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  • 1. Gothóni, Ralf
    et al.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Miten lahjakas kasvaa menestyjäksi2018In: Kanava, ISSN 0355-0303, Vol. 46, no 4, p. 6-10Article in journal (Refereed)
    Abstract [en]

    The article discusses what kind of processes are related to educating or, more precisely, bringing up a talented musician into a successful performing artist. The processes are similar when the giftedness concerns mathematics or other fields who operate with non-concrete content and abstract values.

  • 2.
    Haukkanen, Pentti
    et al.
    University of Tampere, Finland.
    Merikoski, Jorma K.
    University of Tampere, Finland.
    Mattila, Mika
    Tampere University of Technology, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    The Arithmetic Jacobian Matrix and Determinant2017In: Journal of Integer Sequences, ISSN 1530-7638, E-ISSN 1530-7638, Vol. 20, no 9, article id 17.9.2Article in journal (Refereed)
    Abstract [en]

    Let a1,..., am be real numbers that can be expressed as a finite product of prime powers with rational exponents. Using arithmetic partial derivatives, we define the arithmetic Jacobian matrix Ja of the vector a = (a1,..., am) analogously to the Jacobian matrix Jf of a vector function f. We introduce the concept of multiplicative independence of {a1,..., am} and show that Ja plays in it a similar role as Jf does in functional independence. We also present a kind of arithmetic implicit function theorem and show that Ja applies to it somewhat analogously as Jf applies to the ordinary implicit function theorem.

  • 3.
    Haukkanen, Pentti
    et al.
    University of Tampere, Finland.
    Merikoski, Jorma K.
    University of Tampere, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Arithmetic Subderivatives: Discontinuity and Continuity2019In: Journal of Integer Sequences, ISSN 1530-7638, E-ISSN 1530-7638, Vol. 22, no 7, article id 19.7.4Article in journal (Refereed)
    Abstract [en]

    We first prove that any arithmetic subderivative of a rational number defines a function that is everywhere discontinuous in a very strong sense. Second, we show that although the restriction of this function to the set of integers is continuous (in the relative topology), it is not Lipschitz continuous. Third, we see that its restriction to a suitable infinite set is Lipschitz continuous. This follows from the solutions of certain arithmetic differential equations.

  • 4.
    Haukkanen, Pentti
    et al.
    University of Tampere, Finland.
    Merikoski, Jorma K.
    University of Tampere, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    The arithmetic derivative and Leibniz-additive functions2018In: Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132, Vol. 24, no 3, p. 68-76Article in journal (Refereed)
    Abstract [en]

    An arithmetic function 𝑓 is Leibniz-additive if there is a completely multiplicative function ℎ𝑓 such that 𝑓(𝑚𝑛) = 𝑓(𝑚)ℎ𝑓 (𝑛) + 𝑓(𝑛)ℎ𝑓 (𝑚) for all positive integers 𝑚 and 𝑛. A motivation for the present study is the fact that Leibnizadditive functions are generalizations of the arithmetic derivative 𝐷; namely, 𝐷 is Leibnizadditive with ℎ𝐷(𝑛) = 𝑛. We study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function 𝑓 is totally determined by the values of 𝑓 and ℎ𝑓 at primes. We also find connections of Leibniz-additive functions to the usual product, composition and Dirichlet convolution of arithmetic functions. The arithmetic partial derivative is also considered.

  • 5.
    Joutsenlahti, Jorma
    et al.
    Tampereen yliopisto.
    Perkkilä, Päivi
    Jyväskylän yliopisto.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Näytteita murtoluvun käsitteestä eri aikakausien oppikirjoissa: In English: On the concept of fraction in a sample of the Finnish mathematics textbooks from different era2017In: Proceedings of the annual FMSERA symposium 2016 / [ed] Mervi A. Asikainen & Pekka E. Hirvonen, Joensuu, 2017, p. 99-109Conference paper (Refereed)
    Abstract [fi]

