Opettajasuunnan seminaari (MTTMS11)

Julkaistu

 

Vaadittavat opintosuoritukset

Aktiivinen osallistuminen (esitelmät, opponoinnit, itsearviointi, keskustelu). Tarkemmat ohjeet ja tietoa kirjallisuudesta annetaan seminaaritapaamisissa.

Aikataulu

Aikataulu tarkentuu ensimmäisellä tapaamiskerralla, se riippuu osallistujamäärästä. Aikataulua voidaan sovittaa opiskelijoiden tarpeiden mukaan. Poissaoloja voi korvata muilla suorituksilla.

Arvostelu

Hyväksytty/hylätty

Esitelmäaiheita

Esitelmäaiheista sovitaan tarkemmin seminaarissa. Alla on joitakin vaihtoehtoja, mutta omiakin aiheita saa ehdottaa.

  • Trigonometristen funktioiden eri määrittelytavat
  • Kompleksiset trigonometriset funktiot, trigonometristen kaavojen todistuksia
  • Sinin summakaavan todistuksia (geometrinen ja kompleksianalyyttinen)
  • Reaalianalyysiä kompleksianalyysin keinoin, esim. integraali
  • Yleinen mittaintegraali ja Riemannin integraali
  • Analyysiä metrisissä avaruuksissa
  • Arkhimedinen ja ei-arkhimedinen valuaatio (metriikka, ultrametriikka)
  • Funktioiden jatkuvuus ja lukujonon raja-arvo euklidisissa avaruuksissa R ja R^n sekä metrisissä ja topologisissa avaruuksissa
  • Todennäköisyyden aksioomat ja klassinen todennäköisyys
  • Todennäköisyyslaskennan ongelmia, esim. kolmen vangin ongelma
  • Geometrian lauseita: Apolloniuksen lause
  • Säännöllisen monikulmion dihedraaliryhmä  
  • Rationaaliluvun desimaaliesitykset ja ryhmäteoria
  • Algebralliset ja transkendenttiset luvut, algebralliset ja transkendenttiset funktiot
  • Irrartionalisuustodistuksia , esim. luvun  π  irrationaalisuus
  • Transkendenttisuustodistuksia; esim. luvun  e  transkendenttisuus
  • Kompleksiluvuista eteenpäin (kvaterniot, oktoniot)
  • Lukukunnat: sovelluksena nimittäjän rationalisointi
  • Reaalilukujen vastine: p-adiset luvut
  • Matriisin aste ja lineaarisen yhtälöryhmän ratkaisuavaruus
  • Differentiaaliyhtälöiden ratkaisuavaruudet
  • Pääakselilause tasossa ja avaruudessa
  • Äärellinen summa ja tulo, potenssi ja monikerta algebrassa (MAOL:n taulukoiden algebran kaavat yleisissä algebrallisissa struktuureissa)
  • Aritmetiikkaa kunnissa
  • Binomikaavan yleistyksiä: multinomilause, binomityypin polynomit
  • UFD (faktoriaalinen kokonaisalue) ja Eukleideen alueet (kokonaislukujen ja polynomien ominaisuuksien vertailua, jaollisuus kommutatiivisessa renkaassa)
  • Polynomin nollakohdat, erityisesti rationaaliset nollakohdat
  • Algebran peruslauseen vastine eksponenttifunktioille
  • Polynomiyhtälön juuret ja radikaalit
  • Polynomiyhtälön juuret ja symmetriset funktiot (Vietan kaavat ja Newtonin identiteetit)
  • Resultantti ja diskriminantti
  • Eisensteinin jaottomuuskriteeri
  • Descartes’n merkkisääntö ja Sturmin lause
  • 3. asteen yhtälöiden ratkaiseminen
  • 4. asteen yhtälöiden ratkaiseminen
  • Erilaisia keskiarvoja, keskiarvon aksiomatisointi
  • Aksiomaattiset systeemit ja todistaminen kouluopetuksessa
  • Numeerisia menetelmiä (esim. numeerinen integrointi, Newtonin menetelmä, approksimointi, interpolointi)
  • Algebran sovelluksia (esim. koodausteoria)
  • Analyysin sovelluksia (esim. Keplerin lait)
  • Matemaattisia pelejä (esim. Solitaire-peli, Angry Birds)
  • Matematiikka ja teknologia (esim. Google-hakujen tai GPS-paikannuksen matematiikkaa)

