our framework,

7:38

whether I'm acting out the

7:43

problem, to understand it, and

7:43

to solve it, maybe just say to

7:47

understand it, whether acting it

7:47

out of direct modeling, or I'm

7:50

drawing a number bond, a strip

7:50

diagram as getting the

7:53

relationships out of what's

7:53

happening in the problem, we

7:56

would consider all of that the

7:56

model of the situation.

8:01

Right.

8:01

That's sort of the

8:01

beginning of our kind of

8:04

modeling scenario here, or the

8:04

way that we think about

8:07

modeling. So it might be true

8:07

that a student needs to act out

8:11

the problem, directly model the

8:11

problem with actions or drawing

8:15

what's actually happening. And

8:15

that's fine. And we would call

8:18

that a model of the situation.

8:18

But we can also be more abstract

8:22

in general, like the number bond

8:22

and strip or tape diagram, where

8:26

it's more of a graphic of the

8:26

relationships involved, it's

8:29

less the action that's

8:29

happening. Either way, these

8:32

models of the situation set up

8:32

the relationships. What they

8:36

don't do is represent what

8:36

you're actually doing to solve

8:39

the problem. How you're actually

8:39

using the relationships to compute.

8:44

Yeah.

8:44

So let me get a little bit more specific. I know I'm doing a lot of like, not

8:46

very exampled discussion here.

8:49

So let's get examples. Let's get

8:49

to some examples. So what's an

8:53

example of a number bond and how

8:53

it can be helpful, and maybe

8:58

less helpful than we might have

8:58

been led to believe? So here's a

9:04

problem. Let's say, "Hey, we're

9:04

reading. We like to read. Kim

9:07

and I both read, reading's good.

9:07

We're on page 195 of the book.

9:13

And there are 303 pages in the

9:13

book. Hey, how much do we have

9:18

left to read?" That could be a

9:18

great problem, right? There's

9:21

the problem. We need kids to

9:21

first understand the problem. We

9:24

don't want kids flipping a coin,

9:24

and just grabbing the numbers

9:28

195, 303, and like adding them

9:28

or multiplying or dividing. We

9:32

don't want that to happen.

9:33

Sure.

9:33

We want them to actually understand what's happening. So kids could draw a

9:34

number bond or a tape diagram

9:40

with, let's do a number bond

9:40

first. They could put 195 in one

9:43

little circle and 303 in the

9:43

total circle, and then a blank

9:50

in the other sort of

9:50

part-circle, the other smaller

9:52

circle. Like, that's how they

9:52

can sort of say, "Okay, I know

9:55

what's happening. I've got 303

9:55

in the total, that 303 is the

9:58

total number of pages and I'm at

9:58

195. So that's part and then I

10:02

have this other part that's the

10:02

pages I have left to read." I

10:05

could also do that on a tape

10:05

diagram. I could have the whole

10:08

rectangle be 303. And then I

10:08

could have that second rectangle

10:12

where I've cut it and I've put

10:12

195 in part of it and I need to

10:16

find the other part of that

10:16

second rectangle, the empty part

10:19

to find. But now what?

10:21

Yeah.

10:21

We've represented

10:21

the situation, the scenario,

10:23

we've got it, we understand

10:23

what's happening. But now how do

10:26

we find that missing part? Okay,

10:26

so Kim, how might the kids solve

10:30

that problem?

10:31

Right, and here's

10:31

the important part, right? So if

10:33

I'm thinking about 303 minus

10:33

195, I might say 303 minus 200,

10:42

and get 103. And then I've

10:42

subtracted too much. So I can

10:46

add back that five, to get 108.

10:49

Bam, and you use a

10:49

nice over strategy.

10:52

Right.

10:52

You thought about

10:52

that problem as subtraction, and

10:55

you subtracted a bit too much.

10:55

And then you add it back, nice,

10:58

super. That's the strategy.

10:58

We're focusing on how students

11:02

are solving that problem. And

11:02

that was a very nice,

11:05

sophisticated strategy. Super.

11:05

So now as a teacher in our sort

11:08

of way that we think about

11:08

modeling, now it's time for our

11:12

second thing, where I would take

11:12

what Kim just did and I would

11:15

model that. I would make it

11:15

visible. I would draw an open

11:19

number line. And I would start

11:19

at 303. And I would draw this

11:22

big jump back of 200. And I

11:22

would say, "Where did you land

11:26

again?" Land on 103. And then I

11:26

would say, "Kim, what did you do

11:30

again?" Depending on how much

11:30

I'm working with the student.

11:33

And she says, "I subtracted too

11:33

much." And so I'm like, "How

11:37

much too much?" She said, "Five

11:37

too much." So then I would draw

11:40

back. The jumps shouldn't have

11:40

been as long so I would back up

11:44

five? And then okay, so what is

11:44

that? 103 and then up that five.

11:48

So now we're at 108. And I would

11:48

draw that 108 and I would go,

11:52

"Hey, does this represent what

11:52

you just did?" Look at it, check

11:55

it out the relationships you

11:55

just used, now they are visible.