    Tässä artikkelissa kartoitamme sitä, miten murtoluvun käsitettä on opetettueräissä1800–2000-lukujenalaluokkien oppikirjoissa,jamillaisia murtoluvun käsitekuvia tar-kasteluun valituista oppikirjoista välittyy. Osoittautuu, että murtolukua on käsitelty mo-nipuolisestija eri tavoinjo 1800-luvun oppikirjoissa ja osa näistä käsittelytavoista on edelleen käytössä. Uuden matematiikan kirjoissa korostuu jonkin verran konseptuaalinen ajattelu, muuten murtolukujen käsittelyä hallitsee laskemisen näkökulma

  • 6.
    Joutsenlahti, Jorma
    et al.
    University of Tampere.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Matemaattisen ajattelun kielentäminen ja siihen ohjaaminen koulussa2018In: Matematiikan opetus ja oppiminen / [ed] Jorma Joutsenlahti, Harry Silfverberg & Pekka Räsänen, Jyväskylä: Niilo Mäki Institute , 2018, p. 410-430Chapter in book (Refereed)
  • 7. Lehtinen, Matti
    et al.
    Nevanlinna, Heikki
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Matematiikan opetuksen ihanteet2019In: Tieteessä tapahtuu, ISSN 0781-7916, Vol. 37, no 5, p. 3-10Article in journal (Other academic)
    Abstract [fi]

    Useimmilla meistä on pysyviä muistoja joistakin tietyistä oppiaineista ja niiden opettajapersoonista. Eräitä matematiikanopettajia muistellaan vielä vuosikymmeniä heidän kuolemansa jälkeen – niin hyvällä kuin pahallakin. Nämä muistot kertovat myös siitä, millaisia odotamme opettajiemme olevan.

  • 8.
    Mattila, Mika
    et al.
    Tampere University of Technology, Finland.
    Merikoski, Jorma K.
    University of Tampere, Finland.
    Haukkanen, Pentti
    University of Tampere, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Maximal perpendicularity in certain Abelian groups2017In: Acta Universitatis Sapientiae, Mathematica, ISSN 2066-7752, Vol. 9, no 1, p. 235-247Article in journal (Refereed)
    Abstract [en]

    We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (P, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤn, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕⋯. A few such conditions are found and a few conjectured. In studying ℝn, we encounter perpendicularity in a vector space.

  • 9.
    Merikoski, Jorma K.
    et al.
    Faculty of Natural Sciences, University of Tampere, Finland.
    Haukkanen, Pentti
    Faculty of Natural Sciences, University of Tampere, Finland.
    Mattila, Mika
    Department of Mathematics, Tampere University of Technology, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix2018In: Special Matrices, ISSN 2300-7451, Vol. 6, p. 23-36Article in journal (Refereed)
    Abstract [en]

    Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.

  • 10.
    Merikoski, Jorma K.
    et al.
    University of Tampere, Finland.
    Haukkanen, Pentti
    University of Tampere, Finland.
    Mattila, Mika
    Tampere University of Technology, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    The spectral norm of a circulant matrix2018In: JP Journal of Algebra, Number Theory and Applications, ISSN 0972-5555, Vol. 40, no 4, p. 495-500Article in journal (Refereed)
    Abstract [en]

    Let  be the corresponding circulant matrix, and denote the spectral norm. We prove that if the matrix (entrywise), i.e., if

    for all

    where the latter index is modulo n, then

     We apply this result to the case when  is defined by a linear recursion of order two.

  • 11.
    Merikoski, Jorma K.
    et al.
    University of Tampere, Finland.
    Haukkanen, Pentti
    University of Tampere, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Arithmetic subderivatives and Leibniz-additive functions2019In: Annales Mathematicae et Informaticae, ISSN 1216-6014Article in journal (Refereed)
    Abstract [en]

    We introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. In order to generalize these notions a step further, we define that an arithmetic function 𝑓 is Leibniz-additive if there is a nonzero-valued and completely multiplicative function ℎ𝑓 satisfying 𝑓(𝑚𝑛) = 𝑓(𝑚)ℎ𝑓 (𝑛) + 𝑓(𝑛)ℎ𝑓 (𝑚) for all positive integers 𝑚 and 𝑛. We study some basic properties of such functions. For example, we present conditions when an arithmetic function is Leibniz-additive and, generalizing the well-known bounds for the arithmetic derivative, we establish bounds for a Leibniz-additive function.

