“Without the understanding that Piaget deliberately stepped out of

the western philosophical tradition, it is impossible to come

to a comprehensive view of his theory of knowing and the model he built to explain how children

acquire knowledge.”

“…Knowledge, of its nature, cannot have any iconic correspondence

with an ontological reality.”

1) What is Constructivism?

2) What makes radical constructivism different from constructivism?

What is the nature of mathematics in Radical Constructivism?

Bonita is a 9th-grade algebra student. She is asked to solve some problems in an interview setting about a dripping faucet.

Int: Okay. Now let’s suppose that it drips 6 ounces in 8 minutes, and it’s going at the same speed, or same rate. How many ounces do you think it dripped in 4 minutes?

Bonita: Um, can I use your pen?Int: Uh huh. Oh, I hogged both of them.

Bonita: Could I use a calc – Int: You sure can! Now I think this is like

the ones you’ve used in class.

Bonita: Uh huh.Int: I’ll let you turn it on, you probably know

how to use it better than I do.

Bonita: Thank you. It, um, to be able – oh. It would be 3 ounces in 4 minutes.

What does Bonita Understand?

Int: Let me give you one other problem, and

you can use that pen. And let’s just see

if there’s another way to do it. Like, can

you think about it without – the, there’s

nothing wrong with this, you did a great

job with that. But I just want to see if

there’s another way to do it too. So,

this time I’m gonna ask how many

ounces in 40 minutes?

Bonita: So you want me to try different rather

than this? Or…Int: Yeah.Bonita: (Unintelligible) there the same?Int: Can you do it in your head, or is there a

way to just think about it without using

a math formula?Bonita: Uh…Int: It’s okay if there isn’t. I just want you to

think about it a minute.

Bonita: I can’t do it in my head, it’s, it’s…Int: What if this was a 16 instead of a 40?

Could you do it in your head then? Ifit was, um, 6 ounces in 8 minutes, howmany ounces in 16 minutes?

Bonita: Um, I have problem with thinking of these, like, in my head. I don’t come upwith answers really quick, I have to, likeeither, get a calculator to…that’s not…

Int: Can you use a calculator, but not the formula?

Bonita: Hmm.Int: Is there any way to think about it?Bonita: Like without using this? (Points to the

cross-multiplication algorithm)Int: Uh huh.Bonita: Just like putting it in there?Int: Yeah just, just curious.Bonita: I don’t know how to…

What does Bonita Understand?

Int: Now, let’s suppose you have two peopleLet’s put you and me. And we’re

eachwatching a little faucet. And, let’s

say weeach have a container here. And

let’s saythat you collected 3 ounces, and

I’vecollected 3 ounces. Are they

dripping theas fast as each other, or do you

think thatone could be dripping faster than

the other?

Bonita: Um, they’re dripping the same.Int: The same? What if mine took 10 minutes

and yours took 20 minutes. In other words,

you had to watch, sit and watch your clock

for 20 minutes to get the 3 ounces, but

mine filled 3 ounces in only 10 minutes?

Bonita: Um, this one (points to the left one) is going way faster and probably this

(pointsto the right one) is like, I don’t

know it’skind of going, like, fast and slow,

maybe?Int: Hmm. How’d you determine that? What

were you thinking?Bonita: Um.Int: What makes this one faster? (Points to the

left one).

Bonita: Um, makes it faster ‘cause it, it um, it beat

the, the 10 minutes…Int: Okay.Bonita: And this one (points to right one) as I said

it, it was going fast, or, and then slow, or,

the opposite way.

Int: Okay. Let’s put one more person in here.Have you got a friend’s name?

Bonita: Um, Andrea.Int: Andrea. That’s a pretty name. I have no

idea if I spelled it right, but…Bonita: Yeah.

Int: Let’s say that she collected…she stood there the same amount of time as

I did butshe collected 4 ounces instead of

3. Is herfaucet dripping at the same, same

fastnessas mine or is hers faster or slower

thanmine?

Bonita: Um, Andrea’s is faster.

What does Bonita understand?

1) What would an information processor say about Bonita’s math knowledge?2) What would a behaviorist say about Bonita’s math knowledge?3) What would a radical constructivist say about Bonita’s math knowledge?

What is a rate?

What is a Rate?

External perspective• A rate is the relationship of

two measurements with different units, like miles per hour.

• A rate is a “per 1” ratio, like 60 miles per hour.

Radical Constructivist Perspective• “A rate is a reflectively

abstracted constant ratio.” (Thompson & Thompson, 1992, p. 7)

• “A specific ratio in relation to the quantities compared to make it is a mental structure. A rate, as a reflectively abstracted constant ratio, symbolizes that structure as a whole, but gives prominence to the constancy of the result of the multiplicative comparison.”