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.

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.

Let a₁,..., a_m 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 J_a of the vector a = (a₁,..., a_m) analogously to the Jacobian matrix J_f of a vector function f. We introduce the concept of multiplicative independence of {a₁,..., a_m} and show that J_a plays in it a similar role as J_f does in functional independence. We also present a kind of arithmetic implicit function theorem and show that J_a applies to it somewhat analogously as J_f applies to the ordinary implicit function theorem.

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.

In a previous paper, we proved that the arithmetic subderivative D_S 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 D_S is in this sense continuous at the origin if S is finite or p not in S.

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.

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.

This article advances understanding about book reading as a sociocultural phenomenon in the 2020s. We make a contribution to the cultural sociology of reading by investigating Finnish self-identified book readers by analysing the significance of sociodemographic variables (gender, education, age, and place of residence) in terms of reading activity and access to books. Our study is placed in the context of Finnish reading culture that is characterised by a particular appreciation of reading and measures promoting equal access to culture. Based on an online survey of 955 respondents conducted in 2021, our statistical analyses show that the social stratification of book reading activity that is prominent in population level does not recur within the specific group of people who identify themselves as readers. Among Finnish self-identified book readers, education, gender, and place of residence do not induce significant differences in reading activity. Our analysis that foregrounds inclination instead of quantity as a criterion for readers sheds light on reader equality from a different direction than previous research into nationwide reading habits or descriptive studies on avid readers.

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.

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.

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

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. 

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?

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.

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.

Consider the recursion g₀ = a, g₁ = b, g_n = g_n−1 + g_n−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g₀, . . . , g_n−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.

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.

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.

We study the solvability of the congruence xn =-an (mod m), where n, m E Z`, a E Z, and gcd (a, m) = 1. Our motivation arises from computer experiments concerning a geometric property of the roots of the congruence xn + yn = 0 (mod p), where n E Z` and p E P. We encounter several OEIS sequences. We also make new observations on some of them.

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.

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.

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?

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.

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.

Students' motivation plays a crucial role in studying mathematics. A recent Swedish study on engineering students' mathematical self-concept and its dependency on their views of mathematics and study habits showed that engineering students' motivation to study mathematics varies across the years. In order to understand better this variation, we conducted interviews with eleven engineering students from a Swedish university. Our primary focus is to understand the underlying reasons for these changes in engineering students’ motivation throughout their university journey, as well as to explore their opinions on how to sustain motivation to study mathematics. 

This study reports on engineering students (N=369) from two Swedish universities and focuses on their perceived value of proving in learning engineering mathematics and some factors that may explain the observed variation in the perceived value. Our findings show that there is no significant difference in the perceived value between female and male students. In general, proving is not highly valued, and students are not confident in their skills in proving, except for proving by mathematical induction. However, students’ motivation in mathematics correlates with the perceived value and certain study habits are more regular among those students who appreciate proving as a suitable method for learning mathematics. Examples of such study habits include actively communicating with mathematics course teachers and reading the course textbook both before and after lectures.

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.

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.

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.

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.

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.

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. 

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.

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.

We report on three episodes from a case study where upper secondary students numerically explore the definite integral in a Python environment. Our research questions concern how code can mediate and support students' mathematical thinking and what kind of sociomathematical norms emerge as students work together to reach a mutual understanding of a correct solution. The main findings of our investigation are as follows. 1) Students can actively use code as a mediator of their mathematical thinking, and code can even serve as a bridge that helps students to develop their mathematical thinking collaboratively. Further, code can help students to perceive mathematical notions as objects with various properties and to communicate about these properties, even in other semiotic systems than the mathematical language. 2) For the participating students, a common norm was that an acceptable solution is a sufficient condition for the correctness of the solution method although students were aware of a problem in their code, yet also other norms emerged. This demonstrates that learning mathematics with programming can have an effect on what kind of sociomathematical norms emerge in classroom.

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.

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.

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. 

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.