The American Mathematical Monthly 

  • J. P. Jones and S. Toporowski, Irrational Numbers, The American Mathematical Monthly Vol. 80, No. 4 (Apr., 1973), pp. 423-424.
  • N. Calkin and H.S. Wilf, Recounting the Rationals, The American Mathematical Monthly, Vol. 107, No. 4 (Apr., 2000), pp. 360-363.
  • Xiaoshen Wang,  A Simple Proof of Descartes’s Rule of Signs, The American Mathematical Monthly, Vol. 111, No. 6 (Jun. – Jul., 2004), pp. 525-526.
  • Zsolt Lengvárszky, Proving the Pythagorean Theorem via Infinite Dissections, The American Mathematical Monthly, Vol. 120, No. 8 (Oct, 2013), pp. 751-753.
  • Doreen De Leon, Using Undetermined Coefficients to Solve Certain Classes of Variable-Coefficient Equations, American Mathematical Monthly, Vol. 122, No. 3 (March 2015), pp. 246-255

The College Mathematics Journal

  • Xun-Cheng Huang, A Discrete L’Hopital’s Rule, College Mathematics Journal, Vol. 19, No. 4 (Sep., 1988), pp. 321-329
  • Airton von Sohsten de Medeiros: Elementary Linear Algebra and the Division Algorithm, The College Mathematics Journal, Vol. 33, No. 1 (Jan., 2002), pp. 51-53
  • Sharon Barrs, James Braselton and Lorraine Braselton:  A Rational Root Theorem for Imaginary Roots, The College Mathematics Journal, Vol. 34, No. 5 (Nov., 2003), pp. 380-382
  • P. K. Subramanian: Successive Differentiation and Leibniz’s Theorem, The College Mathematics Journal, Vol. 35, No. 4 (Sep., 2004), pp. 274-282
  • Noah Samuel Brannen and Ben Ford: Logarithmic Differentiation: Two Wrongs Make a Right, The College Mathematics Journal, Vol. 35, No. 5 (Nov., 2004), pp. 388-390
  •  Michel Helfgott: Placing the Natural Logarithm and the Exponential Function on an Equal Footing, The College Mathematics Journal, Vol. 35, No. 5 (Nov., 2004), pp. 390-393
  • Ayshhyah Khazad and Allen J. SchwenkSource:  Irrational Roots of Integers, The College Mathematics Journal, Vol. 36, No. 1 (Jan., 2005), pp. 56-57
  • Jennifer Switkes: A Quotient Rule Integration by Parts Formula, The College Mathematics Journal, Vol. 36, No. 1 (Jan., 2005), pp. 58-60
  • Horst Martini and Walter Wenzel: The Computation of Derivatives of Trigonometric Functions via the Fundamental Theorem of Calculus, The College Mathematics Journal, Vol. 36, No. 2 (Mar., 2005), pp. 154-158
  • Jeffrey A. Graham:  Self-Integrating Polynomials, The College Mathematics Journal, Vol. 36, No. 4 (Sep., 2005), pp. 318-320
  • Philip M. Anselone and John W. Lee: Differentiability of Exponential Functions, The College Mathematics Journal, Vol. 36, No. 5 (Nov., 2005), pp. 388-393
  • Robert Dawson: Differentiate Early, Differentiate Often! The College Mathematics Journal, Vol. 36, No. 5 (Nov., 2005), pp. 404-407
  • O-Yeat Chan and James Smoak:  More Designer Decimals: The Integers and Their Geometric Extensions, The College Mathematics Journal, Vol. 37, No. 5 (Nov., 2006), pp. 355-363.
  • David Rose: The Pearson and Cauchy-Schwarz Inequalities, The College Mathematics Journal, Vol. 39, No. 1 (Jan., 2008), p. 64
  • M. Leigh Lunsford, Marcus Pendergrass, Phillip Poplin and David Shoenthal: The Naïve Chain Rule, The College Mathematics Journal, Vol. 39, No. 2 (Mar., 2008), pp. 142-145
  • Carter C. Gay, Akalu Tefera and Aklilu: The Naïve Product Rule for Derivatives, The College Mathematics Journal, Vol. 39, No. 2 (Mar., 2008), pp. 145-148
  • Victor H. Moll: An Elementary Trigonometric Equation, The College Mathematics Journal, Vol. 39, No. 5 (Nov., 2008), pp. 395-399
  • Michael Sheard:  Trick or Technique? The College Mathematics Journal, Vol. 40, No. 1 (Jan., 2009), pp. 10-14
  • Robert D. Poodiack and Kevin J. LeClair: Fundamental Theorems of Algebra for the Perplexes, The College Mathematics Journal, Vol. 40, No. 5 (November 2009), pp. 322-335
  • Greg Oman:  An Independent Axiom System for the Real Numbers, The College Mathematics Journal, Vol. 40, No. 2 (March 2009), pp. 78-86
  • Vania Mascioni: An Area Approach to the Second Derivative, The College Mathematics Journal, Vol. 38, No. 5 (Nov., 2007), pp. 378-380
  • Alan Sultan: CORDIC: How Hand Calculators Calculate, College Mathematics Journal, Vol. 40, No. 2 (March 2009), pp. 87-92

