Research on use of internet ressources in teaching. Can undergraduate students learn mathematics with the internet?

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1 Ca udergraduate studets lear mathematics with the iteret? Research o use of iteret ressources i teachig Geeral Mathematics School Lots of stuff Some Uiversity Quite a lot Much less (~100) Carl Wisløw wislow@id.ku.dk Dias 1 T. Dias March, What s o the web (1995). Dias 3 A very brief review of literature Egelbrecht & Hardig (ESM, 005) preset a survey o the literature o Iteret use i UME up to the, focusig maily o the iteret as a (partial or uique) chael of teachig. They oted some specific problems ad solutios for mathematics: Producig ad exchagig mathematical represetatios Mathematics is a coceptual subject ad a commo opiio is that face-to-face cotact is ecessary to covey these cocepts Other habitual behaviors of studets ad teachers They list a vast variety of modalities of iteret use i UME, ad of ressources (fora, exercise bases, lik collectios &c) available the. More recetly, Gueudet (HDR, 008 ad after) ivestigates the use ad creatio of documets by uiversity mathematics teachers - very ofte supported by olie ressources, ad shared olie - however, for the studets activity, the chages are limited (e.g. a e-exercise eviromet may provide istat feedback) So does iteret use hold o-margial, specific potetials for studet activity i uiversity mathematics? A more geeral problem complex To what extet does the teachig of mathematics (fail to) iduce studets ito a dialectics of study ad research similar to that of the professioal scholarship? What are the mai obstacles? What are, or could be, the cotributios of professioal scholarship (i mathematics ad its didactics) to this effect? Dias 4 1

2 Some verbal hadles o the problem from math didactics Meaigs of scholarship, teachig, research, study, iquiry : Relatios to kowledge (by idividuals, i istitutios) Mediators of the relatio(chevallard, 007): exteral support media (w. itetio to preset kowledge, istruct ) eg text book, professor, joural paper, math video, web page kowledge artefact millieu (w/o such itetio) eg. CAS, paper sheet, exercise kowledge ature A first example Ask studets (who do t kow!): What are complex umbers, ad what are they good for? What might they do? Web search o what are complex umbers RESEARCH: search for kowledge i a millieu STUDY: search for kowledge i a media TEACHING: iduce to research ad/or study (actig as media, proposig millieu), at uiversity preferably close to: PROFESSIONAL SCHOLARSHIP: the study ad research of mathematicias ad other academics Dias 5 Dias 6 Need to study fields first? Are we goig ito poetry? Dias 7 My teacher told me to ever ever Dias 8

3 A first example Ask studets (who do t kow): What are complex umbers, ad what are they good for? What might they do? Web search o what are complex umbers derived questios, such as what ca 1 mea? ca complex umbers be positive? Or ordered? similarities ad differeces with itroducig? idetifyig (a,b) with a+ib, what do operatios mea? what is the use of this ew arithmetic of the plae? what is the polar form [ a+ib = r(cosx+isix) =re ix ] how to remember the additio formulae? rotatig ad oscilatig pheomea (+ a lot of other thigs that may appear rather mysterious ) Directed Dias 9 study paths role of study director A example from Mathematics for teachig (d yr, UCph) Thematic project o liear regressio (U. of Copehage, Ja. 01): explai the theory of simple liear regressio to high school studets Miimise Dias 10 S( a, b) ( ax b k k y k ( x k, yk ) ) y ax b C.F.Gauss BUT WHY? Explai to high school studets Dias 11 Dias 1 3

4 Proof of the formulas for the liear regressio coefficiets Dias 14 The data poits are: (x k, y k ), for 1 k. Defie the equatio of the regressio lie as: Y (x) = a + b x We wat to miimize the sum of squares of the errors, where the errors are the differeces betwee each y k ad the correspodig Y (x k ). This ca be writte as a fuctio of a ad b: f (a, b ) = Σ ( y k - (a + b x k ) ) The, i order to miimize this fuctio, set the derivatives with respect to a ad b equal to zero. f ( y k a b x k ) 0 a f x k ( y k a b x k ) 0 b These reduce to the simultaeous system of equatios i a ad b : Solvig this system gives: a x b y k k x k a x k b x y k a b x k b k y k y b x 1 x k y k x k y k Dias 15 Dias 16 Dias 17 4

5 Dias 18 Dias 19 A crucial coditio for kowledge productio ad for uiversity teachig? Here we get a ad b The existece of a vigorous (ad rigorous) dialectics betwee media ad milieus appears to be a crucial coditio for a study ad research process ot to be reduced to a simple copy of previously elaborated aswers spread over differet social istitutios. (Artigue, Bosch, Gascó, Lefat, 009) Here we get R = B /AC Dias 0 Dias 1 5