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Came2001 1907-2001 CAS in ME: Theory and practise Paul Drijvers Freudenthal Institute Utrecht University P.Drijvers@fi.uu.nl

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Came2001 1907-2001 Theories concerning CAS use 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 …)

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Came2001 1907-2001 1. Local CAS theories 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?)

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Came2001 1907-2001 2. General theories on ME (Socio-) Constructivism (see LALT) Object-process / reification / encapsulation / procept (related to cognitive psychology) RME Semiotics / symbolisation / representation ……

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Came2001 1907-2001 Practice 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.

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Came2001 1907-2001 Assignment 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?

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Came2001 1907-2001 The case of Maurit (1) 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.

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Came2001 1907-2001 The case of Maurit (2) 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?

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Came2001 1907-2001 The case of Maurit (3) 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.

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Came2001 1907-2001 The case of Maurit (4) 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?

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Came2001 1907-2001 The case of Maurit (5) 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.

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Came2001 1907-2001 The case of Maurit (6) 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.

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