On Representations of Intended Structures in Foundational Theories

Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power and meta-theoretic properties when comparing first-order and second-order logic.

COMPUTER PROOFS PRACTICE AND HUMAN UNDERSTANDING: EPISTEMOLOGICAL ISSUES

In recent decades, some epistemological issues have become especially acute in mathematics. These issues are associated with long proofs of various important mathematical results, as well as with a large and constantly increasing number of publications in mathematics. It is assumed that (at least partially) these difficulties can be resolved by referring to computer proofs. However, computer proofs also turn out to be problematic from an epistemological point of view. With regard to both proofs in ordinary (informal) mathematics and computer proofs, the problem of their surveyability appears to be fundamental. Based on the traditional concept of proof, it must be surveyable, otherwise it will not achieve its main goal — the formation of conviction in the correctness of the mathematical result being proved. About 15 years ago, a new approach to the foundations of mathematics began to develop, combining constructivist, structuralist features and a number of advantages of the classical approach to mathematics. This approach is built on the basis of homotopy type theory and is called the univalent foundations of mathematics. Due to itspowerful notion of equality, this approach can significantly reduce the length of formalized proofs, which outlines a way to resolve the epistemological difficulties that have arisen

Black Girls and Mathematics Learning

Though the formal and informal mathematics learning experiences of Black girls are gaining more visibility in the literature, there is still a paucity of research around Black girls’ mathematics learning experiences. Black girls face unique challenges as learners in K–12 educational spaces because of their marginalized racial and gender identities. The interplay of race and racism unfolds in complex ways in Black girls’ learning experiences. This interplay hinders their development as mathematics learners and limits their access to transformative learning. As early as elementary school, Black girls are labeled as having limited mathematics knowledge and are often disproportionately placed in “lower level classrooms” devoid of any rigorous and transformative learning experiences. Teachers spend more time socially correcting Black girls rather than building on their brilliance. Even though Black girls value mathematics more and have higher confidence in mathematics than their White counterparts, they are still held to lower expectations by their teachers and are less likely to complete an advanced mathematics course. Nationally and globally, mathematics serves as an academic gatekeeper into every avenue of the labor market and higher education opportunities. Thus, the lack of opportunities Black girls have to engage in rigorous and transformative mathematics potentially locks them out of higher education opportunities and STEM-based careers. The mathematics learning experiences of Black girls move beyond challenges in K–12 spaces to limiting life choices and individual and community progress. To improve the formal and informal mathematics learning experiences of Black girls, we must understand their unique learning experiences more fully.

Kaitan antara Etnomatematika dan Matematika Sekolah: Sebuah Kajian Konseptual

The awareness that mathematics exists at the respective culture needs to be developed in students’ mind through the integration of ethnomathematics at school. School mathematics is different from pure formal mathematics, while ethnomathematics studies pure informal mathematics as well as applied mathematics. This article contains conceptual reviews focused on the comprehension and scope of ethnomathematics, as well as some views on school mathematics to find out the connection between ethnomathematics and school mathematics, also the role of ethnomathematics in school mathematics. The philosophical foundation that connects ethnomatematics and school mathematics is the nature of mathematics as a social construction. Based on this view, it is identified that there are three connections between ethnomatematics and school mathematics, namely ethnomatematics as a learning object of school mathematics, bridge of informal knowledge leading to formal mathematics, and didactic or pedagogical foundations of school mathematics.

Early Mathematics Counts: Promising Instructional Strategies from Low- and Middle-Income Countries

This occasional paper examines common instructional strategies in early-grade mathematics interventions through a review of studies in classrooms in low- and middle-income countries. Twenty-four studies met the criteria for inclusion, and analyses reveal four sets of instructional strategies for which there is evidence from multiple contexts. Of the 24 studies, 16 involved the use of multiple representations, 10 involved the use of developmental progressions, 6 included supporting student use of explanation and justification, and 5 included integration of informal mathematics. Based on the review, we provide conclusions and recommendations for future research and policy.

Exploring Growth Trajectories of Informal and Formal Mathematics Skills Among Prekindergarten Children Struggling With Mathematics

Growth in two subscales, Informal and Formal Mathematics Skills, of the Test of Early Mathematics Abilitity–3 (TEMA-3) was explored in a sample of 281 children. Children were identified as either typically developing (TYP; n = 205) or having mathematics difficulties (MD; n = 76) based on their total TEMA-3 score at the end of prekindergarten. Their average level of informal and formal mathematics skills, growth rate over time, and rate of acceleration of growth were estimated using conventional growth modeling while controlling for the effects of gender. Results indicated that children with MD had significantly lower informal and formal mathematics knowledge than did TYP children at the end of kindergarten. However, for informal mathematics skills, children with MD grew at a significantly faster rate than did TYP children, and the rate of acceleration was also significantly faster for children with MD. In contrast, both the rate of growth and acceleration of growth in formal mathematics skills were significantly faster for TYP children than they were for children with MD. Implications for early MD identification and interventions are discussed.