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  • 1.
    Bergqvist, Erik
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Tossavainen, Timo
    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.
    An Analysis of High and Low Intercorrelations between Mathematics Self-Efficacy, Anxiety, and Achievement Variables: A Prerequisite for a Reliable Factor Analysis2020In: Education Research International, ISSN 2090-4002, E-ISSN 2090-4010, Vol. 2020, article id 8878607Article in journal (Refereed)
    Abstract [en]

    This paper draws on data from a quantitative study of upper secondary students' general mathematical self-efficacy, anxiety towards mathematics, and their relationship to achievement in mathematics. The main objective of this article is to discuss the type of information that may be lost if potential problems of validity and extreme multicollinearity in exploratory factor analysis would be solved by only removing variables without doing a profound analysis. We also describe a method that treats Likert items in the questionnaire as ordinal variables that may represent the underlying continuous variable. Our study shows, for example, that removal of problematic variables without a profound analysis leads to a loss of significant information about test anxiety. Our qualitative analysis of problematic variables also led to an unexpected finding regarding the relationship between general mathematical self-efficacy and motivational values in mathematics.

  • 2.
    Gothóni, Ralf
    et al.
    Independent, Finland.
    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.

  • 3.
    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, 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.

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  • 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.
    Arithmetic Subderivatives: Discontinuity and Continuity2019In: Journal of Integer Sequences, 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.

  • 5.
    Haukkanen, Pentti
    et al.
    Faculty of Information Technology and Communication Sciences, FI-33014 Tampere University, Finland.
    Merikoski, Jorma K.
    Faculty of Information Technology and Communication Sciences, FI-33014 Tampere University, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Arithmetic Subderivatives: p-adic Discontinuity and Continuity2020In: Journal of Integer Sequences, E-ISSN 1530-7638, Vol. 23, no 7, article id 20.7.3Article in journal (Refereed)
    Abstract [en]

    In a previous paper, we proved that the arithmetic subderivative DS is discontinuous at any rational point with respect to the ordinary absolute value. In the present paper, we study this question with respect to the p-adic absolute value. In particular, we show that DS is in this sense continuous at the origin if S is finite or p not in S.

  • 6.
    Haukkanen, Pentti
    et al.
    Faculty of Information Technology and Communication Sciences, Tampere University, Finland.
    Merikoski, Jorma K.
    Faculty of Information Technology and Communication Sciences, Tampere University, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative2020In: Mathematical Communications, ISSN 1331-0623, E-ISSN 1848-8013, Vol. 25, no 1, p. 107-115Article in journal (Refereed)
    Abstract [en]

    For ∅ ̸= P ⊆ P, let DP be the arithmetic subderivative function with respect to P on Z+, let ζDP be the function defined by the Dirichlet series of DP , and let σDP denote its abscissa of convergence. Under certain assumptions concerning s and P, we present asymptotic formulas for the partial sums of ζDP (s) and show that σDP = 2. We also express ζDP (s), s > 2, using the Riemann zeta function.

  • 7.
    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.

  • 8.
    Haukkanen, Pentti
    et al.
    Faculty of Information Technology and Communication Sciences, FI-33014 Tampere University, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Exploring Perpendicularity and Parallelism Through Play2023In: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980Article in journal (Refereed)
  • 9.
    Johansson, Claes
    et al.
    Luleå gymnasieskola, Luleå, Sweden.
    Juhlin, Alf
    Luleå gymnasieskola, Luleå, Sweden.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Wedestig, Anna
    Luleå gymnasieskola, Luleå, Sweden.
    Nyfikenhet och otillräcklighet: Gymnasielärares erfarenheter av att undervisa matematik med programmering2023In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, Vol. 49, no 3, p. 35-41Article in journal (Other academic)
    Abstract [sv]

    En lärartanke: Det är konstigt att för att få undervisa i programmeringskurser krävs det 90 högskolepoäng men för att tillämpa programmering i matematikkurserna krävs ingen kunskap i programmering.