11:59

We can use this open number line

11:59

to make that visible. Cool. Then

12:03

over time, I can ask that kid,

12:03

"Hey, you've seen me represent

12:06

your thinking, you've seen me do

12:06

that a few times. Now the next

12:10

time you do that over strategy,

12:10

I want you to represent your

12:14

thinking, now you make your

12:14

thinking visible, just like

12:17

we've done, cool." Then over

12:17

time, and with lots of

12:20

experience, that model becomes a

12:20

tool. And kids might actually

12:24

use that model to solve that

12:24

problem. That's our hope. That's

12:27

our modeling sequence. Okay, so

12:27

Kim, what if a kid used a

12:31

different strategy? Same problem.

12:33

Sure. So a couple

12:33

of other strategies, right? So

12:36

here's one. What if I found the

12:36

difference between 195 and 303.

12:42

So that might look like the

12:42

teacher drawing a number line.

12:46

So let me describe what I would

12:46

do first. So if I had 195, then

12:50

I could add five to get to 200.

12:50

And then I could just add 103,

12:56

to get to 303. So add five to

12:56

200, and then add 103 to 303.

13:02

And then that would look like a

13:02

teacher drawing the number line

13:05

with the 195 on one end, and the

13:05

303 on the other end, and making

13:11

those hops that I just

13:11

described, kind of on top of the

13:14

number line to show the plus

13:14

five and the plus 103, which is

13:19

the total of 108.

13:21

Excellent. And you would write all that out. The teacher would model that

13:23

student's thinking, represent

13:27

that student's thinking. Sometimes people call it annotating student's thinking, I

13:29

don't really like that so much.

13:33

Maybe we'll bring that up in a

13:33

minute. I like representing the

13:37

student's thinking using that

13:37

number line, because we know

13:40

that's an important powerful

13:40

tool. But you said a couple

13:43

strategies. All right, give me another one.

13:44

So how about you

13:44

give me one? Maybe two.

13:47

Okay. So if I'm

13:47

thinking about 195 and 303, and

13:52

the distance between them, I

13:52

might choose to find that

13:55

somewhere slightly different. I

13:55

might put them both on the

13:59

number line, and nudge them up

13:59

both five.

14:03

So now I'm finding

14:03

the distance or difference

14:04

Oh, okay. Lots of

14:04

crossy-outies.

14:06

between 200 and 308, because

14:06

I've moved both of them up five,

14:10

because 308 minus 200, bam is

14:10

just 108. And I'm just there.

14:14

And I'm sort of done. And then

14:14

we call that the Constant

14:18

Difference strategy. So as we

14:18

just talked about this kind of

14:22

Yeah, lots of

14:22

crossy-outies. Again, over time,

14:22

sequence of modeling, I have

14:22

gotten to the point now where

14:26

when I solve subtraction

14:26

problems, I think on an open

14:30

number line. It has become a

14:30

tool for reasoning for me. But

14:34

funny, are you ready? When I

14:34

just did that, if you guys could

14:38

have seen me, I literally had my

14:38

hands in the air. And I put one

14:42

hand where I could see 195 and I

14:42

put another hand where I could

14:46

see 303 and I move them both to

14:46

the right up five. So that I

14:51

could see that 200 and see that

14:51

308 and then I could solve the

14:55

problem. Now, it took me way

14:55

longer to discuss what I did,

14:59

than it did for me to just, like

14:59

my hands were there. And it took

15:03

me way shorter to solve that

15:03

problem, way more efficient

15:07

solve that problem than it would

15:07

have - could you imagine the

15:11

traditional algorithm for 303

15:11

minus 195? Wow, all those

15:15

opportunities for error. and with lots of experience,

15:22

that model becomes a tool and

15:26

then kids, and me, actually use

15:26

it to solve problems. Notice

15:31

that the model that we're

15:31

advocating is an open number

15:34

line, that is a tool worth

15:34

building.

15:37

It is. And I'm going to go back for a second because I think you mentioned

15:39

something that we kind of

15:41

glossed over. And the really,

15:41

really important thing that I

15:44

think you just said, was that

15:44

over time you ask a kid, "Hey,

15:48

when your brain does that, it

15:48

can look like this. Does this

15:52

match what your brain did?"

15:52

Because we want to represent

15:56

their thinking. So we just

15:56

talked about a model that we

15:59

love for subtraction and really

15:59

addition too, and that's the

16:02

open number line. But for

16:02

multiplication, division and

16:05

proportional reasoning problems,

16:05

Pam, what's your preferred

16:08

model?

16:09

Yeah, that's a really good question, because we started talking about strip

16:11

diagrams and tape diagrams. And

16:13

that's often where we see tape

16:13

and strip diagrams, I'm saying

16:17

them like, they're two different things. They're just called those so maybe I should choose

16:19

one, strip diagrams. We see

16:22

those often in proportional

16:22

reasoning problems, where we

16:25

sort of set up what's happening

16:25

in the proportion, and then we

16:28

can kind of think about the

16:28

relationship and those ratios,

16:32

and we can kind of solve them.