  • 12.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Matematiikan osaamisen arviointi teknologistuvissa oppimisympäristöissä: haasteita ja mahdollisuuksia2017In: Kriteerit puntarissa / [ed] Virpi Britschgi & Juhani Rautopuro, Jyväskylä: The Finnish Educational Research Association , 2017, p. 119-130Chapter in book (Refereed)
  • 13.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Poimintoja matematiikan oppimateriaalien tutkimuksesta: Title in English: A review of research in mathematics learning materials2019In: Tutkimuskohteena tietokirja: Pirjo Hiidenmaan juhlakirja / [ed] Helena Ruuska, Markku Löytönen, Helsinki: University of Helsinki, 2019, p. 157-167Chapter in book (Other academic)
  • 14.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Tulevaisuuden oppikirja: asiaproosaa vai automaattikaleidoskooppi?2018In: Sähköistyvä koulu: Oppiminen ja oppimateriaalit muuttuvassa tietoympäristössä / [ed] Timo Tossavainen, Markku Löytönen, Helsingfors: Suomen tietokirjailijat ry , 2018, p. 158-170Chapter in book (Other academic)
  • 15.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Faarinen, Ewa-Charlotte
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Swedish Fifth and Sixth Graders’ Motivational Values and the Use of ICT in Mathematics Education2019In: Eurasia Journal of Mathematics, Science and Technology Education, ISSN 1305-8223, Vol. 15, no 12, article id em1776Article in journal (Refereed)
    Abstract [en]

    We study Swedish fifth and sixth graders and the distributions of their motivational values related to studying mathematics with paper and pencil vs. with ICT. Our results show that choosing between these approaches significantly affects the distribution of pupils ’ motivational values; pupils express higher attainment, utility, and cost values when studying mathematics with paper and pencil is concerned. In general, girls express higher motivation in mathematics than boys do. The utility value is the only significant predictor for the view that studying with ICT makes their learning of mathematics qualitatively better. The Swedish fifth and sixth graders also motivated to study mathematics with ICT, yet the distributions of values differ across the genders; boys may benefit more from studying with ICT. Girls seem to need more and a different kind of encouragement than boys to find using ICT in mathematics education meaningful.

  • 16.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Hirsto, Laura
    University of Eastern Finland, School of Applied Educational Science and Teacher Education, Savonlinna, Finland.
    Tablet computers and Finnish primary and lower secondary students' motivation in mathematics2018In: Nordic Research in Mathematics Education, Papers of NORMA 17: The Eighth Nordic Conference on Mathematics Education Stockholm, May 30 - June 2, 2017. / [ed] Eva Norén, Hanna Palmér and Audrey Cooke, Göteborg: Svensk förening för MatematikDidaktisk Forskning - SMDF, 2018, p. 59-68Conference paper (Refereed)
    Abstract [en]

    In this paper, we report in terms of the expectancy–value theory and self-efficacy from the experiences of utilizing tablet computers for the learning of mathematics among primary and lower secondary students (N=256) in one school in Finland. Our main findings are as follows. Using tablet computers seems to increase especially boys' intrinsic values in studying mathematics, yet both boys and girls preferably disagree than agree with the claim that tablet computers have made it easier for them to learn mathematics. Girls clearly prefer to study mathematics with paper and pencil. The utility value of using tablet computers in studying mathematics does not depend on the students' beliefs about their competence in mathematics.

  • 17.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Johansson, Maria
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Faarinen, Ewa-Charlotte
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Klisinska, Anna
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Tossavainen, Anne
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Swedish Primary and Preprimary Student Teachers' Views Of Using Digital Tools in Preprimary Mathematics Education2018In: Journal of Technology and Information Education, ISSN 1803-537X, Vol. 10, no 2, p. 16-23Article in journal (Refereed)
    Abstract [en]

    Our study surveys Swedish primary and preprimary student teachers' (n=94) views of content and methods of mathematics education in preschool and, especially, of using digital tools in preprimary mathematics education. The views related to digital tools turned out to be clearly positive in general. Students who are strongly for using digital tools are also more sure in saying that mathematics education in preschool should be fun. However, they agree less with the claims such as mathematics lessons should be structured, or that the responsibility for the mathematics education of small children belongs mainly to their parents. Those students who were quite strongly for using digital tools agreed less with the claim that mathematics is one of the most important areas of preprimary education. The willingness to take responsibility for children's mathematical education from the parents was the most significant single factor to explain the participants' opinions about using digital tools.