  • 10.
    Johansson, Claes
    et al.
    Luleå gymnasieskola, Luleå, Sverige.
    Juhlin, Alf
    Luleå gymnasieskola, Luleå, Sverige.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Wedestig, Anna
    Luleå gymnasieskola, Luleå, Sverige.
    Programmering på gymnasiet2021In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, no 4, p. 43-48Article in journal (Other (popular science, discussion, etc.))
  • 11.
    Johansson, Maria
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Tossavainen, Timo
    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.
    Tossavainen, Anne
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Student teachers' definitions of the concept “teaching mathematics in preschool”2019In: Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11) / [ed] Uffe Thomas Jankvist, Marja van den Heuvel-Panhuizen, Michiel Veldhuis, ERME , 2019, p. 2277-2284, article id hal-02414896Conference paper (Other academic)
    Abstract [en]

    This article reports on how a group of preschool and primary student teachers define the concept “teaching mathematics in preschool” in the beginning of their studies. The background for this pilot-study is the recent change in the Swedish curriculum, which means a shift from play-based to more teaching oriented activities, and the actual Swedish debate on the role of teaching in preschool.

  • 12.
    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

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  • 13.
    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)
    Abstract [fi]

    Matematiikan opiskelussa on hedelmällistä lähestyä uusia käsitteitä ja algoritmeja tarkastelemalla niiden merkityksiä monipuolisesti. Joustavalla luonnollisen kielen, kuviokielen ja matematiikan symbolikielen käytöllä voidaan opiskeltavista kohteista luoda merkityksellisiä tietorakenteita. Opiskelijalle matemaattisen ajattelun esittäminen (kielentäminen) mainittujen kielten avulla on oiva keino tehdä omaa ajattelua muille opiskelijoille ymmärrettäväksi. Toisaalta opettajalle opiskelijoiden monipuolinen kielten käyttö luo tärkeän pohjan oppimisen arvioinnille ja siten opetuk-sen suunnittelulle. Tarkastelemme matemaattisen ajattelun kielentämistä sekä suullisessa että kirjallisessa työskentelyssä. Esittelemme joitakin tut-kimuspohjaisesti kehitettyjä peruskoulun kielentämisharjoituksia, joita voi soveltaen käyttää korkeakouluopintoihin asti. Lopuksi esittelemme kielentämisen eräänä sovelluksena variaatioteoriaan perustuvan learning study -mallin. 

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  • 14.
    Jumppanen, Pauli
    et al.
    Independent, Finland.
    Nevanlinna, Heikki
    University of Helsinki, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Il­mas­ton­muu­tok­sen ja ener­gian­käy­tön di­dak­tiik­kaa2021In: DimensioArticle in journal (Other academic)
    Abstract [en]

    Ilmastonmuutoksesta ja energiankäytöstä puhutaan paljon mediassa. Pääviesti on jo selvä: maailmanlaajuista energian tuotantoa ja kulutusta on muutettava merkittävästi, jotta ilmaston lämpenemistä pystyttäisiin hillitsemään. Mutta miten näistä asioista pitäisi puhua koulussa?

  • 15.
    Lehtinen, Matti
    et al.
    National Defence University, Finland.
    Nevanlinna, Heikki
    Finnish Meteorological Institute, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Matematiikan opetuksen ihanteet: [On the ideals of mathematics teacher]2019In: 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.

  • 16.
    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.

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  • 17.
    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.

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  • 18.
    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)
  • 19.
    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 1216-6014, Vol. 50, p. 145-157Article 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.

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  • 20.
    Merikoski, Jorma K.
    et al.
    Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland.
    Haukkanen, Pentti
    Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Complete Additivity, Complete Multiplicativity, and Leibniz-additivity on Rationals2021In: Integers: Electronic Journal of Combinatorial Number Theory, E-ISSN 1553-1732, Vol. 21, article id A33Article in journal (Refereed)
    Abstract [en]

    Completely additive (c-additive in short) functions and completely multiplicative (c-multiplicative in short) functions are ordinarily defined for positive integers but sometimes on larger domains. We survey this matter by extending these functions first to nonzero integers and thereafter to nonzero rationals. Then we can similarly extend Leibniz-additive (L-additive in short) functions. (A function is L-additive if it is a product of a c-additive and a c-multiplicative function.) We study some properties of these functions. The role of an L-additive function as a generalized arithmetic derivative is our special interest.

