CAS in ME: Theory and practise. Paul Drijvers Freudenthal Institute Utrecht University [email protected] Theories concerning CAS use. 1. Specific local theories, originating from CAS research 2. Originating from ME research in general, and applied to 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 …)
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.
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?
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.
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,
P: If a = 0, then you get x + 0 times, a whole part.
P: But how much is zero times a whole part?
P: Yes. So what will be dropped?
M: In fact the complete last part?
P: So what will remain?
M: x + a?
P: No, because a = 0, yes, so
P: Yes. Are you guessing now or eh?
M: No, I really think so.
P: OK, I also really think so.
M: Then it is only x.
M: O I understand it, that’s why it is so!
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.
M: Whereas as you draw yourself this never happens.