  • 18.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Leppäaho, Henry
    Matematiikan opettajien ja opettajaksi opiskelevien matemaattisesta osaamisesta2018In: Matematiikan opetus ja oppiminen / [ed] Jorma Joutsenlahti, Harry Silfverberg & Pekka Räsänen, Jyväskylä: Niilo Mäki Institute , 2018, p. 294-304Chapter in book (Refereed)
  • 19. Tossavainen, Timo
    et al.
    Löytönen, Markku
    SÄHKÖISTYVÄ KOULU: Oppiminen ja oppimateriaalit muuttuvassa tietoympäristössä2019Collection (editor) (Other academic)
  • 20.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Rensaa, Ragnhild Johanne
    Department of Electrical Engineering, The Arctic University of Norway, Narvik, Norway.
    Johansson, Monica
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Swedish first-year engineering students’ views of mathematics, self-efficacy and motivation and their effect on task performance2019In: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211Article in journal (Refereed)
    Abstract [en]

    We examine a group (N = 88) of Swedish first-year engineering students, their motivation, self-efficacy, and beliefs about the nature of mathematics, and how these relate to their task performance in mathematics. In our data, engineering students who emphasized the exact reasoning in their view of mathematics performed significantly better in a set of mathematical tasks than those who emphasized the applications of mathematics. Similarly, the higher self-efficacy and the intrinsic and utility values of mathematics relate to better performance in the tasks. In general, the students’ task performance was quite modest in relation to the expressed self-efficacy and motivational values.

  • 21.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching. University of Eastern Finland.
    Viholainen, Antti
    University of Eastern Finland.
    Asikainen, Mervi A.
    University of Eastern Finland.
    Hirvonen, Pekka E.
    University of Eastern Finland.
    Explorations of Finnish mathematics students' beliefs about the nature of mathematics2017In: Far East Journal of Mathematical Education, ISSN 0973-5631, Vol. 17, no 3, p. 105-120Article in journal (Refereed)
    Abstract [en]

    According to certain surveys made in Germany, mathematics can be seen either as a stagnant structure (formalism-related orientation), a collection of rules, formulas and methods (scheme-related orientation), an active and creative construction process (process-related orientation), or as a method of describing the real world (application- related orientation). We examine how a twenty-item test of students’ orientations for students’ beliefs on the nature of mathematics, which was developed in the German context, translates to the Finnish culture remaining validity. Altogether 310 first-year students from three Finnish universities took part in the study. We apply confirmatory factor analysis to reveal that a slightly modified version the translated questionnaire is applicable also in the Finnish context. Using data collected with the translated questionnaire, we construct a binary regression models to predict students’ choosing between a technical and an ordinary university, and how probability that a student chooses teacher’s study programme slightly decreases if the static view grows stronger in his/her view of mathematics. Further, we see that all four orientations exist almost equally strong in the Finnish students’ beliefs.

  • 22.
    Viholainen, Antti
    et al.
    University of Eastern Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Viitala, Hanna
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Johansson, Maria
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    University mathematics students’ self-efficacy beliefs about proof and proving2019In: LUMAT: Luonnontieteiden, matematiikan ja teknologian opetuksen tutkimus ja käytäntö, ISSN 2323-7104, E-ISSN 2323-7112, Vol. 7, no 1, p. 148-164Article in journal (Refereed)
    Abstract [en]

    We examine university students’ motivation and self-efficacy beliefs about proof and proving, i.e., beliefs about personal abilities to understand, construct and present mathematical proofs, as well as their certainty about self-produced proofs. The sample of the study consists of 29 Finnish and Swedish students who were studying mathematics in tertiary level. The results show that the students were highly motivated to learn to understand and construct proofs, but they were more uncertain about their proving skills. Moreover, the study revealed reasons for the uncertainty about proving achievements. Most of the reasons are caused bydeficiencies with respect to knowledge of, understanding about or experience from proof and proving.

1 - 22 of 22
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