International Journal of Mathematical Education in Science and Technology 

  • T. Eisenberg (2003): On an unknown algorithm for computing square roots. International Journal of Mathematical Education in Science and Technology, 34:1, 153–158
  • Kin-Keung Poon & Wai-Chee Shiu (2008): On the Dirichlet’s box principle, International Journal of Mathematical Education in Science and Technology, 39:6, 833-838
  • Jan Benacka (2009): Power series solution to the pendulum equation, International Journal of Mathematical Education in Science and Technology, 40:8, 1109-1117
  • Mohamed Allali (2010): Linear algebra and image processing, International Journal of Mathematical Education in Science and Technology, 41:6, 725-741
  • Manuel A.P. Segurado , Margarida F.B. Silva & Rita Castro (2011): Mathematics in chemistry: indeterminate forms and their meaning, International Journal of Mathematical Education in Science and Technology, 42:5, 664-679
  • Manuel A.P. Segurado , Margarida F.B. Silva & Rita Castro (2011): Mathematics in chemistry: indeterminate forms and their meaning, International Journal of Mathematical Education in Science and Technology, 42:5, 664-679
  • Yukio Kobayashi (2013): Recursion formulae of 1^l+2^l+…+n^l and their combinatoric meaning in terms of the tree diagrams, International Journal of Mathematical Education in Science and Technology, 44:1, 132-142
  • Weng Kin Ho, Foo Him Ho & Tuo Yeong Lee (2013): Exponential function and its derivative revisited, International Journal of Mathematical Education in Science and Technology 44:3, 423-428
  • Jozef Doboš (2013): A continuous, nowhere monotone function, International Journal of Mathematical Education in Science and Technology, 44:4, 617-620
  • V.K. Srinivasan (2013): Normals to a parabola, International Journal of Mathematical Education in Science and Technology, 44:4, 568-579
  • Casey Horgan & Kurt Herzinger (2014) A further investigation of using Theon’s ladder to find roots of quadratic equations, International Journal of Mathematical Education in Science and Technology, 45:1, 150-158 (Ks. myös T. J. Osler, Using Theon’s ladder to find roots of quadratic equations,  Mathematics and Computer Education, 2008)
  • Gavin M Abernethy & Mark McCartney (2016): Cannibalism and chaos in the classroom, International Journal of Mathematical Education in Science and Technology