  • 21.
    Merikoski, Jorma Kaarlo
    et al.
    Faculty of Information Technology and Communication Sciences, FI-33014 Tampere University, Finland.
    Haukkaneny, Pentti
    Faculty of Information Technology and Communication Sciences, FI-33014 Tampere University, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    A Note On Perpendicularities And Inner Products2023In: Applied Mathematics e-notes, E-ISSN 1607-2510, Vol. 23, p. 459-464Article in journal (Refereed)
  • 22.
    Rensaa, Ragnhild Johanne
    et al.
    Department of Electrical Engineering, The Arctic University of Norway, Narvik, Norway.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Norwegian freshmen engineering students' self-efficacy, motivation, and view of mathematics in light of task performance2022In: Nordic Journal of STEM Education, E-ISSN 2535-4574, Vol. 6, no 1Article in journal (Refereed)
    Abstract [en]

    We report on 71 Norwegian freshmen engineering students' self-efficacy and motivation in mathematics. Students' responses to five-point Likert scales were analysed across three groups corresponding to different performance levels on a set of mathematical tasks. The groups were investigated to trace differences in self-efficacy, motivation, and the epistemological beliefs about the nature of mathematics. Results show that the Norwegian first-year engineering students' self-efficacy is closely related to task performance, but there is not a similar correspondence between task performance and the motivational values. The amount of higher performing students who regard mathematics as a set of (ready-made) tools for solving tasks is a little higher than the amount of lower performing students, while in the case of valuing problem-solving processes in mathematics, the distribution of students is opposite with lower performing students being a majority. The task performance levels are a significant predictor of how dynamic the distribution of the epistemological beliefs is.

  • 23.
    Stén, Johan
    et al.
    Department of Mathematics and Statistics, University of Helsinki, Pietari Kalmin katu 5, P.O.B. 68, 00014, Helsinki, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Osmo Pekonen (1960–2022)2023In: The Mathematical intelligencer, ISSN 0343-6993, E-ISSN 1866-7414, Vol. 45, no 1, p. 1-4Article in journal (Other academic)
  • 24.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Cryptography based on algebraic perpendicularities2023In: Mathematical Gazette, ISSN 0025-5572, Vol. 107, no 568, p. 65-69Article in journal (Refereed)
    Abstract [en]

    Cryptography, the mathematics of protecting secret or sensitive information, is a continuously evolving area of interest. A concrete example of this is the fact that worldwide spending on information security and risk management technology and services has been estimated to reach over $150 billion in 2021. Modern cryptography is actually not a single separate domain of mathematics but an advanced encryption scheme can be based on applications of results in number theory, such as the Euler–Fermat Theorem, or involve the discrete logarithm problem using either a primitive root of a large prime or an element of an elliptic curve over a prime field. In the background, probability theory, statistics, studies on computational models and finite geometries, etc. play a major role. Recent research has considered even DNA-based molecular cryptography systems.

  • 25.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Ei ihan minkä tahansa suvun kronikka – Neovius-Nevanlinnan suku on poikinut muun muassa matematiikan professoreita2022In: Tieteessä tapahtuu, ISSN 1239-6540, Vol. 40, no 5Article, book review (Other academic)
  • 26.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Matemaattisia keskusteluja tekoälyn kanssa – Onko ChatGPT:stä oppaaksi matematiikan oppimisessa?2023In: Tieteessä tapahtuu, ISSN 0781-7916, Vol. 41, no 2, p. 39-46Article in journal (Other academic)
    Abstract [fi]

    Tekoäly on tulossa osaksi oppimisen arkea. Kävin keskusteluja tekoälyn kanssa muun muassa eräiden matemaattisten lauseiden todistamisesta selvittääkseni, miten tekoälyä voisi käyttää matematiikan oppimisen tukena. Millaisia mahdollisuuksia tai uhkakuvia tekoälyn käyttöön liittyy?

