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CAS in ME: Theory and practise

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CAS in ME: Theory and practise

Paul Drijvers

Freudenthal Institute

Utrecht University

P.Drijvers@fi.uu.nl

Came2001 1907-2001

1. Specific local theories, originating from CAS research

2. Originating from ME research in general, and applied to CAS use

(or somewhere in between the two …)

Came2001 1907-2001

- White Box - Black Box and the other way around(Buchberger)
- Scaffolding(Kutzler, but the word is also used outside CAS community)
- Instrumentation(also a bit general, originated from cognitive ergonomy, and potentials for application in other IT-use settings?)

Came2001 1907-2001

- (Socio-) Constructivism (see LALT)
- Object-process / reification / encapsulation / procept(related to cognitive psychology)
- RME
- Semiotics / symbolisation / representation
- ……

Came2001 1907-2001

- How can theories help us in interpreting and understanding student behaviour?
- How can we link theory and practice?
- Let us try in the case of a recent observation.

Came2001 1907-2001

- Given are functions y with Here a stands for a number that can also be negative, or a fraction.A. Sketch a ‘comic’ that indicates how the graph of the function changes as a gets bigger.B. What values of a are ‘special’? Why?

Came2001 1907-2001

P:For a = 0 you have a straight line. Can you see this in the formula, too?

M:Eh, no.

P:That’s a pity.

M:Yeah, but with the calculator, I think it is much more clumsy, because normally I understand it very well, but such a formula, I don’t see much in it if I just enter it into the calculator and it draws the graph.

Came2001 1907-2001

P:And if you just look at it, without calculator, you take x, add a times the square root of x^2+1, what happens then if a = 0?

M:Well then it gets straight but I really don’t know why, no idea.

P:What happens with a times that square root if a equals zero?

M:Ehm, well then the square root will be zero as well?

P:Yeah, so what will be left of the formula in fact?

M:x + a times x^2 +1, isn’t it?

P:But a was zero, remember?

Came2001 1907-2001

M:Yes.

P:And in this case

M:Let’s look, well then, … well the square root is then zero en the square, yes zero squared is also zero, so in fact, then I think this complete part is skipped, or not?

P:And what will remain?

M:Eh, x + a times … +1 or something?

P:No x isn’t zero but a equals zero, isn’t it?

M:… O yeah … well then, then I think the square root is dropped.

P:Yes.

Came2001 1907-2001

M:And the rest remains.

P:Yes, and what is the rest then?

M:Well x + a times x^2 +1, .. , or not?

P:But a was zero?

M:O then it is eh x + x^2 +1

P:No, because eh it says, for this a you should read a zero in this case,

M:mmm.

P:If a = 0, then you get x + 0 times, a whole part.

M:Yes.

P:But how much is zero times a whole part?

Came2001 1907-2001

M:Zero.

P:Yes. So what will be dropped?

M:In fact the complete last part?

P:Yes

M:O.

P:So what will remain?

M:x + a?

P:No, because a = 0, yes, so

M:x.

P:Yes. Are you guessing now or eh?

M:No, I really think so.

Came2001 1907-2001

P:OK, I also really think so.

M:Then it is only x.

(…)

M:O I understand it, that’s why it is so!

P:Yes.

M:Yeah but I think it is a bit strange because normally you have a graph and you draw from point to point but here you suddenly have for each a a different graph.

P:Yes.

M:Whereas as you draw yourself this never happens.

Came2001 1907-2001