The Mathematical Gazette 

  • Ann E. Hirst, What shape is an ellipse? The Mathematical Gazette Vol. 83, No. 498 (Nov., 1999), pp. 400-409
  • G. Chambers, An Algorithm for the pth Root, The Mathematical Gazette Vol. 83, No. 497 (Jul., 1999), pp. 258-260
  • Mike Grant and Malcolm Perella, Descending to the Irrational, The Mathematical Gazette Vol. 83, No. 497 (Jul., 1999), pp. 263-267
  • A. Sofo, Generalisation of a Radical Identity, The Mathematical Gazette Vol. 83, No. 497 (Jul., 1999), pp. 274-276
  • Ladislav Beran, The Complex Roots of a Quadratic from a Circle, The Mathematical Gazette Vol. 83, No. 497 (Jul., 1999), pp. 287-291
  • Karel Stroethoff, Heron’s Formula via Complex Numbers, The Mathematical Gazette Vol. 83, No. 497 (Jul., 1999), pp. 292-293
  • Bob Ardler, How the Scalar and Vector Products Are Derived, The Mathematical Gazette Vol. 82, No. 495 (Nov., 1998), pp. 454-456
  • Robert J. Clarke, The Quadratic Equation Formula, The Mathematical Gazette Vol. 82, No. 495 (Nov., 1998), pp. 460-462
  • Mark Harvey, Ever Decreasing Circles and Inversion, The Mathematical Gazette Vol. 82, No. 495 (Nov., 1998), pp. 472-475
  • Murray Humphreys and Nicholas Macharia, Tests for Divisibility by 19, The Mathematical Gazette Vol. 82, No. 495 (Nov., 1998), pp. 475-477
  • Joerg Meyer, A Further Look at Pythagoras, The Mathematical Gazette Vol. 82, No. 495 (Nov., 1998), p. 488
  • Pythagoraan lauseen todistuksia: Esim. Rebecca M. Conley and John H. Jaroma, Pythagoras by the Cross Ratio, The College Mathematics Journal, Vol. 37, No. 1 (Jan., 2006), pp. 50-52.

Mathematics & Computer Education

  • David J. Sprows, Antiderivatives as Inverse Linear Transformations, Mathematics & Computer Education , 44, 1 (2010).

Mathematics Magazine 

  • James E. Ward, Vector Spaces of Magic Squares, Mathematics Magazine, Vol. 53, No. 2 (Mar., 1980), pp. 108-111
  • Hansheng Yang and Heng Yang, The Arithmetic-Geometric Mean Inequality and the Constant e, Mathematics Magazine, Vol. 74, No. 4 (Oct., 2001), pp. 321-323
  • George I. Bell, A Fresh Look at Peg Solitaire, Mathematics Magazine, Vol. 80, No. 1 (Feb., 2007), pp. 16-28
  • JOE DeMAIO and ANDY LIGHTCAP: A Graph Theoretic Summation of the Cubes of the First n Integers, Mathematics Magazine, Vol. 82, No. 5 (December 2009), pp. 363-364
  • Carlos Arcos, Gary Brookfield and Mike Krebs, Mini-Sudokus and Groups, Mathematics Magazine, Vol. 83, No. 2 (April 2010), pp. 111-122
  • Christopher Frayer, More Polynomial Root Squeezing, Mathematics Magazine, Vol. 83, No. 3 (June 2010), pp. 218-221
  • Alexander Kheifets and James Propp, A Counterexample to Integration by Parts, Mathematics Magazine, Vol. 83, No. 3 (June 2010), pp. 222-225
  • John Franks, Cantor’s Other Proofs that R Is Uncountable, Mathematics Magazine, Vol. 83, No. 4 (October 2010), pp. 283-289

Mathematics Teacher

  • John H. Lamb: Angry Birds Mathematics: Parabolas and Vectors, The Mathematics Teacher Vol. 107, No. 5 (December 2013/January 2014), pp. 334-340

PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies)

  • David J. Sprows, Using Linear Algebra to do an Integration by Parts Example, PRIMUS 15, 4 (2005), 303-306.
  • Barbara A. Shipman, A Comparative Study of Definitions on Limit and Continuity of Functions, PRIMUS 22, 8 (2012), 609-633
  • L. Jeneva Moseley (2014) Cartooning in Algebra and Calculus, PRIMUS, 24:3, 232-246.