  • 27.
    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)
  • 28.
    Tossavainen, Timo
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Poimintoja matematiikan oppimateriaalien tutkimuksesta2019In: Tutkimuskohteena tietokirja: Pirjo Hiidenmaan juhlakirja / [ed] Helena Ruuska, Markku Löytönen, Helsinki: University of Helsinki, 2019, p. 157-167Chapter in book (Other academic)
    Abstract [fi]

    Matematiikan opetuksessa oppikirjalla on merkittävä rooli. Silti matematiikan oppikirjojen tutkimus ei ole ollut Suomessa erityisen aktiivista tai määrätietoista. Oppimateriaalien sähköistyminen saattaa kuitenkin muuttaa tilannetta; ainakin opinnäytetöiden tasolla kiinnostus matematiikan oppimateriaaleja kohtaan on kasvussa. Tässä artikkelissa esitellään joitakin ajankohtaisia matematiikan oppimateriaaleihin liittyviä tutkimusaiheita.

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  • 29.
    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)
    Abstract [fi]

    Oppikirjan tulevaisuus on täynnä kysymyksiä. Voidaanko sähköinen oppimateriaali koota olemassa olevista verkkomateriaaleista automaattisesti? Kuka on tällöin oppimateriaalin tekijä?

    Digitaalisen sisällön käyttömahdollisuudet ovat monipuoliset. Oppimisaihioista voidaan koota kullekin oppilaalle juuri hänen tarpeitansa vastaava oppimateriaali, ja oppimisanalytiikan avulla voidaan kerätä tietoa siitä, mitä oppilaat ovat tehneet ja mitkä ovat heidän ongelmakohtiaan. Vaikka tietokonemaailmassa moni asia hoituu kuin itsestään, vielä tarvitaan oppikirjailijaa, joka ammattitaitonsa avulla näkee, mikä kaikessa tiedossa on olennaista.

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  • 30.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Dubois, Francoise
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    A discussion with two five-year-old girls about similarities and dissimilarities in two paintings2021In: For the Learning of Mathematics, ISSN 0228-0671, Vol. 41, no 2, p. 41-44Article in journal (Refereed)
  • 31.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Edholm, 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.
    Lundkvist, Maria
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    An investigation of Swedish pre-service preschool teachers’ knowledge about the negative numbers2020In: Journal of Early Childhood Education Research, E-ISSN 2323-7414, Vol. 9, no 2, p. 420-436Article in journal (Refereed)
    Abstract [en]

    In this paper, we examine the Swedish pre-service preschool teachers’ knowledge about the negative numbers, which is one of the central content areas in their compulsory mathematics course. Our results show that almost a half of the pre-service preschool teachers can give a reasonable definition for the negative numbers, but only a few of them set the negative numbers in relation to the other common number sets. The participants’ performance in a set of mathematical tasks related to the negative numbers follows the qualitative variance in their definitions of the negative numbers, yet this relation is not completely direct; there were also such participants who were not able to give a definition but succeeded well in the tasks. Those who gave a vague definition performed weaker than the others. Although the participants’ knowledge about the negative numbers is, perhaps, not satisfactory with every respect, it does not significantly differ from that of a control group that consists of (non-mathematician) university teachers from their study programme.

  • 32.
    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.

  • 33.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Gothóni, Ralf
    Hochschule für Musik Karlsruhe.
    Ajasta ja mittaamisesta musiikissa ja matematiikassa2021In: Kanava, ISSN 0355-0303, Vol. 49, no 1, p. 28-32Article in journal (Other academic)
    Abstract [en]

    This article discusses a mathematical foundation of the measurement of distances from various perspectives. As an application we consider the measurement of time in music.

