Modernizing mathematics education in Slovenia

Nowadays, we are facing a large number of varied educational projects which aim to direct and modernize mathematics education. Many institutions (from university bodies and institutes to individual secondary and elementary schools, networks of schools and private enterprises) make an appearance on the project market and contribute their ideas. Such quantity can cause confusion among teachers. Encouraged by the article, a mathematician\u27s lament, by Paul Lockhard, we present some simple principles for classroom work that, in our opinion, would enhance the efficiency of mathematics classes in the long term. Thus we try to help mathematics teachers build a strategy for a fruitful approach to the ideas, recommendations and advertisements on the educational market. Showing the respect towards the quality mathematics teachers and avoiding discouraging and confusing them are some of the leading ideas that we should pursue in our attempt to improve mathematics education. At the end, we also offer some recommendations about the teacher training system

Diofantske Steinerjeve trojice in trikotniki pitagorejskega tipa

V članku predstavimo povezavo med diofantskimi Steinerjevimi trojicami (trojicami naravnih števil, ki so povezane s konfiguracijo dveh disjunktnih krogov, od katerih večji vsebuje manjšega, pri kateri nastopi sklenjena Steinerjeva veriga) in trikotniki s celoštevilskimi stranicami in kotom 60°, 90° ali 120°. Dokažemo povezovalno formulo in jo geometrijsko interpretiramo.We present a connection between Diophantine Steiner triples (integer triples related to configurations of two circles, the larger containing the smaller, in which the Steiner chain closes) and integer-sided triangles with an angle of 60, 90 or 120 degrrees. We introduce an explicit formula and provide a geometrical interpretation

Diofantske Steinerjeve trojice

V članku opišemo vse trojice ▫$(R, r, d)$▫, ▫$R > r+d$▫, naravnih števil, za katere ima konfiguracija dveh krogov z radiji ▫$R$▫ in ▫$r$▫ ter razdaljo ▫$d$▫ med središči sklenjeno Steinerjevo verigo. To pomeni, da obstaja cilkično zaporedje ▫$n$▫ krogov, ki se dotikajo začetnih dveh krogov in se dotikajo sosednjih krogov v cikličnem zaporedju. Izkaže se, da je v primeru naravnih vrednosti ▫$R$▫, ▫$r$▫ in ▫$d$▫ dolžina ▫$n$▫ Steinerjeve verige lahko le 3, 4 ali 6.We describe all integer triples ▫$(R,r,d)$▫, ▫$R > r + d$▫, for which a configuration of two circles of radii ▫$R$▫ and ▫$r$▫ with the centers ▫$d$▫ apart possesses a closed Steiner chain. This means that there exists a cyclic sequence of ▫$n$▫ circles which are tangent to the starting two circles, and each of them is tangent to its two neighbors in the sequence. It appears that in the case of integer valued ▫$R$▫, ▫$r$▫ and ▫$d$▫, the only possible values for the lengths ▫$n$▫ of the Steiner chains are 3, 4 or 6

Guidelines for e-learning and E-um

Med oblikami pouka, ki so v zadnjem času vse pogostejše v učni praksi predvsem na višjih stopnjah institucionalnega izobraževanja in na tečajih vseživljenjskega učenja, so tudi e-izobraževanje in različne oblike kombiniranega e-izobraževanja. Hkrati je e-izobraževanje tudi predmet kritičnih znanstvenih raziskovanj. Zaradi relativne mladosti te oblike izobraževanja se pedagogika oziroma didaktika e-izobraževanja šele razvija. V prispevku bomo predstavili nekaj spoznanj in smernic za učinkovitejše e-izobraževanje, ki so nastale na podlagi neposredne učne prakse. Zbrane smernice so lahko dobra osnova za premislek o načrtovanju e-izobraževanja s pomočjo spletnih učnih portalov, kot je na primer portal E-um za pouk matematike v osnovnih in srednjih šolah.E-learning, especially web-based learning and blended e-learning are attractive and handy learning forms for high-school, university and lifelong education in recent time. At the same time e-learning is an important issue for critical researches about its impact on students\u27 learning. Because of newness of e-learning the pedagogy of e-learning is still developing. In this article we are going to present some cognitions and guidelines for effective e-learning based on learning practices. We believe the collected guidelines are appropriate basis for reflecting about planning of e-learning with web-based materials or integrating learning management systems such as E-um for mathematics in elementary and secondary school

The origins, the plans and E-um users feedback

Ustvarjalci projekta E-um smo pripravili spletni portal s preko 1100 interaktivnimi gradivi iz osnovnošolske in gimnazijske matematike, ki je po 20-mesečni aktivnosti sedaj na voljo uporabnikom. Glavni namen prispevka je komentirati odzive uporabnikov in strokovne javnosti na objavljena gradiva in jih soočiti z začetnimi konceptualnimi izhodišči projekta ter iz njih izpeljati nekatere smernice za nadaljnje izboljšave portala. Posebej opozarjamo na eno od izhodišč, ki se nanaša na skrb za poglobljeno predstavljanje snovi. Ker se matematično izobraževanje pri nas vse bolj zožuje le na pripravo na pisne preizkuse znanja, menimo, da bi prav nekatera naša gradiva lahko odigrala pomembno vlogo v promociji ideje, da je matematika mnogo več kot le reševanje rutinskih nalog.E-um portal was designed to become a widely used primary and secondary school e-learning aid for mathematics. Now, after about 20 months ofE-um activities the portal with over 1100 e-learning units is freely accessible. In this paper we comment on feedbacks of users and professionals. Considering these opinions, we revise our conceptual origins and draw some plans for further improvements. One of our main principles in preparation of e-learning units was emphasizing understanding. In their feedbacks users hardly notice that e-learning units provide and stimulate deeper understanding. We assume, this is the consequence of general set of educational values, which emphasize and focus primarily on test productivity, white diminishing the value of understanding. Therefore, we plan to use E-um learning materials also as a promotion of learning focused on understanding