16:32

But how do we solve them, that's

16:34

the rub, right? Like, it's just

16:34

the setup, it's just - the strip

16:38

diagram only helps me get a feel

16:38

for what's happening. It doesn't

16:43

then help me use the numbers in

16:43

the structure to solve the

16:46

problem. It definitely doesn't

16:46

model what I do to solve the

16:49

problem, it just sort of sets up

16:49

the scenario, the situation. So

16:52

in Episode 58, we actually

16:52

walked through kind of a way

16:56

that we use ratio tables, we use

16:56

our modeling paradigm to think

17:01

about setting up a ratio table.

17:01

And I use sticks of gum. So if I

17:05

have sticks of gum in a pack,

17:05

then I can think about how I can

17:09

model that situation by I start

17:09

putting down in a table. Okay,

17:13

I've got one pack to 17 sticks,

17:13

I've got two packs to how many

17:17

sticks. As kids say those

17:17

numbers, I put them in the

17:20

table, and it's a model of the

17:20

situation. And then as they

17:24

solve problems, I model their

17:24

thinking. I represent their

17:27

thinking using scaling, or using

17:27

lines to sort of, I'm adding

17:32

packs together, so I add the six

17:32

together. Or I'm scaling as I

17:35

double the packs. Or as I

17:35

multiply the packs times 10,

17:38

then I multiply the sticks times

17:38

10. And I represent all that

17:41

thinking on the ratio table.

17:41

That's the model of thinking.

17:45

And the more that we do that and

17:45

through time and experience,

17:48

then that tool becomes a model

17:48

for thinking. Well, I can do the

17:53

same kind of a thing with a non

17:53

unit rate scenario. So if I have

17:57

something like four slices of

17:57

pizza for $5. I represent that

18:02

scenario. First it's a model of

18:02

the scenario, of the context,

18:05

where I put four slices of pizza

18:05

for $5. And then as students

18:10

think about different numbers of

18:10

slices, and the cost or

18:14

different amounts of money, and

18:14

the number of slices of pizza I

18:16

get - as they solve those

18:16

problems I represent their

18:20

thinking using that ratio table.

18:20

And our goal is then to

18:23

transition to where that ratio

18:23

table becomes a tool for

18:28

thinking, in order to think. And

18:28

as students when they see a

18:31

proportion, they think to

18:31

themselves "Oh, well, you know,

18:33

how how does that fit in a ratio

18:33

table? Oh, I can use that ratio

18:37

table as a tool for thinking."

18:39

So in a nutshell,

18:39

these other models that we've

18:42

been talking about are not bad

18:42

or wrong. They're just really

18:45

limited. They can help students

18:45

slow down and evaluate what's

18:50

happening. It's just a format,

18:50

though. And in any case, it

18:54

doesn't help you decide what to

18:54

do with the numbers, like you

18:56

said. But teachers think it

18:56

does. It's like manipulatives

19:00

where we think the math is

19:00

embodied in the manipulative but

19:03

that's only because we've

19:03

already schematized it, not

19:07

because we can now just show

19:07

what we worked out and have it

19:10

magically transfer to our students.

19:12

Oh that's so well said.

19:13

Yeah, teachers

19:13

have probably seen this a ton,

19:16

right? You might have seen the

19:16

student who may or may not be

19:21

able to take the numbers from a

19:21

word problem and figure out

19:23

where to place them in a tape

19:23

diagram. Maybe they can do that.

19:27

But even if they're able to,

19:27

many of them get stuck in

19:30

knowing how they want to mess with the numbers.

19:32

Yeah, even if they get to that point. Now they're there. What do we do next? And

19:34

what we haven't done is help

19:38

teachers know what to do next.

19:38

Well, woolah! We are helping you

19:42

hopefully know what to do next.

19:42

Ask lots of good problems, pull

19:46

out that thinking from kids, you

19:46

represent their thinking using a

19:49

very important model that then

19:49

can become a tool for them to

19:52

solve problems with. So it's

19:52

helpful to figure out what's in

19:57

a problem. Okay, you could do it

19:57

in other ways. But tape

20:00

diagrams, number bonds is a fine

20:00

way, that's fine. But don't get

20:04

caught in demanding that from

20:04

students because number bonds

20:08

and tape diagrams are not

20:08

particularly helpful to know how

20:12

to mess with the numbers after

20:12

that. And they're also not the

20:16

models that we believe will

20:16

become the tools we want

20:20

students to become really

20:20

familiar with, so that they can

20:23

use them as tools for thinking

20:23

and reasoning and mathematizing.

20:27

Alright, so if you want to learn

20:27

more mathematics and refine your

20:32

math teaching so that you and

20:32

students are mathematizing more

20:37

and more, then join the Math is

20:37

Figure-Out-Able movement and

20:41

help us spread the word that

20:41

Math is Figure-Out-Able!