  • 34.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Gröhn, Janne
    Department of Physics and Mathematics, University of Eastern Finland .
    Heikkinen, Lasse
    Department of Applied Physics, University of Eastern Finland .
    Kaasinen, Anna
    Department of Applied Physics, University of Eastern Finland .
    Viholainen, Antti
    Department of Physics and Mathematics, University of Eastern Finland .
    University mathematics students' study habits and use of learning materials2020In: LUMAT International Journal of Math, Science and Technology Education, ISSN 2323-7104, Vol. 8, no 1, p. 252-270Article in journal (Refereed)
    Abstract [en]

    In this article, we study a group (N=98) of students from two campuses of one Finnish university and their responses to a questionnaire that surveyed the respondents' study habits and how they use different kinds of learning materials in university mathematics courses. Our results show that the grades in the national matriculation exams in mathematics and mother tongue do not play a significant role in one's study habits or use of learning materials. The older students are more communicative with their teachers, whereas the younger students ask for help more often from fellow students. The sociomathematical norms that constitute the local study culture have a larger impact on the study habits and on the use of learning materials. For example, the use of videos and studying lecture materials before the lectures were clearly more usual at one campus than at the other. We also found some significant differences between the groups that are based on the study programmes. The students of mathematics without an intention to become a teacher were most traditional in their study habits, whereas the students of applied physics were most active to participate in teaching. The student teachers most often lie in the middle in the issues where the other groups differ from one another.

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  • 35.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Haukkanen, Pentti
    Faculty of Information Technology and Communication Sciences, Tampere University FI-33014 Tampere University, Finland.
    Corrigendum to “On the dimension of an Abelian group” [Notes on Number Theory and Discrete Mathematics, 2021, Volume 27, Number 4, Pages 267—275]2022In: Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132, Vol. 28, no 1, p. 20-20Article in journal (Other academic)
  • 36.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Haukkanen, Pentti
    Faculty of Information Technology and Communication Sciences, Tampere University FI-33014 Tampere University, Finland.
    On the dimension of an Abelian group2021In: Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132, Vol. 27, no 4, p. 267-275Article in journal (Refereed)
    Abstract [en]

    We introduce a measure of dimensionality of an Abelian group. Our definition of dimension is based on studying perpendicularity relations in an Abelian group. For G ≅ ℤn, dimension and rank coincide but in general they are different. For example, we show that dimension is sensitive to the overall dimensional structure of a finite or finitely generated Abelian group, whereas rank ignores the torsion subgroup completely.

  • 37.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Haukkanen, Pentti
    University of Tampere, Finland.
    Mattila, Mika
    Tampere University of Technology, Finland.
    Rensaa, Ragnhild Johanne
    The Arctic University of Norway, Narvik.
    Aloittavien diplomi-insinööriopiskelijoiden motivaatio, matematiikkakäsitys ja menestys matemaattisissa tehtävissä2020In: Arkhimedes – Tidskrift för fysik och matematik, ISSN 0004-1920, no 2-3, p. 13-20Article in journal (Refereed)
    Abstract [fi]

    Tässä artikkelissa tarkastellan Tampereen yliopistossa DI-koulutukessa aloittaneiden opiskelijoiden (N=272) matematiikkaan liityvää motivaatiota, käsityksiä oppiaineen luonteesta ja menestystä lukiotasoisissa matematiikan tehtävissä. Kiinostus matematiikkaa kohtaan ja hyvä käsitys itsestä matematiikan oppijana ovat vahvimpia osaamisen selittäjiä. Vaikka erilaisilla käsityksillä matematiikan luonteesta ei ollut suurta vaikutusta testitehtävissä menestymiseen, käsitteiden määritelmien ymmärtämiseen ja päättelytaitojen kehittymiseen panostaminen vaikuttaa paremmalta keinolta kehittää matematiikan opetusta kuin matematiikan hyötyarvon korostaminen tai keskittyminen reseptinomaisiin ohjeisiin matematiikan soveltamisessa. 

