The logarithm function on the positive real numbers can be characterized in several ways. In this paper, we focus on classical characterizations based on the product rule or the power rule, and we improve upon some results by Fubini and Milkman. Our main result can be stated as follows: If a continuous or monotone function on the positive real numbers satisfies a power rule for both a negative and a positive exponent, then it is the logarithm function or the zero function. We also briefly consider the arithmetic logarithm function and its connection to the concept of arithmetic derivative.

This study examines how sustained conceptual instruction influences gender disparities in physics understanding among Swedish upper secondary students. Drawing on Ausubel’s Meaningful Learning Theory and Vergnaud’s Conceptual Field Theory, the research explores how instructional design interacts with students’ cognitive development, prior knowledge, and sociocultural context. Using a quasi-experimental design, 852 students from 38 classes were assessed using a gender-balanced version of the Force Concept Inventory (G-FCI). Normalized change was applied to measure conceptual gains across instructional methods and genders. Findings confirm persistent gender gaps under traditional instruction, favoring male students. However, in cohorts with sustained conceptual teaching across Physics 1 and 2, female students outperformed male peers—a statistically significant shift not observed under traditional methods. While conceptual instruction improved learning for all, its equity potential was most evident when implemented consistently. These results highlight the importance of pedagogical continuity for promoting gender equity in physics education. Nevertheless, instructional change alone is insufficient; broader interventions addressing classroom climate, stereotype threat, and identity development are needed. The study contributes empirical evidence to the international discourse on gender and science education and underscores the value of conceptually grounded pedagogy as part of a systemic strategy for equitable physics learning.

This paper presents a thematic analysis of the experiences of a group of teacher educators and student teachers during an action research project that examined how collaborative learning can be fostered in online mathematics teacher education. Data were collected via written reflections from teacher educators and a questionnaire completed by student teachers. Guided by the Community of Inquiry framework, we investigate the three presences of both students and teacher educators in the online environment. Our results indicate that the course design met participants’ expectations regarding cognitive presence, while teaching presence requires further development. With respect to social presence – particularly, affective expressions – our findings suggest that emotions mediate the relationship between teaching presence and cognitive presence. For example, feelings of curiosity transformed instructional scaffolding into active exploration. Conversely, frustration moderated collaboration: when left unaddressed, it reduced students’ willingness to engage, but when acknowledged by peers or educators, it became a driver for problem-solving.

We determine, for every n ≥ 1, the generalized eigenvalues of an n×n MAX matrixto the corresponding MIN matrix. We also show that a similar result holds for thegeneralized eigenvalues of an n×n LCM matrix to the corresponding GCD matrix whenn ≤ 4, but breaks down for n > 4. In addition, we prove Cauchy’s interlacing theorem for generalized eigenvalues, and we conjecture an unexpected connection between the OEIS sequence A004754 and the appearance of −1 as a generalized eigenvalue in the LCM–GCD setting.

For the purposes of mathematics teacher education, we provide an algebraic and number theoretic explanation of how one can determine, from a and b, whether the decimal expansion of a/b, where gcd(a,b) = 1, contains a pre-period, and if so, how long it is. To that end, we observe that explanation cannot be based solely on group theory because the residue classes of remainders do not form a subgroup of the multiplicative group modulo b. However, the collection of residue classes does form a submonoid of the multiplicative monoid modulo b.

We introduce the notion of perpendicularity of arithmetical functions and discuss a few concrete examples of perpendicularities. Further, we show that the set of arithmetical functions f with f(1) = 1 forms a real vector space with the Dirichlet convolution as addition and real power under the Dirichlet convolution as scalar multiplication. Moreover, we prove that multiplicative functions are perpendicular to antimultiplicative functions with respect to the natural perpendicularity.

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.

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.