  • 38.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Haukkanen, Pentti
    Tampere University, Finland.
    Merikoski, Jorma K.
    Tampere University, Finland.
    Mattila, Mika
    Tampere University, Finland.
    We Can Differentiate Numbers, Too2023In: The College Mathematics Journal, ISSN 0746-8342, E-ISSN 1931-1346Article in journal (Refereed)
    Abstract [en]

    We survey the history of the arithmetic derivative and more recent advances in research on this topic. Among other things, we discuss a few generalizations of the original arithmetic derivative and some arithmetic differential equations that are related to Goldbach’s conjecture and the twin prime conjecture. Our primary purpose is to give an overview of this field, but we also aim at providing supplementary material for an introductory course on discrete mathematics or number theory. Therefore, our survey contains ten exercises.

  • 39.
    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.

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  • 40.
    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.

  • 41.
    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)
    Abstract [fi]

    Tässä luvussa tarkastellaan suomalaisten matematiikan opettajien ja opettajaopiskelijoiden matemaattista osaamista sekä 2010-luvulla tehdyn tutkimuksen että muun käytettävissä olevan aineiston valossa. Lisäksi artikkelissa pohditaan eräitä opettajankoulutuksen järjestämiseen liittyviä mahdollisuuksia vahvistaa peruskoulun matematiikan opettajien matemaattista osaamista.

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  • 42. Tossavainen, Timo
    et al.
    Löytönen, Markku
    SÄHKÖISTYVÄ KOULU: Oppiminen ja oppimateriaalit muuttuvassa tietoympäristössä2019Collection (editor) (Other academic)
  • 43.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Nevanlinna, Heikki
    University of Helsinki, Finland.
    Jumppanen, Pauli
    Independent, Finland.
    To­sie­lä­män fy­siik­kaa – Ener­gia­ta­lou­del­li­nen kat­saus Suo­men ja Eu­roo­pan lä­hi­tu­le­vai­suu­teen2023In: Dimensio, ISSN 2669-8382Article in journal (Other (popular science, discussion, etc.))
    Abstract [fi]

    Energiateollisuus on eräs suurimmista luonnontieteiden sovellusaloista nykymaailmassa. Tämän alan kehitys on ratkaisevassa roolissa myös ilmastonmuutoksen vastaisessa globaalissa taistelussa. Tähän artikkeliin on koottu erityisesti lukion fysiikan opettajille tietopaketti energian tuotannosta ja kulutuksesta. Kirjoituksen lopussa on myös lukiolaisille sopia tehtäviä, joiden avulla voi konkretisoida energiantuotannon suuruusluokkaa, haasteita ja mahdollisuuksia. 

  • 44.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Rensaa, Ragnhild Johanne
    UiT The Arctic University of Norway, Department of Electrical Engineering, Narvik, Norway.
    Haukkanen, Pentti
    Tampere University, Faculty of Information Technology and Communication Sciences, Tampere, Finland.
    Norwegian, Swedish, and Finnish first-year engineering students' motivational values and beliefs about the nature of mathematics2022In: Bringing Nordic mathematics education into the future: Proceedings of Norma 20 The ninth Nordic Conference on Mathematics Education Oslo, 2021 / [ed] Nortvedt, G.A.; Buchholtz, N.F; Fauskanger, J.; Hähkiöniemi, M.; Jessen, B.E.; Naalsund, M.; Nilsen, H.K.; Pálsdóttir, G.; Portaankorva-Koivisto, P.; Radišić, J.; Sigurjónsson, J.Ö.; Viirman, O.; Wernberg, A., Svensk förening för MatematikDidaktisk Forskning , 2022, p. 233-240Conference paper (Refereed)
    Abstract [en]

    We report from a Nordic research project that has investigated first-year engineering students in Norway, Sweden, and Finland, and the relationships between their task performance, motivational values, and beliefs about the nature of mathematics. The present paper focuses on the covariance between their motivational values and beliefs from the perspectives of gender and nationality. The results show that female students' motivational values are more strongly related to their beliefs than is the case for male students. Further, the Finnish students' motivational values are only weakly related to their beliefs, whereas the Norwegian and the Swedish students' motivation to mathematics is quite strongly related to how much they appreciate the applications of mathematics.

  • 45.
    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.
    Haukkanen, Pentti
    Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland.
    Mattila, Mika
    Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland.
    Johansson, Monica
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    First-year engineering students’ mathematics task performance and its relation to their motivational values and views about mathematics2021In: European Journal of Engineering Education, ISSN 0304-3797, E-ISSN 1469-5898, Vol. 46, no 4, p. 604-617Article in journal (Refereed)
    Abstract [en]

    This study investigates Finnish, Norwegian, and Swedish first-year engineering students’ task performance in mathematics and examines how it relates to their motivational values and beliefs about the nature of mathematics. In a set of seven mathematical tasks, female students outperformed male students, for example, in the simplification of symbolic expressions. Our findings also show that female students set more demanding learning goals for themselves. Further, they expressed higher intrinsic motivational values, whereas male students emphasised utility values, which correlated negatively with task performance. Problem-solving and discovering structures and regularities dominate engineering students’ views of mathematics. The ‘toolbox’ view of mathematics is the best predictor of task performance; the stronger this view is, the poorer is the task performance. However, neither motivational values nor views about mathematics are especially strong predictors of task performance. Based on the findings of this study, it is recommended that, in engineering mathematics courses, greater emphasis is placed on learning fundamental reasoning strategies and discussing the theoretical foundations of the main results. This seems to lead to a better result even in applying mathematics than if one focuses only on learning how to use ready-made mathematical tools in concrete examples.

  • 46.
    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 performance2021In: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 52, no 1, p. 23-38Article 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.

  • 47.
    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.

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  • 48.
    Tossavainen, Timo
    et al.
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Sundhäll, Marcus
    Örebro University, SE-701 82 Örebro, Sweden.
    Engineering Students’ Mathematical Self-Concept and its Dependence on Their Study Habits and Views about Mathematics2022In: International Journal of Engineering Education, ISSN 0949-149X, Vol. 38, no 5A, p. 1354-1365Article in journal (Refereed)
  • 49.
    Viholainen, Antti
    et al.
    Department of Physics and Mathematics, University of Eastern Finland, Campus of Joensuu, Finland.
    Tossavainen, Timo
    Luleå University of Technology, Department of Health, Learning and Technology, Education, Language, and Teaching.
    Gröhn, Janne
    Department of Physics and Mathematics, University of Eastern Finland, Campus of Joensuu, Finland.
    Epistemological beliefs, motivational values, and study habits among students of tertiary mathematics2023In: Eurasia Journal of Mathematics, Science and Technology Education, ISSN 1305-8215, Vol. 19, no 5, article id em2269Article in journal (Refereed)
    Abstract [en]

    Epistemological beliefs about the nature of mathematics affect how a student perceives mathematics in general, whereas high motivation for mathematics gives an individual a personal reason to study the subject. Epistemological beliefs and achievement motives are quite stable at individual level, but previous research has shown that there are significant differences across students with respect to them. This study concerns university students from three different study programs and investigates relations between the students' epistemological beliefs, motivational values, and study habits. The data were collected using a questionnaire and consist of responses from altogether 98 students studying mathematics courses at tertiary level. The study revealed that, when students are grouped according to their study programs, their motivational values vary across these groups more than their epistemological beliefs. The stability of epistemological beliefs was also verified in this study as the strengths of the beliefs hardly correlated with the number of passed credits or self-evaluated study success. However, several sum variables representing the motivational values and epistemological beliefs seem to predict certain study habits in linear regression models. These models can help us to design the teaching of mathematics courses favouring these study habits.

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  • 50.
    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 International Journal of Math, Science and Technology Education, ISSN 2323-7104, 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.

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